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DERIVATIVES, RISK MANAGEMENT & VALUE

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DERIVATIVES, RISK MANAGEMENT & VALUE

Mondher Bellalah Université de Cergy-Pontoise, France

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

DERIVATIVES, RISK MANAGEMENT & VALUE Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-283-862-9 ISBN-10 981-283-862-7

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

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DEDICATION

I dedicate this book to the President of the Tunisian Republic, his Excellency Mr. Zine El Abidine Ben Ali, in recognition of his continuous and pivotal support for Science and its men in Tunisia. Over the last twenty-plus years, the scientific achievements of Tunisian researchers in various fields were of high importance. I do believe that the distinction of the scientific research in Tunisia, in relation to other countries in the Euro-Mediterranean region, is due to the particular efforts of Mr. Ben Ali. In the field of Finance, the scientific community and I in Tunisia were lucky to benefit from his particular attention. Indeed, by placing the International Finance Conferences that I organized and headed under his high patronage, they gained a remarkable international reputation. With this opportunity, Ph.D. students, college professors and professionals were able to communicate with professors and experts from the best U.S. universities and institutions, alongside Nobel laureates such as H. Markowitz and J. Heckman. Another important insight of Mr. Ben Ali’s role is related to the setting of national awards to strengthen the mechanics of the scientific research at undergraduate and graduate levels. Such awards boost one’s willingness to improve the Tunisian economy and its ability to meet the new challenges posed by the international context. Furthermore, the creation of Ben Ali’s Chair for the dialogue of civilizations and religions in 2001 had a key role in the enrichment of knowledge and human values in a multi-religions context. In addition, it is considered as the Mecca of researchers from all over the world who are involved in bringing new approaches to make people closer. Mondher Bellalah

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FOREWORD

by Edward C. Prescott (Arizona State University; Federal Reserve Bank of Minneapolis)

This book covers the main aspects regarding derivatives, risk, and the role of information and financial innovation in capital markets and in the banking system. An analysis is provided regarding financial markets and financial instruments and their role in the 2007–2008 financial crisis. This analysis hopefully will be useful in avoiding or at least mitigating future financial crises. The book presents the principal concepts, the basics, the theory, and the practice of virtually all types of financial derivatives and their use in risk management. It covers simple vanilla options as well as structured products and more exotic derivative transactions. Special attention is devoted to risk management, value at risk, credit valuation, credit derivatives, and recent pricing methodologies. This book is not only useful for specific courses in risk management and derivatives, but also is a valuable reference for users and potential users of derivatives and more generally for those with risk management responsibilities.

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FOREWORD

by Harry M. Markowitz (University of California, San Diego)

Herein follows a remarkable volume, suitable as both a textbook and a reference book. Mondher Bellalah starts with an introduction to options and basic hedges built from specific options. He then presents an accessible account of the formulae used in valuing options. This account includes historically important formulae as well as the currently most used results of Black–Scholes, Merton and others. Bellalah then proceeds to the main task of the volume, to show how to value an endless assortment of exotic options. Mondher Bellalah is to be congratulated for this tour de force of the field.

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FOREWORD

by James J. Heckman (University of Chicago and University College Dublin)

Mondher Bellalah offers a lucid and comprehensive introduction to the important field of modern asset pricing. This field has witnessed a remarkable growth over the past 50 years. It is an example of economic science at its best where theory meets data, and shapes and improves on reality. Economic theory has suggested a variety of new and “exotic” financial instruments to spread risk. Created from the minds of theorists and traders guided by theory, these instruments are traded in large volume and now define modern capital markets. Bellalah offers a step-by-step introduction to this evolving theory starting from its classical foundations. He takes the reader to the frontier by systematically building up the theory. His examples and intuition are splendid and the formal proofs are clearly stated and build on each other. I strongly recommend this book to anyone seeking to gain a deep understanding of the intricacies of asset pricing.

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FOREWORD by George M. Constantinides (University of Chicago)

Both the trading of options and the theory of option pricing have long histories. The first use of option contracts took place during the Dutch tulip mania in the 17th century. Organized trading in calls and puts began in London during the 18th century, but such trading was banned on several occasions. The creation of the Chicago Board Options Exchange (CBOE) in 1973 greatly encouraged the trading of options. Initially, trading took place at the CBOE only in calls of 16 common stocks, but soon expanded to many more stocks, and in 1977, put options were also listed. The great success of option trading at the CBOE contributed to their trading in other exchanges, such as the American, Philadelphia and Pacific Stock Exchanges. Currently, daily option trading is a multibillion-dollar global industry. The theory of option pricing has had a similar history that dates to Bachelier (1900). Sixty-five years after Bachelier’s remarkable study, Samuelson (1965) revisited the question of pricing a call. Samuelson recognized that Bachelier’s assumption that the price of the underlying asset follows a continuous random walk leads to negative asset prices, and thus makes a correction by assuming a geometric continuous random walk. Samuelson obtained a formula very similar to the Black–Scholes–Merton formula, but discounted the cash flows of the call at the expected rate of return of the underlying asset. The seminal papers of Black and Scholes (1973) and Merton (1973) ushered in the modern era of derivatives. This is a lucid textbook treatment of the principles of derivatives pricing and hedging. At the same time, it is an exhaustively comprehensive encyclopedia of the vast array of exotic options, fixed-income options, corporate claims, credit derivatives and real options. Written by an expert in the field, Mondher Bellalah’s comprehensive and rigorous book is an indispensable reference on any professional’s desk. xiii

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ABOUT THE AUTHOR

After obtaining his Ph.D. in Finance in 1990 at France’s leading University Paris-Dauphine, Mondher Bellalah began his career both as a Professor of Finance (HEC, INSEAD, University of Maine, and University of CergyPontoise) and as an international consultant and portfolio manager. He started out as a market maker on the Paris Bourse, before being put in charge of BNP’s financial engineering research team as Head of Derivatives and Structured Products. Dr. Mondher has acted as an advisor to various leading financial institutions, including BNP, Rothschild Bank, Euronext, Houlihan Lokey Howard & Zukin, Associés en Finance, the NatWest, Central Bank of Tunisia,DubaiHolding,etc.,andhasbeenChiefRiskOfficer,ManagingDirector inAlternativeInvestments,HeadofCapitalMarketsandHeadofTrading. Dr. Mondher has also enjoyed a distinguished academic career as a tenured Professor of Finance at the University of Cergy-Pontoise in Paris for about 20 years. During this time, he has authored more than 14 books and 150 articles in leading academic and professional journals, and was awarded the Turgot Prize for the best French-language book on risk management in 2005. English-language books co-written/co-edited by Dr. Mondher include Options, Futures and Exotic Derivatives: Theory, Application and Practice published by John Wiley in 1998, and Risk Management and Value: Valuation and Asset Pricing published by World Scientific in 2008. His French-language books include Quantitative Portfolio Management and New Financial Markets; Options, Futures and Risk Management; and Risk Management and Classical and Exotic Derivatives. Dr. Mondher is an associate editor of the International Journal of Finance, Journal of Finance and Banking and International Journal of Business, and has been published in leading academic journals including Financial Review, Journal of Futures Markets, International Journal of Finance, International Journal of Theoretical and Applied Finance as well as in the Harvard Business Review. xv

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CONTENTS Dedication

v

Foreword by Edward C. Prescott Foreword by Harry M. Markowitz

vii ix

Foreword by James J. Heckman Foreword by George M. Constantinides

xi xiii

About the Author

xv

PART I. FINANCIAL MARKETS AND FINANCIAL INSTRUMENTS: BASIC CONCEPTS AND STRATEGIES

1

CHAPTER 1. FINANCIAL MARKETS, FINANCIAL INSTRUMENTS, AND FINANCIAL CRISIS

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Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Trading Characteristics of Commodity Contracts: The Case of Oil . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Fixed prices . . . . . . . . . . . . . . . . . . . . . 1.1.2. Floating prices . . . . . . . . . . . . . . . . . . . 1.1.3. Exchange of futures for Physical (EFP) . . . . . 1.2. Description of Markets and Instruments: The Case of the International Petroleum Exchange . . . . . . . . . . . . . 1.3. Characteristics of Crude Oils and Properties of Petroleum Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Specific features of some oil contracts . . . . . . 1.3.2. Description of Markets and Trading Instruments: The Brent Market . . . . . . . . . . . . . . . . .

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Derivatives, Risk Management and Value

1.4.

Description of Markets and Trading Instruments: The Case of Cocoa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. How do the futures and physicals market work? . . 1.4.2. Arbitrage . . . . . . . . . . . . . . . . . . . . . . . 1.4.3. How is the ICCO price for cocoa beans calculated? 1.4.4. Information on how prices are affected by changing economic factors? . . . . . . . . . . . . . . . . . . . 1.4.5. Cocoa varieties . . . . . . . . . . . . . . . . . . . . 1.4.6. Commodities — Market participants: The case of cocoa, coffee, and white sugar . . . . . . . . . . . . 1.5. Trading Characteristics of Options: The Case of Equity Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Options on equity indices . . . . . . . . . . . . . . 1.5.2. Options on index futures . . . . . . . . . . . . . . 1.5.3. Index options markets around the world . . . . . . 1.5.4. Stock Index Markets and the underlying indices in Europe . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Trading Characteristics of Options: The Case of Options on Currency Forwards and Futures . . . . . . . . . . . . . . . . 1.7. Trading Characteristics of Options: The Case of Bonds and Bond Options Markets . . . . . . . . . . . . . . . . . . . . . 1.7.1. The specific features of classic interest rate instruments . . . . . . . . . . . . . . . . . . . . . . 1.7.2. The specific features of mortgage-backed securities 1.7.3. The specific features of interest rate futures, options, bond options, and swaps . . . . . . . . . . . . . . . 1.8. Simple and Complex Financial Instruments . . . . . . . . . 1.9. The Reasons of Financial Innovations . . . . . . . . . . . . 1.10. Derivatives Markets in the World: Stock Options, Index Options, Interest Rate and Commodity Options and Futures Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1. Global overview . . . . . . . . . . . . . . . . . . . 1.10.2. The main indexes around the world: a historical perspective . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

CHAPTER 2. RISK MANAGEMENT, DERIVATIVES MARKETS AND TRADING STRATEGIES Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Introduction to Commodity Markets: The Case of Oil . . . 2.1.1. Oil futures markets . . . . . . . . . . . . . . . . . . 2.1.2. Oil futures exchanges . . . . . . . . . . . . . . . . 2.1.3. Delivery procedures . . . . . . . . . . . . . . . . . 2.1.4. The long-term oil market . . . . . . . . . . . . . . 2.2. Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. The pricing of forward and futures oil contracts . . 2.2.1.1. Relationship to physical market . . . . . 2.2.1.2. Term structure of prices . . . . . . . . . 2.2.2. Pricing swaps . . . . . . . . . . . . . . . . . . . . . 2.2.3. The pricing of forward and futures commodity contracts: General principles . . . . . . . . . . . . 2.2.3.1. Forward prices and futures prices: Some definitions . . . . . . . . . . . . . . . . . 2.2.3.2. Futures contracts on commodities . . . . 2.2.3.3. Futures contracts on a security with no income . . . . . . . . . . . . . . . . . . . 2.2.3.4. Futures contracts on a security with a known income . . . . . . . . . . . . . . . 2.2.3.5. Futures contracts on foreign currencies . 2.2.3.6. Futures contracts on a security with a discrete income . . . . . . . . . . . . . . 2.2.3.7. Valuation of interest rate futures contracts . . . . . . . . . . . . . . . . . . 2.2.3.8. The pricing of future bond contracts . . 2.3. Trading Motives: Hedging, Speculation, and Arbitrage . . . 2.3.1. Hedging using futures markets . . . . . . . . . . . 2.3.1.1. Hedging: The case of cocoa . . . . . . . 2.3.1.2. Hedging: The case of oil . . . . . . . . . 2.3.1.3. Hedging: The case of petroleum products futures contracts . . . . . . . . . . . . . 2.3.1.4. The use of futures contracts by petroleum products marketers, jobbers, consumers, and refiners . . . . . . . . . . . . . . . .

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2.3.2. Speculation using futures markets . . . . . . . . . 2.3.3. Arbitrage and spreads in futures markets . . . . . 2.4. The Main Bounds on Option Prices . . . . . . . . . . . . . . 2.4.1. Boundary conditions for call options . . . . . . . . 2.4.2. Boundary conditions for put options . . . . . . . . 2.4.3. Some relationships between call options . . . . . . 2.4.4. Some relationships between put options . . . . . . 2.4.5. Other properties . . . . . . . . . . . . . . . . . . . 2.5. Simple Trading Strategies for Options and their Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Trading the underlying assets . . . . . . . . . . . . 2.5.2. Buying and selling calls . . . . . . . . . . . . . . . 2.5.3. Buying and selling puts . . . . . . . . . . . . . . . 2.6. Some Option Combinations . . . . . . . . . . . . . . . . . . 2.6.1. The straddle . . . . . . . . . . . . . . . . . . . . . 2.6.2. The strangle . . . . . . . . . . . . . . . . . . . . . 2.7. Option Spreads . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. Bull and bear spreads with call options . . . . . . 2.7.2. Bull and bear spreads with put options . . . . . . 2.7.3. Box spread . . . . . . . . . . . . . . . . . . . . . . 2.7.3.1. Definitions and examples . . . . . . . . . 2.7.3.2. Trading a box spread . . . . . . . . . . . 2.8. Butterfly Strategies . . . . . . . . . . . . . . . . . . . . . . . 2.8.1. Butterfly spread with calls . . . . . . . . . . . . . . 2.8.2. Butterfly spread with puts . . . . . . . . . . . . . . 2.9. Condor Strategies . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1. Condor strategy with calls . . . . . . . . . . . . . . 2.9.2. Condor strategy with puts . . . . . . . . . . . . . . 2.10. Ratio Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11. Some Combinations of Options with Bonds and Stocks . . . 2.11.1. Covered call: short a call and hold the underlying asset . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.2. Portfolio insurance . . . . . . . . . . . . . . . . . . 2.11.3. Mimicking portfolios and synthetic instruments . . 2.11.3.1. Mimicking the underlying asset . . . . . 2.11.3.2. Synthetic underlying asset: Long call plus a short put and bonds . . . . . . . . . . 2.11.3.3. The synthetic put: put-call parity relationship . . . . . . . . . . . . . . . .

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Contents

2.12. Conversions and Reversals . . . . . . . . . . . . . . . . . . . 2.13. Case study: Selling Calls (Without Holding the Stocks/as an Alternative to Short Selling Stocks/the Idea of Selling Calls is Also an Alternative to Buying Puts) . . . . . . . . . . . . 2.13.1. Data and assumptions . . . . . . . . . . . . . . . . 2.13.1.1. Selling calls (without holding the stock) 2.13.1.2. Comparing the strategy of selling calls (with a short portfolio of stocks): the extreme case . . . . . . . . . . . . . . . . 2.13.1.3. Selling calls (holding the stock) . . . . . 2.13.2. Leverage in selling call options (without holding the stocks) . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.2.1. Selling Call options (without holding the stocks) . . . . . . . . . . . . . . . . . . . 2.13.2.2. Leverage in selling Call options (without holding the stocks): The extreme case . 2.13.2.3. Selling calls using leverage (and holding the stock) . . . . . . . . . . . . . . . . . 2.13.3. Short sale of the stocks without options . . . . . . 2.14. Buying Calls on EMA . . . . . . . . . . . . . . . . . . . . . 2.14.1. Buying a call as an alternative to buying the stock: (also as an alternative to short sell put options) . . 2.14.1.1. Data and assumptions . . . . . . . . . . 2.14.1.2. Pattern of risk and return . . . . . . . . 2.14.2. Compare buying calls (as an alternative to portfolio of stocks) . . . . . . . . . . . . . . . . . . . . . . . 2.14.2.1. Risk return in options . . . . . . . . . . 2.14.3. Example by changing volatility to 20% . . . . . . . 2.14.3.1. Data and assumptions: . . . . . . . . . . 2.14.3.2. Compare buying calls (as an alternative to portfolio of stocks.) . . . . . . . . . . 2.14.3.3. Leverage in buying call options (without selling the underlying) . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Study: Comparisons Between put and Call Options . . . . . . 1. Buying Puts and Selling Puts Naked . . . . . . . . . . . . . 1.1. Buying puts . . . . . . . . . . . . . . . . . . . . . . 1.2. Selling puts . . . . . . . . . . . . . . . . . . . . . .

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2.

Buying and Selling Calls . . . . . . . . . . . . . . . . . . . . 2.1. Buying calls . . . . . . . . . . . . . . . . . . . . . . 2.2. Selling a call . . . . . . . . . . . . . . . . . . . . . 3. Strategy of Buying a Put and Hedge and Selling a Put and Hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Strategy of selling put and hedge: sell delta units of the underlying . . . . . . . . . . . . . . . . . . . . 3.2. Strategy of buy put and hedge: buy delta units of the underlying . . . . . . . . . . . . . . . . . . . . 4. Strategy of Buy Call, Sell Put, and Buy Call, Sell Put and Hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Strategy of Buy Call, Sell Put: Equivalent to Holding the Underlying . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Strategy of Buy Call, Sell Put and Hedge: Reduces Profits and Reduces Losses . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3. TRADING OPTIONS AND THEIR UNDERLYING ASSET: RISK MANAGEMENT IN DISCRETE TIME Chapter Outline . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . 3.1. Basic Strategies and Synthetic Positions . . 3.1.1. Options and synthetic positions . . 3.1.2. Long or short the underlying asset 3.1.3. Long a call . . . . . . . . . . . . . 3.1.4. Short call . . . . . . . . . . . . . . 3.1.5. Long a put . . . . . . . . . . . . . 3.1.6. Short a put . . . . . . . . . . . . . 3.2. Combined Strategies . . . . . . . . . . . . . 3.2.1. Long a straddle . . . . . . . . . . . 3.2.2. Short a straddle . . . . . . . . . . 3.2.3. Long a strangle . . . . . . . . . . . 3.2.4. Short a strangle . . . . . . . . . . 3.2.5. Long a tunnel . . . . . . . . . . . . 3.2.6. Short a tunnel . . . . . . . . . . . 3.2.7. Long a call bull spread . . . . . . . 3.2.8. Long a put bull spread . . . . . . .

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Contents

3.2.9. Long a call bear spread . . . . . . . . . . . . . . 3.2.10. Selling a put bear spread . . . . . . . . . . . . . 3.2.11. Long a butterfly . . . . . . . . . . . . . . . . . . 3.2.12. Short a butterfly . . . . . . . . . . . . . . . . . . 3.2.13. Long a condor . . . . . . . . . . . . . . . . . . . 3.2.14. Short a condor . . . . . . . . . . . . . . . . . . . 3.3. How Traders Use Option Pricing Models: Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Estimation of model parameters . . . . . . . . . 3.3.1.1. Historical volatility . . . . . . . . . . . 3.3.1.2. Implied volatilities and option pricing models . . . . . . . . . . . . . . . . . . 3.3.2. Trading and Greek letters . . . . . . . . . . . . . 3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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161 162 162 163 163 164

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166 167 167

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170 170 174 176 208 175 217

PART II. PRICING DERIVATIVES AND THEIR UNDERLYING ASSETS IN A DISCRETE-TIME SETTING

219

CHAPTER 4. OPTION PRICING: THE DISCRETETIME APPROACH FOR STOCK OPTIONS

221

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The CRR Model for Equity Options . . . . . . . . . . . . . 4.1.1. The mono-periodic model . . . . . . . . . . . . . . 4.1.2. The multiperiodic model . . . . . . . . . . . . . . . 4.1.3. Applications and examples . . . . . . . . . . . . . 4.1.3.1. Applications of the CRR model within two periods . . . . . . . . . . . . . . . . 4.1.3.2. Other applications of the binomial model of CRR for two periods . . . . . . . . . . 4.1.3.3. Applications of the binomial model of CRR for three periods . . . . . . . . . . 4.1.3.4. Examples with five periods . . . . . . .

221 221 222 222 224 226 226 228 231 234

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4.2.

The Binomial Model and the Distributions to the Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. The Put-Call parity in the presence of several cash-distributions . . . . . . . . . . . . . . . . . . . 4.2.2. Early exercise of American stock options . . . . . . 4.2.3. The model . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Simulations for a small number of periods . . . . . 4.2.5. Simulations in the presence of two dividend dates . 4.2.6. Simulations for different periods and several dividends: The general case . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: The Lattice Approach . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246 249 249 249 258

CHAPTER 5. CREDIT RISKS, PRICING BONDS, INTEREST RATE INSTRUMENTS, AND THE TERM STRUCTURE OF INTEREST RATES

259

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Time Value of Money and the Mathematics of Bonds . . . . 5.1.1. Single payment formulas . . . . . . . . . . . . . . . 5.1.2. Uniform-series present worth factor (USPWF) and the capital recovery factor (CRF) . . . . . . . . . . 5.1.3. Uniform-series compound-amount factor (USCAF ) and the sinking fund factor (SFF ) . . . . . . . . . 5.1.4. Nominal interest rates and continuous compounding 5.2. Pricing Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. A coupon-paying bond . . . . . . . . . . . . . . . . 5.2.2. Zero-coupon bonds . . . . . . . . . . . . . . . . . . 5.3. Computation of the Yield or the Internal Rate of Return . . 5.3.1. How to measure the yield . . . . . . . . . . . . . . 5.3.2. The CY . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. The YTM . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. The YTC . . . . . . . . . . . . . . . . . . . . . . . 5.3.5. The potential yield from holding bonds . . . . . . 5.4. Price Volatility Measures: Duration and Convexity . . . . . 5.4.1. Duration . . . . . . . . . . . . . . . . . . . . . . .

237 237 237 238 238 246

259 259 260 261 262 263 265 266 266 267 268 268 269 269 270 270 271 271

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5.4.2. Duration of a bond portfolio . . . . . . . . . . . . 5.4.3. Modified duration . . . . . . . . . . . . . . . . . . 5.4.4. Price volatility measures: Convexity . . . . . . . . 5.5. The Yield Curve and the Theories of Interest Rates . . . . . 5.5.1. The shapes of the yield curve . . . . . . . . . . . . 5.5.2. Theories of the term structure of interest rates . . 5.5.2.1. The pure expectations theory . . . . . . 5.6. The YTM and the Theories of the Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Computing the YTM . . . . . . . . . . . . . . . . 5.6.2. Market segmentation theory of the term structure 5.7. Spot Rates and Forward Interest Rates . . . . . . . . . . . . 5.7.1. The theoretical spot rate . . . . . . . . . . . . . . 5.7.2. Forward rates . . . . . . . . . . . . . . . . . . . . . 5.8. Issuing and Redeeming Bonds . . . . . . . . . . . . . . . . . 5.9. Mortgage-Backed Securities: The Monthly Mortgage Payments for a Level-Payment Fixed-Rate Mortgage . . . . 5.10. Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . 5.10.1. The pricing of interest rate swaps . . . . . . . . . . 5.10.2. The swap value as the difference between the prices of two bonds . . . . . . . . . . . . . . . . . . . . . 5.10.3. The valuation of currency swaps . . . . . . . . . . 5.10.4. Computing the swap . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 6. EXTENSIONS OF SIMPLE BINOMIAL OPTION PRICING MODELS TO INTEREST RATES AND CREDIT RISK Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. The Rendleman and Bartter Model (for details, refer to Bellalah et al., 1998) for Interest-Rate Sensitive Instruments 6.1.1. Using the model for coupon-paying bonds . . . . . 6.2. Ho and Lee Model for Interest Rates and Bond Options . . 6.2.1. The binomial dynamics of the term structure . . . 6.2.2. The binomial dynamics of bond prices . . . . . . .

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273 273 274 275 276 276 276 277 277 278 279 279 279 281 283 286 286 286 287 289 289 290 291

293 293 293 294 296 297 297 298

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6.2.3.

Computation of bond prices in the Ho and Lee model . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. Option pricing in the Ho and Lee model . . . . . . 6.2.5. Deficiency in the Ho and Lee model . . . . . . . . 6.3. Binomial Interest-Rate Trees and the Log-Normal Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. The Black-Derman-Toy Model (BDT) . . . . . . . . . . . . 6.4.1. Examples and applications . . . . . . . . . . . . . 6.5. Trinomial Interest-Rate Trees and the Pricing of Bonds . . 6.5.1. The model . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Applications of the binomial and trinomial models Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Ho and Lee model and binomial dynamics of bond prices . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 7. DERIVATIVES AND PATH-DEPENDENT DERIVATIVES: EXTENSIONS AND GENERALIZATIONS OF THE LATTICE APPROACH BY ACCOUNTING FOR INFORMATION COSTS AND ILLIQUIDITY Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The Standard Lattice Approach for Equity Options: The Standard Analysis . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. The model for options on a spot asset with any pay outs . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2. The model for futures options . . . . . . . . . . . . 7.1.3. The model with dividends . . . . . . . . . . . . . . 7.1.3.1. A known dividend yield . . . . . . . . . 7.1.3.2. A known proportional dividend yield . . 7.1.3.3. A known discrete dividend . . . . . . . . 7.1.4. Examples . . . . . . . . . . . . . . . . . . . . . . . 7.1.4.1. The European put price with dividends 7.1.4.2. The American put price with dividends 7.2. A Simple Extension to Account for Information Uncertainty in the Valuation of Futures and Options . . . . . . . . . . .

298 299 302 303 308 309 313 313 316 318 320 321 325

327 327 327 329 329 330 330 331 331 332 332 333 333 338

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7.2.1.

7.3.

On the valuation of derivatives and information costs . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. The valuation of forward and futures contracts in the presence of information costs . . . . . . . . . . 7.2.2.1. Forward, futures, and arbitrage . . . . . 7.2.2.2. The valuation of forward contracts in the absence of distributions to the underlying asset . . . . . . . . . . . . . . . . . . . . 7.2.2.3. The valuation of forward contracts in the presence of a known cash income to the underlying asset . . . . . . . . . . . . . . 7.2.2.4. The valuation of forward contracts in the presence of a known dividend yield to the underlying asset . . . . . . . . . . . . . . 7.2.2.5. The valuation of stock index futures . . 7.2.2.6. The valuation of Forward and futures contracts on currencies . . . . . . . . . . 7.2.2.7. The valuation of futures contracts on silver and gold . . . . . . . . . . . . . . 7.2.2.8. The valuation of Futures on other commodities . . . . . . . . . . . . . . . . 7.2.3. Arbitrage and information costs in the lattice approach . . . . . . . . . . . . . . . . . . . . . . . 7.2.4. The binomial model for options in the presence of a continuous dividend stream and information costs 7.2.5. The binomial model for options in the presence of a known dividend yield and information costs . . . . 7.2.6. The binomial model for options in the presence of a discrete dividend stream and information costs . . 7.2.7. The binomial model for futures options in the presence of information costs . . . . . . . . . . . . 7.2.8. The lattice approach for American options with information costs and several cash distributions . . 7.2.8.1. The model . . . . . . . . . . . . . . . . . The Binomial Model and the Risk Neutrality: Some Important Details . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. The binomial parameters and risk neutrality . . . 7.3.2. The convergence argument . . . . . . . . . . . . .

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7.4.

The Hull and White Trinomial Model for Interest Rate Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Pricing Path-Dependent Interest Rate Contingent Claims Using a Lattice . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. The framework . . . . . . . . . . . . . . . . . . . 7.5.2. Valuation of the path-dependent security . . . . 7.5.2.1. Fixed-coupon rate security . . . . . . . 7.5.2.2. Floating-coupon security . . . . . . . . 7.5.3. Options on path-dependent securities . . . . . . 7.5.3.1. Short-dated options . . . . . . . . . . . 7.5.3.2. Long-dated options . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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355 355 358 358 359 359 359 359 360 362 362

PART III. OPTION PRICING IN A CONTINUOUS-TIME SETTING: BASIC MODELS, EXTENSIONS AND APPLICATIONS

365

CHAPTER 8. EUROPEAN OPTION PRICING MODELS: THE PRECURSORS OF THE BLACK– SCHOLES–MERTON THEORY AND HOLES DURING MARKET TURBULENCE

367

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Precursors to the Black–Scholes Model . . . . . . . . . . 8.1.1. Bachelier formula . . . . . . . . . . . . . . . . . 8.1.2. Sprenkle formula . . . . . . . . . . . . . . . . . 8.1.3. Boness formula . . . . . . . . . . . . . . . . . . 8.1.4. Samuelson formula . . . . . . . . . . . . . . . . 8.2. How the Black–Scholes Option Formula is Obtained . . 8.2.1. The short story . . . . . . . . . . . . . . . . . . 8.2.2. The differential equation . . . . . . . . . . . . . 8.2.3. The derivation of the formula . . . . . . . . . . 8.2.4. Publication of the formula . . . . . . . . . . . . 8.2.5. Testing the formula . . . . . . . . . . . . . . . 8.3. Financial Theory and the Black–Scholes–Merton Theory 8.3.1. The Black–Scholes–Merton theory . . . . . . . 8.3.2. Analytical formulas . . . . . . . . . . . . . . .

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8.4.

The Black–Scholes Model . . . . . . . . . . . . . . . . . . . 8.4.1. The Black–Scholes model and CAPM . . . . . . . 8.4.2. An alternative derivation of the Black–Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. The put-call parity relationship . . . . . . . . . . . 8.4.4. Examples . . . . . . . . . . . . . . . . . . . . . . . 8.5. The Black Model for Commodity Contracts . . . . . . . . . 8.5.1. The model for forward, futures, and option contracts 8.5.2. The put-call relationship . . . . . . . . . . . . . . . 8.6. Application of the CAPM Model to Forward and Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1. An application of the model to forward and futures contracts . . . . . . . . . . . . . . . . . . . . . . . 8.6.2. An application to the derivation of the commodity option valuation . . . . . . . . . . . . . . . . . . . 8.6.3. An application to commodity options and commodity futures options . . . . . . . . . . . . . 8.7. The Holes in the Black–Scholes–Merton Theory and the Financial Crisis . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1. Volatility changes . . . . . . . . . . . . . . . . . . . 8.7.2. Interest rate changes . . . . . . . . . . . . . . . . . 8.7.3. Borrowing penalties . . . . . . . . . . . . . . . . . 8.7.4. Short-selling penalties . . . . . . . . . . . . . . . . 8.7.5. Transaction costs . . . . . . . . . . . . . . . . . . . 8.7.6. Taxes . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.7. Dividends . . . . . . . . . . . . . . . . . . . . . . . 8.7.8. Takeovers . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The Cumulative Normal Distribution Function . . . Appendix B. The Bivariate Normal Density Function . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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377 377 380 382 383 386 386 388 389 389 390 393 394 394 395 396 396 396 396 396 397 397 398 399 400 401

CHAPTER 9. SIMPLE EXTENSIONS AND APPLICATIONS OF THE BLACK–SCHOLES TYPE MODELS IN VALUATION AND RISK MANAGEMENT

403

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.3.

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Applications of the Black–Scholes Model . . . . . . . . . . . 9.1.1. Valuation and the role of equity options . . . . . . 9.1.2. Valuation and the role of index options . . . . . . 9.1.2.1. Analysis and valuation . . . . . . . . . . 9.1.2.2. Arbitrage between index options and futures . . . . . . . . . . . . . . . . . . . 9.1.3. Valuation of options on zero-coupon bonds . . . . 9.1.4. Valuation and the role of short-term options on long-term bonds . . . . . . . . . . . . . . . . . . . 9.1.5. Valuation of interest rate options . . . . . . . . . . 9.1.6. Valuation and the role of bond options: the case of coupon-paying bonds . . . . . . . . . . . . . . . . . 9.1.7. The valuation of a swaption . . . . . . . . . . . . . Applications of the Black’s Model . . . . . . . . . . . . . . . 9.2.1. Options on equity index futures . . . . . . . . . . 9.2.2. Options on currency forwards and options on currency futures . . . . . . . . . . . . . . . . . . 9.2.2.1. Options on currency forwards . . . . . . 9.2.2.2. Options on currency futures . . . . . . . 9.2.3. The Black’s model and valuation of interest rate caps . . . . . . . . . . . . . . . . . . . . . . . The Extension to Foreign Currencies: The Garman and Kohlhagen Model and its Applications . . . . . . . . . . . . 9.3.1. The currency call formula . . . . . . . . . . . . . . 9.3.2. The currency put formula . . . . . . . . . . . . . . 9.3.3. The interest-rate theorem and the pricing of forward currency options . . . . . . . . . . . . . . The Extension to Other Commodities: The Merton, Barone-Adesi and Whaley Model, and Its Applications . . . 9.4.1. The model . . . . . . . . . . . . . . . . . . . . . . 9.4.2. An application to portfolio insurance . . . . . . . . The Real World and the Black–Scholes Type Models . . . . 9.5.1. Volatility . . . . . . . . . . . . . . . . . . . . . . . 9.5.2. The hedging strategy . . . . . . . . . . . . . . . . . 9.5.3. The log-normal assumption . . . . . . . . . . . . . 9.5.4. A world of finite trading . . . . . . . . . . . . . . 9.5.5. Total variance . . . . . . . . . . . . . . . . . . . . 9.5.6. Black–Scholes as the limiting case . . . . . . . . . 9.5.7. Using the model to optimize hedging . . . . . . . .

404 404 405 405 406 407 408 409 410 411 413 413 414 414 414 415 416 416 416 417 420 420 421 422 422 422 422 423 423 423 424

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Summary Questions Appendix References

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CHAPTER 10. APPLICATIONS OF OPTION PRICING MODELS TO THE MONITORING AND THE MANAGEMENT OF PORTFOLIOS OF DERIVATIVES IN THE REAL WORLD Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Option-Price Sensitivities: Some Specific Examples . . . . . 10.1.1. Delta . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2. Gamma . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3. Theta . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4. Vega . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5. Rho . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.6. Elasticity . . . . . . . . . . . . . . . . . . . . . . . 10.2. Monitoring and Managing an Option Position in Real Time 10.2.1. Simulations and analysis of option price sensitivities using Barone-Adesi and Whaley model . . . . . . . 10.2.2. Monitoring and adjusting the option position in real time . . . . . . . . . . . . . . . . . . . . . . 10.2.2.1. Monitoring and managing the delta . . . 10.2.2.2. Monitoring and managing the gamma . 10.2.2.3. Monitoring and managing the theta . . . 10.2.2.4. Monitoring and managing the vega . . . 10.3. The Characteristics of Volatility Spreads . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Greek-Letter Risk Measures in Analytical Models . . A.1. B–S model . . . . . . . . . . . . . . . . . . . . . . A.2. Black’s Model . . . . . . . . . . . . . . . . . . . . . A.3. Garman and Kohlhagen’s model . . . . . . . . . . A.4. Merton’s and Barone-Adesi and Whaley’s model . Appendix B: The Relationship Between Hedging Parameters . . . Appendix C: The Generalized Relationship Between the Hedging Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D: A Detailed Derivation of the Greek Letters . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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439 439 440 441 441 442 443 444 444 445 445 446 451 451 454 457 458 459 460 461 461 462 463 463 464 465 479 478 489

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PART IV. MATHEMATICAL FOUNDATIONS OF OPTION PRICING MODELS IN A CONTINUOUS-TIME SETTING: BASIC CONCEPTS AND EXTENSIONS

491

CHAPTER 11. THE DYNAMICS OF ASSET PRICES AND THE ROLE OF INFORMATION: ANALYSIS AND APPLICATIONS IN ASSET AND RISK MANAGEMENT

493

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Continuous Time Processes for Asset Price Dynamics . . . 11.1.1. Asset price dynamics and Wiener process . . . . . 11.1.2. Asset price dynamics and the generalized Wiener process . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3. Asset price dynamics and the Ito process . . . . . 11.1.4. The log-normal property . . . . . . . . . . . . . . . 11.1.5. Distribution of the rate of return . . . . . . . . . . 11.2. Ito’s Lemma and Its Applications . . . . . . . . . . . . . . . 11.2.1. Intuitive form . . . . . . . . . . . . . . . . . . . . . 11.2.2. Applications to stock prices . . . . . . . . . . . . . 11.2.3. Mathematical form . . . . . . . . . . . . . . . . . . 11.2.4. The generalized Ito’s formula . . . . . . . . . . . . 11.2.5. Other applications of Ito’s formula . . . . . . . . . 11.3. Taylor Series, Ito’s Theorem and the Replication Argument 11.3.1. The relationship between Taylor series and Ito’s differential . . . . . . . . . . . . . . . . . . . . . . 11.3.2. Ito’s differential and the replication portfolio . . . 11.3.3. Ito’s differential and the arbitrage portfolio . . . . 11.3.4. Why are error terms neglected? . . . . . . . . . . . 11.4. Forward and Backward Equations . . . . . . . . . . . . . . 11.5. The Main Concepts in Bond Markets and the General Arbitrage Principle . . . . . . . . . . . . . . . . . . . . . . . 11.5.1. The main concepts in bond pricing . . . . . . . . . 11.5.2. Time-dependent interest rates and information uncertainty . . . . . . . . . . . . . . . . . . . . . . 11.5.3. The general arbitrage principle . . . . . . . . . . . 11.6. Discrete Hedging and Option Pricing . . . . . . . . . . . . . 11.6.1. Discrete hedging . . . . . . . . . . . . . . . . . . .

493 494 495 495 497 497 499 500 501 501 505 505 508 510 513 513 514 515 517 518 519 519 521 522 523 523

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11.6.2. 11.6.3.

Pricing the option . . . . . . . . . . . . The real distribution of returns and the hedging error . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Introduction to Diffusion Processes . . . . Appendix B: The Conditional Expectation . . . . . . . Appendix C: Taylor Series . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

525

. . . . . . . .

526 526 527 528 529 529 530 532

. . . . . . . .

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. . . . . . . .

. . . . . . . .

. . . . . . . .

CHAPTER 12. RISK MANAGEMENT: APPLICATIONS TO THE PRICING OF ASSETS AND DERIVATIVES IN COMPLETE MARKETS Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Characterization of Complete Markets . . . . . . . . . . . . 12.2. Pricing Derivative Assets: The Case of Stock Options . . . . 12.2.1. The problem . . . . . . . . . . . . . . . . . . . . . 12.2.2. The PDE method . . . . . . . . . . . . . . . . . . 12.2.3. The martingale method . . . . . . . . . . . . . . . 12.3. Pricing Derivative Assets: The Case of Bond Options and Interest Rate Options . . . . . . . . . . . . . . . . . . . . . 12.3.1. Arbitrage-free family of bond prices . . . . . . . . 12.3.2. Time-homogeneous models . . . . . . . . . . . . . 12.3.3. Time-inhomogeneous models . . . . . . . . . . . . 12.4. Asset Pricing in Complete Markets: Changing Numeraire and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1. Assumptions and the valuation context . . . . . . 12.4.2. Valuation of derivatives in a standard Black–Scholes–Merton economy . . . . . . . . . . . 12.4.3. Changing numeraire and time: The martingale approach and the PDE approach . . . . . . . . . . 12.5. Valuation in an Extended Black and Scholes Economy in the Presence of Information Costs . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: The Change in Probability and the Girsanov Theorem

535 535 536 536 538 538 539 541 546 546 547 550 551 551 552 554 560 563 564 564

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Appendix B: Resolution of the Partial Differential Equation for a European Call Option on a Non-Dividend Paying Stock in the Standard Context . . . . . . . . . . . . . . . . . . . . Appendix C: Approximation of the Cumulative Normal Distribution Appendix D: Leibniz’s Rule for Integral Differentiation . . . . . . . Appendix E: Pricing Bonds: Mathematical Foundations . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565 571 572 573 575 580

CHAPTER 13. SIMPLE EXTENSIONS AND GENERALIZATIONS OF THE BLACK–SCHOLES TYPE MODELS IN THE PRESENCE OF INFORMATION COSTS

583

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Differential Equation for a Derivative Security on a Spot Asset in the Presence of a Continuous Dividend Yield and Information Costs . . . . . . . . . . . . . . . . . . . . . . . 13.2. The Valuation of Securities Dependent on Several Variables in the Presence of Incomplete Information: A General Method . . . . . . . . 13.3. The General Differential Equation for the Pricing of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Extension of the Risk-Neutral Argument in the Presence of Information Costs . . . . . . . . . . . . . . . . . . . . . . . 13.5. Extension to Commodity Futures Prices within Incomplete Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1. Differential equation for a derivative security dependent on a futures price in the presence of information costs . . . . . . . . . . . . . . . . . 13.5.2. Commodity futures prices . . . . . . . . . . . . . . 13.5.3. Convenience yields . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: A General Equation for Derivative Securities . . . . . Appendix B: Extension to the Risk-Neutral Valuation Argument . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583 583

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585 588 589 590

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PART V. EXTENSIONS OF OPTION PRICING THEORY TO AMERICAN OPTIONS AND INTEREST RATE INSTRUMENTS IN A CONTINUOUS-TIME SETTING: DIVIDENDS, COUPONS AND STOCHASTIC INTEREST RATES

613

CHAPTER 14. EXTENSION OF ASSET AND RISK MANAGEMENT IN THE PRESENCE OF AMERICAN OPTIONS: DIVIDENDS, EARLY EXERCISE, AND INFORMATION UNCERTAINTY

615

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1. The Valuation of American Options: The General Problem 14.1.1. Early exercise of American calls . . . . . . . . . . . 14.1.2. Early exercise of American puts . . . . . . . . . . . 14.1.3. The American put option and its critical stock price 14.2. Valuation of American Commodity Options and Futures Options with Continuous Distributions . . . . . . . . . . . . 14.2.1. Valuation of American commodity options . . . . . 14.2.2. Examples and applications . . . . . . . . . . . . . 14.2.3. Valuation of American futures options . . . . . . . 14.2.4. Examples and applications . . . . . . . . . . . . . 14.3. Valuation of American Commodity and Futures Options with Continuous Distributions within Information Uncertainty . 14.3.1. Commodity option valuation with information costs 14.3.2. Simulation results . . . . . . . . . . . . . . . . . . 14.4. Valuation of American Options with Discrete Cash-Distributions . . . . . . . . . . . . . . . . . . . . . . . 14.4.1. Early exercise of American options . . . . . . . . 14.4.2. Valuation of American options with dividends . . . 14.5. Valuation of American Options with Discrete Cash Distributions within Information Uncertainty . . . . . . . . 14.5.1. The model . . . . . . . . . . . . . . . . . . . . . . 14.5.2. Simulation results . . . . . . . . . . . . . . . . . . 14.6. The Valuation Equations for Standard and Compound Options with Information Costs . . . . . . . . . . . . . . . . 14.6.1. The pricing of assets under incomplete information 14.6.2. The valuation of equity as a compound option . .

615 616 618 618 620 623 627 627 630 632 634 634 634 638 640 641 642 645 645 647 648 650 650

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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: An Alternative Derivation of the Compound Formula Using the Martingale Approach . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . Option’s . . . . . . . . . . . . . . .

CHAPTER 15. RISK MANAGEMENT OF BONDS AND INTEREST RATE SENSITIVE INSTRUMENTS IN THE PRESENCE OF STOCHASTIC INTEREST RATES AND INFORMATION UNCERTAINTY: THEORY AND TESTS Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1. The Valuation of Bond Options and Interest Rate Options . 15.1.1. The problems in using the B–S model for interest-rate options . . . . . . . . . . . . . . . . . 15.1.2. Sensitivity of the theoretical option prices to changes in factors . . . . . . . . . . . . . . . . . . 15.2. A Simple Non-Parametric Approach to Bond Futures Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1. Canonical modeling and option pricing theory . . . 15.2.2. Assessing the distribution of the underlying futures price . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3. Transforming actual probabilities into risk-neutral probabilities . . . . . . . . . . . . . . . . . . . . . . 15.2.4. Qualitative comparison of Black and canonical model values . . . . . . . . . . . . . . . . . . . . . 15.3. One-Factor Interest Rate Modeling and the Pricing of Bonds: The General Case . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1. Bond pricing in the general case: The arbitrage argument and information costs . . . . . . . . . . . 15.3.2. Pricing callable bonds within information uncertainty . . . . . . . . . . . . . . . . . . . . . . 15.4. Fixed Income Instruments as a Weighted Portfolio of Power Options . . . . . . . . . . . . . . . . . . . . . . . . . 15.5. Merton’s Model for Equity Options in the Presence of Stochastic Interest Rates: Two-Factor Models . . . . . .

654 656 656 657 664

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15.5.1.

The model in the presence of stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2. Applications of Merton’s model . . . . . . . . . . . 15.6. Some Models for the Pricing of Bond Options . . . . . . . . 15.6.1. An extension of the Ho-Lee model for bond options 15.6.2. The Schaefer and Schwartz model . . . . . . . . . 15.6.3. The Vasicek (1977) model . . . . . . . . . . . . . . 15.6.4. The Ho and Lee model . . . . . . . . . . . . . . . . 15.6.5. The Hull and White model . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Government Bond Futures and Implicit Embedded Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Criteria for the CTD . . . . . . . . . . . . . . . . . A.2. Yield changes . . . . . . . . . . . . . . . . . . . . . A.3. The value for a futures position . . . . . . . . . . . A.4. Parallel yield shift . . . . . . . . . . . . . . . . . . A.5. Relative yield shift . . . . . . . . . . . . . . . . . . Appendix B: One-Factor Fallacies for Interest Rate Models . . . . B.1. The models in practice . . . . . . . . . . . . . . . . B.2. Spreads between rates . . . . . . . . . . . . . . . . Appendix C: Merton’s Model in the Presence of Stochastic Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 16. MODELS OF INTEREST RATES, INTEREST-RATE SENSITIVE INSTRUMENTS, AND THE PRICING OF BONDS: THEORY AND TESTS Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1. Interest Rates and Interest-Rate Sensitive Instruments 16.1.1. Zero-coupon bonds . . . . . . . . . . . . . . . 16.1.2. Term structure of interest rates . . . . . . . . 16.1.3. Forward interest rates . . . . . . . . . . . . . 16.1.4. Short-term interest rate . . . . . . . . . . . . 16.1.5. Coupon-bearing bonds . . . . . . . . . . . . . 16.1.6. Yield-to-Maturity (YTM) . . . . . . . . . . . 16.1.7. Market conventions . . . . . . . . . . . . . . .

679 680 681 681 683 683 684 685 686 687 687 688 688 690 691 692 692 693 693 694 701

703 . . . . . . . . . .

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703 704 705 705 705 706 707 707 708 709

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16.2. Interest Rates and the Pricing of Bonds . . . . . . . . . . . 16.2.1. The instantaneous interest rates under certainty . 16.2.2. The instantaneous interest rate under uncertainty 16.3. Interest Rate Processes and the Pricing of Bonds and Options 16.3.1. The Vasicek model . . . . . . . . . . . . . . . . . . 16.3.2. The Brennan and Schwartz model . . . . . . . . . 16.3.3. The CIR model . . . . . . . . . . . . . . . . . . . . 16.3.4. The Ho and Lee model . . . . . . . . . . . . . . . . 16.3.5. The HJM model . . . . . . . . . . . . . . . . . . . 16.3.6. The BDT model . . . . . . . . . . . . . . . . . . . 16.3.7. The Hull and White model . . . . . . . . . . . . . 16.3.8. Fong and Vasicek model . . . . . . . . . . . . . . . 16.3.9. Longstaff and Schwartz model . . . . . . . . . . . . 16.4. The Relative Merits of the Competing Models . . . . . . . . 16.5. A Comparative Analysis of Term Structure Estimation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1. The construction of the term structure and coupon bonds . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2. Fitting functions and estimation procedure . . . . 16.6. Term Premium Estimates From Zero-Coupon Bonds: New Evidence on the Expectations Hypothesis . . . . . . . . . . 16.7. Distributional Properties of Spot and Forward Interest Rates: USD, DEM, GBP, and JPY . . . . . . . . . . . . . . . . . . 16.7.1. Interest rate levels . . . . . . . . . . . . . . . . . . 16.7.2. Interest rate differences and log differences . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: An Application of Interest Rate Models to Account for Information Costs: An Exercise . . . . . . . . . . . . . . . . A.1. An application of the HJM model in the presence of information costs . . . . . . . . . . . . . . . . . . . A.1.1. The forward rate equation . . . . . . . . A.1.2. The spot rate process . . . . . . . . . . . A.1.3. The market price of risk . . . . . . . . . A.1.4. Relationship between risk-neutral forward rate drift and volatility . . . . . . . . . . A.1.5. Pricing derivatives . . . . . . . . . . . . A.2. An application of the Ho and Lee model in the presence of information cost . . . . . . . . . . . . .

710 710 711 712 713 713 713 714 714 716 717 718 718 718 721 721 722 724 726 728 728 731 732 732 732 733 734 735 735 736

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Appendix B: Implementation of the BDT Model with Different Volatility Estimators . . . . . . . . . . . . . . . . . . . . B.1. The BDT model . . . . . . . . . . . . . . . . . B.2. Estimation results . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

737 737 738 739 740

PART VI. GENERALIZATION OF OPTION PRICING MODELS AND STOCHASTIC VOLATILITY

743

CHAPTER 17. EXTREME MARKET MOVEMENTS, RISK AND ASSET MANAGEMENT: GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES, AND INFORMATION COSTS

745

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1. The Jump-Diffusion and the Constant Elasticity of Variance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1. The jump-diffusion model . . . . . . . . . . . . . . 17.1.2. The constant elasticity of variance diffusion (CEV) process . . . . . . . . . . . . . . . . . . . . . . . . 17.2. On Jumps, Hedging and Information Costs . . . . . . . . . 17.2.1. Hedging in the presence of jumps . . . . . . . . . . 17.2.2. Hedging the jumps . . . . . . . . . . . . . . . . . . 17.2.3. Jump volatility . . . . . . . . . . . . . . . . . . . . 17.3. On the Smile Effect and Market Imperfections in the Presence of Jumps and Incomplete Information . . . . . . . 17.3.1. On smiles and jumps . . . . . . . . . . . . . . . . . 17.3.2. On smiles, jumps, and incomplete information . . 17.3.3. Empirical results in the presence of jumps and incomplete information . . . . . . . . . . . . . . . 17.4. Implied Volatility and Option Pricing Models: The Model and Simulation Results . . . . . . . . . . . . . . . . . . . . . 17.4.1. The valuation model . . . . . . . . . . . . . . . . . 17.4.2. Simulation results . . . . . . . . . . . . . . . . . . 17.4.3. Model calibration and the smile effect . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

745 745 747 747 748 749 750 752 753 754 754 759 760 763 763 765 766 767 768 768

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CHAPTER 18. RISK MANAGEMENT DURING ABNORMAL MARKET CONDITIONS: FURTHER GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES, AND INFORMATION COSTS Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1. Option Pricing in the Presence of a Stochastic Volatility . . 18.1.1. The Hull and White model . . . . . . . . . . . . . 18.1.2. Stein and Stein model . . . . . . . . . . . . . . . . 18.1.3. The Heston model . . . . . . . . . . . . . . . . . . 18.1.4. The Hoffman, Platen, and Schweizer model . . . . 18.1.5. Market price of volatility risk . . . . . . . . . . . . 18.1.6. The market price of risk for traded assets . . . . . 18.2. Generalization of Some Models with Stochastic Volatility and Information Costs . . . . . . . . . . . . . . . . . . . . . 18.2.1. Generalization of the Hull and White (1987) model 18.2.2. Generalization of Wiggins’s model . . . . . . . . . 18.2.3. Generalization of Stein and Stein’s model . . . . . 18.2.4. Generalization of Heston’s model . . . . . . . . . . 18.2.5. Generalization of Johnson and Shanno’s model . . 18.3. The Volatility Smiles: Some Standard Results . . . . . . . . 18.3.1. The smile effect in stock options and index options 18.3.2. The smile effect for bond and currency options . . 18.3.3. Volatility smiles: Empirical evidence . . . . . . . . 18.4. Empirical Results Regarding Information Costs and Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1. Information costs and option pricing: The estimation method . . . . . . . . . . . . . . . 18.4.2. The asymmetric distortion of the smile . . . . . . . 18.4.3. Asymmetric Smiles and information costs in a stochastic volatility model . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

771 771 772 774 774 775 777 778 781 782 782 782 784 785 786 788 788 788 789 790 791 791 792 793 795 796 796

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PART VII. OPTION PRICING MODELS AND NUMERICAL ANALYSIS

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CHAPTER 19. RISK MANAGEMENT, NUMERICAL METHODS AND OPTION PRICING

801

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1. Numerical Analysis and Simulation Techniques: An Introduction to Finite Difference Methods . . . . . . . . . . 19.1.1. The implicit difference scheme . . . . . . . . . . . 19.1.2. Explicit difference scheme . . . . . . . . . . . . . . 19.1.3. An extension to account for information costs . . . 19.2. Application to European Options on Non-Dividend Paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1. The analytic solution . . . . . . . . . . . . . . . . . 19.2.2. The numerical solution . . . . . . . . . . . . . . . . 19.2.3. An application to European calls on non-dividend paying stocks in the presence of information costs 19.3. Valuation of American Options with a Composite Volatility 19.3.1. The effect of interest rate volatility on the index volatility . . . . . . . . . . . . . . . . . . . . . . . 19.3.2. Valuation of index options with a composite volatility . . . . . . . . . . . . . . . . . . . . . . . 19.3.3. Numerical solutions and simulations . . . . . . . . 19.4. Simulation Methods: Monte–Carlo Method . . . . . . . . . 19.4.1. Simulation of Gaussian variables . . . . . . . . . . 19.4.2. Relationship between option values and simulation methods . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Simple Concepts in Numerical Analysis . . . . . . . . Appendix B: An Algorithm for a European Call . . . . . . . . . . Appendix C: The Algorithm for the Valuation of American Long-term Index Options with a Composite Volatility . . .

801 801 803 804 806 807 807 808 808 810 811 811 812 813 817 818 818 819 820 820 822 823

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Exercises . . . . . . . . . . . . . . . . . . . Appendix D: The Monte–Carlo Method and Asset Prices . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . the Dynamics of . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 20. NUMERICAL METHODS AND PARTIAL DIFFERENTIAL EQUATIONS FOR EUROPEAN AND AMERICAN DERIVATIVES WITH COMPLETE AND INCOMPLETE INFORMATION Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1. Valuation of American Calls on Dividend-Paying Stocks . . 20.1.1. The Schwartz model . . . . . . . . . . . . . . . . . 20.1.2. The numerical solution . . . . . . . . . . . . . . . . 20.2. American Puts on Dividend-Paying Stocks . . . . . . . . . . 20.2.1. The Brennan and Schwartz model . . . . . . . . . 20.2.2. The numerical solution . . . . . . . . . . . . . . . . 20.3. Numerical Procedures in the Presence of Information Costs: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1. Finite difference methods in the presence of information costs . . . . . . . . . . . . . . . . . . . 20.3.2. An application to the American put using explicit or implicit finite difference methods . . . . . . . . 20.4. Convertible Bonds . . . . . . . . . . . . . . . . . . . . . . . 20.4.1. Specific features of CB . . . . . . . . . . . . . . . . 20.4.2. The valuation equations . . . . . . . . . . . . . . . 20.4.3. The numerical solution . . . . . . . . . . . . . . . . 20.4.4. Simulations . . . . . . . . . . . . . . . . . . . . . . 20.5. Two-Factor Interest Rate Models and Bond Pricing within Information Uncertainty . . . . . . . . . . . . . . . . . . . . 20.6. CBs Pricing within Information Uncertainty . . . . . . . . . 20.6.1. The pricing of CBs . . . . . . . . . . . . . . . . . . 20.6.2. Specific call and put features . . . . . . . . . . . . 20.6.3. The pricing of CBs in two-factor models within information uncertainty . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: A Discretizing Strategy for Mean-Reverting Models . Appendix B: An Algorithm for the American Call with Dividends

826 829 830

833 833 834 835 835 836 837 837 838 839 839 841 842 842 843 845 847 847 850 850 851 851 852 853 861

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Contents

Appendix C: The Appendix D: The Questions . . . . Exercises . . . . References . . . . PART VIII.

Algorithm for the American Put with Dividends Algorithm for CBs with Call and Put Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

EXOTIC DERIVATIVES

CHAPTER 21. RISK MANAGEMENT: EXOTICS AND SECOND-GENERATION OPTIONS Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. Exchange Options . . . . . . . . . . . . . . . . . . . . . . . 21.2. Forward-Start Options . . . . . . . . . . . . . . . . . . . . . 21.3. Pay-Later Options . . . . . . . . . . . . . . . . . . . . . . . 21.4. Simple Chooser Options . . . . . . . . . . . . . . . . . . . . 21.5. Complex Choosers . . . . . . . . . . . . . . . . . . . . . . . 21.6. Compound Options . . . . . . . . . . . . . . . . . . . . . . . 21.6.1. The call on a call in the presence of a cost of carry 21.6.2. The put on a call in the presence of a cost of carry 21.6.3. The formula for a call on a put in the presence of a cost of carry . . . . . . . . . . . . . . . . . . . . . 21.6.4. The put on a put in the presence of a cost of carry 21.7. Options on the Maximum (Minimum) . . . . . . . . . . . . 21.7.1. The call on the minimum of two assets . . . . . . . 21.7.2. The call on the maximum of two assets . . . . . . 21.7.3. The put on the minimum (maximum) of two assets 21.8. Extendible Options . . . . . . . . . . . . . . . . . . . . . . . 21.8.1. The valuation context . . . . . . . . . . . . . . . . 21.8.2. Extendible calls . . . . . . . . . . . . . . . . . . . . 21.9. Equity-Linked Foreign Exchange Options and Quantos . . . 21.9.1. The foreign equity call struck in foreign currency . 21.9.2. The foreign equity call struck in domestic currency 21.9.3. Fixed exchange rate foreign equity call . . . . . . . 21.9.4. An equity-linked foreign exchange call . . . . . . . 21.10. Binary Barrier Options . . . . . . . . . . . . . . . . . . . . . 21.10.1. Path-independent binary options . . . . . . . . . .

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862 864 867 867 872 875

877 877 877 879 880 882 884 885 886 887 888 888 889 889 891 892 892 893 893 894 896 898 898 899 900 901 902

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21.10.1.1. Standard cash-or-nothing options . . . 21.10.1.2. Cash-or-nothing options with shadow costs . . . . . . . . . . . . . . . . . . . 21.10.1.3. Standard asset-or-nothing options . . . 21.10.1.4. Asset-or-nothing options with shadow costs . . . . . . . . . . . . . . . . . . . 21.10.1.5. Standard gap options . . . . . . . . . . 21.10.1.6. Gap options with shadow costs . . . . 21.10.1.7. Supershares . . . . . . . . . . . . . . . 21.11. Lookback Options . . . . . . . . . . . . . . . . . . . . . . 21.11.1. Standard lookback options . . . . . . . . . . . . 21.11.2. Options on extrema . . . . . . . . . . . . . . . . 21.11.2.1. On the maximum . . . . . . . . . . . . 21.11.2.2. On the minimum . . . . . . . . . . . . 21.11.3. Limited risk options . . . . . . . . . . . . . . . . 21.11.4. Partial lookback options . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

902

. .

903 904

. . . . . . . . . . . . . .

905 906 908 908 908 909 909 909 910 910 911 913 914 914

CHAPTER 22. VALUE AT RISK, CREDIT RISK, AND CREDIT DERIVATIVES Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1. VaR and Riskmetrics: Definitions and Basic Concepts . . 22.1.1. The definition of risk . . . . . . . . . . . . . . . . 22.1.2. VaR: Definition . . . . . . . . . . . . . . . . . . . 22.2. Statistical and Probability Foundation of VaR . . . . . . . 22.2.1. Using percentiles or quantiles to measure market risk . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.2. The choice of the horizon . . . . . . . . . . . . . 22.3. A More Advanced Approach to VaR . . . . . . . . . . . . 22.4. Credit Valuation and the Creditmetrics Approach . . . . . 22.4.1. The portfolio context of credit . . . . . . . . . . 22.4.2. Different credit risk measures . . . . . . . . . . . 22.4.3. Stand alone or single exposure risk calculation . 22.4.4. Differing exposure type . . . . . . . . . . . . . .

917 . . . . . .

917 917 919 920 920 921

. . . . . . . .

922 922 923 927 927 927 928 928

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22.5. Default and Credit-Quality Migration in the Creditmetrics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.1. Default . . . . . . . . . . . . . . . . . . . . . . . . 22.5.2. Credit-quality migration . . . . . . . . . . . . . . . 22.5.3. Historical tabulation and recovery rates . . . . . . 22.6. Credit-Quality Correlations . . . . . . . . . . . . . . . . . . 22.7. Portfolio Management of Default Risk in the Kealhofer, McQuown and Vasicek (KMV) Approach . . . . . . . . . . 22.7.1. The model of default risk . . . . . . . . . . . . . . 22.7.2. Asset market value and volatility . . . . . . . . . . 22.8. Credit Derivatives: Definitions and Main Concepts . . . . . 22.8.1. Forward contracts . . . . . . . . . . . . . . . . . . 22.8.2. The structure of credit-default instruments . . . . 22.8.2.1. Total return swaps . . . . . . . . . . . . 22.8.2.2. Credit-default swaps . . . . . . . . . . . 22.8.2.3. Basket default swaps . . . . . . . . . . . 22.8.2.4. Credit-default exchange swap . . . . . . 22.8.2.5. Credit-linked notes (CLNs) . . . . . . . 22.9. The Rating Agencies Models and the Proprietary Models . 22.9.1. The rating agencies models . . . . . . . . . . . . . 22.9.2. The proprietary models . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

932 932 933 933 933 934 934 934 935 935 936 936 936 938 940 941

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I Financial Markets and Financial Instruments: Basic Concepts and Strategies

1

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Chapter 1

FINANCIAL MARKETS, FINANCIAL INSTRUMENTS, AND FINANCIAL CRISIS

Chapter Outline This chapter is organized as follows:

1. Section 1.1 presents the trading characteristics of commodity contracts. The analysis concerns mainly oil markets. 2. Section 1.2 studies the main trading characteristics of commodity markets and instruments. The analysis concerns the instruments in the International Petroleum Exchange. 3. Section 1.3 develops the characteristics of crude oils and the properties of petroleum products. 4. Section 1.4 presents the trading characteristics of another commodity: Cocoa. 5. Section 1.5 illustrates the main trading characteristics of options. The analysis pertains mainly to equity options. 6. Section 1.6 studies the trading characteristics of options on currency forwards and futures. 7. Section 1.7 develops the trading characteristics of options. The analysis pertains mainly to bonds and bond options markets. 8. Section 1.8 provides examples of simple and complex financial instruments. 9. Section 1.9 provides the main reasons behind financial innovations.

3

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Introduction In the last three decades, there has been a wave of financial innovations and structural changes in the securities industry. The three main natural questions which arise are: What are the specific features of the “new” financial contracts? Why are there so many financial contracts? What are the fundamental reasons behind the proliferation of financial assets? A partial answer to such questions is given in the analysis of Miller (1986), Merton (1988), and Ross (1989), among others. These financial instruments are traded either in organized markets or in non-organized markets, known as over-the-counter markets, or OTC markets. These products are presented either in a straight forward form or in a package. They can be used to create several combinations with different risk and reward trade-offs. Financial crisis, subprime and credit crunch in 2008–2009 are exacerbated by the use of derivatives in the areas of mortgages, credit and other areas of finance. What are derivatives? A derivative is a generic term to encompass all financial transactions, which are not directly traded in the primary physical market. It refers to a financial instrument to manage a given risk. The term includes forwards, futures, options, commodity contracts, etc. What is a forward contract They are the simplest and most basic hedging instruments. A forward contract is an agreement between two parties to set the price today for a transaction that will not be completed until a specified date in the future. An example is a forward contract for $1 million, to be delivered in six months, at a price of 5.30 for a dollar. These terms obligate the seller of the contract to deliver $1 million on the company’s account, for the price set today. On the other hand, the buyer has no alternative than to accept delivery under the terms of the contract. The only possibility for the buyer to cancel the contract at a later date, is to enter into a reverse forward contract, with the same bank or another institution, but at the risk of a loss (or a gain) since

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Financial Markets, Financial Instruments, and Financial Crisis

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the new forward rate will be set at a new equilibrium level. Forward rate contracts are flexible and allow for customized hedges since all the terms can be negotiated with the counterparty. However, each side of the contract bears the risk that the other side defaults on the future commitments. That is why futures contracts are often referred to as forward contracts.

What is a futures contract? A future is an exchange-traded contract between a buyer and seller and the clearinghouse of a futures exchange to buy or sell a standard quantity and quality of a commodity at a specified future date and price. The clearinghouse acts as a counterpart in all transactions and is responsible for holding traders’ surety bonds to guarantee that transactions are completed. Like forward contracts, futures contracts are used to lock in the interest rate, exchange rate, or commodity price. But, futures markets are organized in such a way that the risk of default is always completely eliminated. This is possible by trading futures contracts on an organized exchange with a clearinghouse which steps in between a buyer and a seller, each time a deal is struck in the pit. The clearinghouse adopts the position of the buyer to every seller, and of the seller to every buyer, i.e., the clearinghouse keeps a zero net position. This means that every trader in the futures markets has obligations only to the clearinghouse, and has strong expectations that the clearinghouse will maintain its side of the bargain as well. The credibility of the system is maintained through the requirements of margin and daily settlements. The main purpose of the margin is to provide a safeguard to ensure that traders will perform their obligations. It is usually set to the maximum loss a trader can experience in a normal trading day. Daily settlements or marking to market just consists in a transfer of cash from one account to another. The elimination of default risk has a cost: contracts are standardized in order to bring liquidity to the market, there are only a few financial assets which are traded on futures markets and they do not necessarily correspond to the risk to be hedged. Therefore, there is no perfect hedge with futures contracts. The hedgers keep what is called a basis risk and a correlation risk which cannot be fully eliminated.

What are standard options? Options are more flexible than forwards and futures in the sense they provide the buyer with the protection needed, and leave him/her with the

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full benefits associated with a favorable development of the asset price, interest rate, or exchange rate. This nice feature has a price: the option premium. On the contrary, forward and futures prices are set at a level such that the initial price of the contract is exactly zero. A standard or a vanilla option is a security that gives its holder the right to buy or sell the underlying asset within a specified period of time, at a given price, called the strike price, striking price or strike price and no obligation to deliver. A European option can be exercised only on the last day of the contract, called the maturity date or the expiration date. An American option can be exercised at any time during the contract’s life.

What are commodity contracts? Commodity contracts are traded around the world. One of the main examples is oil. Oil has become one of the biggest commodity market in the world. Oil trading evolved from a primarily physical activity into a financial market. The physical oil market trades different types of crude oil and refined products. Prior to 1973, oil trading was a marginal activity and the industry was dominated by large integrated oil companies. The structure of the industry changed in the 1970s with the nationalization of the interests in major oil companies in the Middle East. The driving force behind rapid growth in oil trading is explained by the huge variability in the price of oil. Market participants are exposed to the risk of very large changes in the value of any oil. The emergence of the 15-day Brent market in the 1980s results mainly from the economic features of international trade in oil and the de-integration of the industry in the 1970s. As a consequence of nationalization in the OPEC region, the major companies lost many of the concessions which had provided them with equity oil. The 1979 crisis and de-integration created the necessary conditions for a market to merge. The major development in the late 1970s and early 1980s was the emergence of two systems of oil price determination. There was OPEC fixing at that time a price for a marker crude (Arabian Light) and a market for non-OPEC crudes in which prices were subjected to the pressures of economic forces. The developments spurred significant growth in market activity leading to the emergence of new trading instruments. By the end of 1985, the world market entered a new crisis. The oil shocks ended up with compromises that changed important features of the petroleum world. The international petroleum exchange, IPE, was established by representatives from 28 countries in order to offer the industry the means to manage the

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7

price risk. The first contract, gas oil futures began trading in 1981. A Brent crude futures contract was launched in 1988. A natural gas futures contract is also traded. A network of Quote Vendors relay information on a realtime basis to end users in several countries worldwide. The Brent crude contract and the gas oil contract are used as benchmarks or price references in trading. 1.1. Trading Characteristics of Commodity Contracts: The Case of Oil The oil market is ultimately concerned with the transportation, processing, and storage of a raw material. Since oil is a liquid, it requires specialized handling facilities for transportation, processing, and storage. These elements represent the basic building blocks for the physical oil market. The behavior of prices is influenced by the fundamental forces of demand and supply. The demand of oil depends on the state of the global economy. It is closely linked to the growth of the economy. The oil industry is not properly integrated. In general, oil producers maximize their output, subject to the technical constraints of the field. Since operating costs are lower than the sunk capital costs, oil is produced until its price reaches very low levels. Oil is often viewed as a highly political commodity. The threat of supply disruption remains real and political forces play an important role in the oil market. 1.1.1. Fixed prices Outright prices. A contract for the sale of a cargo of oil must stipulate the basic price, the guaranteed quality, and agreed price adjustment for quality deviations, availability date range, etc. These factors describe the elements of the price of a cargo of oil. Gasoil is sold in Europe at x dollars per metric tonne, based on a specific gravity. The important qualities for crude oil are gravity, measured in API degrees, metals content and sulfur. Crude is in general traded in US dollars per barrel. In Europe, oil products are generally sold in US dollars per metric tonnes. Timing can have an impact on pricing when the market is in backwardation or contango. Backwardation corresponds to a situation in which the price of the commodity available on a prompt basis is higher than that for deferred delivery. Contango corresponds to a situation in which a commodity is cheapest in the prompt position and gets

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progressively more expensive in the future. Hence, the oil price depends not only on its quality but also on the delivery date range. The location of the oil affects its price.

Official selling prices. Until the mid-1980s, traded crude oil, except that of US origin was priced at an official selling price, OSP. Even if an official price is quoted for crude oil, the price actually paid by a refiner is set in general at a premium or a discount to the OSP.

1.1.2. Floating prices As oil prices became volatile, there was increasing uncertainty about the value of oil at the time it was to be loaded or discharged. As the oil market moved away from fixed prices, oil prices reflected the market value of the oil at the time of moving the oil. The growth of the forward market, futures markets, swaps, and options markets developed the pricing mechanisms.

1.1.3. Exchange of futures for Physical (EFP) Exchange of futures for physical (EFP) provide a method of pricing a cargo of oil at a differential to the futures market. The buyer and the seller utilize existing futures positions that match their exposure on the physical oil market. The buyer of a physical cargo transfers ownership to the seller (of the cargo) of a certain number of futures contracts, equivalent to the volume of the cargo of oil. The value of the futures contract is used to calculate the price of the physical oil. The seller becomes long futures contracts and the buyer short futures contracts at the agreed level.

1.2. Description of Markets and Instruments: The Case of the International Petroleum Exchange Crude oil trade is a key nexus between the two main centers of activity: upstream exploration and production and downstream refining, and marketing. In this context, the price of crude oil results from the interaction between the signals provided by product markets and the revenue objectives of producers. The growth of the international spot market in crude oil

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and the ensuing transformation in the structure of oil markets explain the way oil is priced today. The growth in the international spot market during the early 1980s induced the emergence of a market discovery system driven by marginal spot trading, which replaced administered selling prices. Wide swings in prices have fostered the growth of forward and futures markets as well as several risk-management tools. The values of oil grades of crude oil depend on the refined products that can be made from individual grades. Each refined product resulting from a barrel of crude oil has its own separate markets under the law of supply and demand. The upgrading technologies maximize the product output from a barrel of crude oil. The starting point of the process is the distillation of the crude oil. This involves heating the crude oil to gradually higher temperatures giving different types of hydrocarbons. The cracking process allows to break lighter gasoline and gas oil fractions out of heavy gas oil and certain kinds of residue. Refiners and oil-market participants rely on detailed assays of actual cargoes in order to determine the specific features of an individual crude oil. A refiner must evaluate transportation alternatives and the price dynamics of the market. There are more than 100 crude oils in international trade.

1.3. Characteristics of Crude Oils and Properties of Petroleum Products Petroleum or crude oil can be described as a viscous brown to black liquid mixture. Petroleum contains a hydrocarbon mixture and non-hydrocarbon compounds such as sulfur, nitrogen, and oxygen compounds.

1.3.1. Specific features of some oil contracts The IPE Brent Crude futures contract is one of the most important energy price indicators in the world. This contract represents the critical part of the Brent Blend complex which represents the benchmark for two thirds of the world’s internationally traded crude oil.

Brent crude futures There is no maximum price fluctuation imposed upon Brent crude futures. The contract can be settled in cash against a physical price index calculated by the IPE. It can be settled with physical delivery through the EFP

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mechanism. The unit of trading is one or more lots of 1000 net barrels (42,000 US gallons) of Brent crude oil. The contract specifies the current pipeline export quality Brent blend as supplied at Sullom Voe. The contract price is in US dollars and cents per barrel. The minimum price fluctuation is one cent per barrel, which gives a tick value of US$10. All open contracts are marked-to-market daily. Gas oil futures The IPE gas oil contract is a benchmark for the physical market in Europe and beyond. This contract is used as a price basis for most middle distillate spot trades in northwest Europe. Companies can use this contract to evaluate arbitrage, storage, and investment opportunities. Natural gas futures The natural gas futures contract was launched in January 1997. Since the launch, many changes to the contract have been made as the industry has uncovered new needs and oportunities. The IPE natural gas futures (NBP) contracts are traded through the IPE automated energy trading system (ETS) or by the EFPs. The contract size corresponds to a minimum of 5 lots of 1000 therms per lot of natural gas per day during the contract period. The contract price is in Sterling and in pence per therm. If not closed out at expiry, contracts obligate delivery or taking delivery on each day in that contract period of the number of lots remaining open upon expiry. Options The IPE offers American options contracts for Brent crude and gas oil futures. Options enable companies to carry out several complex and hedging techniques. The unit of trading is represented for IPE gas oil options is one IPE gas oil futures contract. The contract price is in US dollars and cents per tonne. The strike price increments are multiples of US$5 per tonne. A minimum of 5 strike prices are listed for each contract month. Due to futures style margining, option premiums are not paid or received at the time of the transaction. Margins are received or paid each day according to the changing value of the option. The total value to be paid or received is only known when the position is closed. This is done by an opposing sale or purchase, the exercise or the maturity of the option. The options can be exercised into gas oil futures contracts.

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1.3.2. Description of Markets and Trading Instruments: The Brent Market Dated Brent and 15-day Brent Two types of transactions can be done in the physical Brent blend market. The first, known as “dated Brent” cargo is a conventional spot transaction which refers to the sale of a specific cargo. The cargo is either available in a specific loading slot or is loaded and in transit to some destination. The second, known as “15-day Brent” is a forward deal, which refers to a standard parcel that will be made available by the seller on an unspecified day of the relevant month. Oil is sold f.o.b. (insurance, freight, and ocean losses are the buyer’s responsibility), but demurrage at the terminal is the seller’s. For 15-day Brent, the contract is a standard telex and there is no exchange to match sellers and buyers. The clearing of the market involves all participants. The clearing consists of two different operations book-outs and the seller’s nominations. A book-out is simply an agreement between some participants to cancel their contracts with a cash settlement procedure for the difference between an agreed reference price and the contract price. The contracts which are not cleared by a book out cancellation are cleared through the nomination process. Sellers through the forward market serve 15-day notices to buyers of cargoes for the relevant month. The 15-day market reveals the buyer’s uncertainty regarding the exact date of delivery and it is characterized by a lack of perfect price/volume transparency.

Spread trading. Forward cargoes can either be traded as single cargoes with an absolute price agreed, or in spread trading. This latter case involves the simultaneous purchase or sale of at least two cargoes and appears in different forms.

Inter-month spreads. Spread trading in the Brent market appears as transactions of the difference in price between Brent for delivery in different months using two Brent cargoes. When trader A buys an April/May spread from trader B, then A has bought an April cargo from B and simultaneously sold a May cargo to B. The inter-month spread is simply a position on the absolute level of the backwardation or contango between the delivery months. In general, a

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contango appears when prices are higher for more distant delivery months. Backwardation is the reverse.

Inter-crude spreads. It is possible to trade the differential against another crude oil as Dubai or WTI. When trader A buys an April Brent-Dubai spread, he buys the April Brent cargo and sells the April Dubai cargo. An inter-crude spread is a position taken on the path of the difference in prices between the crude oils. Trader A will gain if the price of the Brent strengthens relative to that of Dubai.

The box trade. This strategy is implemented by taking a position on the movement of the relative backwardation (or contango) between two crude oil markets. The trader sells and buys simultaneously two inter-crude spreads. The strategy involves the trading of four cargoes. For example, a Brent-Dubai box spread involves the simultaneous purchase of a Brent cargo and the sale of a Dubai cargo for the same delivery month and the purchase of a Dubai cargo for another delivery month.

The IEP Brent futures contract. The International Petroleum Exchange of London (IEP) trades Brent futures contract on cargoes of 500,000 barrels. The physical base of the New York Mercantile Exchange (NYMEX) contract is pipeline scheduling at Cushing Oklahoma of 1000 barrel batches. IPE Brent contracts represent two contracts: one between the buyer and the clearing house and one between the seller and the clearing house. The clearing house is the International Commodities Clearing House Ltd (ICCH).

Exchange of futures for physical (EFPs). An EFP is a physical link between the IPE Brent futures contract and North Sea spot market on the 15-day market. This can be used to exchange an IPE position for a spot cargo. It represents, for example, the exchange of a futures market position of 500 IPE lots, (i.e., 500,000 barrels), for a

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15-day cargo. It can be viewed as a spread between forward Brent and futures Brent.

IPE Brent options. The IPE’s Brent American options contract was launched on 11 May 1989 trading lots of 1000 barrels of Brent blend. The IPE Brent futures contract is the underlying asset. The option is exercised, i.e., transferred to a Brent futures contract at any time before maturity. A call gives its holder the right to buy the underlying futures contract at a strike price defined in multiples of 50 cents per barrel. A put gives its holder the right to sell the underlying futures contract at a strike price defined also in multiples of 50 cents per barrel. Options are also traded on Brent delivery month spreads. The over-the-counter (OTC) market represents a series of personalized bilateral trades and provides tailor-made options on deals of any size. This market is used by several large financial institutions.

Swaps. The swap allows the producer or the consumer of crude oil and oil products to lock in a price or a margin. The main participants are finance houses and the trading departments of large oil companies. A producer can arrange a swap for a given volume over a specified period at a price equating to a “mean” market price over that period. At each agreed settlement period, actual market prices for the agreed volume are compared to the value of that volume under the specified price in the swap transaction. When market prices are higher, the producer pays the swaps provider the difference times the agreed volume. When market prices are lower, the swaps provider pays the producer the difference times the agreed volume. In the swap transaction, there is a physical exchange of oil, but a series of netted transactions or contract for the differences.

1.4. Description of Markets and Trading Instruments: The Case of Cocoa The International Cocoa Organization (ICCO) was established in 1973 to administer the first International Cocoa Agreement, that of 1972 and its successor Agreements, of 1975, 1980, 1986, and 1993. For further information, the reader can refer to [email protected]. The Agreements

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were concluded among the governments of cocoa producing and consuming countries, under the auspices of the United Nations. As on 13 July 1998, the membership of the ICCO comprised a total of 41 members. 1.4.1. How do the futures and physicals market work? There are large differences between physical and futures prices. The cocoa trade is based on the actuals market and the futures market. The actuals market is known as the physical market, the spot market or the cash market. Futures contracts are traded in lots of 10 tonnes. They represent a commitment to deliver or receive the quantity of cocoa implied by the contract at the expiry of the contract term. Any cocoa that has passed tests of quality and bean size through the terminal markets’ grading process can be tendered against contracts. The buyer who takes delivery of cocoa from a terminal market would usually obtain material close to the minimum quality necessary to pass the market’s grading test. In physical contracts, the prices tend to be higher because of a control of the specification of the material. 1.4.2. Arbitrage Arbitrage involves the simultaneous purchase of futures or physical commodities in one market against the sale of the same quantity of futures or physical commodity in a different market. An arbitrage strategy is often implemented to take advantage of differentials in the price of the same instrument on different markets. Cocoa can be traded on CSCE in dollars and LIFFE in pound sterling. The arbitrage price can be derived by subtracting the CSCE price converted to pound sterling from the LIFFE price. In practice, London cocoa sells at a premium over New York cocoa because of a quality in the difference of cocoa. The arbitrage price is affected by the forces of supply and demand and by exchange rates. Arbitrage allows speculation on whether the premium of London cocoa will increase or decrease over New York. 1.4.3. How is the ICCO price for cocoa beans calculated? Is the ICCO price for cocoa beans related to the grade of cocoa? The ICCO prices for cocoa beans are not related to a specific grade of cocoa but to the prices on the London and New York Terminal markets. At LIFFE, the London Terminal market, and at the CSCE, the New York Terminal market, different grades of cocoa are deliverable against

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contracts. However, each lot of cocoa is sampled and graded by Exchange graders.

1.4.4. Information on how prices are affected by changing economic factors? Cocoa price movements can be separated into three categories: long-term, intermediate-term, and short-term. Long-term cocoa price fluctuations are induced by the links between the rate of new planting, production, stocks, and prices. Intermediate-term cocoa price fluctuations, consumption and stocks represent the response of the cocoa industry to annual variations in world cocoa production. Short-term cocoa price fluctuations reflect alternating tides of bullish and bearish speculative enthusiasm in the world’s cocoa markets. 1.4.5. Cocoa varieties The names Criollo, Forastero, and Trinitario refer to the three main types or groups of populations of Theobroma cacao, the cocoa tree. The world cocoa market distinguishes between two broad categories of cocoa beans: “fine or flaviour” cocoa beans, and “bulk” or ordinary cocoa beans. Fine or flaviour cocoa beans are produced from Criollo or Trinitario cocoa-tree varieties, while bulk cocoa beans come from Forastero trees. In 1998, the top producing countries are Cote d’Ivoire, Ghana, Indonesia, Brazil, Nigeria, Cameroon, and Malaysia. The top grindings countries in the world are the Netherlands, United States, Germany, Cote d’Ivoire, Brazil, United Kingdom, and France. 1.4.6. Commodities — Market participants: The case of cocoa, coffee, and white sugar We describe some specific features regarding the cocoa, coffee, and white sugar contracts. The case of cocoa Producer in country of origin. There are different systems of marketing the crop depending on the country.

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Trade house. Buys from country of origin and assumes risks associated with transporting and selling the product to buyers in consuming countries. Processor. Buys cocoa beans and/or produces cocoa liquor, powder, and butter. Manufacturer. Buys beans and/or sell the above products from the trade-houses and or processors. Speculator. The use of the cocoa contract by managed futures funds, who tend to take short-term positions. Institutional investors have a long-term view. The case of coffee Grower in country of origin. There are different systems of marketing the crop depending on the country. Trade house. Buys from country of origin and assumes risks associated with transporting and selling the product to buyers in consuming countries. Roaster. Buys green coffee and roasts it. Manufacturer. Buys beans and/or sell the above products from the trade-houses and or processors. Speculator. The use of the Robusta coffee contract by managed futures funds, who tend to take short-term positions and institutional investors, who have a long-term view.

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The case of White Sugar. Producer in country of origin. There is in general in each country a central sugar marketing organization that negotiates all domestic and international sales. Trade house. Buys from country of origin and assumes risks associated with transporting and selling the product to buyers in consuming countries. Manufacturer. Buys either raw sugar in bulk for further refining or white sugar in clean bags from both country of origin and trade-houses. Speculator. The managed funds are a vital part of daily volumes. 1.5. Trading Characteristics of Options: The Case of Equity Options This section describes the specific features of options markets. A call gives the right to its holder to buy the underlying asset at a given price within or at a specified period of time. A put option gives the right to the buyer to sell the underlying asset at the striking price within or at a specified period of time. Equity warrants are long-term options traded often in securities markets rather than in option markets. Covered warrants are (OTC) long-term options issued by securities houses. 1.5.1. Options on equity indices These options are traded on the major indices around the world. Options on the spot index are cash-settled, i.e., there is no physical delivery of the underlying index. 1.5.2. Options on index futures These options require upon exercise a long (a short) position in the future contract for a call (a put) in the same contract.

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Derivatives, Risk Management and Value Table 1.1. Countries in the FT-actuaries world with listed index options. Country USA Japan France Germany Switzerland Canada the Netherlands Australia

Index S&P 500 and S&P 100 since 1983 Nikkei 225 since 1989 CAC 40 since 1988 DAX since 1991 SMI since 1988 TSE 35 since 1987 EOE since 1978 All ordinaries since 1983

Index options on stock indexes and stock index futures began trading in the United States in 1983 with the introduction of the S&P 100 contract on the Chicago Board Options Exchange. The 10 largest markets in the FT-Actuaries World Index have listed options (See Table 1.1). In these countries, index futures are also traded. In general, combined options and futures volumes exceed trading in the underlying stocks. Volume is concentrated in one-month contracts. The volume in options with longer maturities takes place in the OTC. The OTC options market began to develop in 1988. 1.5.3. Index options markets around the world In North America, options available are traded on several indices: S&P 100, S&P 500, Major Market, S&P MidCap, NYSE Composite, Value Line, Torento 35, etc. Several listed options exist on indices like Value Line (PHX), National OTC (PHX) indexes, etc. In Japan, the main option contract is the Osaka Nikkei. Options exist also on the Singapore International Monetary Exchange (SIMEX), the Tokyo stock price index (TOPIX), etc. The OTC activity appears on the FT-Actuaries Japan Index. In Europe, several options are listed on European indices. In Germany, listed DAX options trade on the Deutsche Terminbourde Bourse (DTB) on a screen-based system. Trading in options started in 1991. Investors use also the OTC market for DAX options with maturities greater than three months. In France, CAC 40 options are traded on the Paris-Bourse. The OTC market is used mainly by large US and European investment banks. In the United Kingdom, FT-SE options are traded on the London International

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Financial Futures Exchange (LIFFE). The market is dominated by major international banks and UK brokers. The OTC market is mainly used by global investment banks and the major UK brokers. In Switzerland, listed SMI options are traded on the Swiss Options and Financial Futures Exchange (SOFFEX) in an electronic-screen system. In the Netherlands, the EOE options market is mainly used by locals and some US banks and Dutch pension funds. In Spain, the main participants in the IBEX-35 options market are the US investment banks. 1.5.4. Stock Index Markets and the underlying indices in Europe The CAC 40 Index The CAC 40 index is computed using 40 stocks in the French market. It does not account for the distributions of dividends. The following formula is used: 1000 N i=1 qi,t Si,t It = Kt CA0 where: t: instant at which the index is computed; N : the number of stocks used (40); qi,t : number of shares of stock i, at date t; Si,t : the price of stock i at date t; CA0 : market capitalization of the sample used in the reference date (December 1987) and Kt : an adjustment coefficient at date t. The STOXXSM 50 and EURO STOXXSM The Dow Jones STOXXSM and Dow Jones EURO STOXSM 50 indices are European indices launched on 26 February 1998 by STOXX limited. These indices provide a representative picture of European equity market performance. These indices are composed of 50 industrial, commercial, and financial blue chips. These indices are available on all major information networks and are disseminated every 15 seconds. The methodology in constructing these indices is based upon a matrix approach that begins with 80% of the investible universe.

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Futures on the Dow Jones STOXXSM and Dow Jones EURO STOXXSM 50 indices allow fund managers to insulate the asset value of their European equity portfolios regardless of their performance benchmark. This can be achieved by selling futures contracts in proportion to the sensitivity of these portfolios to fluctuations in the Dow Jones STOXXSM and Dow Jones EURO STOXXSM 50 indices. Selling derivatives allow market participants to stabilize their portfolios. Anticipating a rise (a fall) in prices, investors can buy (sell) index futures and sell at higher (lower) prices at a later date. Theoretical values of index futures can be calculated at any time using the prices of the equity baskets represented by the Dow Jones STOXXSM and Dow Jones EURO STOXXSM 50 indices. The theoretical value corresponds to the value of the equity basket plus the basis, i.e., the cost of purchasing the index portfolio components, less dividends. Options on Dow Jones STOXXSM and Dow Jones EURO STOXXSM 50 indices can be used as short hedging instruments, through the purchase of puts. Derivatives on Dow Jones STOXXSM and Dow Jones EURO STOXXSM 50 indices give investors the tools to pursue simple strategies based on their expectations of market movements. Investors can buy call (put) options or sell put (call) options. Options can be used in arbitrage transactions allowing strategies to be pursued based on comparative fluctuations between equity markets within the Euro zone and the different countries.

Examples The DOW JONES STOXXSM 50 AND DOW JONES EURO SOXXSM 50 The specific features of the Dow Jones STOXXSM 50 index are as follows. Dow Jones STOXXSM 50 composition: Basket of 50 highly liquid European blue chips (16 countries), belonging to the main business sectors. Calculation method: The index level is given by: I = 1000 (sum of real-time market capitalization for each component stock/adjusted base capitalization). The index is calculated in real-time by STOXX Ltd. Price quotation: The index is disseminated every 15 seconds by ParisBourse.

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Financial Markets, Financial Instruments, and Financial Crisis Table 1.2. Contract specifications Underlying index Trading unit Price quotation Minimum price fluctuation (tick) Contract month Last trading month First trading day Settlement/exercise

Margin Trading hours

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Contract specifications. Dow Jones STOXXSM and Dow Jones EURO STOXXSM futures Dow Jones STOXXSM and Dow Jones EURO STOXXSM indices Contract valued at EURO 10 times the index quoted in the future Index without decimal 1 index point equivalent to EURO 10 3 spot months, 2 quarterly contract months of the March, June, September, and December cycle 3rd Friday of the contract month at 12:00 p.m. First trading day following the last trading day of the previous contract month Cash settlement, expiration settlement price = arithmetic mean (with 2 decimals) of each index value calculated and displayed between 11:50 a.m. and 12:00 p.m. (41 values) Margin information can be obtained from the MATIF/MONEP information department NSC day session: 8:00 a.m.–5:00 p.m. NSC evening session: 5:05 p.m.–10:00 p.m

The specific features of the Dow Jones Euro STOXXSM 50 index are as follows. Dow Jones Euro STOXXSM 50 composition: Basket of 50 highly liquid Euro zone (10 countries) blue chips, belonging to the main business sectors. Calculation method: The index level is given by: I = 1000 (sum of real-time market capitalization for each component stock/adjusted base capitalization). The index is calculated in real-time by STOXX Ltd. Price quotation: The index is disseminated every 15 seconds by ParisBourse (Table 1.2).

1.6. Trading Characteristics of Options: The Case of Options on Currency Forwards and Futures These options are traded in the OTC market. The growth of the OTC market is due to its flexibility. In fact, many banks and financial institutions offer options with tailor-made characteristics in order to match the clients

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needs. Options on the currency futures have been traded since 1982. These options are standardized contracts. 1.7. Trading Characteristics of Options: The Case of Bonds and Bond Options Markets There are several types of bonds and bond options traded in organized and OTC markets. These financial instruments correspond, for example, to zero-coupon bonds, bonds with call provisions, putable bonds, convertible bonds, bonds with warrants attached, exchangeable bonds, etc. 1.7.1. The specific features of classic interest rate instruments Zero-coupon bonds. These bonds are bonds with no periodic coupon payments. The interest due to the bond holder is given by the difference between the maturity value and the purchase price. This class of bonds is referred to as zero-coupon bonds and its price is given by the present value of the maturity value. In the mathematics of bonds, continuous compounded interest rates and/or discrete compounded interest rates can be used. For continuous trading in derivatives, interest rate is often continuously compounded using the factor e−rT to discount US$ 1 payable in T years at a rate r. In this case, the value of a zero-coupon bond is computed by discounting its maturity value at this factor. When interest is accumulated annually for T years, discretely compounded interest rate is computed by using (1/(1 + r )T ). The equivalence between the two fomulas appears when r = log(1 + r ). A coupon-paying bond. This bond is often regarded as a portfolio of several cash flows (the coupons) where each cash flow can be seen as a zero-coupon bond. Hence, a couponpaying bond can be viewed as a package of zero-coupon bonds. The principal amount of a bond issue can decrease or amortize during the life of the interest-sensitive instrument. The principal is paid back gradually at a given rate and interest is paid on the amount of the principal outstanding.

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A corporate bond. This is a bond issued by a firm. The bond obliges the issuer to pay interest rate charges and the principal amount according to a specified schedule. If the bond is guaranteed by some assets of the issuer, it becomes a mortgage bond. If the only guarantee is represented by the credibility of the issuer, the bond is a debenture bond. Each bond issue is accompanied with a document known as indenture. It specifies the main features of the issue. Some bonds are not redeemed before another class of debt. They are referred to as junior or subordinated bonds. The higher priority claims are referred to as senior bonds. A sinking fund provision is often inserted in the bond indenture to describe the way bondholders will be paid. Bonds with specific features. A bond with a call provision gives the right to the issuer to call the issue before the specified redemption date. A bond with a put provision gives its holder the right to put the bond back to the issuer at a fixed price. Indexed bonds. These bonds are useful when the operating profits of a corporation are exposed to the fluctuations of an index, as with a commodity price like oil, aluminium or inflation. The exposure risk can be partially hedged by issuing bonds whose interest rate payments and/or principal repayment is linked to the index, in such a way that the effective cost of debt is reduced when there is an unfavorable movement in the price index, and is increased to the benefit of the investors when the movement is favorable to the firm. Such a bond issue can be split into parts: the bull and bear tranches, so that investors can choose only one side of the risk exposure. These bonds allow some investors to take risky positions which are not directly available to them, or not allowed, on organized markets. Investors are ready to pay a premium for these opportunities which is translated into a reduced financing cost. A convertible bond. Entitles its holder the right to convert the bond into a certain number of units of the equity of the issuing firm or into other bonds.

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A bond with an attached warrant. It is simply a package comprising the bond and a warrant. Most of these bonds are Eurobonds issued in international capital markets. An exchangeable bond. It is similar to a convertible bond, with the exception that it gives its holder the right to exchange the bonds for the equity of another company, etc. When a corporation has a low credit rating and must implement a large investment program to survive, it may well be too costly to issue standard debt, while raising equity might dilute considerably the current shareholders’ position. Then, warrants (bonds with attached warrants) and convertibles become the only affordable financing instruments. Floating Rate Notes, FRN. The value of an FRN depends mainly on the coupon date payment. The coupon is often determined as a mean of the interest rates applied to threemonth treasury bills. A risk premium is added to the mean rate to account for the risk of the issuer. Floating Rate Bonds, FRB. The coupon payments are indexed with reference to a variable interest rate index as the rate on the three-month treasury bills or the rate on 30-year treasury bonds. Several floaters show implicit embedded provisions which have the specific features of call and put options. For example, the provision of the type Floor and Ceiling specifies a minimum coupon rate of x% and a maximum coupon rate of y%. Stripped bonds. The cash flows from treasury bonds can be separated into two assets: an asset corresponding to the principal amount, (principal only, PO), and an asset corresponding to interest rates, (interest only, IO). This type of bond represents a stripped asset and is referred to as treasury-backed stripped. The amount of principal, PO, represents the value of a zero coupon bond. The amount of interest, IO, corresponds to a portfolio of zero coupon bonds. The separation between the cash flows can eliminate the reinvestment risk. It allows the investor to use different IO and PO in hedging strategies.

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In 1985, some American Treasury bonds are traded in the forms of separate trading of registered interest and principal of securities (STRIPS). In the mortgage market, bonds are also traded in the forms of STRIPS as PO STRIP (payment of the principal) and IO STRIP (payment of interest rates). Strips. Correspond to an instrument called “Separate Trading of Registered Interest and Principal of Securities”. The separation between coupons and principal of a bond allows the creation of artificial zero-coupon bonds of longer maturities than would otherwise exist.

1.7.2. The specific features of mortgage-backed securities These securities are linked to the financial crisis in 2007–2008 and mainly to the subprime. The mortgage is a pledge of real estate which is used to secure the payment of a loan originated for the purchase of a real property. The lender, known as the mortgagee, has the right to foreclose on the loan and to seize the property if the borrower (mortgagor) does not satisfy his contracted obligations. A mortgage loan is specified by the interest rate of the loan, the number of years to maturity, and the frequency of payments. The mortgage instrument represents an instrument which is guaranteed by a real asset, a land, a building, etc. Mortgages can be divided into different classes according to the nature of the asset used as a guarantee. Mortgages represent the underlying collateral of mortgage-backed securities. When bonds are guaranteed by the shares of a firm, they are referred to as collateral trust bonds. When the bond is guaranteed by a building for example or other assets of the issuer, it is a mortgage security. If a firm has 100 securities and uses a guarantee of 30, it can issue 30 of mortgage bonds. In the United States, the Federal National Mortgage Association (FNMA), the Government National Mortgage Association, GNMA, and the Federal Home Loan Mortgage Corporation, (FHLMC) play an important role in the mortgage market. The FNMA introduces mortgage-backedsecurities (MBS), which are created by pooling mortgage loans and using this pool of mortgage loans as collateral for the security. The cash flow of an MBS is a function of the cash flows of the underlying mortgage pool.

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If you consider an entity that purchases several loans, pools them, and uses them as collateral for issuance of a security, the security created is referred to as a mortgage pass-through security. The security is guaranteed by the GNMA, the FNMA, and the FHLMC. Pass-through securities can be issued also by private entities. In this case, they are referred to as conventional pass-throughs. When mortgage loans are used in a pool to create a pass-through security, they are said to be securitized. The process of creating the passthrough security is known as the securitization of mortgage loans. The FHLMC introduced in 1983, the collaterized mortgage obligations, (CMO). Since an investor in a pass-through security is exposed to the total pre-payment risk due to the pool of mortgage loans underlying the security, it is possible to create three classes of bonds with different par values. This can be done by indicating how the principal is distributed from the passthrough security. In general, there are three classes: A, B, and C. This mortgage-backed security refers to a CMO. The total pre-payment risk for the CMO remains similar to that of the mortgage loans. The stripped MBS becomes an attractive instrument in managing portfolios of mortgage securities. It is possible to forecast the prepayments from a pass-through security. Therefore, some pre-payment benchmark conventions must be known. In general, the Standard prepayment model, PSA developed by the public securities association can be used. This benchmark is expressed as a monthly series of annual constant pre-payment rates, CPRs. The CPR is converted into a monthly prepayment rate, known as the single monthly mortality rate (SMM) where SMM = 1 − (1 − CPR)1/12 . The PSA model assumes that pre-payment rates will be low for newly originated mortgages. The rate will speed up as the mortgages become seasoned. For more details, see Fabozzi (1993). 1.7.3. The specific features of interest rate futures, options, bond options, and swaps Interest rate futures contracts A futures contract is an agreement between a buyer or a seller and an established exchange to take or make delivery of a given commodity at a specified price at a given delivery or settlement date. An investor can be long (a buyer) or short (a seller). Each investor must deposit an initial margin before trading futures contracts. The margin used to guarantee the transactions can attain a minimum level known as the maintenance

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margin. The margin varies with the variation in the futures price. The futures contract is marked to market at the end of each trading day and is subject to interim cash flows. The main difference between futures contracts and forward contracts is that forward contracts are OTC instruments which are nonstandardized and are subject to counter-party risk. There are several traded interest rate futures contracts. Interest rate futures contracts are traded on treasury bonds, notes, bills and on the LIBOR rate. Interest rate futures options are traded on T-bond futures, T-note futures, Eurodollar futures, etc. Treasury bill futures The underlying asset of this contract is a short-term debt obligation. The treasury bill is quoted in the cash market in terms of the annualized yield on a bank discount basis: 360 D Yd = T t where: D = difference between the face value and the price of a bill maturing in t days, known also as a dollar discount; F = face or nominal value and T = number of days remaining to maturity. The treasury bill futures contract is quoted in terms of an index associated to the yield as follows: Index = 100 − (Yd )(100). Eurodollar futures Eurodollar represent the liabilities of banks outside the United States of America. The London Interbank offered rate, LIBOR is paid in Eurodollars. The underlying asset of the Eurodollar futures contract is the three-month Eurodollar. The contract is settled in cash. Treasury bond futures Treasury bond futures contracts are traded on several exchanges. The underlying asset of the futures contract traded on the Chicago Board of Trade is 100,000 par value of a hypothetical 20-year, 8% coupon-bond. The futures price is quoted in terms of par being 100. The seller of the futures contract can unwind his position before the maturity date by buying back

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the contract. If he decides to make delivery, the seller must deliver some treasury bond chosen from the list of specific bonds published by the CBT. The delivery process allows the seller of the futures contract to choose from one of the acceptable deliverable treasury bonds. The CBT uses conversion factors for the computation of the invoice price of each deliverable treasury. This factor is determined before a contract with a given settlement date begins trading and it remains constant. The invoice price indicates the price paid by the buyer when the treasury bond is delivered. It corresponds to the settlement futures price plus accrued interest and is calculated as follows: Invoice price = Contract size (settlement price of the futures contract × times conversion factor) + accrued interest. The term accrued interest can be defined as follows: Accrued interest Bond market prices are clean prices since they are quoted without any accrued interest. The accrued interest corresponds to the amount of interest since the payment of the last coupon. It is computed as follows: Accrued interest = interest due in full period (N1 /N2 ) with N1 = number of days since the last coupon date and N2 = number of days between coupon payments. The dirty price corresponds to the quoted clean price plus the accrued interest. Upon delivery, the seller will deliver the bond which is cheapest to deliver, also known as the cheapest to deliver (CTD). The seller must compute the return to be earned from buying bonds and delivering them at the settlement date. The return is computed using the price of the treasury issue and the futures price for delivery. This return is referred to as the implied repo rate. The CTD issue corresponds to the issue with the highest implied repo rate since it gives the seller the highest return by buying and delivering the issue. The delivery process gives the contract seller some options. The quality option, also known as the swap option, allows the seller to choose among different acceptable treasury issues. The timing option gives the seller the right to choose the exact time during the delivery month to make delivery. The wildcard option allows the seller to give a notice of intent to deliver up to 8 p.m. Chicago time

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after the closing of the exchange (3:15 p.m.) on the date when the futures settlement price is scheduled. Treasury bond futures The CBT created in 1975 the first financial futures contract a futures for mortgage-backed securities. These securities are issued by the Government National Mortgage Association (GNMA). The underlying asset of a treasury bond futures contracts on the CBT is a 15-year T-bond with a coupon rate of 8%. This rate has changed since then. The invoice price received by the party with a short position in the contract is given by the bond futures settlement price which multiplies the delivery factor for the bond to be delivered plus the accrued interest. For each deliverable bond, there is a delivery factor which is calculated with respect to the coupon rate and the time to maturity of that bond. For example, the conversion factor for a bond with coupon rate rc and a maturity in m years is: CF =

2m j=1

1 rc /2 + (1 + 0.04)j (1 + 0.04)2m

Since there are many bonds that can be delivered in the T-bond futures contract, the CTD is that deliverable issue for which the following difference is minimized: Quoted bond price − settlement futures price (C. factor). The basis or the difference between the spot and futures prices is minimal for the CTD bond or: bit = Bci − ft CF i where: Bci = current price of the ith deliverable bond; ft = bond futures settlement price and CF i = conversion factor for the ith bond. Forward rate agreements A forward rate agreement (FRA) allows a company to reduce interest rate exposure by locking into a rate of interest. In this contract, the parties agree to exchange, at some future date, interest payments on the notional amount of the contract. The buyer of an FRA contract agrees to pay interest at a

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specified rate and to receive interest at a floating rate that prevails at a future date T . Interest rate swaps An interest rate swap is an agreement between two counter-parties to exchange periodic interest payments. These interest payments are determined with reference to a pre-determined principal amount known as the notional principal amount. In general, one party, the fixed rate payer, agrees to pay the other party fixed-interest payments with a given frequency at some specified dates. The other party, the floating rate payer, agrees to pay some interest rate payments that vary according to a reference rate. The London Inter-bank Offered Rate, LIBOR, is often used as the reference rate. Risks in bond investments The buyer of a bond faces different risks: an interest rate risk, a reinvestment risk, a default or credit risk, an inflation risk, an exchange risk, a liquidity risk, etc. The interest rate risk The variations in interest rates modify the bond price. A higher interest rate leads a lower bond value and a lower interest rate produces a higher bond value. The re-investment risk The return for a bond buyer comes from the perceived interest (the coupons), the capital gain (variation in the bond price), and the interest from the placement of the coupons. The credit and default risk This risk accounts for the possibility of the borrower to honor his liabilities: payments of coupons and principal at their exact timing. The credit crunch in 2008 is largely due to this risk. The risk to call The issuer can insert a provision in the debt contract that allows him to buy back his bonds before the maturity date. In this case, the return for

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the bondholder can be different from the return anticipated when buying the bond. The inflation risk The variations in inflation rates affect the return from holding the bond. All bonds are expressed in nominal terms. The difference between a nominal return and the inflation rate gives the return in real terms. The exchange rate risk The price of a bond denominated in a foreign currency is affected by the changes in exchange rates. These rates can affect significantly the return from holding the bond. The liquidity risk The liquidity risk reflects the difficulty in selling the bond at a given market price. This risk can be measured by the spread observed in the market place. The higher the spread, the greater is the liquidity risk. The quality of a bond is denoted by a given letter or rating. Rating agencies like Moody and Standard & Poor give their rating to show the risks associated with investments in bonds. The passage from a letter A to B or C and D reflects a higher risk. The risk premium is higher for bonds of type B than type A. This situation characterises most Islamic bonds or sukuks for which there is often no secondary markets. 1.8. Simple and Complex Financial Instruments Forward-start options These options give an answer to the following question: how much can one pay for the opportunity to decide after a known time in the future, known as “the grant date”, to obtain at the money call with a different time to maturity with no additional cost? Pay-later options For these options, the premium is paid upon exercise. They are contingent options. In fact, the buyer has the obligation to pay upon exercise when the option is in the money regardless of the amount by which the underlying asset price exceeds the strike price.

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Chooser options They are on the holder, immediately after a pre-determined elapsed time, to choose whether the option is to be a call or a put. There are two kinds of chooser options: simple and complex choosers.

Options on the minimum or the maximum of two or more risky assets These options may to be useful in the pricing of a wide variety of contingent claims, traded assets, and financial instruments whose values depend on extreme values. Examples include discount option bonds, compensation plans, risk-sharing contracts, collateralized loans, and growth opportunities among other contracts.

Two-color rainbow options They refer not only to options on the maximum (minimum) of two assets, but also to all options whose pay-off depend on two or more underlying assets: options delivering the best of two assets and cash, spread options, portfolio options, dual-strike options, etc.

Options with extendible maturities They include any financial contract with provisions concerning a rescheduling of payments and a re-negotiation of terms. There are many types of exotic and second generation options which take different forms. They include path-dependent options, lookbacks, partial lookbacks, Asian options, shout options, binaries or digitals, knockouts or barriers, ladder, and cliquet options among other things.

Asian options Asian options have been in popular in the foreign exchange market, interest rate and commodity markets. These financial innovations are traded in OTC markets and allow investors to accomplish several hedging strategies. Examples of these options include commodity-linked bond contracts and average currency options. Commodity-linked bond contracts give the

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right to the holder to receive the average value of the underlying commodity over a certain period or the nominal value of the bond, whichever is higher. Barrier options These belong to the family of path-dependent options. They are in life when they knock-in and are extinguished when they knock-out. They are sometimes referred to as knock-ins or knock-outs when the underlying asset hits (or does not hit) the barrier. The standard form of barrier options refers to European options which appear or disappear (ins and outs) when the underlying asset reaches a certain level known as the barrier. This barrier or knock-out level is set below the strike price for the call and above it for the put. For example, an in barrier option comes into existence whenever the underlying asset value hits a specified level. The right to exercise an out barrier option is forfeited when the barrier is hit. Ratio options These are options on the ratio of two asset prices, index levels, commodities, etc. An example is given by the dollar-denominated European option on the ratio of the Germain DAX stock index to the French CAC index. Innovations in OTC options markets not only involve certain relations between the underlying asset price and the strike price but also on the number of time units for which a certain condition is satisfied. This corresponds, for example, to financial assets which are traded within a specified range. Structured products with embedded digitals are much more interesting than vanilla digitals. There are many types of range structures which may be in the form of range binaries, at maturity range binaries, rebate range binaries, mandarin collars, mega-premium options, limit binary options, boundary options, corridors, wall options, mini-premium options, volatility options, etc. Complex digitals or binaries In their complex forms, complex digitals or binaries may be presented in different forms: compound digitals, boolean digitals, and corridors. Compound digitals obey the same principle as compound options and take different forms: quanto digitals, barriered digitals, and options on digitals.

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1.9. The Reasons of Financial Innovations Financial engineers are working on the design and the valuation of these financial instruments and the new strategies for portfolio and risk management. The main questions are: Why there are so many new financial instruments? Why has the wave of financial innovation not stopped? The most common explanation often advanced by market authorities is that financial markets are incomplete and that these instruments allow us to complete these markets. However, as noted by Ross (1989), this explanation seems awkward. In fact, the success in valuing these financial instruments comes from the fact that they are regarded as contingent claims or derivative securities which are spanned by the underlying assets on which they are traded and a riskless bond. This allows the derivation of simple valuation formulas in complete markets. In reality, markets can never be fully complete but with regard to the price determination, it is often assumed that markets are complete. Otherwise, the pricing of these instruments would be a difficult task. According to Ross (1989), there are two dominant features which contribute to the wave of financial innovation: the role of institutions and the role of marketing. These reasons complement the arguments by Miller (1986) and Merton (1988). Miller’s (1986) analysis is based on the role of taxes and regulations in the innovation process. In his analysis, Miller considers taxes as a source of much of the motivation for financial innovation. Merton (1988) proposes a detailed analysis of the production function underlying the innovation in derivative securities markets. He puts the accent on the role of transaction costs. The analysis by Ross (1989) ignores production costs and is interested mainly in the role of agency costs and marketing costs, which help to shape the form of the new institutional features. Agency costs and restrictions may arise from monitoring and the regulatory environment. They may result either from the natural needs of market relations between institutions and participants or may be imposed by the government. The first reason in Ross’s analysis is that financial markets become institutional markets since institutions are the most significant participants in these markets. This does not mean that financial markets are solely markets where institutions operate, but rather markets where institutions are significant forces.

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Financial institutions range from transparent through translucent to opaque. In this classification, a mutual fund is regarded as a transparent institution and an insurance company is seen as an opaque institution. A pension fund is regarded as a translucent institution. Institutions can be regarded as financial market players whose activities are dominated by agency relations. The second reason in Ross’s analysis of financial innovation is the role of marketing. In perfect and frictionless markets, selling a financial instrument is costless. In reality, the less familiar and the more esoteric the financial instrument, the more costly it is to sell. When the states of nature are exogenously specified, each security can be defined by its payoffs corresponding to the different states. Since uncertainty remains about these pay-offs, new states are generated and are not yet spanned. This uncertainty may be “nearly” spanned. In complete markets, marketing can “explain” the pay-offs in a view where the marginal cost equals the marginal benefit from a transaction. This view of complete markets allows the pricing of financial instruments with a great accuracy. It recognizes the existence of a marketing cost for a new financial instrument or strategy. This instrument or strategy, corresponds in the begining to the needs of some institutions or retail clients. In its mature phase, marketing costs are reduced since the financial instrument or strategy becomes a standard commodity. Buying or selling securities which are standardized and trade in well functioning markets with large volume induce nearly no marketing costs. This is not the case for the tailored and low-grade securities. Ross makes a distinction between marketed and non-marketed securities rather than between high- and low-grade securities. Stocks, for example, are low-grade securities which trade in well organized and competitive makets. Financial futures are examples of low-grade innovations which have evolved into low-cost well-traded commodities. This evolution is costly and the ultimate success relies on the ability to standardize the financial product and to sustain a sufficient volume of trade to justify the initial costs. In Ross’s model, the existence of new financial instruments and strategies and the marketing process are based on the cost structure of the marketing networks and distribution channels. It is the institutional structure of contracts and incentives that allows the process of financial engeneering to continue. Hence, it seems that institutional markets and financial marketing are central to the understanding of financial innovations. For more details, see Ross (1989).

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1.10. Derivatives Markets in the World: Stock Options, Index Options, Interest Rate and Commodity Options and Futures Markets 1.10.1. Global overview Several institutions produce information regarding futures and options around the world. Often, summary statistics on volume and open interest are given for futures and index options. Index options on stock indexes and index futures contracts begin trading in the U.S in 1983. This has been facilitated with the introduction of the SP 100 index contract on the Chicago Board Options Exchange. Today, index futures are traded and are more liquid than index options. 1.10.2. The main indexes around the world: a historical perspective The first options traded on indexes can be traced back to US (SP 500 and SP 100 in 1983), Japan (Nikkei 225 in 1989), UK (FT-SE 100 in 1984), France (CAC 40, 1989), Germany (DAX, 1991), Switzerland, (SMI, 1980), Canada (TSE 35, 1987), Netherlands (EOE, 1978), Australia (All Ordinaries, 1983), ... Options volume in listed markets is mostly concentrated in one month contracts in all markets. For most options, volume with longer maturities take place in OTC markets. In the OTC market, trading began early in 1988. Several investors buy long-term puts to implement portfolio insurance strategies. Today, dealers run large OTC options books. This can reduce or eliminate risk in the market. North America. U.S index options trading appear on listed markets and OTC markets with customized features. Options are traded on SP 100, SP 500, MMI, SPMidCap, options on small capitalization indexes, the NYSE Composite index. Main information used concerns the average daily volume, Average daily dollar volume (in millions) and Index level. Options on SP are preferred by retail investors. MidCap Options and options on SP 500 index attract the interest of institutional money managers and pension funds.

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SP 100 are the mostly traded contracts in the U.S. SP 500 are the have the greatest open interest in the U.S. Hundred billions of dollars are traded. Institutional use of index options: Covered call writing: a call is sold and the underlying asset is held. Long index put strategy and collar positions, which is preferred by institutions. The collar can lead to a skew in index options implied volatilities: out of the money puts have higher volatilities than calls. Options are available on National OTC (PHX) indexes. Stock index markets in North America. The SP 500 index fluctuated in a band. The move gives a volatility in a range of 10%–25%. We can represent a monthly volatility for the year. With its heavier dose of cyclical stocks, the DJIA has been outperforming for some years the broader market. We can compute historical volatility and implied volatility from at the money options. We should compute the spread. The following Tables shows the volume (number of contracts traded) in several countries. Japan. Options exist on Osaka Nikkei, options on TOPIX. Japanese institutions often use for their long term options exposure or customized strike prices fixed income securities with embedded index options. Osaka Nikkei options are used by domestic institutional in short term trading. Regulations by the Ministry of Finance prevent pension funds from completely hedging their portfolios (hedging limit 50%). Hedgers integrated their activities into equity risk management systems. Life insurance companies focus on using options for directional trading. Offshore hedge funds use the Osaka Nikkei options to take outright shortterm trading positions. The Government intervenes to support the market. Foreign institutions act in the OTC market for different reasons: They are restricted by regulation from trading listed options. They do not want to incur the costs of rolling over. Competition among dealers makes this market very competitive. Sector options are popular in Japan. The following Table provides the volume (number of contracts traded) in several countries for index options.

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Stock index options 2004 Exchange

2003

Volume Traded (Number of Contracts)

2003

Americas American SE

40,985,108

33,137,709

89,965

0

Bourse de Montreal

336,544

961,650

35.00%

Chicago Board of Trade (CBOT)

762,007

263,629

289.05%

136,679,303

110,822,096

123.33%

6,451,862

5,168,914

124.82%

40,886,923

23,979,352

170.51%

35,989

0

181,215

110,079

0

0

Pacific SE

14,119,270

15,744,139

89.68%

Philadelphia SE

25,360,908

19,746,264

128.43%

1,589,765

1,600,461

99.33%

BM&F

Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchange (ISE) MexDer New York Board of Trade (NYBOT) Options Clearing Corp.

Sao Paulo SE

123.68%

164.62%

Europe, Africa, Middle East Athens Derivatives Exchange

941,387

1,388,985

67.78%

BME Spanish Exchanges

2,947,529

2,981,593

98.86%

Borsa Italiana

2,220,807

2,505,351

88.64%

1,299

8,440

15.39%

117,779,232

108,504,301

108.55%

Euronext

99,607,852

103,986,651

95.79%

JSE South Africa

11,268,763

10,505,417

107.27%

8,947,439

6,371,381

140.43%

681,783

543,090

125.54%

Tel Aviv SE

36,915,103

29,353,595

125.76%

Warsaw SE

124,392

153,106

81.25%

40,855

27,680

147.60%

794,121

630,900

125.87%

56,046

43

130339.53% 99.20%

Copenhagen SE Eurex

OMX Stockholm SE Oslo Bors

Wiener Börse

Asia Pacific Australian SE BSE, The SE Mumbai Hong Kong Exchanges

2,133,708

2,150,923

2,521,557,274

2,837,724,956

88.86%

2,812,109

1,332,417

211.05%

Osaka SE

16,561,365

14,958,334

110.72%

SFE Corp.

523,428

585,620

89.38%

Singapore Exchange

247,388

289,361

85.49%

43,824,511

21,720,084

201.77%

17,643

98,137

17.98%

3,137,482,893

3,357,354,658

93.45%

Korea Exchange National Stock Exchange India

TAIFEX Tokyo SE Total

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2005 Exchange

2004 Volume Traded

(Number of Contracts)

Americas American SE BM&F Bourse de Montreal Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchange (ISE) MexDer New York Board of Trade (NYBOT) Options Clearing Corp.

8,678,564

7,290,157

6,344

16,485

650,186

336,544

728,349

762,007

192,536,695

136,679,303

15,106,187

6,451,862

4,464,094

83,358

37,346

35,989

217,334

181,215

0

0

Philadelphia SE

6,234,567

5,275,701

Sao Paulo SE

2,257,756

1,589,765

1,163,260

794,121

Asia Pacific Australian SE Bombay SE Hong Kong Exchanges Korea Exchange

100

NA

3,367,228

2,133,708

2,535,201,693

2,521,557,274

National Stock Exchange India

10,140,239

2,812,109

Osaka SE

24,894,925

16,561,365

SFE Corp.

680,303

523,428

Singapore Exchange

157,742

247,388

81,533,102

43,824,511

20,004

17,643

TAIFEX Tokyo SE

Europe, Africa, Middle East Athens Derivatives Exchange BME Spanish Exchanges Borsa Italiana

700,094

941,387

4,407,465

2,947,529

2,597,830

2,220,807

149,380,569

117,779,232

Euronext.liffe

70,228,310

99,607,852

JSE

11,473,116

11,303,311

OMX

12,229,145

8,947,439

Eurex

Oslo Børs

515,538

695,672

Tel Aviv SE

63,133,416

36,915,103

Warsaw SE

250,060

78,752

37,127

40,855

3,203,028,688

3,028,651,872

Wiener Börse Total

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Europe. In Germany. Listed DAX options are done on a screen-based system. Major players in this market are the large U.S and Continental investment banks. In France. Listed CAC 40 options trade on the French options market where trading is dominated by locals taking speculative positions and by large investment banks. Institutional users are French insurance companies and fund managers. Players seek leveraged exposures on the market. Guaranteed funds on the CAC 40 issued by French banks are popular among retail investors. CAC 40 options are used as part of these products. Major participants in the OTC market are large U.S and European investment banks. Stock index markets in France. – An interesting development in the CAC 40 futures is the distribution of open interest across various months. – Institutions have led to move into the quarterly contracts to eliminate the chore of rolling on a monthly basis. The lack of a developed stock-borrowing market can reduce trading in futures. Professional traders can use the futures to hedge OTC options. To hedge collars traders can be short futures. Arbitrageurs (short stock/long futures) can unwind easily their positions. United Kingdom. The market is dominated by major international banks and brokers. Short-term maturities have the most liquidity. End-users are mainly U.K institutions for hedging and guaranteed funds. In OTC markets, the volume is also high because of greater liquidity in the longer-dated contracts. There is flexibility in expiration dates. Switzerland. Options are traded on the SOFFEX in an electronic screen system. Active participants are major Swiss and American Banks. End users are a mixture

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of short-term speculators and international institutions looking for long exposures. The OTC market is important because there is a need for longer-term strategies on the SMI from pension funds. Zero premium collars are very popular. Netherlands. This market is dominated by locals who service a retail base. Users are mainly pension funds who hedge equity portfolios. The index must be compiled using a specific method. The weighting of the index can overweight smaller, domestically oriented stocks and underweight larger, more internationally oriented stocks. For example, stocks can be weighted using a market capitalization and the maximum weight of a stock in the index will not exceed 10%. This puts a cap on some stocks. The following Table gives the notional value (value traded of stocks), the open interest (positions opened and still not unwind) and the option premiums for several countries. The following Tables provide different information for several markets and instruments. The reader can compare the different markets and instruments using these Tables (source: World Federation of Exchanges). DERIVATIVES - 3.1 STOCK OPTIONS

Exchange

2006 2005 Volume Traded (Nber of Contracts)

2006 2005 Notional Value (USD millions)

2006 2005 Open Interest (Nber of Contracts)

2006 2005 Number of Trades

2006 2005 Option Premium (USD millions)

Americas American SE Boston Options Exchange

186,994,609 92,260,125

77,582,231

NA

NA

NA

NA

NA

NA

NA

NA

Bourse de Montreal

12,265,461 49,235,173 390,657,577

10,032,227 92,386,767 275,646,980

68,947 NA 1,960,297

54,904 NA 1,264,511

1,583,405 1,654,931 187,953,281

1,346,141 1,605,194 151,157,355

732,202 NA 25,792,792

554,076 NA 16,820,556

2,212 456 98,751

1,645 547 61,220

Buenos Aires SE Chicago Board Options Exchange (CBOE) International Securities Exchnage (ISE) MexDer Options Clearing Corp. Pacific SE Philadelphia SE Sao Paulo SE

193,086,271

45,779

42,238

NA

NA

4,709,107

7,652,680

NA

NA

583,749,099 448,120

442,387,776 135,931

NA 829

NA 208

NA 0

NA 2,030

NA 62

NA 49

NA NA

NA NA

0 196,586,356 265,370,986

0 144,780,498 156,222,383

NA NA 89,732

NA NA 49,318

220,032,992 NA 8,846,285

181,694,503 NA 8,379,867

NA NA 15,843,704

NA NA 7,190,023

NA NA 89,732

NA NA 49,318

285,699,806

266,362,631

513,350

392,331

1,833,555

1,824,504

6,542,663

5,777,709

9,746

7,909

20,491,483 18,127,353

21,547,732 8,772,393

303,986 88,371

270,423 41,784

1,766,513 2,533,807

1,678,335 1,021,913

1,474,017 399,129

1,418,149 241,785

11,501 2,477

9,057 1,334

1,195 5,214,191 753,937

3,655 5,224,485 1,206,987

41 44,479 NA

11 40,260 NA

50 21,549 22,541

NA 24,181 79,610

NA 4,478,610 4,064

103 4,550,367 5,454

NA 1,254 186

0 1,100 293

1,089,158 190,876

1,018,917 201,798

32 21

79 33

2,797 39,428

3,959 11,906

45,088 NA

126,245 NA

31 21

161 33

Asia Pacific Australian SE Hong Kong Exchanges Korea Exchange National Stock Exchange India Osaka SE TAIFEX Tokyo SE

Europe, Africa, Middle East Athens Derivatives Exchange BME Spanish Exchanges Borsa Italiana Budapest SE Eurex Euronext.liffe JSE OMX Oslo Børs RTS SE Warsaw SE Wiener Börse Total NA : Not Available - : Not Applicable

17,194

21,729

52

60

1,297

2,004

396

397

3

2

12,425,979 16,056,751 350

10,915,227 12,439,716 176

27,775 91,803 5

20,605 67,776 6

2,748,562 1,964,411 NA

2,411,628 1,646,014 NA

75,313 475,942 6

65,136 442,151 8

1,067 2,771 NA

633 1,979 NA

272,543,052 155,552,010 5,751,832 64,514,641

255,918,793 264,714,188 2,539,526 57,138,563

964,097 603,265 312 69,691

752,434 618,732 153 57,580

52,069,011 45,341,415 916,339 8,418,826

53,312,606 55,353,971 564,302 7,404,092

NA 3,272,555 2,835 NA

NA 2,728,180 1,733 NA

59,286 32,141 NA 27,306

38,740 70,685 NA 15,434

5,811,946 10,727,870

3,325,368 7,281,162

NA 11,453

NA 2,797

616,315 1,431,028

364,265 433,158

34,135 150,940

NA 113,317

643 NA

321 NA

10,988 1,053,298

4,372 816,032

98 5,385

29 4,609

162 116,063

413 76,166

5,501 NA

2,642 NA

4 230

1 165

2,653,601,416

2,311,714,514

-

-

-

-

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DERIVATIVES - 3.2 STOCK FUTURES

Exchange



2006

2005 Open Interest (Nber of Contracts)


Americas MexDer

3,000

19,400

21

85

0

3,400

62

17

693,653 958 102,010 100,285,737

490,233 13,069 68,911,754

8,645 4 655 857,436

5,872 77 510,701

124,307 0 4,260 642,395

78,289 1,750 464,559

5,194 NA 9,382 82,217,305

3,308 2,170 56,491,871

2,476,487 21,229,811 7,031,974 919,426 35,589,089 29,515,726 69,671,751 8,459,165 3,626,036 112,824 12,371

1,431,514 18,813,689 5,957,674 740,396 77,802 12,158,093 24,469,988 5,659,823 1,796,570 172,828 23,748

5,543 43,266 49,636 9,052 203,038 344,198 26,288 6,128 3,502 782 180

3,160 31,708 41,798 7,842 NA 64,062 10,223 NA 2,516 845 331

116,576 1,649,184 41,319 65,015 1,459,509 1,489,169 12,027,716 1,764,492 268,572 1,122 1,339

124,815 1,921,717 58,071 24,936 58,107 467,117 1,535,839 1,387,095 126,266 2,928 2,448

285,982 139,441 56,774 92,618 NA 22,948 392,154 NA NA 87,999 NA

167,715 119,499 66,605 81,468 NA 21,006 177,766 NA NA 130,674 NA

279,730,018

140,736,581

-

-

-

-

-

-

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges National Stock Exchange India

Europe, Africa, Middle East Athens Derivatives Exchange BME Spanish Exchanges Borsa Italiana Budapest SE Eurex Euronext.liffe JSE OMX Oslo Børs Warsaw SE Wiener Börse Total

DERIVATIVES - 3.3 STOCK INDEX OPTIONS


Exchange


2006




Americas American SE BM&F Bourse de Montréal Chicago Board of Trade (CBOT)

16% 126% 108% -24%

Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchnage (ISE) MexDer New York Board of Trade (NYBOT) Philadelphia SE

45% 81% 84% 215% -27% 22%

Sao Paulo SE

-19%

Australian SE Hong Kong Exchanges Korea Exchange

-1% 46% -5%

National Stock Exchange India Osaka SE

84% 13%

Singapore Exchange TAIFEX Tokyo SE

-4%

BME Spanish Borsa Italiana Eurex

25% 9% 45%

JSE OMX

2% 11%

Oslo Børs Tel Aviv SE Warsaw SE

8,678,564 101,003 27,897

18,801 4,401 3,477

6,922 3,135 1,527

NA 106,601 1,691

NA 38,382 4,813

123,559 749 4,620

122,714 466 1,648

NA NA 70

NA NA 141

551,190 279,005,803 27,295,611

728,349 192,536,695 15,106,187

NA 17,791,735 6,005,296

NA 11,541,513 3,295,855

21,815 37,749,429 1,527,059

26,794 29,381,746 1,226,413

NA 11,479,090 2,666,446

NA 7,432,423 1,457,075

NA 212,207 NA

NA 141,437 NA

8,212,419 117,568

4,464,094 37,346

NA 23,110

NA 5,048

NA 9,965

NA 3,493

NA 909

NA 459

NA NA

NA NA

159,209 7,625,523 1,818,764

217,334 6,236,922 2,257,756

NA NA 4,303

NA NA 2,773

9,163 NA 146,377

10,904 NA 185,895

NA NA 531,001

NA NA 357,506

NA NA 4,303

NA NA 2,773

1,820,804

1,844,059

108,058

94,089

137,643

193,239

80,637

602,125

2,056

813

4,915,263 2,414,422,955 18,702,248

3,367,228 2,535,201,693 10,140,239

578,927 41,205,406 141,111

304,789 34,652,198 60,025

303,988 3,468,456 154,919

225,654 3,299,722 85,370

1,067,221 NA 5,440,629

728,417 87,656,989 2,749,463

NA 152,013 2,811

NA 137,847 1,022

28,231,169 387,673

24,894,925 157,742

NA 26,111

NA 10,750

695,661 35,458

1,160,453 27,620

1,598,319 NA

1,109,841 NA

24,032 NA

12,943 NA

99,507,934 18,354

81,533,102 20,004

21,492 2,352

20,903 2,102

612,589 2,176

790,814 3,550

16,849,126 NA

15,559,660 NA

21,496 116

40,207 156

670,583 5,510,621

700,094 4,407,465

9,674 83,268

7,745 52,421

11,345 1,235,886

10,820 892,188

74,996 227,616

73,200 86,390

161 2,347

135 1,316

2,819,916 217,232,549 11,801,030 13,613,210 1,320,651

2,597,830 149,380,569 11,605,030 12,229,145 515,538

331,662 9,556,257 13,859 185,555 NA

259,612 5,273,496 7,696 147,261 NA

153,854 32,928,972 1,343,735 985,614 44,194

120,680 24,866,988 1,512,486 973,817 21,405

645,422 NA 13,699 NA 19,409

576,503 NA 10,550 NA NA

3,250 246,120 NA 20,879 176

2,802 140,841 NA 13,001 114

75,539,100 316,840

63,133,416 250,060

1,427,043 3,055

964,607 1,758

436,345 4,347

341,242 6,432

12,917,880 117,266

9,640,727 83,834

15,827 46

11,084 22

146% 22% -8%

Athens

10,050,680 228,254 57,974

156% 20% 27%

2006 Option Trading Volume Growth: Asia Rate of Growth (annual)

160% Singapore Exchange

140% 120% 100% National Stock Exchange India

80% 60%

Hong Kong Exchanges

40%

TAIFEX

20% 0% -20%

Osaka SE

Australian SE Korea Exchange

0

2

4

Tokyo SE

6 Exchange

8

10

September 10, 2009 14:41

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43


DERIVATIVES - 3.4 STOCK INDEX FUTURES

Exchange



2006


2006 2005 Num ber of Trades

Americas 16,940,891

BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Mercantile Exchange (CME) MexDer New York Board of Trade (NYBOT)

6,683,525

293,433

207,990

178,243

301,558

1,464,734

803,605

3,098,659 28,730,906

2,258,404 26,679,733

370,621 NA

245,880 1,501,704

166,640 167,040

110,405 97,208

1,743,005 NA

1,025,432 NA

470,196,436 620,557

378,748,159 410,565

29,270,013 132,292

22,578,526 61,413

47,144,863 30,959

41,786,549 22,130

145,708,814 33,238

122,479,477 24,244

860,539

922,099

NA

NA

71,698

92,485

NA

NA

Asia Pacific Australian SE

6,652,323 1,628,043 19,747,246

5,713,161 1,111,575 13,393,462

613,940 21,153 2,014,834

451,370 13,210 987,256

268,488 24,621 185,262

175,546 17,814 136,465

1,459,407 NA 9,443,472

1,155,276 NA 6,338,836

Korea Exchange National Stock Exchange India Osaka SE

46,696,151 70,286,227

43,912,281 47,375,214

4,283,838 515,354

2,982,607 279,775

91,200 307,761

83,418 234,624

NA 18,792,431

13,557,429 12,771,115

31,661,331

18,070,352

3,560,096

2,068,205

388,666

409,588

3,025,602

949,211

Singapore Exchange TAIFEX

31,200,243 13,930,545

21,725,170 10,104,645

1,660,847 519,019

1,068,947 688,666

499,159 66,980

411,558 63,667

NA 16,864,405

NA 8,464,444

Thailand Futures Exchange (TFEX) Tokyo SE

198,737 14,907,723

12,786,102

2,595 2,074,924

1,510,707

7,601 369,690

385,914

111,214 NA

NA

Budapest SE Eurex Euronext.liffe JSE OMX

2,634,245 8,007,257 5,697,622 1,879,064

2,521,790 6,081,276 4,875,301 529,563

37,971 1,012,015 1,041,826 7,313

27,724 615,976 777,839 5,222

16,159 86,067 15,470 66,747

18,727 75,608 26,348 4,307

454,205 2,889,255 3,763,954 303,992

360,035 1,993,832 2,966,677 182,057

270,134,951 72,135,006 15,506,101 24,374,765

184,495,160 56,092,515 10,663,676 20,259,026

18,565,389 6,318,763 398,761 329,352

10,851,303 4,154,454 224,904 NA

2,790,632 1,166,209 296,485 551,421

2,166,815 1,027,559 289,601 504,687

NA 18,101,967 301,306 NA

NA 13,122,326 445,755 NA

Oslo Børs Tel Aviv SE Warsaw SE Wiener Börse

2,437,118 32,474 6,257,203 154,521

562,591 13,460 5,167,111 104,677

15,616 589 59,920 13,533

8,245 219 34,864 6,981

56,943 2,682 72,706 17,046

13,665 2,315 30,348 13,260

22,816 219 2,121,215 NA

NA 71 1,437,611 NA

1,166,606,884

881,260,593

-

-

-

-

-

-

Bursa Malaysia Derivatives Hong Kong Exchanges


Total

DERIVATIVES - 3.5 SHORT TERM INTEREST RATE OPTIONS

Exchange






Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME)

10,554,948 605,806 9,424,628 2,594 268,957,139

3,052,800 377,370 6,534,587 4,381 188,001,096

11,195 535,720 NA 13 268,957,127

20,940 311,501 32,672,935 14 188,001,090

2,354,423 78,861 1,130,942 343 18,808,764

697,304 44,375 927,916 317 16,325,364

12,853 2,084 NA 288 1,140,562

9,855 1,476 NA 577 951,078

NA 92 NA 1 NA

NA 76 NA 1 NA

206,853 8,700 3,976,697

247,790 0 41,204

156,487 7,091 3,418,070

188,719 0 37,171

59,544 8,700 481,355

54,132 0 32,500

382 NA NA

425 0 NA

NA NA NA

NA 0 NA

92,985,715 95,000

79,482,008 -

104,878,071 NA

89,052,387 -

10,367,389 67,000

9,586,715 -

65,325 NA

76,311 -

NA NA

NA -

386,818,080

277,741,236

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Singapore Exchange Tokyo Financial Exchange

Europe, Africa, Middle East Euronext.liffe OMX Total NA : Not Available - : Not Applicable

DERIVAT IVES - 3.6 SHORT TERM INTEREST RATE FUTURES

Exchange



2006



Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Mercantile Exchange (CME) MexDer

180,822,732 16,702,302 17,833,331 503,729,899 267,450,231

143,655,871 11,157,298 11,602,282 411,706,656 104,339,918

7,353,654 14,770,015 NA 505,339,873 26,564,227

5,538,228 9,209,807 58,011,410 413,781,671 10,348,810

9,784,628 393,078 414,975 9,564,114 44,058,415

7,332,556 331,916 455,444 8,596,023 21,205,907

555,046 825,430 NA 60,357,744 85,227

486,397 724,190 NA 52,168,804 48,626

22,860,491 272,502 14,043 615 3,573,665 40 31,495,084

18,199,674 162,592 25,181 3,308 2,890,729 217 10,977,591

19,823,462 74,545 2,171 187 2,915,805 138 27,070,811

15,665,366 42,963 3,877 622 2,466,068 310 9,903,104

902,397 59,831 1,532 NA 288,215 0 2,326,719

760,267 37,966 1,477 NA 415,431 0 1,418,937

250,184 NA 752 NA NA 72 NA

236,344 NA 1,229 163 NA 217 NA

2,500 767,458 296,008,444 667 8,170,853

1,390 688,831 248,662,893 0 6,315,805

12 937,064 341,274,218 NA NA

3 833,748 280,316,062 NA NA

0 48,307 6,092,072 63 526,914

500 37,838 5,242,458 0 345,833

5 NA 32,413,840 NA NA

16 NA 25,668,450 NA NA

1,349,704,857

970,390,236

-

-

-

-

-

-

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges Korea Exchange Singapore Exchange TAIFEX Tokyo Financial Exchange

Europe, Africa, Middle East Budapest SE Eurex Euronext.liffe JSE OMX Total

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DERIVATIVES - 3.7 LONG TERM INTEREST RATE OPTIONS

Exchange


2006 2005 Notional Value (USD m illions)




Americas Bourse de Montréal Buenos Aires SE Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE)

2,275

7

202

0

0

2

25

NA

0

NA

8,437 95,737,966

86,036 89,888,554

NA NA

NA 8,931,116

0 3,097,170

293 2,517,698

NA NA

NA NA

1 NA

5 NA

18,736

61,245

92

265

2,038

7,465

1,318

5,203

3

13

3,086,456

2,307,659

235,067

175,753

14,733

1,729

11,078

10,494

NA

NA

0 2,060,624

725 1,699,037

0 NA

308 2,120,602

NA 16,987

NA 22,939

NA NA

NA NA

NA 4,306

NA 3,222

76,328,806

58,551,836

10,870,919

8,449,133

1,786,810

1,405,446

NA

NA

NA

NA

2,785

4,831

NA

11

NA

NA

NA

79

NA

NA

177,246,085

152,599,930

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Singapore Exchange Tokyo SE

Europe, Africa, Middle East Eurex JSE Total

DERIVATIVES - 3.8 LONG TERM INTEREST RATE FUTURES

Exchange



2006



Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) MexDer Philadelphia SE

67,301 7,777,098

16,172 4,824,924

4,214 695,280

1,484 398,274

1,731 337,120

181 166,504

1,102 1,005,657

307 772,125

512,163,874 500,479 10

446,065,592 284,460 -

NA 52,437 NA

46,723,075 27,750 -

5,035,467 43,450 0

3,614,314 2,101 -

NA 2,584 10

NA 1,402 -

45,121,853 28,181 0 10,346,884 1,427,462

36,255,583 27,068 1,250 11,223,811 1,241,852

3,413,538 771 0 1,180,451 116,352

2,761,260 715 169 1,208,118 105,758

872,581 0 NA 112,652 40,186

593,812 150 NA 81,407 27,645

671,133 NA 0 NA NA

655,235 NA 50 1,836,163 NA

40,675 13,680 12,149,979

2,887 78,943 9,844,617

6,745 1,176 10,357,258

1,045 7,122 8,881,026

258 300 131,772

22 1,450 116,664

51,878 NA NA

2,348 NA NA

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges Korea Exchange Singapore Exchange TAIFEX Tokyo Financial Exchange Tokyo SE

Europe, Africa, Middle East BME Spanish Exchanges Budapest SE Eurex Euronext.liffe JSE OMX Tel Aviv SE Warsaw SE Total

15 2,500

46 -

2 12

6 -

1 0

2 -

8 5

22 -

654,119,660 23,245,504

599,621,461 19,078,373

92,905,934 4,356,744

85,843,727 3,468,410

3,796,014 360,521

3,357,373 292,141

NA 2,099,645

NA 2,002,722

8,947 4,354,311 25,005

10,362 3,097,742 -

NA NA 562

NA NA -

63 184,780 651

0 140,258 -

NA NA 1,985

NA NA -

12,875

32,362

431

1,028

50

58

164

484

1,271,406,293

1,131,707,505

-

-

-

-

-

-

NA : Not Available - : Not Applicable

DERIVATIVES - 3.9 CURRENCY OPTIONS

Exchange






Americas Bourse de Montréal BM&F Chicago Mercantile Exchange (CME) MexDer New York Board of Trade (NYBOT) Options Clearing Corp. Philadelphia SE

31,262 10,525,832 3,289,498

7,264 6,850,041 3,182,525

277 44,173 451,686

70 36,604 440,565

2,838 927,188 230,426

2,691 799,576 228,288

2,010 30,110 682,415

466 28,340 608,974

3 NA NA

1 NA NA

306 44,322 0 131,508

0 35,970 0 159,748

34 NA NA 149

0 NA NA 166

2 3,690 10,602 10,476

0 1,778 17,330 17,213

9 NA NA 6,370

0 NA NA 8,861

NA NA NA 149

0 NA NA 166

Europe, Africa, Middle East Budapest SE Euronext.liffe Tel Aviv SE Total NA : Not Available - : Not Applicable

1,022,457

258,000

1,323

251

25,500

86,700

1,050

209

NA

NA

733,039 7,447,717

403,957 6,937,575

9,056 74,820

4,728 69,802

52,150 224,904

42,240 217,476

17,712 335,782

23,871 270,799

126 1,456

133 1,597

23,225,941

17,835,080

-

-

-

-

-

-

-

-

September 10, 2009 14:41

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b708-ch01

45


DERIVATIVES - 3.10 CURRENCY FUTURES

Exchange



2006



Americas BM&F Buenos Aires SE Chicago Mercantile Exchange (CME) ROFEX MexDer New York Board of Trade (NYBOT)

1,726,351

1,293,181

2,738,810

1,737,251

677,724

475,755

1,726,351

1,293,181

800 110,338,043

2,416 81,105,391

1 13,399,645

2 9,798,906

NA 1,098,880

NA 711,360

NA 65,453,858

NA 53,154,207

17,936,247 6,077,409

12,932,275 2,934,783

NA 670,393

NA 323,969

196,293 248,205

323,169 134,992

NA 4,415

NA 2,785

3,653,024

3,604,877

NA

NA

149,595

127,497

NA

NA

1,363 3,158,049 0

4,422 2,667,005 600

103 158,463 0

337 133,679 5

0 160,722 NA

37 85,520 NA

370 NA NA

966 633,614 NA

Asia Pacific Australian SE Korea Exchange Tokyo Financial Exchange

Europe, Africa, Middle East Athens Derivatives Exchange Budapest SE Euronext.liffe Turkish Derivatives Exchange Warsaw SE Total

84

21,844

7

1,692

0

80

3

3,861

10,857,327 8,807

7,742,408 7,435

14,535 216

10,698 176

301,032 1,043

406,942 518

30,281 1,221

19,760 1,510

4,598,416 3,144

1,603,797 6,216

NA 34

1,663 65

170,431 68

134,063 84

NA 2,579

NA 5,184

158,359,064

113,926,650

-

-

-

-

-

-


DERIVATIVES - 3.11 COMMODITY OPTIONS

Exchange

2006 2005 Volume Traded (Nber of contracts)


2006 2005 Open Interest (Nber of contracts)



Americas BM&F Chicago Board of Trade (CBOT) Chicago Mercantile Exchange (CME) Mercado a Término de Buenos Aires New York Board of Trade (NYBOT) NYMEX ROFEX

177,719 21,861,340 2,010,226 2,815,000 11,662,056 54,468,396 34,815

195,103 16,353,965 943,377 2,091,500 8,663,470 38,002,895 59,475

194 NA 67,569 NA NA NA NA

194 304,650 34,006 NA 220,560 2,193,391 NA

12,541 2,177,795 307,489 NA 1,146,100 9,297,986 6,039

5,999 900,266 116,431 NA 928,436 NA 4,706

1,354 NA 470,806 NA NA NA NA

1,560 NA 389,526 NA NA NA NA

NA NA NA NA NA NA NA


10,683 27,262

558 27,101

380 NA

72 42

21,264 409

369 288

488 284

49 49

NA NA

NA NA

832 727,190 138,129 512,518 8,412,350

40 444,754 118,476 451,885 8,184,187

13.42 271 NA 1,898,026 NA

0 226 NA 337,671 468,446

260 136,475 23,987 48,568 1,007,248

95 60,129 5,832 57,950 757,837

29 9,257 NA 52,749 NA

3 7,059 NA 40,655 NA

NA 21 NA NA 6,716

NA 11 NA NA 4,397

102,858,516

75,536,786

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Tokyo Grain Exchange

Europe, Africa, Middle East Budapest SE Euronext.liffe ICE Futures JSE London Metal Exchange Total NA : Not Available - : Not Applicable

DERIVATIVES - 3.12 COMMODITY FUTURES

2006 Exchange

2005

Volume Traded (Nber of Contracts)

2006

2005

2006

Notional Value (USD millions)

2005

2006

Open Interest (Nber of Contracts)

2005

Num ber of Trades


1,318,203

1,073,471

12,436

10,106

63,964

50,996

219,847

214,293

118,719,938 17,448,155 11,899,472 28,233,129

76,786,994 11,558,317 11,502,296 24,486,440

NA 613,145 NA NA

1,293,074 394,707 NA 500,155

2,821,951 536,649 NA 1,065,666

1,732,853 387,575 NA 901,038

NA 5,079,223 NA NA

NA 4,212,551 NA NA

178,929,185 116,937

166,608,642 118,973

NA NA

8,893,687 NA

9,326,151 11,984

NA 10,409

NA NA

NA NA

185,349 2,230,340 9,019,416

36,481 1,158,510 33,179,422

3,321 48,051 NA

1,160 21,313 1,943,220

55,600 74,567 117,816

18,010 28,918 182,304

12,295 NA NA

6,150 NA NA

117,681,038 3,158,049 58,106,001

99,174,714 2,667,005 33,789,754

NA 158,463 NA

622,949 133,679 515,274

1,154,982 160,722 196,219

482,979 85,520 154,723

NA NA NA

NA NA NA

35,027 19,106,247 46,298,117

0 25,573,238 28,472,570

2,206 1,302,452 NA

0 406,973 16,166

44 438,435 213,847

0 563,665 452,058

12,724 NA NA

0 NA NA

Asia Pacific Australian SE Bursa Malaysia Derivatives Central Japan Com modity Exchange Dalian Commodity Exchange Korea Exchange Shanghai Futures Exchange TAIFEX Tokyo Grain Exchange Zhengzhou Commodity Exchange


8,750 9,124,195

778 8,054,116

140 119,436

9 85,794

1,093 449,829

601 419,333

1,856 1,257,639

189 906,230

92,582,921 1,436,155 78,527,839

41,936,609 1,335,964 70,444,665

NA 1,864,750 7,146,569

NA 15,158,450 4,045,775

1,389,618 43,295 1,515,663

642,753 51,295 2,411,870

NA 205,430 NA

NA 169,767 NA

794,164,463

637,958,959

-

-

-

-

-

-

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Summary The last three decades have witnessed a proliferation of financial innovations. Roughly speaking, financial innovations seem to belong to two classes. First, there are the new securities and their markets such as traded and OTC equity and interest rate derivative assets. Second, there are dynamic trading strategies using these instruments. Traded derivative assets are standardized contracts which are listed on options exchanges. OTC derivative assets are tailor-made to the investor’s needs and are often written by investment banks. Examples of classic or standard financial assets and commodity contracts include forward rate contracts, futures contracts, swaps, standard calls and puts, traded stock options, equity warrants, covered warrants, options on equity indices, options on index futures contracts, options on currency forwards or currency futures and bond options. Futures and options market enable investors to manage price risk. The market offers an environment that allows all users to control the price risk. The prices of these financial instruments are fully transparent because they are updated second by second as trading occurs. Examples of commodity contracts are oil and cocoa. The oil market is ultimately concerned with the transportation, processing, and storage of a raw material. Crude oil is traded on world markets using the spot asset, physical forward contracts, futures contracts, options on futures contracts, swaps, warrants, etc. Price information can be obtained from oil and energy pricing information such as Reuters, Bridge Telerate, Platt’s, etc. The size and complexity of global crude oil trade are unique among physical commodities. Worldwide crude oil trade in the last 30 years has gone through revolutionary changes that have had large political and economic impact adding to its uniqueness. Each crude oil from each field is unique in quality. The trading instuments apply to some crudes including West Texas Intermediate (WTI), Dubai, Alaska North Slope (ANS) and Brent blend. Each of these crudes or blends define its specific oil market. However, the markets are linked together through arbitrage. The Brent market includes partial forward transactions, a futures contract traded in London at the International Petroleum Exchange (IPE), options on this contract and swap deals. The history of cocoa dates back to the 6th century with its origins in the Amazon Basin. It was first brought to Europe in the 17th century as a luxury drink. Market users include the international cocoa trade, cocoa processors and chocolate manufacturers, managed futures funds, institutional investors and options specialists. Full cocoa-related statistics

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are published in the “Quarterly Bulletin of Cocoa Statistics” in the ICCO publications. For example, the estimate of world cocoa-bean production for the 1996/1997 cocoa year is 2,695,000 tonnes, down 20,000 tonnes from the figure in the June 1997 Newsletter. World grindings of cocoa beans in 1996/1997 were estimated at 2,815,000 tonnes, representing an increase of 12,000 tonnes compared with the previous forecast. The information concerns the gross crop, the net crop, grindings, surplus/deficit, total stocks, and free stocks. The cocoa futures contract was originally launched in 1928. The cocoa traded options contract was launched in 1987 as a means of offering market participants even greater flexibility and choice in their underlying activities. These contracts are traded in London (LIFFE). Index options on stock indices and stock index futures began trading in the United States in 1983 with the introduction of the S&P 100 contract on the Chicago Board Options Exchange. There are several types of bonds and bond options traded in organized and OTC markets. They include zero-coupon bonds, bonds with call provisions, putable bonds, convertible bonds, bonds with warrants attached, exchangeable bonds, etc. The futures contract is marked-to-market at the end of each trading day and is subject to interim cash flows. The main difference between futures contracts and forward contracts is that forward contracts are OTC instruments which are nonstandardized and are subject to counter-party risk. There are several traded interest rate futures contracts. Financial assets may appear in nonstandard fashion, i.e., they can be tailor-made and their pay-offs may be path-dependent or path-independent. A path-dependent contingent claim is an option whose pay-off depends on the history of the underlying asset price. In general, an upward movement of the underlying asset price followed by a downward movement is different from a downward movement followed by an upward movement. This is a main property of path-dependent options. For path-independent contingent claims, an upward movement of the underlying asset price followed by a downward movement is equivalent to a downward movement followed by an upward movement. Examples of nonstandard financial assets include forward start options, pay-later options, chooser options, options on the minimum or the maximum of several assets, two-color rainbow options, options with extendible maturities, ratio options, exotic options, barrier options, Asian options, partial and full lookback options and more sturctured products with embedded digitals such as rebate range binaries, mandarin collars, mega-premium options, and limit binary options. Several authors proposed different explanations for the development of these markets and the proliferation of the new

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financial instruments. For example, according to Ross (1989), the existence of new financial instruments and strategies and the marketing process are based on the cost structure of the marketing networks and distribution channels. It is the institutional structure of contracts and incentives that allows the process of financial engineering to continue. Hence, it seems that institutional markets and financial marketing are central to the understanding of financial innovations.

Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

What are the specific features of options? What are the specific features of futures and forward contracts? What are the trading characteristics of commodity contracts? What are the specific features of the main instruments traded on the International Petroleum Exchange? Describe the specific features of the cocoa market. Describe the specific features of equity options. Describe the specific features of options on currency forwards and futures. Describe the specific features of bonds and bond options markets. Provide some examples of simple and complex financial instruments. Why there are so many new financial instruments? What are the fundamental reasons behind the proliferation of financial assets? Why has the wave of financial innovation not stopped?

Exercises 1. Explain how an investor uses options. • • • • • •

Options are easily bought and sold. Holders can sell or exercise their options at any time. Most options are traded on exchanges and/or on over the counter. At maturity, holders of physical options exercise into actual shares. Holders of cash-settled options choose to sell their options. They involve the purchaser in completing options counterparty documentation.

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2. Explain why institutions and individuals use options. Institutional and individual investors use options to achieve an astonishingly broad spectrum of goals, such as hedging, arbitrage and speculation. Options can be used in several strategies: • • • •

Aggressive strategies Leveraged strategies Protect an existing portfolio Combinations where cash spent is recouped by interest. This prevents from putting capital at risk, etc.

3. Explain exchange traded or listed options and the role of the clearing house. Options are exchange traded contracts (or OTC contracts) for making economic commitments based on the shifting values of stock prices, indices, etc. The clearing house interposes itself in all transactions as the buyer to every seller and the seller to every buyer, so every party is free to liquidate his position at any time by making an offset closing transaction. A committee charged with developing new financial instruments can submit the proposal. Then, we wait for the approval. 4. Provide historical reasons for the development of the option market. The idea of stock options was borne in 1972. The idea of index options and futures was born in 1977. After the success of the initial ‘covered market’ (exercise against shares), banks issued options without having an underlying corporate to provide the hedge. New issues of options can also be cash settled at maturity. The market today provides a range of exposures on most of the world’s significant equity markets. In many markets, local authorities are actively fostering options development. The market today provides options which are highly liquid. For options and warrants, the market provides liquid and less liquid warrants and exotic warrants in emerging markets.

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5. What is the appropriate definition of an option? Options can be calls or puts. A call gives the buyer the right, but not the obligation to exercise, and thereby receive in cash or physical delivery any amount by which the underlying asset is above the strike price. A put gives the buyer the right, but not the obligation to exercise, and thereby receives in cash or physical delivery any amount by which the underlying asset is below the strike price. European options are only exercised at maturity. American options are exercised at any time before maturity.

6. How is an option exercised? An option entitles the right and not the obligation to buy or sell the underlying. This right has value. A call entitles the right to the holder to buy the underlying asset. A put entitles the right to the holder to sell the underlying asset. Options give the holder the right to buy or sell a specific asset at a fixed price on or before a given expiry date. If the right is exercised at any time, this is an American type option. If the right is exercised at maturity, this is a European type option. The value in cash (received or stock) corresponds to the exercise value. Options with exercise value are said to be in the money. Options with no exercise value are out of the money. Options for which the strike price is equal to the underlying asset price is at the money.

7. What happens for buyers and sellers among exercise? For the option buyer, exercise is a right, not an obligation. Sellers have an unconditional obligation to respond whenever the buyer chooses to exercise. It may seem that the buyer has all the advantages, and that the seller assumes nothing but liabilities. That is why the buyer has to pay the seller for the option. The buyer must pay cash to the seller for the option’s full price. Since this is the maximum amount the buyer can lose in a transaction, he is not required to pay any additional security as margin.

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The seller, on the other hand, must post margin with the clearing house as a performance bond or calculated by a formula based on the relationship of the asset price and the strike price. The seller may be required to deposit additional margin if his position moves against him. Options can be exercised in cash against the closing price of the underlying asset (compared with the strike price). They can be exercised also with the requirement of transferring actual shares of stock. 8. How can options be used? • Options can be used as an alternative means of implementing different strategies investors execute directly in the stock market, but with enhanced performance and reduced transaction costs. • Options can be used to structure unique patterns of risks and returns that would have been impossible without them. The cost of an option is significantly less than the price of the underlying asset. This allows for leverage (or Gearing), of the option. (The underlying asset of the option is a single stock, a basket of stocks, an equity index, a currency, etc.) Options can be exercised and settled physically in return for the physical shares. Options can be exercised and settled in cash for an amount equal to their intrinsic value. 9. How is each option contract specified? In selecting the contract that best suits the investment applications, investors can sort through a variety financial instruments: calls, puts, European and American options. Each contract is specified by: • A contract multiplier: the value times which the contract price is multiplied to determine its total value; • Minimum fluctuation: the smallest permissible increment of price change; • Expiration terms dates for expiration; • Trading hours;

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• Position limits: the maximum number of contracts the exchange will permit an investor to control. Investors can contact brokers and the exchanges to obtain the current specifications.

10. Describe cash and margin requirements. Buyers pay the full dollar value of their contracts. The buyer is never required to deposit additional funds if the position moves against him. The option seller is obliged to pay the difference between the option’s strike price and the underlying, a good-faith deposit of cash or securities ensure eventual performance. This is the case for cash setteled options. For traditional stock market trading, the term ‘margin’ suggests a down payment on the full value of securities purchased, with the brokerage firm loaning the investor the balance. For options, the exact amount of margining to be deposited must be determined. Margins can also be different between speculators and hedgers. The margin, for example, for stock (index) options may be a % of the underlying stock (index) plus the option price. Margin requirements can be recalculated each day on a mark-to-market basis. If subsequent calculations show higher requirements, the seller must deposit additional margins.

11. Should investors pay transaction costs? Any time investors trade securities they pay two types of transaction costs. First, they pay an explicit commission to a broker for executing and clearing the trade. Second, they pay an implicit market impact cost because their bids will inevitably drive prices higher when they wish to buy and their offers will drive prices marginally lower when they wish to sell. Commissions are negotiable between the investor and the broker. Like stocks, options commissions are charged on a one way basis.

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12. What about tax treatment? Profits and losses from trading can be treated as long term capitalgains or losses and short term capital gains or losses. 13. What about trading orders? In many ways, trading options is just like trading stocks. Trading orders most commonly used reflect: • Time duration; • Good-Til Canceled (or open); and • Opening only. 14. What are price specifications? Can include: • • • • • •

Limit order (maximum purchase price); Discretion or limit order; Delta; Market order; Market on close order; and Market if touched.

15. What are contingencies? • Contingent: a contingent order is in effect when a specified condition is satisfied; • Stop order; and • Stop limit. 16. What are special instructions? • Immediate or cancel; • All or none; or • Fill or Kill. 17. Explain cancellations. • Straight cancel;

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• If nothing done cancel; or • Cancel former order. 18. How are contracts exercised and assignments conveyed? Orders to exercise must be tendered in writing to the exchanges by a member firm no later than the close of trading on the day of the exercise. Brokerage firms may apply earlier cutoff times for receipt of oral exercise instructions from their customers. At expiration, long customer positions in the money are automatically exercised. Once a contract has been exercised, the clearing corporation assigns it by random lottery among the universe of member brokerage firms carrying matching short positions. Assignment notices are generally conveyed to customers before the opening of the trading on the market day following the exercise date. 19. Explain the world of floor traders. Floor traders are of two basic breeds: market makers and floor/brokers. Floor brokers act as agents executing orders in the crowd on behalf of others. They earn their livelihoods by collecting commissions on the trades they execute. Their income is determined by the volume of transactions they complete. Market makers put their own capital at risk in the trading. Their only source of income is the profit they can derive from their trading and their only limit is the risk they are willing to bear. Before the exchanges admit a new trader as a member, they investigate his background and administer a test. 20. What are some floor strategies? The scalpers: They exploit the fact that the price of any traded asset is quoted as a two sided market comprised of the highest bid and the lowest offer.

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They will simultaneously make bids and offers, indifferent to whether they end up buying or selling. Their concern is that whenever they buy, they buy on the bid side of the market, and whenever they sell, they sell on the offer side of the market, thus their profit earned is the differential between the two. The shooter: He is another type of market maker who tries to make purchases at the bid and sales at the offer. Unlike the scalper, he is willing to inventory positions in anticipation of market moves. The shooter is in the game for the big score. The spreader: He seeks out and exploits minute inefficiencies in the pricing structure of the options markets. 21. Describe a day of trading and how the exchange works. The clearing houses accept as only firms that demonstrate substantial financial strength and business integrity members. They maintain elaborate safeguards against defaults, including special funds to be used in the event of losses, to which member firms must contribute. The process of determining the opening price is an unstructured negotiation that begins several minutes before the official opening. Market makers provide the prices. Exchange employees called pit observers report the transactions to terminal operators who disseminate the transactions to quotation services around the world. Discipline in the pit is provided by pit observers who monitor trading activity for accuracy and fairness throughout the day. Public orders are handled by floor brokers. Market continuity is provided by the presence of competing market makers. In other markets, a staff of exchange employees maintains a public book of limit orders. Customer market orders can be put in computer that randomly assigns them to participating market makers and reports the trades instantaneously. New technologies are used and are integrated into the stock exchange trading process.

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Appendix Derivatives Markets in the World Before and During the Financial Crisis Stock Options, Index Options, Interest Rate and Commodity Options and Futures Markets Global overview Several institutions produce information regarding futures and options around the world. Often, summary statistics on volume and open interest are given for futures and index options. Index options on stock indexes and index futures contracts begin trading in the U.S in 1983. This has been facilitated with the introduction of the SP 100 index contract on the Chicago Board Options Exchange. Today, index futures are traded and are more liquid than index options. The main indexes around the world: a historical perspective The first options traded on indexes can be traced back to US (SP 500 and SP 100 in 1983), Japan (Nikkei 225 in 1989), UK (FT-SE 100 in 1984), France (CAC 40, 1989), Germany (DAX, 1991), Switzerland, (SMI, 1980), Canada (TSE 35,1987),Netherlands(EOE,1978),Australia(AllOrdinaries,1983),.. . Options volume in listed markets is mostly concentrated in one month contracts in all markets. For most options, volume with longer maturities take place in OTC markets. In the OTC market, trading began early in 1988. Several investors buy long-term puts to implement portfolio insurance strategies. Today, dealers run large OTC options books. This can reduce or eliminate risk in the market. North America. U.S index options trading appear on listed markets and OTC markets with customized features. Options are traded on SP 100, SP 500, MMI, SPMidCap, options on small capitalization indexes, the NYSE Composite index. Main information used concerns the average daily volume, Average daily dollar volume (in millions) and Index level. Options on SP are preferred by retail investors. MidCap Options and options on SP 500 index attract the interest of institutional money managers and pension funds.

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SP 100 are the mostly traded contracts in the U.S. SP 500 are the have the greatest open interest in the U.S. Hundred billions of dollars are traded. Institutional use of index options: Covered call writing: a call is sold and the underlying asset is held, Long index put strategy and collar positions, which is preferred by institutions. The collar can lead to a skew in index options implied volatilities: out of the money puts have higher volatilities than calls. Options are available on National OTC (PHX) indexes. Stock index markets in North America: The SP 500 index fluctuated in a band. The move gives a volatility in a range of 10%–25%. We can represent a monthly volatility for the year. With its heavier dose of cyclical stocks, the DJIA has been outperforming for some years the broader market. We can compute historical volatility and implied volatility from at the money options. We should compute the spread. The following Tables shows the volume (number of contracts traded) in several countries. Japan. Options exist on Osaka Nikkei, options on TOPIX. Japanese institutions often use for their long term options exposure or customized strike prices fixed income securities with embedded index options. Osaka Nikkei options are used by domestic institutional in short term trading. Regulations by the Ministry of Finance prevent pension funds from completely hedging their portfolios (hedging limit 50%). Hedgers integrated their activities into equity risk management systems. Life insurance companies focus on using options for directional trading. Offshore hedge funds use the Osaka Nikkei options to take outright shortterm trading positions. The Government intervenes to support the market. Foreign institutions act in the OTC market for different reasons: They are restricted by regulation from trading listed options. They do not want to incur the costs of rolling over. Competition among dealers makes this market very competitive. Sector options are popular in Japan. The following Table provides the volume (number of contracts traded) in several countries for index options.

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Stock index options 2004 Exchange

2003


2003

Americas American SE

40,985,108

33,137,709

89,965

0

Bourse de Montreal

336,544

961,650

35.00%

Chicago Board of Trade (CBOT)

762,007

263,629

289.05%

136,679,303

110,822,096

123.33%

6,451,862

5,168,914

124.82%

40,886,923

23,979,352

170.51%

35,989

0

181,215

110,079

0

0

Pacific SE

14,119,270

15,744,139

89.68%

Philadelphia SE

25,360,908

19,746,264

128.43%

1,589,765

1,600,461

99.33%

BM&F

Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchange (ISE) MexDer New York Board of Trade (NYBOT) Options Clearing Corp.

Sao Paulo SE

123.68%

164.62%

Europe, Africa, Middle East Athens Derivatives Exchange

941,387

1,388,985

67.78%

BME Spanish Exchanges

2,947,529

2,981,593

98.86%

Borsa Italiana

2,220,807

2,505,351

88.64%

1,299

8,440

15.39%

117,779,232

108,504,301

108.55%

Euronext

99,607,852

103,986,651

95.79%

JSE South Africa

11,268,763

10,505,417

107.27%

8,947,439

6,371,381

140.43%

681,783

543,090

125.54%

Tel Aviv SE

36,915,103

29,353,595

125.76%

Warsaw SE

124,392

153,106

81.25%

40,855

27,680

147.60%

794,121

630,900

125.87%

56,046

43

130339.53% 99.20%

Copenhagen SE Eurex

OMX Stockholm SE Oslo Bors

Wiener Börse

Asia Pacific Australian SE BSE, The SE Mumbai Hong Kong Exchanges

2,133,708

2,150,923

2,521,557,274

2,837,724,956

88.86%

2,812,109

1,332,417

211.05%

Osaka SE

16,561,365

14,958,334

110.72%

SFE Corp.

523,428

585,620

89.38%

Singapore Exchange

247,388

289,361

85.49%

43,824,511

21,720,084

201.77%

17,643

98,137

17.98%

3,137,482,893

3,357,354,658

93.45%

Korea Exchange National Stock Exchange India

TAIFEX Tokyo SE Total

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2005 Exchange

2004 Volume Traded (Nber of Contracts)

Americas American SE BM&F Bourse de Montreal Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchange (ISE) MexDer New York Board of Trade (NYBOT) Options Clearing Corp.

8,678,564

7,290,157

6,344

16,485

650,186

336,544

728,349

762,007

192,536,695

136,679,303

15,106,187

6,451,862

4,464,094

83,358

37,346

35,989

217,334

181,215

0

0

Philadelphia SE

6,234,567

5,275,701

Sao Paulo SE

2,257,756

1,589,765

1,163,260

794,121

Asia Pacific Australian SE Bombay SE Hong Kong Exchanges Korea Exchange

100

NA

3,367,228

2,133,708

2,535,201,693

2,521,557,274

National Stock Exchange India

10,140,239

2,812,109

Osaka SE

24,894,925

16,561,365

SFE Corp.

680,303

523,428

Singapore Exchange

157,742

247,388

81,533,102

43,824,511

20,004

17,643

TAIFEX Tokyo SE


700,094

941,387

4,407,465

2,947,529

2,597,830

2,220,807

149,380,569

117,779,232

Euronext.liffe

70,228,310

99,607,852

JSE

11,473,116

11,303,311

OMX

12,229,145

8,947,439

Eurex

Oslo Børs

515,538

695,672

Tel Aviv SE

63,133,416

36,915,103

Warsaw SE

250,060

78,752

37,127

40,855

3,203,028,688

3,028,651,872

Wiener Börse Total

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Europe. In Germany. Listed DAX options are done on a screen-based system. Major players in this market are the large U.S and Continental investment banks. In France. Listed CAC 40 options trade on the French options market where trading is dominated by locals taking speculative positions and by large investment banks. Institutional users are French insurance companies and fund managers. Players seek leveraged exposures on the market. Guaranteed funds on the CAC 40 issued by French banks are popular among retail investors. CAC 40 options are used as part of these products. Major participants in the OTC market are large U.S and European investment banks. Stock index markets in France: – An interesting development in the CAC 40 futures is the distribution of open interest across various months. – Institutions have led to move into the quarterly contracts to eliminate the chore of rolling on a monthly basis. The lack of a developed stock-borrowing market can reduce trading in futures. Professional traders can use the futures to hedge OTC options. To hedge collars traders can be short futures. Arbitrageurs (short stock/long futures) can unwind easily their positions. United Kingdom. The market is dominated by major international banks and brokers. Short-term maturities have the most liquidity. End-users are mainly U.K institutions for hedging and guaranteed funds. In OTC markets, the volume is also high because of greater liquidity in the longer-dated contracts. There is flexibility in expiration dates. Switzerland. Options are traded on the SOFFEX in an electronic screen system. Active participants are major Swiss and American Banks. End users are a mixture

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of short-term speculators and international institutions looking for long exposures. The OTC market is important because there is a need for longer-term strategies on the SMI from pension funds. Zero premium collars are very popular. Netherlands. This market is dominated by locals who service a retail base. Users are mainly pension funds who hedge equity portfolios. The index must be compiled using a specific method. The weighting of the index can overweight smaller, domestically oriented stocks and underweight larger, more internationally oriented stocks. For example, stocks can be weighted using a market capitalization and the maximum weight of a stock in the index will not exceed 10%. This puts a cap on some stocks. The following table gives the notional value (value traded of stocks), the open interest (positions opened and still not unwind) and the option premiums for several countries. The following tables provide different information for several markets and instruments. The reader can compare the different markets and instruments using these tables (source: World Federation of Exchanges). DERIVATIVES - 3.1 STOCK OPTIONS

Exchange






Americas American SE Boston Options Exchange

186,994,609 92,260,125

77,582,231

NA

NA

NA

NA

NA

NA

NA

NA

Bourse de Montreal

12,265,461 49,235,173 390,657,577

10,032,227 92,386,767 275,646,980

68,947 NA 1,960,297

54,904 NA 1,264,511

1,583,405 1,654,931 187,953,281

1,346,141 1,605,194 151,157,355

732,202 NA 25,792,792

554,076 NA 16,820,556

2,212 456 98,751

1,645 547 61,220

Buenos Aires SE Chicago Board Options Exchange (CBOE) International Securities Exchnage (ISE) MexDer Options Clearing Corp. Pacific SE Philadelphia SE Sao Paulo SE

193,086,271

45,779

42,238

NA

NA

4,709,107

7,652,680

NA

NA

583,749,099 448,120

442,387,776 135,931

NA 829

NA 208

NA 0

NA 2,030

NA 62

NA 49

NA NA

NA NA

0 196,586,356 265,370,986

0 144,780,498 156,222,383

NA NA 89,732

NA NA 49,318

220,032,992 NA 8,846,285

181,694,503 NA 8,379,867

NA NA 15,843,704

NA NA 7,190,023

NA NA 89,732

NA NA 49,318

285,699,806

266,362,631

513,350

392,331

1,833,555

1,824,504

6,542,663

5,777,709

9,746

7,909

20,491,483 18,127,353

21,547,732 8,772,393

303,986 88,371

270,423 41,784

1,766,513 2,533,807

1,678,335 1,021,913

1,474,017 399,129

1,418,149 241,785

11,501 2,477

9,057 1,334

1,195 5,214,191 753,937

3,655 5,224,485 1,206,987

41 44,479 NA

11 40,260 NA

50 21,549 22,541

NA 24,181 79,610

NA 4,478,610 4,064

103 4,550,367 5,454

NA 1,254 186

0 1,100 293

1,089,158 190,876

1,018,917 201,798

32 21

79 33

2,797 39,428

3,959 11,906

45,088 NA

126,245 NA

31 21

161 33

Asia Pacific Australian SE Hong Kong Exchanges Korea Exchange National Stock Exchange India Osaka SE TAIFEX Tokyo SE

Europe, Africa, Middle East Athens Derivatives Exchange BME Spanish Exchanges Borsa Italiana Budapest SE Eurex Euronext.liffe JSE OMX Oslo Børs RTS SE Warsaw SE Wiener Börse Total NA : Not Available - : Not Applicable

17,194

21,729

52

60

1,297

2,004

396

397

3

2

12,425,979 16,056,751 350

10,915,227 12,439,716 176

27,775 91,803 5

20,605 67,776 6

2,748,562 1,964,411 NA

2,411,628 1,646,014 NA

75,313 475,942 6

65,136 442,151 8

1,067 2,771 NA

633 1,979 NA

272,543,052 155,552,010 5,751,832 64,514,641

255,918,793 264,714,188 2,539,526 57,138,563

964,097 603,265 312 69,691

752,434 618,732 153 57,580

52,069,011 45,341,415 916,339 8,418,826

53,312,606 55,353,971 564,302 7,404,092

NA 3,272,555 2,835 NA

NA 2,728,180 1,733 NA

59,286 32,141 NA 27,306

38,740 70,685 NA 15,434

5,811,946 10,727,870

3,325,368 7,281,162

NA 11,453

NA 2,797

616,315 1,431,028

364,265 433,158

34,135 150,940

NA 113,317

643 NA

321 NA

10,988 1,053,298

4,372 816,032

98 5,385

29 4,609

162 116,063

413 76,166

5,501 NA

2,642 NA

4 230

1 165

2,653,601,416

2,311,714,514

-

-

-

-

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DERIVATIVES - 3.2 STOCK FUTURES

Exchange



2006



Americas MexDer

3,000

19,400

21

85

0

3,400

62

17

693,653 958 102,010 100,285,737

490,233 13,069 68,911,754

8,645 4 655 857,436

5,872 77 510,701

124,307 0 4,260 642,395

78,289 1,750 464,559

5,194 NA 9,382 82,217,305

3,308 2,170 56,491,871

2,476,487 21,229,811 7,031,974 919,426 35,589,089 29,515,726 69,671,751 8,459,165 3,626,036 112,824 12,371

1,431,514 18,813,689 5,957,674 740,396 77,802 12,158,093 24,469,988 5,659,823 1,796,570 172,828 23,748

5,543 43,266 49,636 9,052 203,038 344,198 26,288 6,128 3,502 782 180

3,160 31,708 41,798 7,842 NA 64,062 10,223 NA 2,516 845 331

116,576 1,649,184 41,319 65,015 1,459,509 1,489,169 12,027,716 1,764,492 268,572 1,122 1,339

124,815 1,921,717 58,071 24,936 58,107 467,117 1,535,839 1,387,095 126,266 2,928 2,448

285,982 139,441 56,774 92,618 NA 22,948 392,154 NA NA 87,999 NA

167,715 119,499 66,605 81,468 NA 21,006 177,766 NA NA 130,674 NA

279,730,018

140,736,581

-

-

-

-

-

-

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges National Stock Exchange India

Europe, Africa, Middle East Athens Derivatives Exchange BME Spanish Exchanges Borsa Italiana Budapest SE Eurex Euronext.liffe JSE OMX Oslo Børs Warsaw SE Wiener Börse Total

DERIVATIVES - 3.3 STOCK INDEX OPTIONS


Exchange


2006




Americas American SE BM&F Bourse de Montréal Chicago Board of Trade (CBOT)

16% 126% 108% -24%

Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchnage (ISE) MexDer New York Board of Trade (NYBOT) Philadelphia SE

45% 81% 84% 215% -27% 22%

Sao Paulo SE

-19%

Australian SE Hong Kong Exchanges Korea Exchange

-1% 46% -5%

National Stock Exchange India Osaka SE

84% 13%

Singapore Exchange TAIFEX Tokyo SE

-4%

BME Spanish Borsa Italiana Eurex

25% 9% 45%

JSE OMX

2% 11%

Oslo Børs Tel Aviv SE Warsaw SE

8,678,564 101,003 27,897

18,801 4,401 3,477

6,922 3,135 1,527

NA 106,601 1,691

NA 38,382 4,813

123,559 749 4,620

122,714 466 1,648

NA NA 70

NA NA 141

551,190 279,005,803 27,295,611

728,349 192,536,695 15,106,187

NA 17,791,735 6,005,296

NA 11,541,513 3,295,855

21,815 37,749,429 1,527,059

26,794 29,381,746 1,226,413

NA 11,479,090 2,666,446

NA 7,432,423 1,457,075

NA 212,207 NA

NA 141,437 NA

8,212,419 117,568

4,464,094 37,346

NA 23,110

NA 5,048

NA 9,965

NA 3,493

NA 909

NA 459

NA NA

NA NA

159,209 7,625,523 1,818,764

217,334 6,236,922 2,257,756

NA NA 4,303

NA NA 2,773

9,163 NA 146,377

10,904 NA 185,895

NA NA 531,001

NA NA 357,506

NA NA 4,303

NA NA 2,773

1,820,804

1,844,059

108,058

94,089

137,643

193,239

80,637

602,125

2,056

813

4,915,263 2,414,422,955 18,702,248

3,367,228 2,535,201,693 10,140,239

578,927 41,205,406 141,111

304,789 34,652,198 60,025

303,988 3,468,456 154,919

225,654 3,299,722 85,370

1,067,221 NA 5,440,629

728,417 87,656,989 2,749,463

NA 152,013 2,811

NA 137,847 1,022

28,231,169 387,673

24,894,925 157,742

NA 26,111

NA 10,750

695,661 35,458

1,160,453 27,620

1,598,319 NA

1,109,841 NA

24,032 NA

12,943 NA

99,507,934 18,354

81,533,102 20,004

21,492 2,352

20,903 2,102

612,589 2,176

790,814 3,550

16,849,126 NA

15,559,660 NA

21,496 116

40,207 156

670,583 5,510,621

700,094 4,407,465

9,674 83,268

7,745 52,421

11,345 1,235,886

10,820 892,188

74,996 227,616

73,200 86,390

161 2,347

135 1,316

2,819,916 217,232,549 11,801,030 13,613,210 1,320,651

2,597,830 149,380,569 11,605,030 12,229,145 515,538

331,662 9,556,257 13,859 185,555 NA

259,612 5,273,496 7,696 147,261 NA

153,854 32,928,972 1,343,735 985,614 44,194

120,680 24,866,988 1,512,486 973,817 21,405

645,422 NA 13,699 NA 19,409

576,503 NA 10,550 NA NA

3,250 246,120 NA 20,879 176

2,802 140,841 NA 13,001 114

75,539,100 316,840

63,133,416 250,060

1,427,043 3,055

964,607 1,758

436,345 4,347

341,242 6,432

12,917,880 117,266

9,640,727 83,834

15,827 46

11,084 22

146% 22% -8%

Athens

10,050,680 228,254 57,974

156% 20% 27%

2006 Option Trading Volume Growth: Asia Rate of Growth (annual)

160% Singapore Exchange

140% 120% 100% National Stock Exchange India

80% 60%

Hong Kong Exchanges

40%

TAIFEX

20% 0% -20%

Osaka SE

Australian SE Korea Exchange

0

2

4

Tokyo SE

6 Exchange

8

10

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DERIVATIVES - 3.4 STOCK INDEX FUTURES

Exchange



2006



Americas 16,940,891

BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Mercantile Exchange (CME) MexDer New York Board of Trade (NYBOT)

6,683,525

293,433

207,990

178,243

301,558

1,464,734

803,605

3,098,659 28,730,906

2,258,404 26,679,733

370,621 NA

245,880 1,501,704

166,640 167,040

110,405 97,208

1,743,005 NA

1,025,432 NA

470,196,436 620,557

378,748,159 410,565

29,270,013 132,292

22,578,526 61,413

47,144,863 30,959

41,786,549 22,130

145,708,814 33,238

122,479,477 24,244

860,539

922,099

NA

NA

71,698

92,485

NA

NA

Asia Pacific Australian SE

6,652,323 1,628,043 19,747,246

5,713,161 1,111,575 13,393,462

613,940 21,153 2,014,834

451,370 13,210 987,256

268,488 24,621 185,262

175,546 17,814 136,465

1,459,407 NA 9,443,472

1,155,276 NA 6,338,836

Korea Exchange National Stock Exchange India Osaka SE

46,696,151 70,286,227

43,912,281 47,375,214

4,283,838 515,354

2,982,607 279,775

91,200 307,761

83,418 234,624

NA 18,792,431

13,557,429 12,771,115

31,661,331

18,070,352

3,560,096

2,068,205

388,666

409,588

3,025,602

949,211

Singapore Exchange TAIFEX

31,200,243 13,930,545

21,725,170 10,104,645

1,660,847 519,019

1,068,947 688,666

499,159 66,980

411,558 63,667

NA 16,864,405

NA 8,464,444

Thailand Futures Exchange (TFEX) Tokyo SE

198,737 14,907,723

12,786,102

2,595 2,074,924

1,510,707

7,601 369,690

385,914

111,214 NA

NA

Budapest SE Eurex Euronext.liffe JSE OMX

2,634,245 8,007,257 5,697,622 1,879,064

2,521,790 6,081,276 4,875,301 529,563

37,971 1,012,015 1,041,826 7,313

27,724 615,976 777,839 5,222

16,159 86,067 15,470 66,747

18,727 75,608 26,348 4,307

454,205 2,889,255 3,763,954 303,992

360,035 1,993,832 2,966,677 182,057

270,134,951 72,135,006 15,506,101 24,374,765

184,495,160 56,092,515 10,663,676 20,259,026

18,565,389 6,318,763 398,761 329,352

10,851,303 4,154,454 224,904 NA

2,790,632 1,166,209 296,485 551,421

2,166,815 1,027,559 289,601 504,687

NA 18,101,967 301,306 NA

NA 13,122,326 445,755 NA

Oslo Børs Tel Aviv SE Warsaw SE Wiener Börse

2,437,118 32,474 6,257,203 154,521

562,591 13,460 5,167,111 104,677

15,616 589 59,920 13,533

8,245 219 34,864 6,981

56,943 2,682 72,706 17,046

13,665 2,315 30,348 13,260

22,816 219 2,121,215 NA

NA 71 1,437,611 NA

1,166,606,884

881,260,593

-

-

-

-

-

-

Bursa Malaysia Derivatives Hong Kong Exchanges


Total

DERIVATIVES - 3.5 SHORT TERM INTEREST RATE OPTIONS

Exchange






Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME)

10,554,948 605,806 9,424,628 2,594 268,957,139

3,052,800 377,370 6,534,587 4,381 188,001,096

11,195 535,720 NA 13 268,957,127

20,940 311,501 32,672,935 14 188,001,090

2,354,423 78,861 1,130,942 343 18,808,764

697,304 44,375 927,916 317 16,325,364

12,853 2,084 NA 288 1,140,562

9,855 1,476 NA 577 951,078

NA 92 NA 1 NA

NA 76 NA 1 NA

206,853 8,700 3,976,697

247,790 0 41,204

156,487 7,091 3,418,070

188,719 0 37,171

59,544 8,700 481,355

54,132 0 32,500

382 NA NA

425 0 NA

NA NA NA

NA 0 NA

92,985,715 95,000

79,482,008 -

104,878,071 NA

89,052,387 -

10,367,389 67,000

9,586,715 -

65,325 NA

76,311 -

NA NA

NA -

386,818,080

277,741,236

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Singapore Exchange Tokyo Financial Exchange

Europe, Africa, Middle East Euronext.liffe OMX Total NA : Not Available - : Not Applicable

DERIVAT IVES - 3.6 SHORT TERM INTEREST RATE FUTURES

Exchange



2006



Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) Chicago Mercantile Exchange (CME) MexDer

180,822,732 16,702,302 17,833,331 503,729,899 267,450,231

143,655,871 11,157,298 11,602,282 411,706,656 104,339,918

7,353,654 14,770,015 NA 505,339,873 26,564,227

5,538,228 9,209,807 58,011,410 413,781,671 10,348,810

9,784,628 393,078 414,975 9,564,114 44,058,415

7,332,556 331,916 455,444 8,596,023 21,205,907

555,046 825,430 NA 60,357,744 85,227

486,397 724,190 NA 52,168,804 48,626

22,860,491 272,502 14,043 615 3,573,665 40 31,495,084

18,199,674 162,592 25,181 3,308 2,890,729 217 10,977,591

19,823,462 74,545 2,171 187 2,915,805 138 27,070,811

15,665,366 42,963 3,877 622 2,466,068 310 9,903,104

902,397 59,831 1,532 NA 288,215 0 2,326,719

760,267 37,966 1,477 NA 415,431 0 1,418,937

250,184 NA 752 NA NA 72 NA

236,344 NA 1,229 163 NA 217 NA

2,500 767,458 296,008,444 667 8,170,853

1,390 688,831 248,662,893 0 6,315,805

12 937,064 341,274,218 NA NA

3 833,748 280,316,062 NA NA

0 48,307 6,092,072 63 526,914

500 37,838 5,242,458 0 345,833

5 NA 32,413,840 NA NA

16 NA 25,668,450 NA NA

1,349,704,857

970,390,236

-

-

-

-

-

-

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges Korea Exchange Singapore Exchange TAIFEX Tokyo Financial Exchange

Europe, Africa, Middle East Budapest SE Eurex Euronext.liffe JSE OMX Total

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DERIVATIVES - 3.7 LONG TERM INTEREST RATE OPTIONS

Exchange






Americas Bourse de Montréal Buenos Aires SE Chicago Board of Trade (CBOT) Chicago Board Options Exchange (CBOE)

2,275

7

202

0

0

2

25

NA

0

NA

8,437 95,737,966

86,036 89,888,554

NA NA

NA 8,931,116

0 3,097,170

293 2,517,698

NA NA

NA NA

1 NA

5 NA

18,736

61,245

92

265

2,038

7,465

1,318

5,203

3

13

3,086,456

2,307,659

235,067

175,753

14,733

1,729

11,078

10,494

NA

NA

0 2,060,624

725 1,699,037

0 NA

308 2,120,602

NA 16,987

NA 22,939

NA NA

NA NA

NA 4,306

NA 3,222

76,328,806

58,551,836

10,870,919

8,449,133

1,786,810

1,405,446

NA

NA

NA

NA

2,785

4,831

NA

11

NA

NA

NA

79

NA

NA

177,246,085

152,599,930

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Singapore Exchange Tokyo SE

Europe, Africa, Middle East Eurex JSE Total

DERIVATIVES - 3.8 LONG TERM INTEREST RATE FUTURES

Exchange



2006



Americas BM&F Bourse de Montréal Chicago Board of Trade (CBOT) MexDer Philadelphia SE

67,301 7,777,098

16,172 4,824,924

4,214 695,280

1,484 398,274

1,731 337,120

181 166,504

1,102 1,005,657

307 772,125

512,163,874 500,479 10

446,065,592 284,460 -

NA 52,437 NA

46,723,075 27,750 -

5,035,467 43,450 0

3,614,314 2,101 -

NA 2,584 10

NA 1,402 -

45,121,853 28,181 0 10,346,884 1,427,462

36,255,583 27,068 1,250 11,223,811 1,241,852

3,413,538 771 0 1,180,451 116,352

2,761,260 715 169 1,208,118 105,758

872,581 0 NA 112,652 40,186

593,812 150 NA 81,407 27,645

671,133 NA 0 NA NA

655,235 NA 50 1,836,163 NA

40,675 13,680 12,149,979

2,887 78,943 9,844,617

6,745 1,176 10,357,258

1,045 7,122 8,881,026

258 300 131,772

22 1,450 116,664

51,878 NA NA

2,348 NA NA

Asia Pacific Australian SE Bursa Malaysia Derivatives Hong Kong Exchanges Korea Exchange Singapore Exchange TAIFEX Tokyo Financial Exchange Tokyo SE

Europe, Africa, Middle East BME Spanish Exchanges Budapest SE Eurex Euronext.liffe JSE OMX Tel Aviv SE Warsaw SE Total

15 2,500

46 -

2 12

6 -

1 0

2 -

8 5

22 -

654,119,660 23,245,504

599,621,461 19,078,373

92,905,934 4,356,744

85,843,727 3,468,410

3,796,014 360,521

3,357,373 292,141

NA 2,099,645

NA 2,002,722

8,947 4,354,311 25,005

10,362 3,097,742 -

NA NA 562

NA NA -

63 184,780 651

0 140,258 -

NA NA 1,985

NA NA -

12,875

32,362

431

1,028

50

58

164

484

1,271,406,293

1,131,707,505

-

-

-

-

-

-


DERIVATIVES - 3.9 CURRENCY OPTIONS

Exchange






Americas Bourse de Montréal BM&F Chicago Mercantile Exchange (CME) MexDer New York Board of Trade (NYBOT) Options Clearing Corp. Philadelphia SE

31,262 10,525,832 3,289,498

7,264 6,850,041 3,182,525

277 44,173 451,686

70 36,604 440,565

2,838 927,188 230,426

2,691 799,576 228,288

2,010 30,110 682,415

466 28,340 608,974

3 NA NA

1 NA NA

306 44,322 0 131,508

0 35,970 0 159,748

34 NA NA 149

0 NA NA 166

2 3,690 10,602 10,476

0 1,778 17,330 17,213

9 NA NA 6,370

0 NA NA 8,861

NA NA NA 149

0 NA NA 166

Europe, Africa, Middle East Budapest SE Euronext.liffe Tel Aviv SE Total NA : Not Available - : Not Applicable

1,022,457

258,000

1,323

251

25,500

86,700

1,050

209

NA

NA

733,039 7,447,717

403,957 6,937,575

9,056 74,820

4,728 69,802

52,150 224,904

42,240 217,476

17,712 335,782

23,871 270,799

126 1,456

133 1,597

23,225,941

17,835,080

-

-

-

-

-

-

-

-

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65


DERIVATIVES - 3.10 CURRENCY FUTURES

Exchange



2006



Americas BM&F Buenos Aires SE Chicago Mercantile Exchange (CME) ROFEX MexDer New York Board of Trade (NYBOT)

1,726,351

1,293,181

2,738,810

1,737,251

677,724

475,755

1,726,351

1,293,181

800 110,338,043

2,416 81,105,391

1 13,399,645

2 9,798,906

NA 1,098,880

NA 711,360

NA 65,453,858

NA 53,154,207

17,936,247 6,077,409

12,932,275 2,934,783

NA 670,393

NA 323,969

196,293 248,205

323,169 134,992

NA 4,415

NA 2,785

3,653,024

3,604,877

NA

NA

149,595

127,497

NA

NA

1,363 3,158,049 0

4,422 2,667,005 600

103 158,463 0

337 133,679 5

0 160,722 NA

37 85,520 NA

370 NA NA

966 633,614 NA

Asia Pacific Australian SE Korea Exchange Tokyo Financial Exchange

Europe, Africa, Middle East Athens Derivatives Exchange Budapest SE Euronext.liffe Turkish Derivatives Exchange Warsaw SE Total

84

21,844

7

1,692

0

80

3

3,861

10,857,327 8,807

7,742,408 7,435

14,535 216

10,698 176

301,032 1,043

406,942 518

30,281 1,221

19,760 1,510

4,598,416 3,144

1,603,797 6,216

NA 34

1,663 65

170,431 68

134,063 84

NA 2,579

NA 5,184

158,359,064

113,926,650

-

-

-

-

-

-


DERIVATIVES - 3.11 COMMODITY OPTIONS

Exchange

2006 2005 Volume Traded (Nber of contracts)


2006 2005 Open Interest (Nber of contracts)




177,719 21,861,340 2,010,226 2,815,000 11,662,056 54,468,396 34,815

195,103 16,353,965 943,377 2,091,500 8,663,470 38,002,895 59,475

194 NA 67,569 NA NA NA NA

194 304,650 34,006 NA 220,560 2,193,391 NA

12,541 2,177,795 307,489 NA 1,146,100 9,297,986 6,039

5,999 900,266 116,431 NA 928,436 NA 4,706

1,354 NA 470,806 NA NA NA NA

1,560 NA 389,526 NA NA NA NA



10,683 27,262

558 27,101

380 NA

72 42

21,264 409

369 288

488 284

49 49

NA NA

NA NA

832 727,190 138,129 512,518 8,412,350

40 444,754 118,476 451,885 8,184,187

13.42 271 NA 1,898,026 NA

0 226 NA 337,671 468,446

260 136,475 23,987 48,568 1,007,248

95 60,129 5,832 57,950 757,837

29 9,257 NA 52,749 NA

3 7,059 NA 40,655 NA

NA 21 NA NA 6,716

NA 11 NA NA 4,397

102,858,516

75,536,786

-

-

-

-

-

-

-

-

Asia Pacific Australian SE Tokyo Grain Exchange


DERIVATIVES - 3.12 COMMODITY FUTURES

2006 Exchange

2005


2006

2005

2006

Notional Value (USD millions)

2005

2006

Open Interest (Nber of Contracts)

2005

Num ber of Trades


1,318,203

1,073,471

12,436

10,106

63,964

50,996

219,847

214,293

118,719,938 17,448,155 11,899,472 28,233,129

76,786,994 11,558,317 11,502,296 24,486,440

NA 613,145 NA NA

1,293,074 394,707 NA 500,155

2,821,951 536,649 NA 1,065,666

1,732,853 387,575 NA 901,038

NA 5,079,223 NA NA

NA 4,212,551 NA NA

178,929,185 116,937

166,608,642 118,973

NA NA

8,893,687 NA

9,326,151 11,984

NA 10,409

NA NA

NA NA

185,349 2,230,340 9,019,416

36,481 1,158,510 33,179,422

3,321 48,051 NA

1,160 21,313 1,943,220

55,600 74,567 117,816

18,010 28,918 182,304

12,295 NA NA

6,150 NA NA

117,681,038 3,158,049 58,106,001

99,174,714 2,667,005 33,789,754

NA 158,463 NA

622,949 133,679 515,274

1,154,982 160,722 196,219

482,979 85,520 154,723

NA NA NA

NA NA NA

35,027 19,106,247 46,298,117

0 25,573,238 28,472,570

2,206 1,302,452 NA

0 406,973 16,166

44 438,435 213,847

0 563,665 452,058

12,724 NA NA

0 NA NA

Asia Pacific Australian SE Bursa Malaysia Derivatives Central Japan Com modity Exchange Dalian Commodity Exchange Korea Exchange Shanghai Futures Exchange TAIFEX Tokyo Grain Exchange Zhengzhou Commodity Exchange


8,750 9,124,195

778 8,054,116

140 119,436

9 85,794

1,093 449,829

601 419,333

1,856 1,257,639

189 906,230

92,582,921 1,436,155 78,527,839

41,936,609 1,335,964 70,444,665

NA 1,864,750 7,146,569

NA 15,158,450 4,045,775

1,389,618 43,295 1,515,663

642,753 51,295 2,411,870

NA 205,430 NA

NA 169,767 NA

794,164,463

637,958,959

-

-

-

-

-

-

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References Fabozzi, F (1993). Fixed Income Mathematics, US: Irwin. Merton, RC (1998). Applications of option pricing theory: twenty five years later. American Economic Review, 88(3), 323–345. Miller, M (1986). Financial innovation: the last twenty years and the next. Journal of Financial and Quantitative Analysis, 21 (December), 451–471. Ross, S (1989). Financial markets, financial marketing and financial innovation. Journal of Finance, 44(3), 541–556.

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Chapter 2 RISK MANAGEMENT, DERIVATIVES MARKETS AND TRADING STRATEGIES

Chapter Outline This chapter is organized as follows: 1. Section 2.1 gives an overview of futures markets and the trading mechanisms in these markets. 2. Section 2.2 presents the main pricing relationships for forward and futures contracts. 3. Section 2.3 develops the main trading motives in futures markets. Several examples explain strategies with reference to hedging, speculation, and arbitrage. 4. Section 2.4 studies the main bounds on option prices. 5. Section 2.5 illustrates some simple trading strategies for options and their underlying assets. 6. Section 2.6 presents some option combinations involving straddles and strangles. 7. Section 2.7 illustrates some option spreads in bull and bear strategies involving calls and puts. 8. Section 2.8 develops butterfly strategies using put and call options. 9. Section 2.9 presents Condor strategies using put and call options. 10. Section 2.10 studies ratio-spread strategies. 11. Section 2.11 illustrates some combinations of options with bonds and stocks and portfolio insurance strategies. 12. Section 2.12 studies conversion and reversal strategies.

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Introduction The first of the modern commodity markets began trading a little over a century ago. Today, futures markets are a direct development of traditional agricultural markets, which were initially located in Chicago and London. Chicago is the largest commodity trading center in the world. The standardized futures markets such as the New York Merchantile Exchange (Nymex), the International Petroleum Exchange (IPE), and the Singapore International Monetary Exchange (Simex), or the forward markets like dated Brent, Littlebrook Lottery, or the Russian Roulette have become an important factor in the pricing of crude oil and refined products. The futures price can be described using different parameters: the spot price, the risk-less interest rate, the cost of carrying the stocks, and the convenience yield. The convenience yield corresponds to a specific interest rate of the commodity. Forward and futures contracts enable firms to determine a price for future delivery. Forward and futures prices can differ from the spot price of the commodity. However, as the expiration date approaches, the forward, futures, and spot prices must converge. The cost of carry model corresponds to the relation between the futures price and the spot price. It is the basis for the valuation of forward and futures contracts. Futures prices and forward prices are often regarded as being equivalent. However, this is true only if the risk-free interest rate is constant or a known function of time. For the valuation of interestrate futures contracts, the theoretical futures price can be determined as a function of the underlying asset price (the bond price), the coupon rate, and the financing rate for borrowing and lending during a given period. The fair price of a forward contract is given by the spot price plus the cost of carry until the maturity date of the bond. Futures markets date back to the medieval marketplaces, but they developed in the United States in the 1800s in response to the nature of agricultural products. In 1848, the Chicago Board of Trade became an organized marketplace for grain transactions. Hedging is a price protection that is used to minimize losses and to protect profits during the production, storage, and marketing of commodities. Hedging is the strategy of taking a position in the futures market as a temporary substitute for the purchase or the sale of a commodity. In general, a perfect hedge is possible when the “basis” or the relationship between the cash market and the futures market is the same when the hedge is removed, as it was when the hedge was implemented. A futures contract is, in general, liquidated by offsetting

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it with a futures contract in the opposite direction. Transactions are done on margin in the futures market. The margin represents a small fraction of the value of the contract. It varies by the type of commodity. Option prices depend on factors that affect the main elements in computing their prices. These factors concern the volatility in the financial markets, the level of interest rates, the option’s maturity date, the exercise price, dividends on the underlying asset, etc. When a derivative asset is about to expire, it is relatively a simple matter to compute its value. In fact, at the option maturity date, the holder can either exercise it or let it expire. Hence, at this date, the option price is given only by the position of the underlying asset price with respect to the option strike price. This position defines the option payoff at this date. At any time, there is a market price for the derivative asset. Models are used to compute the option price at any time. But, this does not mean that the market option price must be equal to the model price. The difference represents the mispricing. Options can be either European or American. A European style-derivative asset cannot be exercised before its maturity date T . An American contract can be exercised at any time t before the maturity date T . At maturity, the European and American derivatives have identical values because it is the last moment to exercise or let expire a contract. A call is in-the-money when the underlying asset price is higher than the strike price. It is out-of-themoney, if the underlying asset price is lower than the strike price. The call is at-the-money if the underlying asset price is equal to the strike price. A put is in-the-money when the underlying asset price is lower than the strike price. It is out-of-the-money if the underlying asset price is higher than the strike price. The put is at-the-money if the underlying asset price is equal to the strike price. These definitions apply at maturity and at each instant before expiration. When a long or a short position is initiated in a derivative contract, the profit or loss is known when an opposite transaction is done or when the option is at its maturity date. The profit or loss is computed with respect to the purchase price or the sale price. It is possible to analyze the profits and losses on options positions by looking at the initial price in the transaction and the maturity price of the derivative contract. This allows the computation of the profit and loss, P&L. This chapter presents in detail the basic theory of commodities, futures, and forward markets. It illustrates the specific features of these markets and the main pricing relationships. It develops the main trading strategies in options markets. In particular, strategies involving calls and puts, straddles, strangles, conversions and reversals, and the box spread are studied. These trading

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strategies can be used for most of the derivative assets in this book, since they can be implemented using options with any particular payoff. 2.1. Introduction to Commodity Markets: The Case of Oil The main commodity exchanges that currently trade oil futures contracts are the Nymex, IPE of London, and the Simex. 2.1.1. Oil futures markets A futures contract is successful when it is based on a volatile price, a standard quality specification, and a wide range of participants in the market. Volatile prices are necessary because they induce investors to use futures markets. In fact, the prime function of a futures market is to provide a hedging mechanism for the related industry. The standard quality specification will attract the whole sector of the industry. The overall volume traded by market participants give a good guide to show how liquid a contract is. A second measure of a successful contract is open interest, i.e., the total number of outstanding bought-and-sold contracts at the close of each trading day. Commercial traders are companies whose business involves handling the physical commodity, i.e., hedgers. Non-commercial traders are mainly financial companies, i.e., speculators. 2.1.2. Oil futures exchanges The most successful oil futures market is the Nymex. The success of this market is largely due to the importance of its West Texas Intermediate (WTI) crude contract. The IPE of London trades only energy contracts. Unlike the Nymex, the IPE has now chosen the cash settlement procedure rather than physical delivery for the Brent crude contract. However, there is a physical delivery option. The Simex offers an oil futures contract. 2.1.3. Delivery procedures In general, futures contracts are almost never delivered since participants prefer to close out (or roll over) their positions before the last trading day for each delivery month. The main reason is that futures contracts are either traded in conjunction with a position in the physical market (hedging) or used in a speculative strategy. Physical delivery ensures that the futures

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prices remain anchored to the underlying physical market. The physical delivery of the standardized commodity specified in the futures contract or its cash equivalent are not the only delivery mechanisms. Other procedures are used like the exchange for physical (EFP), exchange for swaps (EFS), and alternative delivery procedure (ADP). 2.1.4. The long-term oil market The standardized futures markets such as the Nymex, the IPE, and the Simex, or the forward markets like dated Brent, Littlebrook Lottery, or the Russian Roulette have become an important factor in the pricing of crude oil and refined products. The emergence of a long-term oil market allows one to meet specific requirements in the industry. This market provides new opportunities for speculators. The participants in this new over-the-counter (OTC) market are highly ranked oil companies, banks, and traders. 2.2. Pricing Models The futures price, F can be described using different parameters: the spot price S, the risk-less interest rate r, the cost of carrying the stocks b, and the convenience yield cy. The cy represents short- to medium-term effects related to physical supply and demand unbalance. It can be measured using futures prices. For the case of oil, this yield indicates the intrinsic oil interest rate. 2.2.1. The pricing of forward and futures oil contracts Forward and futures contracts enable firms to determine a price for future delivery. Most forward and futures contracts are traded for a few months ahead. In the refined products markets, trading extends to a year. The futures contract, Nymex and WTI can be traded up to four years ahead. 2.2.1.1. Relationship to physical market Forward and futures prices can differ from the spot price of the commodity. However, as the expiration date approaches, the prices must converge. However, there are occasional squeezes on forwards and futures. When the nearby contract is at a premium to the later month, the market is in backwardation. When the nearby month is at a discount, the market is in contango.

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2.2.1.2. Term structure of prices Changes in demand and in stock levels held by the industry affect forward and futures prices. There is a limit on the size of the contango imposed by the cash and carry arbitrage. In fact, buying the oil for one month, paying for it, moving it into storage, insuring it, and delivering it back to the market one month later at profit represent the cash and carry arbitrage. Example. A trader implements the following transactions: • •

Buys heating oil at 60 cents/gallon in March Sells heating oil at 63 cents/gallon in April

In March, he takes delivery of the heating oil futures at 60 cents/gallon He pays storage costs for 6 weeks, at 2.50 cents/gallon The interest costs and product losses are 0.25 cents/gallon The total cost in March is 62.75 cents/gallon In April, he delivers the heating oil at 63 cents/gallon The net profit is 0.25 cents/gallon 2.2.2. Pricing swaps The price of a swap can be determined using the arbitrage relationships between the swap and the forward or the futures markets. A swap agreement can be replicated by a position in a portfolio of futures or forward contracts. Example. Consider a swap agreement between a producer of oil and a swap provider. The swap allows the producer to sell a specified quantity of crude oil at a fixed price over a period of one year. The swap can be reproduced as a portfolio of short futures or forward contracts on the same volume of oil for each delivery month. 2.2.3. The pricing of forward and futures commodity contracts: General principles The proposed relation between the futures price and the spot price, F = SebT is useful. Re-call that F is the future price, S is the spot price, b corrresponds to the carrying cost, and T is the time to maturity. This relationship applies to futures prices and forward prices as well.

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2.2.3.1. Forward prices and futures prices: Some definitions Futures prices and forward prices are often regarded as being equivalent. However, this is true only if the risk-free interest rate is constant or a known function of time. The proof of this equivalence is based on a rollover strategy proposed in Cox et al. (1981). In this strategy, the investor buys every day a specific amount of futures so that he/she holds er futures contracts at the end of the first day of trading (initial time, day 0), e2r futures contracts at the end of the day 1, e3r futures contracts at the end of the day 2, etc., and eir futures contracts at the end of the day i − 1, where r corresponds to the daily interest rate. Since at the beginning of day i, the investor has eir contracts in his/her position, the position on that day shows a profit (loss) of (Fi − Fi−1 )eir . When this amount is invested until the day N corresponding to the contract’s maturity date, where the number of days i is between 0 and N , this amount will be (Fi − Fi−1 )eir e(N −i)r = (Fi − Fi−1 )eN r . The sum of the amounts of profit (loss) from the day 0 until day N turns out to be (FN − F0 )eN r . However, since at the contract’s maturity date, the futures price FN is equal to the spot price, ST , the terminal value of this investment strategy is (FN − F0 )eN r = (ST − F0 )eN r . Using a portfolio which corresponds to this strategy and an investment of an amount F0 in a risk-free bond, gives the following payoff at time T : (ST − F0 )eN r + F0 eN r = ST eN r Since for a strategy in futures contracts no funds are invested, the result ST eN r corresponds to the investment of F0 in the risk-free bond. Another strategy can be constructed to give the same pay-off as the preceding one. In fact, if f0 stands for the forward price at the end of day 0, then a strategy which consists in investing this amount in a risk-less bond and an amount eN r in forward contracts also gives a final payoff at T equal to ST eN r . Since the two strategies require an investment of an amount F0 , (f0 ) and yield the same result, ST eN r , they must have the same value in efficient capital markets. In the absence of profitable arbitrage opportunities, the futures price F0 must be equal to the forward price f0 . Hence, the proposed relation applies for both futures and forward prices and we have F = f = SebT . Some examples are given below to illustrate the use of this relationship in the determination of forward and futures prices on some securities.

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2.2.3.2. Futures contracts on commodities When the cost of carrying an asset refers to the storage costs of a commodity such as silver or gold and when these costs are proportional to the commodity price, the futures price is given by F = Se(r+a)T , where “a” stands for the storage costs. 2.2.3.3. Futures contracts on a security with no income When there are no distributions from the underlying asset, the cost of carry is equal to the risk-less interest rate. The future price is given by F = SerT . More generally, this relation represents the forward or futures price F as a function of the spot price S. It applies to the valuation of forward or futures contracts on a security that provides no income. Example. Consider the valuation of a forward contract on a non-dividend paying stock. Suppose the maturity date is in three months, the current asset price is 100, and the three-month risk-free rate is 7% per annum. In this case, S = 100, T = 0.25 year, r = 0.07, so the futures or forward price is 101.765 or F = 100e(0.07)0.25 = 101.765. 2.2.3.4. Futures contracts on a security with a known income For dividend-paying assets, the cost of carrying the stocks is given by the difference between the risk-less rate and the dividend yield, d. This gives the following relation F = Se(r−d)T . This relationship gives the forward price F as a function of the spot price S for a forward contract on a security that provides a known dividend yield. Example. Consider the valuation of a three-month forward or futures contract on a security that provides a continuous dividend yield of 5% per annum. Suppose that the current asset price is 100 and the risk-free rate is 7% per annum. In this case, S = 100, T = 0.25 year, r = 0.07, and d = 5%, so the futures or forward price is 100.501: F = 100e(0.07−0.05)0.25 = 100.501. Example. Consider the valuation of a three-month forward or futures contract on the CAC 40 stock index. The index provides a continuous dividend yield of 4% per annum. Suppose that the current index price is 1000 and the risk-free rate is 7% per annum. In this case, S = 1000, T = 0.25 year, r = 0.07, d = 4% per annum, so the futures or forward price is 1007.528 F = 1000e(0.07−0.04)0.25 = 1007.528.

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2.2.3.5. Futures contracts on foreign currencies The cost of carrying a foreign currency is given by the difference between the domestic risk-less rate and the foreign risk-less rate, r∗ . This gives the following relation between the futures price and the spot price of the ∗ currency: F = Se(r−r )T . This relationship also gives the forward price F (or foreign exchange rate F ) as a function of the spot price S for a forward contract on a currency. This is often known in international finance as the interest-rate parity theorem. Example. Consider the valuation of a three-month forward or futures contract on a foreign currency. If the spot price is 180, the domestic riskfree rate is 6% per annum and the foreign risk-free rate is 7% per annum, then: S = 180, T = 0.25 year, r = 0.07, and r∗ = 6% per annum. The future or forward price is 180.45. F = 180e(0.07−0.06)0.25 = 180.45 2.2.3.6. Futures contracts on a security with a discrete income When the cost of carrying the commodity is not a constant proportional rate, the formulae discussed above must be slightly modified. In the case of stocks paying known dividends, coupon-bearing bonds and some commodities for which there are storage costs, the formula for future prices becomes F = (S − I)erT . where I is the discounted value of the cash-flow between t and t∗ . It is positive when it corresponds to an income and is negative when it refers to a cost. Example. Consider the valuation of a one-year forward contract on a twoyear bond. The two-year bond’s price is 800, the delivery price is 820, and two coupons of 50 will be paid in 6 and 12 months, respectively. The risk-less interest rate is 8% per annum for 6 months and 9% per annum, for 12 months. In order to apply the formula, the value of I must be discounted to the present at an appropriate interest rate. In this case, I is given by: I = 50e−0.08(0.5) + 50e−0.09(1) = 48.039 + 45.696 = 93.735 and the forward price is: F = (800 − 93.735)e0.09(1) = 706.265e0.09(1) = 772.777.

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Example. Consider now the valuation of a one-year forward contract on a stock with a price equal to 100. When the dividend is two and the interest paid at the end of the year is 10% per annum, the present value of dividends is I = 2e−0.01(1) = 1.8096 and the forward price is given by F = (100 − 1.8096)e0.1(1) = 108.516. Example. Consider the valuation of a one-year futures contract on gold. If the cost of carry is 3 per ounce paid at the end of the year, the spot price is 500, and the risk-free rate is 10% per annum, the value of I is given by I = 3e−0.1(1) = 2.7145 and the futures price is F = (500 + 2.7145)e0.1(1) = 555.585. 2.2.3.7. Valuation of interest rate futures contracts The theoretical futures price can be determined as a function of the underlying asset price (the bond price), the coupon rate, and the financing rate for borrowing and lending during a given period. We denote these by: P: F: T: r: c:

bond price in the cash market; futures price; time to maturity (the delivery date); financing cost or rate and coupon rate divided by the market bond price, known also as the current yield.

Consider the following strategy: sell a futures contract at F , buy the bond at P , and borrow an amount P at the rate r until the date T . At the contract’s delivery date, the investor receives F plus the accrued interest cT P . He must re-pay the loan P and the interest-rate charges, rT P . The profit is given by the difference between the amount received and the outlay or: P rof it = F + cT P − (P + rT P ). At equilibrium, in an efficient market, the fair futures price must be: 0 = F + cT P − (P + rT P ) or: F = P (1 + T (r − c)). This price allows to avoid a cash and carry arbitrage. Consider now the following strategy: buy a futures contract at F , sell (short) the bond at P , and lend (invest) an amount P at the rate r until the date T . At the contract’s delivery date, the investor pays F plus the accrued interest cT P . He/she receives P and the interest earned, rT P . The profit is given by the difference between the amount received and the

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outlay or: P rof it = P + rT P − (F + cT P ) At equilibrium, in an efficient market, the fair futures price must be: 0 = P + rT P − (F + cT P ) Hence: F = P (1 + T (r − c)). If this equation is not satisfied, then a reverse cash and carry arbitrage can be implemented. The difference (r−c) refers to the net financing cost or the cost of carry. The futures price is at a discount with respect to the cash price (F < P ) when (c > r). The futures price is at a premium with respect to the cash price (F > P ) when (c < r). The futures price is equal to the cash price (F = P ) when (c = r). Using the cash and carry arbitrage and the reverse cash and carry arbitrage, it is possible to derive the theoretical fair price of a forward or a futures contract. Our analysis assumes that there is only one deliverable bond. However, in practice, futures contracts on treasury bonds are issued on a number of deliverable issues and the futures price tracks in general, the price of the bond which is the cheapest to deliver. Since the cheapest to deliver is unknown before the delivery date, the futures price must account for the price of the quality option: F = P + P (r − c) — quality option premium. Since the exact delivery date is also unknown and the seller has delivery options, the fair futures price must be: F = P + P (r − c) — delivery option premiums 2.2.3.8. The pricing of future bond contracts The fair price of a forward contract is given by the spot price plus the cost of carry until the maturity date of the bond: F = S + cp In general, the futures contract is traded at a price which accounts for the implicit options. Its price must satisfy the following relationship: F = S + cp − O + e where: F = futures price, S = spot price for the cheapest bond, cp = cost of carry, O = value of the embedded options, and e = a white noise.

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The process for the pricing of a futures contract must account for the following relationship: F (f c) = S + cp − O + e Hence, its theoretical fair price must be: F = (S + cp − O)/(f c) + e 2.3. Trading Motives: Hedging, Speculation, and Arbitrage Futures markets are used for hedging, speculation, and arbitrage motives. The main question for the producer is to implement the appropriate hedging strategies in response to the changes in backwardation or contango. Backwardation: When spot prices are higher than long-term prices, any hedge using a future maturity will be equivalent to a forward sale below the spot price. This can lead to a loss if the market prices do not fall at the same rate. Careful long-term analysis may provide good hedging opportunities. Contango: When spot prices are lower than long-term prices, the producer can sell the futures market at a higher price. So, he/she can fix his/her hedge or future sales at a better price than the spot market. In this case, hedging can generate profits if prices are not increasing at the same rate.

2.3.1. Hedging using futures markets Companies using the physical oil market can hedge themselves against adverse price movements by taking an opposite position on the futures or the forward market. The potential loss in the physical market can be offset by an equivalent gain in the futures or the forward market. In practice, the hedge is rarely perfect. The futures market offers a facility for hedging price risks. Hedging price risk can be regarded as a trading operation allowing one to transform a less acceptable risk into a more acceptable risk by engaging in an offsetting transaction in a similar commodity under roughly the same terms as the original transaction. In this spirit, a futures purchase, which is equal and opposite to the physical trading contract is made with an idea that any loss in the first transaction will be compensated by an equal gain in the offsetting operation.

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2.3.1.1. Hedging: The case of cocoa The selling hedge A company holding stocks of a commodity can protect itself against the risk that the value of the unsold products will depreciate if the commodity price falls. This risk is offset by a forward sale of the same tonnage on the futures market. The hedge is based on the assumption that futures prices decline as physical prices fall. The buying hedge Manufacturers of chocolate, exporters selling cocoa do not actually possess, and manufacturers who fix selling prices and current contracts for future delivery based on the current costs, etc. can implement a buying hedge. These agents would suffer if the costs of the raw material rise. Therefore, they can purchase the required tonnage on the futures market using a long hedge. When the price falls by the time of shipment, the hedger will lose on the futures market and profit from a cheaper purchase on the spot market. The hedging strategy needs some funds to cover for the security deposits and margins and any commission paid to brokers. 2.3.1.2. Hedging: The case of oil Example: A short hedge An oil trader who buys a cargo of physical oil while oil is in transit, can protect himself/herself, by selling an equivalent volume of oil futures or forward contracts. When the physical cargo is sold, the hedger can lift his/her hedge by buying back the futures contract. Example: A short hedge Consider a company A buying 500,000 barrel cargo of crude oil for $28/barrel when futures price is $28.50/barrel. The manager decides to use the futures market to implement a hedging strategy by selling futures contracts. A week later, the manager sells the physical cargo for $27.00/ barrel. He/she decides to lift the hedge by buying back the futures contracts at their market price of $27.40/barrel (Table 2.1). The profit on the futures position is larger than the loss on the physical market. This result can be explained by this basis. It corresponds to the differential between the futures and the spot price.

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Table 2.1.

Short hedge using futures contracts.

Period

Physical market

t=0 t = +6

bought at sold at loss

Table 2.2.

$/bl 28 27.00 −1.00

Futures market futures sold at futures bought at profit

$/bl 28.50 27.40 +1.10

Long hedge using futures contracts.

Period

Physical market

t=1 t=3

sold at bought at loss

$/tonne 200 210 −10

Futures market futures bought at futures sold at profit

$/tonne 199 208 +9

Example: A long hedge Consider a company B which is short of oil in the spot market. A gasoil distributor agrees to sell oil to a customer at a fixed price for some months ahead. The manager decides to use the futures market to implement a hedging strategy by buying futures contracts in order to protect himself/herself against a rise in price (Table 2.2). This result can also be explained by this basis. It corresponds to the differential between the futures and the spot price.

2.3.1.3. Hedging: The case of petroleum products futures contracts When the futures markets tend to move parallel to the spot (cash) market, a hedge can be implemented by buying or selling futures contracts. This is possible because a loss due to an adverse price change in the cash market can be offset by a gain in the futures market. Also, a loss in the futures market can be offset by a gain in the cash market. Using a Short Hedge A short (selling) hedge is implemented when a decline in cash market prices is expected. This is the case for a refiner or a distributor who is holding an inventory. The short hedger forgoes the opportunity of additional profits in a rising cash market.

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Hedging by a petroleum marketer — Hedge of a fixed price purchase commitment Consider a marketer who is committed to buy a certain quantity of a product at a given price. He/she expects a price decline and enters into a short futures contract to protect his/her normal gross profit margin. If the cash price of heating oil declines, the futures price can also decline. Hence, the marketer will realize a loss or less of a gain on physical sale and a profit on his/her futures position. If the oil price increases, the resulting profit from the physical sale will be offset by a loss on the futures market. In both cases, the marketer will have protected his/her normal gross profit margin. Hedging by a petroleum refiner Consider a refiner who has not a firm sales commitment. Since there is a time lag between the time the refiner purchased crude oil and delivers the refined oil to the consumer, the price of oil can decline during this time period. He/she can implement a hedge by selling futures contracts. If the oil price declines, the resulting loss in the value of the inventory will be offset by a gain on the futures market. In the same way, if the oil price increases, the loss on the futures market will be offset by an increase in inventory value. Using a long hedge A long (buying) hedge is implemented when a rise in cash-market prices is expected. This is the case for a consumer of petroleum products. The long hedger gives up the opportunity of increased profits should the price of the physical asset decline. The hedger forgoes this opportunity for the protection of his/her operating margins. Hedging by a petroleum marketer — Hedge of a fixed-price sale commitment Consider a marketer who enters into long-term contracts to deliver products to customers in the futures at a fixed price. His/her profit margin will be at risk since the oil prices may increase before delivery. The marketer can implement a hedge against price risk (a rise in the price in excess of the contracted sale price) by buying futures contracts for the month of delivery.

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He/she expects a price rise and enters into a long futures contract to protect his/her portfolio. If the cash price of heating oil increases, the marketer will realize a loss on the physical sale to customers because he/she will have to buy oil at a higher price in the cash market to satisfy his/her sales commitment. However, the cash loss would be offset by a futures profit on his/her futures position. If the oil price decreases, the resulting profit from the physical sale will be offset by a loss on the futures market. Hedging by a petroleum consumer to protect against rising prices Consider a petroleum consumer who expects an increase in petroleum product prices. He/she can implement a hedge by buying futures contracts. If the oil price increases, the consumer will have to pay the higher market price at the time of purchase. The increased cost can be offset by a gain on the futures market. In the same way, if the oil price decreases, the gain on the physical transaction will be offset by a loss on the futures market. 2.3.1.4. The use of futures contracts by petroleum products marketers, jobbers, consumers, and refiners Example A: Hedge by a petroleum products marketer against a fixed-price purchase commitment This example shows how a marketer can maintain his/her normal gross profits in the physical market, regardless of the dynamics of the oil prices. A marketer is commited to buy a fixed quantity at 90 cents per gallon. He/she expects a price decline and implements a hedge by selling futures contracts. In September, the marketer enters into a contract with a company to supply 420,000 gallons of heating oil for delivery the following December. If the current price of US$ 1 per gallon is expected to prevail at delivery, the marketer expects a profit of 10 cents. The marketer can protect this profit margin against a decline in oil prices by December. He/she can sell 10 heating oil contracts at US$ 1 per gallon for delivery in December (the contract is equivalent to 42,000 gallons). If the cash price declines to 80 cents, the marketer will realize a loss of 10 cents (90–80) on the physical sale.

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However, a profit of 20 cents is achieved in the futures market since the short futures contract is sold at US$ 1 and the investor bought it back at 80 cents. The net effect of 10 cents corresponds to the original expected profit of 10 cents. If oil prices rise, the resulting profit from the physical sale will be offset by an equivalent loss on the futures market.

Example B: Hedge by a petroleum products marketer against a fixed-price sales commitment A marketer is commited to sell to customers a fixed quantity at US$ 1.30 per gallon. In June, the marketer enters into a sales contract to deliver 420,000 gallons of heating oil in December. He/she expects a price decline from US$ 1.3 to US$ 1 in December. The profit margin will be eroded if the marketer is forced to buy the heating oil at a price in excess of the contract price of US$ 1.3. The marketer can protect his/her position against the price risk by buying 10 futures contracts at US$ 1 for delivery in December. In December, he/she can accept the delivery or close out the position by selling an identical futures contract. If the oil price is US$ 1.4 per gallon, the marketer will lose 10 cents on the physical market (to honor his/her sales contract price of 1.3). The marketer can close his/her futures long position by selling a futures contract at 1.4. A profit of 40 cents is achieved in the futures market since the long futures contract is bought at US$ 1 and it is sold at US$ 1.4. The net effect of 30 cents corresponds to the original expected profit of 30 cents. If oil prices decrease to 90 cents, the resulting profit from the physical sale is 40 cents, (selling at 1.3 minus cost of 90 cents).

Example C: Hedge of an existing asset (inventory) position by a refiner This example shows how a refiner can protect his/her inventory from an erosion in value when prices fall, using the futures market. A refiner has a risk resulting from the time lag between the time of buying crude oil and the time of delivery to the consumer. When the oil price declines during this time period, the value of his/her inventory will decline.

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Consider a refiner who holds 420,000 gallons of uncommitted refined inventory at an average cost of 90 cents per gallon at first in September. He/she expects to sell his/her inventory in the cash market in December. Expecting a possible decline below 90 cents, he/she sells 10 futures contracts maturing in December at US$ 1. At maturity, if the price declines to 80 cents, he/she loses 10 cents on his/her inventory. When closing his/her futures position, he/she will realize a gain of 20 cents (US$ 1 less 80 cents). This reduces the basis of his/her inventory to 70 cents. In this context, he/she can sell the inventory in the cash market at a lower cash price, and preserving a profit margin. If the oil prices increase, this would be offset by a corresponding loss on the futures position. 2.3.2. Speculation using futures markets The main objective of speculators is profit. Speculators take on the risk which hedgers try to lay off. Speculators hold onto their positions for a very short time. They are sometimes in-and-out of the market several times a day. Example: Speculation on a price rise Consider a speculator who expects prices to rise and buys consequently IPE Brent futures contracts. A week later, the speculator closes out his/her position in the futures market before the prices can fall back again. The two trades are regarded as purely speculative because there is no physical transaction in the spot market (Table 2.3). 2.3.3. Arbitrage and spreads in futures markets Arbitrage keeps prices in line since the arbitrageur buys the asset in one market and sells it in other market. When prices move out of line, the arbitrageur buys the under-priced asset in one market and sells the overpriced asset in another market. Table 2.3.

Speculating on a price rise.

Period

Physical market

$/bl

t=0 t = +6

nothing nothing

— — —

Futures market futures bought at futures sold at profit

$/bl 26.50 25.00 +1.50

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Risk Management, Derivatives Markets and Trading Strategies Table 2.4.

Period t=0 t = +1

85

Heating-oil arbitrage.

Nymex heating oil

Price cts/gall

Price $/tonne

IPE gasoil

Price $/tonne

Price differential

Buy 3 at Sell 3 at Loss

62 61.50 −0.50

170 169 −1

Sell 4 at Buy 4 at Profit

170.10 164 +6.10

−0.10 +5.00 +5.10

We give an example using IPE gasoil contracts and the Nymex heatingoil contracts. IPE gasoil contracts are 100 tonnes and Nymex heating-oil contracts are 1000 gallons. The relationship shows the trading of three Nymex contracts for every four IPE contracts. Besides, since IPE gasoil prices are quoted in $/tonne and Nymex heating-oil prices are quoted in cents/gallon, a conversion factor must be used. Assuming a specific gravity for gasoil of 0.845 kg/liter and since there are 313 gallons of heating oil in a tonne, the conversion factor is 3.13. Heating-oil arbitrage Consider a trader who expects Nymex heating oil to move to a premium over the IPE. He/she buys the Nymex heating-oil contracts and sells IPE gasoil contracts. The total profit of US$ 5.10/tonne comes from the change in the differential regardless of the market direction (Table 2.4). Other types of spreads can be implemented. The analysis can be extented to the valuation of these contracts in the presence of information costs in Appendices 1 and 2. The financial crisis in 2008 reveals the importance of hedging strategies in a risk framework. 2.4. The Main Bounds on Option Prices The option value before maturity is a function of five parameters: the price of the underlying asset, the strike price, the risk-free rate of interest, the movements in the underlying asset prices (volatility), and the time remaining to maturity. Option prices move at each instant of time as a reaction to the changes in the above parameters. Since we know the option pay-off at maturity, it is possible to determine some main relationships before the maturity date. Hence, we can give minimum and maximum values of calls and puts as well as the behavior of the option price in response to the changing parameter values.

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The two following main relationships apply for the European and American calls (c and C, respectively) and European and American puts payoffs (p and P , respectively) at maturity: CT = cT = max[0, ST − K], PT = pT = max[0, K − ST ] where K is the strike price and T is the option’s maturity date. The boundary space refers to the largest range of possible option prices before expiration.

2.4.1. Boundary conditions for call options When the call holder exercises his/her option, he/she receives the difference between the underlying asset price and the strike price. When the strike price is zero and the maturity date is infinite, the call can be exercised with zero cost. This gives the call holder the right to receive the underlying asset with zero cost. The value of this option must be equal to that of its underlying asset. Hence, the call price must be between these two limits.

2.4.2. Boundary conditions for put options When the put holder exercises his/her option at maturity, he/she receives the difference between the strike price and the underlying asset price. This is the minimum value for the put. When the underlying asset price tends to become zero, the put holder can receive at maximum the value of the strike price. Hence, the maximum value of the American put, which can be exercised at any time before maturity must be the option’s strike price. However, the maximum value for a European put must be the present value of the strike price since the put holder must wait until expiration to exercise his option. Hence, the minimum put price must be max[0, K − ST ]. The maximum price for an American put is K and for a European put is Ke−r(T −t) , where t is the current time and r is the risk-free interest rate. 2.4.3. Some relationships between call options We define some main relationships which hold good for European and American calls.

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The call payoff shows that the lower the strike price, the higher is the call value. This property can be shown using no-arbitrage arguments. Hence, if you consider two calls with different strike prices, then the call with the lower strike price must be at least equal to the call price with the higher strike price. It can also be shown that the call value must be at least equal to the stock price minus the present value of the strike price or Ct ≥ St − Ke−r(T −t) . This relationship holds good at any time before expiration. Example: Consider a call trading at 3 when St = 105, K = 100, r = 7%, and T = 0.5 years. These data violate the previous relationship and give rise to an arbitrage opportunity because the call is undervalued. In this case, an investor can implement the following strategy: sell the underlying asset at 105, buy the option at −3, and buy the bond with the remaining funds at −102. The investor can exercise immediately the option, pay 100, return the stock, and keep 2. The investor can also wait for the maturity date. At this date, the bond price is 102e0.07(0.5) = 105.6332. If the stock price is 97 at expiration, the option expires worthless and the investor pays 97 for the share to re-pay the obligation. The profit in this case is 102e0.07(0.5) − 95 = 10.6332. Even, if we consider other levels of the underlying asset, the investor can re-pay with profit. Hence, the option value must be at least equal to 105 − 100e−0.07(0.5) = 8.435. All the prices below this allow arbitrage profits. The call price is an increasing function of time until expiration. In fact, if we consider two calls with the same characteristics, except for maturity, then the price of the call with a longer maturity must equal or exceed the price of the call with a shorter maturity. If this principle is violated, arbitrage can be implemented with risk-less profits. In the absence of dividend or cash distributions to the underlying asset, there is no reason to exercise a call on a non-dividend paying asset. In fact, a call on a non-dividend paying asset is always worth more than its intrinsic value St − K. Since before expiration, the call value must be at least worth St − Ke−r(T −t) , exercising the call before expiration discards at least the option time value, i.e., the difference between K and Ke−r(T −t) . Early exercise is never optimal for an American call on a non-dividend paying asset. Since European calls cannot be exercised before expiration, this

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makes an equivalence between the European call price and the American call price in the absence of distributions to the underlying asset. Example: Consider a call when St = 110, K = 100, r = 7%, and T = 0.5 years. In this case, the call’s intrinsic value is 10, St −K. But, the call market price must be at least Ct ≥ St −Ke−r(T −t) ≥ 110−100e−0.07(0.5) ≥ 13.4395. The investor can exercise immediately the option of throwing away at least the difference K − Ke−r(T −t) = 100 − 100e−0.07(0.5) = 3.435. The investor discards also the difference between 13.4395−10 = 3.4395.

2.4.4. Some relationships between put options We define some main relationships, which hold good for European and American puts. The put payoff shows that the put must be worth at expiration the difference between the strike price and the underlying asset value. The put payoff reveals that before maturity, the put value must be worth at least the difference between the strike price and the underlying asset value. This is because the American put holder can exercise his/her option at any instant before maturity. It can also be shown that the European put value must be at least equal to the present value of the strike price minus the underlying asset price. Since the American put value must be at least equal to K − St because of the possibility of an early exercise, the European put value must verify the following relationship: pt ≥ Ke−r(T −t) − St This relationship holds good at any time before expiration since the put holder cannot exercise his/her put before expiration. Example: Consider a European put trading at 2.5 when St = 95, K = 100, r = 7%, and T = 0.5 years. Using this information, the put value satisfying the following relationship: pt ≥ Ke−r(T −t) − St = 100e−0.07(0.5) − 95 = 1.5605, must be worth 1.5605. Since the actual price is 2.5, the investor can implement a trading strategy to generate risk-less profits. In this case, he/she can buy the put at 2.5 and buy the stock at 95. The strategy can be implemented by borrowing

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at 97.5 or (95 + 2.5) at 7% for six months. The net cash-flow of this strategy is zero. At maturity, the investor exercises his/her put option, (receives the strike price of 100 and delivers the stock). He re-pays the borrowed amount 97e0.07(0.5). The net cash-flow from this strategy is 100.45511. This is the risk-less arbitrage profit. Hence, with zero investment, the strategy guaranteed a risk-less arbitrage profit. Therefore, the above inequality must hold good. The put price is an increasing function of time until expiration. In fact, if we consider two puts with the same characteristics, except for maturity, then the price of the put with a longer maturity must equal or exceed the price of the put with the shorter maturity. If this principle is violated, arbitrage can be implemented with risk-less profits. The put price is a decreasing function of the strike price. In fact, if we consider two American puts with the same characteristics, except for the strike price, then the price of the put with a higher strike price must be higher than the price of the put with the lower strike price. If we consider two European puts with the same characteristics, except for the strike price, then the price of the put with a higher strike price must be higher than the price of the put with the lower strike price. If we consider two American puts with the same characteristics, except for the strike price, then the difference between their prices must be less than the difference in their strike prices. If we consider two European puts with the same characteristics, except for the strike price, then the difference between their prices must be less than the difference in the present value of their strike prices. 2.4.5. Other properties Interest rate and option prices Call prices are an increasing function of the interest rate. This is not the case for put options. Interest rate and call prices The price of a European or an American call must satisfy the following relationship: Ct ≥ St − Ke−r(T −t)

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This equation shows that the higher the interest rates, the smaller is the present value of the strike price Ke−r(T −t) . Hence, higher interest rates give a higher value for the difference St − Ke−r(T −t). Interest rate and put prices The price of a European put must satisfy the following relationship: pt ≥ Ke−r(T −t) − St Since the put holder receives a maximum value of the strike price (a potential cash inflow), the higher the present value of this cash inflow, the higher is the put value. Therefore, the put must be higher for lower interest rates. The above relationship can be used in this context to show the presence of a profitable arbitrage opportunity when the put fails to adjust to the changing interest rates. Risk and option prices Call and put prices are increasing functions of the risk of the underlying asset. Risk is measured by the standard deviation or the volatility of the underlying asset’s returns. With 25 years of experience with financial markets, I have learnt that options are to be bought, not sold. 2.5. Simple Trading Strategies for Options and their Underlying Assets 2.5.1. Trading the underlying assets The value of any option underlying asset such as a stock, a default-free bond, a futures or a forward contract is determined in a market place. In general, there are two prices quoted: the bid and the ask price. These two prices define a spread. In theory, there is only one price for each asset, but in practice, these two prices are observed in financial markets. An investor can buy (long position) or sell (short position) a financial instrument. It is possible to represent the gain or loss from a transaction at a given date in the future. If one buys a stock at 100 and sells it in a year at 110, the profit is 10. If one buys a bond at 90 and sells it in a year at 100, the profit is 10. The graph of the profit and loss as a function of the stock price at a given

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date (one year, for example) is diagonal. For a bond, the graph of the profit and loss is a straight line because the bond value is known with certainty at maturity. 2.5.2. Buying and selling calls Buying a call gives the right to the option holder to pay the strike price K at maturity and to receive the value of an underlying asset ST . He/she can also let the option expire. In this case, the call option is worth zero. The positive difference between (ST − K) refers to the intrinsic value or exercise value. Hence, at maturity, the value of a European or an American call is given by CT = cT = max[0, ST − K]. This represents the value of a long call position at expiration, where CT and cT refer respectively to the American and European call values. The profit for the option buyer represents exactly the loss of the option seller and vice versa. Example: Consider a call with K = 100 when ST = 90. At maturity, the call buyer does not exercise his/her option because he/she pays 100 and receives an underlying asset that is worth 90. He/she allows the option to expire to avoid losing 10 since the option is worthless. ST − K = 90 − 100 = −10 < 0 Hence, the payoff for the call buyer is: CT = max[0, ST − K] = max[0, 90 − 100] = max[0, −10] = 0 However, if ST = 110, this gives an immediate payoff of 10 exercise, or: CT = max[0, ST − K] = max[0, 110 − 100] = max[0, 10] = 10 The call seller implements a short position in the call. The value of a short call position at expiration is: −CT = −cT = −max[0, ST − K] It is clear that the value of the short position is always negative. This is because at initial time, the option seller receives the premium option. Example: Consider a call with K = 100 when ST = 110. In this case, upon exercise by the call holder, the option seller delivers a stock worth

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ST = 110 and receives the strike price K = 100. Hence, the value of the call position for the seller is −10 or: −CT = −cT = − max[0, ST − K] = − max[0, 110 − 100] = −10 As we compute the payoffs at maturity, we can determine the profit or loss P&L from holding long and short positions in options. If we denote the value of the purchased call at time t or the cost of the call by Ct , then the profit or loss on a long call position held until maturity is: CT − Ct = max[0, ST − K] − Ct In the same way, if we denote the value of the sold call at time t by Ct , then the profit or loss on a short call position held until maturity is: Ct − CT = Ct − max[0, ST − K] Example: Consider a call with K = 100 when ST = 110. The investor paid 10 for this call at time t = 0. In this case, if at the maturity date T , ST ≤ 110, the option buyer loses his/her option premium. However, when 100 < ST < 110, the call holder loses less than the purchase price 10. When ST = 105, the call holder loses 5. In fact, he/she receives 5 upon exercise, i.e., (105 − 100), coupled with the 10 paid for the option, his/her net loss is 5. If ST = 110, he/she receives 10 upon exercise coupled with the 10 paid for the option, gives a zero profit. It is important to note that, in any case, the profits from the buyer and the seller of the option sum up always to zero since the gains for the buyer are the losses of the seller and vice versa. If an investor buys and sells calls on the same underlying asset for the same maturity date, then the profits and losses from the two operations are equal to zero at the call’s maturity date. In fact, derivatives markets are zero-sum games because they do not lead to net profits and losses. What the buyer wins is lost by the seller and vice versa. If one computes the sum of gains and losses from the long and short call positions, the result is zero. In fact: (CT − Ct ) + (Ct − CT ) = 0 since (max[0, ST − K] − Ct ) + (Ct − max[0, ST − K]) = 0

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A trader buys, in general, a call when he/she expects the underlying asset price to rise. He/she shorts a call when he/she expects a stability or a decline in the /sheunderlying asset price.

2.5.3. Buying and selling puts Buying a put gives the right to the option holder to sell at maturity the underlying asset ST at the strike price K. He/she can also let the option expire. In this case, the put option is worth zero. The positive difference between (K − ST ) refers to the intrinsic value or an exercise value. Hence, at maturity, the value of a European or an American call is given by PT = pT = max[0, K − ST ]. The value of a short put position at expiration is −PT = −pT = − max[0, K − ST ]. Example: Consider a put with K = 100 when ST = 105. At maturity, the put buyer does not exercise his/her option because he/she receives 100 and delivers an underlying asset that is worth 105. He/she allows the option to expire to avoid losing 5 since the option is worthless. Hence, the payoff for the put buyer is: PT = max[0, K − ST ] = max[0, 100 − 105] = max[0, −5] = 0 If ST = 100, this gives an immediate payoff of zero upon exercise, or: PT = max[0, K − ST ] = max[0, 100 − 100] = max[0, 0] = 0 The holder of the put receives nothing upon exercise when ST ≥ K. However, if ST < K, the put value is positive for any value of the stock. If ST = 85, we have: PT = max[0, K − ST ] = max[0, 100 − 85] = max[0, 15] = 15 If the put holder had paid 10 at time zero, his/her net profit is 5 or (15−10). The previous analysis shows that the payoffs of calls and puts depend on the position of the option’s underlying asset with respect to the strike price. A trader buys, in general, a put when he/she expects the underlying asset price to fall. He/she shorts a put when he/she expects a stability or an appreciation in the underlying asset price.

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2.6. Some Option Combinations It is possible to use calls or puts in different simple strategies. We can combine calls and puts in order to create some option combinations that are discussed below. 2.6.1. The straddle The straddle is one of the simplest option strategies. An investor buying a straddle buys simultaneously a call and a put on the same underlying asset with the same strike price and maturity date. An investor selling a straddle sells simultaneously a call and a put on the same underlying asset with the same strike price and maturity date. Example: Consider a call and a put on the same underlying asset and maturity date. The call costs 10 and the put’s value is 8. The strike price is 100. The P&L of the straddle corresponds to the combined profits and losses from buying or selling both options. Let us denote the current option prices by Ct and Pt and the option prices at maturity by CT and PT . At maturity, the long straddle position is worth: CT + PT = max[0, ST − K] + max[0, K − ST ] The cost of the long straddle is Ct + Pt . The maximum loss of this strategy corresponds to the costs necessary to its implementation. A trader buys, in general, a straddle when he/she expects erratic movements in the underlying asset price. This refers to a very volatile market for the underlying asset. At maturity, the short straddle position is worth: −CT − PT = − max[0, ST − K] − max[0, K − ST ] A trader shorts a straddle when he/she expects a stability in the underlying asset price. The cost of the short straddle is −Ct − Pt . 2.6.2. The strangle The strangle is one of the simplest option strategies. An investor buying a strangle (long a strangle) buys simultaneously a call and a put on the same underlying asset with different strike prices K1 and K2 at the same maturity date. The call with a strike price K1 and the put with a strike

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price K2 are both out-of-the money with K1 > K2 . An investor selling a strangle (short a strangle) sells simultaneously a call and a put on the same underlying asset with two strike prices. The only difference with the straddle is the presence of two different strike prices. The long strangle is worth at maturity: CT (ST , K1 , T ) + PT (ST , K2 , T ) = max[0, ST − K1 ] + max[0, K2 − ST ] The cost of the long strangle is Ct (St , K1 , T ) + Pt (St , K2 , T ). A trader buys, in general, a strangle when he/she expects erratic movements in the underlying asset price. The short strangle is worth at maturity: −CT (ST , K1 , T ) − PT (ST , K2 , T ) = − max[0, ST − K1 ] − max[0, K2 − ST ] The cost of the short strangle is −Ct (St , K1 , T ) − Pt (St , K2 , T ). A trader sells, in general, a strangle when he/she expects some stability in the underlying asset price.

2.7. Option Spreads 2.7.1. Bull and bear spreads with call options A spread consists in buying an option and selling another option. A bull spread corresponds to a combination of options with the same underlying asset and the same maturity, but different strike prices. When it is implemented with two calls, it is designed to have a profit from a rise in the underlying asset price. This strategy limits the risks and the potential of profits. The bull spread buyer buys an in-the-money call and sells an out-ofthe-money call. At maturity, the long bull spread is worth: CT (ST , K1 , T ) − CT (ST , K2 , T ) = max[0, ST − K1 ] − max[0, ST − K2 ] The cost of the long bull spread is Ct (St , K1 , T ) − Ct (St , K2 , T ). A bear spread corresponds to a combination of options with the same underlying asset and the same maturity, but different strike prices. When it is implemented with two calls, it is designed to profit from falling underlying asset prices. This strategy with calls corresponds to the short position to the bull spread. The bear spread buyer buys the call with the higher strike price and sells the call with a lower strike price. At maturity, the long bear

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spread is worth: −CT (ST , K1 , T ) + CT (ST , K2 , T ) = − max[0, ST − K1 ] + max[0, ST − K2 ] The cost of the long bear spread is −Ct (St , K1 , T ) + Ct (St , K2 , T ). A trader implements a bull spread with calls when he/she expects the underlying asset price to rise. He/she implements a bear spread with calls when he/she expects the underlying asset price to fall.

2.7.2. Bull and bear spreads with put options A bull spread corresponds to a combination of puts with the same underlying asset and the same maturity, but different strike prices. When it is implemented with two puts, the trader buys a put with a lower strike price and sells a put with a higher strike price. The bear spread trader buys a put with a higher strike price and sells a put with a lower strike price. At maturity, the bull spread is worth: PT (ST , K1 , T ) − PT (ST , K2 , T ) = max[0, K1 − ST ] − max[0, K2 − ST ] The cost of the long bull spreads is Pt (St , K1 , T ) − Pt (St , K2 , T ) with K1 < K2 . At maturity, the bear spread with puts is worth: −PT (ST , K1 , T ) + PT (ST , K2 , T ) = − max[0, K1 − ST ] + max[0, K2 − ST ] The cost of the bear spread is −Pt (St , K1 , T ) + Pt (St , K2 , T ). A trader implements a bull spread with puts when he/she expects the underlying asset price to rise. He/she implements a bear spread with puts when he/she expects the underlying asset price to fall.

2.7.3. Box spread 2.7.3.1. Definitions and examples A box spread corresponds to a combination of a bull spreads with calls and a bear spread with puts. It is implemented using two spreads with two strike prices. Since at maturity, the long bull spread is worth: CT (ST , K1 , T ) − CT (ST , K2 , T ) = max[0, ST − K1 ] − max[0, ST − K2 ]

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and at the same date, the bear spread with puts is worth: −PT (ST , K1 , T ) + PT (ST , K2 , T ) = − max[0, K1 − ST ] + max[0, K2 − ST ] then at maturity, the box spread is worth: CT (ST , K1 , T ) − CT (ST , K2 , T ) − PT (ST , K1 , T ) + PT (ST , K2 , T ) This is equal to: max[0, ST − K1 ] − max[0, ST − K2 ] − max[0, K1 − ST ] + max[0, K2 − ST ] This strategy is “neutral” since it produces a return equal to the risk-less rate of interest. Example: Consider the following four legs of a transaction: a long call with K1 = 90, a short call with K2 = 100, a long put with K2 = 100, and a short put with K1 = 90. At maturity, the box spread is worth: max[0, ST − 90] − max[0, ST − 100] + max[0, 100 − ST ] − max[0, 90 − ST ] When the underlying asset ST = 98, the payoff is: max[0, 98 − 90] − max[0, 98 − 100] + max[0, 100 − 98] − max[0, 90 − 98] or 8 − 0 + 2 − 0 = 10. When the underlying asset ST = 70, the payoff is: max[0, 70 − 90] − max[0, 70 − 100] + max[0, 100 − 70] − max[0, 90 − 100] or 0 − 0 + 30 − 20 = 10. Note that in all the cases, the box spread is worth 10. This corresponds to the difference between the strike prices. Hence, the strategy appears equivalent to a risk-less investment. Therefore, to avoid profitable arbitrage opportunities, the box-spread value at time zero must be the discounted value of the difference between the two strike prices. Hence, its initial cost K2 −K1 must be (1+r) ( T −t) .

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2.7.3.2. Trading a box spread The box-spread strategy can be implemented with options on spot or options on futures. In the following discussion, c(K1 ), c(K2 ), p(K1 ), and p(K2 ) denote respectively, the prices of calls and puts with strike prices K1 and K2 with K2 > K1 . Consider a portfolio corresponding to the following strategy: a long bullish call spread: buy c(K1 ) and sell c(K2 ) a long bearish put spread: buy p(K2 ) and sell p(K1 ) This strategy is a box spread which costs c(K1 ) − c(K2 ) − p(K1 ) + p(K2 ). The following non-arbitrage condition must be satisfied: c(K1 ) − c(K2 ) − p(K1 ) + p(K2 ) ≤ (K2 − K1 )e−rT At the maturity date, the result of the strategy is always (K2 − K1 ). In fact, the pay-off of each option is: max[ST − K1 , 0] − max[ST − K2 , 0] − max[K1 − ST , 0] + max[K2 − ST , 0] This shows that the box is worth (K2 − K1 ) at the maturity date. If its value is less than the discounted value of (K2 − K1 ), then risk-less arbitrage would be possible. Consider the two following two relationships between the European options c and p and the American options C and P : C(K1 ) − c(K1 ) ≥ C(K2 ) − c(K2 ), P (K2 ) − p(K2 ) ≥ P (K1 ) − p(K1 ) These relations account for the value of the early exercise premium for calls and puts with different strike prices. If the first condition was not satisfied, then selling the American call and buying the European call (with a strike price K2 ) and buying the American call and selling the European call (with a strike price K1 ), would allow an immediate profit. If the American call with a strike price K2 is not exercised before the maturity date, the position produces a zero cash-flow at this date. If the call with a strike price K2 is exercised, the option with a strike price K1 can be exercised to generate a cash-flow (K2 − K1 ), which will be invested until the maturity date T . If the option with a strike price K2 is exercised before the maturity date at a date t1 < T , the result at maturity is (K2 − K1 )er(T −t1 ) > K2 − K1 . Tests of the box strategy for options traded on the Chicago Board

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Options Exchange (CBOE) over eight years reveal some violations of the non-arbitrage condition. However, the profitable opportunities disappeared when transaction costs were taken into account. Hence, the market is globally efficient. 2.8. Butterfly Strategies 2.8.1. Butterfly spread with calls A butterfly spread corresponds to a combination using three calls with three strike prices K1 < K2 < K3 . The calls have the same underlying asset and maturity date. An investor long the butterfly buys the call with the lowest strike price K1 , buys the call with the highest strike price K3 , and sells two calls with an intermediate strike price K2 . At maturity, the long butterfly spread is worth: CT (ST , K1 , T ) − 2CT (ST , K2 , T ) + CT (ST , K3 , T ) = max[0, ST − K1 ] − 2 max[0, ST − K2 ] + max[0, ST − K3 ] The cost of a long butterfly spread is: Ct (St , K1 , T ) − 2Ct (St , K2 , T ) + Ct (St , K3 , T ) An investor short the butterfly sells the call with the lowest strike price K1 , sells the call with the highest strike price K3 , and buys two calls with intermediate strike price K2 . This strategy tends to be profitable when the underlying asset at maturity is at the intermediate strike prices. At maturity, the short butterfly spread with calls is worth: −CT (ST , K1 , T ) + 2CT (ST , K2 , T ) − CT (ST , K3 , T ) = − max[0, ST − K1 ] + 2 max[0, ST − K2 ] − max[0, ST − K3 ] The cost of the short butterfly spread with calls is: −Ct (St , K1 , T ) + 2Ct (St , K2 , T ) − Ct (St , K3 , T ) A trader implements a long butterfly spread with calls when he/she expects the underlying asset price to be relatively stable. He/she implements a short butterfly spread with calls when he/she expects strong movements in the underlying asset price.

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2.8.2. Butterfly spread with puts A butterfly spread corresponds to a combination using three puts with three strike prices K1 < K2 < K3 . The calls have the same underlying asset and maturity date. An investor long the butterfly buys the put with the lowest strike price K1 , buys the put with the highest strike price K3 , and sells two puts with an intermediate strike price K2 . At maturity, the long butterfly spread with puts is worth: PT (ST , K1 , T ) − 2PT (ST , K2 , T ) + PT (ST , K3 , T ) = max[0, K1 − ST ] − 2 max[0, K2 − ST ] + max[0, K3 − ST ] The cost of a long butterfly spread with puts is: pt (St , K1 , T ) − 2Pt (St , K2 , T ) + Pt (St , K3 , T ) An investor short the butterfly sells the put with the lowest strike price K1 , sells the put with the highest strike price K3 , and buys two puts with an intermediate strike price K2 . At maturity, the short butterfly spread with puts is worth: −PT (ST , K1 , T ) + 2PT (ST , K2 , T ) − PT (ST , K3 , T ) = − max[0, K1 − ST ] + 2 max[0, K2 − ST ] − max[0, K3 − ST ] The cost of the short butterfly spread is: −Pt (St , K1 , T ) + 2Pt (St , K2 , T ) − Pt (St , K3 , T ) A trader implements a long butterfly spread with puts when he/she expects the underlying asset price to be relatively stable. He/she implements a short butterfly spread with puts when he/she expects strong movements in the underlying asset price. 2.9. Condor Strategies 2.9.1. Condor strategy with calls A condor corresponds to a combination using four calls with four strike prices K1 < K2 < K3 < K4 . The calls have the same underlying asset and maturity date. An investor long the condor buys the call with the lowest strike price K1 , sells the call with a somewhat higher strike price K2 , sells the call with the yet higher strike price K3 , and buys the calls

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with the highest strike price K4 . At maturity, the long condor with calls is worth: CT (ST , K1 , T ) − CT (ST , K2 , T ) − CT (ST , K3 , T ) + CT (ST , K4 , T ) = max[0, ST − K1 ] − max[0, ST − K2 ] − max[0, ST − K3 ] + max[0, ST − K4 ] The cost of a long condor with calls is: Ct (St , K1 , T ) − Ct (St , K2 , T ) − Ct (St , K3 , T ) + Ct (St , K4 , T ) An investor short the condor sells the call with the lowest strike price K1 , buys the call with a somewhat higher strike price K2 , buys the call with the yet higher strike price K3 , and sells the calls with the highest strike price K4 . At maturity, the short condor with calls is worth: −CT (ST , K1 , T ) + CT (ST , K2 , T ) + CT (ST , K3 , T ) − CT (ST , K4 , T ) = − max[0, ST − K1 ] + max[0, ST − K2 ] + max[0, ST − K3 ] − max[0, ST − K4 ] The cost of a short condor with calls is −Ct (St , K1 , T ) + Ct (St , K2 , T ) + Ct (St , K3 , T ) − Ct (St , K4 , T ). A trader implements a long condor with calls when he/she expects the underlying asset price to be relatively stable. He/she implements a short condor with calls when he/she expects strong movements in the underlying asset price.

2.9.2. Condor strategy with puts A condor corresponds to a combination using four puts with four strike prices K1 < K2 < K3 < K4 . The puts have the same underlying asset and maturity date. An investor long the condor buys the put with the lowest strike price K1 , sells the put with a somewhat higher strike price K2 , sells the put with the yet higher strike price K3 , and buys the put with the highest strike price K4 .

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At maturity, the long condor with puts is worth: PT (ST , K1 , T ) − PT (ST , K2 , T ) − PT (ST , K3 , T ) + PT (ST , K4 , T ) = max[0, K1 − ST ] − max[0, K2 − ST ] − max[0, K3 − ST ] + max[0, K4 − ST ] The cost of a long condor with puts is: Pt (St , K1 , T ) − Pt (St , K2 , T ) − Pt (St , K3 , T ) + Pt (St , K4 , T ) An investor short the condor sells the put with the lowest strike price K1 , buys the put with a somewhat higher strike price K2 , buys the put with the yet higher strike price K3 , and sells the put with the highest strike price K4 . At maturity, the short condor with puts is worth: −PT (ST , K1 , T ) + PT (ST , K2 , T ) + PT (ST , K3 , T ) − PT (ST , K4 , T ) = − max[0, K1 − ST ] + max[0, K2 − ST ] + max[0, K3 − ST ] − max[0, K4 − ST ] The cost of a short condor with puts is: −Pt (St , K1 , T ) + Pt (St , K2 , T ) + Pt (St , K3 , T ) − Pt (St , K4 , T ) A trader implements a long condor with puts when he/she expects the underlying asset price to be relatively stable. He/she implements a short condor with puts when he/she expects strong movements in the underlying asset price. 2.10. Ratio Spreads A ratio spread is a strategy involving two or more related options in a given proportion. A trader can buy a call with a lower strike price and sell a higher number of calls with a higher strike price. A 2:1 ratio spread corresponds, for example, to a strategy in which the trader buys two options and sells an option. It is possible to use options with different maturities. A spread based on options with different times to maturity is referred to as a calendar spread. The investor can implement different combinations of options with different strikes and time to maturity in order to construct a wide range of profit and loss profiles. These strategies can also be implemented in connection with the underlying assets and in particular with stocks, bonds, foreign currencies, commodities, etc.

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Using simultaneously options and their underlying assets allows one to adjust payoff patterns to fit the attitude of investors toward the risk-return profile. A trader implements ratio spreads with calls and puts in different contexts according to his/her future expectations about the underlying asset price and his/her risk-return profile. 2.11. Some Combinations of Options with Bonds and Stocks 2.11.1. Covered call: short a call and hold the underlying asset This strategy is implemented by selling the call and buying a certain quantity of the underlying asset. This strategy corresponds to a covered strategy since the investor owns the underlying asset. This asset covers the obligation inherent in selling the call. This strategy enhances income. 2.11.2. Portfolio insurance Portfolio insurance is an investment-management technique that protects a portfolio from drops in value. This technique proposes some simple concepts allowing one to insure a stock portfolio. The strategy can be implemented using options, futures contracts, and other financial products. Consider, for example, a well-diversified portfolio of stocks. Portfolio insurance can be implemented in its basic form by buying a put on the owned portfolio of assets. At maturity or the horizon date, the value of the insured portfolio corresponds to the sum of the stock portfolio ST and the put PT written on this portfolio. The value of the insured portfolio can be written as: ST + PT = ST + max[0, K − ST ] The cost of the insured portfolio at initial time t corresponds to the sum of the stock portfolio St and the put price Pt or (St + Pt ). The profit (or loss) on an uninsured portfolio is simply the difference between its final value and initial value or ST − St . The insured portfolio has a superior performance only when: max[0, K − ST ] − Pt − St > 0 This strategy pays when markets are down, as in the year 2008.

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2.11.3. Mimicking portfolios and synthetic instruments A trader can use European options in connection with other instruments to create some specific payoff patterns at expiration. Two main relationships are often used: mimicking portfolios and synthetic instruments. A mimicking portfolio: It shows the same results (profits and losses) as the instrument it mimics, but it might not have the same value. A synthetic instrument: It exhibits the same results and value as the instrument it synthetically replicates. 2.11.3.1. Mimicking the underlying asset An investor who buys a European call and sells a European put on the same underlying asset creates a position that exhibits the same payoff pattern as the underlying asset. At maturity, the payoff of this position is: cT − pT = max[0, ST − K] − max[0, K − ST ] The initial cost of this position is ct − pt . Assume that at time t, K = St . In this setting, the value of the portfolio comprising a long call and a short put is obtained by replacing K with St in the previous equality: max[0, ST − St ] − max[0, St − ST ] If the underlying asset price rises with respect to the initial level St , then ST > St and the call is worth ST − St . The put value is zero. If the underlying asset price falls with respect to the initial level St , then ST < St , the call is worth zero and the put value is St − ST . Note that this result is equivalent to that of the stock portfolio. Hence, the value of a portfolio comprising a long call and a short put is equivalent to that of the underlying asset or portfolio. This result is always verified when the option’s strike price corresponds to the value of the underlying asset when this strategy is implemented. 2.11.3.2. Synthetic underlying asset: Long call plus a short put and bonds A portfolio with a long European call and a short European put shows a profit (loss) pattern that can mimic the result of the underlying asset. Using a risk-free bond (as a proxy for investing a certain amount K at the risk-free interest rate r), it is possible to create a portfolio that synthesizes

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the option’s underlying asset. The synthetic underlying asset has the same value and exhibits the same profit (loss) pattern as the underlying asset (or stock). The value of the synthetic underlying asset at time t can be written as St = ct − pt + Ke−r(T −t) . The investor invests an amount Ke−r(T −t) in risk-free bonds at time t. This amount grows at the rate er(T −t) . At maturity, the value of the investment in risk-free bonds is exactly K. This corresponds also to the face value of the discount bonds at maturity. At the option’s maturity date, the value of the portfolio comprising a long call, a short put, and a certain amount K is: cT − pT + K = max[0, ST − K] − max[0, K − ST ] + K In the absence of an early exercise, this equality is also verified for American options at expiration: CT − PT + K = max[0, ST − K] − max[0, K − ST ] + K At maturity, if ST > K, the call’s value is ST − K. The put value is zero. The value of the portfolio is simply that of the call and the bond or ST − K + K = ST . At maturity, if ST < K, the call’s value is zero and the put’s value is K − ST . The value of the portfolio simply corresponds to that of the short position in the put and the bond or −(K − ST ) + K = ST . Hence, as discussed before, the value of a portfolio comprising a long call and a short put and the bond is equivalent to that of the underlying asset. Buying a call, selling a put, and investing in a risk-free discount bond is a strategy equivalent to an investment in the option’s underlying asset.

2.11.3.3. The synthetic put: put-call parity relationship The put-call parity relationship stipulates simply that buying a call, selling a put, and investing in a risk-free discount bond is a strategy equivalent to an investment in the option’s underlying asset. Hence, using three of these four instruments allows one to synthesize the fourth instrument. The put-call parity relationship is often presented in the following form: pt = ct − St + Ke−r(T −t)

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It stipulates that the put can be duplicated by a short position in the stock, a long position in the call, and an investment in risk-free bonds paying the strike price at the option’s common maturity date. When at initial time t, St = K, then the call price must be higher than the put price. In fact, the relationship can also be presented in the following form: ct − pt = St + Ke−r(T −t). Since St = K, the right-hand side must be positive. Therefore, ct − pt must be positive. Therefore, the call price must be higher than the put price.

2.12. Conversions and Reversals A strategy can be implemented by going long a call and short a put. If the underlying asset price is above the strike price at the option’s maturity date, the put is worthless and the call’s value corresponds to the intrinsic value. The position will behave exactly as the value of the underlying asset. However, if the underlying asset price is below the strike price, the call is worthless and the put’s value is its intrinsic value. The position will again behave exactly as the value of the underlying asset. Buying the call and selling the put is a position equivalent to buying the underlying asset. More generally, the following no-arbitrage between a call c, a put p, the underlying asset price S, and the strike price K must hold: c − p = S − Ke−rT where r stands for the risk-less interest rate and T is the option’s maturity date. A conversion is a strategy based on the above relationship. It can also be written as: short a call + long a put + long the underlying asset = short a synthetic underlying asset + long the underlying asset. A reversal corresponds simply to a reverse conversion: long a call+short a put+short the underlying asset = long a synthetic underlying asset+short the underlying asset. If we substitute the underlying asset by a synthetic underlying asset in the conversion strategy for a different strike price, this eliminates the risks associated with the variations of the underlying asset price and gives the well-known strategy, the box spread. The box spread is simply a strategy equivalent to borrowing or lending money for a certain period. For an analysis of financial markets, volume, volatility, and spreads, readers can refer to Hong and Wang (2000), French (1980), and Gibbons and Hess (1981).

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2.13. Case study: Selling Calls (Without Holding the Stocks/ as an Alternative to Short Selling Stocks/the Idea of Selling Calls is Also an Alternative to Buying Puts) 2.13.1. Data and assumptions We consider a Risk capital of 100,000 to invest for 3 months. When the stock is at 15 and the strike price at 15, using option valuation procedure and assuming a volatility of 20%, the interest rate at 5%, the call price is 0.69. If we change the volatility to 45%, the option price is 1.5. The results show a profit in each scenario. 2.13.1.1. Selling calls (without holding the stock) The investor (DC) sells (short) 6,666 calls. This is a covered position because its aggregate underlying value is no greater than the risk capital that would have gone into buying stocks instead of selling calls: 100,000. Aggregate underlying value × contract multiplier

=

stock price × number of options sold

100,000 = 15 × 6, 666 × 1 The short sale of 6,666 calls would result in proceeds of 4,599.54, which would be invested in T-bills during the holding period: Proceeds of call short sale = call price × number calls × multiplier 4, 599.54 = 0.69 × 6, 666 × 1 The pattern of risk return associated with this strategy is as follows: If the stock price is unchanged or down at expiration, the investment profit is 4,657.03. If the stock is up by 10%, from 15 to 16.5, each option has an exercise value of 1.5 for an aggregate exercise value of 9,999. Offset by the initial proceeds of the short sale and income from T-bills, the investor’s loss is 4,542.05. Recall, if the investor sold a portfolio of stocks short, his loss was 9,750. Comparison with selling short the stock: — When the investor short sells calls, with the stock unchanged, he experiences a profit of 4,657.03 and this is his maximum gain.

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Table 2.5. Results at the maturity date in three months of the strategy of selling 6,666 calls at 0.69 when volatility is at 20%.

Initial proceeds from calls: 6,666 × 0.69 Exercise value of calls Profit (loss) from calls Interest income on T-bills: 5% × 4,599.54 × 0.25 Total profit (loss) Return on risk capital

Stock unchanged: 15

Stock + 10%: 16.5

Stock − 10%: 13,5

4,599.54

4,599.54

4,599.54

0

(9,999) 1.5 × 6,666 (5,399.46) 57.49

0

4,599.54 57.49

4,657.03: maximum gain 4,65%

(4,542.05) (4,54%)

4,599.54 57.49

4,657.03: maximum gain 4,65%

Remark: No initial capital to use. Profit if the stock is less than 15.6986, otherwise losses appear. Table 2.6. Results at the maturity date in three months of the strategy of selling 6,666 calls at 1.5 when volatility is at 47%.

Initial proceeds from calls: 6,666 × 1.5 Exercise value of calls Profit (loss) from calls Interest income on T-bills: 5% × 9,999 × 0.25 Total profit (loss)

Stock unchanged: 15

Stock + 10%: 16.5

Stock − 10%:13,5

9,999

9,999

9,999

0

(9,999) 1.5 × 6,666 (0) 124,98

0

124,98

10,123: maximum gain

9,999 124,98

10,123: maximum gain

9,999 124,98

Remark: No initial capital to use. Profit in each scenario if call sold at this price of 1.5. Profit in every scenario for IC. If the stock is higher than 16.518, losses appear.

No matter how far the underlying asset may fall by expiration, the calls he sold short will still expire worthless and the most he can hope to collect on them is what he initially sold them for 4,599.54, no matter what happens to the stock, the T-bills will still yield 57.49. We can calculate the up-side tolerance point for a short sale of 6,666 calls, 15.6986. Below this stock price level, the premium initially received from the short sale of calls and the interest income from his T-bills is sufficient to repay the exercise value of the short calls.

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Case 1: C = 0,69 Proceeds per contract = (proceeds from calls + T-bills) /(Number contracts × mulplier) 0,6986 = (4,599.54 + 57.49)/(6,666 × 1) Upside tolerance point = Proceeds per contract + strike price 15.6986 = 0.6986 + 15 Above this point, the exercise value of the calls becomes great enough to cause a net loss. The 15.6986 in the price at which the risk return line crosses down through the center of the chart into the region representing losses. Case 2: C = 1, 5 Proceeds per contract = (proceeds from calls + T-bills) /(Number contracts × mulplier) 1,5187 = (9,999 + 124,98)/(6,666 × 1) Upside tolerance point = Proceeds per contract + strike price 16.518 = 1,5187 + 15 Above this point, the exercise value of the calls becomes great enough to cause a net loss. The 16.518 in the price at which the risk return line crosses down through the center of the chart into the region representing losses. 2.13.1.2. Comparing the strategy of selling calls (with a short portfolio of stocks): the extreme case Comparing covered short calls with a short portfolio of stocks, the short calls are less risky. The Extreme case: In theory, there is no limit to how far the stock might rise over a given holding period, but as an extreme example, let’s see what would happen if the stock doubled from 15 to 30. Table 2.7.

Extreme case: Stock up by 100%.

Proceeds from short sale of stocks Repurchase of stocks Loss from stocks Interest income on T-bills 5% × 100,000 × 0.25

100,000, (200,000), (100,000), 1,250

Total loss

(98,750)

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110 Table 2.8.

Stock up by 100%: position in the short calls: case 1.

Proceeds from sale of calls 6,666 × 0.69 Exercise value of calls 6,666 × 15 × 1 Loss from calls Interest income on T-bills 5% × 4, 599.54 × 0.25

4,599.54 (99,990) (95,390.46) 57.49

Total loss

(95,332.97)

Table 2.9.

Stock up by 100%: position in the short calls: case 2.

Proceeds from sale of calls 6,666 × 1.5 Exercise value of calls 6,666 × 15 × 1 Loss from calls Interest income on T-bills 5% × 9,990 × 0.25

9,990 (99,990) (90,000) 124.875

Total loss

(89,875)

Position in calls: shows a net loss of (89,875).

The portfolio would show a loss of 100,000, offset by 1,250 in interest income − a net loss of 98,750. The position with 6,666 calls, fares somewhat better. The investor must pay an aggregate exercise value of 99,990, but this is offset by the initial proceeds from the calls and the interest income — the net loss is (95,332.97), an improvement of 3,417.03. (98,750-95,332.97) 2.13.1.3. Selling calls (holding the stock) The investor sells short 6,666 calls. What must be the position in the stock: To cover the position, we buy Delta units of the underlying asset. Assume that the call’s delta is about 0.55 and that the option is on one share. At time zero, we sell the calls and we buy: 0.55×6,666×15 = 55,000 stocks. The pattern of risk return associated with this strategy is as follows: 2.13.2. Leverage in selling call options (without holding the stocks) If the investor of the previous example were willing to lower his upside tolerance point in exchange for higher profits with the stock unchanged or down, he could sell a larger number of calls.

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Table 2.10. Results at the maturity date in three months of the strategy of selling calls and buying the underlying stock for a hedge: delta stocks: 0.55 stocks. Stock Unchanged

Stock + 10%

Stock − 10%

Total profit (loss) on options Portfolio of Stock’s ending value

4,657.03 55,000

4,657.03 49,500

Profit (loss) from stocks Total profit (loss) from options and stocks Return on risk capital (in stocks)

0 4,657.03

(4,542.05) 60,500 (55,000 × 1,1) 5,500 958

8,46% = 4,657.03/55,000

1,74% = 958/55,000

(1,53%) = (843)/55,000

(5,500) (843)

If we account for dividends of 1%, about 549 dollars, all the results may become positive. Table 2.11. Results at the maturity date in three months of the strategy of selling calls and buying the underlying stock for a hedge: delta stocks: 0.55 stocks. Stock Unchanged

Stock + 10%

Stock − 10%

Profit (loss) from calls Portfolio of Stock’s ending value

9,999 55,000

9,999 49,500

Profit (loss) from stocks Total profit (loss) from options and stocks Return on risk capital (in stocks)

0 9,999

(0) 60,500 (55,000 X1,1) 5,500 5,500

(5,500) 4,499

18% = 9,999/55,000

10% = 5,500/55,000

8,2% 4,499/55,000

2.13.2.1. Selling Call options (without holding the stocks) If the investor wished to commit all of his 100,000 dollars to margining short calls he could sell: Net requirement = 5%of stock price × contract multiplier 0.75 = 5% × 15 × 1 Maximum contracts to Margin = Total Margin deposit/Net requirement per contract 133, 333 = 100,000/0.75 At 0.69, the proceeds from selling 133,333 calls would be: Proceeds of call short sale = call price × number × contract multiplier 91,999 = 0.69 × 133,333 × 1 The patterns of risk and return associated with this strategy shows that with the stock unchanged or down, the investor’s profit would be the initial proceeds from the sale plus interest from the T-bills, a total of 93,148.98.

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Table 2.12. Results at the maturity date in three months of the strategy using leverge in selling calls.

Initial proceeds from calls: 133,333 × 0.69 Exercise value of calls Profit (loss) from calls Interest income on T-bills: 5% × 91,999 × 0.25 Total profit (loss) Return: Profit/initial proceeds

Stock unchanged

Stock + 10%

Stock − 10%

91,999

91,999

91,999

0

0

91,999 1,149,98

(199,999) 1.5 × 133,3333 (100,000) 1,149,98

93,148.98 1,012 = 93,148.98 /91,999

(98,850) (1,074) = (98,850 /91,999)

93,148.98 1.012 = (93,148.98 /91,999)

91,999 1,149,98

Table 2.13. Results at the maturity date in three months of the strategy using leverge in selling calls. Stock unchanged Initial proceeds from calls: 199,999 133,333 × 1.5 Exercise value of calls 0 Profit (loss) from calls Interest income on T-bills: 5% × 199,999 × 0.25 Total profit (loss) Return: Profit/initial proceeds

199,999 2,499.98

Stock + 10%

Stock − 10%

199,999

199,999

(199,999) 1.5 × 133,3333 (000) 2,499.98

0 199,999 2,499.98

202,498 2,499,98 202,498 101,2% = 202,498 1,02% = (2,499.98 101,2% = (202,498 /199,999 /199,999) /199,999)

This will be the maximum profit (as other short positions). But, if the stock rises, he is exposed to unlimited losses. If the stock rises 10%, the aggregate exercise value is (199,999). Offset by the initial proceeds from the sale of calls and the interest on the TB, the investor total loss is (98,850). The investor loss can be more than the 100,000 marging deposit. Losses in short calls are almost limitless. (Unlike the case with short puts, in which losses are limited by the fact that the stock cannot trade below zero.)

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Table 2.14. Case 1. Leverage in selling Call options (without holding the stocks): The extreme case stock up by 100%. Proceeds from sale of calls 133,3333 × 0.69 Exercise value of calls 133,3333 × 15 × 1 Loss from calls Interest income on T-bills 5% × 91,999 × 0.25

91,999 (1,999,990) (1,907,999) 1,149.875

Total loss

(1,906,849)

Table 2.15. Case 2. Leverage in selling Call options (without holding the stocks): The extreme case stock up by 100%: call: 1.5. Proceeds from sale of calls 133,3333 × 1.5 Exercise value of calls 133,3333 × 15 × 1 Loss from calls Interest income on T-bills 5% × 199,999 × 0.25

199,999 (1,999,995) (1,799,996) 2,499.9875

Total loss

(1,7907,496)

2.13.2.2. Leverage in selling Call options (without holding the stocks): The extreme case Extreme case: Stock up by 100% Consider the limiting case in which the stock doubles from 15 to 30, the aggregate exercise value the call seller would have to pay totals (1,999,990). Offset by the initial proceeds from sale of the calls and the interest on the T-bills, the investor’s total loss is (1,906,849). 2.13.2.3. Selling calls using leverage (and holding the stock) The investor sells short 133,3333 calls. What must be the position in the stock? To cover the position, we buy Delta units of the underlying asset. Assume that the call’s delta is about 0.55 and that the option is on one share. At time zero, we sell the calls and we buy stocks: 0.55 × 133,3333 ×15 = 1,099,997. The pattern of risk return associated with this strategy is as follows: 2.13.3. Short sale of the stocks without options An investor (DC) buys stocks in anticipation of a rise in the market. An investor (DC) can sell stocks short in anticipation of a decline.

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Table 2.16. Results at the maturity date in three months of the strategy of selling calls and buying the underlying stock.

Total profit on options (loss) Portfolio of Stock’s ending value Profit (loss) from stocks Total profit (loss) from options and stocks Return on options Return on stocks

Stock unchanged

Stock + 10%

Stock − 10%

93,148.98

(98,850)

93,148.98

1,099,997 0.55 × 133,3333 × 15(1,099,997) 0

1,209,996 0.55 × 133,3333 × 15(1,099,997) × 1,1 109,999

989,997 0.55 × 133,3333 × 15(1,099,997) × 0,9 (109,999)

93,148.98

11,149

(16,851)

100% 0

− 100% 10%

100% (−10%)

Table 2.17. Results at the maturity date in three months of the strategy of selling calls and buying the underlying stock.

Profit (loss) from calls Portfolio of Stock’s ending value

Profit (loss) from stocks Total profit (loss) from options and stocks Return on options Return on stocks

Stock unchanged

Stock + 10%

Stock − 10%

199,999 1,099,997

(000) 1,209,996

199,999 989,997

0.55 × 133,3333 × 15(1,099,997) × 1 0 93,148.98

0.55 × 133,3333 × 15 × 1.1 109,999 109,999

(109,999) 90,0000

100% 0

0% 10%

100% (−10% )

If an investor sells short a stock, another who already owns the stock (IC) has to be willing to lend it to DC to sell. First, if the lender (IC) suddenly requires (DC) to immediately return the borrowed stock, forcing DCG to buy it in the open market, prices may be far from favorable. Second, If IC accepts to lend the stock, DC must put collateral to guarantee the loan (apart from the margin DC must put with broker), tying up some of the proceeds of the sale. This creates an economic asymmetry between buying and short selling. Buying stocks ties up funds that could otherwise be invested to earn interest and although the proceeds from selling stocks one already owns can be fully reinvested to earn interest only on a portion of the proceeds from short-selling stocks can be reinvested.

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Short selling stocks (stock lending).

Stocks

Stock unchanged

Stock + 10%

Stock − 10%

Short sale stocks Repurchase of stocks Profit(loss) from stocks Interest income from T-bills (5%100, 000 × 0.25) Restitution of dividends: 1%

100,000 100,000 0 1,250

100,000 110,000 (10,000) 1,250

100,000 90,000 10,000 1,250

(1,000)

(1,000)

(1,000)

Total profit or loss

250

(9,750)

10,250

Table 2.19.

Short selling stocks (stock lending) Example: double the dividend payment.

Stocks

Stock unchanged

Stock + 10%

Stock − 10%

Short sale stocks Repurchase of stocks Profit(loss) from stocks Interest income from T-bills (5%100,000 × 0.25) Restitution of dividends: 1%

100,000 100,000 0 1,250

100,000 110,000 (10,000) 1,250

100,000 90,000 10,000 1,250

(2,000)

(2,000)

(2,000)

Total profit or loss

(750)

(10,750)

9,250

In the US, New York Exchange rules plus-tick rule, permits short sales to be executed only at prices representing a plus tick from the previous different price. If we sold the portfolio of stocks short, the loss is: As the long stock has a downside tolerance point, the short stock has an upside tolerance point. Net percentage interest income from T-bills: offset by the restitution of dividends/Holding period net interest yield = (Interest income − dividends) Value of T-bills 0.25% = (1, 250 − 1, 000)/100, 000 = 0.0025 The upside tolerance point is: Upside tolerance point = stock price + holding period net interst yield. 15.0025 = 15 + 0.0025 If restitution of dividend payments through expiration is greater than the interest income from the reinvestment of the proceeds of the short sale, the investor will show a loss if the stock is unchanged.

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The short position shows a loss if the index is unchanged, rather than a profit. Performance is also degraded when the stock is down. The question is how low the stock must fall before the position fails to show a loss? Holding period interest yield = (Interest income − dividends)/ Value of T-bills 0.75% = (1, 250−2, 000)/100, 000 = −0.0075 The upside tolerance point is: Downside breakeven point = stock price + holding period interst yield 14.9925 = 15 − 0.0075 A position of $ 100 millions. How options can be used as alternatives to a direct investment in stocks? 2.14. Buying Calls on EMA 2.14.1. Buying a call as an alternative to buying the stock: (also as an alternative to short sell put options) Buying calls can be a unique managerial alternative to buying stocks. Another alternative is to short sell put options. 2.14.1.1. Data and assumptions An investor with 100,000 to invest for 3 months in the call. When the stock is at 15, he can either buy the stock or invest in T-bills. This leads to symmetrical risk and return pattern, profit or loss dollar for dollar plus income from dividends. 2.14.1.2. Pattern of risk and return Consider the pattern of risk and return with a purchase of 3 month call option with a strike price of 15. We selected at the money so that at expirations, appreciation in the underlying will be reflected in the exercise value of calls. To scale the position in calls to the investor Risk capital of 100,000, we begin by calculating the value of one contract.

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The stock price times the multiplier. Call’s underlying value = asset price × contract multiplier = 15 × 1 Number of calls to buy: To determine the number of calls to buy, we divide the risk capital of 100,000 by call’s underlying value of 15: Contracts required = risk capital: call’s underlying value 6,666 = 100000: 15 Valuation: Using option valuation procedure and assuming a volatility of 30%, the interest rate at 5%, the call price is 0.98746. Assuming options can be purchased at theoretical value, the investor would have to spend to buy 6,666 contracts: Investment required to buy = call price × number × contract multiplier $6 583.608 = 0.98764 × 6,666 × 1 The balance of the 100,000 risk capital that could be invested in T-bills would be: Remaining capital in T-bills = total capital − investment required to buy calls 93,416,391 = 100,000 − 6,583 Strategy: invest an amount in options and the balance in T-bills. Buy calls, strike 15, stock initially at 15 at 0.98746. Invest: 93,416,391 in 3 month TB, Hold for 3 months, Collect interest. 2.14.2. Compare buying calls (as an alternative to portfolio of stocks) Here we compare buying calls as an alternative to portfolio of stocks. At maturity After 3 months: If the asset price does not change: Initial cost of calls (6,583.6080) Exercise value: 0 Loss from calls (6,583.6080)

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Interest income on T-bills (5% on 93,416,391 for 3 months: 1,167.704) Total loss: (5,415.903) If we examine the outcomes of investing directly in stock, the return is zero. Return in stock: 0 2.14.2.1. Risk return in options If the investor uses options to take a position in the market, the position is not profitable unless the asset is high enough to give options sufficient exercise value to repay their initial cost. The upside break even point can be computed as follows. Cost per contract = (invt to buy calls-T-bill income) /(number of contracts × multiplier contract) = (6,583.608 − 1,167.704)/(6,666 × 1) 0.8124 = 5, 415.904/6,666 Upside break-even point = Net cost per contract + call’s strike price $15.8124 = 0, 8124 + 15 The Upside break-even point is $15.8124. Each call at this point would has an exercise value of 0,8124. The aggregate exercise value of the position would be 5,415.455 just above the amount required to repay the cost less the offsetting treasury bill income: 0,8124 × 6, 666 = 5,415.455 Exercise value of call = stock price upside — option strike price × contract multiplier 0, 8124 = (15.8124 − 15) × 1 Aggregate exercise value of position = exercise value of call × number of calls in position 5, 415.455 = 0, 8124 × 6, 666 If the stock rallies beyond the upside break even point, the position will show a healthy profit. For example, If the stock rallies by 10%, from 15 to 16.5, each option will have an exercise value of 1.5

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Exercise value of call = (stock − call’s strike) × contract multiplier 1.5 = (16.5 − 15) × 1 Aggregate Exercise value of position = Exercise value of call × number of calls 9,999 = 1.5 × 6,666 Including the interest income from T-bills, the investor’s total profit with the stock up 10% is 4,584.26. At maturity after 3 months: If the asset price up by 10%: Initial cost of calls (6,583.6080) Exercise value: 1.5 × 6,666 × 1 9,999 Profit from calls 3,415.4 Interest income on T-bills (5% on 93,416,391 for 3 months: 1,167.704) 1,167.704 Total profit: 4,584.26 Return from options and T-bills: 4,58% This profit is less than a direct investment in the stock: If the asset price up by 10%: Initial cost of stocks = 100,000 Sale of stocks = 110,000 Profit = 10,000 Return from the stock: 10% When the investor established his position in stocks, he earned a profit of 10,000. With options, he can not break even until the stock rises to $15.8124. With stocks, his profit with the stock up by 10% was 10,000, with options only 4,584.26. Advantages: The advantages gained for these handicaps are that the option buyer is assured that his investment cannot do any worse than a known maximum loss equal to the costs of the options, minus any interest earned on the leftover cash invested in T-bills. Maximum loss in position = cost of calls − interest on T-bills 5, 415.904 = 6583.608 − 1, 167.704 This example is done to replicate a position in the stock.

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When the investor chooses to buy options, he is making a different kind of investment. The tradeoff is: in exchange of a known maximum loss (the option premium), he sacrifies performance. The tradeoff is equitable. This is a managerial choice driven by the attitude toward the pattern of risk and returns. 2.14.3. Example by changing volatility to 20% 2.14.3.1. Data and assumptions: An investor with 100,000 to invest for 3 months in the call. When the stock is at 15. He can either buy the stock or invest in T-bills. This leads to symmetrical risk and return pattern, profit or loss dollar for dollar plus income from dividends. Pattern of risk and return Consider the pattern of risk and return with a purchase of 3 month call option with a strike price of 15. We selected at the money so that at expiration any appreciation in the underlying will be reflected in the exercise value of calls. To scale the position in calls to the investor Risk capital of 100,000, we begin by calculating the value of one contract. The stock price times the multiplier Valuation: Using option valuation procedure and assuming a volatility of 30%, the interest rate at 5%, the call price is 0. 96. Call’s underlying value = asset price × contract multiplier 15 = 15 × 1 Number of calls to buy: To determine the number of calls to buy, we divide the risk capital of 100,000 by call’s underlying value of 15: Contracts required = Risk capital: call’s underlying value 6,666 = 100,000: 15

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Assuming options can be purchased at theoretical value, the investor would have to spend to buy 6,666 contracts: Investment required to buy = call price × number × contract multiplier $4,599 = 0.69 × 6,666 × 1 The balance of the 100 000 Risk capital that could be invested in T-bills would be: Remaining capital in T-bills = total capital − investment required to buy calls 95,400.46 = 100,000 − 4,599 Strategy: Buying calls and investing balance in T-bills Buy calls, strike 15, stock initially at 15 at 0.69 Invest: 95,400.46 in 3 month TB 2.14.3.2. Compare buying calls (as an alternative to portfolio of stocks.) Here we compare buying calls as an alternative to portfolio of stocks. At maturity: After 3 months: If the asset price does not change: Initial cost of calls (4,599) Exercise value: 0 Loss from calls (4,599) Interest income on T-bills (5% on 95,400.46 for 3 months: 1,192.575) Total loss: (3,406.425) Return from investment is stocks: 0% Return using call options: If the investor uses options to take a position in the market, the position is not profitable unless the asset is high enough to give options sufficient exercise value to repay their initial cost. The upside break even point can be computed as follows. Cost per contract = (invt to buy calls − T-bill income)/(number of contracts × multiplier contract) 0.5110 = (4,599 − 1,192.575)/(6,666 × 1)

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Upside break-even point = net cost per contract + call’s strike price $15.511 = 0.5110 + 15 = The Upside break-even point is $15.511. Each call at this point would has an exercise value of 0,511. The aggregate exercise value of the position would be 3,406.42 just above the amount required to repay the cost less the offsetting treasury bill income: 0,511 × 6, 666 = 3,406.42 Exercise value of call = stock price upside-option strike price × contract multiplier 0,5111 = (15.511 − 15) × 1 Aggregate exercise value of position = exercise value of call × number of calls in position: 3, 406.42 = 0, 511 × 6, 666 If the stock rallies by 10%, from 15 to 16.5, each option will have an exercise value of 1.5. Exercise value of call = (stock − call’s strike) × contract multiplier 1.5 = (16.5 − 15) × 1 Aggregate Exercise value of position = Exercise value of call × number of calls 9,999 = 1.5 × 6,666 Including the interest income from T-bills, the investor’s total profit with the stock up 10% is 6,592.57. At maturity after 3 months: If the asset price up by 10%: Initial cost of calls (4,599) Exercise value: 1.5 × 6,666×1 9,999 Profit from calls 5,400

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Interest income on T-bills (5% on 95,400.46 for 3 months: 1,192.575) 1,192.575 Total profit: 6,592.57 Return from options and T-bills: 6,59% This profit is less than a direct investment in the stock: If the asset price up by 10%: Initial cost of stocks = 100,000 Sale of stocks = 110,000 Profit = 10,000 Return from stock: 10% – When the investor established his position in stocks he earned a profit of 10,000. – With options, he cannot break even until the stock rises to $15.511. – With stocks, his profit with the stock up by 10% was 10,000, with options only 6,592.57. Advantages: The advantages gained for these handicaps are that the option buyer is assured that his investment cannot do any worse than a known maximum loss equal to the costs of the options, minus any interest earned on the leftover cash invested in T-bills. Maximum loss in position = cost of calls − interest on T-bills 5,391.04 = 6,583.608 − 1, 192.575 This example is done to replicate a position in the stock. When the investor chooses to buy options, he is making a different kind of investment. The tradeoff is: in exchange of a known maximum loss (the option premium), he sacrifies performance. The tradeoff is equitable This is a managerial choice driven by the attitude toward the pattern of risk and returns. Buy calls, strike 15, stock initially at 15 at 0.69 Invest: 95,400.46 in 3 month TB 2.14.3.3. Leverage in buying call options (without selling the underlying) What happens if the investor devotes the risk capital to buy calls? With this capital, the investor can buy 145,000.

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Table 2.20. Results at the maturity date in three months of the strategy of buying calls and holding the underlying stock.

Investment required to buy calls: 0.69 × 6,666 × 1 Exercise value of calls Profit (loss) from calls Interest income on T-bills: 5% on 95,400.46 Total profit (loss)

Table 2.21.

Stock unchanged

Stock + 10%

Stock − 10%

(4,599)

(4,599)

(4,599)

0 0 1,192.575

9,999 (1.5 × 6,666) 5,400 1,192.575

0 0 1,192.575

(3,406.425)

6,592.57

(3,406.425)

Leverage in buying call options (without selling the underlying).

Asset level

unchanged

Stock + 10%

Stock − 10%

Initial cost of calls Exercise value of calls

(100,000) 0

(100,000) 0

Profit (loss) from calls Total profit and loss Return/initial calls

(100,000) (100,000) (100%)

(100,000) 217,500 1.5 × (145, 000calls) 117,500 117,500 117.5%

(100,000) (100,000) (100%)

To determine the number of calls to buy, we divide the risk capital of 100,000 by call’s underlying value of 15: Number of contracts = risk capital/(cost per contract × contract multiplier) 145,000 = 100,000/(0.69 × 1) If the asset declines or is unchanged, the calls expire worthless. The investor will lose 100,000. In the previous example, with the purchase of 6,666 contracts, the loss was 3,406. This reflects the smaller initial cost of the position and the interest income from T-bill. He can no longer afford not to buy. If the asset rises by 10%, from 15 to 16.5, the investor profit will be 117,500. With options, the investor has the benefit of an assured maximum loss- no matter how much the stock declines, the loss can never be greater than the price paid for options. Using the procedure developed before, the break even point is: Upside break-even point = net cost per contract + call’s strike price $15.69 = 0.69 + 15.

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Hedging: buying call options (selling the underlying).

Asset level

Unchanged Stock + 10% Stock − 10%

Total profit and loss from calls Initial stock position Sell: 0.55 × 15 × 145,000 Ending stock position Profit and loss from stock position

(100,000) 1,196,250

117,500 1,196,250

(100,000) 1,196,250

1,196,250 (000)

1,315,875 (119,625 )

1,076,625 119,625

(2,125)

19,625

Total profit and loss from calls (100,000) and stocks

The investor is assured of a maximum loss of 100,000 with the possibility of upside rewards that could potentially be greater than those of other strategies with other instruments. Summary A forward contract or a futures contract is an agreement between two parties to buy or sell a specific asset at a specified price at a given time in the future. Futures contracts are traded on an exchange and have standardized features. They are settled on a daily basis while the forward contracts are settled at the end of the contract. Besides, for most futures contracts, delivery is never actually made. Futures markets are used for hedging, speculation, and arbitrage motives. Futures and forward contracts are priced using the cost of carry model. Petroleum futures contracts (or other commodity contracts) can be used as specific hedges when they are associated with a planned cash transaction. The benefit to a company using petroleum products futures is to “lock in” profit margins and/or to protect inventory against falling prices. When spot prices are higher than long-term prices, any hedge using a future maturity will be equivalent to a forward sale below the spot price. This can lead to a loss if market prices do not fall at the same rate. When spot prices are lower than long-term prices, the producer can sell on the futures market at a higher price. So, he/she can fix his/her hedge or future sales at a better price than the spot market. Companies using the physical oil market, for example, (or other commodities) can hedge themselves against adverse price movements by taking an opposite position on the futures or the forward market. The potential loss in the physical market can be offset by an equivalent gain on the futures or the forward market. The futures market offers a facility for hedging price risks. Hedging

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price risk can be regarded as a trading operation allowing to transform a less acceptable risk into a more acceptable risk by engaging into an offsetting transaction in a similar commodity under roughly the same terms as the original transaction. Futures markets are often used for speculation. The main objective for speculators is to accomplish profit. Speculators take on the risk, which hedgers try to lay off. Speculators hold onto their positions for a very short time. They are sometimes in and out of the market several times a day. Arbitrage keeps prices in line since the arbitrageur buys the assets in one market and sells it in other market. When prices move out of line, the arbitrageur buys the under-priced asset in one market and sells the over-priced asset in another market. The option value before maturity is a function of five parameters: the price of the underlying asset, the strike price, the risk-free rate of interest, the movements in the underlying asset prices (volatility), and the time remaining to maturity. The value of any option underlying asset such as a stock, a default-free bond, or a futures or a forward contract is determined in a market place. In general, there are two prices quoted: the bid and the ask price. These two prices define a spread. In theory, there is only one price for each asset, but in practice, two prices are observed in financial markets. Buying a call gives the right to the option holder to pay the strike price K at maturity and to receive the value of an underlying asset ST . Buying a put gives the right to the option holder to sell at maturity, the underlying asset ST at the strike price K. The most frequent strategies consist in buying and selling calls and puts with or without using the underlying asset. Several other strategies can be implemented as a function of the risk-reward profile. An investor selling a straddle sells simultaneously a call and a put on the same underlying asset, with the same strike price and maturity date. The strangle is one of the simplest option strategies. An investor buying a strangle (long a strangle) buys simultaneously a call and a put on the same underlying asset with different strike prices K1 and K2 and at the same maturity date. The call with a strike price K1 and the put with a strike price K2 are both out-of-the money with K1 > K2 . A spread consists in buying an option and selling another option. A bull spread corresponds to a combination of options with the same underlying asset and the same maturity, but different strike prices. A bear spread corresponds to a combination of options with the same underlying asset and the same maturity, but different strike prices. When it is implemented

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with two calls, it is designed to profit from falling underlying asset prices. This strategy with calls corresponds to the short position to the bull spread. A box spread corresponds to a combination of a bull spreads with calls and a bear spread with puts. It is implemented using two spreads with two strike prices. A butterfly spread corresponds to a combination using three calls with three strike prices K1 < K2 < K3 . The calls have the same underlying asset and maturity date. A condor corresponds to a combination using four calls with four strike prices K1 < K2 < K3 < K4 . The calls have the same underlying asset and maturity date. A ratio spread is a strategy involving two or more related options in a given proportion. A trader can buy a call with a lower strike price and sell a higher number of calls with a higher strike price. Portfolio insurance is an investment-management technique that protects a portfolio from drops in value. This technique proposes some simple concepts allowing to insure a stock portfolio. An investor who buys a European call and sells a European put on the same underlying asset creates a position that exhibits the same payoff pattern as the underlying asset. A portfolio with a long European call and a short European put shows a profit (loss) pattern that can mimic the result of the underlying asset. This chapter develops some of the most frequent strategies used in the market place.

Questions 1. What are the main specific features of forward and futures markets? 2. What are the main pricing relationships for forward and futures contracts? 3. What are the main trading motives in futures markets? 4. Provide some definitions for hedging, speculation, and arbitrage. 5. What are the main bounds on option prices? 6. Describe the simple trading strategies for options and their underlying assets. 7. How can one implement a straddle? 8. How can one implement a strangle? 9. Describe option spreads in bull and bear strategies involving calls. 10. Describe option spreads in bull and bear strategies involving puts. 11. How can one implement butterfly strategies using put and call options? 12. How can one implement condor strategies using put and call options? 13. Describe ratio-spread strategies.

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14. How can one implement some combinations of options with bonds and stocks and portfolio insurance strategies? 15. How can one implement portfolio insurance strategies? 16. What are the basic synthetic positions? 17. How can one implement a conversion? 18. How can one implement a reverse conversion? 19. What is the main characteristic of a box spread?

CASE STUDY: COMPARISONS BETWEEN PUT AND CALL OPTIONS 1. Buying Puts and Selling Puts Naked We consider a strategy of buying puts using the same data as with calls for the underlying assets. The put price premium is 6% of the underlying asset price.

1.1. Buying puts Allows to cover a position in the underlying asset. The investor pays a premium of let’s say 6 % and hedges the exposure to the underlying asset’s movements. If the market goes down, the investor can earn money. P&L from buy put at Maturity "p" 30.00% 25.00% 20.00% 15.00% P&L from buy put at Maturity "p"

10.00% 5.00%

% 33 % 2. 67 11 % .6 7 20 % .6 7 29 % .6 7 38 % .6 7% -6 .

%

33 5.

-1

33

33

4.

3.

-3

-2

-5.00%

%

0.00%

-10.00% Fig. 2.1.

Buying a put.

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1.2. Selling puts The investor receives the premium of 6 %. He wins if the market goes up: maximum gain limited to the option premium received. The investor loses money if the market goes down. P&L from sell put at Maturity -p 10.00% 5.00% 0.00% -5.00%

1 7 13 19 25 31 37 43 49 55 61 67 73 P&L from sell put at Maturity -p

-10.00% -15.00% -20.00% -25.00% -30.00% Fig. 2.2.

Selling a put.

The graphic shows P&L from the Table for buying and selling puts. Table 2.1.

P&L at maturity for different price of the underlying.

Stock price at maturity −33.33% −32.33% −31.33% −30.33% −29.33% −28.33% −27.33% −26.33% −25.33% −24.33% −23.33% −22.33% −21.33% −20.33% −19.33% −18.33% −17.33%

10 10.15 10.3 10.45 10.6 10.75 10.9 11.05 11.2 11.35 11.5 11.65 11.8 11.95 12.1 12.25 12.4

P&L from buy put at maturity, p 27.33% 26.33% 25.33% 24.33% 23.33% 22.33% 21.33% 20.33% 19.33% 18.33% 17.33% 16.33% 15.33% 14.33% 13.33% 12.33% 11.33%

P&L from sell put at maturity, −p −27.33% −26.33% −25.33% −24.33% −23.33% −22.33% −21.33% −20.33% −19.33% −18.33% −17.33% −16.33% −15.33% −14.33% −13.33% −12.33% −11.33% (Continued)

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Table 2.1.

Stock price at maturity −16.33% −15.33% −14.33% −13.33% −12.33% −11.33% −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.67% 7.67% 8.67% 9.67% 10.67% 11.67% 12.67% 13.67% 14.67% 15.67% 16.67% 17.67% 18.67% 19.67% 20.67% 21.67% 22.67% 23.67% 24.67% 25.67% 26.67% 27.67%

12.55 12.7 12.85 13 13.15 13.3 13.45 13.6 13.75 13.9 14.05 14.2 14.35 14.5 14.65 14.8 14.95 15.1 15.25 15.4 15.55 15.7 15.85 16 16.15 16.3 16.45 16.6 16.75 16.9 17.05 17.2 17.35 17.5 17.65 17.8 17.95 18.1 18.25 18.4 18.55 18.7 18.85 19 19.15

(Continued ).

P&L from buy put at maturity, p 10.33% 9.33% 8.33% 7.33% 6.33% 5.33% 4.33% 3.33% 2.33% 1.33% 0.33% −0.67% −1.67% −2.67% −3.67% −4.67% −5.67% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00%

P&L from sell put at maturity, −p −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% (Continued)

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Stock price at maturity 28.67% 29.67% 30.67% 31.67% 32.67% 33.67% 34.67% 35.67% 36.67% 37.67% 38.67% 39.67% 40.67% 41.67% 42.67%

(Continued ).

P&L from buy put at maturity, p

P&L from sell put at maturity, −p

−6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00% −6.00%

19.3 19.45 19.6 19.75 19.9 20.05 20.2 20.35 20.5 20.65 20.8 20.95 21.1 21.25 21.4

131

6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00%

2. Buying and Selling Calls A strategy of buying and selling calls shows the following P&L on the same stocks. 2.1. Buying calls The investor pays a premium of 10 %. He wins if the market goes up. buy call 35.00% 30.00% 25.00% 20.00% 15.00% 10.00%

buy call

5.00%

Fig. 2.3.

Buying a call.

38.67%

32.67%

26.67%

20.67%

8.67%

14.67%

2.67%

-3.33%

-9.33%

-15.33%

-21.33%

-15.00%

-27.33%

-5.00% -10.00%

-33.33%

0.00%

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132

2.2. Selling a call The investor receives a premium of 10 %. He wins if the market goes down. He loses if the market goes up. sell call 0.15 0.10 0.05

38.67%

32.67%

26.67%

20.67%

14.67%

8.67%

2.67%

-3.33%

-9.33%

-15.33%

-21.33%

-27.33%

-0.10

-33.33%

0.00 -0.05

sell call

-0.15 -0.20 -0.25 -0.30 -0.35 Fig. 2.4.

Selling a call.

3. Strategy of Buying a Put and Hedge and Selling a Put and Hedge Table 2.2.

Buy a put and hedge and sell a put and hedge.

Stock price at maturity −33.33% −32.33% −31.33% −30.33% −29.33% −28.33% −27.33% −26.33% −25.33% −24.33% −23.33% −22.33% −21.33% −20.33% −19.33% −18.33% −17.33%

10 10.15 10.3 10.45 10.6 10.75 10.9 11.05 11.2 11.35 11.5 11.65 11.8 11.95 12.1 12.25 12.4

P&L from buy put at maturity, p −14.00% −13.40% −12.80% −12.20% −11.60% −11.00% −10.40% −9.80% −9.20% −8.60% −8.00% −7.40% −6.80% −6.20% −5.60% −5.00% −4.40%

P&L from sell put at maturity, −p 14.00% 13.40% 12.80% 12.20% 11.60% 11.00% 10.40% 9.80% 9.20% 8.60% 8.00% 7.40% 6.80% 6.20% 5.60% 5.00% 4.40% (Continued)

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Table 2.2.

Stock price at maturity −16.33% −15.33% −14.33% −13.33% −12.33% −11.33% −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.67% 7.67% 8.67% 9.67% 10.67% 11.67% 12.67% 13.67% 14.67% 15.67% 16.67% 17.67% 18.67% 19.67% 20.67% 21.67% 22.67% 23.67% 24.67% 25.67% 26.67%

12.55 12.7 12.85 13 13.15 13.3 13.45 13.6 13.75 13.9 14.05 14.2 14.35 14.5 14.65 14.8 14.95 15.1 15.25 15.4 15.55 15.7 15.85 16 16.15 16.3 16.45 16.6 16.75 16.9 17.05 17.2 17.35 17.5 17.65 17.8 17.95 18.1 18.25 18.4 18.55 18.7 18.85 19

(Continued ).

P&L from buy put at maturity, p −3.80% −3.20% −2.60% −2.00% −1.40% −0.80% −0.20% 0.40% 1.00% 1.60% 2.20% 2.80% 3.40% 4.00% 4.60% 5.20% 5.80% 5.73% 5.33% 4.93% 4.53% 4.13% 3.73% 3.33% 2.93% 2.53% 2.13% 1.73% 1.33% 0.93% 0.53% 0.13% −0.27% −0.67% −1.07% −1.47% −1.87% −2.27% −2.67% −3.07% −3.47% −3.87% −4.27% −4.67%

P&L from sell put at maturity, −p 3.80% 3.20% 2.60% 2.00% 1.40% 0.80% 0.20% −0.40% −1.00% −1.60% −2.20% −2.80% −3.40% −4.00% −4.60% −5.20% −5.80% −5.73% −5.33% −4.93% −4.53% −4.13% −3.73% −3.33% −2.93% −2.53% −2.13% −1.73% −1.33% −0.93% −0.53% −0.13% 0.27% 0.67% 1.07% 1.47% 1.87% 2.27% 2.67% 3.07% 3.47% 3.87% 4.27% 4.67% (Continued)

133

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134

Table 2.2.

Stock price at maturity 27.67% 28.67% 29.67% 30.67% 31.67% 32.67% 33.67% 34.67% 35.67% 36.67% 37.67% 38.67% 39.67% 40.67% 41.67% 42.67%

(Continued ).

P&L from buy put at maturity, p

P&L from sell put at maturity, −p

−5.07% −5.47% −5.87% −6.27% −6.67% −7.07% −7.47% −7.87% −8.27% −8.67% −9.07% −9.47% −9.87% −10.27% −10.67% −11.07%

19.15 19.3 19.45 19.6 19.75 19.9 20.05 20.2 20.35 20.5 20.65 20.8 20.95 21.1 21.25 21.4

5.07% 5.47% 5.87% 6.27% 6.67% 7.07% 7.47% 7.87% 8.27% 8.67% 9.07% 9.47% 9.87% 10.27% 10.67% 11.07%

3.1. Strategy of selling put and hedge: sell delta units of the underlying We win if the stock lies within a given range. We loose if the stock is outside that range on the right or on the left.

P&L , sell put, sell 0.5 Stock -p-0.5S 10.00% 5.00%

38.67%

30.67%

22.67%

14.67%

6.67%

-1.33%

-9.33%

-17.33%

-25.33%

-5.00%

-33.33%

0.00% P&L , sell put, sell 0.5 Stock -p-0.5S

-10.00% -15.00% Fig. 2.5.

Selling a put and sell delta units of the underlying with delta equal 0.5.

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135

3.2. Strategy of buy put and hedge: buy delta units of the underlying Buy put, buy 0.5S p=0.5s 15.00% 10.00% 5.00% Buy put, buy 0.5S p=0.5s

36.67%

29.67%

22.67%

8.67%

15.67%

1.67%

-5.33%

-12.33%

-19.33%

-26.33%

-5.00%

-33.33%

0.00%

-10.00%

Fig. 2.6.

Buy a put and buy delta units of the underlying.

The investor wins if the stock lies outside a given range. The investor loses if the stock is inside that range on the right or on the left. 4. Strategy of Buy Call, Sell Put, and Buy Call, Sell Put and Hedge Table 2.3.

Buy call, sell put, and buy call, sell put and hedge.

Stock price at maturity −33.33% −32.33% −31.33% −30.33% −29.33% −28.33% −27.33% −26.33% −25.33% −24.33% −23.33% −22.33% −21.33% −20.33%

10 10.15 10.3 10.45 10.6 10.75 10.9 11.05 11.2 11.35 11.5 11.65 11.8 11.95

buy call, sell put, c−p

buy call sell put and hedge, c − p = stocks

−37.33% −36.33% −35.33% −34.33% −33.33% −32.33% −31.33% −30.33% −29.33% −28.33% −27.33% −26.33% −25.33% −24.33%

−5.67% −5.62% −5.57% −5.52% −5.47% −5.42% −5.37% −5.32% −5.27% −5.22% −5.17% −5.12% −5.07% −5.02% (Continued)

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136

Table 2.3.

Stock price at maturity −19.33% −18.33% −17.33% −16.33% −15.33% −14.33% −13.33% −12.33% −11.33% −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.67% 7.67% 8.67% 9.67% 10.67% 11.67% 12.67% 13.67% 14.67% 15.67% 16.67% 17.67% 18.67% 19.67% 20.67% 21.67% 22.67% 23.67% 24.67%

12.1 12.25 12.4 12.55 12.7 12.85 13 13.15 13.3 13.45 13.6 13.75 13.9 14.05 14.2 14.35 14.5 14.65 14.8 14.95 15.1 15.25 15.4 15.55 15.7 15.85 16 16.15 16.3 16.45 16.6 16.75 16.9 17.05 17.2 17.35 17.5 17.65 17.8 17.95 18.1 18.25 18.4 18.55 18.7

(Continued ).



−23.33% −22.33% −21.33% −20.33% −19.33% −18.33% −17.33% −16.33% −15.33% −14.33% −13.33% −12.33% −11.33% −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.67% 7.67% 8.67% 9.67% 10.67% 11.67% 12.67% 13.67% 14.67% 15.67% 16.67% 17.67% 18.67% 19.67% 20.67%

−4.97% −4.92% −4.87% −4.82% −4.77% −4.72% −4.67% −4.62% −4.57% −4.52% −4.47% −4.42% −4.37% −4.32% −4.27% −4.22% −4.17% −4.12% −4.07% −4.02% −3.97% −3.92% −3.87% −3.82% −3.77% −3.72% −3.67% −3.62% −3.57% −3.52% −3.47% −3.42% −3.37% −3.32% −3.27% −3.22% −3.17% −3.12% −3.07% −3.02% −2.97% −2.92% −2.87% −2.82% −2.77% (Continued)

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Stock price at maturity 25.67% 26.67% 27.67% 28.67% 29.67% 30.67% 31.67% 32.67% 33.67% 34.67% 35.67% 36.67% 37.67% 38.67% 39.67% 40.67% 41.67% 42.67%

(Continued ).



21.67% 22.67% 23.67% 24.67% 25.67% 26.67% 27.67% 28.67% 29.67% 30.67% 31.67% 32.67% 33.67% 34.67% 35.67% 36.67% 37.67% 38.67%

−2.72% −2.67% −2.62% −2.57% −2.52% −2.47% −2.42% −2.37% −2.32% −2.27% −2.22% −2.17% −2.12% −2.07% −2.02% −1.97% −1.92% −1.87%

18.85 19 19.15 19.3 19.45 19.6 19.75 19.9 20.05 20.2 20.35 20.5 20.65 20.8 20.95 21.1 21.25 21.4

5. Strategy of Buy Call, Sell Put: Equivalent to Holding the Underlying buy call, sell put ,c-p 50.00% 40.00% 30.00% 20.00% 10.00% buy call, sell put, c-p

-10.00%

-3

-20.00%

3. 3 -2 3% 3. 33 -1 % 3. 33 % -3 .3 3% 6. 67 16 % .6 7 26 % .6 7 36 % .6 7%

0.00%

-30.00% -40.00% -50.00%

Fig. 2.7.

Buy a call and sell a put.

137

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138

6. Strategy of Buy Call, Sell Put and Hedge: Reduces Profits and Reduces Losses buy call sell put and hedge c-p=stcocs

36.67%

29.67%

22.67%

15.67%

8.67%

1.67%

-5.33%

-12.33%

-19.33%

-26.33%

-1.00%

-33.33%

0.00%

-2.00% buy call sell put and hedge c-p=stcocs

-3.00% -4.00% -5.00% -6.00% Fig. 2.8.

Buy a call and sell a put and hedge.

Table 2.4. Stock price at maturity −33.33% −32.33% −31.33% −30.33% −29.33% −28.33% −27.33% −26.33% −25.33% −24.33% −23.33% −22.33% −21.33% −20.33% −19.33% −18.33% −17.33% −16.33% −15.33% −14.33% −13.33% −12.33%

10 10.15 10.3 10.45 10.6 10.75 10.9 11.05 11.2 11.35 11.5 11.65 11.8 11.95 12.1 12.25 12.4 12.55 12.7 12.85 13 13.15

Sell call, buy put and hedge.

Sell call, buy put, −c + p 37.33% 36.33% 35.33% 34.33% 33.33% 32.33% 31.33% 30.33% 29.33% 28.33% 27.33% 26.33% 25.33% 24.33% 23.33% 22.33% 21.33% 20.33% 19.33% 18.33% 17.33% 16.33%

Sell call, buy put and hedge 5.67% 5.62% 5.57% 5.52% 5.47% 5.42% 5.37% 5.32% 5.27% 5.22% 5.17% 5.12% 5.07% 5.02% 4.97% 4.92% 4.87% 4.82% 4.77% 4.72% 4.67% 4.62% (Continued)

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Risk Management, Derivatives Markets and Trading Strategies Table 2.4. Stock price at maturity −11.33% −10.33% −9.33% −8.33% −7.33% −6.33% −5.33% −4.33% −3.33% −2.33% −1.33% −0.33% 0.67% 1.67% 2.67% 3.67% 4.67% 5.67% 6.67% 7.67% 8.67% 9.67% 10.67% 11.67% 12.67% 13.67% 14.67% 15.67% 16.67% 17.67% 18.67% 19.67% 20.67% 21.67% 22.67% 23.67% 24.67% 25.67% 26.67% 27.67% 28.67% 29.67% 30.67% 31.67% 32.67% 33.67% 34.67%

13.3 13.45 13.6 13.75 13.9 14.05 14.2 14.35 14.5 14.65 14.8 14.95 15.1 15.25 15.4 15.55 15.7 15.85 16 16.15 16.3 16.45 16.6 16.75 16.9 17.05 17.2 17.35 17.5 17.65 17.8 17.95 18.1 18.25 18.4 18.55 18.7 18.85 19 19.15 19.3 19.45 19.6 19.75 19.9 20.05 20.2

139

(Continued ).

Sell call, buy put, −c + p 15.33% 14.33% 13.33% 12.33% 11.33% 10.33% 9.33% 8.33% 7.33% 6.33% 5.33% 4.33% 3.33% 2.33% 1.33% 0.33% −0.67% −1.67% −2.67% −3.67% −4.67% −5.67% −6.67% −7.67% −8.67% −9.67% −10.67% −11.67% −12.67% −13.67% −14.67% −15.67% −16.67% −17.67% −18.67% −19.67% −20.67% −21.67% −22.67% −23.67% −24.67% −25.67% −26.67% −27.67% −28.67% −29.67% −30.67%

Sell call, buy put and hedge 4.57% 4.52% 4.47% 4.42% 4.37% 4.32% 4.27% 4.22% 4.17% 4.12% 4.07% 4.02% 3.97% 3.92% 3.87% 3.82% 3.77% 3.72% 3.67% 3.62% 3.57% 3.52% 3.47% 3.42% 3.37% 3.32% 3.27% 3.22% 3.17% 3.12% 3.07% 3.02% 2.97% 2.92% 2.87% 2.82% 2.77% 2.72% 2.67% 2.62% 2.57% 2.52% 2.47% 2.42% 2.37% 2.32% 2.27% (Continued)

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140

Table 2.4. Stock price at maturity 35.67% 36.67% 37.67% 38.67% 39.67% 40.67% 41.67% 42.67%

20.35 20.5 20.65 20.8 20.95 21.1 21.25 21.4

(Continued ).

Sell call, buy put, −c + p −31.67% −32.67% −33.67% −34.67% −35.67% −36.67% −37.67% −38.67%

Sell call, buy put and hedge 2.22% 2.17% 2.12% 2.07% 2.02% 1.97% 1.92% 1.87%

References French, K (1980). Stock returns and the weekend effect. Journal of Financial Economics, 8 (March), 55–69. Gibbons, MR and P Hess (1981). Day of the week effects and asset returns. Journal of Business, 54, 579–596. Hong, H and J Wang (2000). Trading and returns under periodic market closures. Journal of Finance, 55(1) (February), 297–354.

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Chapter 3 TRADING OPTIONS AND THEIR UNDERLYING ASSET: RISK MANAGEMENT IN DISCRETE TIME

Chapter Outline This chapter is organized as follows: 1. Section 3.1 develops the basic strategies using calls and puts. 2. Section 3.2 illustrates several combined strategies. 3. Section 3.3 explains the way traders use option pricing models to compute option prices and to estimate the market volatility. Introduction Using the definition of a standard or a plain vanilla option, it is evident that the higher the underlying asset price, the greater the call’s value. When the underlying asset price is much greater than the strike price, the current option value is nearly equal to the difference between the underlying asset price and the discounted value of the strike price. The discounted value of the strike price is given by the price of a pure discount bond, maturing at the same time as the option, with a face or nominal value equal to the strike price. Hence, if the maturity date is very near, the call’s value (put’s value) is nearly equal to the difference between the underlying asset price and the strike price or zero. If the maturity date is very far, then the call’s value is nearly equal to that of the underlying asset since the bond’s price will be very low. The call’s value can not be negative and can not exceed the underlying asset price. Options enable investors to customize cash-flow patterns. We present some of the most common used option strategies, 141

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which apply to options on a spot asset, to options on a futures contract and in general to options with any particular payoff. These strategies are illustrated with respect to the diagram payoffs or the expected return and risk trade-off of standard options. The understanding of option strategies is based on the use of synthetic positions. This chapter provides several illustrations of the main strategies provided in the previous chapter. In particular, we develop the basic strategies and synthetic positions: long or short the underlying asset, long a call, long a put, and short a put. Then, we present some combinations and more elaborated strategies as: long a straddle, short a straddle, long or short a strangle, long a tunnel, short a tunnel, long a call or put bull spread, long a call or a put bear spread, long or short a butterfly, long or short a condor, etc. Finally, we show how traders and market participants use option pricing models and estimate model parameters. We introduce the concepts of Greek letters or the sensitivities of the option price or position with respect to some parameters. 3.1. Basic Strategies and Synthetic Positions This section develops the main option strategies and synthetic positions. 3.1.1. Options and synthetic positions Synthetic positions can be constructed by options on spot assets, options on futures contracts, and their underlying assets. If we use the symbol 0 to denote a horizontal line, the symbol −1 for the slope under 0 and the symbol 1 for the slope above 0, then it is possible to represent the diagram pay offs of a long call by (0, 1), a short call by (0, −1), a long put by (−1, 0), and a short put by (1, 0). Adopting this notation for the basic option pay offs, it is possible to construct all the synthetic positions as well as most elaborated diagram strategies using this representation. We denote by: S, (F ): price of the underlying asset, which may be a spot asset, S (or a futures contract F ); K: strike price; C: call price and P : put price. We use the following symbols: 0,

: 1,

: −1.

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 143

The results of the basic strategies can be represented as follows: , Short a call: (0, −1): , Long a put: (−1, 0): Long a call: (0, 1): , Short a put: (1, 0): . Long the underlying asset: (1, 1): (−1, −1):

, Short the underlying asset:

.

The symbols (−1, 0, 1) refer to a downward movement, (−1), a flat position (0) or an upward movement (1). The risk-return trade-off of the basic strategies can be represented using the different symbols. Using the above notations, it is possible to construct the risk-reward trade-off of any option strategy. For example, long a call (0, 1) and short a put (1, 0) is equivalent to long the underlying asset (1, 1). Also, short a call (0, −1) and long a put (−1, 0) is equivalent to a short position in the underlying asset (−1, −1). We give the basic synthetic positions when the options have the same strike prices and maturity dates. Long a synthetic underlying asset = long a call + short a put. (1, 1) = (0, 1) + (1, 0) Short a synthetic underlying asset = short a call + long a put. (−1, −1) = (0, −1) + (−1, 0) Long a synthetic call = long the underlying asset + long a put. (0, 1) = (1, 1) + (−1, 0) Short a synthetic call = short the underlying asset + short a put. (0, −1) = (−1, −1) + (1.0) Long a synthetic put = short the underlying asset + long a call. (−1, 0) = (−1, 1) + (0, 1) Short a synthetic put = long the underlying asset + short a call. (1, 0) = (1, 1) + (0, −1) The knowledge of synthetic positions is necessary for market participants since it allows the implementation of hedged positions. Hedged

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positions are often implemented by traders and market makers who follow delta-neutral strategies. For example, when managing an option position, buying a call or a put with the same strike price are two equivalent strategies since when buying a call, the trader or the market maker hedges his transaction by the sale of the underlying asset and when buying a put, he hedges his transaction by the purchase of the underlying asset. Buying the call and selling the put is equivalent to a long put with the same strike price. This transaction enables the trader or market maker to make a direct sale of the put since a position in a long call and a short put is equivalent to a long position in the underlying asset.

3.1.2. Long or short the underlying asset The risk-return profile for a position which is long or short the underlying asset (for example a futures contract) shows unlimited profit or loss. If we put on a horizontal line the underlying asset price and on a vertical line the profit or loss, the payoff to a long or a short position in the underlying asset can be easily represented. If the asset price rises or falls by one point, the profit or loss will be of the same amount.

3.1.3. Long a call Expectations: The trader expects a rising market and (or) a high volatility until the maturity date. Definition: Buy a call, c with a strike price K. Specific features: The potential gain is not limited but the potential loss is limited to the option premium. Buying the call at 1.9, reveals the risk-reward profile as shown in Fig. 3.1 at expiration. If S = 111.9 at maturity; (110+1.9), the profit is zero. This is the breakeven point of the position. Beyond this level, the profit is not limited. The maximum loss or performance corresponds to 1.9 or 100% (Table 3.1). In Fig. 3.1, the break-even point is given by the sum of the strike price and the option premium.

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Buy a call 20 18

18.1

16 14

13.1

profit

12 10 8

8.1

6 4

3.1

2 0 -2

1.9-

1.9-

1.9-

1.9-

1.9-

-4 -18

-13

-8

-3

2

7

12

17

22

cours du support Fig. 3.1. Table 3.1. Long a call: S = 102, r = 5%, volatility = 20%, and T = 100 days. Type Break-even point Maximal loss Maximal gain

Point A

Value K+c c Not limited, if S > K

3.1.4. Short call Expectations: The trader expects a falling market and (or) a lower volatility until the maturity date. Definition: Sell a call, c with a strike price K. Specific features: The potential gain is limited to the perceived premium and the potential loss is not limited. The risk-reward trade-off is inverted when selling calls (Table 3.2). The results of the strategy are given in Fig. 3.2.

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146

Table 3.2. Short a call: S = 102, r = 5%, volatility = 20%, T = 100 days. Type

Point

Value

Break-even point Maximal loss Maximal gain

A

K +c Not limited Premium

4 2

1,9

1,9

1,9

1,9

1,9

0 -2 -3,1

profit

-4 -6 -8

-8,1

-10 -12 -13,1 -14 -16 -18

-18,1

-20 90

95

100

105

110

115

120

125

S

Fig. 3.2.

Short a call.

Short a call Strike price = 110 Premium = 1, 9 Profit for a multiple of 10: 19 Break-even point = 111.9 S 90 95 100 105 110 115 120 125 130

Variation (%) −12 −7 −2 3 8 13 18 23 27

Call 1, 9 1, 9 1, 9 1, 9 1, 9 −3, 1 −8, 1 −13, 1 −18, 1

Performance (%) 100 100 100 100 100 −165 −430 −696 −961

130

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In Fig. 3.2, the break-even point is given by the sum of the strike price and the option premium.

3.1.5. Long a put Expectations: The trader expects a falling market and (or) a higher volatility until the maturity date. Definition: Buy a put p with a strike price K Specific features: The potential gain is not limited and the potential loss is limited to the option premium. In Fig. 3.3, the break-even point is given by the algebraic sum of the strike price and the option premium (Table 3.3).

40 37,3 35 30 27,3 25

profit

20 17,3 15 10 7,3 5 0 -2,7

-2,7

-2,7

-2,7

-2,7

-5 60

70

80

90

100

110

S

Fig. 3.3.

Long a put.

120

130

140

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148

Long a put Strike price = 100 Prime = 2, 7 Profit for a multiple of 10:27 Break-even point = 97, 28

S

Variation (%)

60 70 80 90 100 110 120 130 140

−41 −31 −22 −12 −2 8 18 27 37

Put

Performance (%)

37, 3 27, 3 17, 3 7, 3 −2, 7 −2, 7 −2, 7 −2, 7 −2, 7

1371 1003 635 268 −100 −100 −100 −100 −100

Table 3.3. Long a Put: S = 102, r = 5%, volatility = 20%, and T = 100 days. Type

Point

Value

Break-even point Maximal loss Maximal gain

A

K −p Premium Not limited

3.1.6. Short a put Expectations: The trader expects a stable and (or) a rising market. Definition: Sell a put p with a strike price K. Specific features: The potential gain is limited to the option premium and the potential loss is unlimited (Fig. 3.4).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 149 5 2,7

2,7

2,7

2,7

2,7

0 -2,3

profit

-5 -7,3 -10 -12,3 -15 -17,3 -20 80

85

90

95

100

105

110

S Fig. 3.4.

Short a put.

Short a put Strike price Premium Profit Break-even point

S 80 85 90 95 100 105 110 115 120

Variation (%) −22 −17 −12 −7 −2 3 8 13 18

Put −17.3 −12.3 −7.3 −2.3 2.7 2.7 2.7 2.7 2.7

100 2.7 27 97.28

Performance (%) −635 −451 −268 −84 100 100 100 100 100

115

120

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Derivatives, Risk Management and Value Table 3.4. Short a put: S = 102, r = 5%, volatility = 20%, and T = 100 days. Type Break-even point Maximal loss Maximal Gain

Point A

Value K −p Not limited Premium

Figure 3.3 represents in a certain way the opposite of the risk-reward profile in Fig. 3.2. The profit is limited when the underlying asset price increases and the risk is unlimited when the underlying asset price is decreasing (Table 3.4).

3.2. Combined Strategies This section illustrates several combined strategies involving call and put options.

3.2.1. Long a straddle This strategy is perfect during the 2008 financial crisis. Expectations: The trader expects a high volatility until the maturity date. Definition: Buy a call, c and simultaneously buy a put, p on the same underlying (for the same maturity date and the same strike price). Specific features: The initial investment is important since the investor buys simultaneously the call and the put. • •

The loss is limited to the initial cost (c and p). The maximum potential gain is not limited when the market goes up or down.

Buying a straddle needs a simultaneous purchase of call and a put with the same strike price for the same maturity (Fig. 3.5). When the put is worthless, the call is deep-in-the money. When the call is worthless, the put is in-the-money (Table 3.5).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 151 20

15

13.7

13.7

profit

10

8.7

8.7

5

3.7

3.7

0

-1.3

-1.3 Call Put

-5

Straddle

-6.3 -10 80

85

90

95

100

105

110

115

120

S

Fig. 3.5.

Long a call

Long a put

Strategy

100 4.5 45 104.50

100 1.8 18 98.18

6.3 63 5

Strike price Prime Cost Break-even point

S 80 85 90 95 100 105 110 115 120

Variation (%) −22 −17 −12 −7 −2 3 8 13 18

Buying a Straddle.

Call

Put

Straddle

Performance (%)

−4.5 −4.5 −4.5 −4.5 −4.5 0.5 5.5 10.5 15.5

18.2 13.2 8.2 3.2 −1.8 −1.8 −1.8 −1.8 −1.8

13.7 8.7 3.7 −1.3 −6.3 −1.3 3.7 8.7 13.7

216 137 58 −21 −100 −21 58 137 216

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152

Table 3.5.

Long a straddle.

Type

Point

Value

Break-even point

A B C

S = K − (c + p) S = K + (c + p) (c + p) if S = K K − (c + p), if S tends toward 0 Unlimited, if S is beyond the limits

Maximal loss Maximal gain

Notes: The strike price is chosen according to the trader expectations about the future market direction. Simulation: Underlying asset S = 102, Interest rate r (%) = 5%, Volatility (%) = 20%, and Maturity (in days) = 50.

3.2.2. Short a straddle Expectations: The trader expects a low volatility until the maturity date. Definition: • •

Sell a call c and simultaneously. Sell a put, p on the same underlying for the same maturity date and the same strike price.

Specific features: • • •

The initial revenue is limited to the option premiums. The loss is not limited when the market goes up or down. The maximum potential gain is limited to the initial premium (c and p).

When the underlying asset price is expected to be in a specified interval at maturity, the trader can sell simultaneously a call and a put. The profit is limited to the premium received and the risk may be unlimited (Fig. 3.6). If the underlying asset is not expected to move much either side, the investor can sell the put and the call. The maximum profit at expiration is obtained when S is in a given interval (Table 3.6).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 153 10 Call

6.3

Put

5

Straddle

1.3

profit

0

1.3

-3.7

-5

-3.7

-8.7

-10

-8.7

-13.7

-15

-13.7

-20 80

85

90

95

100

105

110

115

120

S

Fig. 3.6.

Table 3.6. S 80 85 90 95 100 105 110 115 120

Variation (%) −22 −17 −12 −7 −2 3 8 13 18

Call 4.5 4.5 4.5 4.5 4.5 −0.5 −5.5 −10.5 −15.5

Short a straddle.

Shorting a straddle. Put

Straddle

Performance (%)

−18.2 −13.2 −8.2 −3.2 1.8 1.8 1.8 1.8 1.8

−13.7 −8.7 −3.7 1.3 6.3 1.3 −3.7 −8.7 −13.7

−216 −137 −58 21 100 21 −58 −137 −216

3.2.3. Long a strangle Expectations: The trader expects a high volatility during the options’ life. Definition: Buy a call with a strike price Kc and buy a put with a strike price Kp where the Kp < Kc .

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154

Specific features • • •

This strategy costs less than the straddle. The maximum loss is limited to the initial cost of (c + p). The net result is a profit only when the market movement is important (Fig. 3.7). In this example, the market must increase by 5%, (107, 49 − 102)/102 or decrease by 9%, (92, 41 − 102)/102 (Table 3.7).

Notes: The trader buys the 105 call and the 95 put. The theoretical prices of these options, respectively are 2.04 and 0.55, or a total of 2.58. The quantity is 10, and the total cost of the strategy is 25.8. 20 Call

17.4

Put Strangle

15

12.4

profit

10 7.4

7.4

5 2.4

2.4

0

-2.6

-5 85

90

95

100

105

110

115

120

S

Fig. 3.7.

Profit (per unit) of a long strangle strategy.

Table 3.7.

Long a strangle.

Type

Point

Value

Break-even point

A B

S = Kp − (c + p) S = Kc + (c + p) (c + p), if Kp < S < Kc

Maximal loss Maximal gain

A B

Kp − (c + p), if S tends toward 0 Unlimited, if S is higher

125

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 155 Table 3.8. S 85 90 95 100 105 110 115 120 125

Profit (per unit) of a long strangle strategy.

Variation (%) −17 −12 −7 −2 3 8 13 18 23

Call

Put

Strangle

Performance (%)

−2 −2 −2 −2 −2 3 8 13 18

9.5 4.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

7.4 2.4 −2.6 −2.6 −2.6 2.4 7.4 12.4 17.4

287 93 −100 −100 −100 93 287 480 674

The two break-even points are computed as follows: • 105 + (2.04 + 0.55) = 107.59 or a variation of 5.38%. • 95 − (2.04 + 0.55) = 92.41 or a variation of −9.40%. If the underlying asset price is between the two strike prices at expiration, the maximum loss is reduced to the initial cost of 25.8. The net result is a loss, if the underlying asset price is between the two breakeven points, 92.41 and 107.59. This loss is less than the initial cost. However, if the underlying asset price is above the break-even points, on either side, the trader benefits from the leverage effect (Table 3.8). For example, if the underlying asset price is 90 at expiration, or a variation of 12%, the net result is 93%. If the underlying asset goes up by 18% to attain a level of 120, the net profit of 12.4, compared to 2.58, represents a performance of 480%. Simulation: The parameters used in the simulation are: S = 102, r = 5%, Volatility = 20%, and Maturity = 50 days.

Strike price Premium Cost for 10 Break-even point

Long a call

Long a put

105 2.04 20.4 107.59

95 0.55 5.5 92.41

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156

3.2.4. Short a strangle Expectations: The trader expects a low volatility during the options’ life. Definition: Sell a call with a strike price Kc and Sell a put with a strike price Kp where the Kp < Kc (Fig. 3.8). Specific features: The maximum gain is limited to the initial premium of (c + p). The strategy can show a loss (Table 3.9). 5

2.6

0 -2.4

-2.4

-5 profit

• •

-7.4

-7.4

-10 -12.4 -15

Call Put

-17.4

Strangle

-20 85

90

95

100

105

110

115

120

125

S

Fig. 3.8.

Table 3.9. S 85 90 95 100 105 110 115 120 125

Variation (%) −17 −12 −7 −2 3 8 13 18 23

Call 2 2 2 2 2 −3 −8 −13 −18

Short a strangle.

Short a strangle. Put −9.5 −4.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Strangle −7.4 −2.4 2.6 2.6 2.6 −2.4 −7.4 −12.4 −17.4

Performance (%) −287 −93 100 100 100 −93 −287 −480 −674

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The reader can make the specific comments by comparing this strategy with the long strangle.

3.2.5. Long a tunnel Expectations: The trader expects a high volatility during the options’ life. Definition: Buy an out-of-the money call and sell out-of-the money put as in Table 3.10 (Fig. 3.9).

Table 3.10. 10 Strike price Premium Cost for 10 Break-even point

Long a tunnel.

Long a call

Short a put

570 22.3 223 592.32

550 15.3 153 534.68

7 −70 10

40 33,0

30 23,0

20 13,0

profit

10 3,0

0 -7,0

-7,0

-10

-7,0 -17,0

-20 -27,0

Call Out Put Out Tunnel

-30 -40 530

540

550

560

570

580

590

600

610

S Fig. 3.9. Long a tunnel (Buy an out-of-the money call and sell an out-of-the money put).

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158

Underlying Variation asset S (%)

Call outof-the money

Put outof-the money

Tunnel

Performance (%)

−22.3 −22.3 −22.3 −22.3 −22.3 −12.3 −2.3 7.7 17.7

−4.7 5.3 15.3 15.3 15.3 15.3 15.3 15.3 15.3

−27 −17 −7 −7 −7 3 13 23 33

−386 −243 −100 −100 −100 43 186 329 471

−5 −4 −2 0 2 4 5 7 9

530 540 550 560 570 580 590 600 610

Simulation: 3.2.6. Short a tunnel This is the opposite of the previous strategy (Fig. 3.10) (Table 3.11). 40 30 27,0

20 17,0

10

profit

7,0

7,0 7,0

0 -3,0

-10 -13,0

-20 -23,0

Call Out Put Out

-30

-33,0

Tunnel -40 530

540

550

560

570

580

590

600

610

S Fig. 3.10. Short a tunnel (Sell an out-of-the money call and buy an out-of-the money put).

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Underlying asset 530 540 550 560 570 580 590 600 610

Variation (%)

Call out

Put out

Tunnel

22.3 22.3 22.3 22.3 22.3 12.3 2.3 −7.7 −17.7

4.7 −5.3 −15.3 −15.3 −15.3 −15.3 −15.3 −15.3 −15.3

27 17 7 7 7 −3 −13 −23 −33

−5 −4 −2 0 2 4 5 7 9

Performance (%) 386 243 100 100 100 −43 −186 −329 −471

Table 3.11. Q = 10

Short a call

Long a put

570 22.3 223 592.32

550 15.3 153 534.68

Strike price Premium Cost Break-even point

7.0 −70 10

3.2.7. Long a call bull spread A strategy can be implemented by buying a call with a lower strike price and selling a call with a higher strike price (Fig. 3.11). If the underlying asset price is below the lower strike price at expiration, the maximum loss is limited to the difference between the two option premiums. If the underlying asset price is above the higher strike price at expiration, the lower strike price call is worth the intrinsic value. This strategy shows a limited profit (a loss) (Table 3.12). Long a Bull Spread with Calls for the following parameters: S = 102, r = 5%, volatility = 20%, T = 100 days.

3.2.8. Long a put bull spread Expectations: Buying a put spread is equivalent to buying the higher strike price put and selling the lower strike price put (Fig. 3.12).

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160 80

60

40

20 profit

11.1

0

-20

-8.9

Call In

-40

Call Out

Spread

-60 490

510

530

550

570

590

610

630

650

S

Fig. 3.11.

Buying a bull spread with calls.

Table 3.12. Bull spread with calls: S = 102, r = 5%, volatility = 20%, and T = 100 days. S 490 510 530 550 570 590 610 630 650

Variation (%) −14 −11 −7 −4 0 3 7 10 14

Call in −19.9 −19.9 −19.9 −19.9 −19.9 0.1 20.1 40.1 60.1

Call out 11 11 11 11 11 11 −9 −29 −49

Spread

Performance (%)

−8.9 −8.9 −8.9 −8.9 −8.9 11.1 11.1 11.1 11.1

−100 −100 −100 −100 −100 125 125 125 125

If the underlying asset is around the lower strike price at maturity, the higher strike price put is worth the intrinsic value and the lower strike price is worthless. The maximum profit is given by the difference between the two option premiums. The strategy is done with a debit (Table 3.13).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 161 80 60 40 11.0

profit

20 0 -20

-9.0

-40 Put Out

-60

Put In Spread

-80 -100 490

510

530

550

570

590

610

630

650

S

Fig. 3.12.

Table 3.13. S 490 510 530 550 570 590 610 630 650

Variation (%) −14 −11 −7 −4 0 3 7 10 14

Buying a bull spread with puts.

Buying a bull spread with puts.

Put out

Put in

66 46 26 6 −14 −14 −14 −14 −14

−75 −55 −35 −15 5 25 25 25 25

Spread −9 −9 −9 −9 −9 11 11 11 11

Performance (%) 82 82 82 82 82 −100 −100 −100 −100

The trader can sell the put spread by selling the higher strike price put and buying the lower strike price put. The strategy is done with a credit. 3.2.9. Long a call bear spread The reader can refer to the previous chapter for more details and make the necessary comments (Fig. 3.13) (Table 3.14).

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162 60 40 20

8.5

profit

0 - 11.5

-20 -40 Call In Call Out

-60

Spread

-80 490

510

530

550

570

590

610

630

650

S

Fig. 3.13.

Selling a call bear spread.

Table 3.14. S 490 510 530 550 570 590 610 630 650

Variation (%) −14 −11 −7 −4 0 4 7 11 14

Selling a call bear spread.

Call in

Call out

Spread

18.8 18.8 18.8 18.8 18.8 −1.2 −21.2 −41.2 −61.2

−10.3 −10.3 −10.3 −10.3 −10.3 −10.3 9.7 29.7 49.7

8.5 8.5 8.5 8.5 8.5 −11.5 −11.5 −11.5 −11.5

Performance (%) 100 100 100 100 100 −134 −134 −134 −134

3.2.10. Selling a put bear spread Refer Fig. 3.14 and Table 3.15. This is best understood from Fig. 3.14 and Table 3.15. (see below). 3.2.11. Long a butterfly Expectations: The reader can refer to the previous chapter for more details and make the necessary comments (Fig. 3.15) (Table 3.16).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 163 100 Put Out

80

Put In Spread

60 40 profit

20

8.7

0 -20

-113.

-40 -60 -80 490

510

530

550

570

590

610

630

650

S

Fig. 3.14.

Selling a put bear spread.

Table 3.15. S 490 510 530 550 570 590 610 630 650

Variation (%) −14 −11 −7 −4 0 4 7 11 14

Selling a put bear spread.

Put out

Put in

Spread

−65.1 −45.1 −25.1 −5.1 14.9 14.9 14.9 14.9 14.9

73.8 53.8 33.8 13.8 −6.2 −26.2 −26.2 −26.2 −26.2

8.7 8.7 8.7 8.7 8.7 −11.3 −11.3 −11.3 −11.3

Performance (%) −77 −77 −77 −77 −77 100 100 100 100

3.2.12. Short a butterfly Expectations: The reader can refer to the previous chapter for more details and make the necessary comments (Table 3.17). 3.2.13. Long a condor Expectations: The reader can refer to the previous chapter for more details and make the necessary comments (Fig. 3.16) (Table 3.18).

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164 20

10 5.5 0.5

profit

0

0.5

-4.5

-10

-4.5

-20 Call Call At Call In Butterfly

-30

-40 80

85

90

95

100

105

110

115

120

S

Fig. 3.15.

Table 3.16. S 80 85 90 95 100 105 110 115 120

Variation (%) −22 −17 −12 −7 −2 3 8 13 18

Long a butterfly.

Long a butterfly.

Call out

Call at

Call in

−12.7 −12.7 −12.7 −7.7 −2.7 2.3 7.3 12.3 17.3

9 9 9 9 9 −1 −11 −21 −31

−0.7 −0.7 −0.7 −0.7 −0.7 −0.7 −0.7 4.3 9.3

Butterfly −4.5 −4.5 −4.5 0.5 5.5 0.5 −4.5 −4.5 −4.5

Performance (%) −20 −20 −20 2 25 2 −20 −20 −20

3.2.14. Short a condor Expectations: The reader can refer to the previous chapter for more details and make the necessary comments (Fig. 3.17) (Table 3.19).

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Trading Options and Their Underlying Asset: Risk Management in Discrete Time 165 Table 3.17. S 80 85 90 95 100 105 110 115 120

Variation (%) −22 −17 −12 −7 −2 3 8 13 18

Call out

Short a butterfly.

Call at −9 −9 −9 −9 −9 1 11 21 31

12.7 12.7 12.7 7.7 2.7 −2.3 −7.3 −12.3 −17.3

Call in 0.7 0.7 0.7 0.7 0.7 0.7 0.7 −4.3 −9.3

Butterfly

Performance (%)

4.5 4.5 4.5 −0.5 −5.5 −0.5 4.5 4.5 4.5

20 20 20 −2 −25 −2 20 20 20

40 Call Call At Call At Call In Condor

30 20

profit

10

3.9

0 -10

-6.1

-6.1

-20 -30 -40 60

70

80

90

100

110

120

130

140

S

Fig. 3.16.

Table 3.18. S 60 70 80 90 100 110 120 130 140

Variation (%) −43 −33 −24 −14 −5 5 14 24 33

Long a condor.

Long a condor.

Call out

Call at

Call at

−16.5 −16.5 −16.5 −16.5 −6.5 3.5 13.5 23.5 33.5

8.2 8.2 8.2 8.2 8.2 −1.8 −11.8 −21.8 −31.8

2.9 2.9 2.9 2.9 2.9 2.9 −7.1 −17.1 −27.1

Call in −0.7 −0.7 −0.7 −0.7 −0.7 −0.7 −0.7 9.3 19.3

Condor −6.1 −6.1 −6.1 −6.1 3.9 3.9 −6.1 −6.1 −6.1

Performance (%) −22 −22 −22 −22 14 14 −22 −22 −22

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166 40 30 20

profit

10

6.1

6.1

0 -3.9

-10 -20

Call Out Call At Call At Call In Condor

-30 -40 60

70

80

90

100

110

120

130

140

S

Fig. 3.17.

Table 3.19. S 60 70 80 90 100 110 120 130 140

Variation (%) −43 −33 −24 −14 −5 5 14 24 33

Call out 16.5 16.5 16.5 16.5 6.5 −3.5 −13.5 −23.5 −33.5

Short a condor.

Short a condor.

Call at

Call at

−8.2 −8.2 −8.2 −8.2 −8.2 1.8 11.8 21.8 31.8

−2.9 −2.9 −2.9 −2.9 −2.9 −2.9 7.1 17.1 27.1

Call in

Condor

0.7 0.7 0.7 0.7 0.7 0.7 0.7 −9.3 −19.3

6.1 6.1 6.1 6.1 −3.9 −3.9 6.1 6.1 6.1

Performance (%) 22 22 22 22 −14 −14 22 22 22

3.3. How Traders Use Option Pricing Models: Parameter Estimation The option price depends on the underlying asset price S, the strike price K, the interest rate r, the time to maturity, T , the volatility σ, and the dividend payouts. The option maturity corresponds to the number of days until expiration. It is often given in a fraction of a year or in days. The dividends must be known or estimated before using an option pricing model.

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3.3.1. Estimation of model parameters We explain briefly how the model parameters are obtained when trading options. The interest rates must be estimated during the option’s life. Volatility is a measure of risk and is a fundamental element in the computation of the option premium. It can be computed using historical prices of the underlying asset. This refers to the historical volatility. It can also be calculated using the observed option prices in the market place and an option pricing model. This refers to the implicit volatility. The effect of the parameters on the option value can be appreciated using Table 3.20. There are at least two ways to estimate the volatility: the historical volatility and the implied volatility. 3.3.1.1. Historical volatility The return on an asset can be computed using three measures for the variations of its price. The first direct measure of return is given by the difference between the asset prices at two dates as: (1)

Ri,t = Si,t+1 − Si,t The second measure allows the computation of a compound return as: (2)

Ri,t = Log(Si,t+1 ) − Log(Si,t ) The third method uses the relative variations in the underlying asset prices. (3)

Ri,t =

Si,t+1 − Si,t Si,t

Table 3.20. The effect of the parameters on option prices. Parameters Underlying asset, S Strike price, K Dividends Interest rates Maturity Volatility

Call

Put

+ − − + + +

− + + − + +

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168

Historical volatility can be estimated using historical returns as follows: Nj ¯ 2 2 τ =1 (Rt−τ − R) (t, Nj ) = σ ˆH Nj − 1 where the mean return is given by, Nj ¯= 1 Rt−τ R Nj τ =1

where Nj corresponds to the number of observations. In general, historical volatilities are computed using a period of nearly 20 days for short-term options and a period of 250 trading days for long-term options. Example: Consider the computation of the historical volatility for the CAC40 (Table 3.21). Computations are done on 11 July 2003. The volatility is computed for a 20-day period using closing prices for the CAC40 over the past 21 days (between lines −1 and −21). The logarithmic formula giving the compound return is used in the computation of the return. For example, the relative variation of prices on 7 July 2003 is computed using the prices of the 4 and 7 July as: 0.45% = Ln(2947,66) − Ln(2934,48) where Ln(.) corresponds to the Neperian logarithm. Other methods can be used in the computation of historical volatilities. These methods use opening prices, closing prices, opening and closing prices, high and low prices during a trading period, etc. The estimated volatility using closing prices is given by the following formula: (f )

σ ˆ02 (t) = (St

(f )

− St−1 )2

where (f ) refers to closing prices. The estimated volatility using opening prices is given by: (o)

(o)

σ ˆ02 (t) = (St+1 − St )2 where (o) refers to opening prices. The estimated volatility using opening and closing prices is given by: (o)

σ ˆ12 (t) ≡

(St

(f )

(f ) (o) − St−1 )2 (S − St )2 + t 2f 2(1 − f )

0 K[1 − e−r(ti+1−ti) ] with (Di +Ri ) ≥ 0. Generally, a put is exercised when it is in the money and the call price is less than the cash amount. Formally, the put is exercised at time ti if: (Di + Ri ) < K[1 − e−r(ti+1−ti) ] with (Di + Ri ) ≥ 0.

4.2.3. The model When there is just one ex-cash income date τ , during the option’s life and k ∆ t ≤ τ ≤ (k + 1)∆t, then at time x, the value of the random component S is: S ∗ (x) = S(x)

when x > τ

S ∗ (x) = S(x) − (Di + Ri )e−r(τ −x)

when x ≤ τ

Assume σ∗ is the constant volatility of S ∗ . Using the parameters p, u, and d, at time t + i∆t, the nodes on the tree define the stock prices: If i∆t < τ : S ∗ (t)uj di−j + (Di + Ri )e−r(τ −i∆t) ∗

j i−j

If i∆t ≥ τ : S (t)u d

j = 0, 1, . . . , i j = 0, 1, . . . , i

4.2.4. Simulations for a small number of periods Example 1. Applications of the CRR model for five periods with dividends Consider the following data for the valuation of European and American call and put options: S = 110, K = 115, r = 10%, σ = 40%, N = 0.5, t = 5 months, ∆t = 1 month. date of dividend: 105 days, dividend amount: D = 10.

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Option Pricing: The Discrete-Time Approach for Stock Options

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239

The European call price Using the above data, the dynamics of the underlying asset are computed using:

u = eσ

√

∆t

,

u = 1.1224,

√ u = e0.4 1/2 , d=

or

1 = 0.8909, u

p=

er∆t − d = 0.5073 u−d

S ∗ = S − D = 110 − 10 = 100 S0,0 = S ∗ u0 d0 + De−r(105/365) = 109.7164 S1,1 = S ∗ u1 d0 + De−r((105/365)−(1/12)) = 100(1.1224) + 10De−r((105/365)−(1/12)) = 122.0377 S1,0 = S ∗ u0 d1 + De−r((105/365)−(1/12)) = 98.8925 S2,2 = S ∗ u2 d0 + De−r((105/365)−(2/12)) = 135.8579 S2,1 = S ∗ u1 d1 + De−r((105/365)−(2/12)) = 109.8797 S2,0 = S ∗ u0 d2 + De−r((105/365)−(2/12)) = 89.2586 S3,3 = S ∗ u3 d0 + De−r((105/365)−(3/12)) = 151.3603 S3,2 = S ∗ u2 d1 + De−r((105/365)−(3/12)) = 122.2024 S3,1 = S ∗ u1 d2 + De−r((105/365)−(3/12)) = 99.0572 S3,0 = S ∗ u0 d3 + De−r((105/365)−(3/12)) = 80.6848 S4,4 = S ∗ u4 d0 = 158.7050, S4,2 = S ∗ u2 d2 = 100.00, S4,0 = S ∗ u0 d4 = 63.0100, S5,4 = S ∗ u4 d1 = 141.3979, S5,2 = S ∗ u2 d3 = 89.0948, S5,0 = S ∗ u0 d5 = 56.1386

S4,3 = S ∗ u3 d1 = 125.782 S4,1 = S ∗ u1 d3 = 79.3788 S5,5 = S ∗ u5 d0 = 178.1305 S5,3 = S ∗ u3 d2 = 112.2400 S5,1 = S ∗ u1 d4 = 70.7224

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The dynamics of the underlying asset are represented in the following way. S 5,5 S 4,4 S 5,4

S 3,3

S 4,3 S 2,2 S 3,2

S 1,1 S 2,1

S 00

S 5,3 S 4,2 S 5,2

S 3,1

S 1,0

S 4,1 S 2,0

S 5,1 S 3,0 S 4,0 S 5,0

The option’s maturity value is computed as: C5,5 = max[0; S5,5 − K] = 63.1305, C5,4 = max[0; S5,4 − K] = 26.3979 C5,3 = max[0; S5,3 − K] = 0, C5,2 = max[0; S5,2 − K] = 0 C5,1 = max[0; S5,1 − K] = 0, C5,0 = max[0; S5,0 − K] = 0 The American call option price is computed as: p · C5,5 + q · C5,4 C4,4 = max ; V I er∆t = max[44.6586; S4,4 − K] = max[44.6586; 43.7050]

C4,3

= 44.6586 where VI stands for the option intrinsic value. p · C5,4 + q · C5,3 = max ; S4,3 − K er∆t = max[13.2805; 10.9782] = 13.2805

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C4,2 = C4,1 = C4,0 = C3,3 =

C3,2 C3,1 C3,0

p · C5,3 + q · C5,2 max ; er∆t p · C5,2 + q · C5,1 max ; er∆t p · C5,1 + q · C5,0 max ; er∆t p · C4,4 + q · C4,3 max ; er∆t

241

max[0; S4,2 − K] = 0 max[0; S4,1 − K] = 0 max[0; S4,0 − K] = 0 S3,3 − K

= max[28.9563; 36.3603] = 36.3603 p · C4,3 + q · C4,2 = max ; S3,2 − K = max[6.6813; 7.2024] = 7.2024 er∆t p · C4,2 + q · C4,1 = max ; S3,1 − K = 0, er∆t p · C4,1 + q · C4,0 = max ; S − K =0 3,0 er∆t

American call option prices are computed as follows: 63.105

44.6586

26.3979

36.3603 21.8117

13.2805 7.2024

12.7437

0

3.6235 7.3019

0 0

0

1.8229

0

0 0

0 0 0

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C2,2

C2,1

C2,0 C1,1

C1,0

C0,0

p · C3,3 + q · C3,2 = max ; S2,2 − K er∆t

= max[21.8117; 20.8579] = 21.8117 p · C3,2 + q · C3,1 = max ; max[0; S2,1 − K] er∆t = max[3.6235; 0] = 3.6235 p · C3,1 + q · C3,0 = max ; max[0; S2,0 − K] = 0 er∆t p · C2,2 + q · C2,1 = max ; max[0; S − K] 1,1 er∆t = max[12.7437; 7.0377] = 12.7437 p · C2,1 + q · C2,0 = max ; max[0; S − K] 1,0 er∆t = max[1.8229; 0] = 1.8229 p · C1,1 + q · C1,0 = max ; max[0; S − K] 0,0 er∆t = max[7.3019; 0] = 7.3019

The American put price Using the above data, the American put price is computed as follows at different nodes. P5,5 = max[0; K − S5,5 ] = 0,

P5,4 = max[0; K − S5,4 ] = 0,

P5,3 = max[0; K − S5,3 ] = 2.76 P5,2 = max[0; K − S5,2 ] = 25.9052,

P5,1 = max[0; K − S5,1 ] = 44.2776

P5,0 = max[0; K − S5,0 ] = 58.8614 Before maturity, the option price is computed as: p · P5,5 + q · P5,4 ; max[0; K − S4,4 ] = 0 P4,4 = max er∆t p · P5,4 + q · P5,3 P4,3 = max ; max[0; K − S4,3 ] er∆t = max[1.3486; 0] = 1.3486

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P4,2 P4,1

P4,0

243

p · P5,3 + q · P5,2 = max ; max[0; K − S4,2 ] = max[14.0461; 0] = 15 er∆t p · P5,2 + q · P5,1 = max ; max[0; K − S ] 4,1 er∆t = max[34.6672; 35.6212] = 35.6212 p · P5,1 + q · P5,0 = max ; max[0; K − S ] 4,0 er∆t = max[51.0360; 51.9900] = 51.9900

Option prices are reported in the following figure. 0 0 0

0.6589

1.3486 4.2441 8.0076

10.0605

15

16.2201

17.0991

2.76

25.9052

24.9513

24.6367

35.6212 33.7211

44.2776 43.3236 51.9900

58.8614

P3,3

P3,2

p · P4,4 + q · P4,3 = max ; max[0; K − S3,3 ] er∆t = max[0.6589; 0] = 0.6589 p · P4,3 + q · P4,2 = max ; max[0; K − S3,2 ] er∆t = max[8.0076; 0] = 8.0076

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P3,1

P3,0

P2,2

P2,1

P2,0

P1,1

P1,0

P0,0

p · P4,2 + q · P4,1 = max ; max[0; K − S3,1 ] er∆t = max[24.9513; 15.9428] = 24.9513 p.P4,1 + q · P4,0 = max ; max[0; K − S3,0 ] er∆t = max[43.3236; 34.3152] = 43.3236 p · P3,3 + q · P3,2 = max ; max[0; K − S2,2 ] er∆t = max[4.2441; 0] = 4.2441 p · P3,2 + q · P3,1 = max ; max[0; K − S2,1 ] er∆t = max[16.2201; 5.1203] = 16.2201 p · P3,1 + q · P3,0 = max ; max[0; K − S2,0 ] er∆t = max[33.7211; 25.7414] = 33.7211 p · P2,2 + q · P2,1 = max ; max[0; K − S1,1 ] er∆t = max[10.0605; 0] = 10.0605 p · P2,1 + q · P2,0 = max ; max[0; K − S ] 1,0 er∆t = max[24.6367; 16.1075] = 24.6367 p · P1,1 + q · P1,0 = max ; max[0; K − S ] 0,0 er∆t = max[17.0991; 5.2836] = 17.0991

Example 2. Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 100, interest rate = 0.1, volatility = 0.4, T = 5 months, N = 5, dividend = 10, and dividend date = 105. In this case, we have: p = 0.5073,

d = 0.8909,

and u = 1.1224.

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Dynamics of the underlying asset for five periods

110

122.0378 98.8925

135.8581 109.8797 89.2584

151.3606 122.2025 99.0571 80.6846

158.7055 125.9784 100 79.3787 63.0098

178.1312 141.3982 112.2401 89.0947 70.7222 56.1384

The valuation of European put options

8.6380

4.2282 13.3256

0

1.2720 7.3442 19.7110

2.6033 12.3506 27.6249

0 0 5.3282 19.7914 36.1603

0 0 0 10.9053 29.2778 43.8616

The valuation of American put options

8.8801

4.3250 13.7214

0

1.2720 7.5423 20.3171

2.6033 12.7561 28.4479

0 0 5.3282 20.6213 36.9902

0 0 0 10.9053 29.2778 43.8616

The valuation of European call options

12.7191

19.7467 5.6987

29.7194 9.8132 1.5587

43.0511 16.4963 3.0982 0

59.5354 26.8082 6.1581 0 0

78.1312 41.3982 12.2401 0 0 0

245

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The valuation of American call options

15.8944

24.6554 7.1430

36.6880 12.6840 1.5587

51.3606 22.2025 3.0982 0

59.5354 26.8082 6.1581 0 0

78.1312 41.3982 12.2401 0 0 0

4.2.5. Simulations in the presence of two dividend dates We consider the valuation of a stock option using the binomial model. The option is priced as of 15/06/2002. The maturity date is 15/06/2004. The following strike prices are used: 22, 23, 24, 25, 26, 27, 28, 29, 30. The following dates and amounts of dividends are available: For 2003, 0.35 and for 2004, 0.35. We used a historical simulation to estimate the volatility parameter. Dividends are distributed at the end of May each year. The interest rate is 5%. The annualized volatility is between 45% and 50%. In this analysis, a dividend rate is used by dividing the dividend amount by the initial underlying asset price. The annualized volatility is 45% (Tables 4.1–4.4). We consider the same valuation problem, except that the volatility used is 50%.

4.2.6. Simulations for different periods and several dividends: The general case Tables 4.5 and 4.6 present the simulation results of the model proposed for American long-term call and put values by taking into account the magnitude of cash distributions and their timing. Simulations are run as of 11/05/2002 for a stock price of 423. Each share entitles the holder on 05/06/2002, a net dividend of 12. The following dates and amounts for the cash distributions are retained: 24/05/2002: −3.08, 24/06/2002: −3.08, 24/07/2002: −2.99, 24/08/2002: −2.98, 24/09/2002: −2.99, 24/10/2002: −2.82, 24/11/2002: −2.82, 24/12/2002: −2.82.

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Option Pricing: The Discrete-Time Approach for Stock Options Table 4.1. Simulations in the presence of two dividends using the binomial model. Call price K K K K K K K K K

= = = = = = = = =

22 23 24 25 26 27 28 29 30

S = 18

S = 19

S = 20

S = 21

3.49 3.25 3.03 2.82 2.60 2.38 2.19 2.06 1.93

4.09 3.76 3.50 3.29 3.07 2.86 2.64 2.42 2.26

4.71 4.38 4.05 3.76 3.55 3.33 3.11 2.89 2.68

5.33 5.00 4.67 4.33 4.03 3.80 3.58 3.37 3.15

Table 4.2. Simulations in the presence of two dividends using the binomial model. Call price K K K K K K K K K

= 22 = 23 = 24 = 25 = 26 = 27 = 28 = 29 = 30

S = 22

S = 25

5.95 5.61 5.28 4.95 4.62 4.29 4.06 3.84 3.62

8.13 7.67 7.21 6.80 6.47 6.14 5.81 5.47 5.14

S = 28 S = 30 10.38 9.90 9.44 8.98 8.52 8.06 7.66 7.33 6.99

12.05 11.46 10.94 10.47 10.01 9.55 9.09 8.63 8.23

Table 4.3. Simulations in the presence of two dividends using the binomial model. Call price K K K K K K K K K

= = = = = = = = =

22 23 24 25 26 27 28 29 30

S = 18

S = 19

S = 20

S = 21

3.99 3.71 3.51 3.30 3.10 2.90 2.70 2.49 2.36

4.61 4.30 4.00 3.78 3.58 3.38 3.18 2.97 2.77

5.23 4.92 4.60 4.29 4.06 3.86 3.66 3.45 3.25

5.86 5.54 5.23 4.92 4.60 4.34 4.14 3.93 3.73

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Derivatives, Risk Management and Value Table 4.4. Simulations in the presence of two dividends using the binomial model.

K K K K K K K K K

= 22 = 23 = 24 = 25 = 26 = 27 = 28 = 29 = 30

S = 22

S = 25

6.48 6.17 5.85 5.54 5.23 4.90 4.60 4.42 4.21

8.68 8.24 7.80 7.41 7.10 6.79 6.47 6.16 5.84

S = 28 S = 30 10.94 10.50 10.06 9.61 9.17 8.73 8.34 8.03 7.72

12.56 12.01 11.56 11.12 10.68 10.23 9.79 9.35 8.96

These amounts correspond to interest rates of 8.75% for options maturing in June, 8.5% for options maturing in September, and 8% for the maturity date of December. For example, when the stock price equals 423, the interest rate is 8.75%, the cash amount is 3.08. Cox et al. (1979) indicates the prices obtained from a modified version of the CRR model. The following dates and amounts for the cash distributions are retained: 24/05/2002: −3.08, 24/06/2002: −3.08, 24/07/2002: −2.99, 24/08/2002: −2.98, 24/09/2002: −2.99, 24/10/2002: −2.82, 24/11/2002: −2.82, 24/12/2002: −2.82. Table 4.5 shows the American call prices for different strike prices varying from 360 to 460. The volatility parameter takes two values 24.4% and 30.4%. Table 4.6 presents the American put prices for the same parameters as in Table 4.5. The number of iterations used for the CRR Table 4.5. Simulations of the American long-term equity call values: S = 423, K = 360 to 460, r = 0.08, D = 12, T = 233 days, number of iterations: 233, and dividend date: 25 days. CRR K

σ = 0.244

σ = 0.304

360 380 400 420 440 460

86.11 70.88 57.22 45.29 35.14 26.82

94.01 80.70 68.58 57.91 48.50 40.29

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Table 4.6. Simulations of the American long-term equity option values. CRR K

σ = 0.244

σ = 0.304

360 380 400 420 440 460

5.52 9.62 15.50 23.42 33.49 45.77

13.71 19.74 27.12 36.10 46.51 58.33

model corresponds to the number of days to maturity (233). The dividend is paid in 25 days. Summary Cox et al. (1979) proposed the first discrete-time model for the pricing of stock options. This binomial model is used for the valuation of options on different underlying assets. Questions 1. Describe the Cox et al. (1979) model for equity options for one period. 2. Describe the Cox et al. (1979) model for equity options for several periods. 3. How can we implement a hedging strategy in this context? 4. What are the valuation parameters in the lattice approach for stock prices? 5. How is an option priced in the lattice approach for stock prices? 6. What modifications are necessary to the standard lattice approach to apply it to American options? 7. What are the effects of cash distributions on the stock price? Appendix: The Lattice Approach The basic lattice approach suggested by CRR considers the situation where there is only one state variable: the price of a non-dividend paying stock. The time to maturity of the option is divided into N equal intervals of length ∆t during which the stock price moves from its initial value S to

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one of its two new values Su and Sd with probabilities p and (1 − p). When u = 1/d, it can be shown that: p=

a−d , u−d

u = eσ

√

∆t

,

d = e−σ

√

∆t

,

a = er∆t .

The nature of the lattice of stock prices is completely specified and the nodes correspond to Suj di−j for j = 0, 1, . . . , i. The option is evaluated by starting at time T and working backward. Let us denote by Fi,j , the option value at time t + i∆t when the stock price is Suj di−j . At time t + i∆t, the option holder can choose to exercise the option and receives the amount by which K (or S) exceeds the current stock price (or K) or wait. The American call is given by: Fi,j = max[Suj d i−j − K, e−r∆t(pFi+1,j+1 + (1 − p)Fi+1,j )] The American put is given by: Fi,j = max[K − Suj di−j , e−r∆t(pFi+1,j+1 + (1 − p)Fi+1,j )] The extension of the lattice approach to the valuation of American options on stocks paying a known cash income is as follows. When there is only one cash income at date, τ , between k∆t and (k + 1)∆t, it is possible to design trees where the number of nodes at time ∆t is always (i + 1). The analysis which parallels that in Hull (2000) and Briys et al. (1998) can be simplified by assuming that the implicit spot stock price has two components: a part which is stochastic and a part which is the present value of all future cash payments during the option’s life.

Exercises Example 1 Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 100, interest rate = 0.05, volatility = 0.2, T = 5 months, and N = 5. In this case, we have: p = 0.5217, d = 0.9439, and u = 1.0594.

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Dynamics of the underlying asset for five periods

100

105.9434 94.3900

112.2401 100

118.9110 105.9434

89.0947

94.3900 84.0965

133.4658 125.9784 112.2401 100 89.0947 79.3787

118.9110 105.9434 94.3900 84.0965 74.9256

Valuation of an European put option

4.3771

2.0783 6.9229

0.6062 3.7021 10.4965

0 1.2727 6.3844 15.0736

0 0 2.6720 10.4895 20.2055

0 0 0 5.6100 15.9035 25.0744

Valuation of an American put option

4.5368

2.1232 7.2092

0.6062 3.7964 10.9947

0 1.2727 6.5824 15.9035

0 0 2.6720 10.9053 20.6213

0 0 0 5.6100 15.9035 25.0744

Valuation of an European call option

6.4389

9.6745 2.9658

14.0885 4.9443 0.8335

19.7409 8.0460 1.6043 0

26.3942 12.6559 3.0878 0 0

33.4658 18.9110 5.9434 0 0 0

251

September 10, 2009 14:41

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Valuation of an American call option

6.4389

9.6745 2.9658

14.0885 4.9443 0.8335

19.7409 8.0460 1.6043 0

26.3942 12.6559 3.0878 0

33.4658 18.9110 5.9434 0 0

0

0

Example 2 Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 100, interest rate = 0.05, volatility = 0.2, T = 5 months, N = 5, dividend = 10, and dividend date: in 105 days. In this case, we have: p = 0.5217, d = 0.9439, u = 1.0594. Dynamics of the underlying asset for five periods

110

115.8418 104.2884

122.1798 109.9397 99.0344

128.8922 115.9246 104.3712 94.0777

133.4658 125.9784 112.2401 100 89.0947 79.3787

118.9110 105.9434 94.3900 84.0965 74.9256

Valuation of European put options

4.3771

2.0783 6.9229

0.6062 3.7021 10.4965

0 1.2727 6.3844 15.0736

0 0 2.6720 10.4895 20.2055

0 0 0 5.6100 15.9035 25.0744

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Valuation of American put options

4.4919

2.1232 7.1149

0.6062 3.7964 10.7967

0 1.2727 6.5824 15.4877

0 0 2.6720 10.9053 20.6213

0 0 0 5.6100 15.9035 25.0744

Valuation of European call options

6.4389

9.6745 2.9658

14.0885 4.9443 0.8335

19.7409 8.0460 1.6043 0

26.3942 12.6559 3.0878 0 0

33.4658 18.9110 5.9434 0 0 0

Valuation of American call options

11.7393

16.6716 6.4618

22.5956 10.3555 2.2710

28.8922 15.9246 4.3712 0

26.3942 12.6559 3.0878 0 0

33.4658 18.9110 5.9434 0 0 0

253

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Appendix Simulations: At the money call and put options using the Black–Scholes Model. Stock price (S) Strike price (K) Interest rate (r) Maturity (t) Volatility 0,05 0,25 N(d1) DELTA N(d2) 0,175000 0,075000 0,569460 0,529893 Put Price Call Price C−P S−Kexp(−rt) 3,3728 4,61499713 1,242220 1,242220

d1

100

d2

100

20%

At the money call (Out of the Money) put options using the Black–Scholes Model. Stock price (S) Strike price (K) Interest rate (r) Maturity (t) Volatility d1

120

d2

100

N(d1)

1,998216 1,898216 Put Price Call Price C−P 0,1060

21,34818885

0,05

0,25 N(d2) 0,977153 0,971166 S−Kexp(−rt) 21,242220 21,242220

20%

Out of the money call (In the Money) put options using the Black–Scholes Model. Stock price (S) Strike price (K) Interest rate (r) Maturity (t) Volatility 0,25 N(d2) −2,056436 −2,156436 0,019870 0,015525 Put Price Call Price C−P S−Kexp(−rt) −18,757780 −18,757780 18,8142 0,056423801

d1

80

d2

100

N(d1)

0,05

20%

Implementing Monte Carlo method for a stock: Computing return, volatility, generating random numbers, assuming an initial portfolio value of 100,000, percentile is 99,816 and Value at Risk is 184. 1 S(t + h) = S(t) exp µ − σ h + σε (h) 2

Last Trade

0,0062

0,0692 4,2988 7,495

0,00

92,038 92,1636 92,1519 92,1212 92,099 92,2151 92,072 92,1333 92,1702 92,2231 92,1246 92,21 92,2093 92,0457 92,0744 92,1415 92,0808 92,0051 92,1574 92,0874 92,0432 92,0442 92,0964 92,1683

100058 99921,5 99934,2 99967,5 99991,6 99865,7 100021 99954,3 99914,3 99857,1 99963,8 99871,3 99872 100049 100018 99945,4 100011 100094 99928,2 100004 100052 100051 99994,4 99916,4

57,8275 −78,5194 −65,8259 −32,5468 −8,4271 −134,2772 20,8743 −45,6531 −85,6764 −142,9266 −36,2167 −128,7412 −127,9926 49,4498 18,3326 −54,5677 11,3346 93,6258 −71,8104 4,1820 52,1802 51,1130 −5,5982 −83,5978

(Continued)

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−1,32735 0,70690 0,51740 0,02069 −0,33920 1,53959 −0,77629 0,21629 0,81376 1,66880 0,07546 1,45689 1,44571 −1,20244 −0,73838 0,34935 −0,63400 −1,86099 0,60674 −0,52731 −1,24315 −1,22724 −0,38141 0,78272

VT−V0

9in x 6in

0,00 −0,0033 0,00 V0 0,00 100000,00 0,00 0,00 Skewness 0,00 Kurtosis 0,00 −0,0109 0,068 0,00 0,00 0,00 0,00 0,00 0,0108 −0,0052 0,00 0,0066 0,0206 −0,0213 0,00 0,0002 −0,0002 0,00

Xt: random T=3 numbers Days simulation VT=

spi-b708

92,30 92,00 92,00 92,00 92,00 92,00 92,00 92,00 91,00 91,00 91,00 91,00 91,00 91,00 91,98 91,50 91,50 92,10 94,00 92,00 92,00 92,02 92,00 92,00

Volatility m−1/2σ 2

255

11/07/2007 12/07/2007 13/07/2007 16/07/2007 17/07/2007 18/07/2007 19/07/2007 20/07/2007 23/07/2007 24/07/2007 26/07/2007 27/07/2007 30/07/2007 31/07/2007 01/08/2007 03/08/2007 06/08/2007 07/08/2007 08/08/2007 10/08/2007 14/08/2007 15/08/2007 16/08/2007 17/08/2007

Mean


Timestamp Close Return

September 10, 2009 14:41

St+h mean PerCentile VaR 92,12478 99816 184,1338

Last Trade Timestamp Close Return

92,1711 91,9484 92,1417 92,1191 92,1302 92,0556 92,0937 92,1185 92,1781 92,0263 92,0566 92,099 92,0522 92,1274 92,1676 92,0896 92,1448 92,1106 92,0719 92,0768 92,0983 92,0451 92,2196

99913,4 100155 99945,3 99969,8 99957,7 100039 99997,4 99970,4 99905,8 100071 100038 99991,6 100042 99960,8 99917,2 100002 99941,9 99979 100021 100016 99992,3 100050 99860,8

−86,6164 155,3205 −54,7312 −30,1938 −42,3065 38,7491 −2,6107 −29,6041 −94,1681 70,5621 37,6191 −8,3890 42,3864 −39,2197 −82,8218 1,8517 −58,1378 −21,0041 21,0686 15,7487 −7,6979 50,1198 −139,2075

(Continued)

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0,82779 −2,78022 0,35179 −0,01442 0,16635 −1,04287 −0,42598 −0,02322 0,94055 −1,51721 −1,02602 −0,33977 −1,09711 0,12028 0,77114 −0,49255 0,40264 −0,15155 −0,77919 −0,69984 −0,35008 −1,21243 1,61324

VT−V0

9in x 6in

0,00 0,00 0,00 0,0022 0,0174 0,0021 0,00 0,00 −0,0155 0,00 −0,0004 0,0022 −0,0009 0,0009 −0,0022 −0,0022 −0,0033 0,00 0,00 −0,0002 0,0002 0,00 0,00

Xt: random T=3 numbers Days simulation VT=

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92,00 92,00 92,00 92,20 93,80 94,00 94,00 94,00 92,54 92,54 92,50 92,70 92,62 92,70 92,50 92,30 92,00 92,00 92,00 91,98 92,00 92,00 92,00



20/08/2007 22/08/2007 23/08/2007 24/08/2007 27/08/2007 28/08/2007 29/08/2007 30/08/2007 31/08/2007 03/09/2007 04/09/2007 05/09/2007 06/09/2007 07/09/2007 10/09/2007 11/09/2007 12/09/2007 13/09/2007 14/09/2007 17/09/2007 18/09/2007 19/09/2007 20/09/2007

Mean

September 10, 2009 14:41

256


Timestamp Close Return 0,0017 −0,0017 0,00 −0,0002 0,0111 −0,0108 0,00 0,00 0,0107 −0,0105 0,00 0,0011 −0,0011 0,0013

Xt: random T=3 numbers Days simulation VT= −0,16898 0,84954 −0,43311 1,74528 0,07017 0,42573 0,39824 −0,83644 −0,36631 1,84818 −2,30586 −0,15914 −0,11042 −1,18806

92,1095 92,1724 92,0932 92,2278 92,1243 92,1463 92,1446 92,0683 92,0973 92,2342 91,9777 92,1101 92,1131 92,0466

99980,2 99911,9 99997,9 99852 99964,1 99940,3 99942,2 100025 99993,4 99845,1 100123 99979,5 99976,2 100048

VT−V0 −19,8356 −88,0730 −2,1327 −148,0461 −35,8621 −59,6846 −57,8435 24,9072 −6,6100 −154,9327 123,4789 −20,4955 −23,7604 48,4856

9in x 6in

92,16 92,00 92,00 91,98 93,00 92,00 92,00 92,00 92,98 92,00 92,00 92,10 92,00 92,12


spi-b708

21/09/2007 24/09/2007 25/09/2007 26/09/2007 27/09/2007 01/10/2007 02/10/2007 03/10/2007 04/10/2007 05/10/2007 08/10/2007 09/10/2007 10/10/2007 11/10/2007

Mean

September 10, 2009 14:41

Last Trade



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257

September 10, 2009 14:41

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References Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654. Boyle, P (1986). Option valuation using a three jump process. International Options Journal, 3, 7–12. Boyle, PP (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23 (March), 1–12. Briys, E, M Bellalah, F de Varenne and H Mai (1998). Options, Futures and Other Exotics. New York: John Wiley and Sons. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach, Journal of Financial Economics, 7, 229–263. Hull, J and A White (1993). Efficient procedures for valuing European and American path dependent options. Journal of Derivatives, 1, Fall 1993, 21–31. Hull, J, A White (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Advances in Futures and Options Research, 3, 29–61. Hull, J (2000). Options, Futures, and Other Derivative Securities. New Jersey: Prentice Hall International Editions. Jarrow, RA and A Rudd (1983). Option Pricing. Homewood, IL: Irwin. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Rendleman, RJ and BJ Barter (1980). The pricing of options on debts securities. Journal of Financial and Quantitative Analysis, 15 (March), 11–24. Rubinstein, M (1994). Implied binomial trees. Journal of Finance, No 3, 771–818. Whaley, RE (1986). Valuation of American futures options: theory and empirical tests. Journal of Finance, 41 (March), 127–150.

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Chapter 5 CREDIT RISKS, PRICING BONDS, INTEREST RATE INSTRUMENTS, AND THE TERM STRUCTURE OF INTEREST RATES

Chapter Outline This chapter is organized as follows: 1. Section 5.1 is an introduction to the main concepts in discounting and factors. 2. Section 5.2 develops the main concepts for the pricing of bonds. 3. Section 5.3 presents some simple measures to calculate the yield on bonds. 4. Section 5.4 develops the main concepts in the analysis of bonds: duration and convexity. 5. Section 5.5 is an introduction to the yield curve and the theories of interest rates. 6. Section 5.6 presents a simple analysis of the yield to maturity and the theories of the term structure of interest rates. 7. Section 5.7 reviews the specific features of spot and forward interest rates. 8. Section 5.8 studies the way bonds are issued and redeemed. 9. Section 5.9 presents a simple analysis of mortgage-backed securities. 10. Section 5.10 is an introduction to swaps. Introduction The price of a bond depends on the future coupons and the notional amount. The price corresponds to the present value of all the future cash flows 259

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discounted to the present at the appropriate discount rate. The yield on a bond can be determined by calculating the interest rate that makes the present value of all future cash flows equal to the initial bond price. The management of a portfolio of bonds needs the knowledge of the main yield measures: the current yield (CY), yield to maturity (YTM), and the yield to call (YTC). The duration of a bond is a measure of the bond price volatility. The concept of duration is introduced by Macaulay (1938) as a proxy for the length of time a bond investment is outstanding. The yield curve corresponds to the graphical relationship between the YTM of a security and the corresponding maturity date. This curve is often constructed using the maturities and the observed yield on Treasury securities. An interest rate swap is an agreement between two counter-parties to exchange periodic interest payments. These payments are computed with reference to a pre-determined amount known as the notional principal amount. In general, one party, the fixed rate payer, agrees to pay the other party fixed-interest payments with a given frequency at some specified dates. The other party, the floating rate payer, agrees to pay some interest rate payments that vary according to a reference rate. The London Interbank Offered Rate (LIBOR), is often used as the reference rate. The risk that one of the parties does not respect his/her obligation in the swap agreement refers to the default risk or the counter-party risk. The credit crunch in 2008 was associated with debt and bonds. 5.1. Time Value of Money and the Mathematics of Bonds We use the following symbols: I: value or sum of money at the present (a present sum); F : value or sum of money in the future (future sum); A: series of equal end-of-period amounts of money (a uniform series); n: number of periods and r: interest rate per interest period. While I and F occur at one time, the fixed amount A occurs at each interest period for a given or a specified number of periods. A net cash flow refers to the difference between receipts (income) and cash disbursements (costs).

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261

Credit Risks, Pricing Bonds, Interest Rate Instruments, and Term Structure

5.1.1. Single payment formulas Simple interest is computed using the principal amount by ignoring interest accrued in previous interest periods. Simple interest is computed using the formula: Interest = I(n)r. Example: When I = 100, r = 10%, and n = 1 year, the formula shows an interest of 10, or 100(0,1)(1). The compound interest is used when more than one interest period is used. When an amount I is invested at time 0, it becomes in one period (a year), F1 or: F1 = I + rI = I(1 + r) At the end of the second period, the amount accumulated corresponds to the amount accumulated after the first period plus the interest from the end of period 1 to the end of period 2, or: F2 = F1 + F1 r = I(1 + r) + I(1 + r)r = I(1 + 2r + r2 ) = I(1 + r)2 It is straight forward to generalize this formula for n periods as: Fn = I(1 + r)n or F = I(1 + r)n

(5.1)

The term (1 + r)n is often referred to as the single-payment compoundamount factor (SPCAF). It is denoted by (F/I, r%, n). It provides the future value F of an initial investment I, after n periods (years) for a given interest rate r. Formula (5.1) can be used to generate the present worth I of a future amount F after n years at the rate r as: I=F

1 (1 + r)n

(5.2)

The term 1/(1 + r)n is referred to as the single-payment present worth factor (SPPWF). It is denoted by (I/F, r%, n). When there is a single payment (or receipt), the formulas allow the derivation of future and present values.

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5.1.2. Uniform-series present worth factor (USPWF) and the capital recovery factor (CRF) The present worth of uniform series can be computed by assimilating each given A as a future worth F in formula (5.2). The present value is: I=A

1 1 1 1 +A +A + ··· + A 2 3 (1 + r) (1 + r) (1 + r) (1 + r)n

or: n 1 1 1 1 1 + + ··· + I=A =A + (1 + r) (1 + r)2 (1 + r)3 (1 + r)n (1 + r)i i=1

Multiplying both sides by (1/(1 + r)) gives: 1 1 1 I 1 + + + · · · + =A (1 + r) (1 + r)2 (1 + r)3 (1 + r)4 (1 + r)n+1 The difference between the last two equations gives: I

1 −1 1+r

=A

1 1 − n+1 (1 + r) (1 + r)

or: −r I =A 1+r

1 −1 (1 + r)n

1 1+r

Dividing by −r/(1 + r) yields: n ((1 + r)n − 1) 1 I=A =A r(1 + r)n (1 + r)i

(5.3)

i=1

The term or factor: n i=1

(1 + r)n − 1 1 1 − (1 + r)−n = = i n (1 + r) r(1 + r) r

is the USPWF. It is denoted by (I/A, r%, n). It allows the computation of the present worth I of an equivalent uniform annual series A that starts at the end of the first period (year 1), for n years.

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263

Equation (5.3) can be written as: n r(1 + r)n r = I A=I (1 + r)i = ((1 + r)n − 1) 1 − (1 + r)−n

(5.4)

i=1

The term

r(1+r)n ((1+r)n −1

corresponds to the capital-recovery factor (CRF).

It is denoted by (A/I, r%, n). It provides the equivalent uniform annual worth A over n years of a given amount I at the rate r. 5.1.3. Uniform-series compound-amount factor (USCAF ) and the sinking fund factor (SFF ) Re-call that the present worth is given by: I=F

1 (1 + r)n

If I is substituted in Eq. (5.4), this yields: 1 r(1 + r)n A=F (1 + r)n ((1 + r)n − 1) or: n (1 + r)i r =F −1 A=F (1 + r)n (1 + r)n i=1

(5.5)

The discounting factor [r/((1 + r)n − 1)] is the sinking fund factor (SFF). It is denoted by (A/F, r%, n). This equation allows the computation of the uniform series A that starts at the end of period 1 and continues through the period of a specified F . Equation (5.5) can also be written in the following way: F =A

((1 + r)n − 1) (1 + r)n = A n i r i=1 (1 + r)

(5.6)

The term in brackets refers to the USCAF. It is denoted by (F/A, r%, n). When this factor is multiplied by a given uniform amount A, this gives the future worth of the uniform series. The factors (a/b, r, n) in formulas (5.1) to (5.6) allows one to find the value of (a) when (b) is given for a specified interest rate at a given number of periods.

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Derivatives, Risk Management and Value Table 5.1.

To find (a) for a given (b) I, F

Computations with reference to standard notations.

Appropriate factor

Equation

Simple

(I/F, r, n): SPPWF

Formula h (5.1): i 1 I = F (1+r) n

discounting h i

Formula (5.1) to (5.6)

1 (1+r)n

F, I

Simple capitalization (1 + r)n

(F/I, r, n): SPCAF

Formula (5.2): F = I(1 + r)n

I, A

Factor

(I/A, r, n): USPWF

providing I

Formula (5.3): h i ((1+r)n −1) I =A r(1+r)n

as a function

=A

of an equivalent annuity. It is denoted sometimes by P 1 I =A n i=1 (1+r)i A, I

Factor of an annuity

(A/I, r, n): CRF

equivalent to I Pn

i=1 (1

A, F

+ r)i

SFF Pn

(A/F, r, n): SFF

Factor providing

(F/A, r, n): USCAF

(1+r)i i=1 (1+r)n

F, A

h

1−(1+r)−n r

Formula h (5.4): n i r(1+r) A = I (1+r)n −1) h i r = J 1−(1+r) −n Formula (5.5): h

A=F

the future value of a constant annuity

i

r ((1+r)n −1)

i

Formula i h (5.6):n ((1+r) −1) F =A r

(1+r)n Pn i i=1 (1+r)

Table 5.1 provides the necessary formulas.

Applications Using Eq. (5.1), when r = 6%, n = 8 years, formula (5.1) gives the factor: (I/F, 6%, 8) = 1/(1 + 0.06)8 = 0.6274126 When F = 1000, the initial value I is 627.4126.

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265

Using Eq. (5.3), when r = 6%(8%, 11%), n = 11(20, 14) years, formula (5.3) gives the value of the uniform series A: (I/A, 6%, 11) = (1 + 0, 06)11 − 1/0.06(1 + 0, 06)1 1 = 7.886869 (I/A, 8%, 20) = (1.08)2 0 − 1/0, 08(1, 08)20 = 9.81815 (I/A, 11%, 14) = (1.11)1 4 − 1/0, 11(1, 11)14 = 6.981866. Using Eqs. (5.4), (5.5), (5.6), we obtain: (A/I, 6%, 5) = 0, 06(1, 06)5/(1, 06)5 − 1 = 0, 2373966 (A/F, 6%, 5) = 0, 06/(1, 06)5 − 1 = 0, 1773966 (F/A, 6%, 5) = (1, 06)5 − 1/0, 06 = 5, 637087. 5.1.4. Nominal interest rates and continuous compounding A nominal interest rate is the usual interest rate which accounts for the effects of inflation. In fact, the real rate plus inflation define the nominal interest rate. When an individual deposits an amount of 1000 in a bank account, for an interest rate of 12% per year (compounded annually), the future worth is: F = 1000(1.12) = 1120 If the bank pays an interest computed for every six months, the future value must account for the interest on the interest earned. When the annual interest rate is 12%, this means that the bank will pay 6% interest two times a year. In the presence of a 6% effective semi-annual interest rate, the future value is: 1000(1 + 0.06)2 = 1123, 6. Hence, the effective annual interest rate is 12.36% rather than 12%. The following relationship shows the link between nominal interest rates and effective interest rates: r = (1 + im )m − 1 or: r = (1 + (12%/2))2 − 1 = 12.36% where r is the effective interest rate per period, i is the nominal interest rate per period, and m stands for the number of compounding periods. This relation represents the effective interest rate equation. This equation

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can be approximated by: r + 1 = (1 + mim ) or r = m(im ) where r is the effective global interest rate. When the number of compounding periods m increases, m approaches infinity, and we must use continuous compounding. Re-call the definition of the natural logarithm base e: Lim (1 + 1/h)h = e

h→∞

When im = (i/m) = (1/h), (m = hi) in Eq. (5.1), we have: Lim r = Lim (1 + i/m)m − 1 = Lim [(1 + 1/h)h ]i − 1

m→∞

m→∞

m→∞

We have: r = ei − 1. Example: If i = 20% (annual), the effective continuous rate is: r = e0,2 − 1 = 22,1408%. For more details, see Blanck and Tarquin (1989) and Bellalah (1991, 1998a, b). 5.2. Pricing Bonds 5.2.1. A coupon-paying bond The price of a bond depends on the future coupons and the notional amount. The price of a bond is given by the present value of all the future cash flows discounted to the present as follows: B=

n t=1

M c + (1 + r)t (1 + r)n

where: B: bond price at time 0; c: coupon payments; r: periodic interest rate; M : par or maturity value of the bond and n: number of periods. The coupon corresponds to the interest rate times, the nominal value of the bond. In several countries, semi-annual coupons are paid every six months. In other countries, coupons are paid annually. For semiannual coupon payments, the periodic interest rate used in the discounting procedure must be the required yield divided by two. Using standard

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267

formulas for the annuity, it is possible to write the bond price in the following form:   1 1 − (1+r) n M + B = c r (1 + r)n When calculating the time value of money, the following equality is often used to facilitate the computations:

1 n 1 − (1+r) n 1 = (1 + r)t r t=1 The interest rate used in the discounting procedure is referred to as the required yield. The following example illustrates the use of the formula in the computation of the bond price. Example: Consider the pricing of a bond in the following context. Time to maturity = 10 years, coupon rate = 10%, maturity value of the bond = 1000, and periodicity of coupons = 20. The annual coupon is 100 or 10% (1000). The semi-annual coupon is 50 or (100/2). Since there are 20 coupons and M is 1000, we need the required yield to compute the bond price. If the yield on a comparable bond (with the same characteristics and risk) is 11%, then the required yield for six months is 5.5% or (11%/2). Using this discount rate, the present value of the coupons is 597,519 or: c({1 − [1/(1 + r)n ]}/r) = 50({1 − [1/(1 + 0,055)20 ]}/0,055) The present value of the nominal amount is 342,298 or 1000/(1,055)20. Hence, the bond price is 940,247 or (597,519 + 342,298). 5.2.2. Zero-coupon bonds A zero-coupon bond pays no periodic coupons. However, the investor gains interest from the difference between the purchase price and the maturity value. Using zero coupons, the bond pricing formula becomes: B=

M (1 + r)n

It shows that the zero-coupon bond price is given by the discounting of its final value to the present.

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5.3. Computation of the Yield or the Internal Rate of Return 5.3.1. How to measure the yield The yield on an investment can be computed by determining the interest rate that makes the present value of all future cash flows equal to the initial price. It is calculated from the following equality: B=

n t=1

CFt (1 + y)t

where CFt refer to the cash flows, y to the yield and t denotes the number of years from year 1 to n. The yield y is calculated in general using a trial and error procedure. Example: Consider a financial instrument which offers the following annual payments in Table 5.2. Suppose that the price of this instrument is 931. What is the yield or the internal rate of return on this instrument? If we use an interest rate of 10%, we have: 120(3,169) + 1000(0,6209) = 1001,18 Euros. If we use an interest rate of 12%, we get: 120(3,0373) + 1000(0,5674) = 931 Euros. This computation is done with reference to the following identity: P =

n t=1

(1 − (1 + y)−n ) 1 = t (1 + y) y

Table 5.2. The promised annual payments for a financial instrument. Number of years from now

Promised annual payments

1 2 3 4 5

120 120 120 120 1000

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Hence, the internal rate of return, y is 12%. In general, people annualize interest rates multiplying by the frequency of payments per year. If, for example, the semi-annual interest rate is 4%, then the annual interest rate is 8% and vice versa. This is not always correct because interest can also be earned when compounding the interest. To obtain the effective annual yield, which is associated with a given periodic interest rate, the following formula can be used: Effective annual yield = (1 + periodic interest rate)m − 1 where m stands for the frequency of payments each per year. Periodic interest rate = (1 + effective annual yield)1/m − 1 The management of a portfolio of bonds requires the knowledge of the main yield measures: CY, YTM, and YTC. Very high yield bonds are regarded as junk bonds. 5.3.2. The CY The CY gives an indication of the relation between the annual coupon interest and the market bond price as: CY = Annual coupon payments/market bond price 5.3.3. The YTM The YTM is calculated in the same way as the internal rate of return for an investor holding the bond until maturity. For a semi-annual coupon paying bond, the YTM is obtained by solving the following equation: B=

n t=1

or:

B=C

M C + (1 + y)t (1 + y)n

{1 − [1/(1 + y)n ]} y

+

M (1 + y)n

where B is the bond price, C is the semi-annual coupon, y is half of the YTM, M is the maturity value of the bond and n the number of periods. For a semi-annual coupon paying bond, it is sufficient to double the interest rate or the discount rate to get the YTM. The YTM accounts for

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the current coupons, their timing and the capital gain or loss from holding the bond until maturity. 5.3.4. The YTC Some bonds have embedded options. These options allow the issuer to call the bond at some specified points in time. The cash flows for the calculation of the yield to call correspond to those appearing before the first call date. Hence, the yield to call can be calculated as the interest rate that would make the present value of the cash flows, if the bond is held until the first call date equal the bond price. The following formula is used to calculate the YTC: B=

n t=1

C CP + t (1 + y) (1 + y)n

where B: the bond price; y: one-half the yield to call; c: semiannual coupon interest; n: the number of periods until the first call and CP : the call price used to redeem the bond. 5.3.5. The potential yield from holding bonds Investors can calculate the potential yield from holding a bond. This is done using the potential sources of dollar return which must be converted in a yield measure. An investor holding a bond portfolio can expect to receive a given return from the coupons perceived, the capital gain or loss and from the re-investment of the periodic coupons. The coupons received by the investor can be re-invested at a given rate, giving rise to a dollar return from coupon interest and interest on interest. This component of return can be calculated using the formula for the future value of an annuity. The interest on interest can be an important source of potential return for an investor holding the bond. Coupon interest and interest on interest = Coupons + interest on interest

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or: Ip =

C[(1 + r)n − 1] r

where the total coupon amount is given by the semi-annual coupon interest times the number of periods or: Total coupon amount = nC The formula for Ip corresponds to the future value of an annuity where C corresponds to the amount of the annuity, r is the annual interest rate divided by the number of payments per year and n corresponds to the number of periods. It is possible to compute just the interest on interest by subtracting from the Ip formula the amount of the total coupon interest as: Interest on interest =

C[(1 + r)n − 1] − nC r

This analysis assumes that re-investment of the coupons is done at the YTM. 5.4. Price Volatility Measures: Duration and Convexity 5.4.1. Duration The duration of a bond is a measure of the bond price volatility. The concept of duration is introduced as a proxy for the length of time, a bond investment is outstanding. It is given by the weighted average termto-maturity of the cash flows of a bond. It is often computed using the first derivative of the bond price with respect to the yield y. Re-call that the bond price is given by the following relationship: B=

c c c M + + ···+ + (1 + y) (1 + y)2 (1 + y)n (1 + y)n

where c and M stand respectively for the coupon amount and the principal. To compute the duration of a bond one needs to examine the sensitivity of the bond price to the yield. This variation is given by: −2 −n −n −1 dB +c +· · ·+c +M =c dy (1 + y)2 (1 + y)3 (1 + y)n+1 (1 + y)n+1

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or: nc nM 1 1c 2c dB + ··· + + (5.7) =− + dy (1 + y) (1 + y) (1 + y)2 (1 + y)n (1 + y)n The term between brackets shows that each cash flow is weighted by its “maturity”. If both sides of Eq. (5.7) are divided by the bond price P , we obtain the percentage price change for a small variation in y: dB dy

1 B

1 (1 + y) nc nM c 1 2c + ···+ + × + (1 + y) (1 + y)2 (1 + y)n (1 + y)n P

=−

The term between brackets divided by the bond price is known as the duration of Macaulay. The amounts of the coupons can be variable with different amounts of ci. In this case, the bond duration can be written as: D=1

c1 (1+y)1

B

+2

c2 (1+y)2

B

+3

c3 (1+y)3

B

+ ··· + n

cn +M (1+y)n

B

Each coupon payment date (including the principal) is multiplied by the present value of the cash flow in period t. The formula for the bond duration multiplies the present value of each cash flow in period t by the period, when the cash flow is expected to be received. The resulting amount is divided by the total present value of the cash flow of the bond using the prevailing yield to maturity. The formula for the duration can be written as: Dm =

1c (1+y)1

+

2c (1+y)2

+ ···+

nc (1+y)n

+

nM (1+y)n

B

or:

n Dm =

tc t=1 (1+y)t

+

nM (1+y)n

B

When the Macaulay duration (in periods) is divided by the number of payments per year (k = 2 for semi-annual-pay bonds), this gives the Macaulay duration in years: Macaulay duration (in years) = Macaulay duration (in periods)/k

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5.4.2. Duration of a bond portfolio A portfolio duration can be computed using the weighted average of the duration of the bonds in the portfolio as: Dp = w1 D1 + w2 D2 + +wn Dn or: Dp =

Di wi

i

where: P vi wi =

i P vi where Dp : Macaulay duration for a portfolio of bonds; N : number of bonds in the portfolio; P vi : present value (market price) of the ith bond and wi : market value of bond i divided by the market value of the portfolio. 5.4.3. Modified duration Substituting for Macaulay duration for a bond gives: 1 dB 1 =− (Macaulay duration) dy B (1 + y)

(5.8)

This relationship allows the definition of the Modified duration: Modified duration =

Macaulay duration (1 + y)

where y is one-half the YTM. Replacing this expression in the previous relationship gives: dB 1 = −Modified duration dy B It is possible to show that the approximate percentage change in price is given by: = −(1/(1 + y)) × Macaulay duration × Yield change = −Modified duration × Yield change

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There is a relationship between duration and the bond price volatility. In fact, using a Taylor series of the price function, it is possible to show that: Approximate percentage change in price = −(1/(1 + y)) × Macaulay duration × Yield change where y corresponds to one-half the YTM. The modified duration is given by: Modified duration = Macaulay duration/(1 + y) which can also be written as: Approximate percentage change in price = −Modified duration × Yield change The modified duration can be used to approximate the percentage change in the bond price per basis point change. The dollar price change per 100 dollars of par value can be computed as: Approximate dollar price change = −(Modified duration) × (Initial price) × (Yield change) The dollar duration is given by: Dollar duration = (Modified duration) (Initial price) So, we have: Approximate dollar price change = −(Dollar duration) (Yield change) 5.4.4. Price volatility measures: Convexity It is possible to use a Taylor series to approximate the price change of a bond for a given change in the required yield by the following relationship: dB =

d2 B dB dy + 0.5 2 (dy 2 ) + dy dy

where is an error term. When both sides of this equality are divided by the bond price, this allows to express the percentage price change as: 1 dB 1 d2 B 2 dB = dy + 0.5 2 (dy ) + dy B dy B B

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The first term of this equation refers to the dollar price change based on dollar duration. The second term corresponding to the second derivative can be used as a proxy for the convexity of the price-yield relationship. The dollar convexity of the bond is given by: Dollar convexity = 0.5

d2 B dy 2

The approximate change in the bond price due to convexity is: dB = (dollar convexity)(dy)2 When the second derivative of the bond price is divided by the price, this gives a measure of the percentage change in the bond price due to convexity: d2 B Convexity = dy 2

1 B

This refers also to convexity. The percentage price change for the bond, which results from its convexity can be written as: dB = (convexity)(dy)2 B In practice, using the second derivative of the bond price with respect to y gives the following equation which is used to compute the convexity: d2 B ct(t + 1) n(n + 1)M = + t+2 dy 2 (1 + y) (1 + y)n+2 t=1 n

For more details, see Fabozzi (1996) and Briys et al. (1998).

5.5. The Yield Curve and the Theories of Interest Rates The yield curve corresponds to the graphical relationship between the YTM of a security and the corresponding maturity date. This curve is often constructed using the maturities and the observed yield on Treasury securities.

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5.5.1. The shapes of the yield curve Different shapes of the yield curve are observed over time. When the yield increases with maturity, the relationship corresponds to an upward sloping or normal yield curve. When the yield decreases with maturity, the relationship corresponds to a downward sloping or an inverted yield curve. When the yield increases and then decreases with maturity, this pattern indicates a humped yield curve. When the yield is constant for all maturities, the yield curve is flat (see Capie, 1991).

5.5.2. Theories of the term structure of interest rates 5.5.2.1. The pure expectations theory This theory asserts that the entire term structure is explained by the market’s expectations of the future short-term rates of different maturities. Hence, a rising term structure can be explained by the fact that the market participants expect a rise in short-term rates in the future. An inverted term structure can be explained by the fact that the market participants expect a decline in short-term rates in the future. A flat term structure can be explained by the fact that the market participants expect constant and stable short-term rates in the future. The pure expectations theory is based on the assumption that forward rates are affected systematically by the expected future short-term rates. The main drawback of the pure expectations theory is that it does not account for the risks inherent in investing in bonds. By assuming that forward rates are perfect predictors of future interest rates, then the bond future prices would be certain. It ignores the uncertainty about the bond’s price at the end of the investment horizon as well as the re-investment risk. The first interpretation of the pure expectations theory, according to Cox et al. (1985a, b), is that the expected return for any investment horizon is the same regardless of the maturity. The second interpretation given by the local expectations theory assumes that the return over very short-term intervals of time must be equal to the short-term riskless rate of interest. The third interpretation of the pure expectations theory is that the return from a roll-over short-term strategy is equivalent to holding a zero-coupon bond with a maturity equal to that of the investment horizon. This is referred to as the return-to-maturity expectations.

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Biased expectations theories The liquidity theory and the preferred habitat theory, referred to as biased expectations theories, assume that other factors affect the forward rates. The liquidity theory: According to this theory, forward interest rates account for interest rate expectations and a liquidity premium or a risk premium. Investors expect to hold bonds for longer maturity dates for higher risk premiums. Hence, the presence of a risk premium implies that the implied forward rates will not be an unbiased estimate of the expectations of future interest rates. The shape of an upward sloping of the yield curve can reflect expectations about a rise, a stable or a fall in the expected future rates, where the liquidity premium can increase fast enough to conserve an upward-sloping yield curve. The preferred habitat theory: This theory shows that the expectations of future interest rates account for a risk premium where this premium (positive or negative) does not necessarily rise uniformly with maturity. According to this theory, for a given imbalance between the supply and demand for funds within a maturity range, investors will not be reluctant to shift their portfolios out of their preferred maturity sector. Investors require a yield premium to move out of their preferred sector. This theory can then explain different shapes of the yield curve. Market segmentation theory: This theory supports the “preferred habitat” and explains the yield curve shape by asset and liability management constraints. This theory assumes that investors are not willing to shift from one maturity to another, to take advantages from the differences in expectations and forward rates. Hence, the yield curve is explained by the supply and demand within each maturity sector (see Carleton and Cooper, 1976).

5.6. The YTM and the Theories of the Term Structure of Interest Rates 5.6.1. Computing the YTM The YTM, refers to the average annual rate of return expected from the purchase of a bond. This is a promised return rather than an actual return. The YTM can be computed using one of the following three methods: the arithmetic mean, the geometric mean, and the internal YTM.

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The arithmetic mean YTM This method computes the arithmetic mean of the expected future rates of return on a bond. It uses the expected rates of return over the next years. The arithmetic mean corresponds to the sum of the expected returns on the bond over its life, which is divided by the number of years until maturity. The resulting figure corresponds to the arithmetic mean YTM or the expected average annual percentage increase in the capital invested. This method assumes that the dollar investment in the bond remains constant.

The geometric mean YTM This method adds 1.00 to each expected annual return on the bond. All the converted returns are multiplied and the nth root of the product is computed, where n refers to the number of years until maturity. Subtracting one from the root gives the geometric mean YTM. This method is based on the assumption that all profits are re-invested by buying more bonds. The geometric mean YTM reflects in this context, the average percentage increase in the capital over the bond’s life.

The internal YTM The equality between the current bond market price and the discounted value of all cash flows of the investment gives an internal YTM. This method assumes the re-investment of coupons at the internal yield without any re-investment risk.

5.6.2. Market segmentation theory of the term structure The market segmentation theory assumes that the market is composed of extremely risk averse investors whose objective is portfolio “immunization”. Immunization is achieved when the effective maturity of assets is perfectly matched up with the effective maturity of liabilities. In this theory, the survival of an institution is an objective function. The yield of each segment in the market corresponds to the intersection of supply and demand.

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5.7. Spot Rates and Forward Interest Rates 5.7.1. The theoretical spot rate The theoretical spot rate for an nth six-month period can be calculated using the following equation: Bn =

C C (C + 100) C + + ···+ + 2 3 (1 + y1 ) (1 + y2 ) (1 + y3 ) (1 + yn )n

where Bn : price of a coupon Treasury security with n periods to maturity (per 100 dollar of value); C: semi-annual coupon for the coupon Treasury security with n periods to maturity (per 100 dollar of par value) and yt : t = 1, . . . , (n − 1): known theoretical spot rates. This equation can also be written as: 

 n1 (C + 100)  −1 yn = 

n−1 1 Bn − C − t=1 (1+yt )t When yn is multiplied by two, this gives the theoretical spot rate on a bond-equivalent basis.

5.7.2. Forward rates Consider the two following investment opportunities: Strategy 1: Invest in a one-year Treasury bill, Strategy 2: Invest in a six-month Treasury bill and buy another six-month Treasury bill at the maturity date of the first Treasury bill. These two strategies are equivalent if they give the same result over the one-year investment horizon. The knowledge of the spot rates on a sixmonth and a one-year Treasury bills allows the computation of the yield on a six-month Treasury bill, six months from now or the forward rate.

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An investor buying a one-year Treasury bill maturing in a year with a maturity value of 100, pays a price B: B = 100/(1 + Y2 )2 where Y2 represents one-half the bond-equivalent yield of the one-year spot rate. If the investor buys the six-month Treasury bill, the value of his investment in six months would be: B(1 + Y1 ) where Y1 represents one-half the bond-equivalent yield of the six-month spot rate. In the same context, if we denote by f2 one-half the forward rate on a six month Treasury bill available six months from now, then the P dollars in one year becomes: B(1 + Y1 )(1 + f2 ) The result of investing in an asset that gives 100 in one year can be written as: B(1 + Y1 )(1 + f2 ) = 100 or: B = 100/[(1 + Y1 )(1 + f2 )] The two strategies are equivalent for the investor if they lead to the same result or: 100/(1 + Y2 )2 = 100/[(1 + Y1 )(1 + f2 )] Hence, the value of the implied forward rate is given by: f2 = [(1 + Y2 )2 /(1 + Y1 )] − 1 Example: When the six-month and the one-year Treasury bill rates are respectively equal to 8.5% and 8.9%, then Y1 = 4,25%, Y2 = 4,45%, and the implied forward rate is 4,65%, or: f2 = [(1,0445)2/1,0425] − 1 = 4,65%

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Hence, the forward rate for a six-month instrument on a bondequivalent basis is 9,3% or 2(4,65%). The formula can be generalized to compute implicit forward rates from theoretical spot rates as follows: 1/t (1 + yn+t )n+t fnt = −1 (1 + y n )n where fnt : forward rate n periods from now for t periods, yn : semi-annual spot rate. The implied forward rate on a bond-equivalent basis is obtained by multiplying fnt by two. 5.8. Issuing and Redeeming Bonds Example: Consider a firm issuing 1000 bonds. The nominal value of each bond is 1000, and the principal amount of the issue is 1,000,000. If the quoted bond price is 100, this means 100% of 1000. If the interest rate falls, the bond price may be 102, i.e., 102% of 1000, or 1020. If the interest rate rises, the bond price can become 98, i.e., 98% of 1000, or 980. If the interest rate is 12% and the coupon is paid the first January each year, an investor buying this bond in France the first May, must pay 1000 plus accrued interest for four months, i.e., 40, 4% of 1000, or 1040. If the coupon is paid each six months, the first July the investor receives 60, corresponding to the 40 paid to the seller and the interest 20 corresponding to two months. The amount of the issue corresponds to the number of bonds (coupures) times the nominal value of each bond. The issue price corresponds to the price paid when buying the bond. Application: Consider the following bond issue: Issuer: Company X; Nominal amount = 10, 000, 000; Nominal value of each bond = 2000; Number of bonds = 5000; Issue price = 980, or 98% of the nominal amount; Maturity date = 10 years; Payment date = 01/01/1994; Maturity date = 01/01/2004; Interest rate = 10% and

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Coupon amounts: the first coupon of 2000 is paid the first January 1995 or (2000)(10%)(1) = 2000. The same coupon amount is paid each year at the same date Payments: 1000 bonds re-paid or 2002; 1000 bonds re-paid or 2006; 1000 bonds re-paid or 2010; 1000 bonds re-paid or 2014 and 1000 bonds re-paid or 2018.

the first January 2000 at 100,1% of the nominal value the first January 2001 at 100,3% of the nominal value the first January 2002 at 100,5% of the nominal value the first January 2003 at 100,7% of the nominal value the first January 2004 at 100,9% of the nominal value

Table 5.3 shows the re-payment of the issue for the borrower. 10000 = 500 × 200 At the time of the issue, the issuer receives 9,900,000 or ((5000)(1980)). Table 5.3. The re-payment profile of the issue for the borrower: issue date, the first January 1994. Years 1 to 10

Years 1 to 10

(1) Number of bonds alive in the beginning of the year in 103

(2) The total amount of coupon payments in 103

(3) The number of bonds paid “dead” at the end the year in 103

5 5 5 5 5 5 4 3 2 1

1000 1000 1000 1000 1000 1000 800 600 400 200

0 0 0 0 0 0 1 1 1 1

(4) The amount of payments in 103

2002 2006 2010 2014 2018

0 0 0 0 0 (2002)103 (2006)103 (2010)103 (2014)103 (2018)103

Years 1 to 10 (5) The annuities in 103 (2)+ (4)

(6) Number of bonds alive at the end of the year in 103 (1) − (3)

1000 1000 1000 1000 1000 3002 2806 2610 2414 2218

5 5 5 5 5 4 3 2 1 0

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Table 5.4. The profile of cash flows for the bondholder (the investor lending money): initial investment at time 0: 19,800 ((1980)(10)). Years 1 to 10 (1) Number of bonds alive in the beginning of the year in 103 10 10 10 10 10 10 10 5 5 5

Years 1 to 10

(2) The total amount of coupon payments in 103

(3) The number of bonds paid “dead” at the end the year in 103

2000 (10)200 2000 2000 2000 2000 2000 2000 1000 (5)200 1000 1000

0 0 0 0 0 0 5 0 0 5

(4) The amount of payments in 103 0 0 0 0 0 0 10,030 5(2006) 0 0 10,090 5(2018)

Years 1 to 10

(5) The annuities in 103 (2) + (4)

(6∗ )

2000 2000 2000 2000 2000 2000 12,030 1000 1000 11,090

10 10 10 10 10 10 5 5 5 0

From the year 1995 to 1999, he pays each year 1,000,000 of coupons. From year 6, (the year 2000), he pays the coupons and makes principal re-payment until year 2004. At this latter date, he pays 200,000 of coupons and re-pays 2,018,000 of the principal. The total amount is 2,218,000. The profile of the cash flows for the bondholder depends on the payment dates. Consider an investor who buys the first January 1994, 10 bonds of the firm X. The issuer makes payment for the first five bonds the first January 2001 and for the other five bonds the first January 2004. The profile of cash flows for the bondholder is given in Table 5.4. With (6∗ ) number of bonds alive at the end of the year in 103 (1)−(6).

5.9. Mortgage-Backed Securities: The Monthly Mortgage Payments for a Level-Payment Fixed-Rate Mortgage When determining the monthly mortgage payments, the formula for the present value of an ordinary annuity is used. The mortgage payment for each month is defined with respect to a level-payment fixed-rate mortgage. The monthly mortgage payment is due in the beginning of each month. It consists of two elements. The first corresponds to interest of 1/12th of the

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fixed annual interest rate times the outstanding mortgage balance amount at the beginning of the previous month. The second element corresponds to the re-payment of a fraction of the outstanding mortgage balance or the principal. The mortgage payment is defined in a way such that after the last fixed monthly payment, the amount of the outstanding mortgage balance is zero. Example: Consider a mortgage loan of 150,000 Euros with maturity in 15 years (180 months), the mortgage rate is 10%. Table 5.5 shows the amortization schedule for a level-payment fixed-rate mortgage for the period of 180 months. Each monthly mortgage payment comprises interest and repayment of principal. The first month, the mortgage balance corresponds to the interest rate for the month on the 150,000 Euros borrowed or (10%/12) 150,000 = 1250 Euros. The monthly mortgage payment that represents re-payment of the principal corresponds to the difference between the monthly mortgage payment and the interest rate. The last monthly mortgage payment is sufficient to pay off the remaining mortgage balance. The following formula is used to compute the monthly mortgage payment for a level-payment fixed-rate mortgage.   1   1 − (1+r) n PV = A   r where: P V : present value of an annuity or the original mortgage balance; A: amount of the annuity (monthly mortgage payment); n: number of periods or months and r: periodic interest rate (annual interest rate/12). The formula corresponds to the present value of an ordinary annuity formula. The equality can also be written as: Monthly payment =

Amount due to be paid Present value of an annuity of 1 Euro/month

Using the above P V formula and the data in the example, we have n = 180, VA = 150,000 Euros, r = 0,1/12, the monthly mortgage payment is

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Table 5.5. Amortization schedule for a level-payment fixed-rate mortgage using the following parameters: term of loan n = 180, mortgage loan = 150,000 Euros, and mortgage rate = 0.1/12.

Month 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ... 74 75 76 77 78 86 87 88 89 90 ... 175 176 177 178 179 180

Beginning mortgage balance

Monthly mortgage payment

1, 50, 000 1, 49, 638.0923 1, 49, 273.1687 1, 48, 905.2041 1, 48, 534.1732 1, 48, 160.0503 1, 47, 782.8097 1, 47, 402.4254 1, 47, 018.8713 1, 46, 632.1209 1, 46, 242.1475 1, 45, 848.9244 1, 45, 452.4244 1, 45, 052.6203 1, 44, 649.4845 1, 44, 242.9892 1, 43, 833.1064 1, 43, 419.8079

1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677

1, 13, 834.9611 1, 13, 171.6781 1, 12, 502.8678 1, 11, 828.484 1, 11, 148.4803 1, 05, 500.4334 1, 04, 767.696 1, 04, 028.8525 1, 03, 283.8519 1, 02, 532.643

1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677

9395.51514 7861.903423 6315.511609 4756.233195 3183.960795 1598.586126

1611.907677 1611.907677 1611.907677 1611.907677 1611.907677 1611.907677

Interest for month

Principal re-payment

Ending mortgage balance (2)−(5)

1250 1246.984103 1243.943073 1240.876701 1237.784776 1234.667086 1231.523414 1228.353545 1225.157261 1221.934341 1218.684563 1215.407703 1212.103537 1208.771836 1205.412371 1202.02491 1198.60922 1195.165066 ... 948.624676 943.0973176 937.523898 931.9040332 926.2373361 879.1702784 873.0641334 866.9071039 860.6987658 854.4386915 ... 78.2959595 65.51586186 52.6292634 39.63527663 26.53300663 13.32155105

361.9076766 364.9235739 367.9646036 371.0309753 374.1229001 377.240591 380.3842626 383.5541314 386.7504158 389.973336 393.2231138 396.4999731 399.8041395 403.1358407 406.495306 409.8827669 413.2984566 416.7426104

1, 49, 638.0923 1, 49, 273.1687 1, 48, 905.2041 1, 48, 534.1732 1, 48, 160.0503 1, 47, 782.8097 1, 47, 402.4254 1, 47, 018.8713 1, 46, 632.1209 1, 46, 242.1475 1, 45, 848.9244 1, 45, 452.4244 1, 45, 052.6203 1, 44, 649.4845 1, 44, 242.9892 1, 43, 833.1064 1, 43, 419.8079 1, 43, 003.0653

663.2830006 668.8103589 674.3837786 680.0036434 685.6703404 732.7373982 738.8435431 745.0005727 751.2089108 757.468985

1, 13, 171.6781 1, 12, 502.8678 1, 11, 828.484 1, 11, 148.4803 1, 10, 462.81 1, 04, 767.696 1, 04, 028.8525 1, 03, 283.8519 1, 02, 532.643 1, 01, 775.174

1533.611717 1546.391815 1559.278413 1572.2724 1585.37467 1598.586126

7861.903423 6315.511609 4756.233195 3183.960795 1598.586126 0

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1611,907677:  

 150000 1−

1

 = 1611,907677.

(1,00833)180 0,008333

It is assumed in the above analysis of fixed-rate mortgages, that the level homeowner would not pay any fraction of the mortgage balance before the fixed date. We refer to the payments made in excess of the fixed principal payments as pre-payments. The amount and the timing of the pre-payment are uncertain and define the pre-payment risk. 5.10. Interest Rate Swaps 5.10.1. The pricing of interest rate swaps Example: Consider the following interest rate swap between two firms A and B in the presence of a commercial bank (Fig. 5.1). The bank pays the six-month LIBOR rate to the firm A and receives a fixed rate of 9.6%. The bank pays to firm B a fixed rate of 9.5% and receives the LIBOR. A notional amount No is used. The bank benefits from a commission of 0.10% per year on the notional amount. If the LIBOR rate is 11%, firm B pays the bank: 1/2(0.11 − 0.095) = 0.0075 No and the bank pays A: 1/2(0.11 − 0.096) = 0.0070 No. The bank gains 0.005 No for six months. This swap can be assimilated to two default bonds since the bank borrows from A at the six-month LIBOR and lends to A at a rate of 9.6%. 5.10.2. The swap value as the difference between the prices of two bonds Let us denote by Sw, the value of the swap for the bank, B1 the value of bond paying a coupon at 9.6% per year, and by B2 the value of a bond paying the six-month LIBOR. The value of the swap for the bank is: Sw = B1 − B2

Firm A

9.6% LIBOR

Commercial bank

Fig. 5.1.

9.5% LIBOR

An interest rate swap.

Firm B

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287

The value of the bond B1 is given by: B1 =

n

Ce−ri ti + N e−rn tn

i=1

where C: semi-annual coupon; Ti : time until the ith coupon payment where 1 ≤ i ≤ n; ri : risk-less rate for ti and N : the notional amount. Since the value of B2 is based on a variable interest rate, its price after each coupon date must be equal to the discounted value of the notional amount and the last coupon. Hence, the value B2 between two payment dates is: B2 = N e−r1 t1 + C1 e−r1 t1 where C1 is the certain coupon at date t1 . The swap as a series of forward contracts A swap position can be seen as a package of forward or futures contracts. In fact, firm A has agreed to pay 9.6% and receive six-month LIBOR. Assuming a 100 million dollars NO, A has agreed to buy the six-month LIBOR for 48 million dollars. This represents a six-month forward contract, where A agrees to pay 48 million in exchange for delivery of six-month LIBOR. An interest rate swap can be assimilated to a portfolio of forward contracts. In fact, firm A receives each six month an amount equal to 0.5 No (LIBOR−0.096) where the LIBOR rate corresponds to the previous period. This is a forward contract on the six-month LIBOR rate. 5.10.3. The valuation of currency swaps Example 1 Consider a firm A borrowing 6,00,000 units of currency A at a fixed rate of 8% for five years. The firm must convert this in a currency B. A firm B enters into a five-year swap with firm A at a rate of 9.5% for an amount of 2,200,000 of currency B. The principal amount is exchanged for an exchange rate equal to 3.666, (the day of the swap implementation).

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At each payment date, firm A pays B interest charges in the currency of B using a rate of 9.5% on the principal amount of 2,200,000 units of currency B. The firm B pays A at a rate of 8% on the principal amount of 6,00,000 units of currency A. Each six months, A pays 1,04,500 units of currency B, or (9.5%) (2,200,000)/2, and B pays 88,000 units of currency B, or (8%) (2,200,000)/2. At the swap term, A pays 2,200,000 units of currency B to B and B pays 6,00,000 units of currency A to A. This swap operation is equivalent to a position in a currency forward contract. Example 2 Two firms A and B are engaged in a swap. The bank receives a commission of 25 basis points on the principal amount denominated in dollars as in Fig. 5.2. Firm A pays in dollars and receives sterling. If the NA is 30 million dollars and 20 million sterling, each year, A pays 2.4 million dollars (8%) (30 million) and receives 2.2 million sterling (11%) (20 million). At the swap term, firm A pays 30 million dollars of principal and 20 million sterling. The swap allows A to convert a fixed-rate loan denominated in sterling, (11% per year) into a fixed-rate loan denominated in dollars (8% per year). Firm B converts a fixed-rate loan in dollars (7.75% per year) into a fixed-rate loan in sterling (11% per year). If, for example, the loan for A (in sterling) is at a rate of 11% and B (in dollars) at 9%, the swap can be presented as in the Fig. 5.3. The loan at 11% for A is converted into a dollar loan at 8% and the loan for B in dollars at 9% is converted into a loan in sterling at 12.25%. If the exchanged amounts are 20 million dollars and 30 million sterling, the

Firm A

(dollars)8% 11%(pound) Fig. 5.2.

Firm A

(dollars)7.75% 11%(pound)

Firm B

Foreign currency swap.

(dollars)8% 11%(sterling) Fig. 5.3.

Bank

Bank

(dollars)9% 12.25%(sterling)

The foreign currency swap.

Firm B

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289

bank loses 3,00,000 dollars per year and gains 2,50,000 sterling. The bank can implement a hedge against exchange rate risk by selling forward the 2,50,000 sterling against dollars.

5.10.4. Computing the swap Since the parties in an interest swap agree to exchange future interest payments with no upfront payment, this means that the present value of the cash flows for the payments must be equal. This equivalence allows the computation of the swap rate. Hence, the swap rate corresponds to an interest rate that makes equal the present value of the payments on the fixed side, with the present value of the payments on the floating rate side. Each cash-flow in a swap must be discounted at a unique theoretical spot rate. This spot rate is obtained from forward rates. The following example illustrates the procedure for the computation of the swap rate.

Summary The yield on an investment can be computed by determining the interest rate that makes the present value of all future cash flows equal to the initial price. The YTM is calculated in the same way as the internal rate of return for an investor, holding the bond until maturity. In bond analysis, the coupons received by the investor can be re-invested at a specified rate giving rise to a dollar return from coupon interest and interest on interest. The concept of duration is introduced by Macaulay (Bellalah et al., 1998) as a proxy for the length of time a bond investment is standing. It is given by the weighted average term-to-maturity of the cash flows of a bond. A portfolio duration can be computed using the weighted average of the duration of the bonds in the portfolio. Several theories are developed to explain the shape of the yield curve. The main theories explaining the behavior of interest rates are the expectations theory and the market segmentation theory. The expectations theory has several variations: the pure expectations theory, the liquidity theory, and the preferred habitat theory. In these theories, the market expectations about future short-term rates can explain the forward rates in current long-term bonds. The YTM, corresponds to the average annual rate of return expected from the purchase of a bond. The YTM refers to a promised return rather

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than an actual return. It can be computed using one of the following three methods: the arithmetic mean, the geometric mean and the internal YTM. The shape of the yield curve is affected by market expectations about future interest rates. The relationship between the yield on zero-coupon Treasury securities and maturity is referred to as the Treasury spot-rate curve where the yield on a zero-coupon bond refers to the spot rate. Several types of bonds and mortgage securities are issued in the market place. When determining the monthly mortgage payments, the formula for the present value of an ordinary annuity is used. The mortgage payment for each month is defined with respect to a level-payment fixed-rate mortgage. A swap position can be seen as a package of forward or futures contracts. Since the parties in an interest swap agree to exchange future interest payments with no upfront payment, this means that the present value of the cash flows for the payments must be equal. This equivalence allows the computation of the swap rate. Hence, the swap rate corresponds to an interest rate that makes equal, the present value of the payments on the fixed side with the present value of the payments on the floating rate side. A swaption gives the right to assume a position in an underlying interest rate swap with a given maturity. In swaptions, the right to pay the fixed component is equivalent to the right to receive the floating component and vice versa. Swaptions are offered as receiver swaptions and payer swaptions. Receiver swaption gives the right to receive a fixed interest rate and payer swaption gives the right to pay a fixed interest rate. Using coupon paying bonds with different maturity dates in the presence of different taxation rates (for capital gains), it is difficult to observe the term structure of interest rates. Therefore, several techniques are proposed to estimate the term structure of interest rates. Several methods are proposed in the literature for the estimation of forward interest rates. Empirical methods are continuous or discrete.

Questions 1. 2. 3. 4.

What are the different types of bonds? What are the specific risks in bond investments? What are the main concepts in the pricing of bonds? What are the main measures that allow investors to calculate the yield on bonds?

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5. 6. 7. 8. 9. 10. 11. 12.

291

What is duration? What is convexity? What are the main theories of interest rates? What are the specific features of spot and forward interest rates? How bonds are issued and redeemed? What are the specific features of mortgage backed securities? What are the specific features of swaps? What are the main techniques used in the estimation models of the term structure?

References Blanck, LT and A Tarquin (1989). Engineering Economy. New York: Mc GrawHill. Bellalah, M (1991). Gestion Quantitative du Portefeuille et nouveaux marchs financiers. Paris: Editions Nathan. Bellalah, M (1998a). Gestion financire: diagnostic, valuation et choix des investissements. Paris: Editions Economica. Bellalah, M (1998b). Finance d’entreprise: stratgies et politiques financires. Paris: Editions Economica. Briys, E, M Bellalah et al. 1998. Options, Futures and exotic Derivatives, en collaboration avec E. Briys, et al., John Wiley & Sons. Cox, J, I Ingersoll and S Ross (1985a). An intertemporal General equilibrium model of Asset prices. Econometrica, 53, 363–384. Cox, J, I Ingersoll and S Ross (1985b). A Theory of the term structure of interest rates. Econometrica, 53, 385–407. Capie, F (1991). Major inflations in History. Vermont: Edward Elgar. Carleton, W and I Cooper (1976). Estimation and uses of the term structure of interest rates. Journal of Finance, 31, 1067–1083. Fabozzi, F (1996). Bond Markets, Analysis and Strategies. New Jersey: Prentice Hall.

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Chapter 6 EXTENSIONS OF SIMPLE BINOMIAL OPTION PRICING MODELS TO INTEREST RATES AND CREDIT RISK

Chapter Outline This chapter is organized as follows: 1. Section 6.1 extends the standard binomial model of Cox et al. (1979) for the valuation of interest-rate sensitive instruments. It develops the Rendleman and Bartter (for details, refer to Bellalah et al., 1998) model for the pricing of bonds and bond options. 2. Section 6.2 studies the Ho and Lee (1986) model for the valuation of bonds and bond options. 3. Section 6.3 shows how to construct interest-rate trees and how to price bonds and options. 4. Section 6.4 presents a simple derivation of the Black-Derman-Toy model. 5. Section 6.5 shows how to construct trinomial interest-rate trees for the pricing of bonds and options. Introduction This chapter extends the basic lattice approach to the pricing of interestrate sensitive instruments and options in the presence of several distributions to the underlying asset. We are interested in the lattice approach pioneered by Cox et al. (CRR) (1979). These authors proposed a binomial model in a discrete-time setting for the valuation of options. Using the risk-neutral framework, their approach is based on the construction of a binomial lattice for stock prices. They applied the risk-neutral valuation argument, pioneered by Black and Scholes (1973), which simply means 293

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that one can value the option at hand as if investors were risk-neutral. With an appropriate choice of the binomial parameters, they established a convergence result of their model to that of Black and Scholes. Since then, the CRR approach was extended and extensively used for the valuation of many contingent claims and options. Rendleman and Bartter (for details, refer to Bellalah et al., 1998) applied this methodology to the pricing of options on debt instruments. They proposed a binomial approach similar to that of the CRR for the pricing of derivative assets. The bond’s value at a given node can be computed using the immediate next two nodes since at a given node, the bond’s value will depend on the future cash flows. The future cash flows correspond to the bond value of one year from now and the coupon payments. The binomial model is based on the recursive procedure starting from the last year and working backward through the tree till the initial time. The value at each node is given by the expected cash flows under the appropriate discount rate. Ho and Lee (1986) proposed a model for the pricing of bonds and options. However, their model allows for the possibility of negative interest rates. Ritchken and Boenewan (1990) developed a simple approach to eliminate the possibility of observing negative interest rates. This analysis is extended by Pederson et al. (for details, refer to Bellalah et al., 1998). Bliss and Ronn (for details, refer to Bellalah et al., 1998) and Hull and White (1988) developed a trinomial model for the pricing of interest-rate sensitive instruments. An alternative to the Ho and Lee model was proposed by Black et al. (1990), and Hull and White (1993) among others. Black et al. (1990) used a binomial tree to construct a one-factor model of the short rate that fits the current volatilities of all discount bond yields as well as the current term structure of interest rates. Rubinstein (1994) developed a new method for inferring risk-neutral probabilities or option prices from observed market prices. These probabilities were used to infer a binomial tree by implementing a simple backward recursive procedure. However, this approach is restricted to the European options, and future research must be done with regard to the pricing of the American options. 6.1. The Rendleman and Bartter Model (for details, refer to Bellalah et al., 1998) for Interest-Rate Sensitive Instruments The CRR (1979) approach can be applied to the valuation of interestrate sensitive instruments. It can be presented in the form developed by

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Extensions of Simple Binomial Option Pricing Models to Interest Rates

295

Rendleman and Bartter (for details, refer to Bellalah et al., 1998). This model is used for the pricing of bonds and interest-rate instruments.1 The dynamics of the interest rate are described by a two-state process with the following parameters: u = eσ

√

dt

,

d = 1/u,

q = e(a−σθ)∆t ,

and p = (q − d)/(u − d)

To illustrate the Rendleman and Bartter (for details, refer to Bellalah et al., 1998) model over a period of six years when the current interest rate is equal to 11%, consider the following data: a = 0,

σ = 0.2,

θσ = −0.03,

n = 6.

∆t = 1,

Using these parameters, we have: u = 1.2214, d = 0.8187, q = 1.03045, and p = 0.5257. Figure 6.1 describes the dynamics of interest rates in this context. Starting from an interest rate of 11%, the next rate in an upstate 13.43 is given by 11 times u or 11(1.2214). The rate 16.41% is given by 13.43 times u, or 13.43(1.2214). In each upstate, this operation is repeated until the rate 36.52% given by 29.9(1.2214) is obtained at the end. Starting from a rate of 11%, the rate 9% is obtained from the product of 11 by d, or 11(1/1.2214). The rate 7.36 corresponds to 9 times d, or 9(1/1.2214). 36.52 29.90 24.48 20.04 16.41 13.43 11

16.41 13.43

11 9

24.48 20.04 16.41 13.43

11 9

7.37

11 9

7.37 6.03

7.37 6.03

4.94

4.94 4.04 3.31

Year 0 Fig. 6.1. 1 In

1

2

3

4

5

6

Rendleman and Bartter model for the dynamics of interest rates.

this model, the dynamics of the short-term rate are described by: dr = (a − θσ)rdt + σrdz

where a, θ and σ are constants and dz is a Wiener process. The term (a − θσ) represents the drift in the short-term rate.

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This operation is repeated till the end. The rate 3.31 is obtained by 4.04(1/1.2214).

6.1.1. Using the model for coupon-paying bonds Consider a 8.5%, six-year bond with a maturity value equal to 1000 paying an annual coupon of 85. In the first step, the bond price must be computed at each node in the binomial tree. The bond value can be calculated using the following relationship: Bij = [pBi+1,j+1 + (1 − p)Bi+1,j + coupon)]/er ij ∆t with: rij : interest rate at position j and time i; Bij : corresponding to the bond price at state j and time i; and p: the probability corresponding to an upstate. The bond price at any position is given by the expected values in an upstate pBi+1,j+1 plus the expected value in a down state (1−p) Bi+1,j , discounted to the present using the corresponding interest rate er ij ∆t. Using this recursive procedure, the bond price at time 0 is 846,869 (Fig. 6.2). Since the bond price at maturity is 1000, the price one period before the maturity date is 804,580. It is calculated using the relation above for

1000 804,580

846,869

727,350 712,44 887,930 734,740 850,100 781,824 856,54 948,590 891,855 944,170 947,400 970,45 991,550 1018,02 1013,19 1055,8 1021,43 1062,34 1041,97

1000 1000 1000 1000 1000 1000

Year 0

1 Fig. 6.2.

2

3

4

5

Dynamics of the bond price.

6

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297

the bond value as follows: [0,5257(1000) + (1 − p)(1000) + 85)]/e0,299(1) = 804,58 The same formula can be applied until the initial time 0. 6.2. Ho and Lee Model for Interest Rates and Bond Options Ho and Lee (1986) assumed the existence of a zero-coupon bond for each (n) maturity, n, (n = 0, 1, 2, . . .). They denoted by Pi (T ) the equilibrium (n) price of a zero-coupon bond at time n and state i. Hence, the price Pi (.) is a function relating each bond price with its corresponding maturity date or a discount function. The price of a bond maturing instantaneously is 1 or: (n)

Pi

(0) = 1

for all i, n

The price of a long-lived bond is nearly zero since: (n)

lim Pi

T →∞

(T ) = 0

for all i, n

6.2.1. The binomial dynamics of the term structure At time zero, the discount function is observed and is denoted by: (0) P (.) = P0 (.) = 1. (1) (1) At time 1, the discount function is specified by two P1 (0) and P0 (0) for an upstate and a down state, respectively. The discount function at the second period between two dates 1 and 2 leads to two possible functions. (1) (2) (2) Hence, at time 1, P1 (.) leads to the functions P2 (.) and P1 (.) at time 2. (n) We denote the discount function at time n by Pi (.) after i ups and (n − i) downs, at period (n + 1) between the dates n and (n + 1) (Fig. 6.3). Given the discount function P (T ), the yield curve can be written as: r(T ) = −ln(P (T ))/T where r(T ) is the continuous yield for a bond maturing in T . (n+1) Pi+1 (.)

: up

(n) Pi (.) (n+1) Pi (.) Fig. 6.3.

: down

The discount function in the Ho-Lee model.

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6.2.2. The binomial dynamics of bond prices When the term structure is described by a binomial model, the same dynamics applies for zero-coupon bonds P (N ) with a maturity date N at an initial time. When it remains at (N − 1) periods, the discounting functions (1) in upstates and down states allow the computation of P1 (N − 1) and 1 P0 (N − 1). This model is similar to the binomial model of CRR (1979) and Rendleman and Bartter (for details, refer to Bellalah et al., 1998). Ho and Lee (1986) introduced two perturbation functions h(T ) and h∗ (T ). (n) The discount function at period n and state i is P1 (.T ). The details are provided in the appendix of this chapter. They developed a non-arbitrage condition, that ensures that: πh(T ) + (1 − π)h∗ (T ) = 1

for n, i > 0

where π corresponds to the implied binomial probability. Hence, we have: (n+1) (n) (n) (n+1) Pi (T ) = πPi+1 (T − 1) + (1 − π)Pi (T − 1) Pi (1) This equation gives the bond price at a given node where the probability is given by: π = (r − d)/(u − d) with: r: return for one period, u return in an upstate, d return in a down state. Ho and Lee (1986) showed that: h(T ) =

1 π + (1 − π)δ T

for T ≥ 0 and h∗ (T ) =

δT π + (1 − π)δ T

where the term δ corresponds to the spread between the two perturbation functions. The dynamics of interest rates is completely specified by π and δ. 6.2.3. Computation of bond prices in the Ho and Lee model Example 1: Consider the following parameters in the Ho and Lee model: n = 4, δ = 0.994, π = 0.5, and t = 365 days. The term structure is specified by the following discount function observed at initial time and applying for periods 1 to 5: P00 (0) = 1, P00 (1) = 0.9826, P00 (2) = 0.9651,

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Extensions of Simple Binomial Option Pricing Models to Interest Rates Table 6.1.

299

Discount function and bond prices in the Ho and Lee model.

Period T Pij (k)

0

1

2

3

4

5

P00 (K) P10 (K) P11 (K) P20 (K) P21 (K) P22 (K) P30 (K) P31 (K) P32 (K) P33 (K) P40 (K) P41 (K) P42 (K) P43 (K) P44 (K)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.98260 0.97923 0.98514 0.9757 0.98164 0.98756 0.97235 0.97822 0.98412 0.99006 0.96915 0.97500 0.98088 0.98680 0.99276

0.96510 0.95837 0.96997 0.95164 0.96316 0.97482 0.94520 0.95664 0.96823 0.97995 — — — — —

0.94740 0.93752 0.95460 0.9278 0.94477 0.96198 — — — — — — — — —

0.92960 0.91687 0.93921 — — — — — — — — — — — —

0.9119 — — — — — — — — — — — — — —

P00 (3) = 0.9474, P00 (4) = 0.9296, and P00 (5) = 0.9119. Results are reproduced in Table 6.1. Example 2: Consider the following parameters to compute the discount functions in the Ho and Lee model: n = 4, δ = 0.994, π = 0.5, and t = 365 days. The present term structure is specified by the following discount function: P00 (0) = 1, P00 (1) = 0.982, P00 (2) = 0.961, P00 (3) = 0.941, P00 (4) = 0.921, and P00 (5) = 0.911. Results are reproduced in Table 6.2. 6.2.4. Option pricing in the Ho and Lee model Consider the pricing of an option on an interest-rate instrument, with a maturity T and a final payoff X(n, i) such that: C(T, i) = X(n, i) 0 ≤ i ≤ T . The underlying asset may be a bond, an interest rate, or a futures contract. The option price at moment n and position i can be located between a lower value L and an upper bound U . The possible prices of the option at instant n and position i satisfy the following relationship: L(n, i) ≤ C(n, i) ≤ U (n, i). The option holder receives at instant n and state i, the payoff X(n, i) for i in the interval 1 ≤ n < T . Ho and Lee (1986) showed that a hedged portfolio comprising options and zero-coupon bonds allowed the elimination

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Discount function and bond prices in the Ho and Lee model.

Period Pij (k)

0

1

2

3

4

5

P00 (K) P10 (K) P11 (K) P20 (K) P21 (K) P22 (K) P30 (K) P31 (K) P32 (K) P33 (K) P40 (K) P41 (K) P42 (K) P43 (K) P44 (K)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.98200 0.97567 0.98155 0.9732 0.97917 0.98508 0.96991 0.97576 0.98165 0.98758 0.9772 0.98313 0.98907 0.99504 1

0.96100 0.95248 0.96604 0.9468 0.95832 0.96992 0.95069 0.96220 0.97385 0.98564 — — — — —

0.94100 0.92941 0.94634 0.9309 0.94786 0.96513 — — — — — — — — —

0.92100 0.91653 0.93886 — — — — — — — — — — — —

0.91100 — — — — — — — — — — — — — —

of profitable arbitrage opportunities and leads to the following equation: C(n, i) = [π{C(n + 1, i + 1) + X(n + 1, i + 1)} (n)

+ (1 − π){C(n + 1, i) + X(n + 1, i)}]Pi

(1)

(n)

where Pi (1) is the price of a zero-coupon bond at the node (n, i) and π corresponds to the implied binomial probability. The option price must satisfy the condition: C(T − 1, i) = max[L(T − 1, i), min(C ∗ (T − 1, i), U (T − 1, i)]. The application of this model to the pricing of any type of interestcontingent claim requires the estimation of the probability π and the spread δ between the two perturbations functions. First, the discount function at the time of pricing must be estimated. Then, the parameters π and δ are estimated using a non-linear procedure like that in Ho and Lee (1986) or Whaley (1986). The estimation approach uses observed contingent claim prices and a pricing model in order to imply the parameters in the same way as we calculate an implied volatility for stock options. Example 3: Consider the pricing of a one-year put option on a Treasury bill. The option’s strike price is 980. The three-month T-bill has a face value of 1000 euros. The term structure is defined by the following parameters: n = 4, δ = 0.99, π = 0.5, and ∆t = 3 months. The discount

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0.9958

1

0.9914 0.9905 0.9860

0.9878 0.9814 0.992

0.9761

0.9779

0.9806

0.9716 0.9664

0.9708

0.9611

Fig. 6.4.

Dynamics of the Treasury bond.

function is: P00 (0) = 1, P00 (1) = 0.992, P00 (2) = 0.975, P00 (3) = 0.957, P00 (4) = 0.939, and P00 (5) = 0.921, where P00 (1) is the price of a defaultfree bond paying 1 in 3 months. Figure 6.4 shows the dynamics of the Treasury bond. At maturity, the option price is calculated using the following condition: P = max[980 − 1000P 4j(1), 0] for j = 0, 1, 2, 3, 4. Using the recursive procedure, the put option price in period 3, position 1 is 4,477, or (0,5(0) + 0,5(9,173))0.976 = 4,477. At period 2, the put price at the pair (2,0) is: (0.5(4,477) + 0,5(13,556))0,9716 = 8,761. As in Fig. 6.5, the option price at initial time is 3.196. Ritchken and Boenawen (1990) showed that the Ho and Lee model does not completely prevent the possibility of negative interest rates. These (n) negative rates disappear when the following constraint is used: Pn (1) < 1. To illustrate this point, consider the following parameters: δ = 0.9, π = 0.5, and n = 4. The observed initial term structure of interest rates for the following five periods is specified by: r0 (1) = 9.531%, r0 (2) = 8.6178%, r0 (3) = 7.696%, r0 (4) = 7.2321%, r0 (5) = 7.2321%, which gives: p(1) = e−r0 (1).1 , p(2) = e−r0 (2).2 , p(3) = e−r0 (3).3 , p(4) = e−r0 (4).4 , and p(5) = e−r0 (5).5

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Derivatives, Risk Management and Value 0

0

0 0 0

1.085 2.197 3.196

4.477

5.358

0

8.762 9.173

13.556

18.882

Fig. 6.5. Dynamics of the option price for the following parameters n = 4, δ = 0.99, π = 0.5, and K = 980.

P(1) = 1.04214

P(1) = 0.90909

P(1) = 0.97457 P(2) = 0.96486 P(3) = 0.9527 P(4) = 0.92531

P(2) = 0.84168

P(5) = 0.69655

P(2) = 1.08026 P(3) = 1.09826 P(1) = 0.98202 P(2) = 0.94322 P(1) = 0.93792

P(3) = 0.79383 P(4 )= 0.74880

P(1) = 1.091135 P(2) = 1.16448

P(2) = 0.87501 P(3) = 0.80063 P(1) = 0.87712 P(2) = 0.78154 P(3) = 0.6945 P(4) = 0.60709

P(1) = 0.88381 P(2) = 0.76401

P(1) = 0.84413 P(2) = 0.70875 P(3) = 0.58366

Fig. 6.6.

P(1) = 0.79543 P(2) = 0.61885

Possibility of negative interest rates in the Ho and Lee model.

Figure 6.6 reveals the possibility of negative interest rates in the Ho and Lee model. 6.2.5. Deficiency in the Ho and Lee model Ho and Lee model presents a deficiency. In fact, the constraints imposed on movements of the entire discount function are not sufficient to eliminate

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negative interest rates. This deficiency has been recognized by Heath et al. (1987), and Ritchken and Boenawen (1990) among others. These authors generalized the Ho and Lee model and provided the necessary adjustments in order to obtain an economically meaningful model. Ritchken and Boenawen (1990) showed that the restrictions imposed by Ho and Lee imply that bonds are priced by a model characterized by a “risk neutral” probability parameter π and an “interest-rate spread”, δ. However, the evolution of the discount function, namely the default-free zero-coupon bond prices, may not be bounded in the interval. To show this, they developed an example where π = 0.4, and δ = 0.8. They generated prices of pure discount bonds at all the vertices and found that some prices exceeded one. This indicates the presence of negative interest rates in the lattice. In fact, if one modifies their δ from 0.8 to 0.9 and generates bond prices, it is clear that some bond prices exceed one. To avoid negative interest rates, the constraint Pnn (1) < 1, must be added for each time period in the lattice.

6.3. Binomial Interest-Rate Trees and the Log-Normal Random Walk Consider a binomial interest-rate tree where the interest rate moves up (u) or down (d). The initial interest rate r0 corresponds to the current one-year interest rate when the length of each period is one year. It also corresponds to the one-year forward rate. The initial rate or the one-year forward rate can take on two possible values in the next period with the same probability of occurring. Hence, one rate results from a rise in rates and the other results from a fall in rates. Assume that the dynamics of interest rates are specified by a lognormal random walk process with a given volatility (Fig. 6.7). For the sake of clarity, let us denote respectively by: • σ: assumed volatility for the one-year forward rates; • r1,u : one-year forward rate, one year from now if rates rise and • r1,d : one-year forward rate, one year from now if rates fall. Hence, in the first year, there are two possible rates. We specify the following relationship between rising and falling rates as follows: r1,u = r1,d e2σ .

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Derivatives, Risk Management and Value r 2,uu r 1,u r0

r 2,ud

r 1,d

r2,dd

Fig. 6.7.

Dynamics of interest rates.

In the second year, there are three possible values for the one-year forward rates: r2,uu , r2,ud , r2,dd where: r2,uu : one-year forward rate in the second year if the interest rate rises in the first and the second year; r2,ud : one-year forward rate in the second year if the interest rate rises in the first year and falls in the second year or vice versa and r2,dd : one-year forward rate in the second year if the interest rate falls in the first and the second year. The same relationship between rising and falling interest rates is maintained. Hence, we have: r2,ud = r2,dd e2σ

and r2,uu = r2,dd e4σ

Let us denote simply by rt the one-year forward rate t years from now if the rates decline. The volatility of the one-year forward rate is equal to r0 σ. It is possible to see this result by noting that e2σ is nearly equal to (1 + 2σ). In this case, the volatility of the one-period forward rate can be written as: (re2σ − r)/2

which is nearly equal to (r + 2rσ − r)/2 or σr.

The process generating the interest-rate tree or the forward rates implies that volatility is measured with respect to the current level of rates. For example, if σ = 20% and the one-year rate is 3%, then the volatility

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of the one-year forward rate is 20% (3%) or 6% or 60 basis points. Hence, when the current one-year rate is 10%, the volatility of the one-year forward rate would be 10% × 20% or 200 basis points. The bond’s value at a given node can be computed using the immediate next two nodes since at a given node, the bond’s value will depend on the future cash flows. The future cash flows correspond to the bond value one year from now and the coupon payments. The binomial model is based on the recursive procedure starting from the last year and working backward through the tree until the initial time. The value at each node is given by the expected cash flows under the appropriate discount rate. The one-year forward rate at this node must be used. We denote these values by: Vu : Vd : C: r:

value of the bond if the one-year interest rate rises; value of the bond if the one-year interest rate falls; amount of the coupon payment and one-year forward rate at the node where valuation is sought.

Hence, the value of the bond at a given node is: Bond value = p(Vu + c)/(1 + r− ) + (1 − p)(Vd + c)/(1 + r− ) How to construct a binomial interest-rate tree? Consider the valuation of a two-year bond with a coupon rate of 4.5% when σ is 10%. Figure 6.8 shows the cash flows at each node. The initial interest rate is equal to 4.5%. Figure 6.9 shows how to calculate the one-year forward rates for the first year by a trial-and-error method. The model is based on an iterative process that allows the computation of the forward rates r1,d and r1,d , which are consistent with the volatility assumption and the observed market value of the bond. This can be done in different steps as shown below. Vu + c: cash flow in a high state

One-year rate at a node where the bond price is V calculated at the rate rVd + c: cash flow in a low state Fig. 6.8.

Computing the bond value at a given node.

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V = 100 C = 4.5 r2, uu = ? V = 99.748576 C = 5.4 r1, u = 4.7634% V = 99.567 C=0 r0 = 4.5%

V = 100 C = 4.5 r2, ud = ? V = 100.5774 C = 4.5 r1, d = 3.9%

V = 100 C = 4.5 r2, dd = ? Fig. 6.9. Computing the one-year forward rates for year 1 using a two-year 4.5% onthe-run issue: first trial.

•

Step 1: A value of r1 (the one-year forward rate, one year from now) is set arbitrarily to 3.9%. • Step 2: Since the one-year forward rate, if rates, rise corresponds to r1 e2σ , then r1,u = r1,d e2σ = 4.7634% = 3.9%e(2×0.1) . • Step 3: The bond’s value is computed for one year from now. The two-year bond’s value is given by its maturity value (100) plus the coupon payment at the same date (4.5), or 104.5. When interest rates rise, the present value is Vu = 99.748576 = 104.5/1.047634. When interest rates fall, the present value is Vd = 100.5774 = 104.5/1.039. At time 0, the present value resulting from a rise in the interest rate is computed as: (Vu + c)/1.045 = (99.748576 + 4.5)/1.045 = 99.7594. The present value resulting from a fall in the interest rate is computed as: (Vd + c)/1.045 = (100.5774 + 4.5)/1.045 = 100.5525.

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Using an equal probability, the value at time 0 is: Value = 0.5(99.113 + 100.021) = 100.15596. Since the observed market value of the bond at time 0 is 100, the computed value of 100.15596 is not the correct one. This means that the one-period forward interest rate used (r1 ) is not consistent with the volatility assumption of 10% and the process used to generate the one-year forward rate. Hence, the 3.9% rate is low and a higher rate must be used in the same procedure. If we use an interest rate of 4% for r1 , this leads to a bond price equal to 100 at time 0. Figure 6.10 is based on a second trial in the computation of the interest rate. • Step 1: A value of r1 (the one-year forward rate one year from now) is set arbitrarily to 4%. • Step 2: Since the one-year forward rate, if rates rise corresponds to r1 e2σ , then r1,u = r1,d e2σ = 4.8856% = 4%e(2×0.1) • Step 3: The bond’s value is computed one year from now. V = 100 C = 4.5 r2, uu = ? V = 99.6323 C = 4.5 r1, u = 4.8856% V = 99.567 C=0 r0 = 4.5%

V = 100 C = 4.5 r2, ud = ? V = 100.46 C = 4.5 r1, d = 4%

V = 100 C = 4.5 r2, dd = ? Fig. 6.10. Computing the one-year forward rates for year 1 using a two-year 4.5% on-the-run issue: second trial.

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The two-year bond’s value is given by its maturity value (100) plus the coupon payment at the same date (4.5), or 104.5. When interest rates rise, the present value is Vu = 99.6323 = 104.5/1.048856. When interest rates fall, the present value is Vd = 100.48076 = 104.5/1.04. At time 0, the present value resulting from a rise in the interest rate is computed as: (Vu + c)/1.045 = (99.6323 + 4.5)/1.045 = 99.6481. The present value resulting from a fall in the interest rate is computed as: (Vd + c)/1.045 = (100.48076 + 4.5)/1.045 = 100.46. Using an equal probability, the value at time 0 is: Value = 0.5(99.6481 + 100.46) = 100. Since the observed market value of the bond at time 0 is 100, the computed value of 100 is the correct one. This means that the one-year forward interest rate used (r1 ) is consistent with the volatility assumption of 10% and the process used to generate the one-year forward rate. Hence, the current one-year forward rate is 4.5% and the forward rate one year from now is 4%. It is possible to extend the analysis to three periods using a 5% three-year bond. The same method can be used in the computation of one-year forward rate, two years from now. This analysis allows to obtain the value of r2 that leads to a bond value of 100. We let this as an exercise for the interested reader.

6.4. The Black-Derman-Toy Model (BDT) This model is consistent with the observed term structure of interest rates. The volatility of the short rate is time dependent. The continuous process of the short rate is given by: d ln r = [θ(t) + (δσ(t)/δt)/σ(t)ln(r)]dt + σ(t)dz where (δσ(t)/δt)/σ(t) corresponds to the speed of mean-reversion.

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The following example calibrates the BDT model to the current term structure of zero-coupon rates and volatilities.

6.4.1. Examples and applications Example 4: Consider the following sample current data (Table 6.3): Consider the following tree for bond prices (Fig. 6.11). The price of a zero-coupon bond maturing in one year is given by: 92.59259 = [100(0.5) + 100(0.5)]/1.08.

Table 6.3.

Current data for zero-coupon bonds.

Years to maturity

Zero-coupon rates (in %)

Zero-coupon volatilities (in %)

1 2 3 4 5

8 8.5 9 9.5 10

20 18 16 14 12

100 Su 4 100

Su 3 Su 3d Su 2 Su2d

100

Su Su 2d 2 S

Sud 100 Su d 2

Sd Sd 2

Sd 3u 100 Sd 3 Sd 4 100

Fig. 6.11.

Binomial tree for bond prices.

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This allows the description of a one-period price tree: 100 92.59259 100

The price today of a two-year zero-coupon bond maturing in two years can be computed as: 84.945528 = 100/1.0852 = 100/1.177225. This second step allows to build a two-period price tree. The second-year bond prices at year one can be computed using the short rates at step one: 100 Su 84.945528

100 Sd 100

The following relationship must be verified: 84.945528 = [0.5Su + 0.5Sd ]/1.08. The standard relationships in the binomial models apply also in the BDT model. In fact, remember that in the standard binomial analysis, we have: u = eσ

√ T /N

,

√ ln(u/d) = 2σ T /N

√

u/d = e2σ T /N , √ and σ = [1/2 T /N ] ln(u/d).

d = 1/u,

In the BDT model, rates follow a log-normal distribution, so we have: √ σN = [1/2 T /N ] ln(ru /rd ) = 0.18 or σN = [1/2] ln(ru /rd ) = 0.18 and Sd = 100/(1 + ru ), Su = 100/(1 + rd ).

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ru4 ru 3 ru 2

ru 3d 2

ru

ru d

r

ru 2d 2

rud rud 2

rd

rd 3u

rd2 rd

3

rd 4 Fig. 6.12.

Binomial tree for interest rates.

Substituting these expressions, it gives: 84.945528 = [0.5{100/(1 + ru )} + 0.5{100/(1 + rd )}]/1.08. We can use the previous equations to determine the two values of the interest rate (Fig. 6.12). Since ru = rd e0.18(2) , this gives the following quadratic equation: 84.945528 = [0.5{100/(1 + rd e0.18(2) )} + 0.5{100/(1 + rd )}]/1.08. Solving this equation gives the following two rates: rd = 7.4%, ru = 10.60%. Since these two rates are known, it is possible to use these rates to compute the corresponding bond prices, i.e., 93.11 = (100/1.074) and 90.41 = (100/1.1060). The tree becomes: 100 90.41 84.11

100 93.11 100

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The next step gives the following exhibit. ruu 10.60% 8%

rud = rdu 7.4% rdd

In order to find the rates at the end of period 2, we use the fact that rates are log-normally distributed and that volatility is time dependent. In this case, we have: 0.5 ln(ruu/rud) = 0.5 ln(rud/rdd)

or rdd = r2 ud/ruu.

As before, the price today of a bond maturing in three years is: 77.22 = 100/(1 + 0.09)3 Using the risk-neutral valuation principle, we have: Suu = 100/(1 + ruu),

Sud = 100(1 + rud),

Sdd = 100/(1 + rdd)

Su = [0.5 Suu + 0.5 Sdd]/(1 + 0.1060), Sd = [0.5 Sdd + 0.5 Sud]/(1 + 0.074), 77.22 = [0.5 Su + 0.5 Sd]/(1 + 0.08). If the bond’s maturity is in two years, its yield must satisfy the following: Su = 100/(1 + yu )2 ,

Sd = 100/(1 + yd )2

or yu =

√ (100/Su) − 1,

yd =

√ (100/Sd) − 1

We have: ln(yu /yd ) = 0.16 or ln(yu /yd ) = 0.32, which gives: yu /yd = e0.32 . Using these last two equations, it is possible to obtain: yu = (yu /yd )yd

√ or yu = e0.32 ( (100/Sd) − 1).

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In this case, Su can be written as: √ Su = 100/[1 + e0.32 ( (100/Sd) − 1)]2 . This equation can be solved using the Newton–Raphson algorithm. The solution gives the values of Su, Sd, rdd, rud, and ruu. 6.5. Trinomial Interest-Rate Trees and the Pricing of Bonds 6.5.1. The model There are three possible movements: up, down, and stay the same. Consider an initial short-term interest rate r of 5% per annum (r = 0.05). The standard deviation is 0.01 per year. The drift pulls the interest rate back to its level of 5% at a rate of 10% per year. Over a short-term interval, the expected increase in the interest rate is written as: 0.1(0.05 − r)∆t. √ The standard deviation is 0.01 ∆t. The interest-rate process is: dr = 0.1(0.05 − r)dt + 0.01dz. The dynamics of the interest rate can be modeled using a grid of equally spaced rates. The probabilities corresponding to up, p1 , down, p2 , and “stay the same”, p3 can be computed in a way to preserve the correct mean and standard deviation at each node. The sum of these probabilities must be equal to 1. Figure 6.13 describes a two-step tree where each interval is one year. The spacing between different rates is 1.5%. The initial value of r is 0.05. Since 0.1(0.05 − r) = 0 for the initial value of r, the expected increase in the interest rate during the first period is zero. The standard deviation is 0.01. The probabilities must verify: p1 + p2 + p3 = 1. The probabilities satisfy the following equation for the expected values: (0.05 + 0.015)p1 + (0.05 + 0)p2 + (0.05 − 0.015)p3 = 0.05 or 0.065p1 + 0.05p2 + 0.035p3 = 0.05.

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r= C

0.05

r=

B

r = 0.05

r= A

r= Fig. 6.13.

The construction of the trinomial tree.

The probabilities also verify the following equation for the variance: 0.0652p1 + 0.052 p2 + 0.0352p3 − 0.052 = 0.012. The solution to the following system gives the appropriate probabilities: p1 + p2 + p3 = 1 0.065p1 + 0.05p2 + 0.035p3 = 0.05 2

0.065 p1 + 0.052 p2 + 0.0352p3 − 0.052 = 0.012 Hence: p1 = 0.222222;

p2 = 0.555556,

and p3 = 0.222222.

For the second period, at the node A, the expected increase in the interest rate is: 0.1(0.05 − 0.035) = 0.0015, so that the expected increase in the interest rate at the end of the year is: 0.035 + 0.0015 = 0.0365. The standard deviation is 0.01. Using a system of three equations allows the computation of the different probabilities. Hence, we have: p1 + p2 + p3 = 1 (0.035 + 0.015)p1 + (0.035 + 0)p2 + (0.035 − 0.015)p3 = 0.0365

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or 0.05p1 + 0.035p2 + 0.02p3 = 0.0365 and 0.052 p1 + 0.0352 p2 + 0.022 p3 − 0.03652 = 0.012 . The solution to the following three equations is: p1 + p2 + p3 = 1 0.05p1 + 0.035p2 + 0.02p3 = 0.0365 2

0.05 p1 + 0.0352p2 + 0.022p3 − 0.03652 = 0.012 is p1 = 0.277222;

p2 = 0.545556,

and p3 = 0.177222

For the second period, at the node B, the expected increase in the interest rate is: 0.1(0.05 − 0.05) = 0.00, so that the expected increase in interest rates at the end of the year is: 0.05 + 0.00 = 0.05. The standard deviation is 0.01. Using a system of three equations allows the computation of different probabilities. Hence, we have: p1 + p2 + p3 = 1 (0.05 + 0.015)p1 + (0.05 + 0)p2 + (0.05 − 0.015)p3 = 0.05 or 0.065p1 + 0.05p2 + 0.035p3 = 0.05 and 0.0652p1 + 0.052 p2 + 0.0352p3 − 0.052 = 0.012 . The solution to these three equations is: p1 = 0.222222,

p2 = 0.555556,

p3 = 0.222222.

For the second period, at the node C, the expected increase in the interest rate is: 0.1(0.05 − 0.065) = −0.0015, so that the expected increase at the end of the year is: 0.065 − 0.0015 = 0.0635. The standard deviation is 0.01.

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Using a system of three equations allows the computation of different probabilities: p 1 + p2 + p3 = 1 (0.065 + 0.015)p1 + (0.065 + 0)p2 + (0.065 − 0.015)p3 = 0.0635 or 0.08p1 + 0.065p2 + 0.05p3 = 0.0635 and 0.082 p1 + 0.0652p2 + 0.052 p3 − 0.06352 = 0.012 . The solution to these three equations is: p1 = 0.177222,

p2 = 0.545556,

and p3 = 0.277222.

Derivative securities can be valued using the recursive method through the trinomial tree in the same way, as in the binomial model. This model applies also to discount bonds. Using bond prices, it is also possible to determine the complete yield curve at any given node of the trinomial lattice. It is convenient to note that the tree is used to model a function or the short rate r, f (r) with the same volatility as r in the small time interval. When the volatility of r is constant, r can be modeled directly. When the volatility of r is proportional to r, as it is the case in the lognormal model, f (r) = log(r) can be modeled directly. When the volatility of r is proportional to rα , f (r) can be r(1−α) . The trinomial model proposed above is based on the following branching process: up, down, and stay the same. This is not the only way to construct the trees. It is also possible to construct a trinomial tree as illustrated in the following figure (Fig. 6.14). The trinomial model allows the user to specify future volatility of the short rate, the current volatility of spot rates, and the volatility associated with the current-term structure of interest rates. It is important to remember that this trinomial approach is a variation of finite difference methods. 6.5.2. Applications of the binomial and trinomial models We consider the valuation of a stock option using the binomial model and the trinomial model (Table 6.4). The option is priced on 15/06/2002.

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Fig. 6.14. Table 6.4.

Possible branching in a trinomial tree.

Pricing a call using the binomial and trinomial models. Call price

S = 18

S = 19

S = 20

S = 21

Binomial Trinomial Binomial Trinomial

2.99 2.99 2.73 2.77

3.50 3.47 3.22 3.21

4.02 4.00 3.73 3.70

4.54 4.54 4.25 4.23

K = 24

Binomial Trinomial

2.55 2.54

2.94 2.98

3.45 3.42

3.96 3.93

K = 25


2.37 2.32 2.19 2.16

2.75 2.76 2.57 2.54

3.16 3.20 2.95 2.97

3.68 3.64 3.39 3.41

K = 27

Binomial Trinomial

2.01 2.01

2.39 2.35

2.77 2.75

3.15 3.19

K = 28


1.83 1.85 1.68 1.70

2.21 2.20 2.03 2.04

2.59 2.54 2.41 2.39

2.97 2.97 2.79 2.75

Binomial Trinomial

1.58 1.57

1.86 1.89

2.23 2.23

2.61 2.58

K = 22 K = 23

K = 26

K = 29 K = 30

317

The maturity date is 15/06/2004. The following strike prices are used: 22, 23, 24, 25, 26, 27, 28, 29, and 30. The following dates and amounts of dividends are available: For 1999, 0.25; for 2000, 0.25; for 2001, 0.25; for 2002, 0.3; for 2003, 0.35 and for 2004, 0.35.

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Pricing a call using the binomial and trinomial models. S = 22

S = 25

S = 28

S = 30


5.08 5.09 4.77 4.77

7.02 6.99 6.60 6.58

9.05 9.08 8.57 8.58

10.58 10.54 10.01 10.02

K = 24

Binomial Trinomial

4.48 4.46

6.18 6.18

8.14 8.09

9.47 9.51

K = 25


4.19 4.16 3.91 3.86

5.77 5.78 5.46 5.47

7.71 7.68 7.29 7.28

9.03 9.01 8.60 8.55

K = 27

Binomial Trinomial

3.62 3.63

5.17 5.16

6.88 6.87

8.18 8.14

K = 28


3.36 3.41 3.17 3.18

4.88 4.85 4.59 4.55

6.46 6.48 6.15 6.16

7.75 7.74 7.34 7.34

Binomial Trinomial

2.99 2.96

4.31 4.28

5.86 5.85

6.92 6.94

K = 22 K = 23

K = 26

K = 29 K = 30

We use a historical simulation to estimate the volatility parameter (Table 6.5). The interest rate is 5%. The annualized volatility is between 45% and 50%. In this analysis, a dividend rate is used by dividing the dividend amount by the initial underlying asset price (Table 6.6). The annualized volatility is 45% (Table 6.7). The annualized volatility is 50%. The reader can compare the differences between both these models. Summary Rendleman and Bartter (for details, refer to Bellalah et al., 1998) developed a similar model for the pricing of interest-rate sensitive instruments. Ho and Lee (1986) extended the binomial model for the valuation of interest-rate options and bond options. This model presents some deficiencies. In fact, the constraints imposed on movements of the entire discount function are not sufficient to eliminate negative interest rates. This deficiency has been recognized by Heath, et al. (1987), Pedersen et al. (for details, refer to Bellalah et al., 1998), and Ritchken and Boenawen (1990) among others. These authors proposed other specific models.

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Extensions of Simple Binomial Option Pricing Models to Interest Rates Table 6.6.

Pricing a call using the binomial and trinomial models. S = 18

S = 19

S = 20

S = 21


3.48 3.45 3.20 3.23

4.00 3.97 3.73 3.69

4.53 4.52 4.26 4.23

5.07 5.07 4.79 4.77

K = 24

Binomial Trinomial

2.99 3.02

3.46 3.47

3.98 3.93

4.51 4.48

K = 25


2.82 2.81 2.65 2.60

3.22 3.26 3.04 3.05

3.71 3.71 3.44 3.50

4.24 4.19 3.96 3.95

K = 27

Binomial Trinomial

2.48 2.45

2.87 2.83

3.27 3.28

3.69 3.74

K = 28


2.31 2.30 2.14 2.16

2.70 2.66 2.53 2.51

3.10 3.07 2.92 2.87

3.49 3.52 3.32 3.31

K = 30

Binomial Trinomial

1.97 2.01

2.36 2.36

2.76 2.72

3.15 3.10

Table 6.7.

Pricing a call using the binomial and trinomial models.

K = 22 K = 23

K = 26

K = 29

S = 22

S = 25

S = 28

S = 30

K = 22

Binomial Trinomial

5.62 5.63

7.60 7.57

9.62 9.65

11.13 11.12

K = 23


5.33 5.33 5.04 5.03

7.19 7.17 6.79 6.79

9.19 9.16 8.78 8.73

10.58 10.62 10.12 10.13

K = 25

Binomial Trinomial

4.77 4.74

6.38 6.40

8.36 8.33

9.70 9.64

K = 26


4.49 4.44 4.22 4.19

6.09 6.10 5.81 5.80

7.96 7.94 7.55 7.55

9.28 9.24 8.87 8.84

K = 28

Binomial Trinomial

3.94 3.98

5.53 5.50

7.15 7.17

8.47 8.45

K = 29


3.72 3.76 3.54 3.55

5.25 5.21 4.98 4.92

6.86 6.86 6.57 6.56

8.06 8.07 7.66 7.68

K = 24

K = 27

K = 30

319

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Trigeorgis (1993) applied the methodology to the valuation of investments with multiple real options and to the pricing of managerial flexibility implicit in investment opportunities. Hull and White (1988,1993) provided a modified binomial lattice model for the valuation of stock options and interest-rate options. Boyle (1986) proposed a trinomial option-pricing model in which the stock price can move either upwards or downwards or stay unchanged at a given time period. In another paper, Boyle (1988) showed how a five-jump, three-dimensional lattice can be used for the valuation of options on two underlying assets. Omberg (1988) studied a family of discrete-time jump processes and applied a Gauss-Hermite quadrature technique to derive the prices of options on options or compound options. Yisong (1993) modified Boyle’s approach by presenting a general methodology that can be applied to any multidimensional lattice approach. He proposed a modified approach to the selection of lattice parameters including probabilities and jumps using additional restrictions. Sandmann (1993) developed a model for the pricing of European options under the assumption of a stochastic interest rate in a discrete time setting. He used a combination of the binomial model for a stock with a binomial model for the spot interest rate. Derivative securities can be valued using the recursive method through the trinomial tree in the same way, as in the binomial model. This model also applies to discount bonds. Using bond prices, it is also possible to determine the complete yield curve at any given node of the trinomial lattice. The trinomial model proposed by Hull and White allows the user to specify future volatility of the short rate, the current volatility of spot rates, and the volatility associated with the current, term structure of interest rates. It is important to remember that this trinomial approach is a variation of finite difference methods.

Questions 1. Describe the Rendleman and Bartter model (1979) for interest-rate sensitive instruments. 2. Describe the Ho and Lee model for interest rates and bond options. 3. Describe the binomial interest-rate trees and the log-normal random walk.

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321

4. Describe the Black-Derman-Toy model. 5. Describe the trinomial interest-rate model and the pricing of bonds. 6. What are the specific features of the Ho and Lee approach in the description of the term structure of interest rates? 7. What are the specific features of the Ho and Lee approach for the valuation of interest-rate-dependent contingent claims? 8. What are the deficiencies in the Ho and Lee model?

Appendix A Ho and Lee model and binomial dynamics of bond prices When the term structure is described by a binomial model, the same dynamics applies for zero-coupon bonds P (N ) with a maturity date N at initial time. When it remains (N − 1) periods, the discounting functions (1) in upstates and down states allow the computation of P1 (N − 1) and P01 (N − 1). This model is similar to the binomial model of CRR (1979) and Rendleman and Bartter. Ho and Lee introduced two perturbation functions, (n) h(T ) and h ∗ (T ). The discount function at period n and state i is Pi (.T ). The discounting function that prevents risk-less profitable arbitrage is known as the implied forward discount function that is specified by (n) Fi (T ), or: (n)

Fi

(n+1)

(T ) = Pi

(n+1)

(T ) = Pi+1

(n)

(T ) =

[Pi

(T + 1)]

(n) [Pi (1)]

for T = 0, 1, . . .

The two functions h(T ) and h∗ (T ) ensure that: (n+1)

Pi+1

(n+1)

Pi

(n)

(T ) = [Pi

(n)

(T ) = [Pi

(n)

(T + 1)/Pi

(n)

(T + 1)/Pi

(1)]h(T )

(A.1)

(1)]h∗ (T )

(A.2)

h(0) = h∗ (0) = 1 The non-arbitrage condition ensures that: πh(T ) + (1 − π)h∗ (T ) = 1 for n, i > 0

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where π corresponds to the implied binomial probability. Hence, we have: (n)

Pi

(n+1)

(T ) = [πPi+1

(n+1)

(T − 1) + (1 − π)Pi

(n)

(T − 1)]Pi

(1).

This equation gives the bond price at a given node where the probability is given by: π = (r − d)/(u − d) with: r: return for one period, u: return in an upstate, and d: return in a down state. (n) Consider the discount function Pi (T ) at state i and time n. Using Eqs. (A.1) and (A.2), we have: (n+2)

Pi+1

(n+2)

Pi+1

(n)

(T ) =

Pi

(n)

Pi (n)

(T ) =

(T + 2)

Pi

(T + 2)

(n)

Pi

(2)

(2)

h(T + 1)h∗ (T ) h(1) h∗ (T + 1)h(T ) . h∗ (1)

The independence condition (reflecting the fact that an upward movement followed by a downward movement is equivalent to a downward movement followed by an upward movement) allows to write: h(T + 1)h∗ (T )h∗ (1) = h∗ (T + 1)h(T )h(1). Elimination of h∗ gives: h(T + 1)[1 − πh(T )][1 − πh(1)] = (1 − π)h(1)h(T )[1 − πh(T + 1)]. Simplifying for T = 1 gives: δ 1 = +Γ h(T + 1) h(T ) where the constants δ and Γ are defined in a way such that: h(1) =

1 ; π + (1 − π)δ

Γ=

π(h(1) − 1) . (1 − π)h(1)

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323


The term δ corresponds to the spread between the two perturbation functions. Ho and Lee showed that: h(T ) =

1 π + (1 − π)δ T

for T ≥ 0

and h∗ (T ) =

δT . π + (1 − π)δ T

The dynamics of interest rates is completely specified by π and δ. Using Eqs. (A.1) and (A.2), it is possible to obtain the discount function for any moment as a function of the initial value, or: Pin (T ) =

P (T + n)h∗ (T + n − 1)h∗ (T + n − 2) · · · h∗ (T + i)h(T + i − 1) · · · h(T ) P (n)h∗ (n − 1)h∗ (n − 2) · · · h∗ (i)h(i − 1) · · · h(1)

or (n)

Pi

(T ) =

P (T + n)h(T + n − 1)h(T + n − 2) · · · h(T )δ T (n−1) . P (n)h(n − 1)h(n − 2) · · · h(1)

(A.3)

Equation (A.3) gives the discount function that can be applied at each time step. When T = 1, the bond price is: (n)

Pi

(T ) =

P (n + 1)δ n−i . P (n)(π + (1 − π)δ n ) (n)

(n)

(n)

And the one-period interest rate ri (1) is: ri (1) = − ln Pi (n)

ri (1) = ln

(1) or

P (n) + ln(πδ −n + (1 − π)) + i ln δ. P (n + 1) (n)

In the presence of a probability q, for each moment n, ri (1) follows a binomial distribution with a mean µ and a variance, σ with: µ = ln[P (n)/P (n + 1)] + ln(πδ −n + (1 − π)) + nq ln δ or: µ = ln[P (n)/P (n + 1)] + ln(πδ −(1−q)n + (1 − π)δ qn ) and: σ = nq(1 − q)(ln δ)2 Do not forget that notional amount of debt has to be monitored carefully after what happened in 2008 with the credit crunch.

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Exercises Example 1. Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 80, interest rate = 0.05, volatility = 0.2, T = 5 months, and N = 5. In this case, we have: p = 0.5217, d = 0.9439, and u = 1.0594. Dynamics of the underlying asset for five periods

80

84.7547 75.5120

89.7921 80 71.2758

95.1288 84.7547 75.5120 67.2772

100.7827 89.7921 80 71.2758 63.5030

106.7726 95.1288 84.7547 75.5120 67.2772 59.9404

Valuation of European put option

18.1946

14.0858 22.8351

9.9155 18.7578 27.4820

5.8694 14.4154 23.6581 31.8929

2.3202 9.7921 19.5842 28.3084 36.0812

0 4.8712 15.2453 24.4880 32.7228 40.0596

Valuation of American put option

20

15.2453 24.4880

10.4136 20 28.7242

6.0675 15.2453 24.4880 32.7228

2.3202 10.2079 20 28.7242 36.4970

0 4.8712 15.2453 24.4880 32.7228 40.0596

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325

Valuation of European call option

0.2564

0.4934 0

0.9498

1.8281 0

0

0

0

0

3.5187 0 0 0 0

6.7726 0 0 0 0 0

Valuation of American call option

0.2564

0.4934 0

0.9498 0 0

1.8281 0 0 0

3.5187 0 0 0 0

6.7726 0 0 0 0 0

References Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654. Black, F, E Derman and W Toy (1990). A one factor model of interest rates and its application to treasury bond options. Financial Analysts Journal, 46 (January–February), 33–39. Boyle, P (1986). Option valuation using a three jump process. International Options Journal, 3, 7–12. Boyle, PP (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23 (March) 1–12. Briys, E, M Bellalah, F de Varenne and H Mai (1998). Options, Futures and Other Exotics. Chichester, UK: John Wiley and Sons. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7, 229–263. Heath, D, R Jarrow and A Morton (1987). Bond pricing and the term structure of interest rate: a new methodology for contingent claims valuation. Working paper, Ithaca, NY: Cornell University (Revised edition, 1989).

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Ho, T and S Lee (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41, 1011–1029. Hull, J (2000). Options, Futures, and Other Derivative Securities. NJ, USA: Prentice Hall International Editions. Hull, J and A White (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Advances in Futures and Options Research, 3, 29–61. Hull, J and A White (1993). Efficient procedures for valuing European and American path dependent options. Journal of Derivatives, 1 (Fall 1993), 21–31. Jarrow, RA and A Rudd (1983). Option Pricing. Homewood, IL: Irwin. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Omberg, E (1988). Efficient discrete time jump process models in option pricing. Journal of Financial and Quantitative Analysis, 23(2), 161–174. Rendleman, RJ and BJ Bartter (1980). The pricing of options on debts securities. Journal of Financial and Quantitative Analysis, 15(March), 11–24. Ritchken, P and K Boenawen (1990). On arbitrage free pricing of interest rate contingent claims. Journal of Finance, 55(1), 259–264. Rubinstein, M (1994). Implied binomial trees. Journal of Finance, 49(3), 771–818. Sandmann, K (1993). The pricing of options with an uncertain interest rate: a discrete time approach. Mathematical Finance, 3(April), 201–216. Trigeorgis, L (1993). The nature of option interactions and the valuation of investments with multiple real options. Journal of Financial and Quantitative Analysis, 28, 1–20. Whaley, RE (1986). Valuation of American futures options: theory and empirical tests. Journal of Finance, 41(March), 127–150. Yisong, T (1993). A modified lattice approach to option pricing. Journal of Futures Markets, 13, 563–577.

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Chapter 7 DERIVATIVES AND PATH-DEPENDENT DERIVATIVES: EXTENSIONS AND GENERALIZATIONS OF THE LATTICE APPROACH BY ACCOUNTING FOR INFORMATION COSTS AND ILLIQUIDITY

Chapter Outline This chapter is organized as follows: 1. Section 7.1 presents the lattice approach and the binomial model for the valuation of equity and futures options. 2. Section 7.2 presents a simple extension of the lattice approach to account for the effects of information costs. 3. Section 7.3 develops some important results regarding the binomial model and the risk neutrality. 4. Section 7.4 presents the Hull and White’s interest-rate trinomial model for the valuation of interest-rate derivatives. 5. Section 7.5 develops a simple context for the pricing of path-dependent interest-rate contingent claims using a lattice.

Introduction This chapter deals with the valuation of derivative assets using the binomial or the lattice approach. The lattice approach was initiated by Cox et al. (CRR) (1979). CRR approach considers the situation where there is only a single underlying asset: the price of a non-dividend paying stock. The time to maturity of the option is divided into several equal 327

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small intervals during which the underlying asset price moves from its initial value to one of the two new values up or down. In a risk-neutral world, it is possible to obtain the values corresponding to an upward or downward movements and the corresponding probabilities. The valuation of options in a binomial framework starts at the maturity date since at this date, the option payoff is known. Then, we proceed backward through the binomial tree from the maturity to the initial time. In a riskneutral world, the option value at each time can be calculated as the expected value at the maturity date discounted at the risk-less rate of interest. The lattice approach can be easily extended to account for the effects of a continuous dividend yield. If a security pays a dividend yield, then the expected return on the underlying asset is given by the difference between the risk-less rate and the continuous dividend yield. The extension of the lattice approach in the presence of discrete dividends to the valuation of options on stocks paying a known dividend can be easily implemented. In the presence of a discrete dividend, the pricing problem can be simplified as in Hull (2000) by assuming that the implicit spot stock price has two components: a part which is stochastic and a part which is the present value of all future cash payments during the option’s life. The lattice approach has also been used by several authors to model the dynamics of the term structure of interest rates and to value bonds and bond options. There have been many attempts and approaches to describe yield-curve movements using a one-factor model. The approach presented by Ho and Lee (1986), in the form of a binomial tree for discount bonds, provides an exact fit to the current-term structure of interest rates. Their model is interesting since it takes the market data such as the current-term structure of interest rates as given. In this respect, it is close to a binomial stock option pricing approach where the current stock price is taken as an input to the model. Unlike most interestrate contingent claims models, this model uses full information on the current-term structure. In fact, using an ingenious discrete-time approach for pricing bonds and interest-rate contingent claims, Ho and Lee (1986) succeeded in incorporating all information about the yield curve in their model. Hull and White (1993) presented a general numerical procedure involving the use of trinomial trees for constructing one-factor models where the short rate is Markovian and the models are consistent with initial market data. Their procedure is efficient and provides a convenient way

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Derivatives and Path-Dependent Derivatives

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329

of implementing models already suggested in the literature. In this respect, their contribution is more a numerical one than a financial one. This chapter covers binomial and more general lattice approaches for the pricing of equity and interest-rate dependent claims. Dharan (1997) developed a simple framework for the valuation of pathdependent interest-rate claims such as index-amortizing swaps or mortgagebacked securities with a simple pre-payment function for which there is no analytical solution. Dharan (1997) showed how to construct a lattice to value a mortgage-backed security when the pre-payment function is linear. The different models are illustrated in detail using several numerical examples. 7.1. The Standard Lattice Approach for Equity Options: The Standard Analysis We re-call the standard lattice approach here in order to allow its extension to account for the effects of information costs on the pricing of derivatives. 7.1.1. The model for options on a spot asset with any pay outs The lattice approach can be introduced by first looking at a stock option whose underlying asset does not pay any dividend. Let T be the option’s maturity date, which is divided into N reasonably small intervals of length ∆t, so that T = N ∆t. In this world, the expected return on the underlying asset in time ∆t is r∆t. The variance of this underlying asset on the same interval is σ2 ∆t. For the valuation of derivative assets, re-call that the expected value of the underlying asset in a risk-neutral world is Ser∆t . In this context, we can write the equality between this expected value and the one given by the binomial model as: pSu + (1 − p)Sd = Ser∆t

or pu + (1 − p)d = er∆t

(7.1)

Since the variance of a variable X is given by E(X 2 ) − E(X)2 , we can write the equality between this variance and S 2 σ 2 ∆t: S 2 σ 2 ∆t = S 2 (pu2 + (1 − p)d2 ) − S 2 (pu + (1 − p)d)2 or σ 2 ∆t = pu2 + (1 − p)d2 − (pu + (1 − p)d)2

(7.2)

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If we use Eqs. (7.1), (7.2) and suppose that u = d1 , then we have: u = eσ

√

∆t

,

d = e−σ

√

∆t

,

and p =

a−d , u−d

a = er∆t .

The expected return on the stock can be written as pSu + (1 − p)Sd = Se . The variance on the stock can be written for the same interval ∆t as: r∆t

pS 2 u2 + (1 − p)S 2 d2 − (Ser∆t )2 . √

√

This last expression can be written as S 2 (er∆t (eσ ∆t + e−σ ∆t) − 1 − e2r∆t ). 2 3 Now, using the expansion of ex in series form as ex = 1+x+ x2 + x6 +· · · It is clear that when terms of order ∆t2 and higher are ignored, the variance of the stock price is S 2 σ 2 ∆t. This shows that we have the appropriate values for u, d, and p. The nature of the lattice of stock prices is completely specified and the nodes correspond to: Suj di−j

where j = 0, 1, . . . , i.

The option is evaluated by starting at time T and working backward. 7.1.2. The model for futures options Merton (1973), Black (1976), and Barone-Adesi and Whaley (1987) among others, showed that futures contracts, stock index options, and currency options may be assimilated to options on a stock that pays a continuous dividend. In a risk-neutral economy, the expected return on an asset paying a continuous dividend yield δ is (r − b) so that we can write e(r−b)∆t = pu + (1 − p)d and a = e(r−b)∆t. Hence, the model for futures options is completely specified using the following equations: u = eσ

√

∆t

,

d = e−σ

√ ∆t

,

p=

a−d , u−d

and a = e(r−b)∆t.

For a futures contract, r = b so that a = 1. 7.1.3. The model with dividends We study three cases: a known dividend yield, a known proportional dividend yield, and a discrete dividend.

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331

7.1.3.1. A known dividend yield The extension of the lattice approach in the presence of a continuous dividend yield is simple. If a security pays a dividend yield q, then the expected return on the underlying asset is (r − q). In this case, we can write the equality between the expected value of the underlying asset and the one given by the binomial model as pSu + (1 − p)Sd = Se(r−q)∆t or pu + (1 − p)d = e(r−q)∆t . In this case, using the same procedure as before, it can be shown that the parameters for the binomial model are: u = eσ

√

∆t

,

d = e−σ

√

∆t

,

p=

a−d , u−d

a = e(r−q)∆t .

The value of a European contingent claim Fi,j at time (t + i∆t) when the underlying asset is Suj di−j can be calculated using the following equation: Fi,j = e(−r)∆t [(pFi+1,j+1 + (1 − p)Fi+1,j )] In the same context, at time t + i∆t, the American call option value is: Fi,j = max[Suj di−j − K, e−r∆t(pFi+1,j+1 + (1 − p)Fi+1,j )] At each node, at time t + i∆t, the American put value is given by, Fi,j = max[K − Suj di−j , e−r∆t(pFi+1,j+1 + (1 − p)Fi+1,j )]. It is convenient to note that these formulas apply to index options, currencies, and futures contracts. 7.1.3.2. A known proportional dividend yield The extension of the lattice approach in the presence of discrete dividends to the valuation of options on stocks paying a known dividend is as follows. Assume that a known proportional dividend yield δ is to be paid at a certain date. When there is only one dividend at the time (t + i∆t), the nodes correspond to the stock prices Suj di−j for j = 0, 1, . . . , i where the time (t + i∆t) is prior to the stock going ex-dividend. √ √ a−d The values of u, d, and p are u = eσ ∆t , d = e−σ ∆t , and p = u−d respectively. When the time (t + i∆t) is after the underlying stock goes ex-dividend, then the nodes give the following prices S(1 − δ)uj di−j for j = 0, 1, . . . , i. The same analysis applies when there are several dividends. In this case, the total dividend yield δi corresponding to all ex-dividend

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dates between t and (t + i∆t) is accounted for using the following values of the stock at different nodes: S(1 − δi )uj di−j for j = 0, 1, . . . , i.

7.1.3.3. A known discrete dividend In general, the dividend amount D is known, rather than the dividend yield. Assume that there is only one dividend date, τ , between an instant k∆t and (k + 1)∆t. When there is just one ex-cash income date τ , during the option’s life, then the nodes on the tree at time (t + i∆t) are Suj di−j for i ≤ k with j = 0, 1, 2, . . . , i. The nodes on the tree at time i = (k + 1) are: (Suj di−j − D)u and (Suj di−j − D)d for j = 0, 1, 2, . . . , i. The analysis can be simplified as before.

7.1.4. Examples In these examples, we use the following parameters for the valuation of a European and an American put option on a stock paying a dividend of 2.05 in three months and a half: S ∗ = 40, S = 42, K = 45, r = 0.1, N = 5, T = 5 months, ∆t = 1 month, and σ = 0.4. The √ first step is√the calculation of the parameters u, d, a, and p using u = eσ ∆t , d = e−σ ∆t , a−d , q = 1 − p, a = er∆t . This gives u = 1.1224, d = 0.8909, a = 1.0084, p = u−d p = 0.5073, and q = 0.4927. Using these parameters, it is possible to generate the dynamics of the underlying asset. The values of the underlying asset at different nodes are: S0,0 = 42; S1,1 = 46.9136,

S1,0 = 37.6556;

S2,2 = 52.4256,

S2,1 = 42.0344,

S2,0 = 33.7859;

S3,3 = 58.6107,

S3,2 = 46.9475,

S3,1 = 37.6893,

S4,4 = 63.4822,

S4,3 = 50.3914,

S4,2 = 40,

S3,0 = 30.3403;

S4,1 = 31.7515,

S4,0 = 25.2039 and S5,5 = 71.2525, S5,1 = 28.2289,

S5,4 = 56.5593,

S5,3 = 44.8960,

S5,2 = 35.6379,

and S5,0 = 22.4554.

The indices (i, j) correspond respectively to the period and the position. The lowest position on a tree is indexed by zero. For example, the values

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of the underlying asset in the nodes following nodes are generated as follows: S0,0 = S ∗ (u0 d0 ) + De−r 12 , τ

S1,1 = S ∗ (u1 d0 ) + De−r

(τ −1) 12

S1,0 = S ∗ (u0 d1 ) + De−r

(τ −1) 12

, S2,2 = S ∗ (u2 d0 ) + De−r

(τ −2) 12

S2,1 = S ∗ (u1 d1 ) + De−r

(τ −2) 12

, S2,0 = S ∗ (u0 d2 ) + De−r

(τ −2) 12

S3,3 = S ∗ (u3 d0 ) + De−r

(τ −3) 12

, S3,0 = S ∗ (u0 d3 ) + De−r

(τ −3) 12

When y > τ , we do not discount the dividends and the above values are generated as follows: S4,4 = S ∗ (u4 d0 ), S4,3 = S ∗ (u3 d1 ), S4,0 = S ∗ (u0 d4 ), and S5,0 = S ∗ (u0 d5 ).

7.1.4.1. The European put price with dividends The values are computed by starting at the maturity date. At this date, the possible option values are: P5,5 = 0, P5,4 = 0, S5,3 = 0.104, P5,2 = 9.3621, P5,1 = 16.7111, and P5,0 = 22.5446. Using the recursive procedure, the European put values are: P4,4 = 0, P4,3 = 0.0508, P4,2 = 4.6266, P4,1 = 12.8751, P4,0 = 19.4226, P3,3 = 0.0248, P3,2 = 2.2861, P3,1 = 8.6183, P3,0 = 15.9673, P2,2 = 1.1294, P2,1 = 5.3610, P2,0 = 12.1375, P1,1 = 3.1876, and P1,0 = 8.6274. The European put price with dividends is equal to 5.8190. For example, at the nodes (5,0) and (4,0), the European put price is computed as: P5,0 = max[0, K − S5,0 ], P4,0 = [pP5,1 + qP5,0 ]/a.

7.1.4.2. The American put price with dividends For the put, we have Pi,j = max[[pPi+1,j+1 + qPi+1,j ]/a; max[0, K − Si,j ]] The values are computed by starting at the maturity date. At this date, the possible option values are P5,5 = 0, P5,4 = 0, S5,3 = 0.104, P5,2 = 9.3621, P5,1 = 16.7111, and P5,0 = 22.5446. These values are the same for both European and American options. Using the recursive procedure, the European put values are P4,4 = 0, P4,3 = 0.0508, P4,2 = 5, P4,1 = 13.2485, P4,0 = 19.7961, P3,3 = 0.0248, P3,2 = 2.4685, P3,1 = 8.9887, P3,0 = 16.3377, P2,2 = 1.2186, P2,1 = 5.6337, P2,0 = 12.5047, P1,1 = 3.3657, and P1,0 = 8.9441.

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For example, the values P4,4 and P4,3 are calculated as follows: P4,4 = max(0.0508; max[0; 45 − 50.3914]) = max(0.0508; 0) = 0.0508. P4,3 = max(4.6266; max[0; 45 − 40]) = max(4.6266; 5) = 5. The price of the American put is 6.0633 when there are dividends. The difference between the American price and the European price corresponds to the early exercise premium. The lattice can be used to estimate the hedge ratio ∆ from the nodes at time t + ∆t as: ∆=

F1,1 − F1,0 . Su − Sd

We can obtain a more accurate estimate at time t − 2∆t by assuming a stock price S at this time. In this case, the ∆ is ∆=

F2,2 − F2,0 Su2 − Sd2

The Γ is given by: Γ=

F2,2 −F2,1 F2,1 −F2,0 Su2 −S − S−Sd2 1 2 2 2 (Su − Sd )

For an introduction to the Greek letters and their use by market participants, the reader can refer to Chapter 3. The Greek-letter delta corresponds to the option partial derivative with respect to the underlying asset price. The gamma indicates the partial derivative of the delta with respect to the underlying asset price. The theta indicates the option partial derivative with respect to time. The vega indicates the option partial derivative with respect to the volatility parameter. The following Tables 7.1–7.8 show the simulations of binomial option prices using 150 periods for different parameters. The reader can see at the same time the sensitivities of the option price to different parameters (the Greek letters). These parameters are provided for illustrative purposes. The binomial model can also be used for the pricing of foreign currency options. In this case, the cost of carry or the risk-free rate is replaced by the difference between the domestic risk-less rate r and the foreign risk-less rate r∗ . In this case, the underlying asset price S indicates the exchange rate (Table 7.9).

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Derivatives and Path-Dependent Derivatives Table 7.1. Simulations of European binomial call prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price 1.43986 2.54367 4.17276 6.29601 8.91144 12.06935 15.63786 19.55770 23.76469

Delta

Gamma

0.18415 −0.00000 0.24220 0.06668 0.38248 −0.00000 0.46056 0 0.61818 −0.00000 0.69165 0 0.75817 0 0.81605 0 0.86445 0

Vega

Theta

0.21391 0.26153 0.34631 0.37528 0.39070 0.37547 0.34667 0.30750 0.26200

−0.00656 −0.00827 −0.01107 −0.01239 −0.01336 −0.01345 −0.01313 −0.01246 −0.01154

Table 7.2. Simulations of American binomial call prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price 1.43986 2.54367 4.17276 6.29601 8.91144 12.06935 15.63786 19.55770 23.76469

Delta

Gamma

0.18415 −0.00000 0.24220 0.06668 0.38248 −0.00000 0.46056 0 0.61818 −0.00000 0.69165 0 0.75817 0 0.81605 0 0.86445 0

Vega

Theta

0.21391 0.26153 0.34631 0.37528 0.39070 0.37547 0.34667 0.30750 0.26200

−0.00656 −0.00827 −0.01107 −0.01239 −0.01336 −0.01345 −0.01313 −0.01246 −0.01154

Table 7.3. Simulations of European binomial put prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

19.45436 15.55817 12.18726 9.31050 6.92594 5.08385 3.65236 2.57220 1.77919

−0.81585 −0.75780 −0.61752 −0.53944 −0.38182 −0.30835 −0.24183 −0.18395 −0.13555

Gamma 0 0.06668 0 −0.00000 0 −0.00000 −0.00000 −0.00000 −0.00000

Vega

Theta

0.21391 0.26153 0.34631 0.37528 0.39070 0.37547 0.34667 0.30750 0.26200

−0.00119 −0.00289 −0.00570 −0.00702 −0.00799 −0.00808 −0.00776 −0.00709 −0.00617

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Derivatives, Risk Management and Value Table 7.4. Simulations of American binomial put prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

20.32877 16.16444 12.58896 9.58087 7.10987 5.19837 3.72566 2.61920 1.80907

−0.88286 −0.79424 −0.65043 −0.55677 −0.43411 −0.31985 −0.24764 −0.18740 −0.13757

0.01314 0.05407 0.00726 0.00338 0.03640 0.00356 0.00126 0.00100 0.00057

0.14678 0.23133 0.32805 0.36977 0.39021 0.37831 0.34991 0.31041 0.26443

−0.00244 −0.00417 −0.00657 −0.00777 −0.00857 −0.00853 −0.00807 −0.00731 −0.00633

Table 7.5. Simulations of European binomial call prices, S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price 0.36079 0.93264 2.00036 3.70780 6.11625 9.25379 12.96246 17.11629 21.61974

Delta

Gamma

0.08693 0 0.16894 −0.00000 0.28849 −0.00000 0.43730 0 0.59557 −0.00000 0.67066 0.06878 0.80000 0 0.85123 0.04165 0.92542 0

Vega

Theta

0.09055 0.15281 0.21932 0.26800 0.27901 0.26813 0.21954 0.18706 0.12028

−0.00526 −0.00903 −0.01322 −0.01658 −0.01794 −0.01807 −0.01604 −0.01475 −0.01144

Table 7.6. Simulations of American binomial call prices, S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price 0.36079 0.93264 2.00036 3.70780 6.11625 9.25379 12.96246 17.11629 21.61974

Delta

Gamma

0.08693 0 0.16894 −0.00000 0.28849 −0.00000 0.43730 0 0.59557 −0.00000 0.67066 0.06878 0.80000 0 0.85123 0.04165 0.92542 0

Vega

Theta

0.09055 0.15281 0.21932 0.26800 0.27901 0.26813 0.21954 0.18706 0.12028

−0.00526 −0.00903 −0.01322 −0.01658 −0.01794 −0.01807 −0.01604 −0.01475 −0.01144

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Derivatives and Path-Dependent Derivatives Table 7.7. Simulations of European binomial put prices, S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price 19.36306 14.93492 11.00263 7.71007 5.11852 3.25606 1.96473 1.11857 0.62201

Delta

Gamma

−0.91307 0 −0.83106 0 −0.71151 −0.00000 −0.56270 0 −0.40443 0 −0.32934 0.06878 −0.20000 −0.00000 −0.14877 0.04165 −0.07458 −0.00000

Vega

Theta

0.09055 0.15281 0.21932 0.26800 0.27901 0.26813 0.21954 0.18706 0.12028

0.00017 −0.00360 −0.00779 −0.01115 −0.01252 −0.01265 −0.01061 −0.00933 −0.00602

Table 7.8. American binomial — CRR, put S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

20.00779 15.33467 11.24360 7.85080 5.20147 3.30000 1.98701 1.13111 0.62757

−0.97909 −0.87409 −0.73921 −0.58226 −0.45140 −0.33283 −0.20271 −0.14941 −0.07518

0.01600 0.01205 0.00890 0.00764 0.03865 0.06361 0.00120 0.04099 0.00028

0.02403 0.11948 0.20431 0.26240 0.27868 0.26892 0.22094 0.18764 0.12100

−0.00077 −0.00456 −0.00862 −0.01179 −0.01303 −0.01297 −0.01083 −0.00943 −0.00609

Table 7.9. CRR binomial call currency price, S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, and σ = 20%. S 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

Price

Delta

Gamma

Vega

Theta

0.05371 0.05826 0.06288 0.06749 0.07210 0.07748 0.08286 0.08823 0.09361

0.45561 0.46131 0.46131 0.46131 0.53768 0.53768 0.53768 0.53768 0.60551

0.50475 0.49915 0.49921 0.49921 0.42285 0.42285 0.42285 0.42285 0.35503

0.00356 0.00374 0.00377 0.00380 0.00382 0.00385 0.00388 0.00391 0.00394

0.00008 0.00008 0.00009 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008

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7.2. A Simple Extension to Account for Information Uncertainty in the Valuation of Futures and Options We can extend the previous analysis to account for the effects of information costs. Information costs can be used in the valuation of futures and options. For introduction, we refer to Appendices A and B. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah et al. (2001a,b), Bellalah and Prigent (2001), Bellalah and Selmi (2001) and so on.

7.2.1. On the valuation of derivatives and information costs An important question in financial economics is how frictions affect equilibrium in capital markets since in a world of costly information, some investors will have incomplete information. Merton (1987) developed a simple model of capital market equilibrium with incomplete information, CAPMI. The CAPMI model can explain several anomalies in financial markets. Merton (1987) advanced the investor recognition hypothesis (IRH) in a mean-variance model. This assumption explains the portfolio formation of informationally constrained investors (ICI). The IRH in Merton’s context states that investors buy and hold only those securities about which they have enough information. Merton (1987) adapted the rational framework of the static CAPM to account for incomplete information. The premise in Merton’s (1987) model and Shapiro’s (2000) extension is that the costs of gathering and processing data lead some investors to focus on stocks with high visibility and also to entrust a portion of their wealth to money managers employed by pension plans. In this context, a trading strategy shaped by real-world information costs should incorporate an investment in well-known, visible stocks, and an investment delegated to professional money managers. In this theory, an investor considers only the stocks visible to him/her, i.e., those about which he/she has sufficient information to implement the optimal portfolio re-balancing. In general, information about larger firms is likely to be available at a lower cost. The claim that large firms are more widely known is consistent with the empirical evidence that large firms have more shareholders as in Merton (1987). For these reasons, it is important to account for information costs in the pricing of assets and derivatives.

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The works of Markowitz (1952), Sharpe (1964), and Lintner (1965) on the capital asset pricing model provided the general equilibrium model of asset prices under uncertainty. This model represents a fundamental tool in measuring the risk of a security under uncertainty. The CAPM model can be applied to the valuation of options and futures contracts. Merton’s model is a two-period model of CAPM in an economy where each investor has information about only a subset of the available securities. The main assumption in the Merton’s model is that an investor includes an asset S in his/her portfolio only if he/she has some information about the first and second moment of the distribution of its returns. In this model, information costs have two components: the costs of gathering and processing data for the analysis and the valuation of the firm and its assets, and the costs of information transmission from an economic agent to another. Merton’s model may be stated as follows: ¯ S − r = βS [R ¯ m − r] + λS − βS λm R where: • • • • • •

¯ S : the equilibrium-expected return on security S; R ¯ Rm : the equilibrium-expected return on the market portfolio; R: one plus the risk-less rate of interest, r; ˜ S /R ˜m ) R βS = cov( ˜ m ) : the beta of security S; var(R λS : the equilibrium aggregate “shadow cost” for the security S and λm : the weighted average shadow cost of incomplete information over all securities in the market place.

The CAPM of Merton (1987), referred to as the CAPMI is an extension of the standard CAPM to a context of incomplete information. Note that when λm = λS = 0, this model reduces to the standard CAPM of Sharpe (1964). Since the publication of the pioneering papers by Black and Scholes (1973) and Merton (1973), three industries have blossomed: an exchange industry in derivatives, an OTC industry in structured products, and an academic industry in derivative research. As it appears in Scholes (1998), derivative instruments provide low-cost solutions to investor problems than that of competing alternatives. Differences in information are important in both financial and real markets (see the models in Bellalah 1999a,b; Bellalah and Jacquillat, 1995).

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7.2.2. The valuation of forward and futures contracts in the presence of information costs 7.2.2.1. Forward, futures, and arbitrage We denote respectively by: • • • • • • • • • •

t: current time; T : maturity date of the contract in years; (T − t): time remaining until the maturity of the contract in years; S: spot price of the asset; K: delivery price for a forward contract; f : value of a long position in a forward contract; F : forward price at time t; r: risk-free rate at time t for maturity T ; rf : risk-free rate at time t for maturity T in a foreign country and λS : information cost for the asset S.

For an introduction to information costs and their use in the valuation of derivatives, we can refer to Bellalah and Jacquillat (1995) and Bellalah (2000a,b, 2001). It is important to make a difference between the price and value of a contract. The forward price of a contract corresponds to its delivery price that would make its value equal to zero. When a contract is initiated, the delivery price is fixed equal to the forward price in such a way that f = 0 and K = F . Bellalah shows that these costs of information explain to a large extent the financial crisis of 2008 because of a lack of information transmission between economic agents. 7.2.2.2. The valuation of forward contracts in the absence of distributions to the underlying asset The underlying asset may be a non-dividend paying stock or a non-couponbearing bond. Consider an investment in two portfolios. Portfolio A corresponds to a long position in a forward contract and a cash amount equal to Ke−r(T −t) . Portfolio B contains the underlying security of the forward contract. At maturity, cash can be used to pay for the security. However, before buying the security, an investor pays information costs and therefore needs an additional return of λS before investing in the asset. To implement arbitrage, the investor must be informed about the markets and pays information costs. Hence, arbitrage considerations imply that the real discount rate must be e−(r+λS )(T −t) rather than e−r(T −t) .

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The equivalence between these values of the two portfolios at each time implies that f + Ke−(r+λS )(T −t) = S

or f = S − Ke−(r+λS )(T −t) .

Re-call that when a contract is initiated, its forward price must be equal to the delivery price, which is chosen in such a way that the contract’s value is zero. The forward price F corresponds to the value of K for which, f = 0. Using the last equation, the forward price F = Se(r+λS )(T −t) . 7.2.2.3. The valuation of forward contracts in the presence of a known cash income to the underlying asset The underlying asset may be a dividend-paying stock or a coupon-bearing bond. Consider an investment in two portfolios. Portfolio A corresponds to a long position in a forward contract and a cash amount equal to Ke−r(T −t) . Portfolio B contains the underlying security of the forward contract plus borrowings of an amount I corresponding to the known income. Since income can be used to re-pay the loan, portfolio B has the same value as one security or portfolio A at T . However, before using the security, an investor pays information costs and therefore needs an additional return of λS before investing in the security. Both portfolios must have the same initial value and f +Ke−(r+λS )(T −t) = S −I or f = S −I −Ke−(r+λS )(T −t) . As before, the forward price F can be computed as the value of K for which, f = 0, or F = (S − I)e(r+λS )(T −t) . 7.2.2.4. The valuation of forward contracts in the presence of a known dividend yield to the underlying asset The underlying asset may be a stock index or a currency. A known dividend yield corresponds to an income expressed as a percentage of the security price. Consider an investment in two portfolios A and B. Portfolio A is conserved and portfolio B contains an amount e−q(T −t) of the security with income re-invested. Hence, the fraction of the security in B will grow as a result of the dividends so that at maturity, one security is held. At T , portfolios A and B have the same value and this must be true at time t to give: f + Ke−(r+λS )(T −t) = Se−q(T −t)

or f = Se−q(T −t) − Ke−(r+λS )(T −t) .

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As before, the forward price F can be computed as the value of K for which f = 0, or F = Se(r−q+λS )(T −t) . 7.2.2.5. The valuation of stock index futures A stock index is constructed using some fixed number of stocks. The stocks can have equal weights or weights that change over time. The stock index can be seen as the price of an asset that pays dividends. In general, it is a reasonable approximation for some indices to assume that they pay a continuous dividend yield. In this case, it is possible to use arbitrage arguments as before to show that the futures price must satisfy the following relationship: F = Se(r−q+λS )(T −t) . 7.2.2.6. The valuation of Forward and futures contracts on currencies Currency forward and futures contracts can be analyzed using arbitrage arguments. Consider as before two portfolios A and B. The first portfolio A corresponds to a long position in a forward contract plus an amount of cash, which is equal to Ke−r(T −t) . Portfolio B contains an amount e−rf (T −t) of the foreign currency. The value of the two portfolios at time T is equivalent to one unit of the foreign currency. Portfolios A and B must have the same value at time t to give: f + Ke−(r+λS )(T −t) = Se−rf (T −t) or f = Se−rf (T −t) − Ke−(r+λS )(T −t) . As before, the forward price F or the forward exchange rate can be computed as the value of K for which f = 0 in this last equation, or: F = Se(r−rf +λS )(T −t) . This corresponds to the interest-rate parity theorem in international finance in the presence of information uncertainty. These last equations are identical to those for stock indices, where q is replaced by rf . This result reveals that a foreign currency is similar to a security paying a continuous dividend yield. The latter corresponds to the foreign risk-free interest rate.

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7.2.2.7. The valuation of futures contracts on silver and gold The underlying assets of these contracts are held by several investors for investment purposes. This is not the case for several other commodities. When storage costs are neglected, silver and gold can be seen as securities paying no income. In this case, the futures price must satisfy the following relationship: F = Se(r+λS )(T −t) where λS corresponds to information costs on the security. We denote by G the present value of the storage costs. In the same context, we have F = (S + G)e(r+λS )(T −t) . When storage costs are proportional to the commodity price, then F = Se(r+g+λS )(T −t) where g corresponds to the storage cost per annum. 7.2.2.8. The valuation of Futures on other commodities Several commodities are held by investors for some reasons other than investment purposes. This is the case for commodities held in inventory because of their consumption values. In this case, investors conserve the commodity and do not buy futures contracts because they cannot be consumed. This is one of the reasons given to explain the decrease in futures prices for longer maturities. In general, holders of commodity positions feel some benefits from holding physical commodities such as the ability to keep a production process running. These benefits are not available for the holder of a futures contract. The benefits are referred to as the convenience yield (cy). In the presence of information uncertainty, the cy can be defined as: F ecy(T −t) = (S + G)e(r+λS )(T −t) . When storage costs are expressed as a constant proportion, g, then y is defined so that: F ecy(T −t) = Se(r+g+λS )(T −t). or F = Se(r+g−cy+λS )(T −t) . When a commodity is held only for investment purposes, the cy is zero and we have F = (S + G)e(r+λS )(T −t) or F = Se(r+g+λS )(T −t) . 7.2.3. Arbitrage and information costs in the lattice approach The lattice approach can be introduced with reference to a stock option in the absence of payments to the underlying asset.

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Let T be the option’s maturity date which is divided into N reasonably small intervals of length, so that T = N ∆t. During each time interval, the stock price moves either upwards from S to Su or downwards from S to dS. This movement in the stock price is binomial with a probability p attached to an upward jump and a probability (1 − p) to a downward movement. The parameters u, d, and p are functions of the mean and variance of the rates of return on S during the interval. The basic lattice approach as suggested by CRR considers the situation where there is only one state variable: the price of a non-dividend paying stock. The time to maturity of the option is divided into N equal intervals of length ∆t during which the stock price moves from its initial value S to one of the two new values Su and Sd with probabilities p and (1 − p), respectively. It is a simple matter to extend the binomial approach to account for information costs. The acquisition of information and its dissimination are central activities in finance, and especially in capital markets. Merton (1987) provided a model of CAPMI to provide some insights into the behavior of security prices. From Merton’s model (1987), it appears that taking into account the effect of incomplete information on the equilibrium price of an asset is similar to applying an additional discount rate to this asset’s future cash flows. In fact, the expected return on the asset is given by the appropriate discount rate that must be applied to its future cash flows. In fact, in a risk-neutral world, it is possible to show that we have the appropriate value for u, d, and p. In this world, the expected return on the underlying asset in time ∆t is (r + λS )∆t. The variance of this underlying asset on the same interval is σ2 ∆t. The term λS indicates the shadow cost of incomplete information for the asset S. Consider the previous analysis in the presence of an information cost λ. This cost can be seen as a shadow cost or a marginal cost reflecting the return required by market participants to include some assets in their portfolios. It is well known that there are thousands of financial assets in financial markets. However, investors decide to include only some of these assets in their portfolios. They suffer “a sunk cost” in collecting data, analysing financial products and implementing financial models. These costs are necessary in the choice of financial assets and the implementation of some strategies. Therefore, investors need to be compensated by a return corresponding to these costs as it appears in Merton’s (1987) model. Consider, for example, a financial institution using a given market. If the costs of portfolio selection and models conception, etc. are computed, then it can require at least a return of say, for example λ = 3%, before acting in

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this market. This cost represents its minimal cost before acting in a given market. It is in some way, its minimal return required before implementing a given strategy. The expected return on the stock can be written as: pSu + (1 − p)Sd = Se(r+λS )∆t . The variance on the stock can be written for the same interval ∆t as: pS 2 u2 + (1 − p)S 2 d2 − (Se(r+λS )∆t )2 . The last expression can be written as: S 2 (e(r+λS )∆t (eσ

√

∆t

+ e−σ

√ ∆t

) − 1 − e2(r+λS )∆t ).

For the valuation of derivative assets, the expected value of the underlying asset in a risk-neutral world of Merton (1987) is Se(r+λS )∆t . In this context, we can write the equality between this expected value and the one given by the binomial model: pSu + (1 − p)Sd = Se(r+λS )∆t or: pu + (1 − p)d = e(r+λS )∆t . If we use the above equations and assume that u = d1 , then we have: u = eσ

√

∆t

,

d = e−σ

√

∆t

,

p=

a−d , u−d

a = e(r+λs )∆t .

In a risk-neutral world, the option value at time T −∆t can be calculated as the expected value at the maturity date T discounted at (r + λc ) for the time ∆t. In the same way, the values of the derivative security can be calculated at time T − 2∆t as the expected value at time T − ∆t discounted at (r + λc ) for the time ∆t and so on. If we define the value of a contingent claim as Fi,j at time t + i∆t when the underlying asset is Suj di−j , then the value of a European option with information costs can be computed using the following equation: Fi,j = e−(r+λc )∆t [(pFi+1,j+1 + (1 − p)Fi+1,j )]. At each node, at time t + i∆t, the American call option value is given by: Fi,j = max[Suj di−j − K, e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )].

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At maturity, the value of a European put on a non-dividend paying asset is: FN,j = max[K − Suj dN −j , 0] This terminal value gives the option values for the (N + 1) terminal nodes. Then at each node, at time t + i∆t, the American put value is given by: Fi,j = max[K − Suj di−j ,

e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )].

7.2.4. The binomial model for options in the presence of a continuous dividend stream and information costs The extension of the lattice approach in the presence of information costs and a continuous dividend yield is simple. If a security pays a dividend yield q, then the expected return on the underlying asset is (r + λc − q). In this case, we can write the equality between the expected value of the underlying asset and the one given by the binomial model as: pSu + (1 − p)Sd = Se(r+λS −q)∆t or: pu + (1 − p)d = e(r+λS −q)∆t . In this case, using the same procedure as before, it can be shown that the parameters for the binomial model are: u = eσ

√ ∆t

,

d = e−σ

√

∆t

,

p=

a−d , u−d

and a = e(r+λs −q)∆t .

The value of a European contingent claim Fi,j at time t + i∆t when the underlying asset is Suj di−j , in the presence of information uncertainty can be calculated using the following equation: Fi,j = e−(r+λc )∆t [(pFi+1,j+1 + (1 − p)Fi+1,j )]. In the same context, at time t + i∆t, the American call option value is given by: Fi,j = max[Suj di−j − K, e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )]. At each node, at time t + i∆t, the American put value is given by: Fi,j = max[K − Suj di−j , e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )].

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7.2.5. The binomial model for options in the presence of a known dividend yield and information costs The lattice or binomial approach can be easily modified when the underlying asset pays a known dividend yield, δ. At an instant t + i∆t, prior to the ex-dividend date, stock prices are given by: Suj di−j for j = 0, 1, . . . , i. At an instant (t + i∆t), just after the ex-dividend date, stock prices are given by: S(1 − d)uj di−j

for j = 0, 1, . . . , i.

7.2.6. The binomial model for options in the presence of a discrete dividend stream and information costs When there is just one ex-cash income date τ , during the option’s life and k∆t ≤ τ ≤ (k + 1)∆t, then at time y, the value of the stochastic component S is given by: S ∗ (y) = S(y) when y > τ, and S ∗ (y) = S(y) − Di e−(r+λc )(τ −y)

when y ≤ τ.

Assume σ ∗ is the constant volatility of S ∗ . Using the parameters p, u, and d, at time t + i∆t, the nodes on the tree define the stock prices: If i∆t < τ : S ∗ (t)uj di−j + De−(r+λc )(τ −i∆t) If i∆t ≥ τ : S ∗ (t)uj di−j

j = 0, 1, . . . , i

j = 0, 1, . . . , i.

7.2.7. The binomial model for futures options in the presence of information costs In a risk-neutral economy, the expected return on an asset paying a continuous dividend yield q is (r − q). In this context, the equations: u = eσ

√

∆t

,

d = e−σ

√

∆t

,

p=

a−d , u−d

are used, except for a, which must be re-written as: a = e(r−q+λc )∆t . For a futures, contract, r + λc = b and a = 1. When information costs are equal to zero, then r = b and a = 1.

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7.2.8. The lattice approach for American options with information costs and several cash distributions 7.2.8.1. The model When the amounts of payments are accounted for, it can be shown that a sufficient condition for optimal exercise is: (Di + Ri ) > K[1 − e−(r+λS )(ti+1 −ti ) ] with (Di + Ri ) ≥ 0. When there are no dividends, the American put may be exercised because interest can be earned on the exercisable proceeds of the option. The put is exercised at time ti if: (Di + Ri ) < K[1 − e−(r+λS )(ti+1 −ti ) ] with (Di +Ri ) ≥ 0. It is a simple matter to extend this approach to account for information uncertainty. When u = 1/d, it can be shown that: p=

a−d , u−d

u = eσ

√

∆t

,

d = e−σ

√

∆t

,

and a = e(r+λS )∆t .

The term λS appears because of the duplication portfolio. The nature of the lattice of stock prices is completely specified and the nodes correspond to: Suj di−j

where j = 0, 1, . . . , i.

The option is evaluated by starting at time T and working backward. Let us denote by Fi,j , the option value at time t + i∆t when the stock price is Suj di−j . At time t + i∆t, the option holder can choose to exercise the option and receives the amount by which K (or S) exceeds the current stock price (or K) or wait. The American call is given by: Fi,j = max[Suj di−j − K, e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )]. The discounting factor is adjusted by information costs in the option market to reflect sunk costs paid in this market. The American put is given by: Fi,j = max[K − Suj di−j , e−(r+λc )∆t (pFi+1,j+1 + (1 − p)Fi+1,j )].

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When there is just one ex-cash income date τ , and k∆t ≤ τ ≤ (k+1)∆t, then at time y, the value of the stochastic component S is given by: S ∗ (y) = S(y) when y > τ S ∗ (y) = S(y) − (Di + Ri)e−(r+λS )(τ −y)

when y ≤ τ

Assume σ ∗ is the constant volatility of S ∗ . Using the parameters p, u, and d, at time t + i∆t, the nodes on the tree define the stock prices: If i∆t < τ : S ∗ (t)uj di−j + (Di + Ri)e−(r+λS )(τ −i∆t) If i∆t ≥ τ : S ∗ (t)uj di−j

j = 0, 1, . . . , i

j = 0, 1, . . . , i.

7.3. The Binomial Model and the Risk Neutrality: Some Important Details Nawalkha and Chambers (1995) re-examine the consistency of the binomial option pricing model with the risk-neutrality argument of Cox and Ross (1976). They show that risk neutrality in discrete time is a consequence of a specific choice of binomial parameters by Cox et al. (CRR) (1979). 7.3.1. The binomial parameters and risk neutrality The original discrete-time binomial model of CRR, can be presented as follows. Consider at time t = 0, a call option with a maturity date T . Let T be divided into N number of sub-intervals. Denote the current time T T (N − 1). At the current time, the option is N periods from the by t = N expiration date. In the next period, the stock S goes up to uS or down to dS with u ≥ d. The probability of an upward movement is q. The probability of a downward movement is (1 − q). As shown by CRR (1979), the call price at the current time is: C = [pu C u + pd C d ]/rh

(7.3)

where, pu =

rh − d u − d

pd =

u − rh u − d

C u = max[0, uS − K] C d = max[0, dS − K] and r = 1+ the risk-less rate over a single period and h =

T N

periods.

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The binomial parameters specified by CRR are: u = exp(σ(T /N )0.5 ) d = exp(−σ(T /N )0.5 ) q = (1/2)[1 + (µ/σ)(T /N )0.5 ). In this expression, the term µ indicates the preference parameter. In this model, the probability q is between 0 and 1, so −σ(T /N )0.5 < µ(T /N )0.5 < σ(T /N )0.5 . Since the expressions of u and d do not contain µ, the risk-neutral probabilities pu and pd and the call prices C u and C d are independent of preferences. The choice of binomial parameters is not unique. For example, Jarrow and Rudd (1983) specify u, d, and q as: u = exp[µ(T /N ) + σ(T /N )0.5 ] 0.5

d = exp[µ(T /N ) − σ(T /N )

]

(7.4) (7.5)

q = (1/2). Jarrow and Rudd (1983) showed that the first three parameters for the stock’s return are consistent with the log-normal process. The three moments are the mean µ(T /N ), the variance σ2 (T /N ), and the skewness, which equal zero. However, the three moments for the log-normal process using the CRR parameters are inconsistent with the corresponding moments of the log-normal process. In fact, using the CRR parameters, the moments are: the mean µ(T /N ), the variance σ2 (T /N ) − µ2 (T /N )2 , and the skewness which equals 2µ[µ2 (T /N )3 − σ 2 (T /N )2 ]. This simple analysis shows that the CRR parameters imply a level of skewness, which is different from zero. The variance and the skewness of the binomial model of CRR converge to the variance and the skewness of the log-normal process only in the continuous time limit. This is the case since (T /N )2 and (T /N )3 become insignificant in comparison to (T /N ) as N tends to infinity. The Jarrow and Rudd (1983) parameters are inconsistent with the risk-neutral approach in discrete time. In fact, if one replaces the values of u and d from Eqs. (7.4) and (7.5) in Eq. (7.3), we see that the probabilities and option values depend on the preference parameter µ. Consider the general form of the binomial model parameters: u = exp[m(T /N ) + σ(T /N )0.5 ] d = exp[m(T /N ) − σ(T /N )0.5 ] q = (1/2)[1 + ((µ − m)/σ)(T /N )0.5 ]

(7.6)

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351

with −σ(T /N )0.5 < (µ − m)(T /N ) < σ(T /N )0.5 . This choice of the binomial parameters leads to the following three moments over a discrete subinterval: a mean µ(T /N ), a variance σ 2 (T /N )− (µ− m)2 (T /N )2 and the skewness which equals 2(µ− m)[(µ− m)2 (T /N )3 − σ 2 (T /N )2 ]. Again, the terms (T /N )2 and (T /N )3 become insignificant in comparison to (T /N ) as N tends to infinity. In this case, the variance and the skewness of the binomial process implied by these parameters converge to the variance and skewness of the log-normal process in the continuous time limit. Equation (7.6) and other equations given under Eq. (7.6) imply that the probabilities and option prices depend upon m. Investors may disagree about the preference parameter µ and agree on m. In the CRR model, the parameter m = 0. In the Jarrow and Rudd (1983) model, the parameter µ = m. A unique equivalent probability measure a` la Harrison and Kreps (1979) exists and implies that the stock price discounted at the risk-less rate is a martingale under this measure. In this setting, the risk-neutral probabilities must be greater than zero in Eq. (7.3). Hence, we must have for the CRR choice that: exp[−σ(T /N )0.5 ] < rh < exp[σ(T /N )0.5 ]. We must have for the Jarrow and Rudd (1983) choice that: exp[µ(T /N ) − σ(T /N )0.5 ] < rh < exp[µ(T /N ) + σ(T /N )0.5 ]. We must have for the revealed preference parameters in Nawalkha and Chambers (1995) choice that: exp[m(T /N ) − σ(T /N )0.5 ] < rh < exp[m(T /N ) + σ(T /N )0.5 ]. The analysis shows that when m = µ, the binomial model is inconsistent with the risk-neutrality argument of Cox and Ross (1976) and the model is preference dependent. When m differs from µ, the risk neutrality can still hold because investors are allowed to disagree with µ. In this case, different values of m can generate different option prices in discrete time. These limitations disappear in the continuous time limit. In fact, when N tends to infinity, the Black and Scholes (1973) formula is

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obtained. Hence, the binomial model is independent of the parameter m only in the continuous time limit.

7.3.2. The convergence argument We use the binomial equation for the general choice of binomial parameters. The call price at time t between 0 and T is given by: u d + pd Ct+h ]/rh Ct = [pu Ct+h

(7.7)

where the different parameters are given in Eq. (7.6). We can write that: u d Ct = C(St , T − t), Ct+h = C(uSt , T − (t + h)), Ct+h = C(dSt , T − (t + h)). By appropriate substitutions, we can write (7.7) as:

rh − exp[mh − σ(h0.5 )] exp[mh + σ(h0.5 )] − exp[mh − σ(h0.5 )]

· C(exp[mh + σ(h0.5 ] · St , T − (t + h)) exp[mh + σ(h0.5 )] − rh + exp[mh + σ(h0.5 )] − exp[mh − σ(h0.5 )] · C(exp[mh + σ(h0.5 ] · St , T − (t + h)) − rh C(St , T − t) = 0 Using Taylor series expansions of C(exp[mh + σ(h0.5 )]St , T − (t + h)) and C(exp[mh − σ(h0.5)]St , T − (t + h)) around the point (St , (T − t)) gives: C(exp[mh + σ(h0.5 )]St , T − (t + h)) = Ct + [exp[mh + σ(h0.5 )] − 1]St

∂Ct ∂St

1 ∂Ct ∂ 2 Ct + ···+ + [exp[mh + σ(h0.5 )] − 1]2 St2 2 + h 2 ∂ St ∂t

(7.8)

A similar expression can be obtained for C(exp[mh − σ(h0.5 )]St , T − (t + h)) with −σ(h0.5 ) instead +σ(h0.5 ). Using Taylor series for: exp[mh + σ(h0.5 )] = 1 + [mh + σ(h0.5 )] 1 + [mh + σ(h0.5 )]2 + · · · + 2

(7.9)

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A similar expression can be obtained for exp[mh − σ(h0.5 )]. A Taylor series expansion of rh gives: 1 rh = exp(h · logr) = 1 + h · logr + (h · logr)2 + · · · + 2

(7.10)

Substituting Eqs. (7.8), (7.9), (7.10) in Eq. (A2) gives: 1 2 2 ∂ 2 Ct ∂Ct ∂Ct σ St 2 h + (logr)S ·h+ · h − (logr)Ct h + Z = 0 2 ∂ St ∂St ∂St where Z contains all terms of higher order of h. Dividing this last equation by h gives: ∂Ct ∂Ct 1 2 2 ∂ 2 Ct σ St 2 h + (logr)S − (logr)Ct + Z/h = 0. + 2 ∂ St ∂St ∂t The solution to this equation depends on the revealed preference parameter contained in the term Z/h. However, when the number of subintervals tend to infinity, the term Z/h goes to zero. This last equation converges then to the Black and Scholes (1973) partial differential equation. This equation is independent of m. 7.4. The Hull and White Trinomial Model for Interest Rate Options Hull and White (1993) presented a general numerical procedure involving the use of trinomial trees for constructing one-factor models, which are consistent with initial market data where the short rate follows a Markovian process. Their procedure is efficient and provides a convenient way of implementing models already suggested in the literature. It should be noted in passing that a one-period trinomial tree is somehow equivalent to a standard two-period binomial tree. For example, they studied the case where the process for r(t) has the general form considered by Hull and White (1990), also called the extended Vasicek model: dr = µ[θ, r, t]dt + σdW (t) In this model, the volatility σ is a known constant and the functional form of µ is known. The value of θt is unknown. The short interest rate r corresponds to the continuously compounded yield on a discount bond maturing at date ∆t. When the tree is constructed,

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the values of r are equally spaced and have the following form: Tθ + j∆t. Note that r0 is the current interest-rate value and j is an integer, which may be positive or negative. Also, the values of ∆t are equally spaced with a positive integer i. As shown in Hull and White (1990), the variables ∆t and √ ∆r must be √ chosen in such a way that ∆r lies in an interval between σ 3∆t and 2σ ∆t. 2 We use the following notation: (i, j): a node on the tree for the values of t = i∆t, and rj = r0 + j∆r with i ≥ 2; R(i): yield at time zero on a discount bond maturing at time i∆t; µ(i, j): drift rate of r at node (i, j) with rj = r0 + j∆r and (i,j) Pk : for k = 1, 2, 3, the probabilities corresponding respectively to the upper, middle, and lower branches emanating from node (i, j). If the tree constructed up to time n∆t, (n ≥ 0) is consistent with the observed R(i) and the interest rate r at time i∆t applies to the interval time between i∆t and (i + 1)∆t, then the tree reflects the values of R(i) for i ≤ (n + 1). Note that between times n∆t and (n + 1)∆t, the value of θ(n∆t) must be chosen in such a way that the tree is consistent with R(n + 2). Once the value of θ(n∆t) is known, then it is possible to calculate the drift rates µn,j for r at time n∆t using: µn,j = µ[θ(n∆t), r0 + j∆r, n∆t)].

(7.11)

The three nodes from node (n, j) are: (n + 1, k + 1), (n + 1, k), and (n + 1, k − 1), where k must be chosen in such a way that rk , (the value of the interest rate reached by the middle branch) is very close to the expected value of the interest rate, rj + µn,j ∆t. Hull and White (1993) gave the following probabilities: P1 (n, j) =

σ 2 ∆t η2 η + + 2 2 2∆r 2∆r 2∆r

P2 (n, j) = 1 − P3 (n, j) =

σ 2 ∆t η2 + 2 ∆r ∆r2

σ 2 ∆t η2 η + − 2 2∆r 2∆r2 2∆r2

with η = µn,j ∆t + (j − k)∆r.

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This general procedure for fitting a one-factor model of the short rate to the initial yield curve using a trinomial interest-rate tree can be used to test the effect of a wide range of assumptions about the interest-rate process on the prices of interest-rate derivatives. However, much research remains to be done to obtain better models for interest rates. Anyway, this model also allows for negative interest rates.

7.5. Pricing Path-Dependent Interest Rate Contingent Claims Using a Lattice Dharan (1997) proposed a framework for the pricing of path-dependent interest rate derivatives such as index-amortizing swaps or mortgagebacked securities with a simple pre-payment function for which there is no analytical solution. He showed how to construct a lattice to value a mortgage-backed security when the pre-payment function is linear. This section develops the models using numerical examples.

7.5.1. The framework Let us denote the interest rate by r. The interest rate goes up to ru with a probability pu or down to rd with a probability pd = 1 − pu . In this case, the interest goes up to ru u with a probability pu u, etc. The superscripts, u and d, represent the sequence of up and down movements through the two-period lattice. If we denote by Ψ(i, j) the Arrow-Debreu price at a node corresponding to the ith period and jth node from the bottom, counting from zero, then: pd pu pd pdd Ψ(0, 0) = 1, Ψ(1, 0) = (1+r) , Ψ(1, 1) = (1+r) , Ψ(2, 0) = (1+r) × (1+r d) , = p Ψ(1, 0) (1+r d) dd

Ψ(2, 1) =

pud pd pdu pu × + × (1 + r) (1 + ru ) (1 + r) (1 + rd )

= Ψ(1, 1) × Ψ(2, 0) =

pud pdu + Ψ(1, 0) × u (1 + r ) (1 + rd )

puu pu × (1 + r) (1 + ru )

= Ψ(1, 1)

puu . (1 + ru )

(7.12)

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Using Eq. (7.12), it is possible to construct a lattice of Arrow-Debreu prices using forward induction. The Arrow-Debreu prices at a node is a summary measure of all paths leading to that node. Consider a path-dependent interest rate derivative. Its original principal is F and its maturity date is T with T = n∆t. The principal is amortized at a given rate a(i, j) that is node dependent. When there are k paths arriving at a node, the principal corresponding to each path is denoted by F (i, j, k). The security pays the difference between the principal in the previous period and its currently amortized value (representing the prepayments for a mortgage-backed security. We denote this difference by P (i, j, k): P (i, j, k) = F (i − 1, l, k) − F (i, l, k) = a(i, j)F (i − 1, l, k)

(7.13)

where l can be (j − 1) or j. At each period, the security pays a cash flow C(i, j, k) calculated as a percentage, c(i − 1, l) of the remaining principal in the previous period. The remaining principal is fully paid at maturity, implying that a total payout is equal to [1 + c(n − 1), l]F (n − 1, l, k) and P (n, j, k) = 0. The next step is to store the principal at each node in terms of its Arrow-Debreu price at each lattice node as follows: F ∗ (0, 0) = F (0, 0) F ∗ (i, 0) = F ∗ (i − 1, 0)[1 − a(i, 0)]

1 − p(i − 1, 0) 1 + r(i − 1, 0)

F ∗ (i, i) = F ∗ (i − 1, i − 1)[1 − a(i, i)] F ∗ (i, j) = F ∗ (i − 1, j)[1 − a(i, j)]

p(i − 1, i − 1) 1 + r(i − 1, i − 1)

1 − p(i − 1, j) 1 + r(i − 1, j)

+ F ∗ (i − 1, j − 1)[1 − a(i, j)]

p(i − 1, j − 1) 1 + r(i − 1, j − 1) for 0 < j < i.

(7.14)

Equation (7.14) use some new notations. Let us denote by r(i, j) and p(i, j), the upward jump in the spot rate and its probability, respectively and by F ∗ (i, j) the Arrow-Debreu price of the principal payments at node (i, j). The second and third lines in Eq. (7.14) represent the values for the nodes located at the lower and upper edges of the lattice. The values for interior nodes are given by the last line.

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The forward induction procedure begins at date 1. C ∗ (i, j) are calculated as follows. 1 − p(i − 1, 0) 1 + r(i − 1, 0)

C ∗ (i, 0) = F ∗ (i − 1, 0)c(i − 1, 0)

C ∗ (i, i) = F ∗ (i − 1, i − 1)c(i − 1, i − 1) C ∗ (i, j) = F ∗ (i − 1, j)c(i − 1, j)

p(i − 1, i − 1) 1 + r(i − 1, i − 1)

1 − p(i − 1, j) 1 + r(i − 1, j)

+ F ∗ (i − 1, j − 1)c(i − 1, j − 1)

p(i − 1, j − 1) 1 + r(i − 1, j − 1) for 0 < j < i.

(7.15)

The values of P ∗ (i, j) at similar nodes are computed as follows: P ∗ (i, 0) = F ∗ (i − 1, 0)a(i, 0)

1 − p(i − 1, 0) 1 + r(i − 1, 0)

P ∗ (i, i) = F ∗ (i − 1, i − 1)a(i, i) P ∗ (i, j) = F ∗ (i − 1, j)a(i, j)

p(i − 1, i − 1) 1 + r(i − 1, i − 1)

1 − p(i − 1, j) 1 + r(i − 1, j)

+ F ∗ (i − 1, j − 1)a(i, j)

p(i − 1, j − 1) 1 + r(i − 1, j − 1) for 0 < j < i.

(7.16)

At time 0, the value of the security V (0, 0) corresponds to the sum of the Arrow-Debreu prices of all cash flows until maturity. It is computed as:

V (0, 0) =

i n−1 i=1 j=0

[C ∗ (i, j) + P ∗ (i, j)] +

n

[C ∗ (n, j) + F ∗ (n, j)]

(7.17)

j=0

This method is exact and does not use any approximation. It is better than the one presented in Hull and White (1993).

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Assuming an initial principal of 1$, the induction method for these prices is: v(n, j) = 1 v(i, j) = [v(i + 1, j)[1 − a(i + 1, j)] + c(i, j) + a(i + 1, j)] ×

1 − p(i, j) + [v(i + 1, j + 1)[1 − a(i + 1, j + 1)] 1 + r(i, j)

+ c(i, j) + a(i + 1, j + 1)] ×

p(i, j) 1 + r(i, j)

for i < n.

(7.18)

This method is similar to that proposed in Hillard et al. (for more details, refer to Bellalah et al., 1998). In the last period in the lattice, the nodes are set to one. Each term in the brackets corresponds to the sum of three values (the price scaled by the amortization rate, the cash flow and the pre-paid principal) from the nodes in the next period. In general, for all k values of the principal, we have: V (i, j, k) = v(i, j)F (i, j, k)

(7.19)

7.5.2. Valuation of the path-dependent security Dharan (1997) developed two examples in the illustration of Eqs. (7.14)– (7.17). The first is a fixed-rate mortgage corresponding to a mortgagebacked security where the underlying pool pays a fixed coupon. The second example is an adjustable-rate mortgage corresponding to a mortgagebacked security based on a pool paying a floating-rate coupon. The examples are based on a specific model for interest rates and the following simple pre-payment function (simlar to that in Hull and White (1993)):  0 r(i, j) ≥ 0.05,      0.05 a(i, j) = 0, 45 −1 0.03 < r(i, j) < 0.05,  r(i, j)     0.3 r(i, j) ≤ 0.03 7.5.2.1. Fixed-coupon rate security The original face value of the security is 100 dollars and the value of coupon is 5%.

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In this case, it is possible to construct a tree for four periods where the cash flows at each node are the sum of the prepayments, the coupons paid, and the remaining principal at maturity. The different cash flows are discounted to the present. Equations (7.14)–(7.16) are used to compute at each node the ArrowDebreu prices of the principal, coupons, and prepayments, i.e., F ∗ (i, j), C ∗ (i, j), and P ∗ (i, j). 7.5.2.2. Floating-coupon security The floating-rate coupon is assumed to be the spot interest rate prevailing in the previous period. Equations (7.14) and (7.16) are used to build the lattice. Equation (7.17) gives the security price. 7.5.3. Options on path-dependent securities Equation (7.18) is used to compute the value of the underlying security for an original principal of one dollar. It is necessary to get the possible values of the principal at maturity. Dharan (1997) proposed two cases: short-dated options for which the number of steps is less than 12 and long-dated options where the difference. 7.5.3.1. Short-dated options Equation (7.19) gives the correct values of the underlying security for each value of the principal at a particular node. The option payoff is given by: V opt (s, j, k) = max(V (s, j, k) − X, 0)

(7.20)

where V opt (s, j, k) corresponds to the option value at maturity s and X is the strike price. The American option price is computed using:

V opt (i, j, k) = max[(V (i, j, k) − X) V opt (i + 1, j + 1, k)p(i, j) + V opt (i + 1, j, k)(1 − p(i, j)) 1 + r(i, j)

(7.21)

7.5.3.2. Long-dated options Let us denote by z the number of values of the principal stored at each node.

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When z = 3, denote the minimum by F0 (s, j), the average by F1 (s, j), and the maximum by F2 (s, j). At maturity, the value of the underlying security corresponding to the three values is computed using Eq. (7.19). Denote these payoffs by V0opt (s, j), V1opt (s, j) and V2opt (s, j). The option payoffs corresponding to the values of the principal at the previous node are computed using the quadratic interpolation method in Hull and White (1993). The computation of V1opt (i, j) corresponding to F1 (i − 1, j) needs the calculation of F1 (i, j) = F1 (i − 1, j)a(i, j). Equation (7.22) that is used for the interpolation is given by: V1opt − V0opt (F1 − F0 )(F1 − F1 ) + F1 − F0 (F2 − F0 ) V opt − V0opt . − 1 F1 − F0

V1opt = V0opt + (F1 − F0 ) ×

×

V2opt − V1opt F2 − F1

The present value option is obtained by repeating the interpolation at each step and discounting the resulting value. This model and several of its applications to the pricing of pathdependent claims are provided by Dharan (1997).

Summary The lattice approach for the pricing of options can be specified with respect to stock options in the absence of payments to the underlying asset. The option’s maturity date is divided into several reasonably small intervals of time. During each time interval, the underlying asset price moves either upward or downward. This movement in the stock price is binomial with a probability p attached to an upward jump and a probability (1 − p) to a downward movement. The parameters corresponding to the up, down, and the probability are functions of the mean and variance of the rates of return on S during the interval. The basic lattice approach suggested by CRR considers the situation where there is only one state variable: the price of a non-dividend paying stock. The time to maturity of the option is divided into equal intervals. It is a simple matter to extend this approach to account for the effects of a continuous dividend yield, a discrete dividend, information costs, etc. The valuation of options in a binomial model is easy to implement since we start at the maturity

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date, where the payoff is known, and then proceed backward through the tree. In a risk-neutral world, the option value at a given time can be calculated as the expected value at the maturity date discounted at the appropriate rate. The valuation of European options is slightly different from that of the American options. For the American options, the holder can exercise his/her option at any instance before the maturity date. Therefore, at each node, the option value must be at least equal to its intrinsic value. The extension of the lattice approach in the presence of information costs and a continuous dividend yield is simple. This chapter shows that the consistency of the discrete-time binomial option pricing model of Cox et al. (1979) with the risk-neutrality argument depends heavily on the appropriate choice of its parameters. In fact, for a certain choice of the parameters, the option prices can depend on investor preferences. This preference dependence diminishes as the number of subintervals become large. This dependence disappears only in the continuous time limit when the binomial model converges to the Black and Scholes (1973) model. Risk neutrality is obtained when the CRR model assumes a very specific behavior regarding the price changes of the underlying asset. Hence, in general, risk neutrality is not obtained in discrete time in some cases. Hull and White (1993) developed a numerical procedure that is based on the use of trinomial trees for constructing one-factor models of interest rates. The models are consistent with initial market data where the short rate follows a Markovian process. The procedure adopted by Hull and White (1993) is efficient and provides a convenient way of implementing several other models in the literature. This chapter presents the basic concepts and techniques underlying the derivative assets pricing problem within the context of binomial models and lattice approaches. The lattice approach is applied to the valuation of European and American equity options when the underlying asset is traded in a spot or in a futures market. The approach is extended to the valuation of options by taking into account several cash distributions to the underlying assets. It is convenient to note that lattice approaches can be easily implemented and adapted to different derivative asset payoffs. This approach is more pedagogical than the continuous time approach. However, it takes some time to offer accurate option prices, which is not a major handicap when there is no closed form or analytic solutions.

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Questions 1. Describe the lattice approach and the binomial model for the valuation of equity options. 2. Describe the binomial model for the valuation of futures options. 3. Describe the extension of the lattice approach to account for the effects of information costs. 4. Describe the Hull and White interest-rate trinomial model for the valuation of interest-rate derivatives. 5. Describe the model of Dharan for the pricing of path-dependent interestrate contingent claims using a lattice.

References Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June 1987), 301–320. Bellalah, M (1999a). The valuation of futures and commodity options with information costs. Journal of Futures Markets, (September), 19, 645–664. Bellalah, M (1999b). Les biais des modeles d’options revisites. Revue Francaise de Gestion, (Juin) 94–100. Bellalah, M (2000a). A risk management approach to options trading on the Paris Bourse. Derivatives Strategy, 5(6), 31–33. Bellalah, M (2000b). Valuation of American CAC 40 index options and wildcard options. International Review of Economics and Finance. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3), 617–635. Bellalah, M and J-L Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bellalah, M, JL Prigent and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Black, F (1976). Studies of stock price volatility changes. In Proc. of the 1976 Meeting of the Business Economics Statistics Section, American Statistical Association, pp. 177–181. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

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Cox, JC and SA Ross (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7, 229–263. Dharan, VG (1997). Pricing path-dependent interest rate contingent claims using a lattice. Journal of Fixed Income, 6(March), 41–49. Harrison, JM and D Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408. Ho, I and S Lee (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41, 1011–1029. Hull, J (2000). Options, Futures, and Other Derivative Securities. 4th Ed. Englewood Cliffs, NJ: Prentice Hall. Hull, J and A White (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592. Hull, J and A White (1993). Efficient procedures for valuing European and American path dependent options. Journal of Derivatives, 1, 21–31 (Fall). Jarrow, RA and A Rudd (1983). Option Pricing, Homewood, IL: Irwin. Merton R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Lintner, J (1965). Security prices, risk and maximal gains from diversification. Journal of Finance, 20, 587–616. Markowitz, HM (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, RC (1987). A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42(3), 483–510. Nawalkha, SK and DR Chambers (1995). The binomial model and the risk neutrality: some important details. Financial Review, 30(3), (August), 605–615. Scholes, M (1998). Derivatives in a dynamic environment. American Economic Review, 88(2), 350–370. Shapiro, A (2000). The investor recognition hypothesis in a dynamic general equilibrium: theory and evidence. Working Paper, New York University: Stern School of Business. Sharpe, WF (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442.

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Part III Option Pricing in a Continuous-Time Setting: Basic Models, Extensions and Applications

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Chapter 8

EUROPEAN OPTION PRICING MODELS: THE PRECURSORS OF THE BLACK–SCHOLES–MERTON THEORY AND HOLES DURING MARKET TURBULENCE

Chapter Outline This chapter is organized as follows: 1. Section 8.1 gives an overview of the option pricing theory in the pre-Black–Scholes period. 2. Section 8.2 presents the story and the main results in the breakthrough work of Black–Scholes for the pricing of derivative assets when the underlying asset is traded in a spot market. It proposes the story until the publication of the original formula. 3. Section 8.3 develops the foundations of the Black–Scholes–Merton Theory. 4. Section 8.4 presents two alternative derivations of the Black–Scholes model. The formula is derived using equilibrium market conditions and arbitrage theory. 5. Section 8.5 reviews the main results in Black’s (1976) model for the pricing of derivative assets when the underlying asset is traded on a forward or a futures market. Some applications of the model are also given. 6. Section 8.6 applies the capital-asset pricing model (CAPM) to the valuation of forward contracts, futures, and commodity options. 7. Section 8.7 studies the “holes” in the Black–Scholes–Merton theory. 9. Appendix 8.A provides an approximation of the cumulative normal distribution function. 367

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10. Appendix 8.B provides an approximation of the bivariate normal density function. Introduction Numerous researchers have worked on building a theory of rational option pricing and a general theory of contingent claims valuation. The story began in 1900, when the French mathematician, Louis Bachelier, obtained an option pricing formula. His model is based on the assumption that stock prices follow a Brownian motion. Since then, numerous studies on option valuation have blossomed. The proposed formulas involve one or more arbitrary parameters. They were developed by Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Thorp and Kassouf (1967), and Chen (1970) among others. The Black and Scholes (1973) formulation, hereafter called as B–S, solved a problem, which has occupied the economists for at least three-quarters of a century. This formulation represented a significant breakthrough in attacking the option pricing problem. In fact, the Black– Scholes theory is attractive since it delivers a closed-form solution to the pricing of European options. Assuming that the option is a function of a single source of uncertainty, namely the underlying asset price, and using a portfolio which combines options and the underlying asset, Black and Scholes constructed a risk-less hedge, which allowed them to derive an analytical formula. This model provides a no-arbitrage value for European options on shares. It is a function of the share price S, the strike price K, the time to maturity T , the risk-free interest rate r, and the volatility of the stock price, σ. This model involves only observable variables to the exception of volatility and it has become the benchmark for traders and market makers. It also contributed to the rapid growth of the options markets by making a brand new pricing technology available to market players. About the same time, the necessary conditions to be satisfied by any rational option pricing theory were summarized in Merton’s (1973) theorems. The post-Black–Scholes period has seen many theoretical developments. The contributions of many financial economists to the extensions and generalizations of Black–Scholes type models has enriched our understanding of derivative assets and their seemingly endless applications. The first specific option pricing model for the valuation of futures options is introduced by Black (1976). Black (1976) derived the formula for futures and forward contracts and options under the assumption that investors create risk-less hedges between options and the futures or forward contracts. The formula relies implicitly on the capital asset pricing model

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(CAPM). Futures markets are not different in principal from the market for any other asset. The returns on any risky asset are governed by the asset contribution to the risk of a well-diversified portfolio. The classic CAPM is applied by Dusak (1973) in the analysis of the risk premium and the valuation of futures contracts. Black’s (1976) model is used in Barone-Adesi and Whaley (1987) for the valuation of American commodity options. This model is referred to as the BAW (1987) model. It is helpful, as in Smithson (1991), to consider the Black–Scholes model within a family tree of option pricing models. This allows the identification of three major tribes within the family of option pricing models: analytical models, analytic approximations, and numerical models. Each analytical tribe can be further divided into three distinct lineages: precursors to the Black–Scholes model, extensions of the Black– Scholes model, and generalizations of the Black–Scholes model. This chapter presents in detail the basic theory of rational option pricing of European options and its applications along the Black–Scholes lines and its extensions by Black (1976) for options on futures, and European commodity options. The question of dividends, stochastic interest rates, and stochastic volatilities are left to other chapters since the main concern in this chapter is about analytical models under the Black and Scholes’ (1973) assumptions. There is usually a difference between model values and options market prices. There are three possible reasons for the difference between the theoretical value and the market price. The first is that the model gives the correct value and the option price is out of line. In this case, it may be possible to trade profitably using this model. The second reason is that the wrong inputs are used in the formula. The main input is the volatility of the underlying asset over the life of the option that must be estimated. The third reason is that the formula is wrong because of its assumptions. These three reasons can explain the difference between model values and market prices. 8.1. Precursors to the Black–Scholes Model The story began in 1900 with a doctoral dissertation at the Sorbonne in Paris, France, in which Louis Bachelier gave an analytical valuation formula for options. 8.1.1. Bachelier formula Using an arithmetic Brownian motion for the dynamics of share prices and a normal distribution for share returns, Bachelier obtained the following

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formula for the valuation of a European call option on a non-dividend paying stock: c(S, T ) = SN

S −K √ σ T

− KN

S−K √ σ T

√ + σ Tn

K −S √ σ T

(8.1)

where S: K: T: σ: N (.): n(.):

underlying common stock price; option’s strike price; option’s time to maturity; instantaneous standard deviation of return; cumulative normal density function and density function of the normal distribution.

As pointed out by Merton (1973) and Smith (1976), this formulation allows for both negative security and option prices and does not account for the time value of money. Sprenkle (1961) re-formulated the option pricing problem by assuming that the dynamics of stock prices are lognormally distributed. By introducing a drift in the random walk, he ruled out negative security prices and allowed risk aversion. By letting asset prices to have multiplicative, rather than additive fluctuations, the distribution of the option’s underlying asset at maturity is log-normal rather than normal.

8.1.2. Sprenkle formula Sprenkle (1961) derived the following formula: c(S, T ) = SeρT N (d1 ) − (1 − Z)KN (d2 ) S S 2 ln K ln K + ρ + σ2 T + ρ− √ √ d1 = , d2 = σ T σ T

σ2 2

T

(8.2)

where ρ is the average rate of growth of the share price and Z corresponds to the degree of risk aversion. As it appears in this formula, the parameters corresponding to the average rate of growth of the share price and the degree of risk aversion must be estimated. This reduces considerably the use of this formula. Sprenkle (1961) tried to estimate the values of these parameters, but he was unable to do so.

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8.1.3. Boness formula Boness (1964) presented an option pricing formula accounting for the time value of money through the discounting of the terminal stock price using the expected rate of return to the stock. The option pricing formula proposed is: c(S, T ) = SN (d1 ) − e−ρT KN (d2 ) S S 2 ln K ln K + ρ + σ2 T + ρ− √ √ d1 = , d2 = σ T σ T

σ2 2

(8.3)

T

where ρ is the expected rate of return to the stock. 8.1.4. Samuelson formula Samuelson (1965) allowed the option to have a different level of risk from the stock. Defining ρ as the average rate of growth of the share price and w as the average rate of growth of the call’s value, he proposed the following formula: c(S, T ) = Se(ρ−w)T N (d1 ) − e−wT KN (d2 ) S S 2 + ρ + σ2 T + ρ− ln K ln K √ √ d1 = , d2 = σ T σ T

σ2 2

T

(8.4) .

Note that all the proposed formulas show one or more arbitrary parameters, depending on the investors’ preferences towards risk or the rate of return on the stock. Samuelson and Merton (1969) proposed a theory of option valuation by treating the option price as a function of the stock price. They advanced the theory by realizing that the discount rate must be determined in part by the requirement that investors hold all the amounts of stocks and the option. Their final formula depends on the utility function assumed for a “typical” investor. Black and Scholes (1973) used a formula that was derived in Thorp and Kassouf (1967), who presented an empirical formula for warrants. This formula determines the ratio of shares of stocks needed to construct a hedged position. This position is constructed by buying an asset and selling another. However, Thorp and Kassouf did not realize that in equilibrium, the expected return on a hedged position must be the return on a risk-less asset. This concept is due to Black and Scholes as we will see in the derivation of their model.

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8.2. How the Black–Scholes Option Formula is Obtained 8.2.1. The short story The main idea behind the Black–Scholes formula is the existence of a relationship among a call price, its underlying asset, the volatility, the strike price, the maturity date, and the interest rate. This relationship indicates how much the option value will change when the underlying asset changes by a small amount within a short time. Assume that the option price increases by $0.5 when the underlying asset increases by $1 and that the option price decreases by $0.5 when the underlying asset decreases by $1. In this context, it is possible to create a hedged position by selling two options and buying one round lot of stock. For small changes in the underlying asset price, the losses on one side will be nearly offset by gains on the other side. Hence, at first, a hedged position is created by selling two options and holding the underlying asset. When the underlying asset price increases, the position shows a loss on the option and a profit on the underlying asset. When the underlying asset price decreases, the position shows a gain on the option and a loss on the underlying asset. A neutral hedge can be maintained by modifying the position in the option, in the stock, or in both. The principle leading to the option formula is that a hedged position should yield an amount equal to the short-term interest rate on close-to-risk-less securities. A “reverse hedge” can also be implemented to generate the Black and Scholes (1973) formula by selling short the stock and buying the option (in the right ratio). The formula can also be derived by assuming that a neutral spread must earn the interest rate. In fact, selling an option and buying another option on the same underlying asset in the right ratio is a neutral spread. The formula can be obtained even without the assumption of hedging or spreading. In this case, a comparison is done between a long stock position with a long option position that has the same action as the stock. In the above example, the investor compares a long position of one round lot of stock and the purchase of twooption contracts. These two positions show the same movements for very small changes in the stock prices, so their returns must differ only by an amount equal to the interest rate times the difference in the total values of both positions. At equilibrium, investors are indifferent between the two positions. This leads to the same valuation formula.

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8.2.2. The differential equation The notion of equilibrium in the market for risky assets implies that riskier securities must have higher expected returns, or investors will not hold them — except that investors do not count the part of the risk they are able to diversify away. Black (1976) applied the CAPM to the valuation of warrants in 1969. The method used at that time was based on the discounted expected value of the warrant at expiration. This method has two problems. First, the warrant price depends on the stock’s expected return. Second, an appropriate discount rate must be chosen. One key to solve this problem is to write the warrant formula as a function of the stock price and other factors. This approach was adopted by Black (1998) and Samuelson and Merton (1969). Black’s unpublished formula shows that the expected return on a warrant depends on the risk of the warrant in the same way that a common stock’s expected return depends on its risk. Black used the CAPM to write down how the discount rate for a warrant varies with time and the stock price. This gave a differential equation for the warrant formula. Black did not recognize the equation as a version of the “heat equation”. Therefore, he did not write down its solution. Besides, he did not note that the warrant value did not depend on the stock’s expected return or any other asset’s expected return. 8.2.3. The derivation of the formula In 1969, Scholes and Black started working on the option problem at Massachusetts Institute of Technology (MIT). They concentrated on the fact that the option price depends on the volatility rather than the expected return. They assumed that the stock’s expected return was equal to the constant interest rate. This assumption is equivalent to the fact that the stock’s beta is zero, so all of its risk can be diversified away. They assumed also that the stock’s volatility was constant and the terminal stock price “fits” into a log-normal distribution. Sprenkle used the same assumptions, except that he allowed the stock to have any constant expected return. The problem was to find the present value of the option rather than its expected terminal value. The main idea to derive the formula was as follows: If the stock had an expected return equal to the interest rate, so would the option. Hence,

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all the stock’s risk could be diversified away, so could all the option’s risk. In other terms, if the stock’s beta were zero, the option’s beta would have to be zero too. Hence, the expected terminal option’s value must be discounted at the constant interest rate. The Black and Scholes (1973) formula can be obtained using Sprenkle’s formula by putting in the interest rate for the expected return on the stock and putting in the interest rate again for the discount rate for the option.

8.2.4. Publication of the formula Several discussions are done with Merton (1973) who was also working on option valuation. Merton (1973) pointed out that assuming continuous trading in the option or its underlying asset can preserve a hedged portfolio between the option and its underlying asset. Merton was able to prove that in the presence of a non-constant interest rate, a discount bond maturing at the option expiration date must be used. Black and Scholes (1973) and Merton (1973) worked separately on the application of the formula to the valuation of risky corporate bonds and common stock. Black and Scholes gave an early version of their paper at a conference on capital market theory during summer 1970. Black and Scholes sent their article to the Journal of Political Economy, the Review of Economics and Statistics and it was rejected even without a review. The paper was sent again in 1971 to the Journal of Political Economy after accounting for the comments and suggestions by Merton, Miller, and Fama. The final draft of the paper dated May 1972 appeared in the May/June 1973 issue of the Journal of Political Economy. However, a paper on the results of some empirical tests appeared in the May 1972 issue of Journal of Finance.

8.2.5. Testing the formula The formula was first tested on warrants. Black and Scholes estimated the volatility of the stock of each of a group of companies with warrants. Black, Scholes, and Merton found the best to buy be National General new warrants. They bought a bunch of these warrants. The formula and the volatility estimates were always based on the information at hand. Using the premiums received by a broker’s option-writing customers in the over-the-counter (OTC) option market for a period of several years, some trading rules were tested by Black and Scholes. The formula was used to find

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out how much money could one have made if he had bought the options whose prices seemed lower than the formula values and sold the options whose prices seemed higher than the formula values. When transaction costs are ignored, this trading rule seemed to generate substantial profits. The second test was to assume buying the underpriced options and selling the overpriced options at the values given by the formula. Galai (for more details, refer to Bellalah et al., 1998) tested the formula using listed options traded in the Chicago Board Options Exchange (CBOE) and found profits which are much larger than those found in the OTC market. He tested the profitability of spreads that are kept neutral continuously. A neutral spread corresponds to a strategy of buying an option and selling another option on the same underlying asset. A neutral spread is maintained when the long or short positions are changed for every change in the underlying asset price and time to maturity. This strategy involves buying one contract of the underpriced option and selling either more or less than one contract of the overpriced option. This strategy seemed to generate a consistent profit if transaction costs and other trading costs were ignored. Today, traders use the formula so much that the market prices are usually close to formula values even in the presence of a cash takeover.

8.3. Financial Theory and the Black–Scholes–Merton Theory 8.3.1. The Black–Scholes–Merton theory Black and Scholes (1973) and Merton (1973) showed that the construction of a risk-less hedge between the option and its underlying asset, allows the derivation of an option pricing formula regardless of investor’s risk preferences. The main intuition behind the risk-free hedge is simple. Consider an at-the-money European call giving the right to its holder to buy one unit of the underlying asset in one month at a strike price of $100. Assume that the final asset price is either 105 or 95. An investor selling a call on the unit of the asset will receive either 5 or 0. In this context, selling two calls against each unit of the asset will create a terminal portfolio value of 95. The certain terminal value of this portfolio must be equal today to the discounted value of 95 at the risk-less interest rate. If this rate is 1%, the present value is (95/1.01). The current option value is (100 − (95/1.01))/2. If the observed market price is above (or below) the theoretical price, it is possible to implement an arbitrage strategy by selling the call and buying

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(selling) a portfolio comprising a long position in a half unit of the asset and a short position in risk-free bonds. The Black–Scholes–Merton model is the continuous-time version of this example. The theory assumes that the underlying asset follows a geometric Brownian motion and is based on the construction of a risk-free hedge between the option and its underlying asset. This implies that the call payout can be duplicated by a portfolio consisting of the asset and risk-free bonds. In this theory, the option value is the same for a risk-neutral investor and a risk-averse investor. Hence, options can be valued in a risk-neutral world, i.e., expected returns on all assets are equal to the risk-free rate of interest.

8.3.2. Analytical formulas The main and general result in the Black–Scholes–Merton theory is that if a risk-free hedge can be implemented using the option and its underlying asset, then risk-neutral valuation may be applied. This means that the theory applies in the presence of simple and complex options payouts. The Black–Scholes and Merton formula for a European call follows directly from the work of Sprenkle and Samuelson. In a risk-neutral world, all assets show an expected rate of return equal to the risk-free interest rate. This does not mean that all assets have the same expected rate of price appreciation. If the asset income which may be a dividend, a coupon, etc. is modeled as a constant continuous proportion of the asset price, then the expected rate of price appreciation must be equal to the interest rate less than the cash distribution rate. The theory covers a wide range of underlying assets. For non-dividend-paying stock options: When there are no dividends on the underlying stock, the expected price appreciation rate of the stock is the risk-free interest rate. For constant-dividend-yield stock options: When stocks pay dividends at a constant and a continuous dividend yield, Merton (1973) derives the option valuation formula. In his formula, the cost of carrying the underlying asset corresponds to the difference between the risk-free rate and the stock’s dividend-yield rate. Futures options: In a risk-neutral world with constant interest rates, the expected rate of price appreciation on a futures contract is zero. This is because the futures contract does not involve any cash outlay.

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Foreign currency options: Garman and Kohlhagen (1983) value options on foreign currency. The expected rate of price appreciation of a foreign currency equals the domestic rate of interest less than the foreign rate of interest. 8.4. The Black–Scholes Model Under the following assumptions, the value of the option will depend only on the price of the underlying asset S, time t, and on other variables that are assumed constants. These assumptions or “ideal conditions” as expressed by Black–Scholes are the following: • The option is European; • The short-term interest rate is known; • The underlying asset follows a random walk with a variance rate proportional to the stock price. It pays no dividends or other distributions; • There is no transaction costs and short selling is allowed, i.e., an investor can sell a security that he/she does not own and • Trading takes place continuously and the standard form of the capital market model holds at each instant. The main attractions of the Black– Scholes model are that their formula is a function of “observable” variables and that the model can be extended to the pricing of any type of option. 8.4.1. The Black–Scholes model and CAPM The CAPM of Sharpe (1964) can be stated as follows: RS − r = βS [Rm − r]

(8.5)

where, RS : Rm : r: ˜ S /R ˜m ) R βS = cov( ˜ ) : var(R m

equilibrium-expected return on security S; equilibrium-expected return on the market portfolio; 1 + the risk-less rate of interest and the beta of security S, that is the covariance of the return on this security with the return on the market portfolio, divided by the variance of market return.

This model gives a general method for discounting future cash flows under uncertainty. Denote the value of the option by C(S, t) as a function

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of the underlying asset and time. To derive their valuation formula, B–S assumed that the hedged position was continuously re-balanced in order to remain risk-less. They found that the price of a European call or put must verify a certain differential equation, which is based on the assumption that the price of the underlying asset follows a geometric Wiener process: ∆S = αdt + σ∆z S

(8.6)

where α and σ refer to the instantaneous rate of return and the standard deviation of the underlying asset, and z refers to Brownian motion. The relationship between an option’s beta and its underlying security’s beta is: CS βC = S βS (8.7) C where, βc : βS : C: CS :

the option’s beta; the stock’s beta; the option value; the first derivative of the option with respect to its underlying asset. It is also the hedge ratio or the option’s delta in a covered position. According to the CAPM, the expected return on a security should be: ¯ m − r] ¯ S − r = βS [R R

¯ m is the expected ¯ S is the expected return on the asset S and R where R return on the market portfolio. This equation may also be written as: ∆S ¯ m − r)]∆t = [r + βS (R E (8.8) S Using the CAPM, the expected return on a call option should be: ∆C ¯ m − r)]∆t = [r + βC (R E C

(8.9)

Multiplying (8.8) and (8.9) by S and C yields: ¯ m − r)]∆t E(∆S) = [rS + SβS (R ¯ m − r)]∆t. E(∆C) = [rC + CβC (R

(8.10) (8.11)

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When substituting for the option’s elasticity from Eq. (8.7), Eq. (8.11) becomes after transformation: ¯ m − r)]∆t E(∆C) = [rC + SCS βS (R

(8.12)

Assuming a hedged position is constructed and “continuously” re-balanced, and since ∆C is a continuous and differentiable function of two variables, it is possible to use Taylor’s series expansion to expand ∆C. ∆C =

1 CSS (∆S)2 + CS ∆S + Ct ∆t 2

(8.13)

This is just an extension of simple results to get Ito’s lemma. Taking expectations of both sides of Eq. (8.13) and replacing ∆S, we obtain: E(∆C) =

1 2 2 σ S CSS ∆t + CS E(∆S) + Ct ∆t 2

(8.14)

Replacing the expected value of ∆S from Eq. (8.8) gives, E(∆C) =

1 2 2 ¯ m − r)]∆t + Ct ∆t σ S CSS ∆t + CS [rS + SβS (R 2

(8.15)

Combining Eqs. (8.11) and (8.15) and re-arranging yields: 1 2 2 σ S CSS + rSCS − rC + Ct = 0. 2

(8.16)

This partial differential equation corresponds to the Black–Scholes valuation equation. Let T be the maturity date of the call and E be its strike price. Equation (8.16) subject to the boundary condition at maturity: C(S, T ) = S − K, C(S, T ) = 0

if S ≥ K

if S < K

is solved using standard methods for the price of a European call, which is found to be equal to: C(S, T ) = SN (d1 ) − Ke−rT N (d2 ) with

√ 1 S d1 = ln σ T, + r + σ2 T K 2

√ d2 = d1 − σ T

and where N (.) is the cumulative normal density function.

(8.17)

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It may be shown that Eq. (8.16) applies to both European and American options.

8.4.2. An alternative derivation of the Black–Scholes model Assuming that the option price is a function of the single source of uncertainty, namely stock price and time to maturity, c(S, t) and that over “short”-time intervals, ∆t, a hedged portfolio consisting of the option, the underlying asset, and a risk-less security can be formed, where portfolio weights are chosen to eliminate “market risk” Black–Scholes expressed the expected return on the option in terms of the option price function and its partial derivatives. In fact, following Black–Scholes, it is possible to 1 create a hedged position consisting of a sale of ∂c(S,t) options against one [ ∂S ] share of stock long. If the stock

price changes by a small amount ∆S, the option changes by an amount ∂c(S,t) ∆S. Hence, the change in value in the ∂S long position (the stock) is approximately offset by the change in

1

[ ∂c(S,t) ] ∂S options. This hedge can be maintained continuously so that the return on the hedged position becomes completely independent of the change in the underlying asset value, i.e., the return on the hedged position becomes certain. The value of equity in a hedged position, containing a stock purchase 1 options is: and a sale of ∂c(S,t) [ ∂S ] C(S, t) S−

∂c(S,t) ∂S

(8.18)

Over a short interval ∆t, the change in this position is: ∆c(S, t) ∆S −

∂c(S,t) ∂S

(8.19)

where, ∆c(S, t) is given by c(S + ∆S, t + ∆t) − c(S, t). Using stochastic calculus for ∆c(S, t) gives, ∆c(S, t) =

∂c(S, t) ∂c(S, t) 1 ∂ 2 c(S, t) 2 2 σ S ∆t + ∆S + ∆t. ∂S 2 ∂S 2 ∂t

(8.20)

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The change in the value of equity in the hedged position is found by substituting ∆c(S, t) from Eq. (8.20) in Eq. (8.19). 2 ∂c(S,t) 1 2 2 ∂ c(S,t) σ S + ∆t 2 2 ∂S ∂t . (8.21) − ∂c(S,t) ∂S

Since the return to the equity in the hedged position is certain, it must be equal to r∆t where r stands for the short-term interest rate. Hence, the change in the equity must be equal to the value of the equity times r∆t, or: 2

∂c(S,t) 1 2 2 ∂ c(S,t) ∆t σ S + 2 2 ∂S ∂t c(S, t) (8.22) − = S − ∂c(S,t) r∆t. ∂c(S,t) ∂S

∂S

Dropping the time and re-arranging gives the Black–Scholes partial differential equation: 1 2 2 ∂ 2 c(S, t) ∂c(S, t) ∂c(S, t) σ S + rS = 0. − rc(S, t) + 2 ∂S 2 ∂t ∂S

(8.23)

This partial differential equation must be solved under the boundary conditions expressing the call’s value at maturity date: c(S, t∗ ) = max[0, St∗ − K] where K is the option’s strike price. For the European put, the above equation must be solved under the following maturity date condition: P (S, t∗ ) = max[0, K − St∗ ]. To solve this differential equation, under the call-boundary condition, Black–Scholes made the following substitution: ∗ S 1 2 σ2 ln − r − σ 2 (t∗ − t) , c(S, t) = er(t−t ) y 2 r − σ 2 K 2

2 1 2 2(t∗ − t) (8.24) r − − σ σ2 2 Using this substitution, the differential equation becomes: ∂ 2y ∂y = . ∂t ∂S 2 This differential equation is the heat transfer equation in physics. The boundary condition is re-written as y(u, 0) = 0, if u < 0 otherwise, „ 1 uσ2 «

y(u, 0) = K e

2 r− 1 σ2 2

−1 .

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The solution to this problem is the solution to the heat transfer equation given in Churchill (1963): 1 y(u, s) = √ 2Π

∞ −u √ 2s

„

K e

1 (u+q√2s)σ2 2 r− 1 σ2 2

«

−1 e

“ ” 2 − q2

dq.

Substituting from this last equation in Eq. (8.24) gives the following solution for the European call price with T = t∗ − t c(S, T ) = SN (d1 ) − Ke−rT N (d2 ) “ ” 2 S ln( K )+ r+ σ2 T √ σ T

√ d1 = , d2 = d1 − σ T where N (.) is the cumulative normal density function given by: 1 N (d) = √ 2Π

d −∞

e

“ ” 2 − x2

dx.

It is important to note that the option value is independent of its underlying asset-expected return. This may sound rather strange. One intuitive way to account for this is to say the expected return on the stock is already embedded into the stock price itself. It is also worth noting that the option price rises when the asset price, the time to maturity, the interest

∂c(S,t) , which is rate, and the variance increase. The partial derivative ∂S equal to N (d1 ) gives the ratio of the underlying asset to options in the hedged

position. It also refers to what is known as the option’s delta. Since ∂c(S,t) S is always greater than one, the option is more volatile than c(S,t) ∂S its underlying asset. The value of the put option can be obtained from that of the call option using the put-call parity relationship. 8.4.3. The put-call parity relationship The put-call parity relationship can be derived as follows. Consider a portfolio A which comprises a call option with a maturity date t∗ and a discount bond that pays K dollars at the option’s maturity date. Consider also a portfolio B, with a put option and one share. The value of portfolio A at maturity is max[0, St∗ − K] + K = max[K, St∗ ]. The value of portfolio B at maturity is max[0, K −St∗ ]+St∗ = max[K, St∗ ]. Since both these portfolios have the same value at maturity, they must have the same initial value at time t, otherwise arbitrage will

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be profitable. Therefore, the following put-call relationship must hold ∗ ct − pt = St − Ke−r(t −t) with t∗ − t = T . If this relationship does not hold, then arbitrage would be profitable. ∗ In fact, suppose for example, that ct − pt > St − Ke−r(t −t) . At time t, the investor can construct a portfolio by buying the put and the underlying asset and selling the call. This strategy yields a result equal to ct − pt − St . If this amount is positive, it can be invested at the risk-less rate until the maturity date t∗ , otherwise it can be borrowed at the same rate for the same period. At the option maturity date, the options will be in-the-money or out-of-the money according to the position of the underlying asset St∗ with respect to the strike price K. If St∗ > K, the call is worth its intrinsic value. Since the investor sold the call, he/she is assigned on this call. He/she will receive the strike price, delivers the stock, and closes his/her position in the cash account. The put is worthless. ∗ Hence, the position is worth K + er(t −t) [ct − pt − St ] > 0. If ST < K, the put is worth its intrinsic value. Since the investor is long the put, he/she exercises his/her option. He/she will receive upon exercise the strike price, delivers the stock, and closes his/her position in the cash account. The call ∗ is worthless. Hence, the position is worth K + er(t −t) [ct − pt − St ] > 0. In both cases, the investor makes a profit without an initial cash outlay. This is a risk-less arbitrage, which must not exist in efficient markets. Therefore, the above put-call parity relationship must hold good. Using this relationship, the European put option value is given by: p(S, T ) = −SN (−d1 ) + Ke−rT N (−d2 ) S 2 + r + σ2 T ln K √ √ , d2 = d1 − σ T , d1 = σ T

(8.25)

where N (.) is the cumulative normal density function. We illustrate by the following examples, the application of the Black and Scholes (1973) model for the determination of call and put prices.

8.4.4. Examples Example 1: When the underlying asset S = 18, the strike price K = 15, the short-term interest rate r = 10%, the maturity date T = 0.25, and the volatility σ = 15%, the call price is calculated as follows.

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First, we compute the discounted value of the strike price, Ke−rT = 15e−0.1(0.25) = 14.6296. Second, the values of d1 and d2 are calculated as: 18 1 1 2 √ ln d1 = + 0.1 + 0.15 0.25 15 2 0.15 0.25 0.21013 = 2.8017 = 0.075 √ d2 = 2.8017 − 0.5 0.25 = 2.7267. Substituting d1 and d2 in the call price formula gives: C = 18N (2.8017) − 15e−0.1(0.25) N (2.7267). Using the approximation of the cumulative normal distribution in the points 2.8017 and 2.7267, the call price is 3.3659 or: C = 18(0.997) − 14.6296(0.996) = 3.3659 The following (Tables 8.1–8.4) provide simulation results for a European call and put prices using the Black–Scholes model. The tables also provide the Greek letters. Table 8.1. Simulations of Black and Scholes put prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

19.44832 15.56924 12.17306 9.29821 6.94392 5.07582 3.63657 2.55742 1.76806

−0.81955 −0.72938 −0.62762 −0.52216 −0.42055 −0.32847 −0.24939 −0.18449 −0.13331

0.01676 0.01967 0.02105 0.02086 0.01933 0.01692 0.01412 0.01130 0.00871

0.21155 0.28240 0.34124 0.37895 0.39156 0.38031 0.35019 0.30789 0.26002

−0.00575 −0.00769 −0.00931 −0.01035 −0.01069 −0.01038 −0.00955 −0.00838 −0.00707

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European Option Pricing Models Table 8.2. Simulations of Black and Scholes call prices, S = 100, K = 100, t = 22/12/2003, T = 22/12/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

1.43382 2.55474 4.15856 6.28371 8.92943 12.06132 15.62208 19.54292 23.75356

0.18045 0.27062 0.37238 0.47784 0.57945 0.67153 0.75061 0.81551 0.86669

0.01676 0.01967 0.02105 0.02086 0.01933 0.01692 0.01412 0.01130 0.00871

0.21155 0.28240 0.34124 0.37895 0.39156 0.38031 0.35019 0.30789 0.26002

−0.00575 −0.00769 −0.00931 −0.01035 −0.01069 −0.01038 −0.00955 −0.00838 −0.00707

Table 8.3. Simulations of Black and Scholes call prices, S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

0.36459 0.93156 1.99540 3.70489 6.12966 9.24535 12.95409 17.12243 21.61673

0.07582 0.15729 0.27358 0.41284 0.55640 0.68669 0.79246 0.87053 0.92356

0.01332 0.02070 0.02656 0.02900 0.02763 0.02332 0.01781 0.01245 0.00807

0.08172 0.14559 0.21252 0.26201 0.27966 0.26377 0.22359 0.17279 0.12322

−0.00442 −0.00791 −0.01158 −0.01429 −0.01526 −0.01439 −0.01218 −0.00939 −0.00668

Table 8.4. Simulations of Black and Scholes put prices, S = 100, K = 100, t = 22/12/2003, T = 22/06/2004, r = 2%, and σ = 20%. S 80 85 90 95 100 105 110 115 120

Price

Delta

Gamma

Vega

Theta

19.36686 14.93383 10.99767 7.70716 5.13193 3.24762 1.95636 1.12471 0.61900

−0.92418 −0.84271 −0.72642 −0.58716 −0.44360 −0.31331 −0.20754 −0.12947 −0.07644

0.01332 0.02070 0.02656 0.02900 0.02763 0.02332 0.01781 0.01245 0.00807

0.08172 0.14559 0.21252 0.26201 0.27966 0.26377 0.22359 0.17279 0.12322

−0.00442 −0.00791 −0.01158 −0.01429 −0.01526 −0.01439 −0.01218 −0.00939 −0.00668

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8.5. The Black Model for Commodity Contracts Using some assumptions similar to those used in deriving the original B–S option formula, Black (1976) presented a model for the pricing of commodity options and forward contracts. 8.5.1. The model for forward, futures, and option contracts In this model, the spot price S(t) of an asset or a commodity is the price at which an investor can buy or sell it for an immediate delivery at current time, t. This price may rise steadily, fall, and fluctuate randomly. The futures price F (t, t∗ ) of a commodity can be defined as the price at which an investor agrees to buy or sell at a given time in the future, t∗ , without putting up any money immediately. When t = t∗ , the futures price is equal to the spot price. A forward contract is a contract to buy or sell at a price that stays fixed until the maturity date, whereas the futures contract is settled every day and re-written at the new futures price. Following Black (1976), let v be the value of the forward contract, u be the value of the futures contract, and c be the value of an option contract. Each of these contracts is a function of the futures price F (t, t∗ ) as well as other variables. So, we can write at an instant t, the values of these contracts, respectively as V (F, t), u(F, t), and c(F, t). The value of the forward contract also depends on the price of the underlying asset, K at time t∗ and can be written V (F, t, K, t∗ ). It is important to distinguish between the price and the value of the contract. The futures price is the price at which a forward contract presents a zero current value. It is written as: V (F, t, F, t∗ ) = 0

(8.26)

Equation (8.26) implies that the forward contract’s value is zero when the contract is initiated and the contract price, K, is always equal to the current futures price F (t, t∗ ). The main difference between a futures contract and a forward contract is that a futures contract may be assimilated to a series of forward contracts. This is because the futures contract is re-written every day with a new contract price equal to the corresponding futures price. Hence, when F rises, i.e., F > K, the forward contract has a positive value and when F falls, F < K, then the forward contract has a negative value. When the transaction takes place, the futures price equals the spot

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price and the value of the forward contract equals the spot price minus the contract price or the spot price: V (F, t∗ , K, t∗ ) = F − K.

(8.27)

At maturity, the value of a commodity option is given by the maximum of zero and the difference between the spot price and the contract price. Since at this date, the futures price equals the spot price, it follows that if F ≥ K, then c(F, t∗ ) = F − K otherwise, c(F, t∗ ) = 0.

(8.28)

In order to value commodity contracts and commodity options, Black (1976) assumed that: • The futures price changes are distributed log-normally with a constant variance rate σ 2 ; • All the parameters of the CAPM are constant through time and • There are no transaction costs and no taxes. Under these assumptions, it is possible to create a risk-less hedge by taking a long position in the option and a short position in the futures contract.

Let ∂c(F,t) be the weight affected to the short position in the futures ∂F

contract, which is the derivative of c(F, t) with respect to F . The change in the hedged position may be written as: ∂c(F, t) ∆F. ∆c(F, t) − ∂F

(8.29)

Using the fact that the return to a hedged portfolio must be equal to the risk-free interest rate and expanding ∆c(F, t) gives the following partial differential equation: ∂c(F, t) 1 2 2 ∂ 2 c(F, t) = rc(F, t) − σ F ∂t 2 ∂F 2 or ∂c(F, t) 1 2 2 ∂ 2 c(F, t) − rc(F, t) + σ F =0 2 ∂F 2 ∂t

(8.30)

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Denoting T = t∗ − t and using Eqs. (8.28) and (8.30), the value of a commodity option is: c(F, T ) = e−rT [F N (d1 ) − KN (d2 )] F σ2 √ + T ln K √ 2 , d2 = d1 − σ T . d1 = σ T

(8.31)

It is convenient to note that the commodity option’s value is the same as the value of an option on a security paying a continuous dividend. The rate of distribution is equal to the stock price times the interest rate. If F e−rT is substituted in the original formula derived by Black and Scholes, the result is exactly the above formula. In the same context, the formula for the European put is: p(F, T ) = e−rT [−F N (d1 ) + KN (−d2 )] F σ2 √ + T ln K √ 2 , d2 = d1 − σ T d1 = σ T

(8.32)

where N (.) is the cumulative normal density function given by: d “ 2” 1 −x N (d) = √ e 2 dx. 2Π −∞ The value of the put option can be obtained directly from the put-call parity. 8.5.2. The put-call relationship The put-call parity relationship for futures options is: p − c = e−rT (K − F ).

(8.33)

This relationship can be explained as follows. Consider a portfolio where the investor is long a future contract, long a put on the future contract, and short a call with the same time to maturity and strike price. Note that the combination of the call and the put is equivalent to a short synthetic future. At expiration, the payoff is given by the difference between the options strike prices and the current futures price. Hence, the current value of this portfolio must be equal to the present value of this difference. Since these options are European, they have the same cash flows as options on the spot asset. This is because at the maturity date, the futures price is

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equal to the spot price. We now give some examples for the calculation of option prices using Black’s formula.

8.6. Application of the CAPM Model to Forward and Futures Contracts The major difficulty arising from the application of the CAPM to the risk-return relationship on a capital asset comes from the definition of the apropriate rate of return. More specifically, the CAPM cannot be directly applied to the futures contract because the initial value of the contract is zero.

8.6.1. An application of the model to forward and futures contracts Following the work of Dusak (1973), when applying the classic CAPM, the percentage change in the futures price is used as a candidate since it can always be computed. However, the percentage change in the futures price ˜ S value, since cannot be interpreted as a rate of return comparable to the R the holder of the position does not invest current resources in the contract. It ˜ S −r) on the underlying spot assets. can be rather seen as a risk premium, (R In fact, the buyer of the futures contract takes the risk and has no capital on his/her own invested. Hence, he/she earns no interest on the capital. The CAPM equilibrium conditions can then be re-stated by saying ˜ S ) = r(1 − βS ) + that the expected return on any asset S, is written as E(R ˜ ˜ βS (E(Rm )) where E(RS ) can be represented in terms of periods 0 and 1 0) . prices for the asset S as: E(SS1 −S 0 The equilibrium risk-return relationship on asset S can then be ˜m ) − r)S0 , where S0 is the price of expressed as S0 = E(S1 ) − βS (E(R asset S in the start of period and S1 is the asset’s price at the end of the period. In present-value form, the above equation is equivalent to ˜ m )−r)S0 (E(R . S0 = E(S1 )−βS(1+r) On the other hand, the current price for asset S under an agreement to buy the asset at time 0 but with payment deferred to time 1 is S0 (1 + r). This can be seen as the current futures price for payment of the spot price in a period. The term E(S1 ) is interpreted as the spot price expected to prevail at time 1. Multiplying the above equation by (1 + r) gives: ˜ m ) − r)S0 . (1 + r)S0 = E(S1 ) − βS (E(R

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Setting F0 = (1 + r)S0 and re-arranging the terms gives: E(S1 ) − F0 ˜ m ) − r). = βS (E(R S0 This equation can be interpreted as expressing the risk premium on the underlying spot asset as the change in the futures price divided by the period zero spot price. The analysis in Black (1976) can be implemented in this context. In fact, let us re-write the CAPM in the following form: ˜S ) − r = βS [E(R ˜ m ) − r]. E(R Writing S0 and S1 as the start and end-of-period prices of an asset i and using the definition of βi , the CAPM can be written as: 0 ˜m cov S1S−S , R (S1 − S0 ) 0 ˜m ) − r]. −r = E [E(R (8.34) ˜m ) S0 var(R If we multiply by S0 , we obtain the expected dollar return on asset i as: E(S1 − S0 ) − rS0 =

˜m ) cov((S1 − S0 ), R ˜m ) − r]. [E(R ˜m) var(R

(8.35)

The price S0 can be set equal to zero since the futures contact’s value at this time is zero. Ignoring daily limits, we set S1 equal to ∆S. In fact, if we apply this equation to a futures contract, the change in the futures price ˜ m) R ˜ over the period is E(∆S) = cov(∆S, ˜ m ) [E(Rm ) − r]. var(R Equation (8.34) can be written for the futures prices as: ˜ m ) − r]. E(∆S) = β ∗ [E(R

(8.36)

Equation (8.35) shows that the expected change in the futures price is proportional to the dollar “beta” of the futures price. The expected change in the futures price may be zero, positive, or negative. Also, the expected change in the futures price is zero when the covariance of the change in the futures price with the market portfolio is zero, i.e., E(∆S) = 0 when ˜m ) = 0. cov(∆S, R 8.6.2. An application to the derivation of the commodity option valuation Black (1976) showed that in the absence of interest-rate uncertainty, a European commodity option on a futures (or a forward) contract can be

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priced using a minor modification of the Black and Scholes (1973) option pricing formula. In deriving expressions for the behavior of the futures price, he assumed that both taxes and transaction costs are zero and that the CAPM applies at each instance of time. Following Black (1976), we assume that the fractional change in the futures price is distributed lognormally, with a known constant variance rate, σ. We also assume that all the parameters of the CAPM are constant through time. Under these assumptions, the value of the futures commodity option, C(S, t), can be written as a function of the underlying futures price and time. In this context, it is possible to create a risk-less hedge by taking a long position in the option and a short position in the futures contract with the same transaction date. Black (1976) assumed that a continuously re-balanced self-financing portfolio of the underlying futures contracts and the riskless asset can be constructed to duplicate the payoff of the futures option. The relationship between a commodity option’s beta and its underlying CS security’s beta is given by βC = S C βS , where βc and βS refer respectively, to the betas of the commodity option and its underlying commodity contract. The expected return on a security in the context of the CAPM is: ¯ S − r = βS [R ¯ m − r] R or ¯ S − r = aβS R

¯ m − r]. with a = [R

This equation can be written for the expected return on the spot asset and the option as: ∆S = r∆t + aβS ∆t E S ∆C = r∆t + aβC ∆t. E C Multiplying this last equation by C and substituting for βC gives: E(∆C) = rC∆t + aSβS CS ∆t. Taking the expected value and replacing E(∆S) gives: 1 E(∆C) = rSCS ∆t + aSβS CS ∆t + Ct ∆t + CSS S 2 σ 2 ∆t. 2

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Making the equality between this equation and E(∆C) = rC∆t + aSβS CS ∆t. Simplification gives: 1 CSS S 2 σ 2 + rSCS − rC + Ct = 0. 2 This equation is the Black and Scholes (1973) equation. Since the value of a futures contract is zero, the equity in the position is just the value of the option. In this context, the system: 1 E(∆C) = rSCS ∆t + aSβS CS ∆t + Ct ∆t + CSS S 2 σ 2 ∆t 2 E(∆C) = rC∆t + aSβS CS ∆t, becomes: 1 E(∆C) = Ct ∆t + CSS S 2 σ 2 ∆t 2 E(∆C) = rC∆t, which gives: 1 CSS S 2 σ 2 − rC + Ct = 0. 2 This equation is the Black (1976) equation. 1 CSS S 2 σ 2 − rC + Ct = 0. 2

(8.37)

Let T be the maturity date of the call and K be its strike price. The equation must be solved under the boundary condition at maturity: C(S, T ) = S − K C(S, T ) = 0

if S ≥ K

if S < K.

There is a simple relationship between the future price and the spot price F = SebT . The value of a European commodity call is: C(F, T ) = e−rT [F N (d1 ) − KN (d2 )] with d1 =

ln

F

+ √ σ T

K

σ2 T 2

,

√ d2 = d1 − σ T .

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The solution for a European futures put is: P (F, T ) = e−rT [−F N (−d1 ) + KN (−d2 )]. The term F N (d1 ) − KN (d2 ) shows that the expected value of the futures call at expiration, is the expected difference between the futures price and the strike price conditional upon the option being in-the-money times the probability that it will be in-the-money. The term e−rT is the appropriate discount factor by which the expected expiration value is brought to the present. It is possible to use directly the put-call parity to write down the put formula. Re-call that the put-call parity relationship between puts and call with identical strike prices is an arbitrage-based relationship, which holds regardless of the distribution of financial asset prices. More general results about calls and puts with different strike prices can be written down for both symmetric and asymmetric processes using the propositions in Bates (1997).

8.6.3. An application to commodity options and commodity futures options A commodity call gives the right to its holder to buy a specific commodity at a specified price within a specified period of time. A commodity put gives the right to its holder to buy a specific commodity at a specified price within a pre-determined period of time. A commodity futures option is an option on the futures contract having a commodity as an underlying asset. The commodity may be a precious metal such as either silver or gold. It may be a financial instrument such as a common stock, a treasury bond, or a foreign currency. For example, if the commodity option is written on a foreign currency, the option refers to a currency option. If the commodity option is written on a stock index, the option is an index option. Since all the analytical models for European options presented in this chapter can be obtained by modifying the parameters in the Merton (1973) and BAW (1987) model, all the applications presented here are also true for this model. For example, when the option’s underlying asset is an index, which is constructed to pay continuous dividends, this version is a particular case of the Merton’s and BAW’s model for commodity options.

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When the continuous dividend yield is d, the formula for European commodity options is re-written for index options with b = (r − d). For a European index call, the formula is: c(S, T ) = Se−dT N (d1 ) − Ke−rT N (d2 )

(8.38)

with d1 =

ln

Se−dT K

√

+

σ2 T 2

σ T

,

√ d2 = d1 − σ T .

For a European index put, the formula becomes: p(S, T ) = −Se−dT N (−d1 ) + Ke−rT N (−d2 ) with d1 =

ln

Se−dT K

√

σ T

+

σ2 T 2

,

√ d2 = d1 − σ T .

8.7. The Holes in the Black–Scholes–Merton Theory and the Financial Crisis Black (1998) examined the assumptions of the Black and Scholes (1973) model and suggested some modifications to improve the model. In the original formula, it is assumed that: • • • • • •

the volatility of the underlying asset is known; investors can either borrow or lend at a single interest rate; the short seller of the underlying asset can use the proceeds of the sale even if they are used as collateral; there are no transaction costs; there are no dividends and no taxes and there are no takeovers or other events that end the life of the option early.

8.7.1. Volatility changes In practice, volatility is not constant. It can change and affect especially far-out-of-the-money options.

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For example, if the formula is computed for an out-of-the-money option with a volatility of 20% when S = 28 and K = 40, the option price is 0.0088. Doubling the volatility to 40%, this can give a new option value of 0.46, which is more than 50 times higher than in the first case. In this case, it is possible to assign probability estimates to various volatility figures and to use the probabilities to weigh the resulting option values. For example, if there is a 0.5 chance that the volatility will be 0.2 and a 0.5 chance that it will be 0.4, then the option value will be 0.23. This procedure increases out-of-the-money option values. It is true that volatility changes in an unexplainable way, but there is a relationship between the stock price and volatility. In general, the stock price and the volatility change in opposite directions, i.e., when the stock price increases, the volatility decreases and vice versa. Cox and Ross (1976) used different formulas to explain the deviations between model values and option prices by accounting for this fact. The Cox and Ross (1976) formula gives lower values for out-of-the-money options than the Black–Scholes formula. Merton (1976) accounted for jumps to increase the relevant values of both out-of-the-money and in-the-money options. His formula decreases at the same time the values of at-the-money-options. Jumps affect the underlying asset price and can be viewed as momentary large increases in the volatility of the underlying asset. The formula handles only jumps and does not account for stock-price related changes in volatility. The changing character of volatility does not lead to a “close-toriskless” hedge. Consider an initial position in which the investor is short two calls and long one round lot of the underlying asset. If the stock moves by $1 and the call by $0.5, the position is protected against stock-price movement but not against changes in volatility. In practice, it is impossible to diversify away this risk so that investors do not care about it.

8.7.2. Interest rate changes As volatility changes over time, interest rates also move with a main difference so that interest rates can be observed. Merton (1973) has shown that the random character of interest rates can be accounted for by substituting the interest rate on a zero-coupon bond and a maturity equal to the option expiration for the short-term interest rate in the formula. This is possible when the volatility is constant. The effect of changing interest rates on option values do not seem as great as the effects of changes in volatility.

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8.7.3. Borrowing penalties In general, investors borrow at a higher rate than the rate at which they can lend. High-borrowing rates may increase option values. This is because options can provide leverage that can substitute for borrowing. 8.7.4. Short-selling penalties Short-selling leads the investor to borrow the stock and to put up a cash collateral with the person who lends the stock. This cash does not produce an interest. In general, professional option traders find that the penalties to writing naked options do not affect them. However, there are often penalties on short selling of the stock. Since buying a put option is equivalent to selling stock short, penalties on short selling of stock can increase the prices of put options. 8.7.5. Transaction costs Transaction costs paid in the form of brokerage charges and clearing charges can affect the hedging strategy. It is not possible to maintain a neutral hedge continuously, by changing the ratio of option position to stock position as the parameters change. Stock prices can also jump without a chance for trades to take place. This makes the strategy impossible to maintain a neutral hedge. 8.7.6. Taxes The existence of different tax rates for institutions and individuals can affect option values. The exact rules used to restrict tax arbitrage will affect option values. For example, index option positions are taxed, in general, partly at short-term capital gains rates and partly at long-term capital gains rates. This depends on whether the position has been closed out each year. If several investors pay taxes on gains and cannot deduct losses, they want to limit the volatilty of their positions and have the freedom to control the timing of their gains and losses. This can affect the use of options and the option values. 8.7.7. Dividends Dividends reduce call-option values and increase put-option values. Several formulas are proposed in the literature to handle dividends, but the exact

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solution, needs the knowledge of how the amount of future unknown dividends depends on the factors affecting the stock price.

8.7.8. Takeovers The Black–Scholes model assumes that the stock will continue trading for the option’s life. If, for example, firm A takes over firm B through an exchange of stock, options on B’s stock will expire early. However, the stock value is higher than before. The premium of the tender offer will increase the call value and reduce the put value.

Summary “Because options are specialized and relatively unimportant financial securities, the amount of time and space devoted to the development of a pricing theory might be questioned”, said Professor Merton (1973), in Bell Journal of Economics and Management Science. Thirty years ago, no one could have imagined the changes that were about to occur in finance theory and the financial industry. The seeds of change were contained in option theory being conducted by the Nobel Laureates Black, Scholes, and Merton. Valuing claims to future income streams is one of the central problems in finance. The first known attempt to value options appeared in Bachelier (1900) doctoral dissertation using an arithmetic Brownian motion. This process amounts to negative asset prices. Sprenkle (1961) and Samuelson (1965) used a geometric Brownian motion that eliminated the occurence of negative asset prices. Samuelson and Merton (1969) proposed a theory of option valuation by treating the option price as a function of the stock price. They advanced the theory by realizing that the discount rate must be determined in part by the requirement that investors hold all the amounts of stocks and the option. Their final formula depends on the utility function assumed for a “typical” investor. Several discussions are done with Merton (1973) who was also working on option valuation. Merton (1973) pointed out that assuming continuous trading in the option or its underlying asset can preserve a hedged portfolio between the option and its underlying asset. Merton was able to prove that in the presence of a non-constant interest rate, a discount bond maturing at the option expiration date must be used. Black and Scholes (1973) and Merton (1973) showed that the construction of a risk-less hedge between the option and its underlying asset, allows the derivation of an option pricing formula

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regardless of investors’ risk preferences. The main attractions of the Black– Scholes model are that their formula is a function of “observable” variables and that the model can be extended to the pricing of any type of option. Using some assumptions similar to those used in deriving the original B–S option formula, Black (1976) presented a model for the pricing of commodity options and forward contracts. Black (1976) showed that in the absence of interest-rate uncertainty, a European commodity option on a futures (or a forward) contract can be priced using a minor modification of the Black and Scholes (1973) option pricing formula. In deriving expressions for the behavior of the futures price, he assumed that both taxes and transaction costs are zero and that the CAPM applies at each instant of time. This chapter presented in detail the basic concepts and techniques underlying rational derivative asset pricing in the context of analytical European models along the lines of Black and Scholes (1973), Black (1976), and Merton (1973). First, an overview of the analytical models proposed by the precursors is given. Second, the simple model of Black and Scholes (1973) is derived in detail for the valuation of options on spot assets and some of its applications are presented. Third, the Black model, which is an extension of the Black–Scholes model for the valuation of futures contracts and commodity options, is analyzed. Also, applications of the model are proposed. Fourth, the basic limitations of the Black–Scholes– Merton theory are studied and the models are applied to the valuation of several financial contracts. The Black–Scholes hedge works in the real, discrete, and frictionful world when the hedger uses the correct volatility of the prices at which he/she actually trades and when the asset prices do not jump too much. The assumptions of the Britten-Jones and Neuberger (1996) model provided a framework in which a trader can avoid jumps and in which total variance can be estimated perfectly. The model transforms the question of pricing and hedging options into how well investors can predict the total variance of returns of the associated hedging strategy. The Black–Scholes formula gives a rough approximation to the formula investors would use, if they knew how to account for the above factors. Modifications of the Black–Scholes formula can move it to the hypothetical perfect formula.

Questions 1. What is wrong in Bachelier’s formula? 2. What is wrong in Sprenkle’s formula?

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3. What is wrong in Boness’s formula? 4. What is wrong in Samuelson’s formula? 5. What are the main differences between the Black and Scholes model and the precursors models? 6. How can we obtain the put-call parity relationship for options on spot assets? 7. How can we obtain the put-call parity relationship for options on futures contracts? 8. Is the Black model appropriate for the valuation of derivative assets whose values depend on interest rates? Justify your answer. 9. Is there any difference between a futures price and the value of a futures contract? 10. What are the holes in the Black–Scholes–Merton theory?

Appendix A. The Cumulative Normal Distribution Function The following approximation of the cumulative normal distribution function N (x) produces values to within 4-decimal place accuracy. 1 N (x) = √ 2π

x

−∞

exp(−z 2 /2)dz

N (x) = 1 − n(x)(a1 k + a2 k 2 + a3 k 3 )

when x 0

1 − n(−x)

when x < 0

where k=

1 , 1 + 0.33267x

a1 = 0.4361836,

a3 = 0.9372980,

a2 = −0.1201676,

2 1 and n(x) = √ e−x /2 . 2π

The next approximation provides the values of N (x) within six decimal places of the true value.

N (x) = 1 − n(x)(a1 k + a2 k 2 + a3 k 3 + a4 k 4 + a5 k 5 ) when x 0 1 − n(−x)

when x < 0

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where 1 , a1 = 0.319381530, a2 = −0.356563782, 1 + 0.2316419x a3 = 1.781477937, a4 = −1.821255978, and a5 = 1.330274429. k=

Appendix B. The Bivariate Normal Density Function F (x, y) =

1 1 2 2 exp − − 2ρxy + y ) . (x 2π(1 − ρ2 ) 2π 1 − ρ2

The cumulative bivariate normal density function The standardized cumulative normal function gives the probability that a specified random variable is less than a and that another random variable is less than b when the correlation between the two variables is ρ. It is given by: 2 a b 1 x − 2ρxy + y 2 . M (a, b; ρ) = exp − 2π(1 − ρ2 ) 2π 1 − ρ2 −∞ −∞ This following approximation produces values of M (a, b; ρ) to within six decimal places accuracy. 5 5 1 − ρ2 φ(a, b; ρ) = xi xj f (yi , yj ), π i=1 j=1 where f (yi , yj ) = exp[a1 (2yi − a1 ) + b1 (2yj − b1 ) + 2ρ(yi − a1 )(yj − b1 )] a , 2(1 − ρ2 ) x1 = 0.24840615

b b1 = 2(1 − ρ2 ) y1 = 0.10024215

x2 = 0.39233107

y2 = 0.48281397

x3 = 0.21141819

y3 = 1.0609498

x4 = 0.033246660

y4 = 1.7797294

x5 = 0.00082485334

y5 = 2.6697604

a1 =

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If the product of a, b, and ρ is nonpositive, we must compute the cumulative bivariate normal probability by applying the following rules: 1. If a 0, b 0, and ρ 0, then: M (a, b; ρ) = φ(a, b; ρ) 2. If a 0, b 0, and ρ 0, then: M (a, b; ρ) = N (a) − φ(a, −b; −ρ) 3. If a 0, b 0, and ρ 0, then: M (a, b; ρ) = N (b) − φ(−a, b; −ρ) 4. If a 0, b 0, and ρ 0, then: M (a, b; ρ) = N (a) + N (b) − 1 + φ(−a, −b; ρ). In cases where the product of a, b, and ρ is positive, compute the cumulative bivariate normal function as: M (a, b; ρ) = M (a, 0; ρ1 ) + M (b, 0; ρ2 ) − δ where M (a, 0; ρ1 ) and M (a, 0; ρ2 ) are computed from the rules, where the product of a, b, and ρ is negative, and: (ρa − b)Sign(a) ρ1 = , a2 − 2ρab + b2 δ=

1 − Sign(a) × Sign(b) , 4

(ρb − a)Sign(b) ρ2 = a2 − 2ρab + b2 +1 when x 0 Sign(x) = −1 when x < 0

References Ayres, HF (1963). Risk aversion in the warrants market. Industrial Management Review, 4 (Fall), 497–505. Bachelier, L (1900). Theorie de la speculation, Ph.D. Thesis in Mathematics, Annales de l’Ecole Normale Superieure, III.17, 21–86. Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June), 301–320. Bates, DS (1997). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. Review of Financial Studies, 9, 69–107.

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Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, (January/March), 167–179. Black, F (1998). The holes in Black–Scholes, Risk (September), 6–8. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Boness, AJ (1964). Elements of a theory of stock option value. Journal of Political Economy, 72 (April), 163–175. Britten-Jones, M and A Neuberger (1996). Arbitrage pricing with incomplete markets. Applied Mathematical Finance, 3(4), 347–363 and 11–13. Chen, A (1970). A model of warrant pricing in a dynamic market. Journal of Finance, 25, 1041–1060. Churchill, RV (1963). Fourier Series and Boundary Value Problems. 2nd Ed. New York: McGraw–Hill. Cox, JC and SA Ross (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166. Dusak, K (1973). Futures trading and investor returns: an investigation of commodity market risk premiums. Journal of Political Economy, 81, 1387–1406. Garman, M and S Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance, 2, 231–237. Merton, R (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, RC (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–144. Samuelson, PA (1965). Rational theory of warrant pricing. Industrial Management Review, 6, 13–31. Samuelson, P and RC Merton (1969). A complete model of warrant pricing that maximises utility. Industrial Management Review, 10, 17–46. Sharpe, WF (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442. Smith, CW (1976). Option pricing: A review. Journal of Financial Economics, 3(January/March 1976), 3–52. Smithson, C (1991). Wonderful life. Risk, 4(9), (October), 50–51. Sprenkle, C (1961). Warrant prices as indications of expectations. Yale Economic Essays, 1, 179–232. Thorp, E and S Kassouf (1967). Beat the Market. New York: Random House.

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Chapter 9 SIMPLE EXTENSIONS AND APPLICATIONS OF THE BLACK–SCHOLES TYPE MODELS IN VALUATION AND RISK MANAGEMENT

Chapter Outline This chapter is organized as follows: 1. Section 9.1 gives some applications of the Black and Scholes (1973) option pricing theory. 2. Section 9.2 presents the main applications of the Black’s (1976) model. 3. Section 9.3 develops the main results in Garman and Kohlhagen’s (1983) model for the pricing of currency options. 4. Section 9.4 presents the main results in the models of Merton (1973) and Barone-Adesi and Whaley (1987) model for the pricing of European commodity and commodity futures options. Some applications of the model are also proposed. 5. Section 9.5 compares the Black–Scholes world with the real world. Introduction Broadly speaking, there are four groups of equity options: traded options, over-the-counter (OTC) options, equity warrants, and covered warrants. Traded options are standardized contracts which are listed on options exchanges. These options are not protected against dividend and their strike prices and maturity dates are set by the exchange. Stock index options and futures markets have experienced remarkable growth rates. Stock index options are of the European or the American type and often involve cash settlement procedure upon exercise. The Black and Scholes (1973) model 403

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can be used in the valuation of options for which the underlying asset is a fixed-income instrument. However, this model presents some limitations in the valuation of interest rate options. The standard binomial models can also be applied for the valuation of interest rate options. Since the Black and Scholes formula is valid for a non-cash paying security, it can be used for the pricing of a zero-coupon bond. The Black and Scholes (1973) model is universally applied by market participants even though several other alternative models exist. The main question is why the Black–Scholes model successful and how to apply the model when the imperfections of the real world loom large. This chapter presents the theory of European options and its applications along the Black–Scholes lines and its extensions by Black (1976) for options on futures, Garman and Kohlhagen (1983) for options on currencies and indirectly by Merton (1973), and Barone-Adesi and Whaley (1987) for European commodity and futures options. These models and especially the Black’s (1976) model apply for commodity options. Commodity markets can be traced back to the corn markets of the Middle Ages when farmers, merchants, and end-users would all meet in a specific place. The term “commodity” refers, today, to a variety of products, ranging from the traditional agricultural crops to oil and financial instruments. Since the main concern in this chapter is about analytical models under the Black– Scholes (1973) assumptions, the question of dividends, stochastic interest rates, and stochastic volatilities are discussed in other chapters. 9.1. Applications of the Black–Scholes Model Equity options can be used in several ways in portfolio management. Buying or selling options involves the payment or the receipt of the option premium at the initial time when the transaction is done. 9.1.1. Valuation and the role of equity options Since the option payoff is asymmetric, this gives rise to an asymmetric distribution of returns. Hence, options can be used in portfolio management to structure the distribution of expected returns. The best-known strategies in portfolio management involve combinations of options. They include vertical spreads, calendar spreads, diagonal spreads, ratio spreads, volatility spreads, and synthetic contracts. The main difference between futures contracts and option contracts is that the investor pays a premium for options and nothing to establish a futures position.

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Applications of Black–Scholes Type Models

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Calls and puts are bought or sold in anticipation of future cash flows, for defensive purposes, and speculative reasons. The investor must choose the appropriate options to be bought or sold. Therefore, the question of the management of an option position is as important as the question of option valuation and strategies. Over-the-counter (OTC) options are tailor-made to the investor’s needs and are often written by investment banks. Equity warrants are long-term options and are often traded in securities markets rather than in option markets. When these options are exercised, new shares are issued by the company. Covered warrants are OTC long-term options issued by securities houses. These equity options can be valued using the Black–Scholes model. However, the following specificities of these instruments imply some extensions of the Black–Scholes model. First, these options are frequently traded on an asset, which distributes dividends and they are of the American type, i.e., they can be exercised before maturity. Second, the assumed diffusion process may not represent reality since equity prices may jump downward or upward in response to either bad or good news. Third, it is more dificult to justify a constant volatility for the underlying asset when the option maturity is long. The same argument applies for the risk-less interest rate. In this chapter, we restrict our analysis to the assumptions of the Black–Scholes model, which will be relaxed afterwards when studying the extensions and generalizations of the model. Many strategies can be implemented with equity derivatives. These strategies are obviously not specific to equities. They also apply to options on other types of underlying assets.

9.1.2. Valuation and the role of index options Stock index options and futures markets have experienced remarkable growth rates. Stock index options are either of the European or the American type and often involve cash settlement procedure upon exercise.

9.1.2.1. Analysis and valuation Stock index options are traded on the major indices around the world. These options are of the European or American-type. Options on the spot

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index are cash-settled and there is no physical delivery of the underlying index, i.e., of a weighted average of prices of the stocks that constitutes the index. There are several weighting schemes. The most commonly used is the market capitalization where each equity price is weighted by the market capitalization of the firm, i.e., the number of shares times the share price. Two alternative methods are sometimes used: equal weighting and price weighting. The last two methods assign greater relative weight to small-company constituents than do capitalization-weighted indices. Index options are also sold in the OTC market as OTC warrants. In this case, they refer to long-term options on the spot index. Because they are traded on OTC markets, they are subject to credit risk. When the option underlying index is constructed to pay continuous dividends, the index price is adjusted by the discounted value of the continuous dividend yield. The appropriately adjusted Black–Scholes (1973) version when the continuous dividend yield d corresponds to the following formula for a European index call is: c(S, T ) = Se−dT N (d1 ) − Ke−rT N (d2 ) −dT 2 + r + σ2 T ln SeK √ √ d1 = , d2 = d1 − σ T σ T where N (.) is the cumulative normal density function given by: d “ 2” 1 −x e 2 dx N (d) = √ 2Π −∞ The European index put formula is given by: p(S, T ) = −Se−dT N (−d1 ) + Ke−rT N (−d2 ) −dT 2 + r + σ2 T ln SeK √ √ d1 = , d2 = d1 − σ T σ T 9.1.2.2. Arbitrage between index options and futures It is convenient to note that the same strategies for stock options can also be used in portfolio management with index options. Also, these options can be used in asset allocation and portfolio insurance. Since these intruments are based on the same underlying index, their prices must be interrelated. If this is not the case, the relative mispricing should instantaneously

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disappear given the variety of cross-market strategies. These strategies include arbitrage between index options and index futures, and between index futures and the stocks comprising the index. Many researchers have studied these arbitrages which imply that significant deviations from prices dictated by the relevant market interrelationships should disappear. It is often found that there are some violations of the no-arbitrage bounds. However, when taking into account the transaction costs, the future price lies often between the no-arbitrage bounds. Even if all tests and published studies are in favor of market efficiency and integrated markets, it is often reported that relative mispricing does exist between index options and futures contracts. For more details, see for example, Evnine and Rudd (1985) and Brennan and Schwartz (1990), among others. On the other hand, despite the controversy about index arbitrage and program trading, these financial intruments are benefical to stock portfolio managers and institutional investors. Before the emergence of these contracts, market participants cannot hedge and control the market risk of their portfolios. Even if there is some evidence that trading in index futures increases cash-market volatility, arbitrage activities via program trading may cause prices to adjust more rapidly to new information. This helps to keep the movements of index futures price and the stock index more synchronous. The deviations of futures prices from their “fair” value result from various considerations including imperfect substitutability between spot and futures markets, the speed with which information is incorporated in prices in the different markets, and market imperfections including transaction costs and regulatory constraints, among other things.

9.1.3. Valuation of options on zero-coupon bonds The Black and Scholes (1973) model can be used in the valuation of options for which the underlying asset is a fixed-income instrument. However, this model presents some limitations in the valuation of interest rate options. The standard binomial models can also be applied for the valuation of interest rate options. Since the Black and Scholes formula is valid for a noncash paying security, it can be used for the pricing of a zero-coupon bond. When using the Black and Scholes formula, the underlying asset is the bond price. The bond price can be observed in the market place, as it can be calculated by discounting its maturity value at the appropriate risk-free rate. The underlying bond price can have a maturity of five years for example and the option time to maturity is in five months. The underlying

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asset (the bond) has, in principle, a maximum value. It is given by the sum of the coupon payments plus the maturity value. If the bond price is greater than this maximum value, this means that interest rates are negative. Re-call that the bond price is higher, if higher is the interest rate. Or, in the Black and Scholes model for stock options, the underlying stock does not have a maximum value. Hence, applying the Black–Scholes model for an interest rate option can lead to nonsensical option prices. Besides, the Black–Scholes model assumes that interest rates are constant during the option’s life. This is clearly an inappropriate assumption for interest rate options since interest rates change every day and affect the option price. Finally, the Black–Scholes model assumes a constant variance for the underlying asset. This assumption is inappropriate for interest rates since the bond-price volatility declines as the bond approaches the maturity date. In fact, the bond price tends to reach its face value at the maturity date. When there is no coupon payment, the Black–Scholes model is applied as follows for bond calls: c(B, T ) = BN (d1 ) − Ke−rT N (d2 ) B 2 ln K + r + σ2 T √ √ d1 = , d2 = d1 − σ T σ T The Black–Scholes model is applied as follows for bond puts: p(B, T ) = −BN (−d1 ) + Ke−rT N (−d2 ) B 2 + r + σ2 T ln K √ √ d1 = , d2 = d1 − σ T σ T 9.1.4. Valuation and the role of short-term options on long-term bonds Short-term options on long-term bonds are often traded on OTC markets. These options may be of the European or the American type. There is a traded option on the Chicago Board Options Exchange (CBOE), which is based upon the yield to maturity on a portfolio of bonds. The yield to maturity is driven by the changes in the term structure of interest rates. The Black–Scholes model is sometimes used to price short-term European options on zero-coupon bonds. In this context, the call’s value is

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given by: c(B, T ) = BN (d1 ) − Ke−rT N (d2 ) where d1 =

ln

B K

+ r+ √ σ T

σ2 2

T

,

√ and d2 = d1 − σ T

where B: price of the bond; K: strike price of the option; T : time to maturity of the option; σ: instantaneous standard deviation of the bond price and r: spot rate on a risk-free investment with a maturity date T . Using the put-call parity relationship, the put’s value is given by: p(B, T ) = −BN (−d1 ) + Ke−rT N (−d2 ) B 2 + r + σ2 T ln K √ √ d1 = , d2 = d1 − σ T σ T However, other European models, which are extensions or generalizations of the Black–Scholes model like Merton’s (1973) model, are more appropriate for the pricing of these options. 9.1.5. Valuation of interest rate options Interest rate options are often used in the management of interest-rate risk in the same way as equity options. A direct implication is that option strategies for equity options apply directly to interest rate options. The most common and specific strategies based on short-term interest rate options are caps and floors. These strategies place either a cap on the future level of interest rates on a floating instrument or a floor on the interest rate receivable on deposits. A cap: It is an option strategy which protects from a rise in interest rates and allows a profit when interest rates are falling. A floor: It is an option strategy which protects from a decrease in interest rates at the time when the deposit rate is re-set.

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A collar: When an investor buys a cap (floor) and sells the floor (cap), this strategy is known as the collar. The cap, the floor, and the collar can be valued using standard formulas for call and put options.

9.1.6. Valuation and the role of bond options: the case of coupon-paying bonds The price of any financial asset is given by the present value of its expected cash flows. The first step in determining the bond’s price is to determine its cash flows, i.e., the periodic coupon interest payments until the maturity date and the par value at maturity. Since the bond price is given by the present value of the cash flows, its price is given by adding all the discounted future payments at the appropriate interest rates. Some bonds do not make any periodic coupon payments and the interest due to the difference between the maturity value and the purchase price given by the bond holder. This class of bonds is referred to as zero-coupon bonds. It is convenient to note that there are several types of bonds: bonds with call provisions, putable bonds, convertible bonds, bonds with warrants attached, exchangeable bonds, etc. A bond with a call provision gives the right to the issuer to call the issue before the specified redemption date. The call price is different from par and is specified at the bond issue. A bond with a put provision gives its holder the right to put the bond back to the issuer at a fixed price. It is a putable bond. A convertible bond entitles its holder the right to convert the bond into a certain number of units of the equity of the issuing firm or into other bonds. This number is often called the rate of conversion which is specified when the bond is issued. A bond with an attached warrant is simply a package comprising the bond and a warrant. It allows the holder to purchase the equity of the issuing firm. Most of these bonds are Eurobonds issued in international capital markets. An exchangeable bond is similar to a convertible bond, with the exception that it gives its holder the right to exchange the bonds for the equity of other company. The call or the put provision in these bonds can be valued using the Black–Scholes model. However, the model is not appropriate if there are many call or put dates and if the embedded options

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in the bonds are of the American type. We give an application of this model to options on zero-coupon bonds and options on coupon-paying bonds under the Black–Scholes assumptions. If the bond is a coupon-paying bond, then the present value of all coupons due during the option’s life must be substracted from the bond’s price. Example Consider a European call for which the underlying asset is a coupon bond with the following characteristics: • • • • • • •

Bond’s price = 96 Euro; One year interest rate = 10%; Time to maturity = 10 years; Volatility of the bond’s price = 8%; Coupon payments = 5 Euros in 3 months and 9 months; Three-month interest rate = 8% and Nine-month interest rate = 8.5%.

The option has a strike price equal to 100 Euro and its maturity date is in one year. The present value of coupon payments is 9.59 Euro, or: 5e−0.25x0.08 + 5e−0.75x0.085 = 4.9 + 4.69 = 9.59. Applying the Black–Scholes formula gives: B d1 d2 c or c

= = = = =

96 − 9.59 = 86.41 86.41 Euro, 1 ln + 0.1 + 0.0032 0.080 100 d1 − 0.08 × 1 = −0.6158 86.41N (−0.5358) + 100e−0.1 N (−0.6158) 1.25 Euro.

It is convenient to note the “incoherence” with this model since it assumes constant interest rates and at the same time a stochastic bond price. 9.1.7. The valuation of a swaption A swaption: It is the right to assume a position in an underlying interestrate swap with a given maturity. In swaptions, the right to pay the fixed component is equivalent to the right to receive the floating component and vice versa. Swaptions are offered as receiver swaptions and payer swaptions.

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Receiver swaption gives the right to receive a fixed interest rate and payer swaption gives the right to pay a fixed interest rate. The buyer of a receiver swaption benefits when the interest rate falls because he/she is guaranteed to receive a fixed rate, which is higher than the floating rate. If interest rates rise, the swaption can be ignored because the buyer can get a higher fixed rate in the market. The seller of the receiver swaption is obliged to pay the fixed rate and to receive the floating rate in the swap context. Most swaptions are of the European style. Interest-rate swap agreements are second-order derivatives. Interestrate swaps reflect an equilibrium rate that equates a floating-rate stream of payments with a fixed-rate stream of payments at the present date. A swaption price can be computed using an option pricing model where the underlying market input is the rate on the swap. The forward rate for the expiry date of the swaption can be used for a swaption with a European exercise. The following formula is often used to determine forward/forward rates: F = (T − t)

(1 + ST )T (1 + rt )

−1

where: F : forward swap rate; S: spot swap rate; r: deposit rate for time t (the time to swaption expiration) and T : term of the swap in years. Swaption pricing is based on a model allowing the computation of the value of the option to exchange one asset for another as given by the formula in Margrabe (1978): W (x1 , x2 , t) = x1 N (d1 ) − x2 N (d2 ) d1 =

(ln(x1 /x2 ) + 0.5σ2 (t∗ − t)) (t∗ − t) d2 = d1 − v (t∗ − t)

σ2 = σ12 + σ22 − 2σ1 σ2 ρ1,2

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where: W : price of receivers/payers option; x1 , x2 : the prices of assets 1 and 2; σ1 (σ2 ): the volatility of assets 1 (asset 2); ρ1,2 : correlation coefficient; t: current date and t∗ : expiration date. This formula is derived in the last chapter (Chapter 8). Tompkins (for more details, refer to Bellalah et al., 1998) proposed the following put-call relationship for swaptions expressed in annual terms: Payer swaption − receiver swaption =

T (prevailing swap rate − swaption strike) (1 + rt )t t=i

where: r: discount rate for the time period t; T : term of the interest rate swap and t: first exchange date of coupons. 9.2. Applications of the Black’s Model 9.2.1. Options on equity index futures Options on index futures require upon exercise the exchange of a long position in the future contract for a call and a short position in the future contract for the put. Hence, a call is exercised into a long position in the future contract and a put is exercised into a short position in the same contract. The underlying futures contract does not require a physical delivery but is rather settled in cash. The amount received corresponds to the difference between the current and the future level of the underlying index. In this context, the futures contract is regarded as an agreement to either pay or receive a cash payment based upon the difference between the current and the future values of a specified index. Options on index futures are often treated as options on an asset paying a continuous stream of dividends, regardless of whether the underlying spot index pays a continuous or a discrete dividend. Since the assumptions used by Black are similar to those in Black– Scholes, some of them are also questionable, such as the constant volatility

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and the certainty of interest rates. In fact, the nonstationarity of volatility causes some problems in the pricing of options on index futures. As we will see later, some extensions on the Black’s model are more appropriate for the valuation of these options. However, when these options are of the European type, Black’s model is often used in the pricing of these options. The various strategies applied with options on individual assets can be used as well for options on index futures. Index futures and options on index futures are often used in asset allocation and portfolio insurance. Asset allocation refers to the structuring of a multiasset portfolio with respect to the type and the weighing scheme of asset classes. Strategic asset allocation is the construction of a portfolio such that long-run objectives are attained when different classes of assets are transacted at their long-run equilibrium values. Tactical asset allocation, also known as market timing involves shortterm allocations toward rising markets and away from falling markets. Portfolio insurance refers to a group of techniques that insure a portfolio against falling in value below a certain specified level, the floor level. This level does not eliminate the potential profits from a rise in the asset value.

9.2.2. Options on currency forwards and options on currency futures 9.2.2.1. Options on currency forwards They are traded in the OTC market. This market is regarded as the major market for currency options. The growth of the OTC market is due to its flexibility, since many banks and financial institutions offer options with tailor-made characteristics in order to match the clients’ needs.

9.2.2.2. Options on currency futures These options have been traded since 1982. These options are standardized contracts and can be inflexible. They are priced off the underlying futures contract. When exercised, the call buyer receives a long position in one futures contract marked-to-market at the current price. In the same way, when exercised, the put holder receives a short position in one futures contract.

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These options must have the same value as European options on the spot currency since they are not exercised before the maturity date, at which the futures price is equal to the spot price. The similarity between forward and futures prices suggests the use of Black’s model for the valuation of these options. Currency futures and currency forwards are often used to hedge currency risks. Options on currency futures are applied in the currency-risk management. The basic strategies of buying and selling calls and puts, the vertical, diagonal, calendar, and volatility spreads are also applied in the currency futures options markets. They can also be used in portfolio insurance. Several other applications of currency futures and options are used to manage currency risks. Examples include basket options, average rate options, cylinder options, and many other exotic options.

9.2.3. The Black’s model and valuation of interest rate caps An interest rate cap: It is defined as an agreement or a contractual arrangement where the seller known as the grantor is obliged to pay cash to the buyer whenever the interest rate exceeds or is less than a pre-specified agreed level at some future time. When the grantor pays cash to the cap holder, this latter’s net position is equivalent to borrowing at a fixed rate at this specified level. Hence, an interest rate cap can be seen as an option where the holder pays a premium upfront. The well-known forms of interest rate cap agreements are the floor and ceiling agreements. The floor holder can establish a minimum level for his/her floating-rate deposits over a given period. If at future dates the interest rate falls below the floor rate, the seller makes good the holder’s interest income shortfall. However, if rates are higher than the floor rate, the buyer receives nothing, but has the possibility to place his/her deposit at a higher market rate. The ceiling agreement gives the right to the buyer to establish a maximum interest-rate level for borrowing over a given period. If rates turn to be higher than the ceiling rate, the buyer receives cash to exactly offset the additional interestrate charge due because of higher rates. However, if rates fall in the future, the holder can borrow at a rate lesser than the ceiling rate. Therefore, the Black’s model can be used for the pricing of the caps. The model requires five variables: time to maturity, the price of the underlying futures, the strike price or cap level, the risk-free rate for the option maturity, and the volatility.

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9.3. The Extension to Foreign Currencies: The Garman and Kohlhagen Model and its Applications Garman and Kohlhagen (1983) provided a formula for the valuation of foreign currency options. These options are traded on the foreign exchange market, which is fundamentally an inter-bank market where transactions are conducted over the telecommunications system. The foreign exchange market called also the FX market operates internationally 24 hours a day where the major participants are commercial banks around the world and treasury departments of large companies. As in other markets, the various participants search for hedging exchange risks, speculation, and the implementation of arbitrage strategies. Foreign currency options satisfy some of the needs of these participants and the important volume of transactions implies the use of an option pricing model. A simple and an interesting analytic model is provided by Garman and Kohlhagen (1983). Foreign currency options are priced along the lines of Black and Scholes (1973) and Merton (1973). Specifically, Garman and Kohlhagen (1983) and Grabbe (1983) presented models for currency options, which are based on the assumption that a risk-less hedge portfolio can be formed by investing in foreign bonds, domestic bonds, and the option.

9.3.1. The currency call formula Using the same assumptions as in the Black and Scholes (1973) model, Garman and Kohlhagen (1983) presented the following formula for a European currency call: ∗

c(S, T ) = Se−r T N (d1 ) + Ke−rT N (d2 ) S 2 √ + (r − r∗ + σ2 )T ln K √ d1 = , d2 = d1 − σ T σ T 9.3.2. The currency put formula The formula for a European currency put is: p(S, T ) = −Se−r d1 =

ln

S K

∗

T

N (−d1 ) − Ke−rT N (−d2 )

+ (r − r∗ + √ σ T

σ2 )T 2

,

√ d2 = d1 − σ T

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Note that the main difference between these formulae and those of B–S model for the pricing of equity options is that the foreign risk-free rate is used in the adjustment of the spot rate. The spot rate is adjusted by the known “dividend”, i.e., the foreign interest earnings, whereas the domestic risk-free rate enters the calculation of the present value of the strike price since the domestic currency is paid over on exercise.

Examples Assume that the US dollar/sterling spot rate is 1.8, the time to maturity is three months, the three-month dollar interest rate is 7%, and the sterling interest rate is 10%. When the volatility is 20%, the option price is 6.3817, or: C = 180e−(0.1)(0.25)N (d1 ) − 180e−(0.07)(0.25) N (d2 ) √ d1 = [ln(180/180) + (0.06 − 0.10 + 0.5(0.2)2 )0.25]/0.2 0.25 = −0.05 √ d2 = d1 − 0.2 0.25 = −0.15 N (d1 ) = 0.4801, N (d2) = 0.4404 C = 84.284 − 77.896 = 6.3817. Note that the value of N (d1 ) is discounted to the present using the foreign interest rate. This is because this rate is assumed to correspond to a continuous dividend stream on the underlying asset. In the same way, we can calculate risk parameters of other options.

9.3.3. The interest-rate theorem and the pricing of forward currency options The interest-rate parity theorem states that the forward rate is equal to the spot rate compounded by the differential between the foreign and domestic interest rates. Using continuously compounded interest rates, the forward exchange rate is: f = Se(r−r

∗

)T

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which means simply that the formula for the pricing of a European call on a spot currency can be re-written as: c(F, T ) = e−rT [f N (d1 ) − KN (d2 )] 2 f + σ2 T ln K √ √ d1 = , and d2 = d1 − σ T σ T The formula for the European currency put is: p(F, T ) = e−rT [−f N (−d1 ) + KN (−d2 )] 2 f + σ2 T ln K √ √ d1 = , and d2 = d1 − σ T σ T where all the parameters have the same meaning as before, except for the spot exchange rate S, which is replaced by the forward exchange rate f . Note that the interest-rate differential is not explicitly taken into account in the above formula. This is because all the available information about spot rates and the interest-rate differential is integrated in the forward exchange rate via the interest-rate parity theorem. The following tables provide simulation results for option prices using the Garman-Kohlhagen model (Tables 9.1–9.4). The tables also give the Greek letters. The reader can make comments about the values of the Greek letters. These tables provide the risk matrix for managing and monitoring options positions. The head of trading and the asset risk officer must continuously control these risk parameters.

Table 9.1. Simulations of Garman-Kohlhagen call prices. S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4% and σ = 20%. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

0.05384 0.05815 0.06266 0.06737 0.07227 0.07736 0.08264 0.08810 0.09375

0.43109 0.45081 0.47042 0.48984 0.50904 0.52798 0.54661 0.56492 0.58286

0.52931 0.50961 0.49003 0.47063 0.45145 0.43253 0.41391 0.39562 0.37769

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00388 0.00387

0.00008 0.00008 0.00009 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008

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Applications of Black–Scholes Type Models Table 9.2. Simulations of Garman-Kohlhagen call prices. S = 1.1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4% and σ = 20%. S

Price

Delta

Gamma

Vega

Theta

1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14

0.10315 0.10988 0.11680 0.12392 0.13123 0.13872 0.14639 0.15423 0.16224

0.61071 0.62921 0.64714 0.66449 0.68123 0.69736 0.71287 0.72774 0.74198

0.34986 0.33137 0.31345 0.29611 0.27937 0.26325 0.24775 0.23289 0.21865

0.00385 0.00382 0.00378 0.00373 0.00368 0.00362 0.00355 0.00347 0.00339

0.00008 0.00008 0.00007 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006

Table 9.3. Simulations of Garman-Kohlhagen put prices. S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4% and σ = 20%. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

0.10195 0.09666 0.09156 0.08666 0.08195 0.07744 0.07311 0.06896 0.06500

−0.52960 −0.50987 −0.49027 −0.47084 −0.45164 −0.43271 −0.41407 −0.39577 −0.37783

0.52931 0.50961 0.49003 0.47063 0.45145 0.43253 0.41391 0.39562 0.37769

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00388 0.00387

0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011

Table 9.4. Simulations of Garman-Kohlhagen put prices and the Greek letters. S = 1.1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4% and σ = 20%. S

Price

Delta

Gamma

Vega

Theta

1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14

0.05904 0.05519 0.05155 0.04810 0.04484 0.04177 0.03887 0.03615 0.03358

−0.34997 −0.33148 −0.31354 −0.29620 −0.27945 −0.26332 −0.24781 −0.23294 −0.21870

0.34986 0.33137 0.31345 0.29611 0.27937 0.26325 0.24775 0.23289 0.21865

0.00385 0.00382 0.00378 0.00373 0.00368 0.00362 0.00355 0.00347 0.00339

0.00011 0.00011 0.00011 0.00011 0.00011 0.00010 0.00010 0.00010 0.00010

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9.4. The Extension to Other Commodities: The Merton, Barone-Adesi and Whaley Model, and Its Applications The model presented in Barone-Adesi and Whaley (1987) is a direct extension of the models presented by Black and Scholes (1973), Merton (1973), and Black (1976). 9.4.1. The model The absence of risk-less arbitrage opportunities imply that the following relationship exists between the futures contract, F , and the price of its underlying spot commodity, S: F = SebT , where T is the time to expiration and b is the cost of carrying the commodity. When the underlying commodity dynamics are given by: dS = αdt + σdW S where α is the expected instantaneous relative price change of the commodity and σ is its standard deviation, then the dynamics of the futures price are given by the following differential equation: dF = (α − b)dt + σdW F Assuming that a hedged portfolio containing the option and the underlying commodity can be constructed and adjusted continuously, the partial differential equation that must be satisfied by the option price, c, is:

∂c(S, t) ∂c(S, t) 1 2 2 ∂ 2 c(S, t) − rc(S, t) + bS σ S + =0 2 ∂S 2 ∂S ∂t This equation first appeared indirectly in Merton (1973). When the cost of carry b is equal to the risk-less interest rate, this equation reduces to that of the equation in B–S (1973) model. When the cost of carry is zero, this equation reduces to that of the equation given in Black (1976). When the cost of carry is equal to the difference between the domestic and the foreign interest rate, this equation reduces to that in Garman and Kohlhagen (1983). It is convenient to note that the short-term interest rate r, and the cost of carrying the commodity, b, are assumed to be constant and proportional rates. Using the terminal boundary condition: c(S, T ) = max[0, ST − K]

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Merton (1973) showed indirectly that the European call price is: c(S, T ) = Se(b−r)T N (d1 ) − Ke−rT N (d2 ) d1 =

ln

S K

+ (b + √ σ T

σ2 )T 2

,

√ and d2 = d1 − σ T

Using the boundary condition for the put p(S, T ) = max[0, K − ST ], the European put price is given by: p(S, T ) = −Se(b−r)T N (−d1 ) + Ke−rT N (−d2 ) S 2 + b + σ2 T ln K √ √ d1 = , d2 = d1 − σ T σ T The call formula provides the composition of the asset-bond portfolio that mimics exactly the call’s payoff. A long position in a call can be replicated by buying e(b−r)T N (d1 ) units of the underlying asset and selling N (d2 ) units of risk-free bonds, each unit with strike price Ke−rT . When the asset price varies, the units invested in the underlying aset and risk-free bonds will change. Using a continuous re-balancing of the portfolio, the pay* outs will be identical to those of the call. The same strategy can be used to duplicate the put’s payoff. 9.4.2. An application to portfolio insurance Dynamic portfolio insurance strategies are based on a dynamic replication argument. The nontrading of the long-term index put options in the 1980s led stock portfolio managers to create their own insurance using a dynamic re-balancing portfolio-containing stocks and risk-free bonds. The portfolio weights in a dynamically re-balancing portfolio are determined using the put option formula: p(S, T ) = −Se(b−r)T N (−d1 ) + KN (−d2 ) d1 =

ln

S K

+ (b + √ σ T

σ2 )T 2

√ and d2 = d1 − σ T

The objective of portfolio insurance is to create an “insured” portfolio whose pay outs mimic the portfolio Se(b−r)T + p. The strategy is equivalent to: Se(b−r)T + p = Se(b−r)T − Se(b−r)T N (−d1 ) + Ke−rT N (−d2 )

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or Se(b−r)T N (d1 ) + Ke−rT N (−d2 ), in this context, a dynamically insured portfolio shows an investment of Se(b−r)T N (d1 ) units of stocks and N (−d2 ) units of risk-free bonds. When the stock price increases, funds are transferred from bonds to stocks and vice versa. 9.5. The Real World and the Black–Scholes Type Models The Black and Scholes (1973) type model is universally applied by market participants even though several other alternative models exist. The main question is why the Black–Scholes model successful and how to apply the model when the imperfections of the real world loom large. 9.5.1. Volatility The historical volatility can be used as a proxy of the future volatility of the underlying asset. For example, it is possible to consider the prices of the asset every day for the last 200 days and compute the standard deviation of log returns on this arbitrary time interval. This gives a good estimate of volatility with a standard error of around 5%. There are several other ways of predicting volatility. In any case, the assumption of a constant volatility is violated in the real world. 9.5.2. The hedging strategy In the Black–Scholes (1973) theory, the more frequently the hedge is adjusted, the more precise is the hedge. This assumes that the volume of trading will also increase without limit as the portfolio is re-balanced more frequently. The model seems to break down when frictions are introduced into the trading process. 9.5.3. The log-normal assumption This model assumes that the log returns are normally distributed. However, for several underlying assets, returns seem to be fat-tailed. Besides, this model assumes that the dynamics of the underlying asset are almost continuous, however prices tend to move discretely and jump. Despite its simplifying assumptions, the model seems to give “good” results because it represents the limiting case for option price bounds that exist in more general models.

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9.5.4. A world of finite trading Britten-Jones and Neuberger (1998) developed a model of finite trading to analyze trading costs, hedging strategies, and the effect of price jumps. Consider the pricing and the hedging of a short position in a European call option on a non-dividend paying asset. The model assumes that the trader can trade when he/she wants and not continuously as in the Black–Scholes model. Denote the net prices by P0 , P1 , . . . , PN (after taking account of market impact, spreads, commissions, etc.) used by the trader to re-balance the hedge. The price P0 corresponds to the initial price when the option is sold and PN is the price at the option maturity. The trader increases the hedge when prices increase and reduces the hedge when the prices decrease. Pi+1 is the total cost of buying (ask price plus commissions) when Pi+1 > Pi and it represents the net revenue from selling (bid price minus commissions) when Pi+1 < Pi . The model assumes that jumps in prices are less than some i with i = 1, . . . , N . amount d with: d ≥ log PPi−1 9.5.5. Total variance

2 Pi The total variance in this model is given by: ν = ΣN . This total 1 log Pi−1 variance is a function of transaction prices. 9.5.6. Black–Scholes as the limiting case Britten-Jones and Neuberger (1996) have shown that when there is an upper bound on total variance and when asset prices do not jump too much, it is possible to place an upper bound on the call price C(ν, d). This upper bound depends on the total variance and the maximum jump size. In the same way, it is possible to place a lower bound on the option price C(ν, d). If the trader has a view that the jump size will not exceed d and the total variance will not be higher than ν, then he/she can sell the option at C(ν, d) confident that at worst he/she will break even. When d tends to zero, the option price in this model tends to be the Black–Scholes price. If the trader is confident that the total variance will lie somewhere between ν and ν and the maximum jump will be less than d, then it is possible to place bounds on the option price C(ν, d) and C(ν, d). These two bounds allow the trader to buy the option for less than the lower bound and to sell it for more than the upper bound. In this limit, when ν is known and d is very small, the

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two bounds coalesce and become equal to the Black–Scholes price. If the trader could predict the total variance exactly and if prices do not jump, the options can be priced exactly.

9.5.7. Using the model to optimize hedging How often a hedge should be re-balanced for a sold call? This question can be turned into an empirical issue about forecasting the total variance of returns. Britten-Jones and Neuberger (1998) give an example to illustrate the answer to this question. Consider a trader who believes that the volatility of the asset price is 15%. Transaction costs are five basis points. If the hedge is re-balanced 10 times for a three-month maturity call, this leads to low transaction costs. Selling an option at a volatility of 15.3% can lead to profits. However, because of infrequent re-balancing, this situation can lead to losing money. The trader can be sure to make money at 99% if the option priced at three standard deviations away from the mean is sold on a volatility of 23%. If the position is re-balanced about 300 times, the investor can make money when the option is sold at a volatility of 18.3%.

Summary Stock index options and futures markets have experienced remarkable growth rates. Stock index options are either of the European or the American type and often involve cash settlement procedure upon exercise. Stock index options are traded on the major indices around the world. These options are of the European or American type. Options on the spot index are cash-settled and there is no physical delivery of the underlying index. The price of any financial asset is given by the present value of its expected cash flows. Options on index futures require upon exercise the exchange of a long position in the future contract for a call and a short position in the future contract for the put. Options on currency forwards are traded in the OTC market. This market is regarded as the major market for currency options. The growth of the OTC market is due to its flexibility, since many banks and financial institutions offer options with tailor-made characteristics in order to match the clients’ needs. Garman and Kohlhagen (1983) provided a formula for the valuation of foreign currency options. These options are traded on the foreign exchange market, which

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is fundamentally an inter-bank market where transactions are conducted over the telecommunications system. The foreign exchange market also called the FX market operates internationally 24 hours a day where the major participants are commercial banks around the world and treasury departments of large companies. Currency options were traded on the spot currency for the first time in 1982 at the Philadelphia Stock Exchange. Since this date, currency options are traded on many other financial places. However, the trading on the OTC market seems to be more important. The strategies discussed for stock options also apply to currency options and currency futures options. This chapter presented in detail the basic concepts and techniques underlying rational derivative asset pricing in the context of the analytical European models along the lines of Black–Scholes and Merton. First, some applications of the Black–Scholes (1973) model are provided. Second, applications of the Black (1976) model are presented. Third, applications of the Garman and Kohlhagen model are presented for the valuation of currency options. Fourth, the Merton (1973) and BaroneAdesi and Whaley (1987) model is proposed for the valuation of European commodity contracts, commodity options, and commodity futures options. Some applications of the model are given. Note that this model reduces to the models in Black–Scholes, Black and Garman and Kohlhagen for some values of its parameters. Since all these models (except Arone-Adesi and Whaley (1987)) are interested in the valuation of European style options in a continuous time framework, (without discrete distributions to the underlying asset), a natural extension of these models must introduce the possibility of an early exercise and discrete distributions. However, before making some extensions of the basic analytical models, it is useful to study in this simple context, the option price sensitivities and the use of these Greek-risk measures in the monitoring and the management of an option position. The question of managing an option position is as important as some issues regarding the option pricing. The Black–Scholes hedge works in the real, discrete, and frictionful world when the hedger uses the correct volatility of the prices at which he/she actually trades and when the asset prices do not jump too much. The assumptions of the Britten-Jones and Neuberger (1998) model provide a framework in which a trader can avoid jumps and in which total variance can be estimated perfectly. The model transforms the question of pricing and hedging options into how well investors can predict the total variance of returns of the associated hedging strategy.

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Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19.

What are the main applications of the Black and Scholes’ model? What are the main applications of the Black’s model? What is meant by the interest-rate parity theorem? What are the main characteristics of currency options and their markets? What is the main difference between Black and Scholes’ model and Black’s model? What is the main difference between Black and Scholes’ model and Garman and Kohlhagen’s model? What is inappropriate in the derivation of Garman and Kohlhagen’s model? What do you think of the assumptions underlying Garman and Kohlhagen’s model? What are the main differences between futures and forward contracts? How can we obtain the formulas in Black and Scholes’ model, Black’s model, and Garman and Kohlhagen’s model using the formula in Merton and BAW? What are the main specificities of index options and their markets? How the Black and Scholes model is adjusted for index options? What are the implications of arbitrage for index option markets and their assets? How indexes are constructed? What is the main difference between zero-coupon bonds and couponpaying bonds? What are the different types of bonds ? What are the main specificities of short-term options on long-term bonds? What are the main specificities of bond options? How can we obtain the put-call parity relationship for futures options from that of European spot options?

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Appendix

Black–Scholes Valuation: Bloomberg

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References Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June), 301–320. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Brennan, MJ and ES Schwartz (1990). Arbitrage in stock index futures. Journal of Business, 63, 7–31. Britten-Jones, M and AF Neuberger (1996). Arbitrage pricing with incomplete markets. Applied Mathematical Finance, 3(4), 347–363 and 11–13. Britten-Jones, M and AF Neuberger (1998). Welcome to the real world. Risk, September 11–13. Evnine, J and A Rudd (1985). Index options: the early evidence. Journal of Finance, 40(3), 743–756. Garman, M. and S Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance, 2, 231–237. Grabe, JO (1983). The pricing of call and put options on foreign exchange. Journal of International Money and Finance, 2, 239–253. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.

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Chapter 10 APPLICATIONS OF OPTION PRICING MODELS TO THE MONITORING AND THE MANAGEMENT OF PORTFOLIOS OF DERIVATIVES IN THE REAL WORLD

Chapter Outline This chapter is organized as follows: 1. In Section 10.1, option price sensitivities are presented and the formulas are applied. 2. In Section 10.2, the Greek-letter risk measures are simulated for different parameters. The question of monitoring and managing an option position in real time is studied for the different risk measures with respect to an option pricing model. 3. In Section 10.3, some of the characteristics of volatility spreads are presented. 4. In Appendix A, we give the Greek-letter risk measures with respect to the analytical models presented in Chapter 9. 5. In Appendix B, we show the relationship between some of these Greekletter risk measures. 6. In Appendix C, we provide a detailed derivation and demonstration of the hedging parameters. 7. In Appendix D, we provide a detailed derivation of the Greek-letter risk measures.

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Introduction As seen during the financial crisis 2008–2009, risk appears as a major concern for the financial system and defies all that has been written by academics in this area. The sensitivity parameters or Greek-letter risk measures are important in managing an option position. The delta measures the absolute change in the option price with respect to a small change in the price of the underlying asset. It is given by the option’s partial derivative with respect to the underlying asset price. It represents the hedge ratio, or the number of options to write in order to create a risk-free portfolio. Call buying involves the sale of a quantity delta of the underlying asset in order to form the hedging portfolio. Call selling involves the purchase of a quantity delta of the asset to create the hedging portfolio. Put buying requires the purchase of a quantity delta of the underlying asset to hedge a portfolio. Put selling involves the sale of delta stocks to create a hedged portfolio. The delta varies from zero for deep out-of-the-money (OTM) options to one for deep in-the-money (ITM) calls. This is not surprising, since by definition, the delta is given by the first partial derivative of the option price with respect to the underlying asset. For example, the value of a deep ITM call is nearly equal to the intrinsic value for which the first partial derivative with respect to the underlying asset is one. Charm is a risk measure that clarifies the concept of carry in financial instruments. The concept of carry refers to the expenses due to the financing of a deferred delivery of commodities, currencies, or other assets in financial contracts. Even though charm is used by market participants as an ad hoc measure of how delta may change overnight, it is an important measure of risk since it divides the theta into its asset-based constituents. The gamma measures the change in delta, or in the hedge ratio, as the underlying asset price changes. The gamma is the greatest for at-the-money (ATM) options. It is nearly zero for deep ITM and deep OTM options. Approximately, the gamma is to the delta what convexity is to duration. The gamma is given by the derivative of the hedge ratio with respect to the underlying asset price. As such, it is an indication of the vulnerability of the hedge ratio. The gamma is very important in the management and the monitoring of an option position. It gives rise to two other measures of risk: speed and color.

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Speed is given by the gamma’s derivative with respect to the underlying asset price. Color is given by the gamma’s derivative with respect to the time remainning to maturity. The theta measures the change in the option price as time elapses since time decays presents a negative impact on option values. Theta is given by the first partial derivative of the option premium with respect to time. The vega or lambda measures the change in the option price for a change in the underlying asset’s volatility. It is given by the first derivative of the option premium with respect to the volatility parameter. The knowledge of the “true” option price is not sufficient for the monitoring and the management of an option position. Therefore, it is important to know the option-price sensitivities with respect to the parameters entering the option formula. We begin our discussion with the delta. In this chapter, we show how to calculate some of these parameters within the context of each analytical model presented in Chapter 9. Also, we develop some examples to show how to use the Greek-letter risk measures in the monitoring and the management of an option position in response to the changing market conditions. 10.1. Option-Price Sensitivities: Some Specific Examples 10.1.1. Delta The delta is given by the option’s first partial derivative with respect to the underlying asset price. It represents the hedge ratio in the context of the Black–Scholes model (B–S model). The call’s delta The call’s delta is given by ∆c = N (d1 ). The use of this formula requires the computation of d1 given by: S + r + 12 σ 2 T ln K √ d1 = σ T Example. Let the underlying asset price S = 18, the strike price K = 15, the short-term interest rate r = 10%, the maturity date T = 0.25, and the volatility σ = 15%, the option’s delta is given by ∆c = N (d1 ).

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To apply this formula, the calculation of d1 is: 1 2 18 1 √ + 0.1 + 0.5 0.25 = 2.8017 ln d1 = 15 2 0.15 0.25 Hence, the delta is ∆c = N (2.8017) = 0.997. This delta value means that the hedge of the purchase of a call needs the sale of 0.997 units of the underlying asset. When the underlying asset price rises by 1 unit, from 18 to 19, the option price rises from 3.3659 to approximately (3.3659 + 0.997), or 4.3629. When the asset price falls by 1 unit, the option price changes from 3.3659 to approximately (3.3659 − 0.997), or 2.3689. The put’s delta The put’s delta has the same meaning as the call’s delta. It is also given by the option’s first derivative with respect to the underlying asset price. When selling (buying) a put option, the hedge needs selling (buying) delta units of the underlying asset. The put’s delta is given by: ∆p = ∆c − 1 = 0.0997 − 1 = −0.003. The hedge ratio is −0.003. When the underlying asset price rises from 18 to 19, the put price changes from 0.0045 to approximately (0.0045 − 0.003), or 0.0015. When it falls from 18 to 17, the put price rises from 0.0045 to approximately (0.0045 + 0.003), or 0.0075. Appendix A provides the derivation of the Greek letters in the context of analytical models. Appendix D provides a detailed derivation of these parameters.

10.1.2. Gamma The option’s gamma corresponds to the option’s second partial derivative with respect to the underlying asset or to the delta partial derivative with respect to the asset price.

The call’s gamma In the B–S model, the call’s gamma is given by Γc = 1 2 with n(d1 ) = √12π e− 2 d1 . Using 2 1 √ 1 e− 2 (2.8017) = 0.09826 6.2831

∂∆c ∂S

=

1√ Sσ T

n(d1 )

the same data as in the example, n(d1 ) =

1√ and Γc = 18(0.15) 0.09826 = 0.0727. 0.25 When the underlying asset price is 18 and its delta is 0.997, a fall in the asset price by 1 unit yields a change in the delta from 0.997 to approximately

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(0.997 − 0.0727), or 0.9243. Also, a rise in the asset price from 18 to 19, yields a change in the delta from 0.997 to (0.997 + 0.0727), or 1. This means that the option is deeply ITM, and its value is given by its intrinsic value (S − K). The same arguments apply to put options. The call and the put have the same gamma. The put’s gamma The put’s gamma is given by Γp = − 12 d21

√1 e 2π

1√ 0.09826 18(0.15) 0.25

∂∆p ∂S

=

1√ Sσ T

n(d1 ) with n(d1 ) =

or Γp = = 0.0727. When the asset price changes by one unit, the put price changes by the delta amount and the delta changes by an amount equal to the gamma. 10.1.3. Theta The option’s theta is given by the option’s first partial derivative with respect to the time remaining to maturity. The call’s theta In the B–S model, the theta is given by: Θc =

−Sσn(d1 ) ∂c √ = − rKe−rT N (d2 ) ∂T 2 T

Using the same data as in the example above, we obtain: Θc = −0.2653 − 1.4571 = −1.1918 When the time to maturity is shortened by 1% per year, the call’s price decreases by 0.01 (1.1918), or 0.011918 and its price changes from 3.3659 to approximately (3.3659 − 0.01918), or 3.3467. The put’s theta In the B–S model, the put’s theta is given by: Θp =

Sσn(d1 ) ∂p =− √ + rKe−rT N (d2 ) ∂T 2 T

or Θp = −0.2653 + 0.0058 = −0.2594

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Using the same reasoning, the put price changes from 0.0045 to approximately (0.0045 − 0.0025), or 0.002. 10.1.4. Vega The option’s vega is given by the option’s price derivative with respect to the volatility parameter. The call’s vega

√ ∂c = S T n(d1 ) In the B–S model, the call’s vega is√ given by vc = ∂σ or using the above data vc = 18 0.25(0.09826) = 0.88434. Hence, when the volatility rises by 1 point, the call price increases by 0.88434. The increase in volatility by 1% changes the option price from 3.3659 to (3.3659 + 1%(0.88434)), or 3.37474. In the same context, the put’s vega is equal to the call’s vega. The put price changes from 0.0045 to (0.0045 + 1%(0.88434)), or 0.0133434. When the volatility falls by 1%, the call’s price changes from 3.3659 to (3.3659 − 1%(0.88434)), or 3.36156 . In the same way, the put price is modified from 0.0045 to approximately (0.0045 − 1%(0.88434)), or zero since option prices cannot be negative. The put’s vega In the B–S model, the put’s vega is given by: vp =

√ ∂p = S T n(d1 ) ∂σ

√ or vp = 18 0.25(0.09826) = 0.88434 and it has the same meaning as the call’s vega. 10.1.5. Rho The call’s rho The option’s rho is given by the option’s first partial derivative with respect ∂c to interest rates. In the B–S model, the call’s rho is given by Rhoc = ∂r = −rT KT e N (d2 ). Using the above data Rhoc = e−0.1(0.25) (0.996)(0.25) = 3.64. The Rho does not affect the call and put prices in the same way. In fact, a rise in the interest rate yields higher call price (positive Rho) and reduces the put price (negative Rho).

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The put’s rho In the B–S model, the put’s rho is given by: Rhop =

∂p = −KT e−rT N (−d2 ) ∂r

or Rhop = −15e−0.1(0.25)(0.996)(0.25) = −3.64. 10.1.6. Elasticity The call’s elasticity For a call option, this measure is given by Elasticity = S ∆c c = using the above data: Elasticity =

S c N (d1 )

or

18 0.997 = 5.3317 3.3659

The elasticity shows the change in the option price when the underlying asset price varies by 1%. Hence, a rise in the asset price by 1%, i.e., 0.18, induces an increase in the call price by 5.33%. The put price decreases by 12%. Hence, when the asset price changes from 18 to 18.18, the call’s price is modified from 3.3659 to approximately (3.3659 (1 + 5.33%)), or 3.545. In the same way, the put price changes from 0.0045 to (0.0045(1 − 12%)), or 0.00396. The put’s elasticity The put’s elasticity is given by: Elasticity = S

S ∆p = [N (d1 ) − 1] p p

18 or Elasticity = 0.0045 [0.997 − 1] = −12. The knowledge of the variations in these parameters is fundamental for the monitoring and the management of an option position. Appendix B provides the relationship between these hedging parameters.

10.2. Monitoring and Managing an Option Position in Real Time Since option prices change in an unpredictible way in response to the changes in market conditions, traders, market makers, and all option users must rely upon some model to monitor the evolution of their profit and loss

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accounts. Such a model allows them to know the variations in option price sensitivities and their risk exposure. With such quantities, the monitoring and the management of option positions are more easily achieved. We illustrate the management of an option position in real time using the model proposed indirectly in Merton (1973) and derived afterwards in Black (1975) and Barone-Adesi and Whaley (1987) for the valuation of European futures options. First, option prices are simulated and the sensitivity parameters are calculated. Second, we study the risk-management problem in real time with respect to option price sensitivities. 10.2.1. Simulations and analysis of option price sensitivities using Barone-Adesi and Whaley model Re-call that the commodity call price and the commodity futures call price in the context of Merton’s (1973) and Barone-Adesi and Whaley’s (1987) model is given by: c(S, T ) = Se(b−r)T N (d1 ) − Ke−rT N (d2 ) S √ ln K + b + 12 σ 2 T √ , d2 = d1 − σ T d1 = σ T where b stands for the cost of carrying the underlying commodity. By the put-call parity relationship or by a direct derivation, the put’s value is given in the same context by: p(S, T ) = −Se(b−r)T N (−d1 ) + Ke−rT N (−d2 ) S √ ln K + b + 12 σ 2 T √ , d2 = d1 − σ T d1 = σ T where all the parameters have the same meaning as before. For a non-dividend paying asset, b = r. For a dividend-paying asset, b = r − d where d stands for the dividend yield. For a currency option, b = r − r∗ , where r∗ stands for the foreign risk-less rate. Tables 10.1 to 10.12 simulate option values and sensitivity parameters for calls and puts in the context of the above model. Sensitivity parameters for call options Table 10.1 gives call prices, delta, gamma, theta, and vega when the underlying commodity price varies from 75 to 110 in steps of 5. For example,

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Table 10.1. Changing the underlying asset prices for European calls. σ = 0.2, r = 0.08, b = 0.10, T = 0.25, and K = 90. Call

Asset

Delta

Gamma

Theta

Vega

0.23 0.88 2.42 5.07 8.76 13.17 17.44 22.37

75 80 85 90 95 100 105 110

0.07 0.21 0.42 0.64 0.82 0.93 0.97 0.99

0.02 0.03 0.04 0.04 0.03 0.02 0.01 0.01

0.05 0.11 0.17 0.17 0.13 0.08 0.04 0.02

0.47 1.10 1.71 1.94 1.78 1.46 1.05 0.89

Table 10.2. Changing the underlying asset prices for European calls. σ = 0.2, r = 0.08, b = 0.10, T = 0.25, and K = 100. Call

Asset

Delta

Gamma

Theta

Vega

0.01 0.09 0.40 1.23 2.92 5.64 8.87 13.11

75 80 85 90 95 100 105 110

0.01 0.03 0.10 0.24 0.44 0.64 0.79 0.90

0 0.01 0.02 0.03 0.04 0.04 0.03 0.02

0.01 0.03 0.08 0.14 0.19 0.19 0.15 0.10

0.06 0.25 0.71 1.37 1.94 2.16 1.89 1.56

when the volatility σ = 20%, r = 8%, b = 10%, and T = 3 months (0.25 year), the price of an ATM call for K = 90 is 5.07. The call has a delta of 0.64, a gamma of 0.04, a theta of 0.17, and a vega of 1.94. Note that an ATM call has more theta and vega than an ITM and an OTM call. Using the same data except for the strike price, which is modified from 90 to 100, Table 10.2 shows that an ATM call (K = 100, S = 100) has more gamma, theta, and vega than ITM, and OTM calls. Table 10.3 shows call prices and sensitivity parameters for European calls when the time to maturity varies from 0.05 to 0.75 year. Note that the call price, the vega, and the theta increase with the time to maturity. However, the delta falls when the time to maturity is longer. Table 10.4 gives call prices and sensitivity parameters for ATM calls when (S = K = 100) and the time to maturity varies from 0.05 to a year. Note that the call price, its delta, and vega are increasing functions of the

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Table 10.3. Changing time to maturity for European calls. σ = 0.2, r = 0.08, b = 0.10, S = 100, and K = 90. Call

Maturity

Delta

Gamma

Vega

Theta

10.47 11.00 12.71 15.44 17.98

0.05 0.10 0.25 0.50 0.75

0.99 0.97 0.91 0.88 0.88

0 0.01 0.02 0.01 0.01

0 0.02 0.08 0.14 0.19

0.83 1.03 1.34 1.45 1.46

Table 10.4. Changing time to maturity for European calls. σ = 0.2, r = 0.08, b = 0.10, S = 100, and K = 100. Call

Maturity

Delta

Gamma

Vega

Theta

2.04 3.05 5.32 8.36 11.04 13.53

0.05 0.10 0.25 0.50 0.75 1.00

0.55 0.58 0.62 0.67 0.71 0.74

0.08 0.06 0.04 0.03 0.02 0.02

0.09 0.12 0.20 0.26 0.31 0.34

2.09 2.09 2.05 1.98 1.91 1.84

Table 10.5. Changing the volatility for European calls. T = 0.25, r = 0.08, b = 0.10, S = 100, and K = 90. Call

Volatility

Delta

Gamma

Vega

Theta

12.29 12.29 12.71 13.78 15.19

0.05 0.10 0.20 0.30 0.40

1 1 0.92 0.85 0.78

0 0.025 0.01 0.01 0.01

0 0.01 0.08 0.13 0.15

0.76 0.78 1.34 2.18 3.02

time to maturity. However, the gamma and the theta are more important on near maturities. Table 10.5 gives ITM call prices (S = 100, K = 90) for different levels of the volatility parameter. When the delta is equal to 1, the gamma is nearly equal to zero. Also, the vega is nearly nil and the theta is weak. Table 10.6 gives the same information as Table 10.5, except that calculations are done for ATM options (S = K = 100).

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Applications of Option Pricing Models Table 10.6. Changing the volatility for European T = 0.25, r = 0.08, b = 0.10, S = 100, and K = 100.

449 calls,

Call

Volatility

Delta

Gamma

Vega

Theta

2.69 3.46 5.32 7.26 9.21

0.05 0.10 0.20 0.30 0.40

0.85 0.70 0.62 0.60 0.59

0.09 0.06 0.03 0.02 0.01

0.12 0.17 0.19 0.19 0.20

0.94 1.28 2.05 2.84 3.64

Table 10.7. Changing the underlying asset prices for European puts. σ = 0.2, r = 0.08, b = 0.10, T = 0.25, and K = 90. Put 13.04 8.61 5.04 2.55 1.12 0.43

Asset

Delta

Gamma

Vega

Theta

75 80 85 90 95 100

−0.89 −0.81 −0.61 −0.38 −0.26 −0.08

0.02 0.03 0.04 0.04 0.03 0.02

0.05 0.11 0.16 0.17 0.13 0.08

0.83 0.96 1.01 0.88 0.61 0.34

Sensitivity parameters for put options Table 10.7 gives put prices and the sensitivity parameters when: σ = 0.2, r = 0.08, b = 0.1, T = 0.25, and K = 90. For an ATM put, (K = 90, S = 90), the gamma and the vega are important. The put’s theta increases when the option tends to parity and decreases afterwards. The same behavior applies for the put’s gamma and vega. Table 10.8 gives the same information as Table 10.7, except for the strike price which is changed from 90 to 100. For ITM puts, the delta approaches −1, the gamma and vega are not important, and the theta is very weak. Table 10.9 gives OTM put prices (S = 100, K = 90) and the put price sensitivities when the time to maturity varies from 0.05 to a year. The values of the delta and gamma are weak. However, there is more vega and theta for longer maturities. Table 10.10 shows the same information for ATM put prices (S = K = 100). Note that the deltas and gammas are decreasing functions of the time

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Table 10.8. Changing the underlying asset prices for European puts. σ = 0.2, r = 0.08, b = 0.10, T = 0.25, and K = 100. Put

Asset

Delta

Gamma

Vega

Theta

22.65 17.69 12.94 8.69 5.26 2.84

75 80 85 90 95 100

−1.00 −0.97 −0.91 −0.73 −0.48 −0.38

0 0.01 0.02 0.03 0.04 0.04

0 0.02 0.07 0.14 0.19 0.19

0.79 0.85 0.97 1.09 1.12 0.98

Table 10.9. Changing time to maturity for European puts. σ = 0.2, r = 0.08, b = 0.10, S = 100, and K = 90. Put

Asset

Delta

Gamma

Vega

Theta

0.01 0.08 0.43 0.91 1.23 1.45

0.05 0.10 0.25 0.5 0.75 1.00

−0.01 −0.04 −0.08 −0.12 −0.13 −0.13

0 0.01 0.02 0.01 0.01 0.01

0 0.02 0.08 0.14 0.19 0.22

0.07 0.22 0.34 0.35 0.32 0.30

Table 10.10. Changing time to maturity for European puts. σ = 0.2, r = 0.08, b = 0.10, S = 100, and K = 100. Put

Asset

Delta

Gamma

Vega

Theta

1.54 2.04 2.84 3.43 3.71 3.83

0.05 0.10 0.25 0.50 0.75 1.00

−0.45 −0.42 −0.38 −0.34 −0.31 −0.28

0.09 0.06 0.04 0.03 0.02 0.02

0.09 0.12 0.14 0.27 0.31 0.34

1.85 1.40 0.98 0.74 0.62 0.54

to maturity. For an ATM put, there is more theta on short maturities and more vega on longer maturities. Table 10.11 gives OTM put prices when the volatility varies from 0.05 to 0.4. When the volatility is respectively equal to 0.05 and 0.1, the put price is nil and so are the sensitivity parameters as well.

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Table 10.11. Changing time to maturity for European puts. T = 0.25, r = 0.08, b = 0.10, S = 100, and K = 90. Put

Volatility

Delta

Gamma

Vega

Theta

0 0.01 0.43 1.50 2.91

0.05 0.10 0.20 0.30 0.40

−0.00 −0.00 −0.09 −0.17 −0.23

0 0 0.02 0.02 0.02

0 0.01 0.08 0.13 0.15

0 0.01 0.34 0.80 1.22

Table 10.12. Changes in volatility, TM puts. T = 0.25, r = 0.08, b = 0.10, S = 100, and K = 100. Put

Volatility

Delta

Gamma

Vega

Theta

0.21 0.98 2.84 4.78 6.73

0.05 0.10 0.20 0.30 0.40

−0.16 −0.30 −0.38 −0.41 −0.41

0.10 0.07 0.04 0.03 0.02

0.12 0.17 0.19 0.19 0.20

0.23 0.55 0.97 1.34 1.69

Table 10.12 shows ATM put prices (S = K = 100) for different levels of the volatility, parameter. Note that the put price, the delta (in absolute value), the vega, and the theta are increasing functions of the volatility parameter. Note that the negative sign for the delta concerns only the put option. 10.2.2. Monitoring and adjusting the option position in real time 10.2.2.1. Monitoring and managing the delta The call’s delta is between zero and one and the put’s delta is between 0 and −1. When the delta is 0.5, the call price rises (falls) by 0.5 point for each increase (decrease) in the asset price by 1 point. A delta of 0.5 corresponds to an ATM call. For a deep ITM call, the variation in the asset price by one unit implies an equivalent variation in the option price. The delta of an OTM call is almost zero and of a deep ITM call is almost 1. Since the put’s delta lies in the interval −1 and 0, a rise in the underlying asset price implies a fall in the put price and vice versa.

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The delta is −0.5 for an ATM put, −1 for a deep ITM put and 0 for a deep OTM put. The call delta is often assimilated to the hedge ratio. Since the underlying asset delta is 1 and that of an ATM call is 0.5, the hedge ratio is (1/0.5) or 2/1. Hence, we need two ATM calls to hedge the sale of the underlying asset. Note that the delta is calculated in practice using the observed volatility or the implicit volatility. Delta-neutral hedging requires the adjustment of the option position according to the variations in the delta. When buying or selling a call (put), the investor must sell or buy (buy or sell) delta units of the underlying asset to represent a hedged portfolio. In practice, the hedged portfolio is adjusted nearly continuously to account for the variations in the delta’s value. An initially hedged position must be re-balanced by buying and selling the underlying asset as a function of the variations in the delta through time. The delta changes as the value of the underlying asset, the volatility, the interest rate, and the time to maturity are modified. Table 10.13 shows how to adjust a hedged portfolio in order to preserve main characteristics of delta-neutral strategies. It is important to note that delta-neutral hedging strategies do not protect completely the option position against the variations in the volatility parameter (Table 10.14). It is also important to note that delta-neutral hedging does not protect the option position against the variations in the time remaining to maturity. The adjustment of the position when the time to maturity changes can be done as explained in Table 10.15. Note that the deltas are additive. For example, when buying two calls having respectively, a delta of 0.2 and 0.7, the investor must sell 0.9 units of the underlying asset in a delta-neutral strategy.

Table 10.13. Adjustment of the hedged portfolio as a function of the underlying asset price. Options

Delta hedging when S rises

Long a call Short a call Long a put Short a put

Delta Delta Delta Delta

increases: short more S increases: buy more S decreases: sell more S decreases: buy more S

Delta hedging when S falls Delta Delta Delta Delta

decreases: buy more S decreases: sell more S increases: buy more S increases: sell more S

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Applications of Option Pricing Models Table 10.14.

Adjustment of a hedged position when the volatility changes.

Options Long a call ITM ATM OTM Short a call ITM ATM OTM Long a put ITM ATM OTM Short a put ITM ATM OTM

Volatility increases

Long a call ITM ATM OTM Short a call ITM ATM OTM Long a put ITM ATM OTM Short a put ITM ATM OTM

Volatility decreases

Delta decreases: buy more S Delta nonadjusted Delta decreases: sell more S

Delta increases: sell more S Delta nonadjusted Delta decreases: buy more S

Delta decreases: re-sell of S Delta nonadjusted Delta increases: buy more S

Delta increases: buy more S Delta nonadjusted Delta decreases: sell more S

Delta decreases: re-sell of S Delta nonadjusted Delta increases: buy more S



Delta increases: sell more S Delta nonadjusted Delta decreases: buy more S

Table 10.15. Options

453

Adjustment of a hedged position as a function of time. Adjustment of the hedged position as a function of time Delta rises: sell more S Delta nonmodified Delta decreases: buy more S Delta rises: buy more S Delta nonmodified Delta decreases: sell more S Delta rises: buy more S Delta nonmodified Delta decreases: sell more S Delta rises: sell more S Delta nonmodified Delta decreases: buy more S

Option market-makers implement often delta-neutral hedging strategies in order to maintain a nil delta (in monetary unit). When the delta of an option position is positive, this means that the market-maker is long the underlying asset. If the asset price rises, he/she makes a profit since he/she will be able to sell it at a higher price. However, if the asset price decreases,

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he/she will lose money since he/she will sell the underlying asset at a lower price. When the delta is positive, the investor is over-hedged with respect to delta-neutral strategies. When the delta (in monetary unit) is negative, the investor is shorting the underlying asset. If the underlying asset price rises, the investor loses money since he/she adjusts his/her position by buying more units of the underlying asset. However, when the asset price falls, he/she makes a profit since he/she pays less for the underlying asset. When a portfolio is constructed by buying (selling) the securities and derivative assets vj, the portfolio value is given by: P = n1 v1 + n2 v2 + n3 v3 + · · · + nj vj where nj stands for the numbers of units of the assets bought or sold. The delta’s position, or its partial derivative with respect to the securities and derivative assets is: ∂v2 ∂v3 ∂vj ∂v1 ∆position = + n2 + n3 + · · · + nj ∂S ∂S ∂S ∂S = n1 ∆1 + n2 ∆2 + n3 ∆3 + · · · + nj ∆j Delta-neutral hedging is convenient for an investor who does not have prior expectations about the market direction. However, if the investor expects a rising market, he/she can have a positive delta, i.e., long the underlying asset, so he/she can sell at a higher price when the market effectively rises. If the investor expects a down market, he/she can have a negative delta, i.e., short the underlying asset, so he/she can buy it at a lower price when the market effectively goes down. 10.2.2.2. Monitoring and managing the gamma The gamma is given by the second derivative of the option price with respect to the underlying asset price. A high value of gamma (either positive or negative) shows a higher risk for an option position. The gamma shows what the option gains (loses) in delta when the underlying asset price rises (falls). For example, when the option’s gamma is 4 and the option’s delta is zero, an increase by 1 point in the underlying asset price allows the option to gain 4 points in delta, i.e., the delta is equal to 4.

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When the delta is constant, the gamma is zero. The gamma varies when the market conditions change. The gamma is the highest for an ATM option and decreases either side when the option gets ITM or OTM. The gamma of an ATM option rises significantly when the volatility decreases and the option approaches its maturity date. When the gamma is positive, an increase in the underlying asset price yields a higher delta. The adjustment of the position entails the sale of more units of the underlying asset. When the asset price falls, the delta decreases and the adjustment of the option position requires the purchase of more units of the underlying asset. Since the adjustment is done in the same direction as the changes in the market direction, this monitoring of an option position with a positive gamma is easily done. When the gamma is negative, an increase in the underlying asset price reduces the delta. The adjustment of a delta-neutral position needs the purchase of more units of the underlying asset. When the asset price decreases, the delta rises. The adjustment of the option position requires the sale of more units of the underlying asset. Hence, the adjustment of the position implies a re-balancing against the market direction which produces some losses (Table 10.16). In general, the option’s gamma is a decreasing function of the time to maturity. The longer is the time to maturity, the weaker is the gamma and vice versa. When the option approaches its maturity date, the gamma varies significantly (Table 10.17). The variations in the gamma for ITM, ATM and OTM options is explained below in Table 10.18. The management of an option position with a positive gamma is simple. When the market rises, the investor becomes long and must sell some quantity of the underlying asset to re-establish his/her delta-neutral position. This produces a gain.

Table 10.16.

The adjustment of a hedged postion and the Gamma.

Position

Gamma

Adjustment of the position

Long options Long options Short options Short options

Positive Positive Negative Negative

Market Market Market Market

up: sell more S down: buy more S up: buy more S down: sell more S

Effect on the position Easy adjustment, yields profits Easy adjustment, yields profits Difficult adjustment, yields losses Easy adjustment, yields losses

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456 Table 10.17.

The variations in gamma and the time to maturity. Longer T

Gamma Gamma Effect on position Effect on position

Low Low Low Low

Table 10.18.

Gamma Gamma Effect

Short T High High Easy adjustment Easy adjustment

Near T High for ATM Very low for OTM Delta is very sensitive to S Gamma is used with care

Effect of the gamma on an option position.

OTM

ATM

Near zero Near zero Weak

High for a shorter T Stable for a longer T Gamma is fundamental

ITM Near zero for a near T Near zero for a near T Weak

When the underlying asset decreases, the investor becomes short and must buy more units of the underlying asset to re-establish his/her deltaneutral position. This yields a profit. The management of an option position with a negative gamma is more difficult when the underlying asset’s volatility is high. When the market rises, the investor becomes short and must buy some quantity of the underlying asset to re-establish his/her delta-neutral position. This produces a loss. When the underlying asset decreases, the investor becomes long and must sell more units of the underlying asset to re-establish his/her delta-neutral position. This yields a loss. In general, one should be careful when adopting a positive gamma since the option position loses from its theta when the market is not volatile. In this context, it is better to have a negative gamma. However, when the market is volatile, a position with a positive gamma allows profits, since the adjustment requires buying the underlying asset when the market falls and selling it when the market rises. The gamma of an option position with several assets is given by: ∂∆position ∂S ∂∆1 ∂∆2 ∂∆3 ∂∆3 = n1 + n2 + n3 + · · · + nj ∂S ∂S ∂S ∂S

Γposition =

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For delta-neutral strategies, a positive gamma allows profits when the market conditions change rapidly and a negative gamma produces losses in the same context. 10.2.2.3. Monitoring and managing the theta The theta is given by the option’s partial derivative with respect to the remaining time to maturity. As maturity date approaches, the option loses value. The theta is often expressed as a function of the number of points lost each day. A theta of 0.4, means that the option loses $0.4 in value when the maturity date is reduced by one day. In general, the gamma and the theta are of opposite signs. A high positive gamma is associated with a high negative theta and vice versa. By analogy with the gamma, as a high gamma is an indicator of a high risk associated with the underlying asset price, a high theta is an indicator of a high exposure to the passage of time. An ATM option with a short maturity loses value much more than a corresponding option on a longer term. The theta of an ATM option is often higher than that of an equivalent ITM or an OTM option having the same maturity date. The option buyer loses the theta value and the option writer “gains” the theta value (Table 10.19). Examples A theta of $1000 means that the option buyer pays $1000 each day for the holding of an option position. This amount profits to the option writer. The theta remains until the last day of trading. When a position shows a positive gamma, its theta is negative. In general, a high gamma induces a high theta and vice versa. For example, when Γ = 1500, theta may be $10,000, i.e., a loss of $10,000 each day for the option position. This loss is compensated by the profits on the positive gamma since the adjustments of the position imply selling (buying) more units of the underlying asset when the market rises (falls). Table 10.19.

Loss in time value Effect on position Effect on position

The option value and the theta for an ATM option.

Longer maturity

Shorter maturity

Near maturity

Low Needs passive monitoring Needs passive monitoring

High Profit for seller Loss for buyer

Very high profit for seller loss for buyer

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When the Γ = −1500 for an option position, theta may be 10,000 i.e., a gain of $10,000 each day. However, the position implies a loss on the underlying asset since the adjustments are done against the market direction when the gamma is negative. The theta of an option position is: ∂v1 ∂v2 ∂v3 ∂vj + n2 − + n3 − + · · · + n1 − Θposition = n1 − ∂t ∂t ∂t ∂t 10.2.2.4. Monitoring and managing the vega The vega is given by the option’s derivative with respect to the volatility parameter. It shows the induced variation in the option price when the volatility varies by 1%. The vega is always positive for call and put options since the option price is an increasing function of the volatility parameter. A vega of 0.6 means that an increase in the volatility by 1% increases the option price by 0.6. For a fixed time to maturity, the vega of an ATM option is higher than that of an ITM or an OTM option. Since all the option pricing parameters are observable, except the volatility, buying (selling) options is equivalent to buying (selling) the volatility. When monitoring an option position, a trade off must be realized between the gamma and the vega. Buying options and hence having a positive gamma is easy to manage. However, when the implicit volatility falls, the investor must adopt one of the two following strategies. He/she can either preserve a positive gamma, if he/she thinks that the loss due to a decrease in volatility will be compensated by adjusting the gamma in the market direction. He/she can sell the volatility (options) and re-establish a position with a negative gamma. In this case, the losses due to the adjustments of the delta must be sufficient to compensate for the decrease in volatility (Tables 10.20 and 10.21). Table 10.20.

Effect of the volatility on a portfolio of options.

Options

Volatility

Long Short

Long Short

Effect Profit (loss) when the volatility rises (falls) Loss (profit) when the volatility rises (falls)

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Applications of Option Pricing Models Table 10.21.

Effect of the vega with respect to time to maturity.

Longer maturity

Shorter maturity

Vega

High

Low

Effect

Very sensitive

Little sensitivity

Table 10.22.

Vega Option position

459

Near maturity Implicit volatility depends on other factors Implicit volatility depends on other factors

Effect of the vega on the option price.

OTM

ATM

ITM

depends on T Low

For a given T , vega is higher High

Depends on T Low

The impact of the vega on ATM option is the highest and is summarized in Table 10.22. When the vega is $50,000, this means that a rise in the volatility by 1% produces a profit of $500. However, when the volatility decreases by 1%, this implies a loss of $500. When the vega is $50,000, an increase in the volatility by 1% implies a loss of $500 and a decrease by 1% yields a profit of $500. The vega of an option position is given by: ∂v1 ∂v2 ∂v3 ∂vj +n2 − +n3 − +· · ·+n1 − (7) Vegaposition = n1 − ∂σ ∂σ ∂σ ∂σ 10.3. The Characteristics of Volatility Spreads As a simple standard option, a spread is also characterized by its delta, gamma, theta, and vega. These sensitivity parameters allow the investor to manage his/her, option positions as a consequence of the changes in the market conditions. The implementation of strategies based on volatility spreads, implies often the use of delta-neutral strategies to be able to predict the variations in the market conditions. When the changes in the underlying asset value give more value to the spread, the gamma is positive. On the other direction, the spread’s gamma exhibits is negative when the variations in the underlying asset price reduce the spread value. Since the effects of these changes in the underlying asset price and the time to maturity operate in the opposite side, a spread with a positive gamma shows a negative theta and vice versa.

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Characteristics of volatility spreads.

Strategy, Position

∆

Γ

Θ

Vega

Short a call ratio spread Short a put spread Long a straddle Long a strangle Short a butterfly Long call ratio spread Long a put ratio spread Short a straddle Short a strangle Long a butterfly

0 0 0 0 0 0 0 0 0 0

+ + + + + − − − − −

− − − − − + + + + +

+ + + + + − − − − −

Table 10.23 summarizes the effect of the sensitivity parameters on various spread strategies. The investor can implement delta-neutral strategies when he/she has no prior anticipations as to where the market is going. However, he/she can resort to bullish and bearish spreads when he/she is confident about the market timing. This leads him/her to be long or short the underlying asset. When the options used are overvalued, (according to the investor), for instance, when the implied volatility is high (with respect to the historical volatility and its normal level), the investor can sell some puts if the market rises and some calls if the market falls. When the options are undervalued, (according to the investor), for instance, when the implied volatility is low (with respect to the historical volatility and its normal level), the investor can be long calls and puts when the market falls. Summary It is important to monitor the variations in a derivative asset price with respect to its determinants or the parameters, which enter the option formula. These variations are often known as Greek-letter risk measures. The most widely used measures are known as the delta, charm, gamma, speed, color, theta, vega, rho, and elasticity. The delta shows the absolute change in the option price with respect to a small variation in the underlying asset price. Charm corresponds to the partial derivative of the delta with respect to time. The gamma gives the change in the delta, or in the hedge ratio as the underlying asset price changes.

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Color corresponds to the gamma’s derivative with respect to the time remaining to maturity. The theta measures the change in the option price as time elapses. The vega or lambda is a measure of the change in the option price for a small change in the underlying asset’s volatility. This chapter presented the main Greek-letter risk measures, i.e., the delta, the gamma, the theta, and the vega in the context of the European analytical models. These risk measures are simulated for different parameters, which enter the option formulas. Then the magnitude of these risk measures is appreciated in connection with the management and the monitoring of an option position. The knowledge of the changes in these risk parameters is necessary for the management of an option position and the determination of the profits and losses associated with the portfolio. To put it differently, the pricing of a European call option can be viewed as requiring inputs (the underlying asset and the Treasury bill) and a production technology (the hedge portfolio and the Greek-letter risk measures). In a B–S world, by tracking continuously the hedge ratio (being delta-neutral), the investor makes sure that the duplicating portfolio does mimic the call option, namely does “produce” the option. In the course of doing so, the investor controls his/her production costs and protects his/her mark up on the option. However, these risk measures depend on the theoretical model used for the valuation and the management of the option position. This is why, one could call such a risk measure as a technological risk. This position must be adjusted nearly continuously, in response to the changes in the market conditions. Appendix A: Greek-Letter Risk Measures in Analytical Models A.1. B–S model Call sensitivity parameters ∆c = N (d1 ),

Γc =

1 ∂∆c √ n(d1 ) = ∂S Sσ T

Sσn(d1 ) ∂c √ − rKe−rT N (d2 ) = ∂T 2 T √ ∂c ∂c = S T n(d1 ), Rhoc = = KT e−rT N (d2 ). vc = ∂σ ∂r

Θc =

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Put sensitivity parameters

∆p = ∆c − 1,

Γp =

1 ∂∆p √ n(d1 ) = ∂S Sσ T

Θp =

Sσn(d1 ) ∂p =− √ + rKe−rT N (−d2 ) ∂T 2 T

vp =

√ ∂p = S T n(d1 ), ∂σ

Rhop =

∂p = −KT e−rT N (−d2 ). ∂r

A.2. Black’s Model The option sensitivity parameters in the Black’s model are presented as follows.

Call sensitivity parameters

∆c = e−rT N (d1 ), Θc = vc =

Γc =

∂∆c e−rT √ n(d1 ) = ∂S Sσ T

∂c Se−rT σn(d1 ) √ + rSe−rT N (d1 ) − rKe−rT N (d2 ) =− ∂T 2 T √ ∂c = Se−rt T n(d1 ) ∂σ


∆p = ∆c − e−rT ,

Γp =

∂∆p 1 √ n(d1 ) = ∂S Sσ T

∂p Sσe−rT n(d1 ) √ − rSe−rT N (−d1 ) + rKe−rT N (−d2 ) = ∂T 2 T √ ∂p vp = = S T n(d1 ) ∂σ

Θp = −

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A.3. Garman and Kohlhagen’s model Call sensitivity parameters −r ∗ T

∆c = e

∗

N (d1 ),

e−r T ∂∆c √ n(d1 ) = Γc = ∂S Sσ T ∗

∗ ∂c Se−r σn(d1 ) √ = r∗ Se−r T N (d1 − rKe−rT N (d2 )) − Θc = ∂T 2 T ∗ ∂c = −T Se−r T N (d1 ) ∗ ∂r ∗ √ ∂c = Se−r T T n(d1 ) vc = ∂σ

Rhoc =


∆p = e

−r ∗ T

∗

[N (d1 − 1)],

e−r T ∂∆p √ n(d1 ) = Γp = ∂S Sσ T ∗

∗ ∂p Sσe−r T n(d1 ) √ Θp = = −r∗ Se−r T N (−d1 ) + rKe−rT N (−d2 ) − ∂T 2 T ∗ √ ∂p = Se−r T T n(d1 ) vp = ∂σ ∗ ∂p Rho = ∗ = T Se−r T N (−d1 ) ∂r

A.4. Merton’s and Barone-Adesi and Whaley’s model Call sensitivity parameters ∆ = e(b−r) N (d1 ),

Γc =

∂∆c e(b−r) √ n(d1 ) = ∂S Sσ T

∂p Se(b−r)T n(d1 ) √ = (r − b)Se(b−r)T N (d1 ) − rKe−rT N (d2 ) − ∂T 2 T ∗ √ ∂c ∂c vc = = Se−r T T n(d1 ), Rhoc = = T Se(b−r)T N (d1 ). ∂σ ∂b

Θc =

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Put sensitivity parameters ∆p = −e(b−r)T [N (−d1 ) + 1] Γp =

∂∆p e(b−r)T √ n(d1 ) = ∂S Sσ T

∂p Se(b−r) σN (d2 ) √ = Se(b−r)N (−d1 ) + rKe−rT N (−d2 ) − ∂T 2 T √ ∂p vp = = Se(b−r)T T n(d1 ) ∂σ ∂p = −T Se(b−r)T N (−d2 ) Rho = ∂b Θp =

Appendix B: The Relationship Between Hedging Parameters Using the definitions of the delta, gamma, and theta, the B–S equation can be written as: ∂c(S, t) 1 2 2 ∂ 2 c(S, t) ∂c(S, t) = rc(S, t) − rS − σ S ∂t ∂S 2 ∂S 2 or 1 −Θ = −rc(S, t) + rS∆ + σ2 S 2 Γ 2 or 1 rc(S, t) = Θ + rS∆ + σ2 S 2 Γ. 2 For a delta-neutral position, the following relationship applies: 1 rc(S, t) = Θ + σ2 S 2 Γ 2 Using the definitions of the hedging parameters, the Black equation can be written as: 1 2 2 ∂ 2 c(F, t) ∂c(F, t) = rc(F, t) − σ F ∂t 2 ∂S 2 or 1 Θ = rc(F, t) − σ2 F 2 Γ. 2

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Using the definitions of the delta, gamma, and theta, the Merton and Barone-Adesi and Whaley (1987) equation can be written as: ∂c(S, t) 1 2 2 ∂ 2 c(S, t) + bS σ S − rc(S, t) + Θ = 0 2 ∂S 2 ∂S or 1 rc(S, t) = Θ + bS∆ + S 2 σ 2 Γ. 2 For a delta-neutral position, the following relationship applies: rc(S, T ) =

1 2 2 S σ Γ+Θ 2

Appendix C: The Generalized Relationship Between the Hedging Parameters We denote respectively by: • • • • • •

Si : price of an asset i; C: value of the contract; σi : volatility of the underlying asset Si ; ρi,j : correlation coefficient between assets Si and Si ; r: instantaneous rate of return on accounting commodity and ri : instantaneous rate of return on the asset i.

If we denote by Hi , the charm asociated with the asset i, the theta given in Garman (1992) is: Θ=

N

S i Hi

i=1 i with Hi = ∂∆ ∂t . The proof of this result relies on the replication equation which states that:

N

Si ∆i = C

i=1

This equation shows that an outright purchase of the security is equivalent to the replicating strategy, which consists of buying or selling the amount dictated by the delta.

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The generalized B–S equation can also be expressed as a function of the greek-letter risk measures as follows: N

N

N

1 σi σj ρij Si Sj Γij + (r − ri )Si ∆i + Θ = rC 2 i=1 j=1 i=1 The use of charm allows to write a potentially more separable equation for derivative assets: N

N

N

1 σi σj ρij Si Sj Γij + Si [Hi − ri ∆i ] = 0 2 i=1 j=1 i=1 Appendix D: A Detailed Derivation of the Greek Letters Re-call that the call option formula is given by: C = Se(b−r)T N (d1 ) − Ke−rT N (d2 ) d1 =

S ln K + (b + 12 σ 2 )T √ , σ T

√ d2 = d1 − σ T

where r stands for the continuous interest rate, τ is the discrete interest rate, and b is the cost of carry. The following relationship applies between interest rates: e−rT =

1 1 + τT

which is equivalent to e−rT = (1 + τ T )−1 or −rT = − ln(1 + τ T ) hence, r=

1 ln(1 + τ T ) T

and T =

Nj base × 100

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The derivation of the call’s delta

∆=

∂d1 ∂d2 ∂C = e(b−r)T N (d1 ) + Se(b−r)T N (d1 ) − Ke−rT N (d2 ) ∂S ∂S ∂S

with: d2 1 1 N (d1 ) = √ e− 2 , 2Π

d2 1 2 N (d2 ) = √ e− 2 2Π

Since ∂d2 1 ∂d1 √ = = ∂S ∂S Sσ T then: d2 1 1 ∂C 1 √ = e(b−r)T N (d1 ) + Se(b−r)T √ e− 2 ∂S 2Π Sσ T d2 1 1 2 √ − Ke−rT √ e 2 2Π Sσ T

or d2 ∂C 1 1 1 = e(b−r)T N (d1 ) + e(b−r)T √ e− 2 √ ∂S 2Π σ T d2 1 S 1 1 −Ke−rT √ e(− 2 +ln K +bT ) √ 2Π Sσ T

The following relationship is used to obtain the desired result: √ √ d22 = (d1 − σ T )2 = d21 − 2d1 σ T + σ 2 T √ S S 1 2 d1 σ T = ln + b + σ T, d22 = d21 − 2 ln − 2bT K 2 K Hence, we have: d2 1 ∂C 1 e(b−r)T − 2 = e(b−r)T N (d1 ) + √ ∂S σ 2ΠT d2 1 S 1 K − √ eln K e((b−r)T − 2 ) S σ 2ΠT

467

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The call’s delta is given by: ∆ = e(b−r)T N (d1 ). The derivation of the put’s delta Re-call that the put’s formula is: P = Ke−rT N (−d2 ) − Se(b−r)T N (−d1 ) with N (−d2 ) = −(N (d2 ) − 1) and N (−d1 ) = −(N (d1 ) − 1) The delta is given by: ∆=

∂P = Ke−rT N (−d2 )(−d2 ) − e(b−r)T N (−d1 ) ∂S − Se(b−r)T N (−d1 )(−d1 )

or d2 1 2 ∆ = −e(b−r)T N (−d1 ) + Ke−rT √ e(− 2 ) (−d2 ) 2π d2 1 1 − Se−(b−r)T √ e(− 2 ) (−d1 ) 2π −rT

d2 d2 e 2 1 +√ Ke(− 2 ) (−d2 ) − SebT e(− 2 ) (−d1 ) 2π d2 e−rT (− d22 ) 1 √ Ke 2 − SebT e(− 2 ) − Sσ 2πT

The following results are used in the computations.

Ke

(−

d2 2 2

)

bT (−

− Se e

d2 1 2

)

= Ke

−

d2 1 2

+ln

S K

+bT

bT

− Se e

−

d2 1 2

and −(−d1 ) = (−d2 ) = −

1 √ Sσ T

Hence, the put’s delta is: ∆ = −e(b−r)T N (−d1 ) or ∆ = e(b−r)T N (d1 ) − e(b−r)T .

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The derivation of the call’s gamma The gamma can be computed as follows: Γ=

∂∆ ∂S

or Γ = e(b−r)T N (d1 )

∂d1 , ∂S

which is equivalent to d2 1 1 1 √ Γ = √ e 2 e(b−r)T 2Π Sσ T

So, we obtain: Γ=

−d2 1 1 √ e(b−r)T e 2 . σS 2ΠT

The derivation of the put’s gamma The gamma is given by: 1 ∂∆ = −e(b−r)T √ e Γ= ∂S 2π or Γ=

1 √ e Sσ 2πT

−

−

d2 1 2

d2 1 2

−

1 √ Sσ T

e(b−r)T .

The derivation of the call’s Vega The following equalities are used to simplify the computations. The partial derivatives with respect to the volatility are: √ √ √ S T ln K + T b + 12 σ 2 T σ2 T T − √ d2 d1 ∂d1 = = T− =− 2 ∂σ σ T σ σ √ √ ∂d1 d2 ∂d2 d1 = − T =− =− − T ∂σ ∂σ σ σ The following equality applies:

√ 2 √ d21 = d2 + σ T = d22 + 2d2 σ T + σ 2 T which is also: d21

=

d22

1 2 S + 2 b − σ T + σ2 T + 2 ln K 2

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S or d21 = d22 + 2 ln K + 2bT

v=

∂d1 ∂d2 ∂C = Se(b−r)T N (d1 ) − Ke−rT N (d2 ) ∂σ ∂σ ∂σ

which is equivalent to:

−d2 σ d2 −d2 √ 1 − 22 −rT √ e − T − Ke σ 2Π

∂C 1 = Se(b−r)T √ e ∂σ 2Π

−

d2 1 2

or d2 d2 ∂C 1 S − 22 (b−r)T − ln K −bT √ − e = Se e e ∂σ σ 2Π d2 2 −d2 √ 1 −K e−rT √ e − 2 − T σ σ 2Π Hence, we have: ∂C Ke−rT e = √ ∂σ 2Π

−

d2 2 2

−d2 σ

Ke−rT e − √ 2Π

−

d2 2 2

d2 √ − − T σ

Finally, we obtain: 1 v = e−rT √ e 2Π

−

d2 2 2

√ T

The derivation of the put’s vega v=

∂P ∂σ

This can be written as: v=

∂P = Ke−rT N (−d2 )(−d2 ) − Se(b−r)T N (−d1 )(−d1 ) ∂σ

or e−rT ∂P =K√ e v= ∂σ 2π

−

d2 2 2

d2 √ − T σ

1 − Se(b−r)T √ e 2π

−

d2 1 2

d2 σ

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and 1 ∂P = √ e−rT v= ∂σ 2π

Ke

d2 − 21

d2 √ − T σ

− SebT e

471

„ d2 − 22 −ln

S K

« −bT

d2 σ

Hence, v=

Ke−rT ∂P = √ ∂σ 2π

2 d2 d2 d2 √ 1 d2 e2 − T − e− 2 σ σ

Finally, we have: v = Ke

−rT −

e

d2 1 2

T 2π

We have used the following results in the derivation. −

∂d1 d2 = , ∂σ σ

−

∂d2 ∂d1 √ d2 √ = − T = − T. ∂σ ∂σ σ

The derivation of the call’s Rho with respect to r The rho corresponds to the option partial derivative with respect to the interest rate: ρ=

∂C . ∂r

When the cost of carry is given by the difference between the domestic and foreign interest rate: b = r − rf . The call formula for a currency option becomes: C = Se−rf T N (d1 ) − Ke−rT N (d2 ) Hence: ρ=

∂C = Se−rf T N (d1 )(d1 ) + T Ke−rT N (d2 ) − Ke−rT N (d2 )(d2 ) ∂r

which is equivalent to: ρ=

1 ∂C = T Ke−rT N (d2 ) + Se−rf T √ e ∂r 2Π d2 2 T 1 √ −Ke−rT √ e − 2 2Π σ T

d2 1 2

T √

σ T

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or 1 ∂C = T Ke−rT N (d2 ) + Se−rf T √ e ρ= ∂r 2Π d2 1 1 T − Se−rT √ e − 2 e(r−rf )T √ 2Π σ T

−

d2 1 2

T √

σ T

Finally, we obtain: ρmon = T Ke−rT N (d2 ). The following equalities are used to obtain the desired result. d1 = or

√ 2 T √ = d22 = d1 − σ T σ T

√ d1 = d21 − 2d1 σ T + σ2 T

and d1 = d21 − 2 ln

S − 2bT K

which is d1 = d21 − 2 ln

S − 2(r − rf )T K

We also use the result: „ « d2 d2 d2 S +(r−rf )T − 21 +ln K 1 S − 22 = e − 2 e(r−rf )T =e e K The derivation of the call’s Rho with respect to rf ρdev =

∂C ∂rf

which is equivalent to: ρdev = −T Se−rf T N (d1 ) + Se−rf T N (d1 )(d1 ) − Ke−rT N (d2 )(d2 ) d2 d2 1 2 T T 1 1 √ − KerT √ e − 2 √ + Se−rf T √ e − 2 2Π σ T 2Π σ T or ρdev = −T Se(b−r)T N (d1 )

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Hence, we have: ρdev = −T Se−rf T N (d1 ). Remark: For a futures contract, b = 0, and the call formula can be written as: C = e−rT (SN (d1 ) − XN (d2 )) The derivation of the put’s Rho with respect to rf ∂P = Ke−rT N (−d2 )(−d2 ) + Se−rf T N (−d1 ) − Se(b−r)T N (−d1 )(−d1 ) ∂rf which is equivalent to: 2 d2 d ∂P e−rT − 22 − 21 −rf T bT Ke = Se N (−d1 ) + √ (−d2 ) − Se e (−d1 ) ∂rf 2π „ « 2 d2 d2 S d − 22 −ln K −bT e−rT − 22 − 21 bT bT − Se e − Se e Ke + √ σ 2πT Hence, the Rho is given by: ρdev = −ST e−rf T (N (d1 ) − 1) The following relation is used in the derivation: T d2 = −d1 = − √ . σ T The derivation of the call’s theta The call’s theta is computed as: θ=

∂C = (b − r)Se(b−r)T N (d1 ) + Se(b−r)T N (d1 )(d1 ) ∂T + rKe−rT N (d2 ) − Ke−rT N (d2 )(d2 )

This is equivalent to: 1 θ = (b − r)Se(b−r)T N (d1 ) + rKe−rT N (d2 ) + Se(b−r)T √ e 2π d2 2 1 −Ke−rT √ e − 2 (d2 ) 2π

−

d2 1 2

(d1 )

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or θ = (b − r)Se(b−r)T N (d1 ) + rKe−rT N (d2 ) d2

d21 2 1 + √ e−rT Se 2 (d1 )ebT − Ke − 2 (d2 ) 2π The following results are used in the computations. √ S + b + 12 σ 2 T b + 12 σ 2 σ T − 12 √σT ln K d1 = σ2 T and d2 = d1 −

1 σ √ 2 T

We use also the fact that: √ √ d21 = (d2 + σ T )2 = d22 + 2d2 σ T + σ 2 T or √ √ d21 = d22 + 2(d1 − σ T )σ T + σ2 T and √ d21 = d22 + 2d1 σ T − 2σ 2 T + σ2 T which is √ d21 = d22 + 2d1 σ T − σ2 T or d21 = d22 + 2 ln

S + 2bT + σ 2 T − σ2 T K

and d21 = d22 + 2 ln

S + 2bT K

Using this last expresion for d21 , the following relation:

1 √ e−rT Se 2π

d2 − 21

„

(d1 )ebT − Ke

d2 2 2

«

(d2 )

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can be written as: 1 √ e−rT 2π

Se

„ d2 − 21 −ln

K = √ e−rT e 2π

−

d2 2 2

S K

« −bT

(d1 )erT

− Ke

−

d2 2 2

K ((d1 ) − (d2 )) = √ e−rT e 2π

(d2 )

−

d2 2 2

1 σ √ 2 T

Hence, the call’s theta is given by: Kσ −rT e θ = (b − r)Se(b−r)T N (d1 ) + rKe−rT N (d2 ) + √ e 2 2πT

−

d2 2 2

The derivation of the put’s theta The put’s theta is given by: θ=

∂P = −rKe−rT N (−d2 ) + Ke−rT N (−d2 )(−d2 ) ∂T − (b − r)Se(b−r)T N (−d1 ) − Se(b−r)T N (−d1 )(−d1 )

or θ = (b − r)Se(b−r)T (N (−d1 ) − 1) + rKe−rT (N (−d2 ) − 1) 2 d2 d e−rT − 22 − 21 bT +√ (−d2 ) − Se e (−d1 ) Ke 2πT « „ d2 d2 S −bT − 22 −ln K 2 e−rT bT (−d1 ) + √ − Se e Ke − 2 (−d2 ) − (−d1 )) 2πT The following relation is used in the derivation: (−d2 ) = −(−d1 ) +

1 σ √ 2 T

Hence, the theta is given by: θ = (b − r)Se(b−r)T (N (d1 ) − 1) + rKe−rT (N (d2 ) − 1) d2 σ 2 Xe−rT e − 2 + √ 2 2πT

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The derivation of the put’s rho The put’s Rho is given by: ∂P = −T Ke−rT N (−d2 ) + Ke−rT N (−d2 )(−d2 ) ∂r − Se(b−r)T N (−d1 )(−d1 ) or ∂P = T Ke−rT (N (−d2 ) − 1) ∂r d2 d2 2 e−rT

1 Ke − 2 (−d2 ) − SebT e− 2 (−d1 ) +√ 2π d2 S e−rT T − d22 2 Ke 2 − SebT e − 2 −ln K −bT + √ σ 2πT Finally, we obtain: v=

∂P = T Ke−rT (N (d2 ) − 1). ∂σ

The call’s partial derivative with respect to the strike price The computation of the partial derivative with respect to K gives: ∂d1 ∂d2 ∂C = Se(b−r)T N (d1 ) − e−rT N (d2 ) − Ke−rT N (d2 ) ∂K ∂K ∂K with d1 =

ln S − ln K + (b + 12 σ 2 )T √ , σ T

√ d2 = d1 − σ T

The partial derivatives of d1 and d2 are: 1 ∂d1 √ , =− ∂K Kσ T

1 ∂d2 √ =− ∂K Kσ T

Hence, we have: d2 ∂C 1 1 1 √ − e−rT N (d2 ) = −Se(b−r)T √ e− 2 ∂K 2Π Kσ T d2 1 1 2 √ + Ke−rT √ e− 2 2Π Kσ T

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or d2 1

d2 2

Se(b−r)T e− 2 e−rT e− 2 ∂C √ − erT N (d2 ) + √ =− ∂K Kσ 2ΠT σ 2ΠT Using the following relationship: d21 = d22 + 2 ln

S + 2bT K

we obtain: S e ∂C =− ∂K K

(b−r)T

„ d2 − 21 −ln

e √ σ 2ΠT

S K

« d2 2

−bT

−e

−rT

e−rT e− 2 N (d2 ) + √ σ 2ΠT

or ∂C = −erT N (d2 ). ∂K The put’s partial derivative with respect to the strike price The partial derivative of the put price with respect to K can be computed in the same way. ∂(−d2 ) ∂P = e−rT N (−d2 ) + Ke−rT N (−d2 ) ∂K ∂K d2 1 1 ∂(−d1 ) 2 √ − Se−(b−r)T N (−d1 ) + Ke−rT √ e− 2 ∂K 2Π Kσ T d2 d2 S 1 S 1 1 S 1 √ e(b−r)T √ e− 2 − √ e(b−r)T √ e − 2 −ln K −bT − Kσ T 2Π Kσ T 2Π Hence, we have: ∂P = e−rT N (−d2 ), ∂K or ∂P = −e−rT (N (d2 ) − 1) ∂K

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Questions 1. What is the option’s delta? Provide a simple derivation of this parameter in the context of analytical models. 2. What is the charm? 3. What is the gamma? Provide a simple derivation of this parameter in the context of analytical models. 4. What does spread mean? 5. What does color mean? 6. What does theta mean? Provide a simple derivation of these parameters in the context of analytical models. 7. What does vega mean? Provide a simple derivation of this parameter in the context of analytical models. 8. What does rho mean? 9. What does elasticity mean? 10. Why the knowledge of these Greek-letter risk measures is important? 11. How does the delta change in response to the changes in the option valuation parameters? 12. How does the gamma change in response to the changes in the option valuation parameters? 13. How does the theta change in response to the changes in the option valuation parameters? 14. How does the vega change in response to the changes in the option valuation parameters? 15. How a hedged portfolio is adjusted in response to the changes in the underlying asset price? 16. How a hedged portfolio is adjusted in response to the changes in the volatility parameter? 17. How a hedged portfolio is adjusted in response to the changes in the time to maturity? 18. What are the main characteristics of volatility spreads?

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References Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June), 301–320. Black, F (1975). The Pricing of Complex Options and Corporate Liabilities. Chicago, IL: Graduate School of Business, University of Chicago. Garman, M (1992). Spread the load. Risk, 5(11), 68–84. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.

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Part IV Mathematical Foundations of Option Pricing Models in a Continuous-Time Setting: Basic Concepts and Extensions

491

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Chapter 11

THE DYNAMICS OF ASSET PRICES AND THE ROLE OF INFORMATION: ANALYSIS AND APPLICATIONS IN ASSET AND RISK MANAGEMENT

Chapter Outline This chapter is organized as follows: 1. Section 11.1 introduces continuous time stochastic processes for the dynamics of asset prices. In particular, the Wiener process, the generalized Wiener process, and the Ito process are presented and applied to stock prices. 2. In Section 11.2, Ito’s lemma is constructed and several of its applications are provided. 3. In Section 11.3, we intoduce the concepts of arbitrage, hedging, and replication in connection with the application of Ito’s lemma. This allows the derivation of the partial differential equation governing the prices of derivative assets. 4. In Section 11.4, forward and backward equations are presented and some applications are given. In particular, we give the density of the first passage time and the density for the maximum or minimum of diffusion processes. 5. In Section 11.5, a general arbitrage principle is provided. 6. In Section 11.6, we introduce some concepts used in discrete hedging. 7. Appendix A is a mathematical introduction to diffusion processes. 8. Appendix B gives the main properties of the conditional expectation operator. 9. Appendix C reminds readers regarding the Taylor series formula. 493

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Introduction This chapter explain how assets are priced with complete and incomplete information. Incomplete information reflects billiquidity, and lack of transparency. This can lead to a loss of confidence, as was the case of the financial crisis in 2008. The reader who is unfamiliar with mathematics can skip this part of the book. Most financial models describing the dynamics of price changes, interest rate changes, exchange rate variations, bond price changes, and derivative asset dynamics among other things, present a term known as a Wiener process. This process is a particular type of a general class of stochastic processes known as Markov stochastic processes. A stochastic process can be defined either in a simple way, as throughout this chapter, or in a more mathematical sense, as in Appendix A. Our presentation is at the same time intuitive and rigorous in order to allow the understanding of the necessary tools in continuous time finance. These tools are not as complicated as an uninformed reader might think. Using the definition of a stochastic process enables us to define the standard Brownian motion and the Ito process. The Ito process allows the construction of stochastic integrals and the definition of Ito’s theorem or what is commonly known as Ito’s lemma. This lemma can be obtained using Taylor’s series expansions or a more rigorous mathematical approach. In both cases, some applications of this lemma to the dynamics of asset and derivative asset prices and returns are provided. The introduction of the concepts of arbitrage, replication, and the hedging argument, which are the basic concepts in finance, allows the derivation of a partial differential equation for the pricing of derivative assets. This equation first appeared in Black and Scholes (1973) and Merton (1973). These authors introduced the arbitrage theory of contingent claim pricing and, using Ito’s theorem, showed that a continuously revised hedge between a contingent claim and its underlying asset is perfect. Since then, Ito’s theorem, the Black and Scholes hedge portfolio, and the concepts of arbitrage and replicating portfolios have been used by many researchers in continuous and discrete time finance. The basic equivalent results in the theory of option pricing in a discrete time setting were obtained by Cox et al. (1979) and Rendleman and Barter (1979). They showed that option values calculated with discrete time models converge to option values obtained by continuous time models. In other words, theoretical work on convergence shows that some discrete time processes converge to continuous time processes. For example, in the context of binomial models, an option can be perfectly hedged using the underlying asset, and Ito’s theorem can be implemented when constructing the hedging portfolio for infinitesimal time intervals.

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The Dynamics of Asset Prices and the Role of Information

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Some important questions regarding the use of Ito’s lemma and the perfect hedge can be studied. The main question is whether a continuously revised hedge is perfect over each revision interval or only when cumulating the hedging error to zero over a large number of revision intervals. In all such cases, these basic arguments lead to a partial differential equation, which must be satisfied by the prices of derivative assets. This can be derived using one of the two definitions of Ito’s lemma: the definition given in the mathematical literature or simply the one obtained by an extension of the Taylor series. 11.1. Continuous Time Processes for Asset Price Dynamics The dynamics of asset prices are often represented as a function of a Wiener process or what is also known as Brownian motion. 11.1.1. Asset price dynamics and Wiener process The Wiener process has some interesting properties and can be introduced with respect to a change in a variable W over a small interval of time t. Wiener process or Brownian motion Let W denote a variable following a Wiener process and ∆W a change in its value over a small interval of time ∆t. The relation between ∆W and ∆t is given by the following equation: √ (11.1) ∆W = ξ ∆t where ξ is a random sample from a normal distribution having a zero mean and a unit standard deviation. If one takes two reasonably short intervals of time, then the values of ∆W are independant. Using these properties, it is clear that ∆W has √ also a normal distribution with a zero mean, a standard deviation equal to ∆t, and a variance of ∆t. Now, if one considers the change in W over a longer time period [0, T ], composed of N periods of length ∆t, i.e., T = N ∆t, then the change in W , from W (0) to W (T ), or W (T ) − W (0), over this period of time is equal to the sum of changes over shorter periods. Hence, one can write: W (t) − W (0) =

N √ ξi ∆t i=1

(11.2)

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Using the independence property, it follows from this last equation that the change W (T ) − W (0), is normally distributed with a √ mean of zero, a variance of N ∆t = T , and a standard deviation equal to T . This is the basic Wiener process with a zero mean or drift rate and a unit variance rate. A mean or a drift rate of zero means that the future change is equal to the current change. A variance rate of one means that the change at time T is 1 × T . Example: Consider a variable W following a Wiener process, starting at W (0) = 20 (in years). This variable will attain in one year, a value which is normally distributed with a mean of 20 and a standard deviation of 1. In two years, its value follows √ a similar type of distribution with mean 20 and a standard deviation of 2. In n years, its value follows the same distribution with mean 20 and a variance of n. What happens if the interval ∆t gets very small, i.e., tends to zero. When ∆t gets close to zero, the analogous to Eq. (11.1) is: √ (11.3) dW = ξ dt. The martingale property and the Brownian motion The notion of martingale is useful in financial models, particularily, when analyzing the concept of arbitrage. A martingale can be defined as follows. Consider a probability space (Ω, F, P ) and a filtration (Ft )t≥0 . An adapted family (Mt )t≥0 of integrable random variables having a finite mean is a martingale when for all s ≤ t, then we have: E(Mt | Fs ) = Ms

(11.4)

where E(.) stands for the mathematical conditional expectation operator. Appendix B gives the main properties of the conditional expectation operator. The notion of martingale asserts that the best approximation of Mt , given all the available information Fs , is Ms . In terms of financial markets, this means that the best way to predict futures prices is to use the current prices. Hence, using current information is equivalent to using all the historical information, since only the most recent information matters. Using this definition, it must be clear that when (Mt )t≥0 is a martingale, then E(Mt ) = E(M0 ). The following result is advanced without proof. If (Mt )t≥0 is an (Ft )t — Brownian standard motion, then (Wt )t is an 1 2 (Ft )t — martingale, Wt2 — t is an (Ft )t — martingale, and e(σWt − 2 σ t) is an (Ft )t — martingale.

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The first and second properties characterize the standard Brownian motion. The third property is useful when studying the dynamics of financial asset prices. In fact, as will be shown later, the price of a stock is often written as: 1 2 St = S0 e(σWt − 2 σ t) .

(11.5)

11.1.2. Asset price dynamics and the generalized Wiener process For a variable X, a generalized Wiener process can be expressed as: dX = adt + bdW

(11.6)

where a and b are constants. This process shows the dynamics of the variable X in terms of time and dW . The first term, adt, called the deterministic term, means that the expected drift rate of X is ‘a’ per unit time. The second term, bdW , called the stochastic component, shows the variability or the noise added to the dynamics of X. This noise is given by b times the Wiener process. When = a. This is equivalent the stochastic component is zero, dX = adt, or dX dt to X = X0 + at. Hence, the value of X at any time is given by its initial value X0 plus its drift multiplied by the length of the time period. Now, it is possible to write the equivalent of Eq. (11.6) using Eq. (11.1) for a longer ∆t: √ ∆X = a∆t + bξ dt. (11.7) Hence, as before, since √ ∆X has a normal distribution, its mean is a∆t, its standard deviation is b ∆t, and its variance is b2 ∆t. 11.1.3. Asset price dynamics and the Ito process An Ito process for a variable X can be written as follows: dX = a(X, t)dt + b(X, t)dW.

(11.8)

The dependence of both the expected drift rate and the variance rate on X and time t, is the main difference from the generalized Wiener process. This process has been extensively used in the finance literature, especially for modeling stock price dynamics.

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Ito process We now give the mathematical definition of an Ito process. Consider a probability space (Ω, Ftt≥0 , P ) with a filtration and (Mt )t≥T an (F )t Brownian motion. An Ito process is a process (Mt )0≤t≤T having its values in R for which for all t ≤ T : Xt = X0 +

t

0

Ks ds +

0

t

Hs dWs

(11.9)

where Kt and Ht are stochastic processes adapted to Ft for which the integral corresponding to the second-order moment is finite. It will be shown later that the second integral in the above expression is a martingale.

The dynamics of stock prices The dynamics of the stock price S are represented by the following Ito process with a drift rate, µS, and a variance rate σ 2 S 2 : dS = µSdt + σSdW.

(11.10)

This process for stock prices, also known as the geometric Brownian motion can be written in a discrete time setting as: √ ∆S = µ∆t + σξ ∆t S

(11.11)

where ξ is a random sample from a normal distribution with a zero mean and a unit standard deviation. When the variance rate of return of the stock price is zero, the expected drift in S over ∆t is: dS = µSdt

or

dS = µdt S

(11.12)

so that S = S0 eµt . When the variance rate is not zero and σ2 S 2 ∆t is the variance of the actual change in S during ∆t, the dynamics of the stock price are given by the expected instantaneous increase in S plus its instantaneous variance times the noise dW . This discrete time version says that the proportional return on the stock S, over a short period of time, is given by an expected return µ∆t and a stochastic return σξ∆dt. Hence, ∆S is normally distributed with a mean µ∆t and a standard S

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√ deviation σ ∆t, or: √ ∆S ∼ N (µ∆t, σ ∆t) S

(11.13)

Some numerical examples Example: Consider the XYZ stock price characterized by an expected return of 14% per annum and a standard deviation or volatility of 35% per annum. The initial stock price is 100 . Using Eq. (11.13), the dynamics of this stock price are given by: dS = 0.14dt + 0.35dW S

(11.14)

or for a small interval, ∆t :

√ dS = 0.14∆t + 0.35ξ ∆t. S

If the time interval ∆t is 2 weeks (or 0.03846 year), then the price increase is given by √ ∆S = 100[0.14(0.03846) + 0.35ξ 0.03846] or ∆S = 100(0.005384 + 0.068639ξ). The price increase is a random sample from a normal distribution with a mean of 0.538 and a volatility of 6.86. Example: Consider the Y Z stock price, having an expected return of 20% per annum and a standard deviation or volatility of 25% per annum. Over is normal with, a time interval of 3 days, or 0.008219178 per year, ∆S S ∆S ∼ N (0.00164, 0.0226) S 11.1.4. The log-normal property Using the previous example, since the change in the underlying asset price between time t and time ti , is normally distributed, with a mean 1 2 µ − 2 σ (ti − t) and a variance σ 2 (ti − t), we have 1 µ − σ2 (ti − t), σ (ti − t) (11.15) ln(Sti ) − ln(St ) ∼ N 2

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or 1 2 µ − σ (ti − t) + ln(St ), σ (ti − t) . ln(Sti ) ∼ N 2 Hence, Sti has a log-normal distribution. Example: If S = 100, the expected return is 15% per annum and the volatility is 30% per annum, the distribution of ln(Sti ) in six months is:

ln(Sti ) ∼ N ln(100) + [0.15 − 0.5(0.09)](0.5), 0.3

6 12

or ln(Sti ) ∼ N [4.6576, 0.212]. 11.1.5. Distribution of the rate of return Let α to be a continuously compounded rate of return. What is the distribution of α? At a future date ti , the stock price can be written as: Sti = St eα(ti −t) and α =

1 ti −t

ln ln

Sti St

. Using the log-normal property, i.e.,

Sti St

1 2 ∼N µ − σ (ti − t), σ (ti − t) 2

then σ 1 2 . µ− σ , 2 (ti − t)

α∼N

(11.16)

Example: What is the distribution of the actual rate of return over two years for a stock having an expected return of 15% per annum and a volatility of 30%? The distribution is normal with a mean of 10.5%; (15% − 0.09 2 ), and a 0.3 √ standard deviation of 2 i.e., 21.21%.

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11.2. Ito’s Lemma and Its Applications Financial models are rarely described by a function that depends on a single variable. In general, a function which is itself a function of more than one variable is used. Ito’s lemma, which is the fundamental instrument in stochastic calculus, allows such functions to be differentiated. First, we derive Ito’s lemma with reference to simple results using Taylor series approximations. We then give a more rigorous definition of Ito’s theorem. The formula for Taylor series is given in Appendix C. 11.2.1. Intuitive form Let f be a continuous and differentiable function of a variable x. If ∆x is a small change in x, then using Taylor series, the resulting change in f is given by: 1 d2 f 1 d3 f df 2 ∆x + ∆x3 + · · · ∆x + (11.17) ∆f ∼ dx 2 dx2 6 dx3 If f depends on two variables x and y, then Taylor series expansion of ∆f is: ∂f 1 ∂ 2f 1 ∂ 2f ∂f 2 ∆x + ∆y + + ∆f ∼ ∆x ∆y 2 ∂x ∂y 2 ∂x2 2 ∂y 2 2 ∂ f + ∆x∆y + · · · (11.18) ∂y∂y In the limit case, when ∆x and ∆y are close to zero, Eq. (11.18) becomes: ∂f ∂f ∆f ∼ dx + dy. (11.19) ∂x ∂y Now, if f depends on two variables x and t in lieu of x and y, the analogous to Eq. (11.18) is, 1 ∂ 2f ∂f 1 ∂ 2f ∂f 2 ∆x + ∆t2 ∆x + ∆t + ∆f ∼ ∂x ∂t 2 ∂x2 2 ∂t2 2 ∂ f ∆x∆t + · · · (11.20) + ∂x∂t Consider a derivative security, f (x, t), whose value depends on time and on the asset price x. Assuming that x follows the general Ito process, dx = a(x, t)dt + b(x, t)dW

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or √ ∆x = a(x, t)∆t + bξ ∆t. In the limit, when ∆x and ∆t are close to zero, we cannot ignore, as before the term in ∆x2 since it is equal to ∆x2 = b2 ξ 2 ∆t + terms in higher order in ∆t. In this case, the term in ∆t cannot be neglected. Since the term ξ is normally distributed with a zero mean, E(ξ) = 0 and a unit variance, E(ξ 2 ) − E(ξ)2 = 1, then E(ξ 2 ) = 1 and E(ξ)2 ∆t is ∆t. The variance of ξ 2 ∆t is of order ∆t2 and consequently, as ∆t approaches zero, ξ 2 ∆t becomes certain and equals its expected value, ∆t. In the limit, Eq. (11.20) becomes: df =

∂f ∂x

∆x +

∂f ∂t

∆t +

1 2

∂ 2f ∂x2

b2 dt.

(11.21)

This is exactly Ito’s lemma. Substituting a(x, t)dt + b(x, t)dW for dx, Eq. (11.21) gives: df =

∂f ∂x

a+

∂f ∂t

+

1 2

∂2f ∂x2

∂f b2 dt + bdW. ∂x

(11.22)

Example: Let us denote by X(t) a standard Brownian motion. Using X 4 (t), show that the following integral is equal to: 0

t

1 3 X (τ )dX(τ ) = X 4 (t) − 4 2 3

t 0

X 2 (τ )dτ.

Solution: We apply Ito’s lemma for a function F (X(t)): dF =

dF dX

dX +

dF dt

dt +

1 d2 F 2 dX 2

dt.

Let, F (X(t)) = X 4 (t) and note that,

t 0

dF = [F (t)]t0 .

In this case, we have:

t 0

dF (X(t)) = X 4 (t) − X 4 (0) = X 4 (t)

as X 4 (0) = 0.

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Applying Ito’s lemma, we obtain: dF = 4X 3 dX + 6X 2 dt since gives:

t

0

dF =

t

0

dF dt

= 0. This

4X 3 (τ )dX(τ ) + 6X 2 (τ )d(τ ) = X 4 (t)

which can also be written as: t t 4X 3 (τ )dX(τ ) + 6X 2 (τ )d(τ ) 0

0

t 6 1 = X 3 (τ )dX(τ ) + X 2 (τ )d(τ ) = X 4 (t) 4 4 0

and we have,

t 0

X 3 (τ )dX(τ ) =

1 4 3 X (t) − 4 2

t

0

X 2 (τ )dτ.

Example: Using Ito’s lemma, show the following result:

t

0

τ 3 X(τ )dX(τ ) = t3 X 2 (t) −

3 2

t

0

1 τ 2 X 2 (τ )dτ − τ 4 8

or, Ito’s lemma will be applied for a function F (X(t), t) = t3 X 2 (t): dF =

dF dX + dX

dF 1 d2 F + dt 2 dX 2

dt.

In this case: dF = 2t3 X(t)dX + (3t2 X 2 (t) + t3 )dt or t 0

3

2τ X(τ )dX(τ )+

since

3 0

t 0

2

2

3τ X (τ )+

t 0

3

τ dτ =

t

0

dF = t3 X 2 (t)−03 X 2 (0)

X 2 (0) = 0. Hence, we have:

t 0

3 τ X(τ )dX(τ ) + 2 3

0

t

1 τ X (τ ) + 2 2

2

t 0

τ 3 dτ = t3 X 2 (t).

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or, re-arranging the formula, we obtain the following result:

t 0

τ 3 X(τ )dX(τ ) =

1 3 2 3 t X (t) − 2 2

t

0

τ 2 X 2 (τ )dτ −

1 2

t 0

τ 3 dτ

or 1 2

t

0

τ 3 dτ =

1 4 τ 8

then: 0

t

τ 3 X(τ )dX(τ ) =

1 3 2 3 t X (t) − 2 2

t

0

1 τ 2 X 2 (τ )dτ − τ 4 . 8

Example: Use Ito’s lemma to show that:

t 0

X 5 (τ )dX(τ ) =

1 5 X (t) − 2 5

t

0

X 3 (τ )dτ.

t Solution: When F = X 5 (t), then 0 dF = X 5 (t) − X 5 (0) = X 5 (t) since X(0) = 0. Applying Ito’s lemma to the function F gives: dF = 5X 4 dX + 10X 3dt. Hence:

t

4

t

5X (τ )dX(τ ) + 0

0

10X 3 (τ )dτ = X 5 (t)

and

t 0

X 4 (τ )dX(τ ) + 2

0

t

X 3 (τ )dτ =

1 5 X (t) 5

or

t 0

1 X (τ )dX(τ ) = X 5 (t) − 2 5 4

0

t

X 3 (τ )dτ

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11.2.2. Applications to stock prices Example: Apply Ito’s lemma to a function f (S, t) when the dynamics of stock prices are described by the following stochastic equation: dS = µSdt + σSdW. Equation (11.22) gives: ∂f ∂f 1 ∂2f ∂f 2 2 σ dt + µS + + σSdW. S df = ∂x ∂t 2 ∂S 2 ∂t (11.23) Example: Apply Ito’s lemma to derive the process of f = ln(S). First calculate the derivatives, 2 ∂f ∂f 1 1 ∂ f = − 2; = ; = 0. 2 ∂S S ∂S S ∂t Then from Ito’s lemma, one obtains ∂f 1 ∂ 2f ∂f ∂f 2 2 σ dt + µS + + σSdW S df = ∂x ∂t 2 ∂S 2 ∂S or

1 df = µ − σ2 dt + σdW. 2

This last equation shows that, f follows a generalized Wiener process with a constant drift of µ − 12 σ 2 and a variance rate of σ 2 . 11.2.3. Mathematical form The following theorem gives Ito’s formula. Theorem: If (Xt)0≤t≤T is an Ito process, i.e., t t Ks ds + Hs dWs Xt = X 0 + 0

(11.24)

0

and f is a continuous function with second-order continuous derivatives, then: t 1 t f (Xs )dXs + f (Xs )dX, Xs (11.25) f (Xt ) = f (X0 ) + 2 0 0

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where by definition, dX, Xt =

t

Hs2 ds.

0

The first integral is given by: 0

t

f (Xs )dXs =

t

0

f (Xs )Ks ds +

0

t

f (Xs )Hs dWs .

The second integral is given by: 0

t

f (Xs )dX, Xs =

t

0

f (Xs )Hs2 ds.

Hence, Ito’s formula is: f (Xt ) = f (X0 ) +

t

+ 0

t 0

f (Xs )Ks ds

1 f (Xs )Hs dWs + f (Xs )Hs2 ds. 2

(11.26)

More generally, if f (x, t) has first-order continuous partial derivatives in t and continuous second-order derivatives in x, then: f (t, Xt ) = f (0, X0 ) + +

1 2

0

t

t

0

fs (s, Xs )ds +

0

t

fx (s, Xs )dXs

fxx (s, Xs )dX, Xs .

(11.27)

Note that if we put Ks = 0 and Hs = 1, in the Ito process, i.e., Xt = X0 +

0

t

Ks ds +

0

t

Hs dWs

(11.28)

then it reduces to Xt = Wt , which is simply the Brownian motion, since W0 = 0. Example: In this example, Ito’s formula is applied to the dynamics of the squared Brownian motion, Wt2 . When f (x) = x2 and Xt = Wt , (the case

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where Ks = 0 and Hs = 1), we have: Wt2 = W02 +

t

0

2Ws dWs +

1 2

t

2dX, Xs .

0

Since f (x) = (x2 ) = 2x, f (x) = (x2 ) = 2 and dX, Xs = ds, then: Wt2 =

t 0

1 2

2Ws dWs +

t

t

2ds = 2 0

0

Ws dWs + t.

t t Since W0 = 0 and 0 ds = t, then Wt2 − t = 2 0 Ws dWs . t 2 Since the expected value of 0 Ws ds is finite, then (Ws2 − t) is a martingale. Example: This application of Ito’s lemma concerns the calculation of an explicit solution to the process describing the dynamics of stock prices. Let us look for the solutions to (St )t≥0 for the following equation: St = S0 +

t

0

Ss [µSdS + σdWs ]

(11.29)

which is often written in the form dSt = St (µdt + σdWt ). Let Yt = ln(St ), where St is solution to the preceding equation. Since St follows an Ito process with Ks = µSs and Hs = σSs , application of Ito’s formula to f (S) = ln(S) gives: ln(St ) = ln(S0 ) +

0

t

1 dSs + Ss 2

t 0

−1 2 2 σ Ss ds. Ss2

(11.30)

Since f (S) = ln(S), f (S) = S1 , f (S) = −1 S 2 , and the term corresponding to the equivalence of dX, Xs is the instantaneous variance of dSs , Var(dSs ) = σ 2 Ss2 ds. Substituting Eq. (11.29) in Eq. (11.30) and simplifying gives: Yt = Y0 +

t 0

t 1 µ − σ2 ds + σdWs . 2 0

(11.31)

t t This result is straightforward since 0 ds = t, 0 dWs = Wt − W0 with W0 = 0. So, we have Yt = ln(St ) = ln(S0 ) + µ − 12 σ 2 t + σWt .

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Applying the exponential to Yt gives the solution to Eq. (11.29) 1 2 St = S0 e(µ− 2 σ )t+σdWt .

This is the explicit formula for the underlying asset price when its dynamics are given by the stochastic differential equation (11.29). The following theorem which is stated without proof, shows that the solution to Eq. (11.29) is unique. Theorem: When (Wt )t≥0 is a Brownian motion, there is a unique Ito process (St)0≤t≤T for which, for every t ≤ T : t Ss [µSdS + σdWs ] (11.32) St = S0 + 0

1 2 This process is given by St = S0 e(µ− 2 σ )t+σdWt . This process is used in the derivation of the Black and Scholes formula and is often referred to as the Black–Scholes process.

Example: The Black and Scholes (1973) model for the valuation of a European options uses two assets: a risky asset with a price St at time t and a risk-less asset Bt at time t. The dynamics of the risk-less asset or bond are given by the following ordinary differential equation: dBt = rBt dt

(11.33)

where r stands for the risk-less interest rate. The bond’s value at time 0, B0 = 1 in a way such that Bt = ert . The dynamics of the risky asset or stock are given by the following stochastic differential equation: dSt = µSt dt + σSt dWt

(11.34)

where µ and σ are constants and Wt is a standard Brownian motion. As we have shown, the solution to Eq. (11.34) is: 1 2 St = S0 e(µ− 2 σ )t+σdWt

where, S0 is the initial asset price at time 0. 11.2.4. The generalized Ito’s formula Ito’s formula can be generalized to the case where the function depends on several Ito processes, which are expressed in terms of standard independent

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Brownian motions. This is the multidimensional Ito’s formula or the vector form of Ito’s lemma. The formula is useful in deriving interest rate models and models of derivative asset pricing with several state variables. Generalization of Ito’s lemma: The first form: The generalization of Ito’s lemma is useful for a function that depends on n stochastic variables xi , where i varies from 1 to n. Consider the following dynamics for the variables xi , dxi = ai dt + bi dzi

(11.35)

Using Taylor series expansion of f gives: ∆f =

∂f 1 ∂2f ∂f ∆t + ∆xi + ∆xi ∆xj ∂xi ∂t 2 i j ∂xi ∂xj i +

∂2f ∆xi ∆t + · · · ∂xi ∂t

(11.36)

√ Equation (11.35) can be discretized as follows ∆xi = ai ∆t+bi i ∆ zi where the term i corresponds to a random sample from a standardized normal distribution. The terms i and j reflecting the Wiener processes present a correlation coefficient ρi,j . It is possible to show that when the time interval tends to zero, in the limit, the term ∆x2i = b2i dt and the product ∆xi ∆xj = bi bj ρi,j dt. Hence, in the limit, when the time interval becomes close to zero, Eq. (11.36) can be written as: df =

∂f 1 ∂2f ∂f dt + dxi + bi bj ρi,j dt. ∂xi ∂t 2 ∂xi ∂xj i

i

j

This gives the generalized version of Ito’s lemma. Using Eq. (11.35) gives:   ∂ 2f ∂f ∂f 1 ∂f + ai + bi bj ρij  + bi dzi df =  ∂xi ∂t 2 ∂xi ∂xj ∂xi i

i

j

(11.37) Generalization of Ito’s lemma: The second form: Consider Wt = (Wt1 , . . . , Wtp ) where (Wti )t≥0 are independent standard Brownian motions

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and Wt is a p-dimensional Brownian motion. The mathematical expression of Ito’s formula is introduced with respect to n Ito processes (Xt1 , Xt2 , . . . , Xtn ) Xti

=

X0i

+ 0

t

Ksi ds

+

p i=1

t 0

Hsij dXsi .

When the function f (.) has second-order partial derivatives in x and firstorder partial derivatives in t, which are continuous in (x, t), then the generalized Ito’s lemma is: t ∂f (s, Xs1 , . . . , Xsn )ds f (t, Xt1 , . . . , Xtn ) = f (0, X01, . . . , X0n ) + ∂s 0 n t ∂f (s, Xs1 , . . . , Xsn )dXsi + ∂x i i=1 0 +

n ∂ 2f 1 t (s, Xs1 , . . . , Xsn )dX i , Xjs 2 i,j=1 0 ∂xi ∂xj

(11.38) with dXsi = Ksi ds +

p

Hsi,j dWsj , dX i , X j s =

j=1

p

Hsi,m Hsj,m ds

m=1

In the financial literature, the notation is more compact than that used in the mathematical literature. The term dX i , Xjs corresponds to the changes in the instantaneous covariance terms Cov(dXsi , dXsj ). 11.2.5. Other applications of Ito’s formula Example: Use Ito’s lemma to show that: t 1 n − 1 t n−2 X n−1 (τ )dX(τ ) = X n (t) − X (τ )dτ n 2 0 0 Solution: When F = X n (t), n ∈ N ∗ then t dF = X n (t) − X n(0) = X n (t) 0

since X(0) = 0.

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Using Ito’s lemma gives: 1 dF = nX n−1 dX + n(n − 1)X n−2 dt. 2 Hence:

t 0

dF =

t

0

1 nX n−1 (τ )dX(τ ) + n(n − 1)X n−2 (τ )dτ = X n (t) 2 ⇐⇒

t

0

1 1 X n−1 (τ )dX(τ ) + (n − 1)X n−2 (τ )dτ = X n (t) 2 n ⇐⇒

t 0

X

n−1

1 (τ )dX(τ ) = X n (t) − n

0

t

n − 1 n−2 (τ )dτ. X 2

Example: Use Ito’s lemma to show that:

t

0

τ m X n−1 (τ )dX(τ ) =

1 m n n − 1 t m n−2 t X (t) − t X (τ )dτ n 2 0 m t m−1 n t X (τ )dτ. − n 0

Solution: When F = X n(t), where n, m ∈ N ∗ then

t 0

dF = tm X n (t) − tm X n (0) = tmX n (t)

since X(0) = 0. Using Ito’s lemma gives: 1 dF = ntm X n−1 dX + n(n − 1)X n−2dt + mtm−1 X n (t)dt. 2

511

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Hence, we have: t t t 1 dF = n τ m X n−1 (τ )dX(τ ) + n(n − 1) τ m X n−2 (τ )dτ 2 0 0 0 t τ m−1 X n (τ )dτ = tm X n(t) +m 0

⇐⇒

t 0

1 τ m X n−1 (τ )dX(τ ) + (n − 1)τ m X n−2 (τ )dτ 2 t 1 m τ m−1 X n (τ )dτ = tm X n (t) + n 0 n ⇐⇒

t 0

m

τ X

n−1

t 1 m n n − 1 m n−2 τ X (τ )dX(τ ) = t X (t) − (τ )dτ n 2 0 m t m−1 n τ X (τ )dτ − n 0

Example: We consider a function f (t) which is continuous and bounded on the interval [0, t]. Using the integration by parts, we want to show that: t t f (τ )dX(τ ) = f (t)X(t) − X(t)df (τ ) 0

0

Solution: Consider the following function F = f (t)X(t). In this case, we have t dF = f (t)X(t) − f (0)X(0) = f (t)X(t) 0

since X(0) = 0 and by Ito’s lemma: dF = f dX + Xdf. Therefore,

t 0

f (τ )dX(τ ) +

t 0

X(τ )df (τ ) = f (t)X(t)

which can be written as: t t f (τ )dX(τ ) = f (t)X(t) − X(τ )df (τ ). 0

0

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11.3. Taylor Series, Ito’s Theorem and the Replication Argument Let us denote by c(S, t) the option value at time t as a function of the underlying asset price S and time t. Assume that the underlying asset price follows a geometric Brownian motion: dS = µdt + σdW (t) S

(11.39)

where µ and σ 2 correspond, respectively to the instantaneous mean and the variance of the rate of return of the stock.

11.3.1. The relationship between Taylor series and Ito’s differential Using Taylor series differential, it is possible to express the price change of the option over a small interval of time [t, t + dt] as: dc =

∂c ∂S

dS +

∂c ∂t

dt +

1 2

∂2c ∂2S

(dS)2

(11.40)

where the last term appears because dS 2 is of order dt. The last term in Eq. (11.40) appears because the term (dS)2 is of order dt. Omberg (1991) makes a decomposition of the last term in Eq. (11.40) into its expected value and an error term. This allows one to establish a link between Taylor’s series (dc) and Ito’s differential dcI as: dc =

∂c ∂S

dS +

∂c ∂t

dt +

1 2

∂2c ∂2S

σ2 S 2 dW 2 + de(t)

which can be written as the sum of two components corresponding to the Ito’s differential dcI and an error term de(t): dc = dcI + de(t)

(11.41)

where dcI =

∂c ∂S

dS +

∂c ∂t

1 dt + 2

∂ 2c ∂ 2S

σ 2 S 2 dt

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and 1 de(t) = 2

∂2c ∂2S

σ 2 S 2 dW 2 − dt .

11.3.2. Ito’s differential and the replication portfolio A replication argument is often used in financial theory. It means simply that in complete markets, the payoff of an asset can be created or duplicated using some other assets. A combination of these assets provides a similar payoff as that of the original asset. The standard case: In general, the payoff of any derivative asset can be created by an investment of a certain amount in the underlying risky asset and another amount in risk-free discount bonds. Also, the payoff of a derivative asset can be created using the discount bond, some options, and the underlying asset. The portfolio which duplicates the payoff of the asset is called the replicating portfolio. When using Ito’s lemma, the error term de(t) is often neglected and, the equation for the option is approximated only by the term dcI . The quantity dcI is replicated by QS units of the ∂c underlying asset

2 and an amount of cash Qc with QS = ∂S and Qc = 1 ∂c 1 ∂ c 2 2 S where r stands for the risk-free rate of return. r ∂S + 2 ∂ 2 S σ Hence, the dynamics of the replicating portfolio are given by: dΠR =

∂c ∂S

dS + rQc dt

(11.42)

where ΠR refers to the replicating portfolio. An extension to account for information costs: Consider, for example, a financial institution using a given market. If the costs of portfolio selection and models conception, etc., are computed, then it can require at least a return of say, for example λ = 3%, before acting in this market. This cost is in some sense the minimal cost before acting in a certain market. It represents somehow, the minimal return required before implementing a given strategy. For the analysis of information costs and valuation, we can refer to Bellalah (1999, 2000, 2001), Bellalah et al. (2001a,b), Bellalah and Prigent (2001), Bellalah and Selmi (2001), and soon. If you consider the above replicating strategy, then the returns from the replicating portfolio

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must be at least

dΠR =

with 1 Qc = r+λ

∂c ∂S

∂c ∂S

dS + (r + λ)Qc dt

1 + 2

∂ 2c ∂ 2S

2

σ S

2

(11.43)

where ΠR refers to the replicating portfolio. This shows that the required return must cover at least the costs necessary for constructing the replicating portfolio plus the risk-free rate. In fact, when constructing a portfolio, some money is spent and a return for this must be required. Hence, there must be a minimal cost and a minimal return required for investing in information at the aggregate market level. For this reason, the required return must be at least λ plus the risk-less rate. 11.3.3. Ito’s differential and the arbitrage portfolio If one uses arbitrage arguments, then the option value must be equal to the value of its replicating portfolio. The standard analysis: Using arbitrage arguments, we must have c = Qs S + Qc . or

c=

∂c ∂S

1 S+ r

∂c ∂t

1 + 2

and 1 ∂2c 2 2 ∂c σ S + rc − r S+ 2 ∂2S ∂S

∂2c ∂2S

∂c ∂t

2

σ S

2

(11.44)

= 0.

This equation is often referred to, in financial economics, as the Black– Scholes–Merton partial differential ∂c equation. Note that the value of the replicating portfolio is ΠR = ∂S S + Qc . It is possible to implement a hedged position by buying the derivative asset and selling delta units of the underlying asset: ∂c ΠH = c − S = Qc (11.45) ∂S where the subscript H refers to the hedged portfolio.

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A hedged position or portfolio is a portfolio whose return at equilibrium must be equal in theory to the short-term risk-free rate of interest. This is the main contribution of Black–Scholes (1973) to the pricing of derivative assets. Merton (1973) uses the same argument as Black–Scholes (1973) by implementing the concept of self-financing portfolio. This portfolio is also constructed by buying the option and selling the replicating portfolio or vice versa. The condition on the self-financing portfolio is: ΠA = c −

∂c ∂S

S − Qc = 0

where S refers to the self-financing portfolio. The ommited error term in the above analysis, de(t), can reflect a replication error, a hedging error or an arbitrage error. It can have different interpretations. The term de(t) is neglected or omitted when the revision of the portfolio is done to allow for the replicating portfolio to be self-financing. When this term is positive, this may refer to an additional cash that must be put in the portfolio. When it is negative, a withdrawal of cash from the portfolio is possible. An extension to account for information costs: In the same way, the previous analysis can be extended to acount for information costs. In this context, we must have: c = QS S + Qc or c=

∂c ∂S

S+

1 (r + λ)

∂c ∂t

+

1 2

∂ 2c ∂ 2S

σ2 S 2

(11.46)

and 1 2

∂c ∂2c 2 2 ∂c + (r + λ)c − (r + λ) σ S + = 0. S ∂2S ∂S ∂t

This equation corresponds to an extended version of the well-known Black– Scholes–Merton partial differential equation accounting for the effect of information costs. For the sake of simplicity, we assume that information costs are equal in both markets: the option market and the underlying asset market. Or in practice, institutions and investors support these costs on both markets. Therefore, a more suitable analysis must account for two

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costs: cost λc on the option market and a cost λS on the underlying asset market. In this case, we obtain the following more general equation as in Bellalah and Jacquillat (1995) and Bellalah (1999): ∂c ∂c 1 2 2 ∂2c σ S 2 + (r + λc )c − (r + λS ) S+ = 0. 2 ∂ S ∂S ∂t

(11.47)

11.3.4. Why are error terms neglected? Now, we give an answer to the following question: If each portfolio revision is self-financing and the hedging error de(t) tends to zero as the interval of time becomes extremely small, is there a mathematical or an economic “rationale” in ignoring the error term? In the Black and Scholes (1973) theory, the term de(t) is ignored because it is not correlated with the underlying asset price in the context of the capital asset pricing model (CAPM). In this context, it is regarded as a diversifiable risk. This justification is referred to as the “equilibrium option pricing theory”. However, if one uses the Black and Scholes theory with respect to the implementation of Ito’s lemma, then ignoring the error term, refers to a pure-arbitrage result. Omberg (1991) proposed two explanations for pure arbitrage results with respect to the two following assumptions H1 and H2 . According to H1 , the error term is of order o(dt), which is a higher order than dt: H1 : de(t) = o(dt).

(11.48)

According to H2 , the error term is of order Odt: H2 : de(t) = O(dt)

(11.49)

and

τ 0

de(t) = 0

for τ > 0.

In H1 , the error is neglected because in the limit, the hedging error disappears more quickly than the risk-free return. This is because the riskfree return is of order dt. In this context, the replicating portfolio is perfect because the smallness of the interval justifies the disparition of the hedging error.

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In H2 , the price dynamics are represented by the Taylor’s series differential rather than Ito’s differential. In this case, cumulation of hedging errors over several time intervals is necessary. Merton (1973) considered that the error term is O(dt) and neglected it in the context of the law of large numbers. Omberg (1991) showed that the hedging error is zero over each revision interval in two cases. The first case corresponds to the limit of the binomial discrete option pricing model of Cox et al. (1979). The second case concerns a stochastic revision strategy, which succeeds with a certain probability of one-over-one revision interval. However, it remains unproved that the strategy succeeds over a high number of revision intervals. For other cases, the hedging error over one revision interval is of the same order as the risk-free return and cannot be eliminated.

11.4. Forward and Backward Equations When the asset price dynamics are described by the following Markov diffusion process: dSt = µ(S, t)dt + σ(S, t)dWt the probability density function for S at time t conditional on St0 = S0 , denoted by f (S, t; S0 , t0 ), satisfies the partial differential equations of motions, which are the backward and the forward Kolmogorov equations. The backward Kolmogorov equation is given by: 1 2 σ (S0 , t0 ) 2

∂ 2f ∂S 2

+ µ(S0 , t0 )

∂f ∂S0

+

∂f ∂t0

= 0.

(11.50)

The forward Kolmogorov or Fokker–Planck equation is given by: 1 2

∂2f ∂S 2

[σ 2 (S0 , t0 )f ] −

∂f ∂S0

[µ(S, t)f ] +

∂f ∂t

= 0.

(11.51)

These equations can be solved under the condition f (S, t0 ; S0 , t0 ) = δS0 (S), i.e., at the initial time, S is equal to S0 and δS0 is the Dirac measure at S0 . It is defined by δS0 (S)([a, b]) = 1, if S0 is ∈ [a, b] and zero else where. Since we are interested only in time-homogeneous processes for which σ = σ(S) and µ = µ(S), two types of constraints are sometimes imposed to obtain an absorbing barrier and a reflecting barrier when the drift is finite.

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An absorbing barrier means simply that once the process attains a certain value, this value will be conserved for all subsequent instants. A reflecting barrier means that when the process hits a certain level, he/she will return from the direction from which it comes. When we constrain f (S1+ , t) to take the value zero, then S1 is an absorbing barrier from above (+). The intuition of this result is that when S = S1 at t1 , then S will conserve this value for all the subsequent instants t after t1 . When we constrain the term, 1 2

∂ 2 f (S, t)σ2 (S) ∂S 2

− [µ(S)f (S, t)]

(11.52)

to zero at S = S1 , then S1 is a reflecting barrier. It is convenient to note that when σ(S1 ) = 0, then the value at which σ is zero represents a natural barrier. When µ(S1 ) is zero, we have a natural absorbing barrier and when µ(S, t) is strictly positive (or negative), we have a natural reflecting barrier from above (or below). Example: When µ(S) = µ and σ(S) = σ, then S is a Brownian motion with drift. It is possible to verify that the solution to Eqs. (11.50) and (11.51) is: f (S, t; x0 , t0 ) = N

S − x0 − µ(T − t0 ) √ σ T − t0

(11.53)

where N [.] stands for the density of the standard normal distribution. 11.5. The Main Concepts in Bond Markets and the General Arbitrage Principle This section presents the main concepts in bond markets and the general arbitrage principle.

11.5.1. The main concepts in bond pricing The yield to maturity (YTM) Consider a zero-coupon bond B(t, T ) at time t maturing in T years. The present value of one-dollar received at time T is B(t, T ) = e−y(T −t) . Hence, B the YTM is given by y = − log T −t . The price of a coupon-paying bond corresponds to the present value of all its cash flows N + 1: N coupons

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Ci and principal payment P as follows: V = P e−y(T −t) +

N

Ci e−y(ti −t) .

(11.54)

i=1

When the market bond price is set equal to this equality, this gives the YTM or internal rate of return. Duration If we derive the expresion of the bond price with respect to the YTM, this gives the slope of the price/yield curve, N

dV = −(T − t)P e−y(T −t) − (ti − t)Ci e−y(ti −t) . dy i=1

Macaulay duration is given by: −1 dV . V dy

(11.55)

If the discrete compounded rate is used, this quantity refers to the modified duration. Duration is often computed for small movements in the yield in order to examine the change in the price of the bond. Convexity We denote by δy, the change in the yield y. Using a Taylor’s series expansion of V gives: dV 1 dV 1 d2 V = δy + [δy]2 + · · · V V dy 2V dy 2

(11.56)

Convexity is often computed for large movements in the yield in order to examine the change in the price of the bond with respect to yield. The dollar convexity is defined with respect to the bond price as: N d2 V 2 −y(T −t) = (T − t) P e − (ti − t)2 Ci e−y(ti −t) . dy 2 i=1

Convexity corresponds simply to

1 d2 V V dy 2

.

(11.57)

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11.5.2. Time-dependent interest rates and information uncertainty When the spot interest rate is only a known function of time, then B = B(t). Consider the change in the value of a zero-coupon bond paying 1 at t = T over a time step dt from t to (t + dt). The change in the zero-coupon bond . In this case, this return must be equal to the price can be written as dB dt = r(t)B. return from a bank deposit r(t) or dB dt However, if this equality is set by arbitrage considerations, then investors must suffer sunk costs to get informed about these opportunities. In fact, it is well known in practice to find only one interest rate for each maturity. This is more difficult to assert in an international context. There will be also some risks related to the re-investments of the coupons and investors must also pay about future investment opportunities. Merton (1987) showed that information costs are specific to each market and are paid when investors want to get informed about investment opportunities. In this case, they require that the return on the bond must be equal to the risk-less rate plus an “additional return” corresponding to information costs. Hence, we have dB = (r(t) + λB )B. dt In the standard literature, the additionel return λB = 0 and we have: dB = r(t)B. dt The price of the zero coupon satisfyingRthis equation in the presence of an T “additional return” λB is B(t; T ) = e− t (r(τ )+λB )dτ . In the standard literature, the additional return λB = 0 and we have: B(t; T ) = e−

RT t

(rτ )dτ

.

For a coupon-paying bond, when coupons C(t)dt are paid dB in the time interval [t, t + dt], then the holdings change by an amount dt + C(t) dt. Again, if investors pay sunk costs to get informed about the bond and the coupons, then we have:

dB + C(t) dt = (r(t) + λB )B. dt

(11.58)

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We obtain the standard case when λB = 0. The solution to this differential equation is given by: B(t) = e−

RT t

(r(τ )+λB )dτ

T

1+ t

C(t )e

RT t

(r(τ )+λB )dτ

dt

(11.59)

+ V (t− c ) = V (tc ) + Cc .

In the standard case, the solution becomes: −

B(t) = e

RT t

r(τ )dτ

T

C(t )e

1+ t

RT t

rτ dτ

dt

(11.60)

11.5.3. The general arbitrage principle Rogers (1997) considered the general principles of financial modeling in the light of his 1997 approach to interest rate modeling. The general arbitrage principle shows that an asset with price YT at time T , is represented at time t < T by:

Yt = Et

exp −

T

rs ds YT

t

(11.61)

where (rt )t≥0 is the spot-rate process and Et is the conditional expectation in a risk-neutral measure P . The price of a zero-coupon bond at time t with maturity T is:

P (t, T ) = Et

exp −

t

T

rs ds

.

Rogers (1997) showed that it is a good solution to keep the same model all the time and to express the new assets in terms of this same model. To have another look on interest rate modeling, Rogers (1997) assumed a reference probability P˜ with respect to which the risk-neutral probability has a density: ρt =

dP dP˜ F

t

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on the collection Ft of events. In this context, the bond prices can be written as:

T rs ds P (t, T ) = Et exp − t

˜t =E

ρT exp −

T

t

rs ds

˜t [ζT ] ζt ρt = E

where ζt is a state-price density process defined by: t ζt = exp − rs ds ρt . 0

The final expression for the bond price corresponds to a different approach in the modeling of interest rates called by Rogers (1997), as the “potential approach”. 11.6. Discrete Hedging and Option Pricing The Black–Scholes model is based on the assumption of continuous hedging. This is impossible in practice since hedging is done in a discrete way. In general, re-hedging is done regularly at times separated by a constant interval referred to as the length of the hedging period, δt. This period can vary from a few minutes to some weeks depending on the market activity. It is very short for a market maker and longer for a trader. 11.6.1. Discrete hedging It is possible to use Taylor’s series expansions to find a better hedge than the Black–Scholes. This hedge results from the use of a certain number of the underlying assets that minimizes the variance of the hedged portfolio. This allows also to find an option-adjusted value. Following the analysis in Willmott (1998), consider the discrete model for the underlying asset, S = ex with

δx =

σ2 µ− 2

(11.62)

1

δt + σφδt 2

(11.63)

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where φ is a random variable drawn from a standardized normal distribution 1 and φδt 2 corresponds to the Wiener process in the continuous time version. Now, consider the construction of a hedged portfolio by a long position in the derivative security and a short position in ∆ units of the underlying asset S, Π = V − ∆S.

(11.64)

If we determine the right hedge, it is relatively a simple matter to price the option. Using Taylor’s series expansion for the portfolio gives: 1

3

δΠ = δt 2 A1 (φ, ∆) + δtA2 (φ, ∆) + δt 2 A3 (φ, ∆) + δt2 A4 (φ, ∆) + · · · where A1 is given by:

A1 (φ, ∆) = σφS

and A2 is, A2 (φ, ∆) =

∂V +S ∂t

(11.65) ∂V −∆ , ∂S

∂ 2V 1 ∂V 1 −∆ µ + σ2 (φ2 − 1) + σ2 φ2 S 2 ∂S 2 2 ∂S 2

In this context, the ∆ must be chosen in such a way so as to minimize the variance of the change in the portfolio’s value. The option is valued by setting the expected return on the portfolio equal to the risk-less rate plus information costs. Using Eq. (11.65), the variance of the change in the portfolio value can be computed as: Var[δΠ] = E[δΠ2 ] − (E[δΠ])2

(11.66)

The value of ∆ minimizing the variance is found using: ∂ Var[δΠ] = 0 ∂∆ Hence, the optimal ∆ is: ∆=

∂V + δt(. . .) ∂S

(11.67)

The first term is the Black–Scholes hedge ratio in a context of continuous trading. The second term indicates a reduction in risk.

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The leading-order random term in the portfolio is given by 2 2 , which can also be written as 12 σ 2 S 2 ∂∂SV2 + 12 (φ2 − 1)σ 2 S 2 ∂∂SV2 . The second term in this equation corresponding to the hedging error is random. The hedging errors at each re-hedge can add up giving the total hedging error. The latter has a zero mean and a standard deviation 1 of O(δt 2 ). 1 2 2 2 ∂2V σ φ S ∂S 2 2

11.6.2. Pricing the option The determination of the right hedge ratio allows the computation of the option price. Since the expected return on the discretely hedged portfolio must be equal to the risk-less rate plus the information costs, we have: E[δΠ] =

1 (r + λV )δt + (r + λV )2 δt2 + · · · V 2 1 − (r + λS )δt + (r + λS )2 δt2 + · · · ∆S. 2

(11.68)

If we substitute Eqs. (11.64) and (11.65) in Eq. (11.68), one obtains: ∂ 2V 1 ∂V ∂V + σ2S 2 − (r + λV )V + δt(. . .) = 0. + (r + λS )S ∂t 2 ∂S 2 ∂S

(11.69)

When information costs are zero, the first term reduces to the Black– Scholes equation. The second term is a correction to allow for the imperfect hedge. Solving Eqs. (11.67) and (11.69) iteratively as in Willmott (1998), the option price must satisfy. 1 ∂V ∂2V ∂V + σ2 S 2 − (r + λV )V + (r + λS )S 2 ∂t 2 ∂S ∂S ∂ 2V 1 + δt(µ − r)(r − µ − σ2 )S 2 =0 2 ∂S 2

(11.70)

and the “better” delta is given by: ∆=

1 ∂ 2V ∂V + δt(µ − r + λS + σ2 )S . ∂S 2 ∂S 2

This equation shows the growth rate of the asset and information costs on the option and its underlying asset. The second derivative terms in

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Eq. (11.70) correspond to a constant times the squared value of S. Hence, the corrected option price needs the adjustment of the volatility and using the value σ ∗ with: δt σ ∗ = σ 1 + 2 (µ − r)(r − µ − σ2 ) . 2σ 11.6.3. The real distribution of returns and the hedging error Several studies have shown that the empirical distribution of the underlying assets have a higher peak and fatter tails than the normal distribution. Hence, how this distribution affects the hedging arguments and the hedging errors? Consider the following return for the underlying asset at time δt δS S = ψ, where the distribution of the random variable ψ is determined empirically. Now, following the analysis in Willmott (1998), we study the change in the value of the hedged portfolio under the assumption that the option component obeys the Black–Scholes equation with an implied volatility of σi . The random return in excess of the risk-free rate for the hedged portfolio is: ∂V (rδt − ψ) δΠ − rΠδt = S ∆ − ∂S 1 ∂ 2V 2 + S2 (ψ − σi2 δt) + · · · (11.71) 2 ∂S 2 If a delta hedge is implemented as in Black–Scholes, this gives: δΠ − rΠδt =

1 2 ∂ 2V 2 S (ψ − σi2 δt). 2 ∂S 2

In an economy with information costs, a factor reflecting these costs must be added to the interest rate r. Summary This chapter contains the basic and general material for the dynamics of financial assets and derivative asset prices in a continuous time framework. The presentation is made as simple as possible. The aim is to allow nonfamiliar readers with these concepts to follow without difficulties the basic

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methods in continuous time finance. Each concept is proposed in two forms: an intuitive version and a more rigorous mathematical version. The Wiener and Ito processes are used to model the dynamics of asset prices. Ito’s lemma is proposed to differentiate a function of one and several stochastic variables. It is illustrated through several examples in different contexts. The Kolmogorov forward and backward equations are presented as well as the first passage time and the maximum (minimum) of a diffusion process. These tools allow the pricing of standard options and more complex derivative assets. The principal of arbitrage simply states that financial assets having identical characteristics must trade at the same price. If this principle is not respected, then selling the highpriced asset and buying the low-priced asset allows a risk-free profit. This principle is used to determine the fair price of a security or a derivative asset. The analysis by Omberg (1991) reveals that the hedging error is zero over each revision interval in the following two cases. The first case represents the limit of the classical binomial discrete option pricing model. The second case corresponds to a stochastic revision strategy, which succeeds with a certain probability of one-over-one revision interval. For other cases, the hedging error over one revision interval is of the same order as the risk-free return and cannot be eliminated. This chapter is necessary for the understanding of the basic techniques behind the theory of rational option pricing in a continuous time framework. However, it is not necessary for the use and applications of all the formulas presented in this book.

Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

How the dynamics of asset prices are modeled in the literature? What is a Wiener process or a Brownian motion? What is the martingale property? What is an Ito process? What is the log-normal property? Define the simple form and the generalized Ito’s formula. What do you understand from the replication argument? Why error terms are neglected? What are the main concepts in bond markets? What stipulates the general arbitrage principle? What are the specific features of diffusion processes?

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Appendix A: Introduction to Diffusion Processes The notion of a stochastic process can be introduced with respect to the notion of vector of variables. Using the notations: Ω: the space of all possible states ω; Ft : σ-algebra defined over Ω; a class of partitions of Ω. X(ω) is said to be a random variable when it is a measurable application from (Ω, Ft ) to R. A vector of random variables X(ω) = [X1 (ω), . . . , Xn (ω)] is a measurable application from (Ω, Ft ) into Rn . The notion of a vector of random variables is similar to that of n ordinary variables defined on the same probability space. A stochastic process is the extension of the notion of a vector of variables when the number of elements becomes infinite. It is a family of random variables, Xt (ω), t ∈ T when the index varies in a finite or an infinite group. When ω = ω0 , X(ω0 , t) is a function of t called a realization or a path of the process. When t = t0 , X(ω, t0 ) is a vector of variables. A stochastic process will be denoted by X(t). Definition: A continuous time stochastic process having its values in a space E with a tribe, Ft is a family of random variables, (Xt )t≥0 is defined on the probability space (Ω, Ft ; P ) taking its values in (E, Ft ). Definition: A stochastic process X(t) for which the changes in its values over successive intervals are random, independent, and homogeneous is said to have no “memory”. The process has no derivative and its path can be represented by a continuous curve. The Wiener-Levy process is the most regular process among the processes for which the changes are independant and homogeneous. Definition: A filtration (Ft )t≥0 is an increasing family of sub-tribes of Ft in the probability space (Ω, Ft , P ). An example of stochastic processes often used in continuous time finance when deriving models for the valuation of financial assets is the Brownian motion.

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Appendix B: The Conditional Expectation Consider a probability space (Ω, Ft ; P ) and let B be is a sub-tribe of F . The following theorem allows the definition of the conditional expectation. Theorem: For any integrable random variable X, there is a unique associated random variable Y such that for all element in B, E(X1(B) ) = E(Y 1(B) ) Y is known as the conditional expectation of X given B, or E(X/B). The conditional expectation operator obeys the following properties: If X is B-mesurable, then E(X | B) = X. E(E(X | B)) = X. For any random variable Z, measurable with respect to B, E(ZE(X | B)) = E(ZX). E(αX + µY | B) = αE(X | B) + µE(Y | B). If X > 0 then E(X | B) ≥ 0. If C is a sub-tribe of B, then E(E(X | B) | C) = E(X | C). If X is independant of B, then E(X | B) = E(X). Appendix C: Taylor Series If a function f has derivatives in the region (x, x+h), then the development of this function around x gives: 1 1 f (x + h) = f (x) + f (x)h + f (x)h2 + · · · + f n (x)hn . 2 n! If the function f and its derivatives up to order n exist in the same region, then using Taylor series, we have: 1 1 f (n−1) (x)h(n−1) f (x + h) = f (x) + f (x)h + f (x)h2 + · · · + 2 (n − 1)! +

1 n ∗ n f (x )h n!

where x∗ is in the region [x, x + h].

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A function of two variables x and y is represented in Taylor series expansions by: ∂f (x, y) h f (x + h, y + k) = f (x, y) + ∂x ∂f (x, y) 1 ∂ 2 f (x, y) + h2 k+ ∂y 2 ∂x2 2 1 ∂ 2 f (x, y) ∂ f (x, y) 2 + + k hk 2 ∂y 2 ∂x∂y n 1 ∂ 2 f (x, y) 1 1 ∂ 2 f (x, y) h k . +···+ n! 2 ∂x 2 ∂y Similar expressions can be derived for functions of three or more variables. Exercises Exercise: Find the values of u(w, t) and v(w, t) where dW (t) = udt+vdx(t) in the presence of the following functions for W (t). W (t) = X β (t) W (t) = 1 + tα + etX(t) W (t) = f (t)α X β (t) with αβ ∈ N ∗ . Solution: Ito’s lemma can be used for a function W (X(t), t): dW =

∂W 1 ∂ 2W ∂W dX + dt + dt. ∂X ∂t 2 ∂X 2

For the first function W (t) = X β (t), we have: dW = βX β−1 dX +

β−1 β(β − 1) β−2 β(β − 1) β−2 dt = βW β dX + X W β dt. 2 2

Hence, u(W, t) = βW

β−1 β

;

v(W, t) =

β(β − 1) β−2 W β . 2

For the second function W (t) = 1 + tα + etX(t) , we have: 1 dW = tetX(t)dX + αtα−1 + X(t)etX(t) dt + t2 etX(t) dt 2

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or etX(t) = W (t) − 1 − tα hence: dW = t(W (t) − 1 − tα )dX + (αtα−1 + X(t)(W (t) − 1 − tα ))dt 1 + t2 (W (t) − 1 − tα )dt 2 and finally, we have u(W, t) = t(W (t) − 1 − tα ), 1 v(W, t) = αtα−1 + X(t) + t2 (W (t) − 1 − tα ) . 2 For the third function, W (t) = f (t)β X α (t), we have ∂f 1α(α − 1) β α−2 dt dt + f X ∂t 2 α−2 W (t) W (t) ∂f α(α − 1) W (t) dW = α dX + β dt + dt. X(t) f (t) ∂t 2 f (t) dW = αf β X α−1 dX + βX α f β−1

Hence, u(W, t) = α

W (t) = αf β X α−1 X(t)

and α(α − 1) W (t) ∂f + v(W, t) = β f (t) ∂t 2

W (t) f (t)

α−2

with X α (t) =

W (t) f β (t)

Exercise: Consider an underlying asset S whose price follows a log-normal random walk. 1. Apply Ito’s lemma for the following functions f (S) and g(S): f (S) = AeS + B, where A and B are constants.

g(S) = eS

n

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Solution: Re-call that Ito’s lemma for a function f (S) is given by: ∂f ∂2f ∂f 1 df = σS dX + µS + σ2 S 2 2 dt ∂S ∂S 2 ∂S or ∂ 2f = AeS ∂S 2 1 2 2 S S df = Ae σSdX + Ae µS + σ S dt. 2 ∂f = AeS , ∂S

The application of the lemma to g(S) gives: n ∂g = nS n−1 eS ∂S n ∂ 2g = ((nS n−1 )2 + n(n − 1)S n−2 )eS ∂S 2

hence, dg = nS n−1 eS σSdX 1 2 2 n−1 S n n−1 2 n−2 S n e µ + σ S ((nS ) + n(n − 1)S )e + SnS dt 2 n

then n Sn

dg = nS e

1 2 n Sn n 2 n Sn dt σ σdX + nS e µ + ((nS ) + n(n − 1)S )e 2

or (nS n )2 + n(n − 1)S n = nS n (nS n + n − 1) then n Sn

dg = nS e

1 2 n σdX + µ + σ (nS + n − 1) dt. 2

References Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September), 645–664. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3) (August), 617–635.

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Bellalah, M and J-L Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bellalah, M, JL Prigent and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Risk Premiums (1973). Journal of Political Economy, 1387–1404. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7, 229–263. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, RC (1987). A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42(3), 483–510. Omberg, E (1991). On the theory of perfect hedging. Advances in Futures and Options Research, 5, 1–29. Rendleman, RJ and BJ Barter (1979). The pricing of options on debts securities. Journal of Financial and Quantitative Analysis, 15 (March) 11–24. Rogers, C (1997). One for all. Risk, 10(3) (March) 56–59. Willmott, P (1998). Derivatives. New York: John Wiley and Sons.

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Chapter 12 RISK MANAGEMENT: APPLICATIONS TO THE PRICING OF ASSETS AND DERIVATIVES IN COMPLETE MARKETS

Chapter Outline This chapter is organized as follows: 1. In Section 12.1, we give some definitions and characterize complete markets. 2. In Section 12.2, equity options are priced with respect to the partial differential equation method and the martingale approach. Both methods are applied to the valuation of equity options. 3. In Section 12.3, bond options and interest rate options are studied and valued. Several models are proposed for the dynamics of interest rates. 4. In Section 12.4, the main techniques are proposed for the pricing of assets in complete markets using the change of numeraire and time. 5. In Section 12.5, the main results in Section 12.4 are extended to account for the effects of information uncertainty. 6. Appendix A presents the change in probability and the Girsanov theorem. 7. Appendix B gives in great detail the resolution of the partial differential equation under the appropriate condition for a European call option. 8. Appendix C gives two approximations of the cumulative normal distribution function. 9. Appendix D states Leibniz’s rule for integral differentiation.

535

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Introduction Modern finance allows to quantify risk and its remuneration. The pioneering papers of Arrow (1953) and Merton (1973) assumed that markets are complete. Prices of contingent claims are presented in the form of solutions to partial differential equations (PDE). Prices are also represented as conditional expectation of functionals of stochastic processes. These representations provide solutions to the main pricing problems in modern finance. The pricing of derivative assets is usually based upon two methods which use the same basic arguments. The first method involves the resolution of a PDE under the appropriate boundary conditions corresponding to the derivative asset’s payoffs. This is often referred to as the Black– Scholes method. The second method uses the martingale method, initiated by Harrison and Kreps (1979) and Harrison and Pliska (1981), where the current price of any financial asset is given by its discounted future payoffs under the appropriate probability measure. The probability is often referred to as the risk-neutral probability. Both methods are illustrated in detail for the pricing of European call options. Black and Scholes (1973) and Merton (1973) provide the PDE for derivatives and its solutions. The probabilistic method known also as the martingale method, initiated by Cox and Ross (1976), Harrison and Kreps (1979), and Harrison and Pliska (1981) allows to compute the prices of derivatives under a risk-neutral probability, under which the discounted price of any financial claim is a martingale. It must be clear that both methods lead to the same results. Application of the Feynman–Kac formula allows a switch from price as a solution of a PDE to a probability representation. Unfortunately, for most problems in financial economics, and in particular for the pricing of American options, there is often no closed form solutions and option prices must be approximated. Therefore, financial economists often resort to numerical techniques.

12.1. Characterization of Complete Markets In financial markets, there are two classes of financial assets: securities and derivative assets. Securities correspond fundamentaly to common stocks and bonds. Derivative assets are contingent claims characterized by their intermediate and final payoffs. There are several definitions of a complete market. The idea of complete markets is proposed first by Arrow (1953) and

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Applications to the Pricing of Assets and Derivatives in Complete Markets

537

Debreu (1954). They defined a complete market with respect to state securities or state contingent claims. A state security or an Arrow-Debreu security is a security which pays off one dollar if and only if a given state of the nature occurs. A state of nature or a possible state in the economy is said to be an insurable state when it is possible to construct a portfolio of assets which has a non-zero return in that state. In this economy, the price of each traded asset at the begining of the period is ps . For an economy where every state is insurable, a price vector can be completely determined with unique state prices. This implies the absence of arbitrage opportunities. A contingent claim is attainable, if there is a strategy that gives at the derivative asset’s maturity date the same value as the contingent claim terminal payoff. Hence, a complete market can be defined as a market in which all the contingent claims are attainable, i.e., all the contingent claims can be reached or obtained when implementing a replication or a duplication strategy. A complete market can be defined with respect to the concept of a viable financial market. A viable financial market is a market where there is no profitable risk-less arbitrage. The absence of risk-less arbitrage or arbitrage opportunities means that a strategy which is implemented at the initial time with a zero-cost must have a nil final or terminal value. It is important to note that there is a relationship between the notion of arbitrage and the martingale property of security prices. The martingale property for security prices means simply that the best estimation of the future price is the last information. Hence, when historical data of security prices are used to predict the future price, only the most recent information matters, i.e., the last price. This concept defines also that of an efficient market. In mathematical terms, a financial market is viable if and only if there is a probability P ∗ which is equivalent to a probability P , under which the discounted asset prices have the martingale property. This is done under the theorems of the change in probability and in particular, the Girsanov theorem. (See Appendix A for more detail). A viable financial market is complete, if and only if there is a probability P ∗ equivalent to the probability P under which the martingale property is satisfied by security prices. It can be shown that this probability P ∗ is unique.

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If one considers a contingent claim specified by its final payoff h, which is of the form h = (ST − K)+ for a European call and h = (K − ST )+ for a European put, then an attainable strategy Φ simulates or duplicates the option price when its payoff at the maturity date T is equal to h, or VT (Φ) = h. The sequence of the random derivative asset prices between the initial time 0 and the option’s maturity date T is a martingale under the unique probability P ∗ . Hence, we have: V0 (Φ) = E ∗ [VT (Φ)] and V0 (Φ) = E ∗ [h | ST ]. This gives the general result in complete markets: Vt (Φ) = St E ∗ [h | ST | Ft ] for t : 0, . . . , T. If at the initial time a derivative asset is sold at its expected price, E ∗ [h | ST ], then an investor following a replication strategy can obtain the exact payoff h at time T . He is said to follow a full hedging strategy. 12.2. Pricing Derivative Assets: The Case of Stock Options Since each financial asset is specified by its intermediate and terminal payoffs, option pricing consists in finding the fair price at the initial time when the derivative asset is bought or sold. There is a unique approach for the pricing of derivative assets. 12.2.1. The problem The value of each option is given by its expected terminal and intermediate payoffs discounted to the present. However, there are two methods for the pricing of options. The first was initiated by Black and Scholes (1973) and Merton (1973). The second is the martingale approach due to Harrison and Kreps (1979) and Harrison and Pliska (1981). The first method, known as the Black–Scholes method, is based on the resolution of the following PDE: ∂c ∂c 1 2 2 ∂2c σ S 2 + rS + − rc = 0 2 ∂ S ∂S ∂t under the appropriate boundary conditions. The second is based on the use of martingale techniques. In either the first or the second approach, the boundary conditions are the same. These conditions refer to the appropriate payoff conditions corresponding to the value of European and American contingent claims. For a European call, the final payoff is given by c(S, T ) = max[0, ST − K] where K is the option strike price. For a European put, the

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final payoff is given by p(S, T ) = max[0, K − ST ] where K is the option strike price. For an American call, the additional following condition must be satisfied by the PDE C(S, t) ≥ max[0, St − K] where K is the option strike price. The difference from the condition for a European call is that the American call holder can exercise his option at each instant. This condition indicates that at each instant, the American call value must be at least equal to or greater than the intrinsic value, which corresponds to the value of a European call at maturity. For an American put, the additional following condition must be satisfied since the put holder can exercise his option at each instant P (S, t) ≥ max[0, K − ST ] where K is the option strike price. This condition shows that at each instant, the American put value must be at least equal to the intrinsic value. In the presence of dividend distributions, some other conditions must be imposed. All these conditions apply in the absence of dividends. When there are distridutions to the underlying asset, there is in general no explicit solutions to these problems and numerical methods are often used. First, we illustrate the PDE method for the valuation of European call options. Second, we illustrate the use of the martingale approach for the pricing of European calls. The reader can verify that the price of the call is the same under both methods. Third, since it is difficult to get closed form solutions for American options with and without distributions to the underlying asset, financial economists often use numerical methods. Therefore, we develop the principles of these techniques in the last section of this chapter. 12.2.2. The PDE method The standard analysis Consider the search for the solution of the following PDE: ∂c ∂c 1 ∂2c 2 2 σ S + rS + − rc = 0 2 ∂2S ∂S ∂t for a European call with the payoff c(S, T ) = max[0, ST − K] where K is the option strike price. Using the appropriate change of variables, it can be shown that the call price is: c = SN (d1 ) − Ke−r(T −t) N (d2 ) S S + r + 12 σ 2 (T − t) + r − 12 σ 2 (T − t) ln K ln K √ √ , d2 = d1 = σ T −t σ T −t

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where 1 N (d) = √ 2Π

“ ” 2 − x2

d

−∞

e

dx

A detailed resolution of this system is given in Appendix B. The payoff of a European put option is p(S, T ) = max[0, K −ST ]. Using the appropriate change of variables, it can be shown that the put price is:

d1 =

ln

S K

p = −SN (−d1 ) + Ke−r(T −t) N (−d2 ) S + r + 12 σ 2 (T − t) + r − 12 σ 2 (T − t) ln K √ √ , d2 = . σ T −t σ T −t

The Analysis in the presence of information costs Consider the search for the solution of the following PDE: 1 2

∂ 2c ∂ 2S

σ 2 S 2 + (r + λS )S

∂c ∂S

+

∂c ∂t

− (r + λc )c = 0

for a European call with the payoff c(S, T ) = max[0, ST − K] where K is the option strike price. Using the appropriate change of variables, it can be shown that the call price under information uncertainty is given by: c = Se−(λS −λc )(T −t) N (d1 ) − Ke−(r+λc )(T −t) N (d2 ) S + r + λS + 12 σ 2 (T − t) ln K √ d1 = , σ T −t S + r + λS − 12 σ 2 (T − t) ln K √ d2 = σ T −t where: 1 N (d) = √ 2Π

2 x dx exp − 2 −∞

d

The payoff of a European put option is p(S, T ) = max[0, K − ST ]. Using the appropriate change of variables, it can be shown that the put price is

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given by: p = Se−(λS −λc )(T −t) N (−d1 ) + Ke−(r+λc )(T −t) N (−d2 ) This model appears in Bellalah (2001). 12.2.3. The martingale method The Black and Scholes (1973) pricing of options requires first the knowledge of the probability under which the asset price St is a martingale. The standard analysis In the context of the Black–Scholes model, there is an equivalent probability to P under which the discounted expected stock price, St∗ = e−rt St is a martingale. In fact, if one uses the stochastic differential equation for the stock price, we have: dSt∗ = −re−rt St dt + e−rt dSt or dSt∗ = St∗ [(µ − r)dt + σdWt ].

If we put the change in variables Wt∗ = Wt + µ−r t then dSt∗ = σ (µ−r) ∗ ∗ St σdWt . Using the Girsanov theorem, Θt = σ , there is a probability P ∗ equivalent to P under which (Wt∗ )0≤t≤T is a standard Brownian motion. Hence, under P ∗ , we deduce from Eq. (12.12) that St∗ is a martingale and ∗

1

St∗ = S0∗ eσWt − 2 σ

2

t

The option price in the Black and Scholes (1973) economy can be computed using its discounted expected terminal value under the appropriate probability P ∗ as: ct = E ∗ [e−r(T −t) h | Ft ] with h(ST − K)+ = f (ST ). The option price at time t can be expressed as a function of time and the underlying asset price. In fact, ct = E ∗ [e−r(T −t) h | Ft ] or ∗ ∗ 1 2 ct = E ∗ e−r(T −t) f St e−r(T −t) eσ(WT −Wt )− 2 σ (T −t) |Ft .

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Derivatives, Risk Management and Value ∗

1

2

Since for all t, St = S0 eσWt − 2 σ t , and at the option’s maturity date ∗ 1 2 ST = S0 eσWT − 2 σ T the ratio of the stock price between two dates is: ∗ ∗ 1 2 ST = e[σ(WT −Wt )− 2 σ (T −t)] St

Since the random stock price St is Ft measurable and (WT∗ − Wt∗ ) is independent of Ft under P ∗ , it can be shown that ct = H(t, St ) with: ∗ ∗ σ2 H(t, S) = E ∗ e−r(T −t) f Ser(T −t) eσ(WT −Wt )− 2 (T −t)

(12.1)

Underthe probability P ∗ , the quantity√(WT∗ − Wt∗ ) follows a Gaussian law, N (0, (T − t). When (WT∗ − Wt∗ ) = Y T − t and Y follows a N (0, 1), then ∞ √ 1 −y2 σ2 H(t, S) = e−r(T −t) f Se(r− 2 )(T −t)+σy T −t √ e 2 dy 2Π −∞ When Y follows N (0, 1), under P ∗ , then h(y) −y2 Ep∗ [h(Y )] = √ e 2 dy, 2Π since √ H(t, S) = e−r(T −t) Ep∗ G( T − t )Y where √ √ σ2 G( T − t)Y = f Seσ( T −t)Y +(r− 2 )(T −t) 1

2

and Y follows N (0, 1) with density √12Π e− 2 y under P ∗ . If one replaces the call’s pay off function, then using Eq. (12.1) gives: + ∗ ∗ σ2 H(t, S) = E ∗ e−r(T −t) Se(r− 2 )(T −t)+σ(WT −Wt ) − K or √ + σ2 H(t, S) = E ∗ Se(σY τ − 2 τ ) − Ke−rτ

where τ = T − t.

(12.2)

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Equation (12.2) is equivalent to Eq. (12.1) where f (z) is replaced by (zT − K)+ . Now, using the notation, S √ + r + 12 σ 2 τ ln K √ , d2 = d1 − σ τ d1 = σ τ we consider again the equation + ∗ ∗ σ2 H(t, S) = E Se(r− 2 )(T −t)+σ(WT −Wt ) − K Using the following lemma E[(Z − K)+ ] = E[[(Z − K)]IZ≥K ] with IZ≥K =

1, if Z ≥ K 0 else

the condition (Z ≥ K) is equivalent to: Se(σY

√ τ − 12 σ2 τ )

≥K

or σY

√ 1 τ − σ2 τ ≥ ln 2

Hence Y ≥

ln

K S

K S

.

− r − 12 σ 2 τ √ σ τ

or Y ≥ −d2

or Y + d2 ≥ 0.

Using this remark, H(t, S) can be written as: √ σ2 H(t, S) = E Se σY τ − 2 τ − Ke−rτ IY +d2 ≥0 or

H(t, S) =

∞

−d2

Se(σY

√

2

τ − σ2 τ )

1 1 2 − Ke−rt √ e(− 2 y ) dy 2Π

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which is equivalent to: H(t, S) =

d2

(−σY √τ − σ2 τ ) 1 1 2 2 − Ke−rt √ e(− 2 y ) dy Se 2Π −∞

This integral can be written as the difference between two integrals H(t, S) =

d2

−∞

−

Se(−σY

√

2

τ − σ2 τ )

1 1 2 √ e(− 2 y ) dy 2Π

d2

1 2 1 Ke−rτ √ e(− 2 y ) dy 2Π −∞

The second integral is equal to −Ke−rτ N (d2 ). Using the change in variable √ √ z = y + σ τ or y = z − σ τ , the first integral can be written as:

√ d2 +σ τ

−∞

Se(−σ(z−σ

√

√ 2 τ ) τ − σ2 τ )

√ 2 1 1 √ e(− 2 (z−σ τ ) ) dz 2Π

Simple computation of this integral gives exactly SN (d1 ). The sum of both integrals corresponds to the Black–Scholes formula: H(t, S) = SN (d1 ) − Ke−rτ N (d2 )

(12.3)

with d1 =

ln

S K

+ r + 12 σ 2 (T − t) √ , σ T −t

d2 =

ln

S K

+ r − 12 σ 2 (T − t) √ . σ T −t

The analysis in the presence of information costs It is possible to use the martingale method for the pricing of derivative claims in a Black–Scholes context in the presence of information uncertainty. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah et al. (2001a,b), Bellalah and Prigent (2001) and Bellalah and Selmi (2001) etc. In this context, we can show that there is an equivalent probability to P under which the discounted expected value of the underlying asset given by St∗ = e−(r+λS )t St is a martingale. In this expression, the term λi refers to information costs.

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Consider the stochastic differential equation for the underlying asset: dSt∗ = −(r + λS )e−(r+λS )t St dt + e−(r+λS )t dSt or the following dynamics: dSt∗ = St∗ [(µ − r − λS )dt + σdWt ] If the following change in variables is used

(µ − r − λS ) t Wt∗ = Wt + σ then dSt∗ = St∗ σdWt∗ . Using the Girsanov theorem, Θt =

(µ − r − λS ) , σ

there is a probability P ∗ equivalent to P under which Wt∗ is a standard Brownian motion. Hence, under P ∗ , we deduce from this last equation that St∗ is a martingale and:

1 2 ∗ ∗ ∗ St = S0 exp σWt − σ t . 2 The option price in the Black–Scholes economy can be computed using its discounted expected terminal value under the appropriate probability P ∗ as ct = E ∗ [e−(r+λc )(T −t) h/Ft ] with h(ST − K)+ = f (ST ) The option price at time t for a maturity date is given by:

ct = E ∗ e−(r+λc )(T −t) f St e−(r+λS )(T −t) 1 2 ∗ ∗ × exp σ(WT − Wt ) − σ (T − t) Ft 2 ∗ 1 2 Since for all t, we have: St = S0 e[σWt − 2 σ t] and the value of the underlying asset at the option’s maturity date is ST = S0 exp σWT∗ − 12 σ 2 T , the ratio

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of the stock price between two dates is given by:

1 2 ST ∗ ∗ = exp σ(WT − Wt ) − σ (T − t) St 2 It can be shown that the option value is given by ct = H(t, St ) with: ∗ ∗ σ2 (12.4) H(t, S) = E ∗ e−(r+λc )(T −t) f Se(r+λS )(T −t) eσ(WT −Wt )− 2 (T −t) Under√the probability P ∗ , the quantity WT∗ −√Wt∗ follows a Gaussian law, N (0, t). When the difference WT∗ − Wt∗ = Y T − t and Y follows an N (0, 1), then ∞ √ 1 −y2 σ2 H(t, S) = e−(r+λc )(T −t) f Se(r+λS − 2 )(T −t)+σy T −t √ e 2 dy 2Π −∞ When Y follows N (0, 1), under P ∗ , then h(Y ) −y2 Ep∗ [h(Y )] = √ e 2 dy, 2Π since √ H(t, S) = e−(r+λc )(T −t) Ep∗ [G( T − t)Y ] where

√ √ σ2 (T − t) G( T − t)Y = f S exp σ( T − t)Y + (r + λS ) − 2 and Y follows N (0, 1) with density

1 2 √1 e− 2 y 2Π

under P ∗ .

12.3. Pricing Derivative Assets: The Case of Bond Options and Interest Rate Options The value of a zero-coupon bond at maturity is B(T, T ) = 1. 12.3.1. Arbitrage-free family of bond prices The concept of arbitrage is central to the valuation of financial assets. Musiela (1997) gives the following definition for the arbitrage family of bond prices.

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Definition: Consider a process r defined on a filtered probability space (Ω, F, P ). An arbitrage-free family of bond prices relative to the interest rate r when B(T, T ) = 1 for every T in [0, T ∗ ], and when there is a probability P ∗ on (Ω, F) equivalent to P such that the relative bond price Z ∗ (t, T ) = B(t, T )/Bt ,

∀ t ∈ [0, T ]

is a martingale under P ∗ . The probability P ∗ is a martingale measure for the family B(t, T ). Hence, for any P ∗ of an arbitrage-free family of bond prices, we have: RT B(t, T ) = EP ∗ e− t ru du | Ft , ∀ t ∈ [0, T ] (12.5) The reader can refer to Appendix E for more details. 12.3.2. Time-homogeneous models The Vasicek (1977) model The diffusion process proposed in the Vasicek (1977) model is a meanreverting version of the Ornstein–Uhlenbeck process. In this model, the short-term interest rate follows the following dynamics: drt = (a − brt )dt + σdWt∗ where a, b, and σ are positive constants. This Gaussian model allows for the possibility of negative interest rates. Consider a security paying a continuous dividend at a rate (h, rt , t) whose payoff is a function of the interest rate r, FT = f (rT ) at time T . The price process Ft of this security has the representation Ft = v(rt , t) where v is solution to the following PDE: 1 ∂ 2v ∂v ∂v (r, t) + σ 2 2 (r, t) + (a − br) (r, t) − rv(r, t) + h(r, t) = 0, ∂t 2 ∂r ∂r (12.6) The terminal boundary condition is: v(r, T ) = f (r)

(12.7)

When h = 0 and f (r) = 1, the price of a zero-coupon bond is: B(t, T ) = v(rt , t, T ) = em(t,T )−n(t,T )rt ,

(12.8)

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where n(t, T ) =

1 (1 − e−b(T −t) ) b

and σ2 m(t, T ) = 2

T t

2

n (u, T )du − a

t

T

n(u, T )du

It is possible to verify this result using Eq. (12.8) with m(T, T ) = n(T, T ) = 0. Using the PDE and seperating the terms in r gives the following system: nt (t, T ) = bn(t, T ) − 1,

n(T, T ) = 0,

1 mt (t, T ) = an(t, T ) − σ2 n2 (t, T ), 2

m(T, T ) = 0

(12.9)

which leads to the above expressions. We can check that we have: dB(t, T ) = B(t, T )(rt dt + σn(t, T )dWt∗ ) where the bond price volatility equals b(t, T ) = σn(t, T ). Using the expression (12.9) for the bond price, the bond’s yield is: Y (t, T ) =

n(t, T )rt − m(t, T ) , T −t

Since this yield is an affine function and therefore the categories of similar models are known as affine models of the term structure. Jamshidian (for more details, refer to Bellalah et al., 1998) gives a closed-form solution for a European call on a bond (with and without coupons) with a H-maturity in the context of this model using: Ct = B(t, T )EQ (ξη − K)+ | Ft (12.10) where η =

B(t,H) ; B(t,T )

K: the strike price; Q: a probability measure equivalent to P ∗ and ξ: a random variable with a variance: 2 (t, T ) vH

= t

T

2

|b(t, T ) − b(t, H)| du

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or 2 vH (t, T ) =

σ2 (1 − e−2b(T −t) )(1 − e−b(H−t) )2 2b3

The formula for the call on a bond is given by: Ct = B(t, H)N (d1 (t, T )) − KB(t, T )N (d2 (t, T )) with d1,2 (t, T ) =

2 ln (B(t, H)/B(t, T )) − ln K ± 12 vH (t, T ) vH (t, T )

for every t less than T and H. This formula does not show the coefficient a and accounts for the bond volatility. The Cox, Ingersoll and Ross (CIR) (1985) model In their general equilibrium approach, CIR (1985) use the familiar square root process for the dynamics of interest rates: √ drt = (a − brt )dt + σ rt dWt∗ where a, b, and σ are positive constants. This process does not allow for negative interest rates because of the square root in the diffusion process. The price process of a standard European option Ft = v(rt , t) must satisfy the following partial differential equation: ∂v ∂2v ∂v 1 (r, t) + σ2 r 2 (r, t) + (a − br) (r, t) − rv(r, t) + h(r, t) = 0, ∂t 2 ∂r ∂r under the boundary condition: v(r, T ) = f (r, T ). CIR (1985) provide the following closed-form solutions for the price of a zero-coupon bond. Γebr/2 2a m(t, T ) = 2 ln σ Γ cosh Γr + 12 b sinh Γr and n(t, T ) =

sinh Γr , Γ cosh Γr + 12 b sinh Γr

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with τ = T − t, 2Γ =

(b2 + 2σ 2 ).

They give also solutions in closed forms for options on zero-coupon and coupon-bearing bonds. Longstaff (1990) proposes closed-form formulas for European options on yields in the context of the CIR (1985) model. The yield on a zero-coupon bond is a linear function of the short-term rate in the CIR model since: Y (t, t + τ ) = Y˜ (rt , τ ) = m(τ ˜ )+n ˜ (τ )rt , The yield at time t for a zero-coupon bond with a maturity τ for a current level rt = r is Y˜ (t, τ ). The payoff of a yield European call is given by: CTY = (Y˜ (rT , τ ) − K)+ , where K is the fixed level of the yield. The Longstaff model Longstaff (1989) uses the following dynamics for the short-term rate: √ √ drt = a(b − c rt )dt + σ rt dWt∗ (12.11) referred to as a double square root process. In this model, the price of a zero-coupon bond is given by: B(t, T ) = v(rt , t, T ) = em(t,T )−n(t,T )rt −p(t,T )

√ rt

where m, n, and p are known functions. In this model, the yield of the bond is a non-linear function of the short-term rate. 12.3.3. Time-inhomogeneous models It is important to note that the Vasicek (1977) and the CIR (1985) models are special cases of the following process: drt = a(b − crt )dt + σrtβ dWt∗ where β is a constant between zero and one. Hull and White use the following dynamics for the short-term interest rate: drt = (a(t) − b(t)rt )dt + σ(t)rtβ dWt∗ where β is a positive constant, W ∗ a Brownian motion and a, b, and σ are locally bounded functions. This model appears as a generalization of

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the Vasicek (1977) and the CIR (1985) model. When β = 0, this gives a generalized Vasicek model: drt = (a(t) − b(t)rt )dt + σ(t)dWt∗ . When β = 0.5, this gives a generalized CIR model: √ drt = (a(t) − b(t)rt )dt + σ(t) rt dWt∗ 12.4. Asset Pricing in Complete Markets: Changing Numeraire and Time This section introduces two mathematical instruments: the change of numeraire and the change of time. These tools are very effective in solving derivative asset pricing problems.

12.4.1. Assumptions and the valuation context Consider an economy in which transactions take place instantaneously without transactions costs and taxes. The interest rate follows a Gaussian process. The short-term risk-less interest rate r(t) under the risk-neutral probability Q is governed by the following stochactic differential equation dr(t) = a(t)[b(t) − r(t)]dt + σ(t)dW1 (t) where the different parameters a(t), b(t) and σ(t) are deterministic functions. Under the risk-neutral probability Q, the dynamics of return on zerocoupon bonds in the absence of default are given by: dP (t, T ) = r(t)dt − σp (t, T )dW1 (t) P (t, T ) where P (t, T ) is the price of a zero-coupon bond at time t for a maturity date T . The volatility in the process describing the dynamics of zero-coupon bond prices is given by: σp (t, T ) = σ(t)

T t

exp −

t

u

a(s)ds du

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Consider the value at time t of a portfolio which corresponds to an investment of one dollar at time t = 0 at the rate r(t): B(t) = e

Rt 0

r(u)du

This portfolio corresponds to a capitalization factor. Consider the following dynamics for a risky asset St : dSt = r(t)dt + σS ρdW1 (t) + 1 − ρ2 dW2 (t) St under the risk-neutral probability Q. The coefficient ρ ∈ [−1, 1] introduces a correlation between the risky asset and the interest rate. The absence of arbitrage opportunities gives the price of any contingent claim under the risk-neutral probability Q as

h(T ) V0 = EQ , B(T ) where EQ corresponds to the conditional expectation under the probability Q. 12.4.2. Valuation of derivatives in a standard Black–Scholes–Merton economy Consider the dynamics of the option’s underlying stock dSt = rdt + σS dW2 (t) St Using arbitrage arguments, Black and Scholes derived the following PDE: ∂C ∂C ∂ 2C 1 + σS2 S 2 2 + rS − rC = 0 ∂t 2 ∂S ∂S

(12.12)

Consider a replicating portfolio V consisting of a long position in the ∂C option and a short position portfolio ∂C in ∂S units of the underlying asset. The results from investing ∂S in the risky asset and an amount C − S ∂C ∂S in the risk-free asset. Its dynamics are given by: ∂C ∂C dV = C − S rdt + dS (12.13) ∂S ∂S

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Using Ito’s lemma gives: ∂C 1 2 2 ∂2C ∂C dt + + σS S dS dV = 2 ∂t 2 ∂S ∂S

553

(12.14)

The solution for a European call is: C(S0 , T, σS , r) = S0 N (d1 ) − Ke−rt N (d2 ) σ2 ln SK0 + r + 2S T √ √ d1 = = d2 + σS T σS T Now, consider the following change of variables in Eq. (12.12) ft = St er(T −t) ,

Cf = Cer(T −t)

The two changes of variables correspond to a change of numeraire giving respectively the value of a forward contract as a function of its underlying asset and the cost of carry and the value of a forward call as a function of a spot option. Consider also the following time change in Eq. (12.12) τ = σS2 t and ∗ τ = σS2 T . The change of underlying asset f in Eq. (12.12) allows to write this equation as: ∂ 2C 1 ∂C + σS2 f 2 2 − rC = 0 ∂t 2 ∂f

(12.15)

Equation (12.15) corresponds to the Black (1976) equation providing the value of an option on a forward contract f . A simple comparison between has disappeared. This Eqs. (12.15) and (12.12) reveals that the term rS ∂C ∂S is because the forward positions cost nothing to initiate and to adjust. Therefore, the dynamics of the replicating portfolio can be written as: dV = rCdt +

∂C df ∂f

Using Ito’s lemma dV =

1 2 2 ∂ 2C ∂C + σA f ∂t 2 ∂f 2

gives the dynamics of C in Eq. (12.15).

dt +

∂C df ∂f

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When Cf = Cer(T −t) , this change of variables gives: ∂Cf 1 2 2 ∂ 2 Cf f =0 + σA ∂t 2 ∂f 2 In the same way, no initial investment is required for the call on a forward contract, since it represents a forward option. In this case, the ∂C dynamics of the replicating portfolio are given by dVf = ∂ff df and Ito’s lemma gives: ∂Cf 1 2 2 ∂ 2 Cf ∂Cf dt + + σS f df dVf = 2 ∂t 2 ∂f ∂f It is possible to use a change of variables for τ and τ ∗ to simplify the equation to the following form: 1 ∂ 2 Cf ∂Cf + f2 =0 ∂τ 2 ∂f 2

(12.16)

Equation (12.16) is useful for the valuation of a forward option on a forward contract maturing in τ ∗ for which the volatility is equal to one. In this case, the solution to Eq. (12.16) for a forward call on a forward contract is given by: Cf (f0 , τ ∗ ) = C(f0 , τ ∗ , 1, 0) = f0 N (d1 ) − KN (d2 ) since

ln

f0 K

√

+ τ∗

τ∗ 2

ln =

S0 K

+ r+ √ σS T

2 σS 2

T

A simple comparison of Eqs. (12.16) and (12.12) reveals that the option is priced as if the interest rate were zero. Besides, the volatility is replaced by 1 and the maturity T by τ ∗ . This result comes from the changes in numeraires but, in practice, the interest rate enters the cost of carry and the time change comprises the volatility of the forward contract. 12.4.3. Changing numeraire and time: The martingale approach and the PDE approach The change of numeraire and the change of time appear in several papers including Merton (1973), Jamshidian (for details, refer to Bellalah et al., 1998), El Karoui and Geman (1993) etc. In a martingale approach, the change of numeraire and time is useful from a computational point of view.

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We introduce these concepts and make the analogy with partial differential equations approach.

Change of numeraire A numeraire corresponds in general to a process X(t) for which there exists a probability measure QX which is defined by its Radon–Nikodym derivative with respect to Q as: dQX X(T ) B(0) = dQ X(0) B(T ) in such a way that the relative price of the asset compared with the numeraire X is a QX martingale. In the above expression B(t) corresponds to the capitalization factor and Q is the risk-neutral probability. In fact, as we know, in the martingale approach, the price of any derivative asset with a payoff h(T ) can be computed using the absence of arbitrage opportunities argument. This allows the computation of the price of any claim under the risk-neutral probability Q as contingent h(T ) , where EQ corresponds to the conditional expectation V0 = EQ B(T ) under the probability Q. This conditional expectation can also be written as:

EQ

h(T ) h(T ) = X(0)EQX B(T ) X(T )

As an example, consider the price of a risk-less zero-coupon bond P (t, T ) as a numeraire in an economy with stochastic interest rates. Using this numeraire, it is possible to define for its process a new probability QP by giving its Radon–Nikodym derivative with respect to Q as: P (T, T ) B(0) dQP = dQ P (0, T ) B(T ) 1 1 = P (0,T since by definition, the prices which is also equal to dQ dQ ) B(T ) P (T, T ) and B(0) are equal to 1. Using the two previous equations, it is evident that:

h(T ) EQ = P (0, T )EQX [h(T )] B(T ) P

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The effect of this change of numeraire can be appreciated using the Black–Scholes context for the pricing of a European call on an asset S in the presence of a stochastic interest rate. We use the following dynamics for the underlying asset: dSt = rdt + σS dW2 (t) St and for the short-term interest rate: dr(t) = a(t)[b(t) − r(t)]dt + σ(t)dW1 (t) In the presence of stochastic interest rates, the option price can be computed as:

CS = EQ

ST − K B(T )

IST ≥K

The indicator function IE for E is the real-valued random variable defined by IE (ω) = 1 for all ω ∈ E and zero otherwise. Equivalently, under the probability QP , the value of the call can be written as:

K ST − I ST ≥K CS = P (0, T )EQP (12.17) P (T ,T ) P (T, T ) P (T, T ) where the zero-coupon bond value is by definition equal to one at maturity. Equation (12.17) allows us to avoid the stochastic character of interest rates. A comparison can be done with respect to the PDE approach. In the presence of stochastic interest rates, the call price is a function of the underlying asset S, the interest rate r and time t. It is also possible to consider the option price as a function of the prices of the underlying asset, the price of zero-coupon bonds, P and time. In this case, it is possible to show that the European call price must satisfy the following partial differential equation: 1 1 ∂ 2 CS ∂ 2 CS ∂ 2 CS ∂CS + σS2 S 2 + σp2 P 2 − ρσS σP SP 2 2 ∂t 2 ∂S 2 ∂P ∂S∂P ∂CS ∂CS + rS + rP − rCS = 0 ∂S ∂P

(12.18)

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This equation must be solved under the terminal condition CS = max[S − K, 0]. Since the call price CA is first-degree homogeneous in the underlying asset and the discount bond, or CS = S

∂CS ∂CS +P ∂S ∂P

Equation (12.18) can be written, after simplification as: ∂CS ∂ 2 CS 1 1 2 2 ∂ 2 CS ∂ 2 CS + σS2 S 2 σ =0 + P − ρσ σ SP S P ∂t 2 ∂S 2 2 p ∂P 2 ∂S∂P

(12.19)

Now, a change of variables can be made using the bond price P (t, T ) as a new numeraire. We denote by Ft the price of a forward contract on St and by CF the the underlying asset S for delivery at time T , Ft = P (t,T ) price of a European call on the forward contract CF = CPS . Using this new change of variables, Eq. (12.19) can be written as: 1 ∂ 2 CF ∂CF + σF (t, T )2 F 2 =0 ∂t 2 ∂F 2

(12.20)

where the instantaneous variance of the forward contract in this stochastic interest rate economy is σF (t, T )2 = σS2 + σP (t, T )2 + 2ρσS σP (t, T ) When the interest rate is constant, the correlation coefficient is zero, and the bond’s price volatility is also zero. In this case, the volatility of the forward contract is equivalent to that of the underlying spot asset or σF (t, T ) = σS . The option payoff corresponds to that of a European option on a forward contract CF = max[F − K, 0]. As it is shown, the change of numeraire P under the martingale approach amounts to a change of variables in the partial differential equation method. The main difference between this economy and the Black–Scholes economy appears in the presence of stochastic interest rates. The intuition of the change in numeraire appears in Merton’s (1973) paper for the pricing of a stock option in the presence of stochastic interest rates. Merton (1973) uses a change P t corresponds to a “forwardof variables KPS(t,T ) . The new probability Q neutral” probability. It is possible to simplify the analysis by considering the underlying asset itself as a numeraire. In this case, a new probability

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measure QS can be defined such that: ST P (0, T ) dQS = dQP S0 P (T, T ) where, as before, the value of P (T, T ) is equal to one by definition. When this change of probability is used in the right-hand side of Eq. (12.17), the European call price is given by: S P CS = S0 E Q I P (T ,T ) ≤K − KP (0, T )E Q I ST ≥K P (T ,T )

ST

The change of time The main idea behind the change of numeraire in the partial differential equation approach is to operate an appropriate change of variables that facilitates the search of a solution to the price of a derivative contract in the standard Merton (1973) approach. The main idea behind the change of numeraire in the martingale approach is to express under an appropriate probability measure, the relative price of a security as a martingale. Consider a continuous P -martingale defined by the following stochastic differential equation: dMt = x1 (t)dW1 (t) + x2 (t)dW2 (t) where the coefficients x1 and x2 are deterministic functions. In this context, it is possible to show the existence of a unique onedimentional standard Brownian motion Y such that: Mt =

2 i=1

t 0

xi (u)dWi (u) = YMt

(12.21)

where M t =

2 i=1

0

t

x2i (u)du.

Consider again the pricing of a European call in the presence of stochastic interest rates. Using two changes of numeraire, the call price is: S P (12.22) CS = S0 E Q I P (T ,T ) ≤K − KP (0, T )E Q I ST ≥K ST

P (T ,T )

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) St The quantities P (t,T and P (t,T are respectively QP and QS martin) S gales and satisfy t P (t, T ) St = ln = [σS2 + σp (u, T )2 + 2ρσp (u, T )σS ]du ln P (t, T ) t St 0 t

Now, it is possible to introduce the time change τ defined as t t 2 2 [σS + σp (u, T ) + 2ρσp (u, T )σS ]du = σF (u, T )2 du τ (t) = 0

0

with

∗

τ = τ (T ) =

t 0

[σS2 + σp (u, T )2 + 2ρσp (u, T )σA ]du.

We can use Eq. (12.21) to show the existence of two standard onedimensional Brownian motions Y P and Y S under the appropriate probabilities QP and QS such that P (t, T ) P (0, T ) A 1 = eYτ (t) − τ (t). At A0 2

St S0 Y p − 1 τ (t) , = e τ (t) 2 P (t, T ) P (0, T )

Using these two last expressions and Eq. (12.22), the European call price is: CS = S0 EQA IY S ≤ln S0 + τ − KP (0, T )EQS IY P ≥ln S0 + τ ∗ τ∗

KP (0,T )

τ∗

2

KP (0,T )

2

The computation of this equation is straightforward. Hence, the European option price is:     S0 τ∗ τ∗ 0 ln + − ln KPS(0,T ) 2 KP (0,T ) 2  − KP (0, T )N   √ √ CS = S0 N  τ∗ τ∗ (12.23) We can compare this martingale approach with the PDE approach. Consider the following change of variable in Eq. (12.20) t τ (t) = σF (u, T )2 0

In this case, the forward option price CF as a function of F and τ satisfies the equation: 1 ∂ 2 CF ∂CF =0 + F2 ∂τ 2 ∂F 2

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under the appropriate terminal condition CF = max[F −K, 0]. The equation gives the price of a European option of a forward contract with a strike price K, a maturity date τ ∗ and a unit volatility for the underlying asset. The solution as given by the martingale approach (12.23) can also be stated as the solution to the standard Black–Scholes–Merton (1973) equation: CS = P (0, T )CF (F0 ) = P (0, T )C

S0 , τ ∗ , 1, 0 P (0, T )

where C is the extended Black–Scholes equation defined by: C(S0 , T, σS , r) = S0 N (d1 ) − Ke−rtN (d2 ) σ2 ln SK0 + r + 2S T √ √ d1 = = d2 + σS T . σS T The above analysis shows that the change of numeraire absorbs the stochastic character in the pricing of stock options when interest rates are random. This allows to transform the economy into a new economy in which the forward volatility is unity.

12.5. Valuation in an Extended Black and Scholes Economy in the Presence of Information Costs Consider the dynamics of the option’s underlying stock dSt = (r + λS )dt + σS dW2 (t) St where λS corresponds to the shadow cost of incomplete information on asset S. We have shown by arbitrage arguments that the extended Black– Scholes equation in the presence of information costs is given by: ∂C ∂C ∂2C 1 + σS2 S 2 2 + (r + λS )S − (r + λc )C = 0 ∂t 2 ∂S ∂S

(12.24)

The dynamics of this portfolio are given by: ∂C ∂C dV = C(r + λc )dt − S (r + λc )dt + dS ∂S ∂S

(12.25)

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Using Ito’s lemma gives: ∂2C ∂C 1 ∂C + σS2 S 2 2 dt + dS dV = ∂t 2 ∂S ∂S

561

(12.26)

The difference between Eqs. (12.26) and (12.25) is the partial differential Eq. (12.24). The solution for a European call is: C(S0 , T, σS , r) = S0 e−(λS −λc )t N (d1 ) − Ke−(r+λc )t N (d2 ) σ2 ln SK0 + r + λS + 2S T √ √ = d2 + σS T d1 = σS T Now, consider the following change of variables: ft = St e(r+λS )(T −t) ,

Cf = Ce(r+λc )(T −t)

Consider also the time change in Eq. (12.24) τ = σS2 t,

τ ∗ = σS2 T.

The change of underlying asset f in Eq. (12.24) gives: ∂C 1 ∂ 2C + σS2 f 2 2 − (r + λc )C = 0 ∂t 2 ∂f

(12.27)

This last equation corresponds to the extended Black (1976) equation in the presence of information costs. It gives the value of an option on a forward contract f . The dynamics of the replicating portfolio can be written as: dV = C(r + λc )dt +

∂C df ∂f

Using Ito’s lemma dV =

∂C 1 2 2 ∂ 2C f + σA ∂t 2 ∂f 2

+

∂C df ∂f

gives the dynamics of C. When Cf = Ce(r+λc )(T −t) , this change of variables accounts for the following expression: 1 2 2 ∂ 2 Cf ∂Cf + σA f =0 ∂t 2 ∂f 2

(12.28)

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The dynamics of the replicating portfolio are given by dVf = Ito’s lemma gives: ∂Cf ∂Cf 1 2 2 ∂ 2 Cf dt + + σS f df dVf = 2 ∂t 2 ∂f ∂f

∂Cf ∂f

df and

It is possible to use a change of variables for τ and τ ∗ to simplify Eq. (12.28) to the following form: ∂Cf 1 ∂ 2 Cf + f2 = 0. ∂τ 2 ∂f 2

(12.29)

This last equation is useful for the valuation of a forward option on a forward contract maturing in τ ∗ for which the volatility is equal to one. In this case, the solution for a forward call on a forward contract is given by: Cf (f0 , τ ∗ ) = C(f0 , τ ∗ , 1, 0) = f0 N (d1 ) − KN (d2 ). since ln

f0 K

√

+ τ∗

τ∗ 2

ln =

S 0

K

+ r + λS + √ σS T

2 σS 2

T

A simple comparison of Eqs. (12.29) and (12.24) reveals that the option is priced as if the interest rate were zero and there are no information costs in the economy. Besides the volatility is replaced by 1 and the maturity T by τ ∗ . Changing numeraire and time We know that:

EQ

h(T ) = P (0, T )EQX [h(T )] B(T )

The effect of this change of numeraire can be appreciated using the Black– Scholes context for the pricing of a European call option on an asset S in the presence of a stochastic interest rate. Let us use the following dynamics for the underlying asset dSt = (r + λS )dt + σS dW2 (t) St

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It is possible to show that the European call price in the presence of information costs must satisfy the following partial differential equation: 1 ∂ 2 CS ∂ 2 CS ∂ 2 CS 1 ∂CS + σp2 P 2 − ρσS σP SP + σS2 S 2 2 2 ∂t 2 ∂S 2 ∂P ∂S∂P ∂CS ∂CS + (r + λS )S + (r + λP )P − (r + λc )CS = 0. ∂S ∂P

(12.30)

This equation must be solved under the following terminal condition CS = max[S − K, 0] Equation (12.30) can be written, after simplification as Eqs. (12.19) and (12.20). When the change of probability is used as before, the European call price is given by: S P CS = S0 E Q I P (T ,T ) ≤K − KP (0, T ) E Q I ST ≥K ST

P (T ,T )

The change of time Consider a continuous P -martingale and the Eqs. (12.21) to (12.23). Using the same procedure, the solution for a call as given by the martingale approach can also be stated as the solution to the standard Black–Scholes– Merton (1973) S0 , τ ∗ , 1, 0 CS = P (0, T )CF (F0 ) = P (0, T )C P (0, T ) where C is the extended Black–Scholes equation C(S0 , T, σS , r) = S0 e−(λS −λc )t N (d1 ) − Ke−(r+λc )t N (d2 ) σ2 √ ln SK0 + (r + λS + 2S )T √ = d2 + σS T d1 = σS T The above analysis shows that the change of numeraire absorbs the stochastic character in the pricing of stock options when interest rates are random. This allows to transform the economy into a new economy in which the forward volatility is unity. Summary This chapter contains the basic material for the pricing of derivative assets in a continuous-time framework. The presentation is made as simple as

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possible in order to enable uninformed readers to understand the main derivations. First, we present in detail the search for an analytic formula for European call option within the partial differential equation method. Second, we illustrate in detail the martingale method for the derivation of a European call formula. Third, we develop the foundations of some interest rate models. Fourth, we present two mathematical tools, the change of numeraire and time change, in order to facilitate the pricing of options. These two transformations have an economic significance and can simplify considerably some valuation problems. While this chapter is necessary for the understanding of the basic techniques of option pricing theory, it is not necessary for the use and applications of all the formulas presented in this book. Questions 1. Provide a definition of a complete market. 2. How equity options are priced with respect to the partial differential equation method? and the martingale approach? 3. How equity options are priced with respect to the martingale approach? 4. How bond options and interest rate options are valued? 5. Describe the main techniques for the pricing of assets in complete markets using the change of numeraire and time. 6. Describe the resolution of the partial differential equation under the appropriate condition for a European call option. Appendix A: The Change in Probability and the Girsanov Theorem The equivalent probability Consider a certain probability space (Ω, F, P) in the presence of a probability P , and a probability Q which is continuous with respect to P . The following theorem gives the equivalent probability. Theorem: A probability Q is strictly continuous with respect to probability P, if and only if there exists a random variable Z taking on positive values on (Ω, F) such that for all A in Z(w)dP (w). F : Q(A) = A

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where Z corresponds to the density of Q with respect to P . It is often . These two probabilities P and Q are equivalent when they denoted by dQ dP are continuous with respect to each other.

The Girsanov theorem Consider the probability space (Ω, F, F , P). If (Θt )0≤T is an adapted process such that:

T 0

Θ2s ds < ∞

and the following process (Lt )0≤T is a martingale t t 2 Lt = exp − Θs dWs − Θs ds 0

then Wt∗ = Wt +

t 0

0

Θs ds is a standard Brownian motion.

Appendix B: Resolution of the Partial Differential Equation for a European Call Option on a Non-Dividend Paying Stock in the Standard Context The problem is to search for the solution to the following partial differential equation: 1 2 2 σ S 2

∂ 2c ∂S 2

+ rS

∂c ∂S

+

∂c ∂t

− rc = 0

(B.1)

under the following terminal condition which must be satisfied by the call price at its maturity date c(S, t∗ ) = max[0, St∗ − K]. Let us try a change of variables and postulate that the solution is of the following form: c(S, t) = f (t)y(u1 , u2 )

(B.2)

where f (t) and y(u1 , u2 ) are unknown functions. We begin by calculating the different partial derivatives in the partial differential equation with respect to time and to the underlying asset price.

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The partial derivative ∂c ∂t is given by: ∂y ∂u1 ∂c ∂f = y(u1 , u2 ) + f (t) ∂t ∂t ∂u1 ∂t ∂u2 ∂y + f (t) ∂u2 ∂t ∂c The partial derivative ∂S is given by:

∂c ∂u1 ∂u2 ∂y ∂y = f (t) + ∂S ∂u1 ∂S ∂u2 ∂S

The partial derivative

∂2c ∂S 2

(B.3)

(B.4)

∂2 c ∂S 2

is: 2 2 ∂y ∂ 2y ∂u1 ∂u1 + = f (t) 2 ∂u1 ∂S ∂u1 ∂S 2 2 2 ∂u2 ∂u2 ∂2y ∂y + f (t) + 2 ∂u2 ∂S ∂u2 ∂S 2

∂u1 ∂2y ∂u2 + 2f (t) ∂u1 ∂u2 ∂S ∂S

(B.5)

These quantities for the partial derivatives are substituted in the partial differencial Eq. (B.5) 2 ∂2y ∂2y 1 2 2 ∂u1 ∂u2 ∂u1 + 2 0 = σ S f (t) 2 ∂u21 ∂S ∂u1 ∂u2 ∂S ∂S 2 2 2 2 ∂u1 ∂ y ∂u2 ∂u2 ∂y ∂y + + + ∂u1 ∂S 2 ∂u22 ∂S ∂u2 ∂S 2

∂y ∂u1 ∂u2 ∂y + rSf (t) + ∂u1 ∂S ∂u2 ∂S ∂f y(u1 , u2 ) − rf (t)y(u1 , u2 ) + ∂t

∂y ∂u1 ∂u2 ∂y + (B.6) + f (t) ∂u1 ∂t ∂u2 ∂t This last equation can be solved, if we refer to the heat transfer equation ∂y 1 ∂2y (B.7) = ∂u2 2 ∂ 2 u1

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We can search for the functions f , u1 (S, t) and u2 (S, t). Assume that −rf (t)y(u1 , u2 ) +

∂f ∂t

y(u1 , u2 ) = 0

or rf (t) =

∂f ∂t

. ∗

Hence, the function f (t) is given by f (t) = er(t−t ) . We denote by a2 =

1 2 2 σ S 2

∂u1 ∂S

2 .

Simplifying by f (t) and re-writing the partial differential equation to identify the different terms, it appears that the following quantity: 2 2 ∂ y 1 2 2 ∂u1 σ S ∂u21 2 ∂S 2 ∂u2 1 2 2 ∂u2 ∂u2 ∂y + rS σ S + + ∂u2 2 ∂S ∂S ∂t is equal to zero if 1 a = σ2 S 2 2 2

∂2y ∂u21

∂u22 ∂S 2

+ rS

∂u2 ∂S

+

∂u2 ∂t

Assuming that

∂u2 ∂S

=

∂u22 ∂S 2

=0

it is possible to show that when u2 (t) = −a2 then u2 = −a2 (t − t∗ ) The Black–Scholes equation is reduced to: 1 2 2 σ S 2

∂u1 ∂S

2 + rS

∂u1 ∂S

+

∂u1 ∂t

=0

(B.8)

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Using the expresion for a2 , we have:

∂u1 ∂S

=

√ a 2 σS

or √ a S u1 = 2 ln + b(t) σ K Replacing this in Eq. (B.8) gives: 2 √ a √ a 1 1 1 2 2 σ S − + b (t) + rS 2 =0 2 2 S σ σ S which is equivalent to √ a 1 √ =0 aσ 2 + b (t) + r 2 2 σ Since the term b (t) is equal to: √

√ a a 2 σ2 1 √ = −r b (t) = aσ 2 − r 2 2 σ σ 2

then the term b(t) is given by b(t) =

√

a 2 σ2 − r (t − t∗ ). σ 2

This allows to write the value of u1 (t) as:

2 √ a S σ ∗ u1 (t) = 2 ln + − r (t − t ) σ K 2

(B.9)

This analysis concerns the partial differential equation. Now, we turn to the call’s maturity condition. Since at the option’s maturity date, the call value can be written as c(S, t∗ ) = f (t)y(u1 , u2 ) and since

∂y ∂u2

=

∂ 2y ∂u22

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then, we can write the maturity condition as: ∗

√ a 2 S u2 (S, t )] = y ln ,0 σ K

∗

∗

c(S, t ) = y[u1 (S, t ),

(B.10)

Re-call that the solution to the heat transfer equation is given by: y(u1 , u2 ) = √

1 2Πu2

+∞ −∞

U0 (ε)e

2

ε (− 2u ) 2

dε

(B.11)

where the limit as u2 approaches zero of y(ε, u2 ) is equal to U0 (ε). The following condition must be satisfied by the option price function at the maturity date ( √kσ ) c(S, t∗ ) = y(k, 0) = K e 2a − 1 when k is positive or 0, if k is negative with √ S 2a ln k= σ K or ln

S K

σk =√ . 2a

Hence S = Ke

( √σk ) 2a

.

When this condition is applied, Eq. (B.11) becomes y(u1 , u2 ) = √

1 2Πu2

u1 −∞

K

ε2 1 1 2 σ r σ 2 e(u−ε) − 1) e(− 2 ) dε 2 2

(B.12)

If we make a change in variables, Eq. (B.12) can be written as: 1 y(u1 , u2 ) = √ 2Π

∞

√ −u1 / 2s

K

σ √ a 2

e(u1 +q

√

2s)

q2 − 1 e(− 2 ) dq

(B.13)

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The term

u√ 1 u2 2

is given by:

√ 2

a 2 1 u1 S σ ∗ √ = ln + − r (t − t ) √ σ K 2 u2 2 2a(t − t∗ ) or u1 √ =− u2 2

ln

S K

+

σ2 2

− r (t − t∗ )

√ σ t∗ − t

and we obtain the following result: √ √ √ S σ2 σ a 2 √ ln + − r (t − t∗ ) + qa 2 t∗ − t σ K 2 a 2 2 √ σ S + − r (t − t∗ ) + qσ t∗ − t = ln K 2 If we denote this last term by 2 √ σ S + − r (t − t∗ ) + qσ t∗ − t d = ln K 2 and substitute these values in the integral solution of the heat equation, we obtain: ∞ ln( S )+( σ2 −r)(t−t∗ )+qσ√t∗ −t q2 K 2 − 1 e(− 2 ) dq (B.14) y(u, t) = √ e K 2Π d Equation (B.14) can also be written as: ∗

Ker(t−t ) y(u, t) = √ 2Π

∞ ∞ √ q2 q2 ∗ S σ2 ∗ e(qσ t−t ) e(− 2 ) dq e( 2 −r)(t−t ) − e(− 2 ) dq × K d d (B.15) If we make a change of variables and set p = −q. Equation (B.15) can be written as: ∗ ∗ σ2 Ker(t−t ) S e( 2 −r)(t−t ) y(u, t) = √ K 2Π −d −qσ √ q2 ∗ × e t∗ −t e(− 2 ) dq − Ker(t−t ) N (d2 ) (B.16) −∞

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Using the fact that √ √ 2 1 2 1 q + σ t∗ − t + q − σ t∗ − t = q2 + σ2 (t∗ − t) 2 2 √∗ and letting p = q + σ t − t the integral can be written as:

√ −d+σ t∗ −t

−∞

−d

= −∞

=e

e(−

q2 2

√ )(−qσ t∗ −t)

2

e

(− q2 − √ qσ ∗

t −t

2

( σ2 )(t∗ −t)

−d

−∞

)

dq

dq 1

e(− 2 (q+σ

√

t∗ −t)2 )

dq

which is equivalent to e

2

( σ2 )(t∗ −t)

√ −d+σ t∗ −t

∞

1

2

e(− 2 p ) dp.

After these computations, Eq. (B.16) becomes: ∗

y(u, t) = er(t−t ) e[(

σ2 2

2

−r)(t−t∗ )] [ σ2 (t−t∗ )]

e

∗

SN (d1 ) − Ker(t−t ) N (d2 )

The European call price is given by: ∗

c(S, t) = SN (d1 ) − Ker(t−t ) N (d2 ) with d1 =

ln

S K

+

σ2 2

√

+ r (t∗ − t)

σ t∗ − t

,

d2 =

ln

S K

+

√

The cumulative normal distribution given by

d

−∞

e(−

σ t∗ − t

Appendix C: Approximation of the Cumulative Normal Distribution

1 N (d) = √ 2Π

r−σ2 2

x2 2

)

dx

(t∗ − t)

.

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is often used in option pricing. Two approximations are provided for this function. The first approximation has a precision of 10−3. If x ≥ 0, then with coefficients a1 = 0.196854,

a2 = 0.115194,

a3 = 0.000344,

a4 = 0.019527

the formula is 1 N (x) = 1 − (1 + a1 x + a2 x2 + a3 x3 + a4 x4 )−4 2 if x < 0, then N (−x) = 1 − N (−x). The second approximation has a precision of 10−7 . If x ≥ 0, then with coefficients p = 0.0.2316419,

b1 = 0.319381530,

b3 = 1.781477937,

b2 = −0.356563782,

b4 = −1.821255978,

b5 = 1.330274429,

c=

1 (1 + px)

the formula is 1 x2 N (x) = 1 − √ e 2 (b1 c + b2 c2 + b3 c3 + b4 c4 + b5 c5 ). 2π Appendix D: Leibniz’s Rule for Integral Differentiation Consider the following integral I(x) =

B(x)

A(x)

F (x, t)dt

If the function F and its derivatives are continuous in t in the interval [A, B] and x takes all its values in the interval [a, b], then the derivative of this integral is given by: B ∂I(x) ∂F (x, t) = dt + F (x, B)B (x) − F (x, A)A (x) ∂x ∂x A If A and/or B are infinite, then this rule is applied only if the absolute value of the derivative of F with respect to t is less than a certain . value C(t) for all x in [a, b] and t in [A, B]. Also, the indefinite integral . C(t)dt must be convergent in the interval [A, B].

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Appendix E: Pricing Bonds: Mathematical Foundations Using the definition in Section 12.3, for any P ∗ of an arbitrage-free family of bond prices, we have: RT B(t, T ) = EP ∗ e− t ru du | Ft ,

∀ t ∈ [0, T ]

(E.1)

Expectations Hypotheses The local expectations hypothesis (L-EH) of Cox et al. (1981) or the riskneutral expectations hypothesis shows that the current bond price is equal to its expected value discounted at the current short-term rate. Under the actual probability measure P , the bond price is given by: RT B(t, T ) = EP e− t ru du | Ft ,

∀ t ∈ [0, T ]

According to the return-to-maturity expectations hypothesis (RTM-EH), the return from holding a bond to maturity is given by the return expected from a roll-over strategy of a series of a single-period bonds or: RT 1 = EP e t ru du | Ft , B(t, T )

∀ t ∈ [0, T ],

According to the yield-to-maturity expectations hypothesis (YTM-EH), or the unbiased expectations hypothesis, the yield from holding a bond to maturity is given by the yield expected from a roll-over strategy of a series of a single-period bonds or: B(t, T ) = exp −EP

T

t

ru du | Ft

,

∀ t ∈ [0, T ]

This formula can also be written as: 1 EP Y (t, T ) = T −t

T t

ru du | Ft

,

or f (t, T ) = EP (rT | Ft ),

∀ t ∈ [0, T ]

In this context, the forward interest rate, f (t, T ) is an unbiased estimate of the future short-term interest rate. The Ito process is often used to model

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the dynamics of the short-term interest rate. When the dynamics of r is given by: drt = µt dt + σt · dWt ,

r0 > 0

or in an equivalent form: rt = r0 +

0

t

µu du +

0

t

σu · dWu ,

∀ t ∈ [0, T ]

the underlying probability is the actual probability. To construct the martingale probability, re-call that any probability Q equivalent to P is given by the Radon–Nikodym derivative dQ = ET ∗ dP

.

0

αu · dWu

= ηT ∗ ,

P − a.s.

(E.2)

for some process where: ηt = Et

.

0

αu · dWu

t

= exp 0

αu · dWu −

1 2

t 0

2

| αu | du

(See Musiela, 1997 for details). Using the Girsanov theorem, the process: Wtα = Wt −

0

t

αu du,

∀ t ∈ [0, T ∗]

follows a d-dimensional Brownian motion under P α which denotes the probability measure in Eq. (E.2). In this context, Musiela (1997) gives the following proposition: Proposition: When the short rate follows an Ito process under P, then for any martingale measure P ∗ = P α , the process r satisfies under P α : drt = (µt + σt · αt )dt + σt · dWtα and there exists a process bα (t, T ) such that: dB(t, T ) = B(t, T )(rt dt + bα (t, T ) · dWtα )

(E.3)

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Hence, the bond price is given by: B(t, T ) = B(0, T )Bt exp

0

t

1 2

bα (u, T ) · dWuα −

t 0

2

|bα (u, T )| du

The corollary of the above proposition is that in the presence of two probability measures equivalent to P , when the bond price is given by Eq. (E.1), then: R T∗ R T∗ RT α ˜ 2 1 B(t, T ) = EP α˜ e− t ru du e t (αu −α˜ u )·dWu − 2 t |αu −α˜ u | du |Ft .

The implications of the above results is that for any probability measure P ∗ = P γ equivalent to P , the bond price can be defined by: RT α B(t, T ) = EP ∗ e− t ru du | FtW ,

∀ t ∈ [0, T ]

Using Eq. (E.3), the bond price satisfies under the actual probability P : dB(t, T ) = B(t, T )((rt − αt · bγ (t, T ))dt + bα (t, T ) · dWt ). This result indicates that the instantaneous returns from holding a bond are in general different from the short-term rate by an additional term reflecting the risk premium or the market price for risk. Using the process for the short-term interest rate and a probablity measure P ∗ the initial term structure is determined by the following formula: RT B(0, T ) = EP ∗ e− 0 ru du ,

∀ T ∈ [0, T ∗ ]

Exercises Consider the following dynamics of a share price dS = A(S, t)dX + B(S, t)dt where the functions A(S, t) and B(S, t) depend on S and t. Consider a derivative security f (S, t). 1. Apply Ito’s Lemma or Taylor’s Theorem to the function f (S, t). 2. Can you choose the values of A and B so that a function g(S) has a zero drift, but non-zero variance?

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Solution First method: Application of Ito’s Lemma df =

∂f 1 ∂2f ∂f (dS)2 dS + dt + ∂S ∂t 2 ∂S 2

we will replace dS by its equivalent indicated in the previous equation. ∂f ∂f 1 ∂2f 2 ∂f A (dX)2 AdX + B dt + dt + ∂S ∂S ∂t 2 ∂S 2

df = with

(dS)2 = A2 dX 2 + B 2 (dt)2 + 2ABdXdt or dX 2 ∼ dt, (dt)2 = 0;

dXdt = 0

Hence, df = A

∂f ∂f ∂f 1 ∂ 2f dX + B + + A2 2 dt ∂S ∂S ∂t 2 ∂S

Second method: Application of Taylor’s theorem Taylor’s theorem can be applied to a function f (S + δS, t + δt) over a small time step. It can be written as: f (S + δS, t + δt) = f (S, t) + +

∂f 1 ∂ 2f ∂f δS + δt + (δS)2 ∂S ∂t 2 ∂S 2

1 ∂2f ∂ 2f δSδt + · · · (dt)2 + · · · ∂S∂t 2 ∂t2

Since δS is given by δS = σSδX + µSδt, it can be replaced to give: δf =

∂f ∂f 1 ∂2f ∂f σSδX + µSδt + δt + (δS)2 ∂S ∂S ∂t 2 ∂S 2 +

1 ∂ 2f 1 ∂2f δtδS + (δt)2 + · · · 2 ∂S∂t 2 ∂t2

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577

We denote by A = σS and B = µS. In this case, δf can be written as: δf = (AδX + Bδt)

3 ∂f 1 ∂ 2f 2 ∂f A (δX)2 + O(δt 2 ) + δt + ∂S ∂t 2 ∂S 2 3

where the terms of order O(δt 2 ) and smaller can be neglected. When δt → 0, we can replace δX by dX and (δX)2 by dt to obtain the following stochastic differential equation that must be satisfied by f (S, t): ∂f ∂f 1 2 ∂2f ∂f dX + B + + A df = A dt ∂S ∂S ∂t 2 ∂S 2 2. The function g(S) with a zero drift, but non-zero variance must satisfy the following equation: ∂g 1 2 ∂ 2g ∂g dt dX + B + A dg = A ∂S ∂S 2 ∂S 2 with the restriction that g(S) has a zero drift and Var(g(S)) = 0 i.e., B

1 ∂g ∂ 2g + A2 2 = 0, ∂S 2 ∂S

or ∂2g 2B ∂g =0 + 2 ∂S 2 A ∂S with A2 = 0. It is possible to find a solution to the last equation. Let us denote by ∂2g = y ∂S 2

et

∂g = y. ∂S

Hence, we have y (S) +

2B y = 0. A2

To solve the following equation dy 2B(S, t) = 2 d(S, t), y A (S, t)

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let (S, t) = x hence

Log|y| =

−

We can denote by

2B(x) dx + K A2 (x)

F (x) = −

2B(x) dx A2 (x)

and y = eK eF (x) . The solution to this problem can be found when independent of time.

2B A2

is

Exercise Consider the following dynamics for the stocks S1 and S2 : dS1 = µ1 S1 dt + σ1 S1 dX1 dS2 = µ2 S2 dt + σ2 S2 dX2 The correlation coefficient between the two processes is denoted by ρ. What is the stochastic differential equation that must be satisfied by the function f (S1 , S2 )? Taylor’s Theorem can be used to find the change in the function f over short-time intervals. Solution The application of Taylor’s theorem over a short-time interval gives: f (S1 , δS1 , S2 + δS2 ) = f (S) + δS1 + δS1 δS2

1 ∂f ∂f ∂ 2f + δS2 + (δS1 )2 2 ∂S1 ∂S2 2 ∂S1

1 ∂ 2f ∂ 2f + (δS2 )2 2 + · · · ∂S1 ∂S2 2 ∂S2

When the terms in δS1 and δS2 are substituted in this equation and the 3 terms of O(δt 2 ) and smaller are neglected, we obtain: δf = σ1 S1

∂f ∂f ∂f ∂f δX1 + σ2 S2 δX2 + µ1 S1 δt + µ2 S2 δt ∂S1 ∂S2 ∂S1 ∂S2

1 ∂2f ∂2f 1 + (σ1 )2 (S1 )2 2 (δX1 )2 + (σ2 )2 (S2 )2 2 (δX2 )2 2 ∂S1 2 ∂S2 + σ1 σ2 S1 S2

∂2f 3 δX1 δX2 + O(δt 2 ) ∂S1 ∂S2

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579

When the term δt → 0, δt becomes equivalent to dt, δX1 becomes equivalent to dX1 , δX2 becomes equivalent to dX2 and the term (δX1 )2 becomes equivalent to dt, and finally (δX2 )2 becomes equivalent to dt; δX1 δX2 dX2 becomes equivalent to ρdt. This gives the stochastic differential equation that must be satisfied by f (S1 , S2 ) ∂f ∂f ∂f ∂f dX1 + σ2 S2 dX2 + µ1 S1 + µ2 S2 df = σ1 S1 ∂S1 ∂S1 ∂S1 ∂S2 1 1 ∂ 2f ∂2f ∂ 2f + σ12 S12 2 + ρσ1 σ2 S1 S2 + σ22 S22 2 dt 2 ∂S1 ∂S1 ∂S2 2 ∂S2 Exercise Consider the following dynamics of the underlying asset: dS = µSdt + σSdX. Find the stochastic differential equation that must be satisfied by f (S) = log(S n ) n ∈ N. Solution When Ito’s lemma is applied for a function f (S), we have: df =

∂f 1 ∂2f ∂f dS + dt (dS)2 + ∂S 2 ∂S 2 ∂t

Since we have the following partial derivatives then: ∂f = 0, ∂t

∂f 1 =n , ∂S S

∂2f 1 = −n 2 2 ∂S S

(dS)2 = µ2 S 2 dt2 + 2µS 2 σdtdX + σ 2 S 2 (dX)2 (dt)2 = 0;

dtdX = 0;

(dX)2 = dt

Hence, df = µS

∂ 2f ∂f ∂f 1 dt + σS dX + σ2 S 2 2 dt ∂S ∂S 2 ∂S

∂f 1 2 2 ∂ 2f ∂f dt dX + µS + σ S df = σS ∂S ∂S 2 ∂S 2

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Replacing gives:

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1 2 dt. df = nσdX + n µ − σ 2

Since this stochastic differential equation for log(S n ) has constant coefficients, S satisfies a log-normal random walk. References Arrow, KJ (1953). Le rˆ ole des valeurs boursières pour la rèpartition la meilleur des risques. International Colloquium on Econometrics, 1952, CNRS, Paris. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 79(3), 167–179. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian Distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M, JL Prigent and C Villa (2001a). Skew without Skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Cox, JC and SA Ross (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–66. Cox, J, J Ingersoll and S Ross (1981). A reexamination of traditional hypothesis about the term structure of interest rates. Journal of Finance, 36, 769–799. Cox, J, J Ingersoll and S Ross (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407. Debreu, G (1954). Representation of a preference ordering by a numerical function. In Decision Processes, R Thrall, C Coombs and R Davis (eds.), pp. 159–165, New York: Wiley. El Karouin N and H Geman (1993). A probabilistic approach to the valuation of general floating rate notes with an application to interest swaps. Unpublished working paper. Harrison, JM and D Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408. Harrison, JM and S Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their applications, 11, 215–260.

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Longstaff, FA (1990). Time varying term premiums and traditional hypothesis about the term structure. Journal of Finance, 45, 1307–1314. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Musiela, M (1997). Martingale Methods in Financial Modelling: theory and Applications. Berlin: Springer. Vasicek, O (1977). A equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188.

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Chapter 13 SIMPLE EXTENSIONS AND GENERALIZATIONS OF THE BLACK–SCHOLES TYPE MODELS IN THE PRESENCE OF INFORMATION COSTS

Chapter Outline This chapter is organized as follows: 1. Section 13.1 provides a simple derivation of the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. 2. Section 13.2 presents the valuation of securities dependent on several variables in the presence of incomplete information. 3. Section 13.3 provides the general differential equation for the pricing of derivatives. 4. Section 13.4 gives the extension of the risk-neutral argument in the presence of information costs. 5. Section 13.5 provides an extension of the analysis to commodity futures prices in the presence of information costs. 6. Appendix A provides a general equation for derivative securities. 7. Appendix B gives an extension to the risk-neutral valuation argument. Introduction This chapter develops a general context for the analysis and valuation of options and futures contracts in the presence of one or several state variables and information uncertainty. We provide a simple derivation of the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. Then, we extend this 583

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analysis to account for several state variables. This allows the derivation of the valuation equation for securities dependent on several variables in the presence of incomplete information. When a variable does not indicate the price of a traded security, its market price of risk corresponds to the market price of risk of a traded security, whose price is a function only on the value of the variable and time. The value of the market price of risk of the variable is the same at each instant of time. We also show, how to extend the risk-neutral argument in the presence of information costs and how to apply this analysis for the valuation of commodity futures within incomplete information.

13.1. Differential Equation for a Derivative Security on a Spot Asset in the Presence of a Continuous Dividend Yield and Information Costs We denote by f the price of a derivative security on a stock with a continuous dividend yield q. The dynamics of the underlying asset are given by: dS = µSdt + σSdz where the drift term µ and the volatility σ are constants. Using Ito’s lemma for the function f (S, t) gives: dV =

∂V dV 1 ∂ 2V 2 2 ∂V dt + µS + + σSdz. σ S dS dt 2 ∂S 2 ∂S

It is possible to construct a portfolio Π by holding a position in the derivative security and a certain number of units of the underlying asset: Π = −V +

∂V S. dS

Over a short time interval, the change in the portfolio value can be written as: 1 ∂ 2V 2 2 ∂V σ S ∆t. − ∆Π = − ∂t 2 ∂S 2 Over the same time interval, dividends are given by qS ∂V ∂S ∆t.

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Simple Extensions and Generalizations

585

We denote by ∆W , the change in the wealth of the portfolio holder. In this case, we have: 1 ∂ 2f 2 2 ∂V ∂V − ∆t. σ S + qS ∆W = − ∂t 2 ∂S 2 dS Since this change is independent of the Wiener process, the portfolio is instantaneously risk less and must earn the risk-free rate plus information costs or: ∂V ∂V ∂V 1 ∂2f 2 2 − σ S + qS − ∆t = −(r + λV )V ∆t + (r + λS )S ∆t 2 ∂t 2 ∂S ∂S ∂S where λi refers to these costs. This gives: ∂V 1 ∂ 2V 2 2 ∂V + (r + λS − q)S + σ S = (r + λV )V. ∂t ∂S 2 ∂S 2

(13.1)

Equation (13.1) must be satisfied by the derivative security in the presence of information costs and a continuous dividend yield. 13.2. The Valuation of Securities Dependent on Several Variables in the Presence of Incomplete Information: A General Method When a variable does not indicate the price of a traded security, the pricing of derivatives must account for the market price of risk. The market price of risk γ for a traded security is given by: γ=

µ−r−λ σ

(13.2)

where µ indicates the expected return from the security. This equation can also be written as: µ − r − λ = γσ.

(13.3)

The excess return over the risk-free rate in the presence of shadow costs on a security corresponds to its market price of risk multiplied by its volatility. When γ > 0, the expected return on an asset is higher than the risk-free rate plus information costs. When γ = 0, the expected return on an asset is exactly the risk free rate plus information costs. When γ < 0, the expected return on an asset is less than the risk free rate plus information costs.

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When a variable does not indicate the price of a traded security, its market price of risk corresponds to the market price of risk of a traded security whose price is a function only on the value of the variable and time. The value of the market price of risk of the variable is the same at each instant of time. In fact, we can show as shown by Hull (for details, refer to Bellalah et al., 1998) that two traded securities depending on the same asset must have the same price of risk, i.e., that Eq. (13.2) must be verified. Consider the following dynamics for a variable θ, which is not a tradable asset: dθ = µ(θ, t)dt + s(θ, t)dz. θ We denote this by V1 and V2 , respectively the prices of two derivative securities as a function of θ and t. The dynamics of these derivatives can be written as: dV1 = µ1 dt + σ1 dz V1

(13.4)

dV2 = µ1 dt + σ2 dz. V2

(13.5)

These two processes can be written in discrete time as: ∆V1 = µ1 V1 ∆t + σ1 V1 ∆z

(13.6)

∆V2 = µ2 V2 ∆t + σ2 V2 ∆z.

(13.7)

It is possible to construct a portfolio Π which is risk free using σ2 V2 of the first derivative security and −σ1 V1 of the second derivative security. Π = σ2 V2 V1 − σ1 V1 V2 . The change in the value of this portfolio can be written as: ∆Π = σ2 V2 ∆V1 − σ1 V1 ∆V2 . Using Eqs. (13.6) and (13.7), the change in the portfolio value can be written as: ∆Π = µ1 σ2 V1 V2 − µ2 σ1 V1 V2 ∆t. Since the portfolio Π is instantaneously risk less, it must earn the risk free rate plus information costs on both markets. Hence, we must have: µ1 σ2 − µ2 σ1 = (r + λ1 )σ2 − (r + λ2 )σ1

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587

µ2 − (r + λ2 ) µ1 − (r + λ1 ) = . σ1 σ2

(13.8)

or

The term µ−(r+λ) must be the same for all securities that depend on time σ and the variable θ. It is also possible to show that V1 and V2 must depend positively on θ. Since, the volatility of V1 is σ1 , it is possible to use Ito’s lemma for σ1 to obtain: σ1 V1 = sθ

∂V1 . ∂θ

Hence, when V1 is positively related to the variable θ, the σ1 is positive and corresponds to the volatility of V1 . But, when f1 is negatively related to the variable θ, σ1 is negative and Eq. (13.4) can be written as: dV1 = µ1 dt + (−σ1 )(−dz). V1 This indicates that the volatility is −σ1 rather than σ1 . The result in Eq. (13.3) can be generalized to n state variables. Consider n variables, which are assumed to follow Ito diffusion processes, where for each state variable i between 1 and n, we have: θi = mi θi dt + si θi dzi where dzi are Wiener processes. The terms mi and si correspond to the expected growth rate and the volatility of the θi with (i = 1, . . . , n). The price process for a derivative security that depends on the variables θi can be written as: n

dV = µdt + σi dzi V i=1

(13.9)

where µ corresponds to the expected return from the security and σi is its volatility. The volatility of V is σi when all the underlying variables except θi are kept fixed. This result is obtained directly using an extension of the generalized version of Ito’s lemma in its discrete form. We show in Appendix A that: µ − r − λV =

n

γi σi

i=1

where γi indicates the market price of risk for the variable θi .

(13.10)

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Equation (13.10) shows that the expected excess return on the security (option) in the presence of shadow costs depends on γi and σi . When γi σi > 0, a higher return is required by investors to get compensated for the risk arising from the variable θi . When γi σi < 0, a lower return is required by investors to get compensated for the risk arising from θi . 13.3. The General Differential Equation for the Pricing of Derivatives We denote by • • • • • •

θi : value of ith state variable; mi : expected growth in ith state variable; γi : market price of risk of ith state variable; si : volatility of ith state variable; r: instantaneous risk-free rate and λi : shadow cost of incomplete information of ith state variable,

where i takes the values from 1 to n. Garman (1976) and Cox et al. (1985) have shown that the price of any contingent claim must satisfy the following partial differential equation: ∂V 1 ∂2f ∂V θi (mi − γi si ) + ρi,k si sk θi θk = rf + ∂t ∂θi 2 ∂θi ∂θk i i,k

where ρi, k stands for the correlation coefficient between the variables θi and θk . We show in Appendix A, how to obtain a similar equation in the presence of incomplete information. In this context, the equation becomes: 1 ∂V ∂V ∂2V θi (mi −γi si )+ ρi,k si sk θi θk = (r+λ)V. + ∂t ∂θi 2 ∂θi ∂θk i

(13.11)

i,k

In the presence of a single state variable, θ, Eq. (13.11) becomes: ∂V ∂ 2V ∂V 1 +θ (m − γs) + s2 θ2 2 = (r + λ)V. ∂t ∂θ 2 ∂ θ

(13.12)

For a non-dividend paying security, the expected return and volatility must satisfy: m − r − λ = γs and m − γs = r + λ. In this case, Eq. (13.12) becomes the extended Black–Scholes equation in the presence of information costs.

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For a dividend-paying security at a rate q, we have: q + m − r − λ = γs or m − γs = r + λ − q. In this case, Eq. (13.12) becomes (13.1) ∂V ∂ 2V 1 ∂V + (r + λs − q)S + s2 S 2 2 = (r + λV )V. ∂t ∂S 2 ∂ S 13.4. Extension of the Risk-Neutral Argument in the Presence of Information Costs We know that the market price of risk is given by γ=

µ−r−λ σ

or µ − r − λ = γσ.

Appendix B shows how to price a derivative as if the world were risk neutral. This is possible when the expected growth rate of each state variable is (mi − γi si ) rather than mi . For the case of a non-dividend paying traded asset, using Eq. (13.5), we have mi − r − λi = γi si

or mi − γi si = r + λi .

This result shows that a change in the expected growth rate of the state variable from θi to (mi − γi si) is equivalent to using an expected return from the security equal to the risk-less rate plus shadow costs of incomplete information. For the case of a dividend-paying traded asset, we have qi + mi − r − λi = γi si

or mi − γi si = r + λi − qi .

This result shows that a change in the expected growth rate of the state variable from θi to (mi − γi si) is equivalent to using an expected return (including continuous dividends at a rate q) from the security equal to the risk-less rate plus shadow costs of incomplete information. This analysis allows the pricing of any derivative security as the value of its expected payoff discounted to the present at the risk-free rate plus the information cost on this security or: ˆ T] f = e−(r+λV )(T −t) E[f

(13.13)

ˆ is to the where VT corresponds to the security’s payoff at maturity T and E expectation operator in a risk-neutral economy. This refers to an economy

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where the drift rate in θi corresponds to (mi − γi si). When the interest rate r is stochastic, it is considered as the other underlying state variables. In this case, the drift rate in r becomes γr sr where γr refers to the market price of risk related to r. The term sr indicates its volatility. In this case, the pricing of a derivative is given by the discounting of its terminal payoff at the average value of r as: ˆ −(¯r+λV )(T −t) VT ] V = E[e

(13.14)

where r¯ corresponds to the average risk-free rate between current time t and maturity T .

13.5. Extension to Commodity Futures Prices within Incomplete Information Consider the pricing of a long position in a commodity forward contract with delivery price K and maturity T .

13.5.1. Differential equation for a derivative security dependent on a futures price in the presence of information costs Assume that the relationship between the futures price F and the spot price S is given by F = Seα(T −t) , where α depends only on time. In this case, if the volatility of the underlying asset is constant, then the volatility of F is constant and equal to that of S. The dynamics of the spot asset are given by dS = µSdt + σSdz, where µ is the instantaneous expected return, σ is the instantaneous volatility, and dz is an increment of a Wiener process. Using Ito’s lemma, the volatility of the futures price σF is given by: σF F = σS

∂F = σSeα(T −t) = σF. ∂S

Hence, the volatility of the futures price is equal to the volatility of the spot price and σF = σ. Now, consider the following dynamics for the futures price: dF = µF F dt + σF dz.

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Since the derivative asset price f (F, t) is a function of the futures price F and time t, using Ito’s lemma gives: f=

1 ∂2f 2 2 ∂f ∂f ∂f µF F + + σF dz. σ F dt + ∂F ∂t 2 ∂F 2 ∂F

Consider the value of a portfolio comprising the derivative security and a position in a certain number of the underlying futures contracts. The total change in wealth for the portfolio holder over a short time interval can be written as: ∆W =

∂f ∆F − ∆f. ∂F

The discrete versions of the previous equations for dF and df can be written as: ∆F = µF F ∆t + σF ∆z and ∂f ∂f 1 ∂ 2f 2 2 ∂f σ F ∆t + σF µF F + + ∆z ∆f = − 2 ∂F ∂t 2 ∂F ∂F √ where ∆z = ∆t. Hence, we have: ∆W =

−

1 ∂2f 2 2 ∂f − σ F ∆t. ∂t 2 ∂F 2

Since this change in value is risk free, it must earn the risk free rate plus information costs or: 1 ∂2f 2 2 ∂f σ F ∆t = −(r + λf )f ∆t − − ∂t 2 ∂F 2 and this gives ∂f 1 ∂ 2f 2 2 + σ F = (r + λf )f. ∂t 2 ∂F 2 This equation must be satisfied by a derivative security dependent on a futures price in the presence of information uncertainty.

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13.5.2. Commodity futures prices We denote by S the spot price of the commodity, F , the futures price, µ the growth rate of the commodity price, σ its volatility, and Γ its market price of risk. Using the extension of the risk-neutral valuation principle, the price is given by ˆ T ] − K) ˆ T − K] or f = e−(r+λf )(T −t) (E[S V = e−(r+λV )(T −t) E[S (13.15) ˆ refers to the expected value in a risk-neutral economy. where E(.) The forward or futures price F corresponds to the value of K that makes the value of the contract f equal to zero in Eq. (13.15). So, we have: ˆ T ]. F = E[S

(13.16)

Equation (13.16) shows that the futures price corresponds to the expected spot price in a risk-neutral world. If (γσ) is constant and the drift µ is a function of time then: ˆ T ] = E[ST ]e−γσ(T −t) E[S where E corresponds to the real expectations or the expectations in the real world. Using Eq. (13.16), we have: F = E[ST ]e−γσ(T −t) .

(13.17)

If γ = 0, the futures price is an unbiased estimate of the expected spot price. However, if γ > 0, the futures price corresponds to a downwardbiased estimate of the expected spot price. If γ < 0, the futures price is a downward-biased estimate of the expected spot price. 13.5.3. Convenience yields When the drift or the expected growth in the commodity price is constant, we have E(ST ) = Seµ(T −t) , Using Eq. (13.17), we have F = Se(µ−γσ)(T −t) . This last equation is consistent with the cost of carry model, when the convenience yield y satisfies the relationship: µ − γσ = r + λ + g − y.

(13.18)

This equation shows that the commodity can be assimilated to a traded security paying a continuous dividend yield equal to the convenience yield

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593

y minus the storage costs u. Hence, the convenience yield must satisfy the relationship: y = g + r + λ − µ + γσ. When the convenience yield is zero, we have µ − γσ = r + λ + g. This result shows that some commodities can be assimilated to traded securities paying negative dividend yields, which are equal to storage costs. The main derivations in this chapter appear in Bellalah (1999).

Summary This chapter develops a general context for the pricing of derivative assets. First, we derive the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. Second, we provide the valuation of securities dependent on several variables in the presence of incomplete information. Third, we propose the general differential equation in the same context. Fourth, we show how to extend the risk-neutral argument in the presence of information costs. Fifth, we extend the analysis to the valuation of commodity futures contracts within incomplete information.

Questions 1. Provide a simple derivation of the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. 2. Describe the valuation of securities dependent on several variables in the presence of incomplete information. 3. How can one extend the risk-neutral argument in the presence of information costs? 4. How can one price commodity futures contracts within incomplete information?

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Appendix A: A General Equation for Derivative Securities Consider a derivative security whose price depends on n state variables and time t. The security can be priced under the standard Black–Scholes assumptions. The state variables are assumed to follow Ito diffusion processes, where for each state variable i between 1 and n, we have: θi = mi θi dt + si θi dzi where the growth rate mi and the volatility si can be functions of any of the n variables and time. Let us denote respectively by, • Vj : price of the jth traded security for j between 1 and n + 1; • r: risk-free rate; • λj : information cost for the jth traded security for j between 1 and n+ 1 and • ρi,k : correlation coefficient between dzi and dzk . Since, the (n + 1) traded securities depend on θi , then using Ito’s lemma, we have: fj = µj Vj dt + σi,j Vj dzi (A.1) i

where µj Vj =

∂Vj ∂Vj 1 ∂ 2f m i θi + ρi,k si sj θi θj ∂θk + ∂t ∂θi 2 ∂θi i

(A.2)

i,k

σij fj =

∂fj si θi . ∂θi

(A.3)

In this context, it is possible to construct a portfolio using the (n+1) traded securities. We denote by aj , the amount of the jth security in the portfolio Π so that Π = j aj Vj , where the aj is chosen in a way to eliminate the stochastic components of the returns. Using Eq. (A.1), we have: aj σij fj = 0 (A.4) j

for i between 1 and n. The instantaneous return from this portfolio can be written as dΠ = j aj µj Vj dt, where the cost of constructing this portfolio is j aj Vj .

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If this portfolio is riskless, it must earn the risk-less rate plus information costs corresponding to each asset in the portfolio:

aj µj fj =

j

aj fj (r + λj )

(A.5)

j

which is equivalent to:

aj fj (µj − r − λj ) = 0.

(A.6)

j

Equations (A.4) and (A.6) are consistent only if: fj (µj − r + λj ) =

γi σij fj

(A.7)

i

or µj − r − λj =

γi σij

(A.8)

i

where for γi , i is between 1 and n. Using Eqs. (A.2) and (A.3) and replacing in Eq. (A.7) gives the following equation: ∂Vj 1 ∂Vj ∂Vj ∂ 2 Vj + m i θi + ρik si sk θi θk − (r + λj )Vj = γi si θi . ∂t ∂θi 2 ∂θi ∂θk ∂θi i

i

i,k

This last equation reduces to: 1 ∂ 2 Vj ∂Vj ∂Vj θi (mi − γi si ) + ρik si sk θi θk = (r + λj )Vj . + ∂t ∂θi 2 ∂θi ∂θk i i,k

Hence, any security f contingent on the state variables θi and time must satisfy the following second-order differential equation: 1 ∂2V ∂V ∂Vj + θi (mi − γi si ) + ρik si sk θi θk = (r + λ)V ∂t ∂θi 2 ∂θi ∂θk i

i,k

(A.9)

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Appendix B: Extension to the Risk-Neutral Valuation Argument When the variable θi is not a traded asset, it is possible to assume the existence of a traded asset θ˜i paying a continuous dividend qˆ where: qˆ = r + λi − mi + γi si . The values of θî and θi must be equal and the following differential equation must be verified: ∂V 1 ∂ 2V ∂V + (r + λi − qî ) + ρi,k si sk θî θˆk = (r + λ)V. θî ∂t 2 ∂ θî ∂ θî ∂ θˆk i

i,k

This equation is independent of risk preferences. Since (r + λi − q˜) = mi − γi si , the derivative security can be valued in a risk-neutral economy if the drift term in θi is modified from mi to mi − γi si . Exercises Exercise 1 1) Can you verify that the following terms are solutions of the extended Black–Scholes equation derived by Bellalah (1999)? 1.a) V (S, t) = S 1.b) V (S, t) = e(r+λv )t where r refers to the risk-less interest rate and λv corresponds to the information cost regarding the security V . Solution Re-call that the extended Black–Scholes equation in the presence of shadow costs of incomplete information can be written as: 1 ∂V ∂V ∂2V + σ2 S 2 − (r + λv )V = 0 + (r + λs )S ∂t 2 ∂S 2 ∂S 1.a) We can compute the partial derivatives with respect to S and replace in the PDE: ∂V = 0, ∂t

∂V = 1, ∂S

∂ 2V = 0. ∂S 2

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Hence, the equation becomes: (r + λs )S − (r + λv )V = 0 or V (S, t) = S

and, in this case λv = λs .

then V (S, t) = S verify B-S equation. 1.b) For the second function, the calculations of the partial derivatives gives ∂V = (r + λv )e(r+λv )t , ∂t

∂V =0 ∂S

and

∂ 2V = 0. ∂S 2

This leads to the extended Black–Scholes equation: (r + λv )e(r+λv )t − (r + λv )e(r+λv )t = 0 then V (S, t) = e(r+λv )t verify the extended Black–Scholes equation.

Exercise 2: The Black–Scholes Model in the presence of information costs Can you find the most general solution to the extended Black–Scholes equation with each of the following forms: (1) V (S, t) = A(S) (2) V (S, t) = B(S)C(t)

Solution 1) When V = A(S) is replaced in the extended Black–Scholes equation, this gives: 1 2 2 ∂ 2A ∂A + (r + λs )S σ S − (r + λA )A = 0. 2 ∂S 2 ∂S

(13.19)

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We can try a solution of the form A(S) = S n because the powers of S in the equation match the order of the derivatives. This gives: 1 2 2 σ S (n(n − 1))S n−2 + (r + λs )nS n−1 S − (r + λA )S n = 0 2 ⇐⇒ 1 2 n σ S (n(n − 1)) + (r + λs )nS n − (r + λA )S n = 0 2 ⇐⇒ 1 2 σ n(n − 1) + (r + λs )n − (r + λA ) = 0 2 ⇐⇒ 1 2 2 1 σ n + (r + λs − σ2 )n − (r + λA ) = 0 2 2

(13.20)

Equation (13.20) is of order 2 in n if and only if 12 σ 2 = 0. The following cases must be studied. a) When 1 2 σ = 0; 2

(r + λA ) = 0.

Hence, r = −λA

and r + λs =

1 2 σ . 2

This gives 1 1 2 2 σ n − (r + λA ) = 0 ⇐⇒ σ2 n2 = (r + λA ) 2 2 For this equation, we have: n2 = and the roots are

2(r + λA ) n1 = |σ|

2(r + λA ) σ2

2(r + λA ) and n2 = − . |σ|

The general solution for V (S, t): V (S, t) = CS n1 + BS −n1 , where C and B are constants.

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b) When we have 1 2 σ = 0, 2

r + λA = 0,

1 r + λs − σ2 = 0. 2

In this case, we compute the ∆ of Eq. (13.20) is defined by: ∆ = b2 − 4ac. Hence, ∆=

2 1 1 r + λs − σ2 + 4 σ2 (r + λA ) 2 2

or 1 ∆ = (r + λs )2 + σ4 − σ2 (r + λs ) + 2σ2 r + 2σ2 λa 4 which is equivalent to ∆=

1 4 σ + r2 + 2rλs + λ2s − σ 2 r − λs σ 2 + 2σ 2 r + 2σ 2 λA 4

or ∆=

1 4 σ + r2 + λ2s + σ2 r + 2rλs + 2σ 2 λA − λS σ 2 4

or 1 ∆ = r2 + (2λs + σ 2 )r + 2σ 2 λA + λ2s + σ 4 − λs σ 2 . 4 So, we have 2 σ2 σ2 2 r + 2λA σ + λs − ∆ = r + 2 λs + 2 2 2

We can compute ∆ for Eq. (13.21) as follows: 2 σ 2 1 − 2λA σ 2 − λs − σ2 = 2σ 2 (λs − λA ) ∆ = λs + 2 2 b.1) When λs = λA ⇐⇒ ∆ = 0

(13.21)

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In this case, Eq. (13.21) has a double solution: σ2 r1 = r2 = − λs + < 0. 2 b.1.1) ∀ r ∈ −{r1 = r2 },

∆>0

and Eq. (13.20) has two solutions: 2 2 2 r + λs + σ2 + 2σ 2 (r + λs ) − r + λs + σ2 + n1 = σ2 or 2 2 σ2 r + λs + σ2 − r + λs + 2 + n1 = σ2 which is also equivalent to: 1

=

−r − λs +

σ2 2

+ |r + λs + σ2

σ2 | 2

In fact,

1 2

1

if r + λs + σ = r + λs + σ2 2 2

then n1 = 1.

or, when

r + λs + 1 σ2 = −r − λs − 1 σ2

2 2 then, n1 =

−2 (r + λs ) σ2

and n2 =

− r + λs − 12 σ 2 − |r + λs + 12 σ 2 | . σ2

Hence, n2 = 1

or n2 =

−2 (r + λs ) . σ2

Finally, the solution can be written as: V (S, t) = HS n1 + KS n2

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where H and K are arbitrary constants or (S, t) = HS + KS

−2(r+λs ) σ2

σ2 . ∀ r ∈ − r1 = − λs + 2

,

b.1.2) When σ2 r = − λs + ; 2

∆ = 0.

In this case, Eq. (13.20) has a double solution: n1 = n2 = n =

−r − λs + 12 σ 2 −r + λs 1 = + σ2 σ2 2

as 1 r = −λs + σ2 2 Hence, n=

1 1 + =1 2 2

and the result is: V (S, t) = BS;

B = constant.

b.2) It is the case when λA > λs , ∆ < 0 and Eq. (13.21) does not have a solution and ∆ > 0, since the equation has the same sign as r2 . In this case, we have: −(r + λs ) + 12 σ 2 + m1 = σ2 m2 =

−(r + λs ) + 12 σ 2 − σ2

√ √

∆ ∆

, and V (S, t) = LS m1 + GS m2

where G and L are arbitrary constants.

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b.3) When λA < λs ;

∆ > 0.

Equation (13.20) has two solutions: 2 − λs + σ2 − |σ| 2(λs − λA ) r1 = 0

In this case, Eq. (13.20) has two solutions: √ √ − r + λs − 12 σ 2 + ∆ − r + λs − 12 σ 2 − ∆ α1 = , α2 = . σ2 σ2 Hence: V (S, t) = NS α1 + MS α2 where N and M are arbitrary constants. b.3.2) When r1 = −λs −

σ2 − |σ| 2(λs − λA ), 2

∆ = 0.

In this case, Eq. (13.20) has a double solution: β= or β=

− −λs −

σ2 2

−(r + λs − 12 σ 2 ) σ2 − |σ| 2(λs − λA ) + λs − 12 σ 2 σ2

which is equivalent to: 2(λs − λA ) −σ 2 − |σ| 2(λs − λA ) β=− =1+ σ2 |σ|

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and finally: V (S, t) = ES β where E is a constant. b.3.3) When r2 = −λs −

σ2 + |σ| 2(λs − λA ), 2

∆=0

then Eq. (13.20) has a double solution: r + λs − 12 σ 2 2(λs − λA ) =1− . g 1 = g2 = g = − 2 σ |σ|

Hence, we have V (S, t) = F S g , where F is a constant. c) When (r + λA ) = 0 then r = −λA . Using Eq. (13.20), we have: n

1 1 2 σ n + r + λs − σ2 2 2

= 0.

Hence: n = 0 or n =

2(λs − λA ) −2(r + λs ) + σ 2 =− +1 σ2 σ2

and finally, we have: V (S, t) = C + DS −

2(λs −λA ) +1 σ2

where C and D are constants. 2) Show that when V (S, t) = B(S)C(t) is replaced in the extended Black and Scholes equation, this gives: B

∂C 1 ∂ 2B ∂B + σ2 S 2 C − (r + λv )B(S)C(t) = 0. + (r + λs )SC ∂t 2 ∂S 2 ∂S

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In fact, since B(S) is two times derivable, we have: ∂V ∂C = B(S) , ∂t ∂t

∂V ∂B = C(t) , ∂S ∂S

∂ 2B ∂2V = C(t) . ∂S 2 ∂S 2

This gives 1 ∂C 1 1 2 2 ∂2B ∂B =− σ S − (r + λ + (r + λ )S )B(S) . s v C ∂t B 2 ∂S 2 ∂S Note that the left-hand side C1 ∂C ∂t of the equation is a function of time t. The right-hand side, ∂B 1 1 2 2 ∂2B σ S − (r + λv )B(S) + (r + λs )S − B 2 ∂S 2 ∂S is a function of S. Both sides of the equation must be equal to a constant, 1 ∂C C = = k. C ∂t C Therefore, we have: LogC(t) = kt and C(t) = C0 ekt . Using the right-hand side of the equation, we have: ∂B 1 1 2 2 ∂ 2B σ S − (r + λv )B(S) = k + (r + λs )S B 2 ∂S 2 ∂S or ∂B 1 2 2 ∂ 2B σ S − (r + λv )B(S) = −Bk + (r + λs )S 2 ∂S 2 ∂S which is equivalent to: ∂B 1 2 2 ∂2B + (r + λs )S σ S − (r + λv − k)B(S) = 0. 2 2 ∂S ∂S It is possible to try the following form B(S) = S n which gives: 1 2 n σ S n(n − 1) + (r + λs )S n n − (r + λv − k)S n = 0 2 or 1 2 σ n(n − 1) + (r + λs )n − (r + λv − k) = 0 2

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and finally 1 1 2 2 σ n + r + λs − σ 2 n − (r + λv − k) = 0 2 2

(13.22)

Now, we can look for the solutions to this quadratic equation.

First case If 1 r + λs − σ2 = 0; 2

r=

1 2 σ − λs ; 2

1 2 σ = 0, 2

r + λv − k = 0,

then 2 2(r + λv − k) n = = σ2 2

1 2

σ 2 − λs + λv − k . σ2

The quadratic equation for n has two roots n1 and n2 : n1 =

2(r + λv − k) |σ|

and n2 = −

2(r + λv − k) |σ|

Since n1 = −n2 ,

n1 =

2 12 σ 2 + λv − λs − k |σ|

then we have, V (S, t) = (A1 S n1 + A2 S n2 )ekt for two arbitrary constants A1 and A2 . Second case If r − k − λv = 0,

r = k − λv ,

1 2 1 σ = 0 and r + λs − σ2 = 0, 2 2

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then the quadratic equation for n becomes: 1 1 2 σ n + k − λv − σ 2 + λ s = 0 n 2 2 and the solutions n1 or n2 are given by: n1 = 0

or n2 =

−2(k − λv + λs ) + σ 2 −2 = 2 (k − λv + λs ) + 1. σ2 σ

The final solution is:

V (S, t) = (D1 + D2 S n2 )ekt where D1 and D2 are arbitrary constants. Third case If 1 r + λs − σ2 = 0, 2 then

k − r − λv = 0,

σ2 = 0 2

2 1 1 ∆ = r2 + 2 λs + σ2 r + λs σ2 + 2σ 2 (λv − k) 2 2 ∆ = 2σ 2 (λs − λv − k).

In this case, three situations must be analyzed. First situation If ∆ < 0;

λs − λv − k < 0;

∆ has no roots and when ∆ > 0, Eq. (13.20) has two solutions n1 and n2 where: √ √ −(r + λs − 12 σ 2 ) − ∆ −(r + λs − 12 σ 2 ) + ∆ n1 = , n2 = . σ2 σ2 The solution is given by: V (S, t) = (ES n1 + F S n2 )ekt with F and E are constants.

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Second situation If ∆ = 0 ⇐⇒ λs = λv + k ⇐⇒ λs − λv = k, then ∆ shows a double solution σ2 . r1 = r2 = r = − λs + 2 Hence: Consider in this second situation, the case: ∀r ∈ −r1 ; ∆ > 0. In this case, Eq. (13.20) has two distinct solutions:

m1 =

m2 =

2 − r + λs − 12 σ 2 − r + λs − 12 σ 2 − 2σ 2 (λs − r − 2λv ) − r + λs − 12 σ 2 +

σ2

r + λs − 12 σ 2

2

− 2σ 2 (λs − r − 2λv )

σ2

.

Finally, the solution is given by: (S, t) = (GS m1 + HS m2 )e(λs −λv )t where G and H are constants. Consider in this second situation the case: when σ2 r1 = r2 = − λs + ; ∆ = 0. 2 Equation (13.20) shows a double solution. m1 =

−(r + λs − σ2

σ2 2 )

= 1.

Hence, the final solution is V (S, t) = ISe kt = IS 1 e(λs −λv )t , where I is a constant.

Third situation ∆ > 0 ⇐⇒ λs > λv + k.

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In this situation, the ∆ of the equation shows two distinct roots: σ2 − |σ| 2(λs − λv − k) a1 = − λs + 2 σ2 + |σ| 2(λs − λv − k) a2 = − λs + 2 and ∀ r ∈] − ∞, a1 [∪]a2 , +∞[ ⇐⇒ ∆ > 0. Equation (13.20) has two distinct roots given by: −((r + λs ) − α= σ2

σ2 2 )

+

√

∆

,

−((r + λs ) − β= σ2

The solution is given by V (S, t) = (LS α + JS β )ekt where L and J are constants. When r = a1 , the ∆ = 0, there is a double solution: α1 = 1 +

2(λs − λv − k) . |σ|

Hence, the solution is given by V (S, t) = ekt (M S α1 ) where M is a constant. When r = a2 , the ∆ = 0 and the solution is: γ = 1−

2(λs − λv − k) . |σ|

Hence, the solution is given by V (S, t) = NS γ ekt where N is a constant.

σ2 2 )

−

√

∆

.

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Exercise 3 When a change of time variable is used, the equation can be reduced to the ∂2 u diffusion equation, C(τ ) ∂u ∂τ = ∂x2 , when C(τ ) > 0. Consider the extended Black–Scholes equation with a constant volatility and interest rate. Can this equation be reduced to the diffusion equation in this case?

Solution We can try a change of variable of the form u(x, τ ) = v(x, τ˜), where τ˜ = F (τ ) corresponds to a certain function of τ . In this case, we have: ∂u ∂u dF (τ ) = . ∂τ ∂ τ˜ dτ The partial differential equation becomes C(τ )

∂ 2 v(x, τ˜) dF (τ ) ∂v(x; τ˜) = . dτ ∂ τ˜ ∂x2

If you choose the function F (τ ) such that: C(τ )

dF (τ ) = 1, dτ

where F (τ ) =

dS C(S)

then this leads to the following equation: ∂v ∂ 2v . = ∂ τ˜ ∂x2 Now, it is possible to solve the extended Black–Scholes equation: 1 ∂2V 2 2 ∂V ∂V + − (r(t) + λv (t))V = 0. σ S + (r(t) + λs (t))S ∂t 2 ∂S 2 ∂S We use the following transformation S = Eex

and V (S, t) = Ev(x, t)

and compute the following partial derivatives: ∂V ∂v =E , ∂t ∂t

∂V ∂v ∂x E ∂v = = , ∂S ∂x ∂S S ∂x

E ∂v ∂2V E ∂ 2v =− 2 + 2 2. 2 ∂S S ∂x S ∂x

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This leads to ∂v 1 2 + σ (t) ∂t 2

∂v ∂ 2v − 2 ∂x ∂x

+ (r(t) + λs (t))

∂v − (r(t) + λv (t))v = 0 ∂x

⇐⇒ ∂ 2v 2 2 ∂v 2 ∂v + − (r(t) + λv (t))V = 0. + (r(t) + λs (t) − 1) 2 σ 2 (t) ∂t ∂x2 σ (t) ∂x σ 2 (t) Now, we can choose a new time variable τ such that: ∂ 2 ∂v(x, t) = v(X, τ ) σ 2 (t) ∂t ∂τ i.e., 1 τ =− 2

0

t

σ 2 (s)ds.

The equation is reduced to: ∂2v 2 2 ∂v ∂v = (r(t) + λ − (r(t) + λv (t)) 2 v. + (t)) − 1 s ∂τ ∂x2 σ 2 (t) ∂x σ (t) If we set v(x, t) = eαx+βτ w(x, τ ) and choose the values of: α=−

1 2 (r(t) + λ (t) − 1 s 2 σ2

and β=

1 4

2 2 2 (r(t) + λ (t) + 1 + 2 (λs (t) + λv (t)) s σ2 σ

we obtain, ∂2w ∂w = . ∂τ ∂x2

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Exercise 4 For the following problem: ∂ 2u ∂u ; = ∂τ ∂x2

−∞ < x < ∞, τ > 0

with u(x, 0) = u0 (x) > 0

it is possible to show that u(x, τ ) > 0 ∀ τ. This result can be used to show that an option will always have a positive value.

Solution The solution to this general problem: ∂u ∂ 2u = ; −∞ < x < ∞, τ > 0 ∂τ ∂x2 with u(x, 0) = u0 (x) > 0 and u ⇐⇒ 0

as |x| ⇐⇒ ∞

is 1 u(x, τ ) = √ 2 Πτ

∞

−∞

u0 (S)e

−(x2 −s)2 4τ

ds

if u0 (x) > 0,

then u0 (s)e

−(x2 −s)2 4τ

> 0 and u(x, τ ) > 0.

For an option with a positive payoff, we want to solve the following extended Black–Scholes equation: ∂V ∂V 1 ∂ 2V 2 2 S σ + (r + λs )S + − (r + λv )V = 0 2 ∂t 2 ∂S ∂S under the condition that V (S, T ) > 0.

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We apply the same transformation as before and in particular, we apply the first transformation to obtain: 2 ∂v 2 ∂2v ∂v + 2 (r + λs ) − 1 = − (r + λv ) 2 v 2 ∂τ ∂x σ (t) ∂x σ (t) and v(x, 0) =

1 V (Eex , T ) > 0. E 2

∂ u The second transformation gives ∂u ∂τ = ∂x2 with initial data u(x, 0) = e−αx v(x, 0) > 0. Hence, we show that u(x, τ ) > 0 leads to v(x, τ ) = eαx+βτ u(x, τ ) > 0 and V (S, t) = V (Eex , T − σ2τ2 ) = Ev(x, t) > 0. Finally, the option value is always positive and the result is independent of the term λv .

References Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September) 645–664. Briys, E, M Bellalah et al. (1998). Options, Futures and Exotic Derivatives. En collaboration avec E. Briys, et al., John Wiley & Sons. Cox, JC, JE Ingersoll and SA Ross (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407. Garman, M (1976). A general theory of asset valuation under diffusion state processes, Working Paper, No. 50, Berkeley: University of California.

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Part V Extensions of Option Pricing Theory to American Options and Interest Rate Instruments in a Continuous-Time Setting: Dividends, Coupons and Stochastic Interest Rates

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Chapter 14 EXTENSION OF ASSET AND RISK MANAGEMENT IN THE PRESENCE OF AMERICAN OPTIONS: DIVIDENDS, EARLY EXERCISE, AND INFORMATION UNCERTAINTY

Chapter Outline This chapter is organized as follows: 1. Section 14.1 is an introduction to the general context for the pricing of American options with and without distributions to the underlying asset. 2. Section 14.2 studies the valuation of American spot and futures options in the context of a constant proportional rate. 3. Section 14.3 deals with the valuation of American spot and futures options in the context of a constant proportional rate within incomplete information. 4. Section 14.4 studies the valuation of American options when there are discrete distributions to the underlying asset. 5. Section 14.5 deals with the valuation of American options when there are discrete distributions to the underlying asset within incomplete information. 6. Section 14.6 is devoted to the valuation of compound options within incomplete information. 7. Appendix A presents an alternative derivation of the compound option’s formula using the martingale approach.

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Introduction There are several extensions of the basic Black and Scholes (1973) model. We have already seen the first extension by Black (1976) who takes into account the specificities of futures contracts. The second extension was made by Garman and Kohlhagen (1983) who derive analytical valuation formulas for European options on currencies. In the same year, Grabbe (1983) implemented a similar approach to that used in Black (1976) to provide the value of options on foreign currencies. Merton (1973) indirectly and Barone-Adesi and Whaley (1987), hereafter Barone-Adesi and Whaley (1987) provided the values of European commodity options and commodity futures options. All these models apply only to European options. Since, all the proposed analytical models deal with the pricing of the European options in the absence of discrete distributions to the underlying assets, the extensions of the analytical models to the valuation of American options are proposed in this chapter. The two important extensions and contributions to the literature on the valuation of American options are those of Merton (1973) and Geske (1979). The pricing of American options is first analyzed by Merton (1973) who showed the difficulties in obtaining closed-form solutions when there are discrete distributions to the underlying asset. However, he provided closedform solutions for American options when the time to maturity is infinite. Geske (1979) provided analytical formulas for the valuation of options on options or compound options. He used a valuation by duplication technique for the pricing of American options. A compound option can be defined as an option on the firm’s equity. For a levered equity firm (a firm with debt in its capital structure), equity can be seen as a call on the value of the assets. This was first noted by Black and Scholes (1973); B–S, who considered the option on equity as an option on the value of the firm’s asset. Geske’s approach is interesting since it allows the valuation of American options and includes the question of dividend. American option pricing models have been proposed by several authors. However, given the difficulties in obtaining closed-form solutions, it became quite a natural way to resort to analytical approximation models, binomial methods, and numerical techniques. When there are continuous distributions to the underlying asset, the literature on the valuation of American options provides some analytical approximation formulas. Barone-Adesi and Whaley (1987) presented simple analytic approximations for the pricing

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Extension of Asset and Risk Management in the Presence of American Options 617

of American options on commodity futures contracts. These formulas are accurate and computationally more efficient than compound-option pricing models, binomial models, and finite difference methods. The approach relies on a quadratic approximation method similar to that used by McMillan (1986) for the valuation of the American put option on a nondividend-paying stock. However, the formulas apply only for a constant proportional rate. When there are discrete distributions to the underlying asset, the valuation of American options is more complex. In this context, models were proposed by several authors including Roll (1977), Geske (1979), Whaley (1981), and Whaley (1986) for call options. Put formulas were proposed by Johnson (1983), Geske and Johnson (1984), Geske and Shastri (1985), Blomeyer (1986), and Barone-Adesi and Whaley (1987), among others. Most of these models, if not all, are based on the concept of compound options. The compound option or an option on an option has been studied in a context of complete information by several authors. The concept of an option on an option is important in the study of several opportunities with a sequential nature where some of them are available only if earlier opportunities are undertaken. Black and Scholes (1973), Black and Cox (1976), Galai and Masulis (1976), and Geske (1979) show that several corporate liabilities may be considered as options. In a context of complete information, they studied the pricing of a firm’s common stock and bonds by considering the stock as an option on the firm’s value. They showed that corporate investment opportunities may be analyzed as options and compound options. However, their analysis does not account for information uncertainty. We use arbitrage arguments to derive the formula in a Black and Scholes (1973) economy. Such a formula might be applied to the valuation of equity in the capital structure of the firm. The information uncertainty about the firm and its cash flows reflects the agency costs and the asymmetric information problems. By assuming the stock as an option on the value of the firm, the value of the call as a compound option can be derived as a function of the firm’s value by accounting for information costs and the effects of leverage. We present two alternative derivations for the value of the firm’s stock as a compound option in the presence of information costs. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah et al. (2001a,b), Bellalah and Prigent (2001), Bellalah and Selmi (2001) etc.

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14.1. The Valuation of American Options: The General Problem Two problems arise in the valuation of American options. The first is associated with the possibility of early exercise. The second is linked to the distributions to the underlying asset. The nonexistence of a put call parity theorem for American options implies a distinct treatment for call and put options. The problem is different according to the pattern of distributions to the underlying asset. In fact, if the dividend stream is continuous, it may not induce the optimal early exercise of American options. However, when the dividends are discrete, this may induce early exercise of American options. 14.1.1. Early exercise of American calls We deal with the main reasons behind rational exercise of American options. Exercise without distributions When there are no distributions to the underlying asset, there is no incentive to exercise an American call option before its maturity date. Hence, the value of an American call is equal to that of a European call. This result is due to Theorems 1 and 2 of Merton (1973) where it is shown that in the absence of dividend payments, an American call will never be exercised prior to expiration. In this context, the value of an American call is equivalent to that of a European call when the interest rates are constant. The intuition of this result is simple. In the absence of dividends, the option is worth more “alive” than “dead” because of its time value. Killing the option would mean pocketing its intrisic value while losing its speculative. The investor is better off in selling the option rather than killing it. Early exercise with continuous distributions When dividends are paid continuously at a constant rate of d dollars per unit time, and when the interest rate is constant, a sufficient condition for no premature exercise is that: K>

d r

(14.1)

This condition shows that the call strike price must be greater than the ratio of the continuous dividend rate to the short-term risk-less rate.

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Early exercise with discrete distributions When the amounts of dividends per share dj are paid at different dates τj , for j = 1 to n, are known and the interest rate is constant, then a sufficient condition for no premature exercise is: p P (τj ) d(τj ) (14.2) K> 1 − P (τn ) j=1

where P (τ ) is the price of a discount bond paying one-Dollar in τ years. This condition shows that the net present value of future dividends must be less than the present value of earnings from investing the strike price for τ periods. The intuition of this result is simple. In fact, if the losses from dividends are less than the gains from investing the required funds to exercise the option and hold the stock, the option is exercised. When the dividend per share is D(S, t), the B–S differential equation is slightly modified. In fact, the instantaneous rate of return is no longer µSdt dt. This gives the following partial differential equation, but rather µ−D(S,t) S which was formulated first by Merton (1973): 1 2 2 ∂ 2C ∂C ∂C σ S + [rS − D] + − rC = 0 (14.3) 2 ∂ 2S ∂S ∂t where C(S, τ, K) is the American call value. This partial differential equation must satisfy the following boundary conditions: C(0, τ, K) = 0

(14.4)

C(S, 0, K) = max[0, S − K]

(14.5)

C(S, τ, K) ≥ max[0, S − K]

(14.6)

Equation (14.4) shows that the option is worthless when the underlying asset is zero for any time to maturity τ . Equation (14.5) indicates that the option price is equal to the greater of zero and its intrinsic value at the option maturity date. Equation (14.6) is an arbitrage condition. It indicates the American option value at any time during the option’s life. It shows also that at each instant τ , there is a positive probability of early exercise. This implies that there exists a certain level I(τ ) of the underlying asset price such that for each S > I(τ ), the option is worth more “dead” than in “life”. Since the value of an immediate exercise is (S − K), the structure of the problem implies the additional condition C(I(τ ), τ, K) = I(τ ) − K = h, where C(I(τ ), τ, K) satisfies the partial differential equation for

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0 ≤ S ≤ I(τ ). Since I(τ ) is an unknown function of time, the structure of the problem is complex. Indeed the boundary conditions are semi-infinite and time dependent. The value of I(τ ) must be determined as part of the solution. As defined by Samuelson (1972) and Merton (1973), this is rather a difficult problem since the unknown function I(τ ) must be determined from the behavior of the call’s holder, who maximizes the option value at each instant, by adding a condition of regularity. To make this point explicit, consider a function f (S, τ ; K, I(τ )), which is a solution to this problem for a given I(τ ), or: C(S, τ, K) = max(I) f (S, τ ; K, I). The optimal I(τ ) is independent of the current underlying asset price, and the following condition, which is “high contact” at the boundary ),τ,K) = 1. This means that the option’s must be accounted for ∂C(I(τ ∂S partial derivative with respect to the underlying asset price is equal to one. Intuitively, this condition corresponds to the point of tangency between the call function and its intrinsic value, since only in-the-money options are exercised. The proof of this condition is relatively a simple matter. In fact, for the function f (x, I), (differentiable and concave with respect to its second argument), the total derivative with respect to I along the boundary x = I is given by: df dh = = f1 (I, I) + f2 (I, I). dI dI When the function has a maximum for a level I = I ∗ , f2 (x, I ∗ ) = 0. df ∗ ∗ ∗ ∗ = dh Since, dI dI = f1 (I , I ) and h = I − K, so f1 (I , I ) = dh dI = 1. This is the proof that the derivative must be equal to one. The solution to this problem gives the value of the American call when there are dividends. Samuelson (1972) and Merton (1973) analyzed this “free boundary” problem and did not provide solutions for a finite-life option. However, a solution for an infinite-life option (a perpetual warrant) was provided by Merton (Eq. (46), p. 172). 14.1.2. Early exercise of American puts The valuation of American calls is simpler than the valuation of American puts with and without discrete cash distributions to the underlying asset. While European and American calls have the same value when there are no distributions to the underlying asset, this result does not hold for European

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and American puts. While there is no incentive to exercise an American call option before its maturity date in the absence of dividends, there is always a probability of pre-mature exercise of an American put. This is due to the first arbitrage condition, according to which the value of an American put must be greater than its intrinsic value at each instant, and second, to the fact that the value of a European put with an infinite time to maturity or a perpetual put is zero. Exercise without distributions Since there is no put-call parity relationship for American options, American puts must be treated separately. The problem with the rational pricing of European or American put options is somewhat different from that of European or American calls. This results from the work of Merton (1973) who showed that an American put option can be exercised early even in the absence of distributions to the underlying asset. The American put must satisfy the Black and Scholes (1973) or Merton’s (1973) partial differential equation: ∂P ∂P 1 2 2 ∂ 2P σ S + rS + − rP = 0 2 ∂ 2S ∂S ∂t under the following boundary conditions: P (∞, τ, K) = 0

(14.7)

P (S, 0, K) = max[0, K − S]

(14.8)

P (S, τ, K) ≥ max[0, K − S]

(14.9)

Equation (14.7) shows that the put is worthless when the underlying asset price tends to infinity. This result is rather intuitive. In fact, the put intrinsic value is given by the difference between a constant (the strike price) and the infinite underlying asset price. Equation (14.8) is standard and gives the put value at the option’s maturity date. Equation (14.9) shows that it is sometimes optimal to exercise the put before its maturity date. This happens when the underlying asset price tends to zero. In this situation, it is interesting to exercise the put as soon as possible to benefit from the investment of the strike price until the option’s maturity date. From the analyses of McKean (1969), Samuelson (1972), and Merton (1973), there is no-closed form solution for a finite-life put when the above boundary conditions are applied to the partial differential equation. However, for an

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infinite time to maturity, a closed-form solution is given by Merton (1973), (Eq. (52), p. 174). The structure of this problem implies the existence of a certain level of the underlying asset I(t), referred to as the critical asset price level, for which the exercise depends on the maximization behavior of the put’s holder. The determination of this critical level implies adding the following condition on the put’s derivative with respect to the underlying ∗ , ∞, K) asset ∂P (I ∂t = −1. This condition is the Samuelson’s “high contact” boundary condition. The analysis of this problem (the free boundary problem) by Samuelson and McKean (for details, refer to Bellalah et al., 1998) and Merton (1973) allows the derivation of the put’s value when the time to maturity is infinite. Early exercise with continuous distributions When there are continuous distributions to the underlying asset, the pricing of American puts is rather difficult. However, an interesting approach was proposed by Barone-Adesi and Whaley (1987) for the valuation of American options. This approach is referred to as the quadratic approximation method. Early exercise with discrete distributions When there are no dividends, the American put option may be exercised early. In fact, since interest income can be earned on the exercisable proceeds of the option when exercised, this is sufficient to justify early exercise. When the American option holder delays exercise, this means that he forgoes the interest income on the exercisable proceeds. When there are dividends, the American put option holder must compare the effect of dividends on the put value and the interest income. If the holder does not exercise his put, he forgoes the interest income. If he exercises before the ex-dividend date, he will not profit from the increase in the exercisable proceeds when the underlying stock goes ex-dividend. There is a trade-off between the interest income and the dividend. Hence, between dividend dates, the option holder is always in a dilemma. Some interesting approximations of American values are given in the literature. These approximations include, Johnson (1983), Geske and Johnson (1984), Geske and Shastri (1985), Blomeyer (1986), and Barone-Adesi and Whaley (for details, refer to Bellalah et al., 1998) among others. However, most of them work less well than binomial models and finite difference methods.

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14.1.3. The American put option and its critical stock price We provide an expression for the critical stock price for the American put. The put price is expressed in an integral form which involves first passage probabilities. Using the fact that: • the put ceases to depend on time when the critical stock price is reached and • the result that an American put corresponds to the value of a European put plus the early exercise premium, Bunch and Johnson (2000) provide a formula for the critical price. The first-passage approach and the perpetual put option case The American put price P is exercised when the stock price S hits the critical stock price Sc . When the stock price and its critical value have no discontinuities, we can write: T P = max e−rt (K − Sc )f dt, (S Sc ) (14.10) Sc

0

where T is the maturity date and K is the strike price. The first factor in the integral corresponds to a discount factor. The second factor corresponds to the payoff among exercise. The third factor f is the first-passage probability, i.e., the probability that the stock price declines from its value S to the critical value Sc for the first time t. The maximization concerns all possible functions Sc (τ ) where τ = T −t. For values of the underlying asset less than the critical price, the put value is simply its intrinsic value. For an infinite time to maturity, the critical asset price is constant and can be determined using the first-order condition. The first-passage probability is provided by Feller (1971). First, define: 1 2 S 1 + r− σ t log Z= σ St 2 where σ corresponds to the volatility of the underlying asset. Second, define: S 1 1 log a= + r − σ2 t σ Sc 2

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This term can be seen as a standardized measure of how the put is from early exercise. When γ = σ2r2 = 1, so that a is constant, Eq. (14.10) requires an expression for the first-passage time of a standardized Brownian motion. Feller (1971) shows that the following density function must be used: f (x)d(x) = √

1 2πx3

e−1/(2x) dx.

The American put value can be computed using the appropriate density obtained via the transformation x = at2 so that f (t)d(t) = √

a 2πt3

e−a

2

/(2t)

dt

in Eq. (14.10). It is possible to choose the constant in this last equation to normalize f (t), so the integral of f (t) from zero to infinity is one. Since the critical price is constant in this case, we have: K − Sc P = √ 2π

∞ 0

2

ae−rt e−a t3/2

/(2t)

dt.

When γ = 1, a is constant: a = σ1 log(S/Sc ) Using the Laplace transform Table of Abramowitz and Stegun (1972, p. 1026), the result is ∞ √ 2 k √ e−k /(4t) e−st dt = e−k s . 3 2πt 0 When s = r and k 2 = a2 , we have:

√ P = (K − Sc ) exp −( 2r/σ) log

S Sc

which is equivalent to: P = (K − Sc )

S Sc

(14.11)

A similar result is obtained by Merton (1973). In fact, the Black and Scholes (1973) partial differential equation, ∂2P ∂P 1 ∂P = rP − rS − σ2S 2 2 ∂t ∂S 2 ∂S simplifies to an ordinary equation for an infinite maturity.

(14.12)

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When γ = 1, by substituting Eq. (14.11) into: rP − rS

∂P ∂ 2P 1 − σ2 S 2 2 ∂S 2 ∂S

(14.13)

we can show that Eq. (14.11) satisfies Eq. (14.13). Equation (14.11) can be written as: 2 Sc Sc P =K . (14.14) −S S S ∂P 2 2 Since in Eq. (14.11) ∂P ∂S = −(K − Sc )Sc /S , we have ∂S = −(Sc /S) . Equation (14.14) shows that the American put price corresponds to the exercise price times the expected discount factor plus the stock price times the hedge ratio. This is similar to a European put given in Black and Scholes (1973) as:

p = Ke−rT N (−d2 ) − SN (−d1 ).

(14.15)

It is possible to obtain expressions when γ is different from one. Bunch and Johnson (2000) show that: γ Sc (14.16) P = (K − Sc ) S Equations (14.11) and (14.16) do not contain normal density functions. This is the exact result and the exact density is 2

f (t) =

log SSc e−a /(2t) √ . 2πσt3/2

Now, we consider the more difficult case of finite-lived puts. The critical stock price function for a finite time to maturity When the time to maturity is finite, the critical stock price is no longer constant. The first-passage probabilities may be estimated using the tangent approximation: ˙

log SS + SSc t −a2 /(2t) e f= √ c 2πσt3/2 where the dot indicates the time derivative. The variables Sc and a are general functions of t. Bunch and Johnson (2000) derive an expression for

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626

the critical stock price using the tangent approximation: e−β2 τ −α1 γ + Sc /X = 1+γ 1+γ

√

τ

where β2 =

σa 3 a2 ar + 2 −r− − 2t 2t 2t σt

and α1 is a positive constant. = 0. When S = Sc , we have ∂P ∂τ It is possible to use the equation providing a relationship between the American put, P and the European put p as: T rKe−rtN (−d2 (S, Sc , t))dt P =p+ 0

where, d2 (S, Sc , t) =

log

S Sc

+ (r − 12 σ 2 )t √ . σ t

The second term corresponds to the early exercise premium. Bunch and Johnson (2000) shows that: √ 2 Sc = e−(r+(1/2)σ )τ −gσ τ K

(14.17)

where, g=±

2 log

σ2 2r √ x log xe−α(r+(1/2)σ2 )2 τ /(2σ2 ) α

where, x =

K . Sc

(14.18)

The function g has typically a value about 1.5. Equation (14.17) has the appropriate asymptote for very large τ when, 1 1+γ 1 g = √ log (14.19) − r + σ2 τ . τ γ 2 It is important to know, when g = 0 and the corresponding time τ0 by setting g = 0 in Eq. (14.19): τ0 ≈

log 1+γ γ

r + 12 σ 2

.

(14.20)

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The exact expression can be obtained when g = 0 in Eq. (14.18) to: √

2 α 2 log γ(1+γ)σ 2τ 0

τ0 = 2 α σ (1 + γ) 1 − 4 (1 + γ)

This equation can be solved iteratively. Bunch and Johnson (2000) shows that 2 γ A 1 α=1− . , where A = 2 2 1+γ 1 + (1+γ) γ 2τ 4 The approximation for g corresponding to a small τ is: σ2 g = log 4er2 τ /α

(14.21)

The analysis in Bunch and Johnson (2000) shows that for typical puts, the values of g are between one and two. Equation (14.21) shows that when r = 0, there is no reason to early exercise, so Sc should be zero. In fact, when the interest rate is zero in Eq. (14.21), g tends to infinity and the critical stock price is worthless. 14.2. Valuation of American Commodity Options and Futures Options with Continuous Distributions Most options written on commodities and commodity futures are of the American type. Hence, an early exercise premium is embedded in American call and put prices. Analytical solutions for the American option pricing problems with several dividends have not yet been found. The quadratic approximation method used by Barone–Adesi and Whaley (1987), is computationally efficient. It provides an accurate, inexpensive method for the valuation of American call and put options traded on commodities and commodity futures. 14.2.1. Valuation of American commodity options The following analysis applies to commodity options for which the cost of carrying the underlying commodity is a constant proportional rate, to commodity futures options and to stock options. The model used here is

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628

an extension of the Merton (1973) and Barone–Adesi and Whaley’s (1987) model to American options. The pricing of American options involves the valuation of the early exercise premium attached to the possibility of early exercise. It is possible to show that, under certain conditions, American options should be exercised early. In fact, when the cost of carrying the underlying commodity, b, is less than the risk-less interest rate of interest r, and the option becomes deep-in-the money, the values of N (d1 ) and N (d2 ) approach 1. The European call value tends toward the quantity Se(b−r)T − Ke−rT . Since an American option may be exercised for its intrinsic value, (S − K), this value may be higher than Se(b−r)T − Ke−rT . When b = r, there is no possibility of early exercise for an American call option. This argument does not hold for American put options since for puts, as shown by Merton (1973), there is always some probability of early exercise. Since the cost of carrying any futures position is nil, i.e., b = 0, then all formulas proposed here can be used to price commodity futures options by substituting the futures price F for the commodity price S and setting b = 0. Also, when the option’s underlying asset is a non-dividend paying stock, b = r, and the formulas can be applied to stock options. European and American option values must satisfy the following partial differential equation, ∂C(S, t) ∂C(S, t) 1 2 2 ∂ 2 C(S, t) σ S + bS + − rC(S, t) = 0. 2 ∂ 2S ∂S ∂t

(14.22)

This equation applies to American options, European options, and the early exercise premium. This premium is given by the difference in value between the American and European option values, i.e., c = C(S, T ) − c(S, T ) and p = P (S, T ) − p(S, T ). Using this notation, Eq. (14.22) can be re-written for the early exercise premium as: ∂(S, t) ∂(S, t) 1 2 2 ∂ 2 (S, t) σ S + bS + − r(S, t) = 0. 2 ∂2S ∂S ∂t Let M = S2

2r σ2 , N

=

2b σ2 , τ

(14.23)

= T − t. Multiplying Eq. (14.23) by ( σ22 ) gives:

∂ 2 (S, t) ∂(S, t) M ∂(S, t) + NS − − M (S, t) = 0. ∂ 2S ∂S r ∂τ

(14.24)

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Now, define the early exercise premium as (S, k) = k(τ )f (S, k). Hence,

∂ 2 (S, t) ∂ 2S

=k

∂2f ∂ 2S

and

∂(S, t) ∂τ

= kτ f + kkτ

∂f ∂k

.

When substituting into Eq. (14.24), we obtain: S2

∂2f ∂ 2S

+ NS

∂f ∂S

kτ k ∂f − Mf 1 + 1+ = 0. rk f ∂k

If you choose k(τ ) = 1 − e−rτ , this equation becomes, S2

∂ 2f ∂2S

+ NS

∂f ∂S

−

Mf − M (1 − k) k

∂f ∂k

= 0.

(14.25)

Now, the quadratic approximation ∂fto is applied to Eq. (14.25) in order . In fact, as τ approaches 0, (∞), eliminate the term M (1 − k) ∂f ∂k ∂k approaches 0 (k approaches 1). Hence, Eq. (14.25) becomes a second-order differential equation S

2

∂ 2f ∂2S

+ NS

∂f ∂S

−

Mf = 0. k

The above equation presents two linearly independent solutions of the form f (S) = a1 S q1 + a2 S q2 . Solving this equation, the American call option value presented by Barone-Adesi and Whaley (1987) is: C(S, T ) = c(S, T ) + A2

S S∗

C(S, T ) = S − K

q2

when S < S ∗

when S ≥ S ∗

with S∗

1 − e(b−r)T N (d1 (S ∗ )) q2 1 M −(N − 1) + (N − 1)2 + 4 q2 = 2 k A2 =

N=

2r , σ2

M=

2b , σ2

k = 1 − e−rT .

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In this formula, S ∗ stands for the critical commodity price, which can be determined iteratively using the following system: S ∗ − K = c(S ∗ , T ) +

S ∗ [1 − e(b−r)T N (d1 (S ∗ )] . q2

The value of S ∗ corresponds to the value of S above which the call’s value is equal to its exercisable proceeds, (S − K). This value can be determined by a Newton–Raphson procedure or using the efficient algorithm presented by Barone-Adesi and Whaley (1987). When b is less than r, the American call value is given by the above formula. Otherwise, when b is greater or equal to r, C(S, T ) = c(S, T ) since the call will never be exercised early. In the same context, the American commodity option value P (S, T ) is given by: q1 S P (S, T ) = p(S, T ) + A1 when S > S ∗ S∗ P (S, T ) = K − S

when S ≤ S ∗

with, −S ∗

1 − e(b−r)T N (−d1 (S ∗ )) q1 1 M 2 q1 = −(N − 1) − (N − 1) + 4 , 2 k A1 =

N=

2r , σ2

M=

2b , σ2

k = 1 − e−rT .

The critical underlying commodity price is given by an iterative procedure from the following equation K − S ∗ = p(S ∗ , T ) −

S ∗ [1 − e(b−r)T N (−d1 (S ∗ ))] . q1

14.2.2. Examples and applications Tables 14.1 and 14.2 provide the values of European and American call options and the critical underlying asset price level for different parameters. Tables 14.3 and 14.4 provide the values of European and American put options and the critical underlying asset price level for different parameters.

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Extension of Asset and Risk Management in the Presence of American Options 631 Table 14.1. European and American call option prices. K = 100, r = 0.08, T = 0.25, b = 0.04, and σ = 0.2. S 90 100 110 120

c

C

S∗

0.84 4.439 11.451 20.891

0.849 4.431 11.662 20.898

214.952 214.952 214.952 214.952

Table 14.2. European and American call option prices. K = 100, r = 0.08, T = 0.25, b = 0.08, and σ = 0.2. S 90 100 110 120

c

C

S∗

0.698 3.909 10.737 19.748

0.705 3.934 10.823 20.009

120.323 120.323 120.323 120.323

Table 14.3. European and American put option prices. S = 100, r = 0.08, T = 0.25, b = −0.04, and σ = 0.2. S 90 100 110 120

p

P

S∗

9.765 3.455 0.777 0.112

10.180 3.544 0.798 0.118

87.438 87.438 87.438 87.438

Table 14.4. European and American put option prices. K = 100, r = 0.08, T = 0.25, b = 0.08, and σ = 0.2. S 90 100 110 120

p

P

S∗

10.500 3.909 0.935 0.144

10.565 3.921 0.938 0.145

83.501 83.501 83.501 83.501

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14.2.3. Valuation of American futures options From the analysis by Barone-Adesi and Whaley (1987), American futures call option formula when there are no carrying costs is: C(F, T ) = c(F, T ) + A2 (F/F ∗ )q2 C(F, T ) = F − K

when F < F ∗

when F ≥ F ∗

with F∗

1 − e−rT N (d1 (F ∗ )) q2 2r 1 M q2 = 1+ 4 , M = 2 , k = 1 − e−rT . 2 k σ A2 =

When there are carrying costs, the formula becomes: C(F, T ) = c(F, T ) + A2 (F/F ∗ )q2 C(F, T ) = F − K

when F < F ∗

when F ≥ F ∗

with, F∗ 1 − e(b−r)T N (d1 (F ∗ )) q2 1 M −(N − 1) + (N − 1)2 + 4 q2 = 2 k A2 =

N=

2r , σ2

M=

2b , σ2

k = 1 − e−rT .

The price of an American call futures option C(F, T ), is equal to the price of a European futures option c(F, T ), plus a term corresponding to the probability of early exercise A2 (F/F ∗ )q2 . The critical futures price, F ∗ , is calculated using the following equation:

F ∗ 1 − e(b−r)T N (d1 (F ∗ ) ∗ ∗ . F − K = c(F , T ) + q2 When the futures price is below F ∗ , the American option price is given by the European price plus the early exercise premium. When the futures price is above F ∗ , the American option price is given by the intrinsic value, (F − K).

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In the absence of carrying costs, the formula for the American futures put is: P (F, T ) = p(F, T ) + A1 (F/F ∗ )q1 P (F, T ) = K − F

when F > F ∗

when F ≤ F ∗

with, −F ∗∗

1 − erT N (−d1 (F ∗ )) q1 2r 1 M q1 = 1− 4 , M = 2 , k = 1 − e−rT . 2 k σ A1 =

When there are carrying costs, the formula for the American put is: P (F, T ) = p(F, T ) + A1 (F/F ∗ )q1 P (F, T ) = K − F

when F > F ∗

when F ≤ F ∗

with, −F ∗

1 − e(b−r)T N (−d1 (F ∗ )) q1 1 M 2 q1 = −(N − 1) − (N − 1) + 4 , 2 k A1 =

N=

2r , σ2

M=

2b , σ2

k = 1 − e−rT .

The critical futures price corresponding to an optimal early exercise is calculated using the following equation:

F ∗ 1 − e(b−r)T N (−d1 (F ∗ )) . K − F = p(F , T ) − q1 ∗

∗

When the futures price is above the critical price, the American futures option price is given by the sum of the European price and the early exercise premium A1 (F/F ∗ )q1 . When the futures price is above the critical futures price, the American futures option value is equal to its intrinsic value, (K − F ).

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634

Table 14.5. European and American futures call prices. K = 100, r = 0.08, T = 0.25, b = 0, and σ = 0.2. S 90 100 110 120

N (d1 )

N (d2 )

c

C

0.1624 0.5199 0.8428 0.9710

0.1417 0.4801 0.8173 0.9635

0.4360 3.900 10.7600 19.7750

0.4402 3.9251 10.8380 20.0250

Table 14.6. European and American futures put prices. K = 100, r = 0.08, T = 0.25, b = 0, and σ = 0.2. S 90 100 110 120

N (−d1 )

N (−d2 )

p

P

0.8522 0.5199 0.1826 0.0364

0.8375 0.4801 0.1571 0.0289

10.2300 3.9087 0.9607 0.1714

10.6500 4.0199 0.9807 0.1795

14.2.4. Examples and applications Tables 14.5 and 14.6 provide the values of European and American call and put futures options for different parameters. The difference between European and American call option prices correspond to the early exercise premium attached to American options. 14.3. Valuation of American Commodity and Futures Options with Continuous Distributions within Information Uncertainty This section is devoted to the valuation of American commodity options within information uncertainty. 14.3.1. Commodity option valuation with information costs Following Black (1976), we assume that the fractional change in the futures price is distributed log-normally, with a known constant variance rate, σ. We also assume, that all the parameters of the CAPM of Merton (1987), CAPMI, are constant through time. In this context, it is possible to create a risk-less hedge by taking a long position in the option and a short position in the futures contract with the same transaction date. Black (1976) assumed that a continuously re-balanced self-financing portfolio of the underlying

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futures contracts and the risk-less asset can be constructed to duplicate the payoff of the futures option. The absence of costless arbitrage oppotunities implies the following relationship between the futures or the forward price and its underlying asset: F = Se(b+λS )T where F is the current forward price, T is the option’s maturity date, b is the constant proportional cost of carrying the commodity, and λS is the information cost on the spot asset. When a hedged position is constructed and “continuously” re-balanced, using limiting arguments as in Omberg (1991), yields: 1 2 2 σ S CSS + (b + λS )SCS − (r + λC )C + Ct = 0. 2 When λS and λC are set equal to zero, this equation collapses to that in Barone-Adesi and Whaley (1987). Let T be the maturity date of the call and K be its strike price. This equation must be solved under the call boundary condition at maturity. The value of a European commodity call by Bellalah (1999) is: C(S, T ) = Se((b−r−(λC −λS ))T ) N (d1 ) − Ke−(r+λC )T N (d2 ) with

d1 = ln

S K

√ 1 2 σ T, + b + σ + λS T 2

√ d2 = d1 − σ T

and where N (·) is the cumulative normal density function. When λS and λC are equal to zero and b = r, this formula is the same as that in Black and Scholes.1 If besides, the cost of carrying the 1 In

fact, to obtain this equation, the CAPMI can be written as: ˆ ˜ ¯ S − r − λS = βS R ¯ m − r − λm , R

or ¯ S − r − λS = aβS R

˜ ˆ ¯ m − r − λm . with a = R

This equation can be written for the expected return on the spot asset and the option as: « „ ∆S = (r + λS )∆t + aβS ∆t E S « „ ∆C = (r + λC )∆t + aβC ∆t E C Multiplying this last equation by C and substituting for βC gives: E(∆C) = (r + λC )C∆t + aSβS CS ∆t.

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636

commodity is zero, this equation is equivalent to that in Black. Using the futures price F = Se(b+λS )T instead of the spot price, the equation becomes for a European futures call as: C(F, T ) = e−(r+λC )T (F N (d1 ) − KN (d2 )) with

d1 = ln

F K

1 + σ2 T 2

√ σ T,

√ d2 = d1 − σ T .

The solution for a European futures put option in the same context is: P (F, T ) = e−(r+λC )T (−F N (−d1 ) + KN (−d2 )) The term FN (d1 ) − KN (d2 ) shows that the expected value of the futures call at expiration, is the expected difference between the futures price and the strike price conditional upon the option being in-the-money times the probability that it will be in-the-money. The term e−(r+λC )T is the appropriate discount factor within a framework of incomplete information by which the expected expiration value is brought to the present. The following equation (with information costs) is the analogous of that as given by Black (1976)2 : 1 CSS S 2 σ 2 − (r + λC )C + Ct = 0. 2 Taking the expected value b and replacing E(∆S) gives: E(∆C) = (r + λS )SCS ∆t + aSβS CS ∆t + Ct ∆t +

1 CSS S 2 σ2 ∆t. 2

Making the equality between this equation, and E(∆C) = (r + λC )C∆t + aSβS CS ∆t, and simplifying gives: 1 CSS S 2 σ2 + (r + λS )SCS − (r + λC )C + Ct = 0. 2 If λS = λC = 0, this equation is the Black and Scholes equation. is possible to use the previous footnote to see this result. In fact, since the value of a futures contract is zero, the equity in the position is just the value of the option. In this context, the system: 2 It

E(∆C) = (r + λS )SCS ∆t + aSβS CS ∆t + Ct ∆t +

1 CSS S 2 σ2 ∆t 2

E(∆C) = (r + λC )C∆t + aSβS CS ∆t,

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This valuation equation applies to forward contracts, European, and American options. The American commodity option value CA (S, T ) is given by: CA (S, T ) = C(S, T ) + A2 CA (S, T ) = S − K

S S∗

q2

when S < S ∗

when S ≥ S ∗

with S∗ (1 − e(b+λS −r−λC )T N (d1 (S ∗ ))) q2 1 M −(N − 1) + (N − 1)2 + 4 q2 = 2 k A2 =

N=

2(r + λC ) , σ2

M=

2(b + λS ) , σ2

k = 1 − e−(r+λC )T .

The critical underlying commodity price is given by an iterative procedure from the following equation: ∗

S −K =C

S ∗ 1 − e(b+λS −r−λC )T N (d1 (S ∗ )) (S , T ) + . q2 ∗

The American commodity option value PA (S, T ) is: PA (S, T ) = P (S, T ) + A1 PA (S, T ) = K − S

S S∗

q1

when S > S ∗∗

when S ≤ S ∗∗

becomes: E(∆C) = Ct ∆t +

1 CSS S 2 σ2 ∆t 2

E(∆C) = (r + λC )C∆t, which gives: 1 CSS S 2 σ 2 − (r + λC )C + Ct = 0. 2 If λC = 0, this equation is the Black equation with information costs.

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638

with: −S ∗∗ (1 − e(b+λS −r−λC )T N (−d1 (S ∗∗ ))) q1 2(r + λC ) 1 M 2 −(N − 1) − (N − 1) + 4 , N= , q1 = 2 k σ2 A1 =

M=

2(b + λS ) , σ2

k = 1 − e−(r+λC )T .

The critical underlying commodity price is given by an iterative procedure from the following equation3 : K − S ∗ = P (S ∗ , T ) −

S ∗ (1 − e(b+λS −r−λC )T N (−d1 (S ∗ ))) . q1

A similar algorithm as the one developed by Barone-Adesi and Whaley (1987) can be used to determine the critical underlying price.

14.3.2. Simulation results Table 14.7 presents a sensitivity analysis of the theoretical European futures call and put values as given by Black’s model and the model derived in the previous section. The calculations are based on two alternative assumptions regarding information costs, 1% and 5%. When F = K = 100, T = 0.25, σ = 0.2, r = 0.08, and b = 0, results show that an increase in information costs from 1% to 5% reduces call values respectively from 3.8990 to 3.8602. The simulations show that model prices are less than Black’s prices for outof-the-money and in-the-money calls and are nearly equal to Black’s prices for at-the-money calls. This result shows that our model explains at least the relative mispricing of out and at-the-money calls, which is an advantage with respect to Black’s model. It is important to note that the differences in option values increase with the interaction between information costs and the other option valuation parameters. When the information costs vary from 1% to 5%, a decrease is observed in put values. The simulations performed indicate that model prices are 3 The

above solutions are written as in Barone-Adesi and Whaley (1987). They can be re-written as a function of the futures price as given by Whaley (1986) using the cost-of carry model: the relationship between the futures price and the spot price.

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Extension of Asset and Risk Management in the Presence of American Options 639 Table 14.7. Simulations of Futures call (put) option values using Black’s model, CBlack , (PBlack ) and the proposed model, CM , (PM ) for the following parameters. K = 100, T = 0.25, σ = 0.2, r = 0.08, and b = 0. Panel A, F

CBlack

CM,λC =0.01

CM,λC =0.05

PBlack

PM,λC =0.01

PM,λC =0.05

80 90 100 110 120

0 0.6963 3.9087 10.7427 19.7554

0 0.6353 3.8990 10.7358 19.7260

0 0.6310 3.8602 10.6290 19.5297

19.6611 10.4365 3.9087 0.9487 0.1414

19.6203 10.3128 3.8990 0.9483 0.1370

19.4549 10.3112 3.8602 0.9482 0.1363

T = 0.25

σ = 0.2

r = 0.12

0 0.6820 3.8698 10.6356 19.5487

0 0.6810 3.8602 10.6290 19.5297

0 0.6767 3.8217 10.5237 19.3354

19.4461 10.4356 3.8698 0.9412 0.1497

19.4249 10.4112 3.8602 0.9408 0.1493

19.2912 10.3106 3.8217 0.9393 0.1476

T = 0.5

σ = 0.2

r = 0.08

0 1.6913 5.4161 11.7325 19.9157

0 1.6833 5.3890 11.6939 19.8562

0 1.6520 5.2823 11.4235 19.4630

18.1417 11.3091 5.4161 2.1246 0.6999

18.0566 11.2433 5.3870 2.1139 0.6961

17.6991 11.1226 5.2823 2.0916 0.6916

T = 0.5

σ = 0.4

r = 0.08

2.6940 6.1331 10.8051 16.7917 19.9557

2.6806 6.1025 10.7512 16.7080 19.8562

2.6275 5.9817 10.5383 16.6123 19.4630

21.9098 15.7410 10.8151 7.1838 4.6800

21.8005 15.6625 10.7512 7.1480 4.6567

21.3688 15.3524 10.5383 7.0165 4.5645

Panel B 80 90 100 110 120 Panel C 80 90 100 110 120 Panel D 80 90 100 110 120

nearly equivalent to Black’s prices for at and out-of-the money puts. However, model prices are less than those reported in Black’s model for inthe-money puts. Since our model eliminates the overvaluation bias of these puts, this is an advantage with respect to Black’s model. The calculations performed are based on three alternative assumptions regarding information costs for European and American call options when the futures price varies from 90 to 120. Table 14.8 shows that our model prices are most of the time less or equal to Barone-Adesi and Whaley’s model prices for call options. Table 14.9 provides the theoretical values for European and American futures put options.

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640

Table 14.8. Comparisons of European and American Futures call option values using the proposed models, CEuropean , CAmerican , CBAW for the following parameters. K = 100, T = 0.25, σ = 0.2, r = 0.08, b = 0, and λC = 0.01. CEuropean

CAmerican

FCritical

CBAW

90 100 110 120

0.6950 3.8000 10.7427 19.7500

0.7407 3.9380 10.8158 20.0270

121.770 121.758 121.778 121.770

0.70 3.93 10.81 20.02

Panel B

T = 0.25

σ = 0.2

r = 0.08

λC = 0.02

90 100 110 120

0.6340 3.8800 10.7000 19.6700

0.6419 3.9201 10.8186 20.0230

121.629 121.612 121.629 121.624

0.70 3.93 10.81 20.02

Panel C

T = 0.25

σ = 0.2

r = 0.08

λC = 0.05

90 100 110 120

0.6310 3.8602 10.6290 19.5297

0.6478 3.910 10.8056 20.0183

121.3529 121.3110 121.3244 121.3240

0.70 3.93 10.81 20.02

Panel D

T = 0.5

σ = 0.2

r = 0.08

λC = 0.01

90 100 110 120

1.5830 5.3800 11.6930 19.8560

1.61087 5.46840 11.9060 20.3413

128.240 128.290 128.136 121.139

1.72 5.48 11.90 20.34

Panel E

T = 0.5

σ = 0.2

r = 0.08

λC = 0.02

90 100 110 120

1.5750 5.3600 11.6300 19.7500

1.61220 5.4639 11.8896 20.3423

127.920 127.929 127.935 127.922

1.72 5.48 11.90 20.34

Panel F

T = 0.5

σ = 0.2

r = 0.08

λC = 0.05

90 100 110 120

1.6520 5.2823 11.4235 19.4630

1.6265 5.4647 11.8724 20.3218

127.5502 127.5533 127.5430 127.5470

1.72 5.48 11.90 20.34

Panel, A, F

14.4. Valuation of American Options with Discrete Cash-Distributions Most of the traded stock and index options around the world may be exercised before their expiration dates and are therefore of the American type.

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Extension of Asset and Risk Management in the Presence of American Options 641 Table 14.9. Comparisons of European and American Futures put option values using the proposed models, PEuropean , PAmerican , PBAW for the following parameters. K = 100, T = 0.25, σ = 0.2, r = 0.08, b = 0, and λC = 0.01. PEuropean

PAmerican

FCritical

PBAW

90 100 110 120

10.3128 3.8990 0.9584 0.1710

10.6440 4.0130 0.9850 0.1800

86.780 86.132 86.770 86.710

10.58 3.93 0.94 0.15

Panel B

T = 0.25

σ = 0.2

r = 0.08

λC = 0.02

90 100 110 120

10.1870 3.8890 0.9450 0.1370

10.6310 4.00600 0.98420 0.17950

86.821 86.190 86.817 86.756

10.58 3.93 0.94 0.15

Panel C

T = 0.25

σ = 0.2

r = 0.08

λC = 0.05

90 100 110 120

10.3112 3.8602 0.9482 0.1363

10.6050 3.99600 0.98000 0.17900

86.9420 86.3589 86.940 86.889

10.58 3.93 0.94 0.15

Panel D

T = 0.5

σ = 0.2

r = 0.08

λC = 0.01

90 100 110 120

11.2433 5.3890 2.1139 0.6961

11.7240 5.5846 2.21400 0.7757

82.251 82.144 82.190 82.228

11.48 5.48 2.15 0.70

Panel E

T = 0.5

σ = 0.2

r = 0.08

λC = 0.02

90 100 110 120

11.0870 5.3620 2.1232 0.7325

11.69200 5.5656 2.20680 0.77000

82.3570 82.2556 82.299 82.336

11.48 5.48 2.15 0.70

Panel F

T = 0.5

σ = 0.2

r = 0.08

λC = 0.05

90 100 110 120

11.1226 5.2823 2.0916 0.6916

11.6400 5.5209 2.1843 0.77100

82.6640 82.5700 82.614 82.640

11.48 5.48 2.15 0.70

Panel, A, F

14.4.1. Early exercise of American options Early exercise of these options is often induced by the payment of dividends on the underlying asset. For stock options, it may be optimal to exercise a call on the last cum-dividend day to receive the dividend. In this case,

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the adapted versions of the Black–Scholes model proposed by Roll (1977), Geske (1979), and Whaley (1981) can be used. For an index option, when the dividends are distributed fairly evenly over time, B–S type models may be used. However, when the dividends tend to be clustered, early exercise may be triggered at many points in time and B–S type models can not be applied. Another potential dividendinduced reason of early exercise is due to asset carrying costs. When these costs are different from the risk-less interest rate, early exercise is also possible. Hence, for many reasons, the B–S model does not price adequately American commodity and futures options, since by definition, the model is “reserved” to European options on non-dividend paying stocks. Harvey and Whaley (1992) showed that ad-hoc valuation procedures sometimes produce large pricing errors because of the discrete and seasonal pattern of dividends on some indexes like the S&P 100 index portfolio. The most commonly used methods are the B–S model adjusted for dividends because of its ease of computation and the American-style option pricing approximations with a constant proportional dividend yield. These include the quadratic approximation of Barone-Adesi and Whaley (1987) and the Cox et al. (1979) binomial method under the assumption of a constant proportional dividend yield rate. The appropriate approach for the valuation of American options when there are discrete dividends is based on the compound option approach. Several models are proposed in this context. Given the contribution of the compound option approach to the pricing of American options, this approach is presented in detail.

14.4.2. Valuation of American options with dividends The valuation of American stock options is equivalent to the valuation of a portfolio which contains a commodity option on a stock with some distributions. These distributions are reflected in the assumed proportional rate of carrying the stock, y, and the amount of cash dividend paid once a year. For the sake of simplicity, let us use the context in Roll (1977), Geske (1979), and Whaley (1981) and assume that: • Investors can borrow and lend without restrictions at the short-term instantaneous risk-less rate r during the option’s life T . • The stock pays a cash income at some dates ti , (ti < T ). • The stock pays a dividend D and the ex-dividend instant is t, (t < T ).

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• The stock price net of the discounted dividend, S, is given by: Sτ = Pτ − De−r(t−τ ) Sτ = Pτ

for τ < t

for τ ≥ t

where P is the stock price cum-dividend. The price dynamics of the stock S are given by the stochastic differential equation: dS/S = µdt + σdz where µ and σ stand respectively for the instantaneous expected rate of return and the standard deviation of the stock price return and dz is a standard Wiener process. We can show that there exists some finite ex-dividend stock price St above (under) which the early exercise of an American call (put) may be optimal. We call this value of St , St∗ , which is easily calculated by an iterative procedure. Once this critical value is determined, the valuation by duplication technique can be implemented. Consider the following portfolio of options: (a) The purchase of a European call having a strike price K and a maturity date T ; (b) The purchase of a European call with a strike price St∗ and a maturity date (t − ) and (c) The sale of a European call option on the option defined in (a) with a strike price (St∗ + D − K) and a maturity date (t − ). The contingent payoff of this portfolio of options is identical to that of an American call. In a perfect capital market, the absence of costless arbitrage opportunities ensures that the American call value is identical to that of this portfolio. The American call value must be equal to the algebraic sum of the three options in the portfolio. The option described in (a), Ca , can be valued using the commodity option formula. Its value is given by: ca = Se(y−r)T N1 (a1 ) − Ke−rT N1 (a2 ) √ √ 1 S σ T , a2 = a1 − σ T + y + σ2 T a1 = ln K 2 where N1 (·) stands for the cumulative normal distribution.

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644

The option described in (b/), Cb can be priced using an extension of Merton’s (1973) formula for which the strike price is St∗ . Its value is given by: Cb = Se(y−r)t N (b1 ) − St∗ e−rt N (b2 ) √ √ 1 2 ∗ b1 = ln(S/St ) + y + σ t σ t, b2 = b1 − σ t 2

where St∗ is the solution to the following equation, c(St∗ , T − t, K) = St∗ + D − K. The option described in (c/) can be priced using the compound option formula proposed in Geske (1979). Its value is given by: t (y−r)T N2 a1 , b1 , cc = Se − Ke−rT N2 T ti − (St∗ + D − K)e−rtN1 (b2 ) × a2 , b2 , T where, √ 1 2 T σ T, σ a1 = ln(S/K) + y + 2 √ 1 σ t, b1 = ln(S/St∗ ) + y + σ2 t 2

√ a2 = a 1 − σ T √ b2 = b1 − σ t.

Where N2 (. . .) is the bivariate cumulative normal density function with upper integral limits a and b and a correlation coefficient ρ. Since the value of the American call is equivalent to the algebraic sum of the three options in the portfolio, we have: C = ca + cb − cc . Using the properties of the bivariate cumulative normal density function, little algebra allows the derivation of the following formula for the American call on a stock, t C = S e(y−r)t N1 (b1 ) + e(y−r)T N2 a1 , −b1 , − T t −rt −rT − K e N1 (b2 ) + e N2 a2 , −b2 , − + De−rtN1 (b2 ) T

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Extension of Asset and Risk Management in the Presence of American Options 645 Table 14.10. dividends. S 82 85 87 90 92 95 97 100 102 105 107 110 112 115 117 120 122 125

American call prices in the presence of Scr

Scr

C(S, T, K)

77.196 80.196 82.196 85.196 87.196 90.196 92.196 95.196 97.196 100.196 102.196 105.196 107.196 110.196 112.196 115.196 117.196 120.196

123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818

3.8050 4.8175 5.5758 6.8389 7.7645 9.2759 10.3636 12.1113 13.3506 15.3155 16.6922 18.8509 20.3486 22.6759 24.2774 26.7476 28.4363 31.0255

Simulations The formula obtained by Whaley (1981) is used to generate call option prices. Using the following parameters: Strike price, K = 100, Ex-dividend date, t = 6 months, Time to maturity, T = 1 year, Interest rate for 6 months, r = 0.04, Volatility of the stock, (6 months), σ = 0.2, and Dividend, D = 5. Option prices are reported in Table 14.10. The different applications presented for European options can be used with American options. Besides, the compound option approach apply to the valuation of wildcard options, some exotic and complex options. These applications will be studied in this book.

14.5. Valuation of American Options with Discrete Cash Distributions within Information Uncertainty 14.5.1. The model We use a similar context as that in Roll (1977), Geske (1979), and Whaley (1981) and assume besides that the stock price net of the discounted

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dividend, S, is: Sτ = Pτ − De−(r+λS )(t−τ ) Sτ = Pτ

for τ < t

for τ ≥ t

where P is the stock price cum-dividend and λS is the information cost regarding the underlying asset price. If we account for information costs as given by Bellalah (1999), then the cost of carry model is F = Se(y+λS )T where F is the current forward price. The option Ca is given by: ca = Se((y−r−(λC −λS ))T N1 (a1 ) − Ke−(r+λC )T N1 (a2 ) √ S 1 σ T + y + λS + σ2 T a1 = ln K 2 √ a2 = a1 − σ T where λC stands for information cost on the option market. The option described in (b/), Cb can be priced using Bellalah’s (1999) formula: Cb = Se(y−r−(λC −λS ))t N (b1 ) − St∗ e−(r+λC )t N (b2 ) √ 1 2 S + y + σ t, σ b1 = ln + λ S t ∗ St 2 √ b2 = b1 − σ t where St∗ is the solution to the following equation: c(St∗ , T − t, E) = St∗ + D − K. The option described in (c/) can be priced using: t ((y−r−(λC −λS ))T cc = Se N 2 a1 , b 1 , T ti −(r+λC )T − (St∗ + D − K)e−(r+λC )t N1 (b2 ) − Ke N 2 a2 , b 2 , T with:

√ 1 S σ T, + y + σ 2 + λS T a1 = ln K 2 √ 1 2 S + y + σ + λS t σ t, b1 = ln St∗ 2

√ a2 = a1 − σ T √ b2 = b 1 − σ t

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where N2 (. . .) is the bivariate cumulative normal density function with upper integral limits a and b and a correlation coefficient ρ. The American call is given by: t ((y−r−(λC −λS ))t ((y−r−(λC −λS ))T N1 (b1 ) + e N2 a1 , −b1 , − C=S e T t − K e−(r+λC )t N1 (b2 ) + e−(r+λC )T N2 a2 , −b2 , − T + De−(r+λC )t N1 (b2 ).

14.5.2. Simulation results Table 14.11 gives the computation of the American call value referred to as call, the option ca , the option cb , the option cc , the CRR price, the Table 14.11. Simulations of option values for the continuous-time model and the discrete time model using the following parameters. S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0, and λs = 0. Strike

Call

ca

cb

cc

CRR

ca + cb − cc

S∗

Call-CRR

100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 240

76.03 71.06 66.09 61.13 56.16 51.19 46.22 41.26 36.30 31.37 26.50 21.78 17.31 13.25 9.72 6.82 4.56 2.91 1.77 1.03 0.57 0

74.42 69.46 64.51 59.55 54.59 49.63 44.67 39.72 34.79 29.90 25.11 20.52 16.23 12.38 9.07 6.37 4.27 2.74 1.68 0.98 0.55 0

74.25 69.11 63.96 59.07 53.65 48.48 43.32 38.49 32.99 27.87 22.85 18.05 13.64 9.78 6.63 4.22 2.52 1.41 0.74 0.36 0.16 0

72.65 67.51 62.37 57.49 52.07 46.92 41.76 36.94 31.47 26.39 21.46 16.80 12.56 8.92 5.98 3.77 2.23 1.24 0.64 0.31 0.14 0

75.81 70.85 65.89 60.93 55.97 51.01 46.05 41.09 36.16 31.25 26.42 21.77 17.37 13.39 9.91 7.03 4.77 3.10 1.92 1.13 0.64 0

76.03 71.06 66.09 61.13 56.16 51.19 46.23 41.26 36.30 31.37 26.50 21.78 17.31 13.25 9.72 6.82 4.56 2.91 1.77 1.03 0.57 0

100.02 105 110 115 120 125 130 135 140 145 150 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200 240

0.22 0.21 0.20 0.19 0.19 0.18 0.17 0.16 0.14 0.11 0.07 0.01 −0.05 −0.13 −0.19 −0.20 −0.21 −0.18 −0.14 −0.10 −0.06 −0

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Table 14.12. Simulations of option values for the continuous-time model and the discrete time model using the following parameters. S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0.001, and λs = 0.01. Strike 100 120 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

CRR

ca + cb − cc

S∗

Call-CRR

76.14 56.427 36.41 31.48 26.61 21.89 17.42 13.34 9.80 6.89 4.62 2.95 1.80 1.05 0.58 0

74.56 54.73 34.93 30.04 25.25 20.65 16.35 12.49 9.16 6.44 4.33 2.78 1.71 1 0.56 0

74.34 53.73 33.06 27.94 22.92 18.11 13.67 9.83 6.66 4.24 2.53 1.42 0.74 0.36 0.17 0

72.76 52.19 31.57 26.49 21.55 16.87 12.61 8.97 6.02 3.80 2.25 1.25 0.65 0.32 0.14 0

75.94 56.11 36.30 31.39 26.56 21.90 17.49 13.50 10.01 7.10 4.84 3.14 1.95 1.16 0.65 0

76.14 56.27 36.41 31.48 26.61 21.89 17.42 13.34 9.80 6.89 4.62 2.95 1.80 1.05 0.58 0

100.02 120 140.01 145 150 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200 240

0.19 0.16 0.11 0.09 0.05 −0.01 −0.07 −0.15 −0.34 −0.21 −0.22 −0.19 −0.14 −0.10 −0.06 −0

algebric sum of the three options (ca + cb − cc ), the critical underlying asset price, and the difference between our call formula and the benchmark. Table 14.12 uses the same data except for information costs. Information costs are set equal to λS = 0.01 and λC = 0.001. Table 14.13 uses the same parameters except for the information costs which are set equal to λS = 0.1 and λC = 0.05. The results reveal that the difference between our model prices and the CRR prices are very small. Information costs offer a simple way to calibrate model prices to market data as shown in Bellalah and Jacquillat (1995) and Bellalah (1999, 2001). If we invert the information costs as in Table 14.14, we see that the mispricing bias in inverted. Table 14.14 gives the results for the different parameter values, except for information costs, which are set respectively equal to λS = 0.05 and λC = 0.1.

14.6. The Valuation Equations for Standard and Compound Options with Information Costs We denote by S(V, t) the value of the option as a function of the underlying asset and time.

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Extension of Asset and Risk Management in the Presence of American Options 649 Table 14.13. Simulations of option values for the continuous-time model and the discrete time model using the following parameters. S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0.05, and λs = 0.1. Strike 100 120 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

CRR

ca + cb − cc

S∗

Call-CRR

76.92 57.12 37.32 32.40 27.54 22.80 18.29 14.15 10.52 7.49 5.09 3.31 2.05 1.22 0.69 0

75.55 55.79 36.06 31.18 26.38 21.75 17.39 13.43 9.97 7.11 4.85 3.16 1.97 1.18 0.67 0

74.95 54.31 33.59 28.45 23.40 18.55 14.08 10.15 6.90 4.41 2.65 1.49 0.78 0.38 0.18 0

73.58 52.98 32.33 27.23 22.25 17.51 13.18 9.42 6.36 4.04 2.41 1.34 0.70 0.34 0.16 0

76.92 57.17 37.43 32.53 27.70 23.01 18.56 14.47 10.86 7.81 5.40 3.56 2.25 1.35 0.78 0

76.92 57.12 37.32 32.40 27.54 22.80 18.29 14.15 10.52 7.49 5.09 3.31 2.05 1.22 0.69 0

100.02 120 140.01 145 150 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200 240

−0 −0.05 −0.10 −0.13 −0.16 −0.21 −0.26 −0.32 −0.34 −0.32 −0.30 −0.25 −0.19 −0.13 −0.08 −0

Table 14.14. Simulations of option values for the continuous-time model and the discrete time model using the following parameters. S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0.1, and λs = 0.05. Strike 100 120 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

CRR

ca + cb − cc

S∗

Call-CRR

76.10 56.36 36.63 31.73 26.89 22.18 17.71 13.63 10.07 7.11 4.80 3.09 1.90 1.12 0.63 0

74.52 54.85 35.21 30.35 25.59 21 16.70 12.82 9.46 6.69 4.53 2.93 1.81 1.07 0.60 0

74.25 53.73 33.15 28.03 23.04 18.24 13.82 9.93 6.75 4.31 2.58 1.45 0.76 0.37 0.17 0

72.67 52.22 31.72 26.66 21.74 17.07 12.81 9.12 6.15 3.89 2.31 1.29 0.67 0.33 0.15 0

75.90 56.22 36.57 31.70 26.90 22.25 17.85 13.84 10.32 7.37 5.05 3.30 2.07 1.23 0.70 0

76.10 56.36 36.63 31.73 26.89 22.18 17.71 13.63 10.07 7.11 4.80 3.09 1.90 1.12 0.63 0

100.02 120 140.01 145 150 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200 240

0.20 0.13 0.06 0.03 −0.01 −0.07 −0.13 −0.21 −0.25 −0.25 −0.24 −0.21 −0.16 −0.11 −0.07 −0

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14.6.1. The pricing of assets under incomplete information Using the same assumptions as in the seminal paper of Black and Scholes and the additional assumption of information uncertainty on the option S(V, t) and its underlying asset, V , it is possible to construct a hedged position which contains one share of stock long and S1V options short. The term SV is the first derivative of the option with respect to its underlying asset. Since the return in the hedged position is certain, the return must be equal to (r + λV )∆t for the underlying asset and (r + λS )∆t for the option where λV and λS refer respectively to the information costs on the underlying asset and the option. The differential equation for the value of the option: 1 2 2 σ V SV V + (r + λV )V SV − (r + λS )S + St = 0. 2

(14.26)

Let T be the maturity date of the call and M be its strike price. Equation (14.26) subject to the following boundary condition at maturity: S(V, T ) = VT − M

if VT ≥ M

S(V, T ) = 0 if VT < M is solved using standard methods for the price of a European call, which is found to be equal to: S(V, T ) = V e−(λS −λV )T N1 (d1 ) − M e−(r+λS )T N1 (d2 ) with:

(14.27)

√ √ 1 V d1 = ln σ T d2 = d1 − σ T + r + σ 2 + λV T M 2

and where N1 (·) is the univariate cumulative normal density function. 14.6.2. The valuation of equity as a compound option Following Geske (1979), consider a levered firm for which the debt corresponds to pure discount bonds maturing in T years with a face value M . Under the standard assumptions of liquidating the firm in T years, payingoff the bondholders and giving the residual value (if any) to stockholders, the bondholders have given the stockholders the option to buy back the assets at the debt maturity date. In this context, a call on the firm’s stock is a compound option, C(S, t) = f (g(V, t), t) where t stands

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for the current time. The return on the firm’s assets follows the stochastic differential equation: dV /V = αv dt + σv dzv where αv and σv refer to the instantaneous rate of return and the standard deviation of the return of the firm per unit time, and dzv is a Brownian motion. Using the definition of the call C(V, t), its return can be described by: dC/C = αc dt + σc dzc where αc and σc refer to the instantaneous rate of return and the standard deviation of the return on the call per unit time, and dzc is a Brownian motion. Using Ito’s lemma as before, the dynamics of the call can be expressed as: dC =

1 Cvv σv2 V 2 dt + Cv dV + Ct dt. 2

It is possible to create a risk-less hedge with two securities, between the firm and a call to get the partial differential equation: 1 2 2 σ V Cvv + (r + λv )V Cv − (r + λC )C + Ct = 0 2 v

(14.28)

where λv in an information cost relative to the firm’s value. At the option’s maturity date, t = T0 , the value of the call option on the firm’s stock must satisfy the following condition: CT0 = max[ST0 − K, 0] where K stands for the strike price. Since the stock is viewed as an option on the value of the firm, re-call the value of ST0 : S(VT0 , T0 ) = VT0 e−(λS −λV )(T −T0 ) N1 (d1 ) − M e−(r+λS )(T −T0 ) N1 (d2 ) with: V 1 σ (T − T0 ) d1 = ln + r + σ2 + λV (T − T0 ) M 2 d2 = d1 − σ (T − T0 ). It is convenient to note that Eq. (14.28) is more difficult to solve than Eq. (14.26) because its boundary condition is a function of the solution to Eq. (14.26). At date T0 , the value of the firm making the holder of the call on the stock indifferent with regard to the exercise decision is solution to ST0 − K = 0 where ST0 is given by the last formula. Following the methodology in Geske (1979), the compound call option value with

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information costs is: C0 = V0 e

−(λc −λv )T0 −(λs −λv )(T −T0 )

e

N2 h + σv

− M e−(r+λs)T e−(λc −λs )(T0 ) N2

h, k,

T0 T

√

T0 , k + σv T ,

T0 T

. − Ke−(r+λc )T0 N1 (h).

With: 1 2 + r + λv − σv T0 σv T 0 h = ln 2 √ V0 1 k = ln σv T + r + λv − σv2 T M 2

V0 V¯

The value V¯ is determined by the following equation: ST0 − K = V¯ e−(λs −λv )(T −T0 ) N1 (k(V¯ ) + σv

T − T0 )

− M e−(r+λs)(T −T0 ) N1 (k(V¯ )) − K = where: ¯ √ V 1 2 ¯ σv T k(V ) = ln + r + λv − σv T M 2 and N2 x, y, TT0 is the bivariate cumulative normal distribution with upper integral limits x and y and TT0 is the correlation coefficient. A first special case is obtained when information costs regarding the call are equal to information costs for the stocks, i.e., λc = λs . In this case, the investors suffer sunk costs to get informed about the equity and the assets of the firm. The costs regarding the equity and the firm’s cash flows reflect the agency costs and the asymmetric information costs. The formula is: √ T0 C0 = V0 e−(λc −λv )T N2 h + σv T0 , k + σv T , T T0 −(r+λc )T − Ke−(r+λc )T0 N1 (h). − Me N2 h, k, T

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The value V¯ is determined by the following equation: ST0 − K = V¯ e−(λc −λv )(T −T0 ) N1 (k + σv T − T0 ) − M e−(r+λc)(T −T0 ) N1 (k) − K = 0. A second special case of the compound option formula is obtained when the incurred information costs are equal to a same value λ for the stock, the firm’s value and the call. In this context, the compound option formula becomes: √ T0 C0 = V0 N2 h + σv T0 , k + σv T , T T0 −(r+λ)T − Me N2 h, k, − Ke−(r+λ)T0 N1 (h) T where the value V¯ is determined from the following equation: ST0 − K = V¯ N1 k + σv T − T0 − M e−(r+λ)(T −T0 ) N1 (k) − K = 0. If the information cost is zero, this compound option pricing formula becomes the one as given by Geske (1979). The following tables present the simulation results of the above models. Table 14.15 gives the call equity values using the Black and Scholes model and our model. The following parameters are used for the simulations: M = 100, r = 0.08, T = 0.25, and σv = 0.4. The following information costs are used to generate option values: (λs = λv = 0) which correspond to the Black and Scholes case, and (λs = 0.1%, λv = 1%), (λs = 0%, λv = 1%), and (λs = 0.1%, λv = 3%). The results show that in all cases our option values are less than those Table 14.15. Simulation and comparison of the Black and Scholes model and our model for the values of European equity options. M = 100, r = 0.08, T = 0.25, and σv = 0.4. V

λs = λv = 0

λs = 0.1%, λv = 1%

λs = 0%, λv = 1%

λs = 0.1%, λv = 3%

70 80 90 100 110 120

0 0.9800 4.1206 8.9163 15.6302 23.8003

0 0.9829 4.1189 8.9085 15.6119 23.7660

0 0.9775 4.1103 8.8941 15.5912 23.7409

0 0.9780 4.0984 8.8641 15.5340 23.6489

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Table 14.16. Simulation and comparisons of European equity values as compound options in the presence of information costs using Geske’s model and our model. K = 20, M = 100, r = 0.08, T = 0.25, T0 = 0.125, and σv = 0.4.

V0

λc = 0% λs = 0% λv = 0%

λc = 2% λs = 2% λv = 2%

λc = 1% λs = 1% λv = 2%

λc = 1% λs = 2% λv = 2%

110 120 130

6.82 15.17 26.52

7.13 15.65 27.16

7.16 15.70 27.25

7.14 15.67 27.20

reported for the Black and Scholes model. The difference between the two models depends on the magnitude of information costs. Since the Black and Scholes model overvalues call option prices, our model reduces the amount of this mispricing bias. Gives the simulation results for the compound option formula with information costs and the Geske’s compound call formula for the following parameters: K = 20, M = 100, r = 0.08, T = 0.25, T0 = 0.125, and σv = 0.4. The parameters used for information costs in Table 14.2 are: Case Case Case Case

a: (λc = λs = λv = 0%); b: (λc = λs = λv = 2%); c: (λc = λs = 1%, λv = 2%) and d: (λc = 1%, λs = λv = 2%).

In case (a), we have exactly the same values as those generated by the formula given by Geske (1979). This case is a benchmark for the comparisons of our results. The table shows that the compound option price is an increasing function of the firm’s assets V . This result is independent of the values attributed to information costs. Note also that the compound option price is an increasing function of the information costs regarding the firm’s assets, λv . When λv is fixed, this allows the study of the effects of the other information costs on the option value. In this case, the option price seems to be a decreasing function of the two information costs λc and λs . When comparing cases (b) and (c) on one hand and the cases (c) and (d) on the other hand, we observe this decreasing feature. Summary The option pricing literature has evolved since the work of Black–Scholes (1973) and Merton (1973). Even if models now exist for the valuation of

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European and American call and put options on a variety of underlying assets and commodities including common stocks, bonds, agricultural futures contracts, and financial futures, there is still no-known analytical solutions for options written on commodities that have known discrete cash payments during the option’s life. Bunch and Johnson (2000) use the first-passage probability approach to derive exact expressions for the critical stock price function and for the American put price. The analysis considers the infinite or perpetual put and the finite case. The derivation is based on the fact that exercise occurs when the interest rate effect is exactly offset by the volatility effect. A similar result can be obtained, if we account for the effects of shadow costs of incomplete information since these costs are added to the interest rate. The method in Bunch and Johnson (2000) can be extended to currency options, options on futures, to exotic options, etc. This chapter presents in detail the basic concepts and techniques underlying rational pricing of American options. This is done in the context of analytical European models along the lines of Black–Scholes (1973), Merton (1973), Black (1976), Garman and Kohlhagen (for details, refer to Bellalah et al., 1998), and Barone-Adesi and Whaley (1987). Our attention is focused on the question of dividend and the privilege feature of early exercise and information uncertainty. When there are no distributions to the underlying asset, there is no incentive to exercise an American call option before its maturity date. Hence, the value of an American call is equal to that of a European call. The valuation of American calls is simpler than the valuation of American puts with and without discrete cash distributions to the underlying asset. While European and American calls have the same value when there are no distributions to the underlying asset, this result does not hold for European and American puts. The nonexistence of a put-call parity theorem for American options imply a specific treatment for put options. The pricing of American puts is rather a difficult task, even in the absence of distributions to the underlying asset. However, some interesting results are presented in the absence of dividends and when there is a continuous dividend rate. Most options written on commodities and commodity futures are of the American type. Hence, an early exercise premium is embedded in American call and put prices. Analytical solutions for the American option pricing problems with several dividends have not yet been found. The quadratic approximation method used by Barone-Adesi and Whaley (1987) is computationally efficient. It provides an accurate, inexpensive method for the valuation of American call and put options traded on commodities and

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commodity futures. In the case of discrete distributions to the underlying asset, the formulas proposed in the literature are not so efficient as those proposed for American calls. The valuation of American stock options is equivalent to the valuation of a portfolio which contains a commodity option on a stock with some distributions. The general problem of valuing American options is analyzed in three different contexts: when there are no distributions to the underlying asset, when there is a constant proportional distribution rate, and in the presence of discrete cash distributions. The analysis covers American options on spot assets and American futures options. The main results reported in the literature regarding the pricing of American calls are reviewed. Some models are presented in detail and simulations are run. Information costs play a central role in the analysis and the valuation of financial assets, firms and their cash flows. We present an arbitrage argument to derive an option pricing formula within a context of information uncertainty. The formula collapses to that in Black and Scholes in the absence of these costs. By analogy between the option theory and the assets in the capital structure of the firm, the formula can be used to value equity in a levered firm. The compound option formulas derived in this chapter shed light on the effects of leverage and information uncertainty in the valuation of corporate assets. Since several corporate liabilities can be valued with the compound option approach, our results are useful in the pricing of the capital structure of the firm.

Questions 1. 2. 3. 4.

What is the general problem in the valuation of American options? When American calls are exercised? When American puts are exercised? When American futures calls and puts are exercised?

Appendix A: An Alternative Derivation of the Compound Option’s Formula Using the Martingale Approach Using similar arguments as before, we obtain: C0 = e−(r+λC )T0 EQ [CT0 ].

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But, VT0 = V0 eσ

√ T0 ξ+(r+λV −1/2σ 2 )T0 ]

where the distribution of ξ is the standard Gaussian law. Letting g(x) = √ 2 V0 eσ T0 x+(r+λV −1/2σ )T0 ] , then: −(r+λC )T0 C0 = e (e−(λS −λV )(T −T0 ) g(x)N (d1 (x)) − Me with:

d1 (x) =

ln

−(r+λS )(T −T0 )

2 e−x /2 N (d2 (x)) − K) √ dx 2π

V

√ + σ T0 x + (r + λV )T + 1/2σ 2 T − σ2 T0 σ (T − T0 ) d2 (x) = d1 (x) − σ (T − T0 ).

0

M

Then by applying standard integral calculation (change of variables for example), we deduce the result. Exercises Exercise Show that the following bounds apply for the valuation of European call options on an underlying asset S for a maturity date T . 1. Prove that C ≤ S. 2. Prove that C ≥ max[S − Ee(r+λs )(T −t) , 0]. 3. Show that: 0 ≤ C1 − C2 ≤ (E2 − E1 )e−(r+λc1 +λc2 )(T −t) . Solution 1. To show that the call value is less than the underlying asset price, we construct the following portfolio, Π which comprises the underlying asset S and the call C: Π=S−C

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At the maturity date T , the value of the portfolio is given by: Π(T ) = S − max(S − E, 0) ≥ 0 Hence, the absence of arbitrage opportunities in efficient markets implies that: Π(t) = S − C ≥ 0. Therefore, we must have: S ≥ C. 2. As before, we can construct the following portfolio, Π which comprises the underlying asset S and the call C: Π = S − C. Note that Π(T ) ≤ E. The absence of arbitrage opportunities in efficient markets implies that Π(t) = S − C ≤ Ee−(r+λs +λc )(T −t) . Hence, we have: C ≥ S − e−(r+λs +λc )(T −t) . Since C ≥ 0, then: C ≥ max S − e−(r+λs +λc )(T −t),0 3. We construct a portfolio with two calls with two different strike prices on the same underlying asset, Π = C1 − C2 . When t = T , the value of the portfolio at maturity is given by, Π(T ) = max(S − E1 , 0) − max(S − E2 , 0) this allows to write: 0 ≤ Π(T ) ≤ E2 − E1 . Hence, the absence of arbitrage opportunities in efficient markets implies that 0 ≤ Π(T ) ≤ (E2 − E1 )e−(r+λc1 +λc2 )(T −t)

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Finally, we have: 0 ≤ C1 − C2 ≤ (E2 − E1 )e−(r+λc1 +λc2 )(T −t) . Exercise Show that the following bounds and relationships apply for the valuation of European put options on an underlying asset S with a maturity date T . 1. P ≤ Ee−(r+λp )(T −t) . 2. P ≥ Ee−(r+λp +λs )(T −t) − S. 3. 0 ≤ P2 − P1 ≤ (E2 − E1 )e−(r+λp1 +λp2 )(T −t) . Solution The arbitrage argument can be used to prove the different results. 1. To show that the put value is less than the discounted value of the strike price, we construct the following portfolio, Π which comprises the underlying asset S and the put P : Π = P − E at the maturity date, t = T , the value of the portfolio can be written as: Π(T ) = max(E − S, 0) − E ≤ 0. Hence, the absence of arbitrage opportunities in efficient markets implies that: Π(T ) = P − Ee−(r+λp +λs )(T −t) ≤ 0. Hence, P ≤ Ee−(r+λp +λs )(T −t) . 2. To prove the desired result, we construct a portfolio which comprises the put and the underlying asset Π(T ) = P + S. At the option’s maturity date t = T and the portfolio value is given by: Π(T ) = S + max(E − S, 0) ≥ E. The absence of arbitrage opportunities in efficient markets implies that: Π(t) = S + P ≥ Ee−(r+λp +λs )(T −t) .

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Hence, P ≥ Ee−(r+λp +λs )(T −t) − S. 3. We construct a portfolio comprising two puts with two different strike prices, Π = P2 − P1 . At the option’s maturity date t = T , the portfolio value is given by: Π(T ) = max(E2 − S, 0) − max(E1 − S, 0). This gives: 0 ≤ Π(T ) ≤ E2 − E1 . The absence of arbitrage opportunities in efficient markets implies that: 0 ≤ Π(T ) ≤ (E2 − E1 )e−(r+λp1 +λp2 )(T −t) hence, 0 ≤ P2 − P1 ≤ (E2 − E1 )e−(r+λp1 +λp2 )(T −t) . Exercise We consider a call C(S, t) and a put option P (S, t) on the same underlying asset S and the same strike price. The maturity date is T . 1. Since, each of these options satisfies the extended Black–Scholes equation, can you prove that a portfolio with a long call and a short put verifies also the Black–Scholes equation? 2. What are the boundary and final conditions that must be satisfied for this portfolio? Solution Since the call and the put satisfy the extended Black–Scholes equation, we have: 1 ∂ 2C 2 2 ∂C ∂C + − (r + λc )C = 0 σ S + (r + λs )S 2 ∂t 2 ∂S ∂S

(14.29)

1 ∂2P 2 2 ∂P ∂P + − (r + λp )P = 0. σ S + (r + λs )S ∂t 2 ∂S 2 ∂S

(14.30)

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The difference between the two equations gives: ∂P 1 ∂ 2C ∂2P ∂C − + − σ2 S 2 ∂t ∂t 2 ∂S 2 ∂S 2 ∂C ∂P + (r + λs )S − − (r + λc )C + (r + λp )P = 0 ∂S ∂S

(14.31)

which can be written as: ∂C ∂ 2P ∂P 1 ∂ 2C − σ2 S 2 − + ∂t ∂t 2 ∂S 2 ∂S 2 ∂P ∂C − − (r(C − P ) + λc C − λp P ) = 0. + (r + λs )S ∂S ∂S We denote by V (S, t) = C(S, t) − P (S, t). In this case, the portfolio with a long call and a short put satisfies the extended Black–Scholes equation. Hence, we have: 1 ∂ 2V 2 2 ∂V ∂V + − (r + λv )V = 0 σ S + (r + λs )S ∂t 2 ∂S 2 ∂S where, (r + λv )V = (r(C − P ) + λc C − λp P ). This can be written as: r(C − P ) + λv (C − P ) = r(C − P ) + λc C − λp P or, λv (C − P ) = λc C − λp P. Hence, λv (C − P ) = λc (C − P ) + λp (C − P ) + λc P − λp C = (λ+ λp )(C − P ) + λc P − λp C By identification, the information cost on the portfolio corresponds to the sum of both costs on the call and the put option, λc + λp = λv Hence, λc P = λp C then: C λc = . λp P

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The equation can be written as: 1 ∂2V 2 2 ∂V ∂V + − (r + λv + λp )V = 0 σ S + (r + λs )S ∂t 2 ∂S 2 ∂S with, λc P = λp C. Hence, the portfolio V (S, t) = C(S, t) − P (S, t) satisfies the extended Black–Scholes equation. 2. We can show that the boundary boundary conditions for the portfolio are, V (0, t) = C(0, t) − P (0, t) and

∂V − (r + λv + λp )V = 0 ∂t

then 1 ∂V = r + λc + λp V ∂t and, we have: V (0, t) = −Ee−(r+λc +λp )(T −t) . Another boundary condition is: V (S, t) = C(S, t) − P (S, T ) when, S → +∞. The terminal condition at the maturity date T is: V (S, T ) = C(S, T ) − P (S, T ) = max(S − E, 0) − max(E − S, 0) = S − E. Exercise We look for a random walk followed by a European option V (S, t). The extended Black–Scholes equation can be used to simplify the equation for dV .

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Solution Consider the following dynamics of the underlying asset. dS = µSdt + σSdX. Using Ito’s lemma for the function V (S, t) gives: ∂V 1 ∂ 2V 2 2 ∂V dt. dS + + σ S dV = ∂S ∂t 2 ∂S 2

(14.32)

According to the extended Black–Scholes equation, we have ∂V 1 ∂ 2V 2 2 ∂V + = 0. σ S = (r + λv )V − (r + λs )S ∂t 2 ∂S 2 ∂S

(14.33)

Replacing Eq. (14.33) in Eq. (14.32) gives: dV = Hence: dV =

∂V ∂V dS + (r + λv )V dt − (r + λs )S dt. ∂S ∂S

∂V ∂V ∂V dS + r V − S dt + λv V − λs S dt ∂S ∂S ∂S

or, dV = (r + λv )V dt + σS

∂V ∂V dX + (µ − (r + λs ))S dt ∂S ∂S

and finally:

∂V ∂V dV = σS dX + (r + λv )V + (µ − rλs ) S dt. ∂S ∂S

Exercise We consider the extended Black equation for the pricing of futures options and the Black–Scholes equation with a constant continuous dividend yield D = r. How futures options are priced when we know the value of an option with the same payoff for the spot asset? Solution We denote by W (F, t) the value of an option as a function of the futures price F and time t. This option must satisfy the following equation: ∂W 1 ∂ 2W 2 2 σ F − (r + λw )W = 0. + ∂t 2 ∂F 2

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We denote by V (S, t) the value of a spot option whose value depends on the underlying spot asset S, time t, and a continuous dividend yield D. The value of this option must satisfy the extended Black–Scholes equation: ∂V 1 ∂ 2V 2 2 ∂V + − (r + λv )V = 0. σ S + (r + λs − D)S 2 ∂t 2 ∂S ∂S When D = r, the first equation becomes: 1 ∂ 2W 2 2 ∂W + σ F − (D + λw )W = 0 ∂t 2 ∂F 2 and the second equation becomes: 1 ∂2V 2 2 ∂V ∂V + − (D + λv )V = 0. σ S + λs S 2 ∂t 2 ∂S ∂S Both equations are identical only when λs = 0 and λw = λv . This means that the knowledge of the method to price the spot option allows to price the futures option when the asset pays a dividend equal to the interest rate by accounting for the effect of information uncertainty. References Abramowitz, M and IA Stegun (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, D.C.: U.S. Government Printing Office. Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June), 301–320. Bellalah, M. (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September), 645–664. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3) (August), 617–635. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bellalah, M, JL Prigent and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973.

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Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Black, F and J Cox (1976). Valuing corporate securities: some effects of bond indenture conditions. Journal of Finance, 31 (May), 351–367. Blomeyer, EC (1986). An analytic approximation for the American put price for options on stocks with dividends. Journal of Financial and Quantitative Analysis, 21 (June), 229–233. Bunch, DS and H Johnson (2000). The American put option and its critical stock price. Journal of Finance, 55(5), 2333–2357. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7, 229–263. Feller, W (1971). An Introduction to Probability Theory and Its Applications. New York: John Wiley. Galai, D and R Masulis (1976). The option pricing model and the risk factor of stock. Journal of Financial Economics, 3, 53–81. Garman, M and S Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance, 2, 231–237. Geske, R (1979). The valuation of compound options. Journal of Financial Economic, 7, 375–380. Geske, R and HE Johnson (1984). The American put option valued analytically. Journal of Finance, 39 (December), 1511–1524. Geske, R and K Shastri (1985). Valuation by approximation: a comparison of alternative option valuation techniques. Journal of Financial and Quantitative Analysis, 20 (March), 45–71. Grabe, JO (1983). The pricing of call and put options on foreign exchange. Journal of International Money and Finance, 2, 239–253. Harrison, JM and S Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications, 11, 215–260. Harrison, JM and D Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408. Harvey, C and R Whaley (1992). Dividends and the S-P 100 index option valuation. Journal of Futures Markets, 12(2), 123–137. Johnson, HE (1983). An analytic approximation to the American put price. Journal of Financial and Quantitative Analysis, 18(March), 141–148. McKean, Jr (1969). Stochastic Integrals. New York: Academic press. McMillan, L (1986). Analytic approximation for the American put option. Advances in Futures and Options Research, 1, 119–139. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, R (1987). An equilibrium market model with incomplete information. Journal of Finance, 42, 483–511. Omberg (1991). On the theory of perfect hedging. Advances in Futures and Options Research, 5, 1–29.

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Roll, R (1977). An analytical valuation formula for unprotected American call options with known dividends. Journal of Financial Economics, 5, 251–258. Samuelson, P (1972). The Collected Scientific Papers of Paul Samuelson. RC Merton (ed.). Cambridge, MA: MIT Press. Whaley, R (1981). On the valuation of American call options on stocks with known dividends. Journal of Financial Economics, 9, 207–211. Whaley, RE (1986). Valuation of American futures options: theory and empirical tests. Journal of Finance, 41 (March), 127–150.

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Chapter 15 RISK MANAGEMENT OF BONDS AND INTEREST RATE SENSITIVE INSTRUMENTS IN THE PRESENCE OF STOCHASTIC INTEREST RATES AND INFORMATION UNCERTAINTY: THEORY AND TESTS

Chapter Outline This chapter is organized as follows: 1. Section 15.1 develops the main concepts for the valuation of bond options and interest-rate options. 2. Section 15.2 presents a simple non-parametric approach to bond futures option pricing. 3. Section 15.3 provides one-factor interest-rate modeling and the pricing of bonds in a general case by accounting for the effects of information uncertainty. 4. Section 15.4 shows how fixed income instruments can be valued as a weighted portfolio of power options. 5. Section 15.5 develops in detail the Merton’s model for equity options in the presence of stochastic interest rates. 6. Section 15.6 provides some models for the pricing of bond options. 7. Appendix A presents in detail an analysis of Government bond futures and their implicit embedded options. 8. Appendix B shows how one-factor interest-rate models are used in practice. 9. Appendix C presents in detail the derivation of Merton’s model in the presence of stochastic interest rates. 667

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Introduction Several authors have derived alternative formulas to the basic Black and Scholes (1973) model (B–S) for the pricing of stock options, index options, bond options, and foreign currency options, when interest rates are not constant. However, until 1989, all the proposed models, except Merton (1973), used the assumption of a constant free rate prevailing during the option’s life, i.e., the effects of the interest rate’s variance and covariance with the underlying asset’s return on the option’s price were precluded from models other than that of Merton (1973). Options on the Chicago Board of Trades’ (CBOT) Treasury bond futures were introduced in October 1982. It is well known that interest-rate options are more difficult to value than stock options, currency options, index options, and most futures options. The Black (1976) model applied to fixed-income futures options is based on the assumption of a known constant interest rate, while long-term rates and their associated note and bond futures prices are uncertain. The most popular alternatives to the Black model depend in general on an ad hoc dynamic assumption about the dynamics of the short-term interest rate, which in turn constrains their comovements with long-term interest rates. One-factor term structure models are widely used in the pricing of interestrate derivatives that cannot be accomodated by Black’s (1976) model. In these models, changes in the yields of all maturities are perfectly correlated, at least instantaneously. This is the case for affine one-factor term structure models as the Ho and Lee (1986), Hull and White (1990), and Cox et al. (1985). In these models, the yield on each zero-coupon bond is a linear function of the short-term interest rate. This implicit assumption affects the volatility of forward interest rates and option prices. The standard models of interest rates proposed by Vasicek (1977), Cox et al. (1985), Brennan and Schwartz (1977) etc., have two major disadvantages. First, they have a significant number of parameters which are not observable. Second, they are not often consistent with the current term structure of interest rates. This chapter presents some models for the analysis and valuation of derivative assets whose values depend on stochastic interest rates. Since Merton’s model represents a convenient starting point for most of the complete option pricing models, it deserves a special treatment in this chapter. We present specific models for the valuation of bonds and contingent claims whose values depend directly on the dynamics of interest rates. Positions in government bond futures have several implicit embedded options. These positions corresponding to several specifications of the day (delivery date and end-of-month options), the time of delivery (wildcard

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Risk Management of Bonds and Interest Rate Sensitive Instruments

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options), the specific bond in a deliverable basket (the delivery switch options) etc. The nominal short-term interest rate cannot be negative since people can hold currency at a zero nominal rate. The real interest rate can be negative since low-risk investment opportunities do not lead necessarily to positive returns. In the same context, the inflation rate can also be negative during periods of great depressions. However, at the microeconomic or the macroeconomic levels, all the costs of information relative to immaterial investments among others are ignored in financial economics. If we admit that the nominal short-term rate is the “shadow real interest rate, plus a “shadow” cost of incomplete information and the expected inflation, or zero, then the nominal short-term rate can be seen as an option as in Black (1995). 15.1. The Valuation of Bond Options and Interest Rate Options Several models are proposed in the finance literature for the estimation of the option fair price. Most models are based on an arbitrage or risk-less valuation argument. The most well-known model is the Black and Scholes (1973) model. 15.1.1. The problems in using the B–S model for interest-rate options To illustrate the problems in using this model, consider a six-month European call on a six-year zero-coupon bond. The bond’s value at maturity is 100. The strike price is 130. Since this bond’s value never exceeds 100, the option will not be exercised and its value must be zero. However, if you consider a given interest rate and volatility, this option has some value in the Black and Scholes (1973) model. Since the model is based on an assumption of a log-normal distribution for the underlying zero-coupon bond, this means that there is a probability that the asset takes on any positive value. This means that the bond price can be above 100. This situation is also possible in the presence of negative interest rates. In the same vein, the value of a coupon-paying bond cannot be greater than the sum of the coupon payments plus the maturity value. Hence, the use of such values can lead to nonsensical option prices. The B–S model assumes also that the short-term interest rate is constant. However, a change in the short-term interest rate modifies the rates along the yield curve. Hence, it is inappropriate to assume constant interest rates in the pricing of interest

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rate options. The third assumption regarding a constant variance for prices is also inappropriate. Therefore, it is appropriate to use models which account for the yield curve and do not allow arbitrage opportunities: yield curve option pricing model and arbitrage for option pricing models. Option pricing models use six parameters: the current price of the underlying bond, the strike price, the short-term risk-free rate, the coupon rate, the time to maturity, and the expected interest rate volatility. This last factor is unknown and must be estimated. This parameter can be estimated by assuming that the option is priced correctly and to imply the interest rate volatility from the option pricing model. 15.1.2. Sensitivity of the theoretical option prices to changes in factors The shape of the curve between the theoretical call option price and the underlying bond is convex. The change in the call price with respect to a small change in the underlying bond price refers to the option delta. The lambda refers to the percentage change in the call price with respect to the percentage change in the underlying bond price. For example, a lambda of 1.25 shows that the call price will change by 1.25% for a change in the underlying bond price by 1%. The measure of convexity for call options is defined by the gamma or the change in delta with respect to a change in the underlying bond price. The theta is given by the change in the option price with respect to a decrease in the time to expiration. The vega also known as the kappa gives the change in the option price with respect to a 1% change in the expected interest rate volatility. As we define the modified duration of a bond as an indicator of its price sensitivity to variations in interest rates, it is possible to define the modified duration for an option as follows: modified duration for an option = modified duration of the underlying asset×(delta)×(price of the underlying asset option price). It is also possible to use for interest-rate options, the put-call parity relationship which can be written for a coupon-bearing bond as: put = call + present value of the strike price + present value of the coupons − price of the underlying bond. 15.2. A Simple Non-Parametric Approach to Bond Futures Option Pricing Options on the CBOTs’ Treasury bond futures were introduced in October 1982. It is well known that interest-rate options are more difficult to value than stock options, currency options, index options, and most futures

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options. The Black (1976) model applied to fixed-income futures options is based on the assumption of a known constant interest rate while longterm rates and their associated note and bond futures prices are uncertain. The most popular alternatives to the Black’s model depend in general on an ad hoc dynamic assumption about the dynamics of the short-term interest rate, which in turn constrains their comovements with long-term interest rates. Stutzer and Chowdhury (1999) proposed a combination of the riskneutral valuation framework and the flexibility of practitioner methods in the pricing of options. A canonical valuation model is a simplified, riskneutral valuation method allowing the model user to specify an individual assessment of the distribution of the underlying asset at the option maturity date. This distribution is used to estimate risk-neutral probabilities needed to value the option. Stutzer and Chowdhury (1999) showed that the canonical model predicts that in the historically typical range of bond futures prices, the Black’s model implied volatility of in-the-money calls must be higher than of other calls. They showed that this pattern is more pronounced for short-term options. They found that the canonical model predicts that implied volatilities should, in general, be much higher when the futures price is near historic lows. The cannical model seems to outperform the Black’s model.

15.2.1. Canonical modeling and option pricing theory Black’s model can be used for European futures options in the following form: Call = e−rT [F N (d1 ) − KN (d2 )] F 1 2 log K + 2 σ T √ d1 = , σ T

(15.1)

√ d2 = d1 − σ T

where F is the current underlying futures price and X is the option’s strike price. This model and any other parametric model, determine implicit estimates of both actual and risk-neutral probabilities. By contrast, canonical valuation gives the user the possibility to specify a particular assessment about the actual distribution on the underlying asset at expiration. This provides the basis for risk-neutral probabilities. Hence, the futures price growth rate until T does not have to be normal or to conserve the same distribution for each possible current futures price, as it is assumed in Black’s model.

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Stutzer and Chowdhury (1999) used the history of bond prices to form a catalog of histograms of futures price growth rates. This allows to determine non-normal distributions of the futures price growth rate, which varies with the current futures price. 15.2.2. Assessing the distribution of the underlying futures price Daily closing prices or the CBOT bond futures are used for the year 1996. The ratio of two prices is recorded as the price relative or gross “return” RhF . The T-ahead futures price is denoted by Ph (T, F ) = F RhF . Assigning equal probabilities gives a simple non-parametric estimate of the T -ahead probability distribution of the futures price, conditional on the current futures price, F . The N -month-ahead probability distributions are illustrated by histograms. The results show substantial skewness of the returns distribution when the current futures price is relatively low, i.e., when bond yields are high. This means that the probability of unusually large, positive increases in F is higher when the starting F is relatively low. Also, the results show a higher volatility associated with the returns in periods of low futures prices, i.e., high bond yields. These properties are absent in the Black’s model. 15.2.3. Transforming actual probabilities into risk-neutral probabilities A canonical risk-neutral probability distribution is a distribution satisfying the following martingale constraint: the risk-neutral expected value of the time T -ahead futures price must equal the current futures price. For each combination of T and F , the following convex problem is solved: ˆ = arg P max Π h

Π(h)=1

−

H

Π(h) log Π(h)

(15.2)

h=1

subject to the martingale constraint requiring that F equals the risk-neutral probability Π(h) weighted average of the T -ahead futures price Ph (T, F ): Π(h)Ph (T, F ) = F h

The maximand in Eq. (15.2) corresponds to the Shannon entropy of the risk-neutral distribution. The risk-neutral probabilities obtained by

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ˆ solving Eq. (15.2) are known as the canonical probability distribution Π(h), h = 1, . . . , H. The canonical model value of a call is computed as: Call =

H

ˆ Π(h) max[Ph (T, F ) − X, 0]e−rT

(15.3)

h=1

15.2.4. Qualitative comparison of Black and canonical model values The results show that the Black’s model has a tendency to relatively and absolutely underprice in-the-money calls. Also, the relative mispricing of in-the-money calls persists even when using an implied volatility. The most comprehensive empirical studies of the CBOT bond futures options reveal that the Black’s model of futures options suffers from moneyness bias, which is similar to the one documented for B–S model of stock index options. Stutzer and Chowdhury (1999) used a canonical model to value CBOT bond futures without any assumption implying a specific parametric form for the underlying futures price distribution. They observed that Black’s model underpriced in-the-money calls relative to others and that the implied volatilities are inversely related to the strike price. This mispricing bias does not seem to appear in the canonical model. 15.3. One-Factor Interest Rate Modeling and the Pricing of Bonds: The General Case The interest rate that applies to the shortest possible deposit is known as the spot interest rate. The dynamics of the spot rate can be modeled by the following stochastic differential equation dr = u(r, t)dt + w(r, t)dX where the functional forms u(r, t) and w(r, t) can be specified in different contexts. 15.3.1. Bond pricing in the general case: The arbitrage argument and information costs Consider the pricing of an interest-rate sensitive instrument V (r, t, T ) when the interest rate is stochastic. Since the interest rate is not a traded asset, the pricing of bonds is different from the pricing of traded assets. If V (r, t, T ) refers to the price of a bond, then the implementation of a hedging strategy in a B–S context needs the use of another bond with a different maturity as a hedging instrument. Consider the following portfolio comprising a long

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position in a bond V1 (r, t, T ) and a short position in ∆ units of the bond V2 (r, t, T ) Π = V1 − ∆V2

(15.4)

It is possible to apply Ito’s lemma to find the change in the portfolio value over a short interval of time dt. dΠ =

∂V1 1 ∂ 2 V1 ∂V1 dt dt + dr + w2 ∂t ∂r 2 ∂r2 ∂V2 ∂V2 1 2 ∂ 2 V2 −∆ dt dt + dr + w ∂t ∂r 2 ∂r2

(15.5)

In order to construct a hedged portfolio, we have to eliminate the risk component in the dr terms. In this case, the hedge ratio must be equal to 1 2 / ∂V . Using arbitrage arguments as in the B–S context, the return ∆ = ∂V ∂r ∂r on the hedged portfolio must be equal to the risk-free rate or the spot rate plus information cost on each market (or an asset). This gives:

∂V1 ∂V2 ∂V2 1 2 ∂ 2 V2 dΠ = dt + w ∂r ∂r ∂t 2 ∂r2 ∂V1 ∂V2 V2 dt = (r + λV1 )V1 dt − (r + λV2 ) ∂r ∂r ∂V1 1 ∂ 2 V1 − + w2 ∂t 2 ∂r2

Collecting the terms in V1 and V2 and re-arranging this equation gives: ∂V1 ∂t

2

+ 12 w2 ∂∂rV21 − (r + λV1 )V1 ∂V1 ∂r

=

∂V2 ∂t

2

+ 12 w2 ∂∂rV22 − (r + λV2 )V2 ∂V2 ∂r

.

Since this equation has two unknowns, and the right and left-hand sides differ by the maturity, this means that we can write this without the maturity. In this case, we have: ∂V ∂t

2

+ 12 w2 ∂∂rV2 − (r + λV )V ∂V ∂r

= a(r, t).

It is possible to write the function a(r, t) as follows: a(r, t) = w(r, t)γ(r, t) − u(r, t)

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In this context, the bond pricing equation is given by: 1 ∂ 2V ∂V ∂V + w2 2 + (u − γw) − (r + λV )V = 0 ∂t 2 ∂r ∂r

(15.6)

where λV stands for the information cost in the bond market and γ(r, t) corresponds to the market price of risk. Since the zero-coupon bond price at maturity T is V (r, T ; T ) = 1. This gives the final condition that must be imposed in the search for the solution to this equation. In the presence of a coupon-paying bond, the coupons must be integrated in the equation as follows: 1 ∂ 2V ∂V ∂V + w2 2 + (u − γw) − (r + λV )V + C(r, t) = 0 ∂t 2 ∂r ∂r If the coupon is paid discretely, then the following condition must be satisfied: + V (r, t− c ; T ) = V (r, tc ; T ) + C(r, tc ),

where tc refers to the coupon-payment date, t− c the instant just before the instant just after the coupon payment. the coupon payment, and t+ c This condition reflects the jump in the bond price at the coupon date. The difference between the bond price before and after the coupon date corresponds to the coupon payment. In order to explain the market price of risk, consider an investor who holds an unhedged position in a bond maturing in T years. The change in the bond value over a small interval of time dt is given by: ∂V 1 2 ∂ 2V ∂V ∂V dX + + w dt dV = w +u ∂r ∂t 2 ∂r2 ∂r Using the previous bond pricing equation, this may be written as: ∂V ∂V dV = w dX + wγ + (r + λV )V dt ∂r ∂r or in an equivalent way: dV − (r + λV )V dt = w

∂V (dX + γdt) ∂r

(15.7)

The term dX in this equation reveals that this portfolio is not riskless. The deterministic term in γ in the right-hand side can be seen as the excess

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return above the risk-free rate. The extra risk is compensated by an extra γdt per unit of extra risk, dX. The solution of the bond pricing equation can be seen as the expected value of all cash flows where the expectation is taken with respect to the risk-neutral variable rather than the real variable. In fact, the drift in the equation does not correspond to the drift of the real spot rate u, but rather to the drift of a “risk-neutral spot rate”. The drift of this rate is (u − γw) and its dynamics are given by dr = (u − γw)dt + wdX. 15.3.2. Pricing callable bonds within information uncertainty A callable bond is a bond with a call provision that allows the issuer to call back on specified dates for a given amount of the issue. The price of a callable bond satisfies the following equation: ∂V 1 ∂2V ∂V + w2 2 + (u − γw) − (r + λV )V = 0 ∂t 2 ∂r ∂r where λV corresponds to the information cost related to the debt market. At maturity, the callable bond price converges to V (r, T ) = 1. At a couponpayment date, the following relationship is applied: If the coupon is paid discretely, then the following condition must be satisfied: + V (r, t− c ) = V (r, tc ) + C + where t− c refers to the instant just before the coupon payment and tc refers to the instant just after the coupon payment. This condition reflects the jump in the bond price at the coupon date. If the call price is Cp , then the bond price must satisfy the following condition V (r, t) ≤ Cp .

15.4. Fixed Income Instruments as a Weighted Portfolio of Power Options Hart (1997) showed that a fixed income instrument which is either convex or concave with respect to the interest rate can be viewed as a weighted portfolio of power options on interest rates through a polynomial tansformation of the option payout. This allows the pricing of swaps (on Eurodollar futures) in a Black and Scholes context. If swaps are viewed as bonds, the swap convexity can be valued using Black’s (1976) model. There is a relationship between swaps, bonds, and the forward rate agreements (FRAs). A plain vanilla swap of the type “receive fixed, pay floating” can

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be assimilated to a long position in a coupon-paying bond and a short position in a floating-rate note. It can also be viewed as a series of FRAs where the holder of the “receive fixed” position pays the difference between the observed three-month LIBOR rate and the fixed swap coupon at each re-set date. Consider a two-year swap, starting one year forward. In the presence of annual re-sets, the value of the swap in one year is the sum of the cash flows multiplied by their respective forward discount functions: C exp(−Ra ) + (C + 100) exp(−2Rb ) where c: fixed swap coupon; Ra : forward rate for one year and Rb : two-year rate (in a year). This value of a two-year swap in one year is equivalent to the value of a bond paying an annual coupon. It is also equivalent to the value of two zero-coupon bonds at time one year, one paying c in two years and the other paying (c + 100) in three years. Consider one of the two zero-coupon bonds (the second), and denote by Vt the value of the cash flow in t years as: V1 = X exp(−2Rb ) where X = (c + 100). This relationship is exponentially decreasing and represents the value of an option as a function of the rate Rb , which is the underlying asset. This reasoning also applies for the first bond. More generally, for a forward contract on a zero-coupon bond (with maturity T , yield Y for purchase at t), the value of the bond at time t is an exponentially decreasing function of its yield: e−Y (T −t) per dollar face value. In the above example, T = 3, t = 1, and Y is the forward rate which will give the value of the bond at t = 1. Using Taylor’s expansion, Hart (1997) provided the exponential curve as a weighted portfolio of simpler curves of increasing power. This portfolio can then be valued using a series of Black and Scholes power options using the results in Hart and Ross (1994). The value of a European power 2 call is: √ √ S 2 exp(t(r − 2d + σ 2 ))N [x + σ t ] − K exp(t(−r))N [x − σ t ] S ln K + (r − d)t 1 √ √ + σ t x= 2 σ t

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where S: K: r: d:

underlying asset price; strike price; risk-less rate to t and continuous dividend yield or the risk-less foreign rate. The generalization for any power n gives:

√ √ S n exp(−t(r − n(r − d + σ2 /2) − (nσ2 )/2))N [x − σ t + nσ t ]

(15.8)

Using Taylor’s expansion of e−Y (T −t) with Eq. (15.8) gives an iteration of the following sum for exponentials: inf

1/n!(−Y τ )n exp(−t(r − n(r − d − σ2 /2) − (nσ2 )/2))

n=0

where τ = T − t. The first four terms of this expression are: 1 exp(−rt) + (−Y τ ) exp(−dt) + (−Y τ )2 exp(−t(−r + 2d − σ 2 )) 2 1 + ((−Y τ )3 exp(−t(−2r + 3d − 3σ2 )), . . . 6 Using the Black framework for r = d gives: exp(−rt) + (−Y τ ) exp(−rt) + 1/2(Y τ )2 exp(−t(r − σ2 )) +

1 (−Y τ )3 exp(−t(r − 3σ 2 )), . . . 6

This result gives the value of a forward zero-coupon swap in a Black–Scholes economy with volatility σ and Eurodollar-implied yield Y that includes a convexity. This result is similar to the time value of an option. This formula can be used for a range of 5–15 terms. 15.5. Merton’s Model for Equity Options in the Presence of Stochastic Interest Rates: Two-Factor Models Consider the basic case of a European option with no payouts to the underlying asset. Assume that there are no transaction costs that trading takes place continuously and that no restrictions are imposed on borrowing and short selling. Assume also that the borrowing rate is equal to the lending rate.

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15.5.1. The model in the presence of stochastic interest rates Let C(S, P, τ, K) be the option price function depending on the stock price S, a bond price P , the strike price K, and the time remaining to maturity τ with τ = T − t, where T is the maturity date and t is the current time. The solution presented in Merton (1973) for the call price is: 1 [xerf c(h1 ) − erf c(h2 )] 2

y(x, T ) = where h1 = −

ln(x) + 12 T √ , 2T

and

T =

0

τ

h2 = −

ln(x) − 12 T √ 2T

[σ2 + δ 2 − 2ρσδ]du

and where erf c(.) is the error complement function 2 erf c(h) = 1 − √ 2Π

0

h

e−

w2 2

dw.

In the special case when r = 0, σ2 = 1, K = 1, Eq. (15.21) is identical to the Black–Scholes equation. The call price in the context of Merton’s model can be written in a B–S form as: c = SN (d1 ) − P (τ )KN (d2 ) S √ ln K − ln(P ) + 12 στ2 τ √ , d2 = d1 − στ τ d1 = στ τ In the same context, the put price is given by: p = −SN (−d1 ) + P (τ )KN (−d2 ) S √ − ln(P ) + 12 στ2 τ ln K √ , d2 = d1 − στ τ d1 = στ τ with P (τ ) = e

−rτ

and

στ2

1 = τ

τ 0

[σ 2 + δ 2 − 2ρσδ]du.

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15.5.2. Applications of Merton’s model Options on yields of short-term financial instruments are traded in the over-the-counter (OTC) market. These financial products are in the form of bank deposits, certificates of deposit, Treasury bills, commercial papers and so on. Options on bank deposit rates and on treasury bill yields Treasury securities are backed by the faith and credit of the government. They are regarded as having no credit risk. The interest rates on Treasury securities are often used as a benchmark in national and international capital markets. There are two categories of government assets: discount and coupon assets. Treasury bills are securities issued with maturities of a year or less as discount securities, which do not make periodic payments. All the assets having a longer maturity are issued as coupon securities. When the maturity is between 2 and 10 years, the issued security is a note. When the maturity is greater than 10 years, the Treasury security is a bond. Re-call that a fundamental property of a bond is that its price moves in the opposite side with respect to the required yield, i.e., when the required yield increases, the present value of the par value and the coupon decreases and vice versa. In general, the price yield relationship is convex. The yield on any investment corresponds to the interest rate which makes the present value of the cash flows equal to the cost of the investment. The yield is simply an implicit interest rate. In practice, two measures of yield are used: the current yield and the yield to maturity. The current yield gives the relationship between annual coupon interest and the market price. It is simply given by the ratio of the annual dollar coupon interest to the market price. The yield to maturity is the implicit interest rate that makes the present value of the future cash flows until the bond’s maturity date equal to the bond’s price. Merton’s (1973) model can be applied to the valuation of these options and is more appropriate than the Black and Scholes (1973) model since interest rates are stochastic. Short-term options on long-term bonds Transactions on short-term options on long-term bonds take place often in the OTC market. Some options are also traded on organized markets. These options may be either European or American. A European short-term option on a long-term bond is traded on the Chicago Board Options

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Exchange (CBOE). This option is similar to the option on yield of shortterm financial instruments traded on the CBOE and is cash settled. The interest rate corresponds to the average of the yields to maturity of the two most recently auctioned 7-year Treasury notes and two most recently auctioned 10-year Treasury notes and also the two most recently auctioned 30-year Treasury bonds. The average of the yield to maturity of these six instruments gives the interest rate on which the option is priced. This option can be priced using Merton’s model which is more appropriate than the B–S model in the context of stochastic interest rates.

15.6. Some Models for the Pricing of Bond Options Hull and White (for details, refer to Bellalah et al., 1998) developed a unifying context for the pricing of all interest-rate dependent securities based on the term structure of interest rates and their volatilities. Bond options are often priced using a version of the B–S model. Caps, collars, and floors are often priced by assuming that forward interest rates are lognormal. The main difference between the dynamics of bonds and stocks is that the bond price tends to reach its face value at maturity. This is not the case for the stock price. European options can be valued using the Black’s (1976) model by assuming that forward interest rates are lognormal. In this case, volatility is a decreasing function of the option time to maturity. It is important to recognize that when forward bond prices are log-normal, this does not mean that forward interest rates are also lognormal. The standard models of interest rates proposed by Vasicek (1977), Cox et al. (1985), Brennan and Schwartz (1977) and so on have two major disadvantages. First, they have a significant number of parameters which are not observable. Second, they are not often consistent with the current term structure of interest rates.

15.6.1. An extension of the Ho-Lee model for bond options Ho and Lee model allows the pricing of several interest rate derivatives in a consistent way. The main drawbacks of this model are that actual changes in interest rates are normally distributed and that all spot interest rates and forward interest rates are equally variable. However, these disadvantages can be overcome as shown in Heath et al. (for details, refer to Bellalah et al., 1998), Jamshidian (1989), and Hull and White (1990). In fact, the short-term interest rate process can be adjusted to eliminate the possibility

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of negative interest rates. In this case, the Ho-Lee model can be written as: √ dr = θ(t)dt + σ rdz. It is also possible to introduce mean reversion in the Ho-Lee model using some possible models suggested in Hull and White (for details, refer to Bellalah et al., 1998). The two possible models accounting for mean reversion are: dr = (θ(t) + a(t)(b − r))dt + σdz and √ dr = (θ(t) + a(t)(b − r))dt + σ rdz where b is constant and a is a function of time. Hull and White (1990) derived simple formulas for the pricing of European interest rate options. They considered the following dynamics for the short-term interest rate r: dr = (θ(t) + a(t)(b − r))dt + σdz, which can also be written as: dr = (φ(t) + a(t)r)dt + σdz. We denote by A(t, T )e−B(t,T )r the price at time t of a discount bond with a maturity T . The bond price volatility is σB(t, T ). For a period between T1 and T2 , the volatility of a forward rate F is given by: σ(B(t, T2 ) − B(t, T1 )) . F (T2 − T1 ) In this model, B(0, T ) can be obtained using the current term structure of volatilities and A(0, T ) from the term structure of interest rates. Hull and White (1990) showed that: B(t, T ) =

(B(0, T ) − B(0, t)) , (δB(0, t)/δt)

a(t) = −

(δ 2 B(0, t)/δt2 ) (δB(0, t)/δt)

Hull and White showed that the price of a European call option maturing at time T on a discount bond maturing at time s is: call = P1 N (h) − XP2 N (h − v)

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with h=

log PP21X v

v + , 2

v 2 = (B(0.τ ) − B(0, T ))2

T

0

σ δB(0, τ )/δτ

2 dτ

where P1 and P2 refer to the prices of discount bonds maturing at times τ and T , respectively. 15.6.2. The Schaefer and Schwartz model Schaefer and Schwartz (for details, refer to Bellalah et al., 1998) assumed that the bond’s price volatility is proportional to the bond’s duration. Schaefer and Schwartz (for details, refer to Bellalah et al., 1998) developed a modified B–S model for the pricing of bond options. In their model, the price volatility of a bond increases with duration. The call price is given by: c = Se(b−r)T N (d1 ) − Xe−rT N (d2 ) S √ + (b + 12 σ 2 T ) ln X √ , d2 = d1 − σ τ , d1 = σ τ σ0 σ = D(KS (α−1) ), K = D.S (α−1) where: D: D∗ : α: σ0 :

duration of the bond after the option’s maturity date; duration of the bond today; a constant which may be equal to 0.5 and the observed price volatility of the bond.

15.6.3. The Vasicek (1977) model The dynamics of interest rates in Vasicek (1977) model are given by: dr = κ(θ − r)dt + σdz Bond prices The time t price of a discount bond maturing in T is given by: P (t, T ) = A(t, T )e−B(t,T )r(t) and B(t, T ) =

[1 − e−κ(T −t) ] κ

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with A(t, T ) = e

[(B(t,T )−T +t)(κ2 θ−0.5σ2 )] κ2 −σ2 B(t,T )2 /4κ

.

Prices of European options The European call price is: c = P (t, τ )N (h) − XP (t, T )N (h − σp ) where: 1 P (t, τ ) h= ln + 0.5σp , σp P (t, T )X

σp = B(t, τ )

σ 2 [1 − e−2κ(T −t) ] 2κ

The European put price is: p = XP (t, τ )N (−h + σp ) − P (t, τ )N (−h) When κ = 1.5%, θ = 2%, σ = 8%, r = 5%, P = 100, X = 100, t = 0, T = 2, and τ = 3, we have: B(t, T ) = 1.97029776, A(t, T ) = 1.00778006, and P (t, T ) = 0.91323235. For the valuation of options, we have: σp = 33.48%, h = 97.98%, B(t, τ ) = 2.993350121, A(t, τ ) = 1.38806545, and P (t, τ ) = 1.19869799. The call price is 32.632 and the put price is 4.085. When κ = 1.5%, θ = 2%, σ = 8%, r = 5%, P = 100, X = 100, t = 0, T = 2, and τ = 2, we have: B(t, T ) = 1.97029776, A(t, T ) = 1.00778006, and P (t, T ) = 0.91323235. For the valuation of options, we have: σp = 33.48%, h = 16.74%, B(t, τ ) = 1.97029776, A(t, τ ) = 1.00778006, and P (t, τ ) = 0.91323235. The call price is 12.141 and the put price is 12.141. 15.6.4. The Ho and Lee model The dynamics of the spot rate in the Ho and Lee (1986) model are given by: dr = θ(t)dt + σdz. In this model, bond prices are given by: P (t, T ) = A(t, T )e−r(t)(T −t) and

ln A(t, T ) = ln

P (0, T ) P (0, t)

− (T − t)δ ln P (0, t)/δt − 0.5σ2 t(T − t)2 .

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European options The European call price is: c = P (t, τ )N (h) − XP (t, T )N (h − σp ) where: h=

1 P (t, τ ) + 0.5σp , ln σp P (t, T )X

σp = σ(τ − T )T − t]

The European put price is: p = XP (t, T )N (h − σp ) − P (t, τ )N (h) 15.6.5. The Hull and White model The dynamics of interest rates in this model are given by: θ(t) − r dt + σdz dr = κ κ where

θ(t) κ

is a time-dependent mean-reversion level.

Bond prices The time t price of a discount bond maturing in T is given by: P (t, T ) = A(t, T )e−B(t,T )r(t) and B(t, T ) =

[1 − e−κ(T −t) ] κ

with ln A(t, T ) = e

[(B(t,T )−T +t)(κ2 θ−0.5σ2 )] κ2 −σ2 B(t,T )2 /4κ

where ν(T, t)2 =

σ2 2κ3 (e−κT − e−κT )2 (e2κT − 1)

.

685

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Prices of European options The European call price with maturity date T on a zero-coupon bond maturing in t is: c = P (0, τ )N (h) − XP (0, T )N (h − ν(T, t)) where: h=

P (0, τ ) 1 ln + 0.5ν(T, τ ). ν(T, t) P (0, T )X

Summary Several models are proposed in the finance literature for the estimation of the option fair price. Most models are based on an arbitrage or risk-less valuation argument. The most well-known model is the Black and Scholes (1973) model. The most comprehensive empirical studies of the CBOT bond futures options reveal that the Black’s model of futures options suffers from moneyness bias, which is similar to the one documented for B–S model of stock index options. Stutzer and Chowdhury (1999) used a canonical model to value CBOT bond futures without any assumption implying a specific parametric form for the underlying futures price distribution. They observed that Black’s model underpriced in-the-money calls relative to others and that the implied volatilities are inversely related to the strike price. This mispricing bias does not seem to appear in the canonical model. Hart (1997) showed that a fixed income instrument, which is either convex or concave with respect to the interest rate can be viewed as a weighted portfolio of power options on interest rates through a polynomial transformation of the option payout. This allows the pricing of swaps (on Eurodollar futures) in a Black and Scholes context. If swaps are viewed as bonds, the swap convexity can be valued using Black’s (1976) model. In this chapter, the model developed by Merton (1973) for the pricing of stock options in the context of stochastic interest rates is presented in great detail. This model represents a starting point towards the theory of option pricing when interest rates are uncertain. The main ideas in Merton’s model were used later by many authors for the pricing of a large number of interest-rate sensitive claims. The literature on the pricing of bonds and bond options is concerned mainly with the stochastic process describing the dynamics of interest rates. Since there are several approaches for the modeling of term structure dynamics, we give the main results in the work

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of Vasicek and Cox et al. We also present an interesting model of interest rates, in Heath et al. which accounts for several reasons of the term structure movements. Much work is currently done on the term structure modeling. The introduction of information costs in the analysis of interest rates and the valuation of financial assets is equivalent to applying an additional discount rate for the computation of the present value of future risky cash flows. The shadow costs are introduced in Merton’s (1987) model of capital market equilibrium with incomplete information. The concept of information costs is used to illustrate the limiting case of money demand behavior (the liquidity trap) within an option framework. When the shortterm rate is close to zero, this context allows to illustrate the effect of the “currency” option embedded into nominal interest rates on savings, investments, and the yield curve. Our analysis can be extended to develop formal models of interest rates and derivatives.

Questions 1. 2. 3. 4. 5.

What are the different categories of government assets? What is a Treasury security? What is the main property of a bond? What are the different measures of yields? Which of the models presented in this chapter is appropriate for the pricing of short-term options on long-term bonds? 6. What are the valuation parameters in Merton’s model? 7. What are the valuation parameters in Heath, Jarrow, and Morton’s model? 8. What are the specificities of the Heath, Jarrow, and Morton’s model?

Appendix A: Government Bond Futures and Implicit Embedded Options Positions in government bond futures have several implicit embedded options. These options corresponding to several specifications of the day (delivery date and end-of-month options), the time of delivery (wildcard options), the specific bond in a deliverable basket (the delivery switch options) etc. Dabansens and Bento (1997) approximated the delivery switch option as a simple combination of European calls and puts for which the underlying asset corresponds to the cheapest to deliver (CTD) bond yield and the deliverable bond yield spread. These options present the same

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time to maturity and different strike prices. They defined simple creteria for the CTD under yield changes, quantified the value for a short position, and presented the European call and put options values on the bond yield. A.1. Criteria for the CTD The CTD is defined as the bond with the lowest basis net of carry B or the lowest implied futures price (IFP). The basis net of carry corresponds to the cost of a cash and carry trade on the bond: B = basis-carry B = (P (t) + AI(t))(1 + m/36,000) − [Crec + Al(t ) + F (t)CF ] where t: t : P (t): AI(t): n: Crec : CF :

(A.1)

value date for each cash bond settlement; delivery date of the futures contract; clean price of the bond; interest accrued at time t; number of days between the cash and the futures settlement dates; coupon received in the holding period (per face value) and conversion factor. This criteria is used in the determination of the CTD.

The implied futures price for each bond is defined by: IF P =

P (t) CF

(A.2)

where IF P : implied futures price; P (t): clean price of the bond and CF : conversion factor. A.2. Yield changes When market conditions change, this makes one bond referred to as the new CTD cheaper than the original CTD. This produces CTD switches. A new bond becomes the CTD when the difference between its B and that of the original CTD tends to become zero. Let us consider in a first step the case of parallel yield shift and write the modified duration M D of a bond as: dP = −P.M D.dy

(A.3)

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where y refers to the bond yield. The CTD bond is determined by comparing the B of various bonds for each of the N bonds in the basket using the variable: ∆BN = BN − BCT D . The change in the yield (referred to as Switchpar (N ) which makes the bond N the CTD is equivalent to: dBN (Switchpar (N )) = −∆BN

(A.4)

When Eq. (A.1) is used to get the first derivative of BN with respect to dy, this gives: dBN dPN dF = (1 + m/36,000) − dy dy dy

(A.5)

Using Eqs. (A.3) and (A.5), Dabansens and Bento (1997) gave the following expression for the yield switch for bond N : Switchpar (N ) =

∆BN (1 + m/36,000)

× [M DN (t)PN (t) − (CFN /CFCT D )M DCT D (t)PCT D (t)]−1 In this context, the futures price related to this yield shift (F utSwpar (N )) is: F (t) − F utSwpar (N ) =

PCT D M DCT D CFCT D

(1 + m/36,000)Switchpar (N )

Let us consider in a second step the case of a relative yield shift. Assuming no change in the yields of other bonds, the yield change for the bond N , Switchrel (N ) can be calculated using a similar condition as in the previous case: dBN (Switchrel (N )) = −∆BN . The relative yield switch for bond N to become the CTD is: Switchrel (N ) =

∆BN (1 + m/36,000)(M DN (t)PN (t))

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A.3. The value for a futures position How the holder of a cash and carry position on the CTD bond profits from a switch in the CTD? By construction, we have: IF PN (TS ) = IF PCT D (TS ), where Ts refers to the time when the CTD switches from the original CTD bond to bond N from the basket. In this context, the expression of the profit at futures settlement is given by: P rofN = max (0, CFCT D (dIF PCT D − dIF PN )) CFCT D = max 0, dPCT D − dPN CFN where: dIF PN , (dIF PCT D ): change in IFP of bond N (or the CTD) between the switch and futures settlement and dPN , (dPCT D ): changes in bond price (or the CTD) between the switch and futures settlement. We denote by dyN , (dyCT D ), the yield change of bond N , (CTD) between switch and futures settlement. Using modified duration for firstorder changes of bond price allows to write the profit as:

P rofN = max 0, −PCT D (TS )M DCT D (TS )dyCT D +

PN (TS )M DN (TS )dyN CFCT D CFN

We simplify the notation and define: ACT D = PCT D (TS )M DCT D (TS ) and XN =

PN (TS )M DN (TS )CFCT D CFN

in order to write the profit as: P rofN = max[0, −ACT D .dyCT D + XN .dyN ]. Now, it is possible to answer the following question: What is the fair value of this switch under different market scenarios? The values of delivery options can be approximated under a parallel yield shift and a relative yield shift.

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A.4. Parallel yield shift Under a parallel yield shift, the time of switch to CTD is TSpar dyCT D = dyN and the profit is: P rofN = max(0, Kpar .dyN ) with a strike price: Kpar = −ACT D (TSpar ) + XN (TSpar ) Let us denote by: Optpar (N ): the value of the delivery switch option related to bond N ; and P yN (.): the probability distribution function of yN . The value of the option is studied in two cases. In the first case, if: Kpar ≥ 0(XN ≥ ACT D ) the profit: P rofN = Kpar max(0, dyN ),

dyN ≥ 0

is positive when dyN > 0 or: yN ∈ (yN 0 + Switchpar (N ), +∞), where yN 0 corresponds to the present yield of bond N . In this case, the option value is given by: ∞ Optpar (N ) = Kpar P (yN = y)(y−yN 0 −Switchpar (N ))dy yN 0 +Switchpar (N )

In the second case, if: Kpar ≤ 0(XN ≤ ACT D ),

P rofN = −Kpar max(0, −dyN )

then the option value is given by: Optpar (N ) = −Kpar

yN 0 +Switchpar (N )

0

+ Switchpar (N ) − y)dy

P yN (yN = y)(yN 0

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A.5. Relative yield shift Under relative yield shift, the time of switch to CTD is TSrel dyCT D = 0 and yN represents the yield spread between bond N and the CTD. The profit is: P rofN = Krel max(0, dSprN ) with: yN − yCDT = SprN ,

Krel = XN (TSrel )

The value of the delivery switch option related to bond N is given by: ∞ PSN (SprN = S) Optpar (N ) = −Krel sprN O+Switchrel (N )

× (SprN − SprN O − Switchrel (N ))ds where PSN (.) is the probability distribution of SprN . This framework allows the valuation of delivery switch options in cases of both parallel and non-parallel yield shifts. The above analysis can be implemented to quantify the cheapness of a futures contract relative to the CTD. It can also be used in the selection of a cash and carry or a reverse cash and carry trade with a bond other than the CTD. For further details, see the original paper by Dabansens and Bento (1997). Appendix B: One-Factor Fallacies for Interest Rate Models One-factor term structure models are widely used in the pricing of interest rate derivatives that cannot be accommodated by Black’s (1976) model. In these models, changes in yields of all maturities are perfectly correlated, at least instantaneously. This is the case for affine one-factor term structure models as the Ho and Lee (1986), Hull and White (1990), and Cox et al. (1985). In these models, the yield on each zero-coupon bond is a linear function of the short-term interest rate. This implicit assumption affects the volatility of forward interest rates and option prices. Using time series data, for the estimation of model parameters, the volatility of forward interest rates will be underestimated, which in turn, yield to lower option prices. Besides, when parameters are estimated by calibrating the model to market data, implied volatilities of zero-coupon

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interest rates will be higher than in reality. Hence, it is risky to use onefactor models in the pricing of instruments other than the ones on which the model is calibrated. Using market data, Klaassen et al. (1998) showed when the deficiencies of one-factor term structure models lead to large errors. B.1. The models in practice Since it is nearly impossible to obtain the equality between model prices and market prices, (a perfect fit), model parameters are chosen to minimize a certain measure of discrepancy between model and market prices. This is referred to as calibration. It is often observed that model parameters resulting from the calibration on prices of caps and floors differ significantly from those obtained from the calibration on swaptions. In fact, using one-factor models, Klaassen et al. (1998) found that the cap/floor volatility curve lies about 10% higher than the swaption volatility curve. This difference seems to persist over time. This difference can be understood with reference to spreads between rates and the calculations of correlations. B.2. Spreads between rates Caps, floors, and swaptions are viewed as options on forward interest rates. Klaassen et al. (1998) showed that a forward interest rate can be written as a weighted difference between two zero-coupon interest rates to account for the fact that volatility depends on the correlation between zero-coupon rates. Let us denote by f t1 t2 the forward rate between two instants as: er1 t1 eft1 t2 (t2 − t1 ) = er2 t2 where r1 , (r2 ) refers to the continuously compounded zero-coupon rate for maturity r1 , (r2 ). This forward rate can be written as a weighted average between two zero-coupon rates: ft1 t2 =

r2 t2 − r1 t1 t2 − t1

In this context, the volatility of the forward rate is given by:

t21 σ(r1 )2 + t22 σ(r2 )2 − 2t1 t2 ρ(r1 , r2 )σ(r1 )σ(r2 ) σ(ft1 t2 ) = t2 − t 1

(B.1)

(B.2)

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where ρ(r1 , r2 ) refers to the correlation between the two rates and σ corresponds to the volatility. This relation shows the effect of the correlation between zero-coupon rates on the standard deviation of a forward rate. Denoting by ∆ = t2 − t1 and deriving σ(ft1 t2 ) with respect to ρ gives: δσ(ft1 ,t2 ) t1 (t1 + ∆) σ(r1 )σ(r2 ) =− δρ(r1 , r2 ) ∆2 σ(ft1 ,t2 )

(B.3)

Klaassen et al. (1998) estimated correlations between zero-coupon interest rates from daily movements of Deutschmark swap rates for the period 1993– 1998. They found that when volatilities of zero-coupon rates equal their empirical values, the volatilities of forward rates (and option prices) are severely underestimated by one-factor models. When market prices of caps, floors, and swaptions are used in the calibration, model parameters must be chosen such that the forward-rate volatility implied by the option price is matched by the model very closely. Hence, volatilities of zero-coupon rates must be chosen higher than that they are in reality. The analysis reveals how the implicit assumption of perfect correlation between zero-coupon interest rates in one-factor models affects the pricing of options.

Appendix C: Merton’s Model in the Presence of Stochastic Interest Rates Let C(S, P, τ, K) be the option price function depending on the stock price S, a bond price P , the strike price K, and the time remaining to maturity τ with τ = T − t, where T is the maturity date and t is the current time. The stock price dynamics are represented by the stochastic differential equation: dS = αdt + σdW S

(C.1)

where α: instantaneous expected return on the common stock and σ 2 : instantaneous variance of return, restricted to be a known function of time. The bond price dynamics are given by the following equation: dP = µ(τ )dt + δ(τ )dq(t, τ ) P

(C.2)

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where P (τ ): µ(τ ): δ 2 (τ ): dq(t, τ ):

695

price of a discounted loan with maturity τ satisfying P (0) = 1; instantaneous expected return; instantaneous variance with δ(0) = 0 and a standard Gauss–Wiener process.

Assume that there is no serial correlation among the returns on any of the assets: dq(s, τ )dq(t, T ) = 0 for s#t; dq(s, τ ))dW (t) = 0

for s#t

and dq(t, τ )dq(t, T ) = ρτ T dt where ρτ T may be less than 1 for τ #t. When the interest rate is constant over time, this means that δ = 0, µ = r, and p(t) = e−rτ . Assuming that investors agree on the values of (δ, σ), Ito’s lemma gives the change in the option price over time. ∂C ∂C ∂C dS + dP + dτ ∂S ∂P ∂τ 2 2

2 1 ∂ C ∂ C ∂ C 2 2 + + + 2 (dS) (dP ) dSdP 2 ∂ 2S ∂2P ∂S∂S

dC =

(C.3)

Using the properties of stochastic calculus, the values of (dS)2 , (dP )2 , and (dS)(dP ) are given by: (dS)2 = (αSdt + σSdW )2 = σ2 S 2 dt

(C.4)

(dP )2 = (µP dt + δP dq)2 = δ 2 P 2 dt

(C.5)

(dS)(dP ) = (αSdt + σSdW )(µP dt + δP dq) = σδSP (dW )(dq) = δρσSP dt where ρ corresponds to the instantaneous correlation coefficient between the returns on the stock and on the bond.

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Substituting from Eqs. (C.1) and (C.2), Eq. (C.3) is re-written as:

∂C ∂C ∂C dC = [αSdt + σSdW ] + [µP dt + δP dq] + dτ ∂S ∂P ∂τ 2 2

2 ∂ C ∂ C ∂ C 1 2 2 2 2 S dt + P dt + 2 σ δ σδρSP dt + 2 ∂ 2S ∂ 2P ∂S∂S which is equivalent to:

2 ∂C ∂C 1 ∂C ∂ C + (µP ) − + σ2 S 2 dC = (αS) ∂S ∂P ∂τ 2 ∂2S 2 2 ∂ C 1 ∂ C + σδρSP dt + δ2P 2 2 2 ∂ P ∂S∂S ∂C ∂C + σSdW + δP dq ∂S ∂P or in a simple form: dC = βCdt + γCdW + ηCdq

(C.6)

with

∂C ∂C ∂C 1 ∂ 2C 1 (αS) + (µP ) − + σ2 S 2 β= C ∂S ∂P ∂τ 2 ∂ 2S 2 1 ∂2C ∂ C 2 2 δ P + σδρSP + 2 ∂2P ∂S∂S S ∂C S ∂C γ=σ , η=δ C ∂S C ∂S The expression for β represents the instantaneous expected return on the option. Consider now a hedged portfolio with the underlying asset, the option and risk-less bonds where the portfolio weights w1 , w2 , and w3 are chosen to eliminate market risk. The aggregate investment in the portfolio y is zero when investors are allowed to use proceeds from short sales and to borrow without restrictions to finance long positions, so that: w1 + w2 + w3 = 0. In this context, the instantaneous dollar return

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dy, may be written as: dy = w1

dP dS + w2 [βdt + γdW + ηdq] + [−w1 − w2 ] S P

or dy = w1 [αdt + σdW ] + w2 [βdt + γdW + ηdq] + [−w1 − w2 ][µdt + δdq] Re-arranging these terms yields: dy = [w1 (α − µ) + w2 (β − µ)]dt + (w1 σ + w2 γ)dW + [ηw2 − (w1 + w2 )δ]dq

(C.7)

Now consider a strategy where the weights wj = wj∗ are chosen so that the stochastic terms in Eq. (C.7) affecting dW and dq are always zero. The expected return on this strategy must be zero since it requires zero investment. Hence, w1∗ (α − µ) + w2∗ (β − µ) = 0 w1∗ σ + w2∗ γ = 0

(C.8)

−w1∗ δ + w2∗ (η − δ) = 0 This linear system presents a solution, if and only if: (β − µ) γ (δ − η) = = (α − µ) σ δ

(C.9)

η γ = 1− σ δ

(C.10)

If Eq. (C.9) holds, then:

This implies from the definition of γ and Eq. (C.6) that: S ∂C P ∂C =1− C ∂S C ∂P or

C =S

∂C ∂S

+P

∂C ∂P

Using Eq. (C.17) gives: (β − µ) =

γ(α − µ) σ

.

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Re-call the definitions of β and γ in (C.6), we have: 0=

∂ 2C 1 2 2 ∂ 2C P + δ ∂ 2S 2 ∂ 2P 2 ∂C ∂ C + (µS) + σδρSP ∂S∂S ∂S ∂C ∂C − − µC + (µP ) ∂P ∂τ 1 2 2 σ S 2

(C.11)

Using Eq. (C.6) and re-arranging terms, Eq. (C.11) can be re-written as: 0=

2

2 1 2 2 ∂2C ∂ C ∂ C ∂C 2 2 P + 2σδρSP σ S + δ − 2 ∂2S ∂2P ∂S∂S ∂τ (C.12)

This is a second-order linear partial differential equation of the parabolic type. The price of any option in the Merton’s economy must satisfy Eq. (C.12). In particular, the price of a European call must satisfy this equation and the following boundary conditions: C(0, P, τ, K) = 0 and C(S, 1, 0, K) = max[0, S − K]. Using the change in variables, x = KPS(τ ) , let us re-write the change in variables as follows: x=

S u = ; KP (τ ) v

dS = αdt + σdz S and dP = µ(τ )dt + δ(τ )dq(t, τ ). P The system can also be written in a matrix form. Taking the partial derivatives with respect to u, v, uu, vv, uv, gives 1 1 ∂f −1 1 ∂f = = , = 2 = , ∂u v P (τ ) ∂u∂v v P (τ )2 2 2 −S ∂ f ∂ f 2u 2S ∂f −u , = = , = 2 = = 0. ∂2v v KP (τ ) ∂v v P (τ )2 ∂ 2u

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Applying Ito’s lemma for x = gives: dx =

S KP (τ )

699

and replacing each partial derivatives

−S 1 2S 2 1 2 dS + (δP ) dP + dt − σSδpηdt KP 2 KP KP 2 KP 2

or dx =

−S 1 S 1 dS + (δ)2 dt − σSδP ηdt dP + 2 KP KP KP KP

If we substitute for dS and dP in this last equation, we have: dx =

1 −S [µ(τ )P dt + δ(τ )P dq] [αSdt + σSdz] + KP KP 2 S [(δ)2 − σδpη]dt + KP

which can be written as: dx =

σS −S −Sδ S S 1 αSdt + dz + µdt + dq + (δ)2 dt − σδηdt KP KP KP KP KP KP

This last equation can be written using the definition of x =

S KP

dx = αxdt + σxdz − µxdt − δxdq + δ 2 xdt − xσδηdt or dx = αdt + σdz − µdt − δdq + δ 2 dt − σδηdt x Isolating the drift terms gives: dx = (α − µ + δ 2 − σδη)dt + σdz − δdq x It is clear that the expected instantaneous return on x is a function of S and P and that the variance ν 2 (τ ) = σ 2 + δ 2 + 2σδη depends only on τ . It is possible to obtain a solution to this last equation using: h(x, τ ; K) = C(S, P, τ, K)

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and using (H, x) in lieu of (H, S) to get the following system: 1 2 2 ν x h11 − h2 = 0, 2

h(0, τ, K) = 0,

h(x, 0, K) = max[0, x − 1]

We denote by T =

τ −0

ν 2 (s)ds

and y(x, T ) = h(x, τ ) This last system can be written as 1 2 x y11 − y2 = 0, 2

y(0, T ) = 0,

h(x, 0) = max[0, x − 1]

Using the inverse change of variables and simplifying gives:

t S ν 2 (s)ds C(S, P, τ, K) = KP (τ )y ; KP (τ ) −0 The solution presented in Merton (1973) for the call price is: y(x, T ) =

1 [xerf c(h1 ) − erf c(h2 )] 2

(C.13)

where h1 = −

ln(x) + 12 T √ , 2T

h2 = −

ln(x) − 12 T √ 2T

and T =

τ 0

[σ 2 + δ 2 − 2ρσδ]du

and where erf c(.) is the error complement function. 2 erf c(h) = 1 − √ 2Π

0

h

e−

w2 2

dw

In the special case, when r = 0, σ2 = 1, K = 1, Eq. (C.13) is identical to the B–S equation. The call price in the context of Merton’s model can be

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written in a B–S form as: c = SN (d1 ) − P (τ )KN (d2 ) S √ ln K − ln(P ) + 12 στ2 τ √ , d2 = d1 − στ τ d1 = στ τ In the same context, the put price is given by: p = −SN (−d1 ) + P (τ )KN (−d2 ) S √ − ln(P ) + 12 στ2 τ ln K √ , d2 = d1 − στ τ d1 = στ τ with P (τ ) = e−rτ

and στ2 =

1 τ

τ 0

[σ 2 + δ 2 − 2ρσδ]du.

References Briys, E, M Bellalah et al. (1998). Options, Futures and Exotic Derivatives. En collaboration avec E. Briys, et al., John Wiley & Sons. Black, F (1995). Interest rates as options. Journal of Finance, 50 (December), 1411–1416. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Brennan, MJ and E Schwartz (1977). Saving bonds, retractable bonds, and callable bonds. Journal of Financial Economics, 5, 67–88. Cox, J, J Ingersoll and S Ross (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407. Dabansens, F and F Bento (1997). Swiching on to bonds. Risk, 10(1), (January), 68–73. Hart, I (1997). Unifying theory. Risk, 10(2) (February), 54–55. Hart, I and M Ross (1994). Striking continuity. Risk, 7(6), 50–51. Hull, J and A White. Coming to terms. Black holes. Ho and Lee (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41, 1011–1029. Hull, J and A White (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592. Jamshidian, F (1989). An exact bond option formula. Journal of Finance, 44 (March), 205–209. Klaassen, P, E Van Leewen and B Schreurs (1998). One-factor fallacies. Risk, 11 (December) 1998, 56–59.

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Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, R (1987). An equilibrium market model with incomplete information. Journal of Finance, 42(3), 483–511. Stutzer, M and M Chowdhury (1999). A simple non-parametric approach to bond futures option pricing. Journal of Fixed Income, 8(4), 67–76. Vasicek, O (1977). A equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188.

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Chapter 16 MODELS OF INTEREST RATES, INTEREST-RATE SENSITIVE INSTRUMENTS, AND THE PRICING OF BONDS: THEORY AND TESTS

Chapter Outline This chapter is organized as follows. 1. Section 16.1 provides simple examples of interest-rate sensitive instruments and explains the main concepts in bond pricing. 2. Section 16.2 studies interest rates and the pricing of bonds under certainty and uncertainty. 3. Section 16.3 develops some standard models for the pricing of bonds and bond options. 4. Section 16.4 discusses the relative merits of the competing models. 5. Section 16.5 provides a comparative analysis of term structure estimation models. 6. Section 16.6 presents some new evidence on the expectations hypothesis. It gives some term premium estimates from zero-coupon bonds. 7. Section 16.7 studies the distributional properties of spot and forward interest rates using the following currencies: USD, DEM, GBP, and JPY. 8. Appendix A provides some applications of interest-rate models to account for the effects of information costs. 9. Appendix B shows how to implement the Black–Derman and Toy model (BDT) with different volatility estimators.

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Introduction The fixed income market refers to the global financial market where various interest-rate sensitive products are traded. The management of interest-rate risk refers to the control of changes in value of the cash flows due to the changes in interest rates. Market conventions can vary from one country to another. The Treasury bill or T-bill is issued in general for a short-term period and corresponds to a discount bond for which the buyer receives the face value at maturity with no coupons. The pricing of bonds and interest-rate options needs a specification of the dynamics of interest rates under certainty and uncertainty. Several interest-rate models are based on the spot rates where the spot interest rate is the underlying state variable. This is the case for the Vasicek (1977) model, Hull and White (1987, 1988, 1990, 1993), Cox et al. (1985) and so on. These models require as inputs different parameters in the description of the possible future paths of the spot rate. This implies the search for parameter values which causes the calculated zero-coupon bond prices to be close to the market prices. Many other popular spot rates can be expressed in a simple way under the Heath, Jarrow and Morton (1992) (HTM) model. The Ho and Lee (1986) model is useful in the pricing of interest-rate options with respect to the observed initial term structure of interest rates. However, the model is based on a simplifying assumption, that all rates along the yield curve fluctuate to the same degree. This assumption is not supported in practice. Hull and White (1992) defines a yield-curve-based interest-rate model as a model for the dynamics of the current term structure of interest rates. The model should allow a correct valuation of bonds giving a price equivalent to the market price. Hull and White (1992) compared three models describing the process for the short rate: Ho and Lee (1986), Black, Derman and Toy (1990) BDT, and Heath, Jarrow and Morton (1990). Empirical work by Fama (1984a, b, 1986), Fama and Bliss (1987) and Froot (1989) document term premiums at the short end of the term structure. Longstaff (1990) finds that even shortterm premiums may be simply a function of the time-varying nature of bond returns. Fama (1984b) and Fama and Bliss (1987) find that forward rates are poor forecasters of future short-term interest rates. However, forecasts improve for longer forecast horizons. This chapter presents some of the popular models of the short-term interest rate in a continuous-time setting. Several authors have shown the existence of an arbitrage free family of bond prices associated with a given short-term rate process. In the presence of a one-dimensional diffusion process, it is possible to obtain analytic results or closed-form solutions for zero-coupon bonds and some European options.

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16.1. Interest Rates and Interest-Rate Sensitive Instruments 16.1.1. Zero-coupon bonds A zero-coupon bond or a discount bond of maturity “T ” refers to a financial security, which pays one unit of cash (a dollar for example) at a future prespecified date T . Hence, the bond’s nominal value (principal or face value) is one dollar. We denote by B(t, T ) the price at time t of a zerocoupon bond of maturity T . At the maturity date, B(T, T ) = 1. Since the maximum value of this bond is unity, the bond trades at a discount (a value less than the principal).

16.1.2. Term structure of interest rates Consider a zero-coupon bond with a maturity date T less than T ∗ . The simple rate of return from holding the bond until the maturity date can be calculated as the difference between the final value and the current value is divided by the current value or: 1 − B(t, T ) 1 = − 1. B(t, T ) B(t, T ) Using continuous compounding, the equivalent rate of return or the yield to maturity (YTM) on a bond is given by: Y (t, T ) = −

1 ln[B(t, T )], T −t

∀ t ∈ [0, T ).

(16.1)

The term structure of interest rates or the yield curve describes the relationship between YTM and the maturity T . Given the dynamics of the YTM, it is possible to show that the bond price process is determined by the following formula: B(t, T ) = e−(Y (t,T ))(T −t) ,

∀ t ∈ [0, T ].

(16.2)

In this expression, the bond price is related to its maturity by a discount function. At the initial time, the term structure of interest rates may be described by the current bond prices B(0, T ) or by the initial yield curve: B(0, T ) = e−Y (0,T )T ,

∀ T ∈ [0, T ∗].

(16.3)

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This term structure is estimated in practice using different instruments and methodologies. The yield curve exhibit different shapes. It may be flat, upward sloping, downward sloping, or humped.

16.1.3. Forward interest rates Let us denote by f (t, T ) the forward interest rate at date t < T for risk-free operations (borrowing and lending) at date T . The instantaneous continously compounded forward rate corresponds to a “non observable” interest rate over very short-time intervals [T, T +dT ] as viewed from time t. HJM used an exogenous specification of a family of forward rates f (t, T ) and has defined the bond prices as: B(t, T ) = exp −

T

t

f (t, u)du ,

∀ t ∈ [0, T ].

(16.4)

This allows to compute the implied instantaneous forward interest rate as: f (t, T ) = −

∂ ln[B(t, T )] . ∂T

(16.5)

The instantaneous forward rate is also viewed as a limit case of a forward rate f (t, T, H) that prevails at t for risk-less operations over the time interval [T, H]. In the same way, using two zero-coupon bonds with different maturities T and H, the discounting factor contains the forward rate as: B(t, H) = e−f (t,T,H)(H−T ) , B(t, T )

∀ t ≤ T ≤ H.

This expression gives the forward rate as: f (t, T, H) =

ln B(t, T ) − ln B(t, H) . H −T

(16.6)

Since, the strategy of investing at t in T -maturity bonds is equivalent to the strategy of lending over the interval [t, T ], it is clear that Y (t, T ) = f (t, t, T ). In real markets, interest rates are quoted on an actuarial basis. For example, the actual rate referred to also as the effective rate ra (t, T ) at t for maturity T is given by: (1 + ra (t, T ))T −t = ef (t,t,T )(T −t) = eY (t,T )(T −t) ,

∀t ≤ T

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where the standard bond price is given by: B(t, T ) =

1 , (1 + ra (t, T ))T −t

∀ t ≤ T.

In the same context, the forward actuarial rate prevailing at t for the interval [T, H] must satisfy the following relationship: (1 + ra (t, T, H))H−T = exp[f (t, T, H)(H − T )] = B(t, T )/B(t, H). 16.1.4. Short-term interest rate The short-term interest rate or the instantaneous interest rate rt is often modeled as a price process of a risk-free asset: t ru du , Bt = exp 0

∀ t ∈ [0, T ∗ ]

(16.7)

where the function B(t) solves the differential equation: dBt = rt Bt dt with the initial condition B0 = 1. Since Bt corresponds to the cash accumulated up to t using one unit cash at time 0, and rolling over successive periods, the process is also known as the accumulation factor or a savings account.

16.1.5. Coupon-bearing bonds A coupon-bearing bond or coupon-paying bond BC (T ) gives its holder different cash-flows (c1 , c2 , . . . , cm ) at different dates (T1 , T2 , . . . , Tm ). The price of this bond is obtained by discounting all the future cash flows at the appropriate rates. Bc (t) =

m

cj B(t, Tj ).

(16.8)

j=1

A real bond pays in general a fixed coupon c and re-pays the principal amount N . Hence, the last cash flow is cm = (c + N ). The total return on

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a coupon-bearing bond is uncertain because of the re-investment risk since interest rates are not constant. Bonds with different coupons and time to maturity are in general not directly comparable. To overcome this difficulty, the concept of YTM is used.

16.1.6. Yield-to-Maturity (YTM) The price of a bond with fixed equal coupons and an uncertain YTM at time 0, Y˜c (0) is given by: Bc (0) =

m j=1

N c + . (1 + Y˜c (0))j (1 + Y˜c (0))m

Using the coupon rate c = rc N , the bond price is: Bc (0) =

m j=1

N rc N + . j (1 + Y˜c (0)) (1 + Y˜c (0))m

When coupon rate rc = Y˜c (0), the bond price is priced at par since its price equals its face value. The bond is priced at a discount (below par) when Bc (0) < N or rc < Y˜c (0). The bond is priced at a premium (above par) when Bc (0) > N or rc > Y˜c (0). For continuous compounding, using the current market price of the bond shows that the corresponding YTM, Yc (0) satisfy: Bc (0) =

m

ce−jYc (0) + N e−mYc (0) .

j=1

For the case of zero-coupon bonds, the knowledge of its YTM allows the computation of its price in a discrete setting as: B(0, m) =

1 . (1 + Y˜ (0, m))m

In a continuous setting, the initial price of a zero-coupon bond with maturity T and a YTM, Y (0, T ) is: B(0, T ) = e−Y (0,T )T . The discretely compounded YTM at time i < m, Y˜c (i) on a coupon bond with “m” cash

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flows can be calculated using the formula: m

Bc (i) =

j=i+1

cj . ˜ (1 + Yc (i))j−i

(16.9)

The bond price does not account for the coupon at time i since we use the price Bc (i) after the coupon at time i has been paid. The continuously compounded YTM at time t < Tm ,

Yc (t) = Yc (t; c1 , . . . , cm , T1 , . . . , Tm )

on a coupon bond with “m” cash flows can be calculated using the formula: Bc (t) = cj e−Yc (t)(Tj −t) . (16.10) Tj >t

The bond price does not account for the coupon at time t since we use the price Bc (t) after the coupon at time t has been paid. Note that the bond price moves inversely to its YTM. In general, the decrease in yields raises bond prices more than the same increase lowers bond prices. This reflects the convexity. 16.1.7. Market conventions Market conventions can vary from one country to another. In the United States, the US Treasury issues bonds, notes and bills. The Treasury bill or T -bill is issued in general for a short-term period and corresponds to a discount bond for which the buyer receives the face value at maturity with no coupons. The T -bonds and T -notes are coupon paying securities, which differ in general by the maturity date. The maturity for T -bonds is more than 10 years and it is longer than that of T -notes. A m-year government bond pays coupons semi-annually at time Tj = 0.5j, for j = 1, 2, . . . , 2m. The YTM on a government bond, Yˆc (0) called also a bond equivalent yield is calculated from a bond price as follows: Bc (0) =

2m−1 j=1

(1 + rc /2)N rc N/2 + , j ˆ (1 + Yc (0)/2)) (1 + Yˆc (0)/2)2m

where: rc : coupon rate; N : bond face value and Yˆc (0): an annualized interest rate with no compounding.

(16.11)

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This formula can also be written as: Bc (0) =

rc N N (1 − rc /Yˆc (0)) . + ˆ (1 + Yˆc (0)/2)2m Yc (0)

In this context, the YTM on a bond at time “i” can be found using the following relationship: Bc (i) =

2m−1 j=i+1

(1 + rc /2)N rc N/2 + , j ˆ (1 + Yc (i)/2) (1 + Yˆc (i)/2)2m

after the ith coupon payment. To obtain the compounded annualized yield, or the effective annual yield, the following equality is used: Yˆce (0) = (1 + Yˆc (0)/2)2 − 1. 16.2. Interest Rates and the Pricing of Bonds The pricing of bonds and interest-rate options needs a specification of the dynamics of interest rates under certainty and uncertainty.

16.2.1. The instantaneous interest rates under certainty When an investor borrows $1 at time t, he must pay F (t, T ) at time “T ” where T is the maturity date of the debt. This amount corresponds to an average interest rate R(t, T ) which applies over the period of [t, T ]. It is given by: F (t, T ) = eR(t,T )(T −t) . In the context of certainty, when the interest rates are known over the period, the function F is given at each instant by: F (t, s) = F (t, u)F (u, s) for all t < u < s. Using this equality and the fact that F (t, t) = 1, allows one to show that there is an interest rate r(t) such that in the absence of arbitrage opportunities, the amount F (t, T ) is given by: F (t, T ) = e(

RT t

r(s)ds)

.

(16.12)

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So, 1 R(t, T ) = (T − t)

T

t

r(s)ds .

(16.13)

Consider a zero-coupon bond, which is a bond paying $1 with certainty at its maturity date, P (T, T ) = 1. The price of this bond at time t, P (T, t) in the context of certainty (in an economy with no risk, where all future interest rates are known) is given by: P (t, T ) = e(−

RT

r(s)ds)

t

.

(16.14)

In a context of uncertainty, the future interest rates, R(u, T ) are unknown and the models of interest rates must be established to find the relationships between these rates. 16.2.2. The instantaneous interest rate under uncertainty Under uncertainty, the instantaneous interest rate r(t) is a stochastic process between times t and t + dt. If we consider a risk-less asset, then its price is given by: B(0, t) = e(−

Rt 0

r(s)ds)

.

Denote by H the following assumption. Consider a process P (t, u)0≤t≤u satisfying the boundary condition P (u, u) = 1. Then, as for the processes of stock prices, it can be shown, with no arbitrage opportunities, that there is a probability P ∗ equivalent to the probability P under which the process Pˆ (t, u) = e(−

Rt 0

r(s)ds)

P (tu)

is a Martingale for each u in time interval [0, T ]. This is an interesting assumption since, under the new probability P ∗ , we have: Pˆ (t, u) = EP ∗ (Pˆ (u, u) | Ft ) = EP ∗ (e−

Ru 0

rs ds

| Ft )

or P (t, u) = EP ∗ (e−

Ru t

rs ds

| Ft ).

(16.15)

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If one compares this formula with formula (16.14), we notice that the prices of zero-coupon bonds depend only on the process P (t, u)0≤t≤u under the probability P ∗ . Assumption H allows the identification of the probability density of P ∗ , denoted, LT , with respect to P . In this context, it is possible to show that there exists a stochastic process q(t)0≤t≤T such that for all t in the interval [0, T ] Lt = e(−

Rt 0

q(s)dWs − 12

Rt 0

q 2 (s)ds)

.

Using the property: EP ∗ (X | Ft ) = E(XLT | Ft )/Lt , it is possible to show that the price at time t of a zero-coupon bond with a maturity date u is: P (t, u) = E exp −

t

u

r(s)ds +

t

u

q(s)dWs −

1 2

u t

q 2 (s)ds | Ft

.

(16.16) The probability is often referred to as the risk-neutral probability.

16.3. Interest Rate Processes and the Pricing of Bonds and Options Several interest-rate models are based on the spot rates where the spot interest rate is the underlying state variable. This is the case for the Vasicek (1977) model, Hull and White (1987, 1988, 1990, 1993), Cox et al. (1985) and so on. These models require as inputs different parameters in the description of the possible future pats of the spot rate. This implies the search for parameter values, which cause calculated zero-coupon bond prices to be close to market prices. Many other popular spot rates can be expressed in a simple way under the Heath et al. (1992) model. Once these problems are solved, the option value can be calculated using the appropriate boundary conditions. A European bond option with a maturity date on a zero-coupon bond with a maturity date τ is an option characterized by its terminal payoff. For a call option, this payoff is given by c = (P (τ, T )−K)+ For a put option, this payoff is given by c = (K − P (τ, T ))+.

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16.3.1. The Vasicek model The presentation here uses Eqs. (16.15) and (16.16). Vasicek (1977) used the following process for the dynamics of the interest rate r(t) dr(t) = a[b − r(t)]dt + σdWt

(16.17)

where a, b and σ are constants. If we assume that the process q(t) is a constant equal to −γ, then: ˆ t, dr(t) = a[b∗ − r(t)]dt + σdW with b∗ = b −

γσ a

and ˆ t = Wt + γt. W The process in the Eq. (16.17) can be shown as an Ornstein–Uhlenbeck process. The valuation of bond options in this context, can be easily done since, the Ornstein–Uhlenbeck process is a Gaussian process. 16.3.2. The Brennan and Schwartz model The model proposed in Brennan and Schwartz (1982) is a two-factor model where the dynamics of the short-rate are given by: dr = (a1 + b1 (r − l))dt + σ1 rdX1 . The long-term rate dynamics are given by: dl = l(a2 + b2 r + c2 l)dt + σ2 ldX2 . The drift terms have mean-reversion features. The parameters have to be selected using empirical data. However, this model can allow for infinite interest rates. 16.3.3. The CIR model In the context of this model, the dynamics of the instantaneous interest rate are described by the following equation: dr(t) = [a − br(t)]dt + σ r(t)dWt

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where a and σ are positive and b is a real number. In this context, the process q(t) takes the form q(t) = −α r(t) where α is a real number. 16.3.4. The Ho and Lee model The Ho and Lee (1986) model corresponds to a new generation of interestrate option pricing model. This model exists in a discrete-time version and in a continuous time version. The continuous-time version of the model can be written as: dr = θ(t) + σdW . This model belongs to the class of term structure option models which try to model the behavior of the entire yield curve. These models eliminate in general risk-less arbitrage opportunities that arise from inconsistencies between the model and the observed yield curve. The Ho and Lee (1986) one factor model is based on an implicit assumption that bond prices of all maturities are perfectly correlated. Besides, it assumes a constant volatility regardless of the interest rate level. This allows for the existence of negative interest rates with a positive probability. These deficiencies are avoided in the HJM model. In this model, the initial yield curve is endogenous since it is derived from the specified interest rate process. Therefore, the parameters driving the interest rate process must be chosen in a way to fit the initial yield curve. The Ho and Lee (1986) model is useful in the pricing of interest-rate options with respect to the observed initial term structure of interest rates. However, the model is based on a simplifying assumption, that all the rates along with the yield curve fluctuate to the same degree. This assumption is not supported in practice.

16.3.5. The HJM model The main drawbacks of the Vasicek (1977) and the CIR (1985) models are that these models do not account for the observed term structure of interest rates. Ho and Lee (1986) proposed an interesting discrete-time model, which is appropriate for the description of the whole term structure of interest rates. The same ideas were used in a continuous-time version of this model by Heath et al. (1989), hereafter, HJM. The HJM, interest-rate models are based on a new approach to interest-rate option pricing. As in the Black– Scholes (B–S) model, the HJM model requires the underlying asset (the initial term structure) and a measure of its volatility as the only inputs. The HJM model uses all the available information in the term structure. The model accounts for several reasons of the term structure movements.

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It uses a methodology, which is often applied in multifactor models of the term structure risk. The HJM, model is not based on the assumption of perfect correlations between bond prices of all maturities and does not allow for negative interest rates. The main features of the HJM model is the use of an exogenous approach for the initial yield curve, the multifactor modeling and the volatility functions. In this model, the initial yield curve is exogenously given, which eliminates the parameter estimation. Since, the model is very general, it contains the standard models of Ho and Lee (1986), Vasicek (1977), Hull and White (1990), and Cox et al. (1985) as special cases. The HJM model uses volatility functions to conform the empirical volatility behavior by making the volatility of a given forward rate as a decreasing function of the time to its effective date. This type of models forces the long-term rates to fluctuate less than the short-term rates. The use of the volatility functions allow it to overcome the problem of negative interest rates. In the continuous-time model of HJM, the returns of zero-coupon bonds of different maturities are not perfectly correlated and volatility functions are directly calculated from data of changes in the term structure of interest rates. Since, forward rates are more stable than the prices of zero-coupon bonds, they are used in a two-factor model. The two factors are the changing level and the changing slope of the term structure. The two stochastic processes corresponding to these two factors prevent the perfect correlation between bond prices or forward rates. The model has many similarities with the B–S model since, it only needs the knowledge of the underlying term structure and its associated volatilities. The use of multiple volatility functions gives the model a certain flexibility since, it can accomodate the volatility of the term structure resulting from changes in the level, the slope and the curvature of the term structure. In this model, option values depend only on s. This gives a situation, which is analogous to the B–S. More formally, denoted by f (t, u) the forward interest rate, i.e., the rate at which an investor can contract at date t to borrow and lend for a short period at a future date u. The initial forward rate curve is taken from market data and used as an input in the HJM model. The volatility of each forward rate is specified in the model. The forward rate for date u can change over the instant of time from t to (t + dt) in the following way f (t + dt) = f (t, u) + σ(.)dW + drif t(.)dt. The volatility function σ(.) of each forward rate describes the dynamics of the term structure. When σ(.) is a constant, all the forward rates have the same volatility and the entire forward curve is submitted to parallel shocks. In this context, the HJM model is the continuous time limit of the Ho and Lee model.

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Consequently, the model shows the same drawbacks as that of Ho and Lee (1986), namely, the possibility of negative interest rates. A more interesting model is obtained by HJM when σ(.) is a function of forward rate f (t, u)’s maturity (u − t). For example, when σ(u − t) = σ0 e(−a(u−t)) with a > 0 and σ0 > 0, then the model proposed in Vasicek and studied in Hull and White is obtained. Example: t ˆ f (t, T ) = f (0, T ) + σ t T − + σ W (t) . 2 2

In the HJM model, the price of a European bond is given by: C(t) = P (t, T )N (h) − P (t, t∗ )KN (h − q)

P (t,T ) 1 2 ln KP ∗ √ (t,t ) + 2 q h= , q = σ(T − t∗ ) t∗ − t q

(16.18)

and where P (t, T ) is the bond with an exercise price K and a maturity t∗ with: 0 ≤ t ≤ t∗ ≤ T. For more details and other applications of the model to the pricing of an entire book of options and volatility estimation, see Heath et al. (1992), and Spindel (1992). 16.3.6. The BDT model The Black, Derman and Toy model (1990), can be written as: dr = −α(t)dt + σ(t)dW. r In the BDT model, the long-term yields are assumed to reflect the expectations of the market regarding the future short-term rates. In this context, if the market expects future short rates to be high (low), then there is a tendency for current long-term yields to be higher (lower). The model gives in a valuation formula current expected long-term yields, using the expected future short rates. It also offers a sensitivity formula, that predicts the expected changes in the current long-term yields when the short rate changes.

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16.3.7. The Hull and White model The Hull and White one-factor model (1990) tries to overcome some of the main shortcomings in the Ho and Lee (1986) model. The dynamics of the short-term interest rate are given by: dr = [θ(t) − αr]dt + σ(t)dW where: r: short-term interest rate; θ: drift factor which is a function of time; α: reversion rate or a mean reverting factor; σ: standard deviation and dW : “white noise”. This model is a simple extension of the Vasicek model. Since, the drift factor is a function of time, this model guarantees consistency with the initial term structure of interest rates. The mean reverting factor allows the long-term rates to show lower volatility than short-term rates. This specificity is confirmed in practice and represents an improvement on the Ho and Lee (1986) and the Vasicek (1977) models. The volatility function in this model reflects the fact that short-term interest rates fluctuate more than long-term rates because of the reversion parameter in the spot rates. Using this model, it is possible to obtain an analytic solution for the pricing of European options. The price of a European call on a discount bond at time zero is given by: c = P (0, T )N (d1 ) − KP (0, t∗ )N (d2 ) with d1 =

1 P (0, T ) 1 ln + v(t∗ , T ), v(t∗ , T ) P (0, t∗ )K 2 v(t, T ) =

where: t∗ : K: T: P (t1 , t2 ): v(t, T ):

d2 = d1 − v(t∗ , T )

1 2 −αT σ (e − e−αt )(e2αT − 1) 2α3

option maturity date; strike price; maturity date of the underlying discount bond; price at time t1 of a bond maturing at time t2 and volatility as a function of the reversion rate.

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The reader can note the analogy between the Hull and White and the HJM formulas.

16.3.8. Fong and Vasicek model The Fong and Vasicek model proposed in Fong and Vasicek (1991) is a two-factor model where the dynamics of the risk-adjusted spot rate are given by: dr = a(¯ r − r)dt +

√

σdX1

and the square root of the volatility of the spot rate are: √ dσ = b(¯ σ − σ)dt + c σdX2 . This formulation allows the derivation of some simple pricing formulas for interest-rate sensitive instruments.

16.3.9. Longstaff and Schwartz model The Longstaff and Schwartz model proposed in Longstaff and Schwartz (1992) is a two-factor model where the dynamics of the risk-adjusted variable is: dX = a(¯ x − x)dt +

√

xdX1

and dy = a(¯ y − y)dt +

√ ydX2

where the spot rate is given by r = cx + dy. 16.4. The Relative Merits of the Competing Models Hull and White (1992) defines a yield-curve-based interest rate model as a model for the dynamics of the current term structure of interest rates. The model should allow a correct valuation of bonds giving a price equivalent to the market price. Hull and White (1992) compared three models describing the process for the short rate: Ho and Lee (1986), Black, Derman and Toy (1990), BDT, and Heath et al. (1990). The process for the short rate in the

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Ho and Lee (1986) model is given by: dr = θ(t)dt + σdz. The process for the short rate in the BDT model is given by: σ (t) d log r = θ(t) + log r dt + σ(t)dz. σ(t) This model assumes that interest-rate changes are log-normal. The process for the short rate in the Hull and White (1990) model is: dr = (θ(t) − φ(t)r)dt + σ(t)dz. This model assumes that interest-rate changes are normally distributed. HJM, shows that the model for the short rate is completely determined using the term structure of interest rates and the volatilities of all forward rates. The function θ(t) can provide a perfect fit to the current term structure of interest rates. In the Ho and Lee model, the volatility of the term structure is described by the constant σ. Hence, the model ignores the differences between the volatility of short and long-term spot rates. In practice, long-term spot rates are less volatile than short spot rates. In the BDT model, the volatility of the term structure is described by σ(t). Hence, the model can provide a perfect fit to the current term structure of spot rate volatilities. In the Hull and White model, the volatility structure is described by two functions of time: σ(t) and φ(t). The second function provides an additional degree of freedom with respect to the BDT model. Mean reversion comes from the fact that there is a “link” between current long-rate volatilities and future short-rate volatilities. It refers to the tendency for short rates to be pulled back to some long-run average over time. Mean reversion explains also the fact, that long-rate volatilities are less than short-rate volatilities. When the interest rates are high, it is more probable that they decrease. When they are low, it is more probable that they increase. In economic theory, when interest rates rise, the economy slows down since, loans are less demanded. When interest rates fall, the economy heats up since loans are more demanded. Mean reversion exists in the BDT model and Hull and White model, but not in the Ho and Lee model. Hull and White find that their model is appropriate for the pricing of at-the-money options. The log-normal assumption for interest rates seems to be better than the normal assumption since it does not allow for negative interest rates. However, the analytic tractability of the option pricing model can be lost in this context. It is also possible to use the normal assumption

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and adjust the volatilities of out-of-the-money and in-the-money options, in order to account for the skewness of the interest rate distribution. Leong (1998) reviews the relative merits of the competing models in the pricing of interest rate sensitive instruments. He uses six criteria, which are necessary in the choice between different models. The first criteria concerns arbitragefree pricing. In this context, a correct model must at least price correctly, an instrument relative to the prices of the other related instruments. In this sense, to avoid arbitrage opportunities, the model must preserve the put-call parity and must be consistent with the observed term structure of interest rates. When comparing Hull and White’s model and HJM model, Leong (1998) finds that both models price options “correctly” with respect to the initial yield curve. In fact, these models preserve the put-call parity. When comparing Black’s (1976) model to the HJM model, HJM shows that Black’s model is inconsistent when used in the pricing of caps of different terms because of the use of different volatilities. However, this same problem appears as well when using the HJM model. In general, dealers adopt a volatility matrix approach to pricing and portfolio revaluation. This matrix is based on different estimates for options with different time to maturities. This practical approach leads to inconsistencies and incoherence regardless of the model used. The second criteria corresponds to the computation speed. Traders, market makers, and risk managers resort to option pricing models to price the deals, to re-value on a mark-to-market basis, a trading book, to calibrate the models to observed data, to manage the risks of a trading position, and so on. Since, the model may be used several times during a trading day, it must respond to a certain computation speed. The main drawbacks of the HJM two factor model is that it is too slow. In general, the speed problem in deal pricing is translated into serious problems in model calibration. Therefore, as a book of interest-rate sensitive instruments using a two-factor model may or may not be viable. The slower computation speed causes the two-factor models to be at a disadvantage with respect to one-factor models regardless of the state of the technology. Leong (1998) finds that the pricing of a five-year cap on Sun Sparc two station using the two-factor HJM model is 17 times as long as for a onefactor model. The third criteria concerns the hedge effectiveness or the model’s ability to suggest hedge ratios, which contribute to reduce the risk exposure of the model user. The multifactor approach allows the model to explain more of the variance in yield curve changes. When compared to a one-factor approach, the multifactor appraoch suggests hedges, which are more robust to nonparallel yield curve changes. The one-factor approach

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ignores in general the lack of perfect correlations between different rates in the yield curve. Since two-factor models compute two deltas for each instrument, and lead to using two hedging instruments, this refinement can lead to better hedge recommandations. However, the two-factor HJM model may not be perfect in hedging yield curve risk. The fourth criteria refers to the marginal utility. The “marginal utility of complexity” refers to the practical benefit that can be achieved by adding another dimension of complexity. A “good” model can do practically an equivalent task with the least complexity. The fifth criteria refers to versatility and coherence. A versatile model is able to price several instruments with the same analytical framework. This framework is fundamental for integrated risk management. The sixth criteria concerns with the fitting error. The theoretical value generated by a model must be close to the market value. Hence, a good model must be easily calibrated to reflect the general market level.

16.5. A Comparative Analysis of Term Structure Estimation Models Ferguson and Raymar (1998) used the bond prices to estimate six discount functions: a six-degree polynomial, four-and-six-parameter Vasicek and Fong (1982) models (VF4 and VF6); three four-parameter analytic techniques of Nelson and Siegel, NS (1987); the Vasicek (1977) bond price equation and the Cox et al., CIR (1985) bond equation. They compared the six methods with respect to accuracy in a simulation framework. They found that a simple OLS bond price application of the six parameter Vasicek–Fong (1982) model is the most accurate and robust. The polynomial estimator is accurate only when bond maturities are sampled at uniform intervals. They show that the Nelson–Siegel model is also reliable and accurate, and that the Vasicek (1977) and Cox et al. (1985) models are not “as good” as the other methods.

16.5.1. The construction of the term structure and coupon bonds Ferguson and Raymar (1998) generated realistic term structures using a forward rate generator (the following equation) to specify a set of crosssectional rate changes in the log of the forward rate: √ z ∆ ln(f ) = a[m − ln(f )]∆t + σ ∆t˜

(16.19)

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where: f: a: m: σ: z:

forward rate; reversion rate; forward rate reverts to this mean m; volatility rate and a standard normal variate.

The study uses three-month time intervals over a 30 year horizon. Hence, a single trial corresponds to 120 random draws z allowing the definition of a forward rate curve. The initial forward rate f0 is 7% and σ = a = 15%. The values of m are set to 7%, 10% and 4% to get respectively flat, upward, and downward sloping spot curves. They generate for each yield curve (shape), 50 forward rate curves. Using a forward curve, it is simple to compute the discount function, the spot rate curve and the coupon bond prices.

16.5.2. Fitting functions and estimation procedure Using a set of 15 bond prices, Ferguson and Raymar (1998) estimated a continuous discount function out to 30 years by each of six fitting functions. They used a discount curve to get “fitted prices” for the bonds for which they calculated simulated prices. Then, they minimize the sum of the squared errors over the 15 true and fitted bond prices to get the parameters of the continuous discount functions. Ferguson and Raymar (1998) used a constrained OLS for three of the estimates: a six-degree polynomial and four-and-six-parameter Vasicek-Fong (1982) models (VF4 and VF6). Nonlinear least square methods are used in the other estimation approaches. The parameters of the polynomial and Vasicek-Fong models are estimated subject to a constraint to price an instantaneously maturing $1 face value discount bond at $1. The six term structure estimates or fitting functions are defined as follows. Following Ferguson and Raymar (1998), we have denoted by: Dt : present value at time 0 of a bond that matures at time t; ai : parameters used to fit the term structure in the presence of the constraint that D0 = 1; R: yield to maturity (YTM) on the longest bond used when fitting the term structure and rt : spot rate for a bond with a maturity date t.

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The models of spot interest-rate processes are written using the same notation employed for the estimation of discount price functions. Each fitted function gives a discount bond pricing equation Dt = e−rt t . This equation allows the computation of the spot rate for maturity t. In this context, the polynomial model is given by: Dt = a0 + a1 t1 + a2 t2 + a3 t3 + a4 t4 + a5 t5 . The constraint equation in this context corresponds to a0 = 1. This model is standard. The Vasicek–Fong (1982) component functions are specified by VF (1982) as appropriate. The Vasicek–Fong (1982) model (VF6) is given by: Dt = a0 + a1 e−Rt + a2 e−2Rt + a3 e−3Rt + a4 e−4Rt + a5 e−5Rt . The constraint equation in this context corresponds to Σai = 1. The Vasicek–Fong (1982) model (VF4) is given by: Dt = a0 + a1 e−Rt + a2 e−2Rt + a3 e−3Rt . The constraint equation in this context corresponds to Σai = 1. The Nelson–Siegel (1987) model is given by: Dt = e−rt t with rt = a0 + (a1 + a2 )

1 − e−a3 t a3 t

− a2 e−a3 t .

The constraint equation in this context is endogenous. This model is presented in terms of the spot rates and the discount function. The Vasicek (1977) model is given by: Dt = A(t)e−Bt a0

with B(t) =

1 − e−a1 t . a1

a2 a2 ln(A(t)) = (B(t) − t) a2 − 32 − 3 (B(t))2 . 2a1 4a1 The constraint equation in this context is endogenous.

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The short-rate process used is: da0 = a1 (a2 − a0 )d3 t + a3 dz. The CIR (1985) model is given by: Dt = A(t)e−Bt a0 with: 2we0.5(a1 +w)t , ((a1 + w)(ewt − 1) + 2w 2(ewt − 1) B(t) = , w = a21 + a23 . ((a1 + w)(ewt − 1) + 2w A(t) = h

2a1 a2 a2 3

,

h=

The constraint equation in this context is endogenous. The short-rate process used in this context is: √ da0 = a1 (a2 − a0 )dt + a3 a0 dz. The simulations conducted show that the Vasicek–Fong model (VF6) is the best of the models above since, it provides fast and reliable fit for the full simulated term structure. 16.6. Term Premium Estimates From Zero-Coupon Bonds: New Evidence on the Expectations Hypothesis The concept of a liquidity premium implicit in the term structure of interest rates was first proposed by Hicks (1946). Empirical work by Fama (1984a, 1984b, 1986), Fama and Bliss (1987) and Froot (1989) document term premiums at the short end of the term structure. Longstaff (1990) finds that even short-term premiums may be simply a function of the time-varying nature of bond returns. Fama (1984b) and Fama and Bliss (1987) found that forward rates are poor forecasters of future short-term interest rates. However, forecasts improve for longer forecast horizons. Dhillon and Lasser (1998) reexamined the presence of term premiums in the term structure and the issue of forecasting future interest rates from current forward rates. They used a unique data set of zero-coupon stripped Treasury securities. Since, Treasury bond issues have different characteristics regarding the maturities and the coupons, Fama (1984) constructs portfolios of bonds for maturities longer than a year. Fama and Bliss (1987) implemented

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an interpolation method for forward rates and returns from existing onethrough five-year bond issues. Froot (1989) extracted premiums on different fixed-income securities from survey expectations. Dhillon and Lasser (1998) estimated term-structure premiums from a yield curve, consisting of zerocoupon stripped US Treasury securities. Using the stripped Treasury bond yield curve, they found a monotonically increasing term premium. This evidence is not consistent with the expectations hypothesis. They found that the current forward rates can be used to forecast both short-term and long-term interest rates. However, Fama (1984a,b) and Fama and Bliss (1987) found that short-term rates cannot be forecasted using forward rates. Dhillon and Lasser (1998) used quarterly US coupon strip prices from August 1986 to May 1997. Coupon strips existed at every 3-month interval. The quarterly return from time t to t + 31 on a zero-coupon bond with τ months to maturity at time t is given by: B(τ − 1)t+1 (16.20) Rτt+1 = ln Bτt where: Bτt : price at time t of a zero-discount bond maturing at month t + τ and B(τ − 1)t+1 : price of a zero-discount bond at quarter t + 1. The premium in the quarterly return is given by the difference between the quarterly return Rτt+1 and the quarterly spot rate, St+1 : P τt+1 = Rτt+1 − St+1 . The price of a zero-coupon maturing in τ months at any time t is given by: Bτt+1 = exp(−St+1 − F 2t − · · · − F τt ) where F τt corresponds to the forward rate for quarter t + τ observed at time t. This forward rate can be determined using: B(τ − 1)t . F τt = ln Bτt The expectations hypothesis suggests that forward rates contain forecasts of the future spot rates. The following two time series regressions are used to determine whether the current forward-spot differential (F τt − St+1 )

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represent a predictor of either the future quarterly premium P τt+1 or the future change in spot rates (St+τ − St+1 ): P τt+1 = α1 + β1 (F τt − St+1 ) + εt+1 St+τ − St+1 = α2 + β2 (F τt − St+1 ) + nt+τ −1 .

(16.21)

When β1 is positive and significant, then forward rates contain information about the premium in three months. Also, when β2 is positive and significant, then the forward rate at time t is a good predictor of the future quarterly spot rate at time t + τi − 3. Results show strong evidence for liquidity premiums in the term structure as well as a monotonically increasing relationship between liquidity premium and term to maturity. Results show that current forward rates can be used to forecast quarterly interest rates. This evidence is in contrast with the findings in Fama and Bliss (1987) where it is shown that short-term rates cannot be forecast using forward rates. 16.7. Distributional Properties of Spot and Forward Interest Rates: USD, DEM, GBP, and JPY Using the Kernel density estimation method, Lekkos (1999) estimates the distributions of spot and forward interest rates in levels and differences. He studies normal and log-normal distributions as well as a mixture of two distributions in the characterization of the distribution of interest-rate changes. The database comprises daily money market and swap market rates of the USD, the DEM, the GBP, and the Yen. Lekkos (1999) uses money market and swaps market data to infer the prices and the yields of zero-coupon bonds. He uses the bootstrap method to get daily spot and forward term structures of interest rates from the rates and maturities in the database. Lekkos (1999) uses money market and swaps market data to infer the prices and yields of zero-coupon bonds. Money market rates data (short rates with less than one year to maturity) are used to price zero-coupon bonds. Swap market data are used for longer maturities. Money market rates corresponding to the rates at which banks lend or borrow money for three, six, and 12-months. The discount bond price for maturities upto one year are estimated using: B0,t =

1 1 + rt αo,t

(16.22)

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where: B0,t : price of a pure discount bond paying $1 at t; rt : money market rate for a loan of an identical maturity and αo,t : accrual factor. For USD, DM, and JPY, this factor is αi−1,i =

30 . 360

For GBP, this factor is αi−1,i =

ti − ti−1 . 365

The bootstrap method is applied to obtain the prices of discount bonds implied by the swap market as follows: B0,t =

t−1 1 − st i=1 αi−1,i B0,i 1 + st αt−1,t

where: B0,t : price of a pure discount bond that pays $1 at time t; st : swap rate and αi−1,i : accrual factor which refers to a certain number of days. The accrual factor used in the study is For the GBP, the accrual factor is αi−1,i =

30 360

for USD, DEM, and Yen.

(ti − ti−1 ) . 365

The above equation allows the computation of zero bond prices by the bootstrap method. Using these prices, it is straight forward to estimate the implied annualized yields as: R0,t =

1 B0,t

−α0,t

−1

where the accrual factors referring to bonds are: αi−1,i =

(ti − ti−1 ) . 365

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The one-year forward rates are estimated using the following equation: −αt,t+12m B0,t − 1. f0,t,t+12m = B0,t+12m The Kernel method used by Ait–Sahalia (1996) and Lekkos (1999) allows the identification of the “true” non-parametric distribution of interest rates. The estimated density is constructed by centering around each observation of a kernel function K(u) and averaging the values of the kernel function at any given rate. Lekkos (1999) uses the following estimator for the density function: fˆ(r) = (T h)−1

T t=1

K

r − rt h

where: T : number of observations; h: a smoothing parameter and K(.): Gaussian Kernel. The estimated probability densities of spot and forward interest-rate levels show that interest rates are not generated from a normal or a lognormal distribution. They are generated from a mixture of distributions with different means and standard deviations. 16.7.1. Interest rate levels The data shows for most spot and forward rates, the presence of multimodality. This reveals that interest rates are not generated from a univariate distribution (normal or log-normal). They are generated from a mixture of distributions with different means and volatilities. These shapes can be explained by regime shifts or jumps in interest rates. Similar patterns are observed in Bellalah and Prigent (2002) for stock and index options. 16.7.2. Interest rate differences and log differences The models of Vasicek (1977), Ho and Lee (1986) and Hull and White (1990) assume that the distribution of interest rate differences is normal with a constant volatility. Black et al. (1990) and Black and Karasinski (1991) assume that the distribution of interest rates is log-normal.

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The tests of Lekkos (1999) show that the most characteristic feature of interest rates is the high kurtosis. The results indicate that neither the normal distribution nor the log-normal is adequate for the description of the distribution of interest rates. The most characteristic feature of interest-rate changes is the high kurtosis, which seems to be higher than three. A mixture of two normal distributions seems to give a better fit to the data than the normal or the log-normal distribution. Lekkos (1999) assumes a mixture of normal distributions with N mixing components. The probability density function of each observation is described by: N

f (∆rt ; Π, σ, µ) =

Πi gi (∆rt ; σi , µi )

(16.23)

i=1

where Π = (Π1 , Π2 , . . . , ΠN −1 ) represent the N − 1 independent mixing proportions of the mixture and are such that: 0 < Πi < 1,

ΠN = 1 −

N −1

Πi

i=1 2 and µ = (µ1 , . . . , µN ) and σ = (σ12 , . . . , σN ). For the case of a mixture of normal distributions, the probability density function of the ith component distribution is given by: 1 −(∆rt − µi )2 . gi (∆rt ; σi , µi ) = √ exp 2σi2 2Πσi

The parameter vector of the mixture, 2 , Π1 , Π2 , . . . , ΠN ) ϑ = (µ1 , µ2 , . . . , µN , σ12 , . . . , σN

can be estimated by maximizing the log-likelihood function: L(ϑ) =

T

log f (∆rt , ϑ).

(16.24)

t=1

Since the maximization of the log-likelihood function L(ϑ) is not evident with standard methods, the parameter vector was estimated via the expectation maximization (EM) algorithm proposed by Dempster et al. (1977). This algorithm is also used in Bellalah and Lavielle (2002). The EM algorithm assumes the presence of the components in a fixed proportion

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in the mixture. Then, it determines the posterior probability of that observation which belongs to the component distribution s as: P (s | ∆rt ) =

Πs gs (∆rt ; σs , µs ) f (∆rt ; Π, σ, µ)

(16.25)

ˆ is a solution to the following system The maximum likelihood estimate of ϑ of non-linear equations: T s = 1 Π P (s | ∆rt ) s = 1, 2, 3, . . . , N T t=1

µ s =

T 1 P (s | ∆rt )∆rt s TP

s = 1, 2, 3, . . . , N

(16.26)

(16.27)

t=1

σs =

T 1 s )2 . P (s | ∆rt )(∆rt − µ T Pîs

(16.28)

t=1

This system can be solved via an iterative procedure. Given initial values of P (s | ∆r ) for t = 1, . . . , T can be evaluated. These can be inserted in Eqs. (16.26) to (16.28) to produce revized parameter estimates. This procedure stops when some convergence criteria is satisfied. The EM ˆ but does not give the number of algorithm is useful for the estimation of ϑ, component distributions N . This number can be computed empirically. We denote by Li (ϑ) the log-likelihood function when N = i. Lekkos (1999) compares two specifications with i and j components using: log Λij = log Li (ϑ) − log Lj (ϑ)

for i ≤ j.

(16.29)

He makes inferences about which distribution is more likely to have generated in each observation. Given ϑ, when the number N = 2, then: P (s = 1 | ∆r ) =

Π1 g1 (∆rt ; σ1 , µ1 ) f (∆rt ; Π, σ, µ)

(16.30)

gives the probability that each ∆rt comes from the first component distribution. He finds that all interest rate changes have nearly a zero mean and a very small variance. He shows that only rare events produce large changes in interest rates, which in turn create the second component that is responsible for excess kurtosis and skewness.

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Summary Identification of the stochastic process describing the dynamics of the options underlying asset is an important key in asset valuation. When the movements of interest rates are well described by an interest-rate model, this allows the derivation of accurate bond and option prices. Theoretical interest-rate models depend on the knowledge of statistical properties of the “stylized features” of the term structure. The identification of the stochastic properties of the dynamics of interest rates is of great interest in theory and practice. The statistical properties of the term structure of interest rates are fundamental to the development of theoretical interestrate models. Each interest-rate model has its merits and dismerits. The use of a one-factor model offers the ease of computation (the speed). The use of a multifactor model has an advantage in hedging the yield curve. Hence, from a practical point of view a trade-off must be done: more robust management of yield curve risk against speed. The choice between the two approaches depends on the situation in which a decision must be made. What is certain is that the choice of the model depends on the users’ needs. Ferguson and Raymar (1998) study several popular term structure estimation models and the methodological issues in estimating the term structure. They consider six methods to derive a cross-sectional discount function. They generate a “true forward rate curve” to obtain the associated spot curve and discount function. Lekkos (1999) present an in-depth analysis of the distributional properties of interest rates. He conducts an indirect test of the validity of the assumptions in interest rate models of Vasicek (1977), Cox et al. (1985), Ho and Lee (1986), Hull and White (1990), Black et al. (1990) and Black and Karasinski (1991). The differences between these models result from the assumptions imposed on the stochastic process. The statistical analysis in Lekkos (1999) shows that the kernel density estimation method provides estimates of the probability densities of interest-rate levels. The density estimates show a great degree of multimodality. He rejects the normal and the log-normal distributions of interest rates because of the excess kurtosis in all interest rates. He finds that a better fit to the data is achieved using a mixture of two normal distributions. The models of Vasicek (1977), Ho and Lee (1986) and Hull and White (1990) assume that the distribution of interest-rate differences is normal with a constant volatility. BDT (1990) and Black and Karasinski (1991) assume that the distribution of interest rates is lognormal. The tests of Lekkos (1999) show that the most characteristic feature

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of interest rates is the high kurtosis. The results indicate that neither the normal distribution nor the log-normal is adequate for the description of the distribution of interest rates. A mixture of two normal distributions seems to give a better fit to the data, than the normal or the log-normal distribution. Bali and Karagozoglu (1999) use different estimators in the pricing of interest rate sensitive options with the single BDT (1990) model. The BDT model is often implemented with a constant volatility estimators (historical or implied volatility). These volatilities estimators do not allow the volatility of the short rate to vary over time. The choice of a class of volatility estimation model for short rates is crucial since, the short-term rate drives the changes in the term structure in the BDT (1990) model. Appendix A: An Application of Interest Rate Models to Account for Information Costs: An Exercise As an exercise, this section introduces a parameter reflecting information costs in interest-rate models. This allows a presentation of the old formulas in a new form. A.1. An application of the HJM model in the presence of information costs The HJM Model is different from standard models since it starts with a model for the whole forward rate curve. A.1.1. The forward rate equation We denote by: F (t; T ): forward rate curve at time t; Z(t; T ): price of a zero-coupon bond at t maturing at T and λ: Information cost regarding the zero-coupon bond. The price of a zero-coupon bond is given by: Z(t; T ) = e−

RT t

(F (t;s)+λ)ds

.

(A.1)

Consider the following dynamics for zero-coupon bond prices dZ(t; T ) = µ(t, T )Z(t; T )dt + σ(t, T )Z(t; T )dX.

(A.2)

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Since the price of a zero-coupon bond maturing instantaneously is one, then σ(t, t) = 0. Using (A.1), we have: F (t; T ) = −

∂ log Z(t; T ) + λ. ∂T

If we differentiate this equality and substitute from (A.2), this gives the dynamics of the forward curve: ∂ dF (t; T ) = ∂T

1 2 ∂ σ (t; T ) − µ(t, T ) dt − σ(t; T )dX. 2 ∂T

(A.3)

A.1.2. The spot rate process The spot rate corresponds to the forward rate for a maturity equal to the current date: r(t) = F (t; t) − λ. Wilmott (1998) shows how the HJM approach is slow in the pricing of derivatives. We will reproduce their argument here in the presence of information costs. Consider the current time t∗ and assume that the whole forward rate curve is known today, F (t∗ , T ). The spot rate can be written for any time t as: r(t) = F (t; t) − λ = F (t∗ ; t) +

t

t∗

dF (s; t) − λ.

Using Eq. (A.3) gives: t t ∂σ(s, t) ∂σ(s, t) ∂µ(s, t) σ(s, t) r(t) = F (t ; t)+ − ds− dX(s)−λ. ∂t ∂t ∂t t∗ t∗ ∗

Differentiating this last expression with respect to time t gives:

2 t ∂σ(s, t) ∂F (t∗ ; t) ∂µ(t, s) ∂ 2 σ(s, t) + + dr = σ(s, t) − ∂t ∂s s=t ∂t2 ∂t t∗ t 2 ∂ 2 µ(s, t) ∂σ(t, s) ∂ σ(s, t) − dX(s) dt − dX. ds − ∂t2 ∂t2 ∂s s=t t∗

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It is important to note that the underlined term depends on the history of σ and the stochastic increments dX. This model makes the dynamics of the spot rate non-Markov. A.1.3. The market price of risk In the HJM context, the changes in the forward rate curve are perfectly correlated. Therefore, it is possible to hedge one bond with an other bond maturing at a different date. An initial portfolio can be constructed using a long position in a bond maturing in T1 and a short position in another bond maturing in T2 as follows: Π = Z(t; T1 ) − ∆Z(t; T2 ). Over a short interval of time, the change in the portfolio’s value can be written as: dΠ = dZ(t; T1 ) − ∆dZ(t; T2 ) or dΠ = Z(t; T1 )(µ(t, T1 )dt + σ(t, T1 )dX) − ∆Z(t; T2 )(µ(t, T2 )dt + σ(t, T2 )dX). When ∆ is chosen in a way such that: ∆=

σ(t, T1 )Z(t; T1 ) σ(t, T2 )Z(t; T2 )

then the portfolio is hedged and is risk-free. Hence, the return on this portfolio must be the risk-less rate plus the information costs, which are necessary to get informed about the market and the arbitrage opportunities. In this context, we have: µ(t, T1 ) − (r(t) + λ) µ(t, T2 ) − (r(t) + λ) = . σ(t, T1 ) σ(t, T2 ) When both sides are independent of the maturity T , we have: µ(t, T ) = (r(t) + λ + γ(t)σ(t, T ) where γ(t) stands for the market price of risk.

(A.4)

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Real and risk neutral In a risk-neutral world, the return on any traded asset must be the rate r(t) + λ rather than µ. This implies that the dynamics of all zero-coupon bonds are given by: dZ(t; T ) = (r(t) + λ)Z(t; T )dt + σ(t, T )Z(t; T )dX. A.1.4. Relationship between risk-neutral forward rate drift and volatility It is possible to write the dynamics of the forward rate in a risk-neutral world as: dF (t; T ) = m(t, T )dt + ν(t, T )dX where from Eq. (A.3), the value of the forward rate volatility ν(t, T ) is given by ν(t, T ) = −

∂ σ(t, T ) ∂T

and the drift rate is computed as: T 1 2 ∂ ∂ σ (t, T ) − µ(t, T ) = ν(t, T ) µ(t, T ). ν(t, s)ds − ∂T 2 ∂T t Since in a risk-neutral world the value of µ(t, T ) = r(t) + λ, the drift of risk-neutral forward rate curve is linked to the volatility by: T m(t, T ) = ν(t, T ) ν(t, s)ds. (A.5) t

A.1.5. Pricing derivatives Since the HJM model is characterized by its non-Markov nature, it is not possible to obtain a finite-dimensional partial differential equation for a derivative price. The model can be simulated or implemented via a tree structure. Multi-factor HJM Several authors develop multifactor models for the pricing of derivative assets. If the risk-neutral forward rate curve follows a stochastic differential

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equation in N -dimensions, then the dynamics of the forward rate can be written as: dF (t, T ) = m(t, T )dt +

n

νi (t, T )dXi

i=1

where the different processes are dXi are uncorrelated. In this case, the drift term is written as: T n m(t, T ) = νi (t, T ) νi (t, s)ds. i=1

t

A.2. An application of the Ho and Lee model in the presence of information cost The dynamics of the spot rate in the Ho and Lee model are given by: dr = η(t)dt + cdX where c is a constant. The prices of zero-coupon bonds satisfy the following equation: ∂Z 1 ∂ 2Z ∂Z + c2 2 + η(t) − (r + λZ )Z = 0 ∂t 2 ∂r ∂r

(A.6)

with the condition that the zero-coupon bond price at maturity equals to one: Z(r, T ; T ) = 1. In this context, the zero-coupon bond price is given by: T 1 2 3 c (T − t) − Z(r, t; T ) = exp η(s)(T − s)dS − (T − t)(r + λZ ) . 6 t The drift in the Ho and Lee model must fit the yield curve at time t∗ . Hence, the forward rate can be written as: 1 F (t∗ ; T ) = r(t∗ ) − c2 (T − t∗ )2 + 2

T

t∗

and so η(t) =

∂F (t∗ ; t) + c2 (t − t∗ ). ∂t

η(s)ds

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For any time later than t∗ , the forward rate can be written as: 1 F (t; T ) = r(t) − c2 (T − t)2 + 2

T t

η(s)ds.

Hence, it is possible to show that: dF (t; T ) = c2 (T − t)dt + cdX

Appendix B: Implementation of the BDT Model with Different Volatility Estimators Bali and Karagozoglu’s (1999) model both the instantaneous and time properties of the interest rate in pricing Eurodollar futures options. When testing the predictive power of the volatility estimation models, they show that the forecasting ability of a moving average volatility estimators is inferior to that of the time series volatility models. They provide evidence on the sensitivity of derivatives pricing to volatility estimators in predicting option prices.

B.1. The BDT model In the BDT’s model, the long-term yields are assumed to reflect the expectations of the market regarding the future short-term rates. In this context, if the market expects future short rates to be high (low), then there is a tendency for current long-term yields to be higher (lower). The model gives in a valuation formula current expected long-term yields using the expected future short rates. The model also offers a sensitivity formula that predicts the expected changes in the current long-term yields when the short rate changes. Bali and Karagozoglu (1999) determine the value of the Eurodollar options within the BDT’s model using the yield curve and the volatility curve, referred to as the term structure of interest rates. Since BDT’s model assumes that changes in short-term interest rates are log-normal, then the evolution of the short rate in a discrete time is described by: ∆ ln r(t) = [δ0 (t) − δ1 (t) ln r(t)]θ + σ(t)∆W (t)

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where: ln r(t): natural logarithm of the short rate at t; ∆ ln r(t) = ln r(t + θ) − ln r(t): change in the log interest rate; θ: length of the time interval; δ0 (t), δ1 (t): time-varying parameters to be estimated; σ(t): conditional volatility of log interest rate changes at t; W (t): a standard Brownian motion and ∆W (t): a random variable normally distributed with zero mean and a variance θ. The evolution of the short rate in a discrete time can be approximated using a binomial representation: ln r(t + θ) − ln r(t) √ [δ0 (t) − δ1 (t) ln r(t)]θ + σ(t)√θ; with probability 1/2 = . [δ0 (t) − δ1 (t) ln r(t)]θ − σ(t) θ; with probability 1/2

(B.1)

In this context, the term δ1 (t) can be seen as a measure of the speed of mean reversion in the log rate levels. The drift of the logarithm of the short rate µ(r, t) = δ0 (t) − δ1 (t) ln r(t) is added to the log interest rate in upward or downwoard movements of the log interest rate. Using Eq. (B.1), ln r(t) will be one period later: √ [ln r(t + θ)up − ln r(t)] = [δ0 (t) − δ1 (t) ln r(t)]θ + σ(t) θ

(B.1a)

√ [ln r(t + θ)down − ln r(t)] = [δ0 (t) − δ1 (t) ln r(t)]θ − σ(t) θ.

(B.1b)

or

The difference between these two last equations gives the sensitivity formula of the BDT’s model. The formula gives at time t + θ, the spread of two log interest rates as a function of the volatility. B.2. Estimation results Bali and Karagozoglu (1999) uses data daily on Eurodollar spot rates, futures and futures options from 1987 to 1996. They use a rolling regression

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procedure and generate one-step-ahead forecasts by estimating the timevarying drifts parameters. They construct time-varying forecasts of the variance of log interest-rate changes using the above models. The time varying parameters are estimated by a maximum likelihood estimation technique. The valuation formulas are used to generate a binomial tree of expected future short rates using at the same time the yield curve and each of the four volatility curves as inputs. They compare the estimated prices with the actual values and employ the mean square error for forecast evaluation MSE: n

MSE =

1 (Pi − Pî )2 n i=1

(B.2)

where Pi and Pî correspond to the actual and estimated prices of the options. This statistic depends on the scale of the option price. A lower value of MSE shows that the estimated option prices are close to actual prices. They use also the Theil inequality coefficient (TIC) which is invariant to scale: n 1 ˆ 2 i=1 (Pi − Pi ) n , 0 ≤ T IC ≤ 1. (B.3) TIC = n n 1 1 2+ ˆ2 P P i=1 i i=1 i n n This statistic is between zero and one where zero corresponds to a perfect fit. Bali and Karagozoglu (1999) shows that the time series volatility estimates lead to more accurate predictions of option prices than the moving average models. In particular, the GARCH and the integrated GARCH models give more accurate representation of the volatility structure than moving average models. Questions 1. Can you describe with simple examples some interest rate sensitive instruments? 2. How can we price bonds under certainty and uncertainty? 3. How to develop some standard models for the pricing of bonds and bond options? 4. What are the relative merits of the competing models? 5. Can you provide a comparative analysis of term structure estimation models? 6. What are the distributional properties of spot and forward interest rates?

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References Ait-Sahalia, Y (1996). Testing continuous-time models of the spot interest rate. Review of Financial Studies, 9, 385–426. Bali, T and A Karagozoglu (1999). Implementation of the BDT model with different volatility estimators. The Journal of Fixed Income, 8, March, 25–33. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. Black, F and P Karasinski (1991). Bond pricing and option pricing when short rates are lognormal. Financial Analysts Journal, 47, 52–59. Black, F, E Derman and W Toy (1990). A one factor model of interest rates and its application to treasury bond options. Financial Analysts Journal, 46(1), January–February, 33–39. Brennan, MJ and ES Schwartz (1982). The case for convertibles. Chase Financial Quarterly, 1(3), Spring, 27–46. Cox, J, J Ingersoll and S Ross (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407. Dhillon, US and DJ Lasser 1998. Term premium estimates from zero-coupon bonds: new evidence on the expectations hypothesis. Journal of Fixed Income, 8 (June), 53–58. Fama, EF (1984a). The information in the term structure. Journal of Financial Economics, 13, 509–528. Fama, EF (1984b). Premiums in bond returns. Journal of Financial Economics, 13, 529–546. Fama, EF (1986). Term premiums and default premiums in money markets. Journal of Financial Economics, 17, 175–196. Fama, EF and RR Bliss (1987). The information in long maturities forward rates. American Economic Review, 77, 680–692. Ferguson, R and S Raymar (1998). A comparative analysis of several popular term structure estimation models. Journal of Fixed Income, 8 (March), 17–33. Froot, KA (1989). New hope for the expectations hypothesis of the term structure of interest rates. Journal of Finance, 44, 283–305. Heath, D, R Jarrow and A Morton (1989). Contingent claims valuation with a random evolution of interest rates. Review of Futures Markets, 9(1), 54–76. Heath, D, R Jarrow and A Morton (1992). Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60, 77–105. Ho, T and S Lee (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41, 1011–1029. Hull, J and A White (1987). The pricing of options on assets with stochastic variables. The Journal of Finance, 42, 281–300. Hull, J and A White (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Advances in Futures and Options Research, 3, 29–61. Hull, J and A White (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592. Hull and White (1992).

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Hull, J and A White (1993). Efficient procedures for valuing European and American path dependent options. Journal of Derivatives, 1, Fall, 21–31. Lekkos, I (1999). Distributional properties of spot and forward interest rates: USD, DEM, GBP, and JPY. Journal of Fixed Income, 8, March, 35–54. Leong, K (1998). Model choice. Risk, September, 19–22. Longstaff, FA (1990). Time varying term premiums and traditional hypothesis about the term structure. Journal of Finance, 45, 1307–1314. Nelson, C and A Siegel (1987). Parsimonious modeling of yield curves. Journal of Business, 60(4), 475–489. Spindel, M (1992). Easier done than said. Risk, 5(9), October, 77–80. Vasicek, O (1977). A equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. Vasicek, O and G Fong (1982). Term structure modeling using exponential splines. Journal of Finance, 37(2), 339–348. Willmott, P (1998). Derivatives, Chicheslec: John Wiley and Sons.

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Part VI Generalization of Option Pricing Models and Stochastic Volatility

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Chapter 17

EXTREME MARKET MOVEMENTS, RISK AND ASSET MANAGEMENT: GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES, AND INFORMATION COSTS

Chapter Outline This chapter is organized as follows: 1. Section 17.1 presents briefly the main results in the Merton’s (1976) jump-diffusion model and the Cox and Ross (1976) constant elasticity of variance model. 2. Section 17.2 develops the main results regarding the pricing and hedging of options in the presence of jumps and information costs. 3. Section 17.3 reviews the models for the valuation of options within the presence of jumps and information costs. It also calibrates the model to market data and shows some properties of the smile. 4. Section 17.4 studies implied volatility functions and option pricing models. Introduction Since the path-breaking contribution by Black and Scholes (1973) was published, many papers have tried to relax its most stringent assumptions. In particular, the introduction of a stochastic volatility has been considered by many authors. Much of the known bias reported in empirical studies based on the B–S formula has something to do with this assumption. The biases reported in Rubinstein (1994) for stock options, Melino and Turnbull (1990), and Knoch (1992) for options on other underlying assets, 745

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seem to be more pronounced for foreign currency options. These biases are not surprising since the model assumes a log-normal distribution for the underlying asset with known mean and variance. Using S&P 500 options, Dumas et al. (1998) studied the predictive and hedging performance of a deterministic volatility function option pricing model. They found that some option pricing models were not better than an ad hoc procedure that merely smooths Black and Scholes (1973) implied volatilities. They concluded that “simpler is better”. Using a different approach, we investigate the performance of a simple model for the valuation of options within a context of “implied liquidity premiums”. Rubinstein (1994) documented the main following bias when testing the Black–Scholes (1973) model. • When pricing out of the money calls, (puts), Black–Scholes model tends to overvalue (undervalue) these options with respect to market values. • When pricing in the money calls (puts), Black–Scholes model tends to undervalue (overvalue) these options with respect to market values. • The degree of mispricing is a function of the option’s moneyness. Dumas et al. (1998) documented a smile effect and specified four different models regarding the estimation of the volatility function. However, when measuring the prediction errors, they used an “Ad hoc” strawman as a proxy for a benchmark. Re-calling the fact that several market makers smoothed the implied volatility relation across exercise prices to price options, they fit the Black and Scholes model to the observed options prices. This method allows them to operationalize the practice. This procedure is internally inconsistent since the Black and Scholes model depends on the assumption of a constant volatility. This “ad hoc” strawman procedure is tested in Dumas et al. (1998), who found that it performs marginally better than other models. The fact that asset prices do not move continuously but rather jump from time to time, led Cox and Ross (1976) to price options for alternative stochastic processes. In the same way, Merton (1976) used a combination of a jump process and a diffusion process. The simple analytic formulas given by Bellalah and Jacquillat (1995) and Bellalah (1999) explained some of the biases reported in the literature, and in particular, the smile effect. We present a simple option pricing model when markets can make sudden jumps. The option value depends upon the probability and magnitude of jumps and a continuous volatility. The model is useful in explaining the smile effect. Using the market prices of at least two options on the same

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Extreme Market Movements, Risk and Asset Management

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underlying asset and maturity with different strike prices, the model can be used to extract the market implied volatility and information regarding the implied jumps. The model can be applied to hedging strategies for different strike prices and can be used for the valuation of different types of options. It can also be used in the identification of mispriced options. Some simulations are run with and without shadow costs of incomplete information. The option valuation model proposed in Bellalah and Prigent (2001) and Bellalah (2002) might help to understand why the Black and Scholes model, leads to theoretical prices which are systematically biased and why implied volatilities differ from one strike price to another for the same underlying asset. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah et al. (2001a), Bellalah et al. (2001b), Bellalah and Prigent (2001), Bellalah and Selmi (2001) and so on. 17.1. The Jump-Diffusion and the Constant Elasticity of Variance Models This section presents Merton’s (1976) model, which is a combination of a jump process and a diffusion process. It also develops the results in the Cox and Ross (1976) constant elasticity of variance diffusion model. 17.1.1. The jump-diffusion model Merton (1976) used a combination of a jump process and a diffusion process assuming that after each jump in the underlying asset price, a diffusion process is used. By constraining the jumps in such a way that they are distributed log-normally, Merton (1976) presented the following formula for the pricing of European call options, C=

∞ 1 −γ(1+h)T e [γ(1 + h)T ]n [SN (d1 ) − Ke−r T N (d2 )] n! n=1

with d1 =

ln

S K

+ r + 12 σ 2 (T ) √ , σ T

d2 =

ln

S K

+ r − 12 σ 2 (T ) √ σ T

where: r = r − γh +

n ln(1 + h) , T

σ 2 = σ 2 +

n 2 σ , T j

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where γ: rate at which jumps occur; h: average jump size measured as a proportional increase in the stock price and σj2 : variance in the distribution of jumps. For non-familiar readers, this is an introductive definition of the Poisson process. Definition: The Poisson process If (Nt )t≥0 is a Poisson process with an intensity γ, then for all t > 0, the random variable Nt satisfies P (Nt = n) =

1 −γt e (γt)n . n!

In particular, E(Nt ) = γt,

Var(Nt ) = (Nt2 ) − (Nt )2 = γt.

Also, when s > 0,

(sNt ) = e(γt(s−1)) .

17.1.2. The constant elasticity of variance diffusion (CEV) process The family of CEV processes are described by the following stochastic differential equation: θ

dS = µSdt + δS 2 dW with µ, θ, and δ > 0 and W is a Wiener process. θ In the above expression, δS 2 is the instantaneous variance of the stock price where θ is the elasticity of this variance with respect to S. In this equation, the instantaneous variance of the return σ 2 is given by: σ 2 = δ 2 S θ−2 . When θ < 2, this variance is a decreasing function of the asset price. When θ = 2, the instantaneous variance of returns is δ2 and the process reduces to that used in Black and Scholes (1973). The variance is independent of

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the asset price level. The formula presented in Cox and Ross (1976) for the valuation of a call at any instant of time is: ∞ S E 1 g C=S G + n+1 n+1 2−θ n=0

S 1 E −rT g −Ke + G n+1 2−θ n+1 where,

2re−rT (2−θ) [e−rT (2−θ) − 1] S =S δ 2 (2 − θ)

−rT (2−θ) − 1) 2−θ 2r(e E =E δ 2 (2 − θ)

2−θ

where the gamma density function, g(x, m) = and

G(x, m) =

e−x xm−1 Γ(m) ∞

x

g(y, m)dy

where K is the strike price, r is the risk-less interest rate, and T is the time to expiration. This model can be easily simulated. In fact, given a daily variance of return and a value of θ, δ can be chosen to satisfy σ 2 = δ 2 S θ−2 . 17.2. On Jumps, Hedging and Information Costs It is well known that sudden movements in asset prices appear at random discrete times in financial markets. In general, when there is a jump in the underlying asset price, it is very difficult to implement a hedge in the standard sense. Financial asset prices can be modeled by a jump-diffusion process, which corresponds to the standard diffusion plus a jump component as follows: dS = µSdt + σSdX + (J − 1)Sdq. The above equation shows that the dynamics of the underlying asset price corresponds to the standard diffusion process dS = µSdt + σSdX plus a

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Poisson component. The term dq corresponds to the Poisson process, which can be defined as follows:

0 with probability 1 − γdt dq = 1 with probability γdt. This means that in the presence of a jump, dq = 1, otherwise it is equal to zero. The probability of a jump in q over the interval dt is γdt, where γ refers to the intensity of the Poisson process. When there is a jump, dq = 1 and S goes immediately to JS, where J refers to the jump size. The jump size J can be a constant or a random variable. If J is random, it is often assumed that it is drawn from a distribution with a probability density function P (J) and that is independent of the Poisson process and the Brownian motion. The random walk in the logarithm of the underlying asset follows from the dynamics of S: 1 2 d(log S) = µ − σ dt + σdX + (log J)dq. 2 This represents a jump-diffusion version of Ito. 17.2.1. Hedging in the presence of jumps As in the original Black and Scholes (1973) context, it is possible to construct a portfolio with a long position in the derivative security and a short position in ∆ units of the underlying asset: Π = V (S, t) − ∆S. The change in the portfolio’s value over a small interval of time can be written as: ∂ 2V ∂V 1 ∂V + σ2 S 2 2 dt + − ∆ dS dΠ = ∂t 2 ∂S ∂S + [V (JS, t) − V (S, t) − ∆(J − 1)S]dq This represents also a jump-diffusion version of Ito. In the absence of a jump at time t, dq = 0 and the elimination of risk can be done using the standard ∆ = ∂V ∂S . When there is a jump, dq = 1 and the portfolio changes by an amount O(1) that cannot be diversified away. In order to implement the hedging argument for the diffusion, consider the change in the portfolio

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over a small time interval (with no stochastic term dS): ∂V ∂ 2V ∂V 1 dΠ = + σ2 S 2 2 dt + V (JS, t) − V (S, t) − (J − 1)S dq. ∂t 2 ∂S ∂S The above equation shows that the value of the portfolio in a deterministic context comprises a “deterministic component” and a “non-deterministic jump” component. Merton (1976) showed that in the absence of a correlation between the jump component and the “market portfolio”, the diversifiable risk should not be paid for. Hence, in the absence of market price for risks (jumps are not priced), the expected return on the above portfolio must be the risk-less rate plus the information costs. Following this approach, this gives the following equation: ∂V ∂2V 1 ∂V + (r + λS )S + σ2 S 2 − (r + λV )V ∂t 2 ∂S 2 ∂S ∂V SE[(J − 1)] = 0 + γE[V (JS, t) − V (S, t)] − γ ∂S where λS and λV correspond to the information costs on the underlying asset market and the option market, respectively. The expectation in this equation is taken with respect to the jump size J and can be written as follows: E[x] = xP (J)dJ where P (J) corresponds to the probability density function for the jump size J. In the absence of jumps γ and information costs λS and λV , this equation becomes the standard Black and Scholes (1973) equation. It is possible to obtain a closed-form solution to this equation when the logarithm of J is normally distributed with a volatility σ . In this case, the European option price is given by: ∞ 1 −γ (T −t) e (γ (T − t))n VBS (S, t; λS , λV , σn , rn ). n! n=1

with k = E[J − 1],

γ = γ(1 + k),

σn2 = σ2 +

and rn = r + λs − γk +

n log(1 + k) T −t

nσ2 T −t

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where VBS corresponds to the standard Black and Scholes (1973) price in the absence of jumps within a context of incomplete information. This formula shows that the option price corresponds to a sum of individual prices where it is assumed that there are n jumps and where each price is weighted by the probability of the occurrence of n jumps before the option maturity date. 17.2.2. Hedging the jumps It is possible to “hedge” the portfolio in the presence of jumps by choosing “the hedge ratio” that minimizes the variance of the hedged portfolio as shown in Willmott (1998). In this context, the change in the portfolio value with any ∆ can be written as: ∂V − ∆ dS + (−∆(J − 1)S + V (JS, t) − V (S, t))dq + · · · dΠ = ∂S The risk in this change of the portfolio value can be computed as: 2 ∂V − ∆ σ 2 S 2 dt Var[dΠ] = ∂S + γE[(−∆(J − 1)S + V (JS, t) − V (S, t))2 ]dt + · · · Differentiating with respect to ∆ and setting the resulting equation equal to zero, gives the ∆ that minimizes the variance: ∆=

γE[(J − 1)(V (JS, t) − V (S, t))] + σ 2 S ∂V ∂S γSE[(J − 1)2 ] + σ 2 S

Under this minimizing strategy, the discounted expectation of the option must satisfy the following equation: 1 σ2 ∂V ∂ 2V ∂V + σ2 S 2 2 + S µ− (µ + γk − (r + λS )) − (r + λV )V ∂t 2 ∂S ∂S d

J −1 = 0, + γE (V (JS, t) − V (S, t)) 1 − (µ + γk − r + λS ) d where d = γE[(J − 1)2 ] + σ 2 . In the absence of information costs and jumps, this equation reduces to the classic Black–Scholes equation. When the volatility σ is zero, (the case of

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no diffusion), the ∆ is given by ∆=

E[(J − 1)(V (JS, t) − V (S, t))] SE[(J − 1)2 ]

and the equation becomes: ∂V ∂V + µS − (r + λV )V ∂t ∂S

J −1 = 0. + γE (V (JS, t) − V (S, t)) 1 − (µ + γk − (r + λS )) d 17.2.3. Jump volatility Volatility can jump from one state to another from time to time. Consider the volatility in one of the two states σ− or σ + with σ− < σ + . The volatility can jump from lower to higher value in the presence of a Poisson process with intensity γ + or γ − in one direction or the other. As given in Willmott (1998), if a hedge portfolio is constructed and real expectations are taken, then the return on the portfolio must be equal to the risk-less rate plus information costs. This leads to the following equation for the value V of the option V + in the presence of the volatility σ + : ∂V + ∂V + 1 +2 2 ∂ 2 V + + (r + λ ) + σ S − (r + λV )V + + γ − (V − − V + ) = 0. S ∂t 2 ∂S 2 ∂S The same methodology leads to the following equation for the value V − of the option in the presence of the volatility σ− : ∂V − ∂V − 1 −2 2 ∂ 2 V − + σ S − (r + λV )V − + γ + (V + − V − ) = 0. + (r + λ ) S ∂t 2 ∂S 2 ∂S Jump volatility with exponential decay After a jump, the volatility shows an exponential decay and can be presented as follows: σ(τ ) = σ − + (σ + − σ − )e−ντ with τ : time since the last jump in volatility and ν: a decay parameter.

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The volatility is described in the presence of a Poisson process with intensity γ and can jump from its present level to σ + . In this context, the option price V (S, t, τ ) is a solution to the following equation: ∂V 1 ∂V ∂V ∂2V + + σ(τ )2 S 2 − (r + λV )V + (r + λS )S 2 ∂t ∂τ 2 ∂S ∂S + γ(V (S, t, 0) − V (S, t, τ )) = 0. 17.3. On the Smile Effect and Market Imperfections in the Presence of Jumps and Incomplete Information This section develops a simple option pricing model when markets can make sudden jumps. The option value depends upon the probability and magnitude of jumps and a continuous volatility. The model is useful in explaining the smile effect. 17.3.1. On smiles and jumps Using the market prices of at least two options on the same underlying asset and maturity with different strike prices, the model can be used to extract the market implied volatility and information regarding the implied jumps. The model can be applied to hedging strategies for different strike prices and can be used for the valuation of different types of options. It can also be used in the identification of mispriced options. Some simulations are run with and without shadow costs of incomplete information. The smile effect and the S&P 500 index in the presence of jumps Consider the implied volatilities on June 21, 1991 for the European-style July S&P 500 index options expiring in 28 days. Table 17.1 shows the implied volatilities and the deltas of S&P calls and puts using the Black and Scholes (1973) model. Table 17.1 is reproduced from Derman et al. (1991). Note that the − sign refers to the put’s delta and the sign + refers to the call’s delta. It is important to note that options with strike prices below the index price or out-of-the-money (OTM) puts with low deltas are traded at higher implied volatilities than options with strike prices above the asset price which correspond to OTM calls with low deltas. The presence of different

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Table 17.1. Implied volatilities and the deltas of S&P calls and puts using the Black and Scholes (1973) model when the index was at 377.25, r = 0.059, and the dividend yield is 2%. The index forward price is 378.33. Strike

Type

σImplied (in %)

∆ (in %)

345 350 355 360 365 370 375 380 385 390 395 400

put put put put put put put call call call call call

21.5 19.3 18.4 17.3 16.2 15.3 13.3 13.6 12.3 12.5 12.5 13.1

−5.7 −6.8 −10.1 −14.5 −20.5 −29.2 −39.8 46.1 31.0 19.5 11.0 6.4

implied volatilities for different strike prices refers to the well-known smile. This may be viewed as an “anomaly” in the Black–Scholes model since when using their formula, one must adjust the volatility as the strike price changes. Besides, the fact that implied volatilities seem to be higher for puts than calls may be a “strange” result. In fact, if market participants believe that the underlying asset is driven by a continuous random walk, then the volatility must be independent of the strike price. This strange result can be explained by the fact that market participants expect an occasionally sharp downward jump in the underlying asset price. If it were the case, then OTM puts could show a higher probability of paying off than OTM calls. In this case, the smile can be explained by a jump-diffusion process. This process corresponds to a continuous diffusion which is accompanied occasionally by a jump. The use of the Black and Scholes (1973) model assumes that all future variations in the underlying asset value is attributed to the continuous diffusion and none to the discontinuous jump. The jump-diffusion process is defined by a diffusion volatility and a probability and magnitude for the discontinuous jump. The diffusion volatility characterizes the continuous diffusion. A small probability of a jump of the underlying asset price in the direction of the strike price can affect the value of an OTM option. In the presence of such process, at least two options are used to extract information about the implied volatility and the implied jump.

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Valuing options when markets can jump Consider a simple model. The underlying asset price at time zero today is S. In the next instant, the underlying asset price can jump up by u% to Su with probability w or down by d% to Sd with probability (1 − w). The probability w is expected to be close to zero or one. This means that either a jump up or a jump down predominates. After the first jump, the underlying asset will diffuse with volatility σ as in the Black and Scholes (1973) model. No other jumps will occur. The value of any security in this model can be computed as the average of its payoffs over the scenarios where the undelying asset jumps up or down. Hence, the option value is given by: option = wBS(Su , K, σ, r, δ, t) + (1 − w)BS(Sd , K, σ, r, δ, t)

(17.1)

where BS(S, K, σ, r, δ, t) is the formula in Black and Scholes (1973) and δ refers to the continuous dividend yield. The values used for the underlying asset are: Su = S(1 + u),

Sd = S(1 − d).

The current value of the underlying asset also corresponds to an average value after a jump up and a jump down. Hence, the jump up and the jump down are related by: d(1 − w) = wu. Using the model for calls when the index can jump up Consider the trading of index options when the current index level is 100, r = 10%, and T = 1. Table 17.2 gives market prices and the corresponding implied volatilities. Table 17.2. Estimation of the implied probability of an upward jump and a diffusion volatility. Market call prices and implied volatilities in the presence of an upward jump for the following parameters: S = 100, r = 0.1, δ = 0, and T = 1. Strike

Call

110 120

4.83 2.41

σBlack−Scholes 12.1% 14.09%

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Using the model needs the knowledge of the upward jump and the diffusion volatility, which are consistent with these prices. Using a simple algorithm and formula, as in Eq. (17.1), it is possible to verify that the upward jump u = 171.8% and the diffusion volatility is σ = 10%. The implied probability of an upward jump to 271.8 is 1%. The downward jump probability to 98.26 is 99%. The subsequent diffusion volatility is 10%. Note that the Black–Scholes implied volatility in Table 17.2, σBlack−Scholes = 12.1% is higher than the implied diffusion volatility σ = 10%. Using Eq. (17.1), Table 17.3. gives the details of the computation of call values after the jump. Knowing the values of w, σ, u, and d that fit the data, it is possible to use Eq. (17.1) to value options on the same underlying and maturity for different strike prices. Table 17.4 gives some option values and their corresponding deltas using the model. ∗ The value of σBlack−Scholes is the implied Black–Scholes volatility corresponding to the model value. It corresponds to the implied volatility necessary to have the Black–Scholes formula reproduce the model price. Note that the OTM call deltas are less than those predicted by the Table 17.3.

Model values in the presence of jumps.

Strike

Index–jump

BS–call

Probability

Model–contribution

110 110

271.8 98.26

171.8 3.14

0.01 0.99

1.72 3.11

120 120

271.8 98.26

162.71 0.79

Total

4.83

0.01 0.99

1.63 0.78

Total

2.41

Table 17.4. A comparison between Black–Scholes and model call values and deltas in the presence of jumps.

Strike

Call

∆ (in %)

∗ σBlack−Scholes (in %)

∆Black−Scholes (in %)

90 100 110 120 130 140

18.3 10.26 4.83 2.41 1.67 1.46

90 80 47 18 6.2 3.2

10.7 11.2 12.1 14.1 17.3 21.0

97 82 52 29 19 15

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Black–Scholes model. The model seems to be insensitive to the exact value of the jump probability w. Using the model for puts when the index can jump down Consider the trading of index options using the values as shown in Table 17.1. The OTM puts with strike price of 370 and 365 are used to fit the model for different levels of jump probabilities w. Table 17.5 gives the implied percentage jump and the implied diffusion volatility using Eq. (17.1) and the two option prices. Table 17.5 shows for example that a 2% probability of an immediate down jump in the underlying asset by 18.6% and a subsequent diffusion of 12.9% can exactly reproduce the put values with strike prices 370 and 365. Table 17.6 shows the Black–Scholes implied volatilities and the corresponding prices for the model in the presence of jumps for a jump probability of 5%. Table 17.6 also gives the market prices and Black–Scholes implied volatilities of Table 17.1. For example, with a 5% probability of a 10.7% downward jump, the model fits the values of OTM puts. The model can be implemented in trading by choosing from the possible jumps that fit the initial two options by using the trader perception of what is a reasonable jump probability and magnitude. Table 17.5. Estimation of the implied jump magnitude and diffusion volatilities that match the S&P puts struck at 365 and 370. Jump–probability, w (in %)

Implied–jump–down, d (in %)

Implied–volatility (in %)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

31.4 18.6 14.3 12.1 10.7 9.8 9.1 8.6 8.2 7.8 7.5 7.2 7.0 6.8 6.7

13.1 12.9 12.7 12.5 12.2 12.0 11.8 11.6 11.4 11.2 11.0 10.7 10.5 10.2 9.9

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Extreme Market Movements, Risk and Asset Management Table 17.6.

759

Black–Scholes implied-volatilities for 5% jump probability.

Strike

Model σ (in %)

Market σ (in %)

Option

Model–price

Market–price

345 350 355 360 365 370 375 380 385 390 395 400

20.6 19.7 18.5 17.4 16.2 15.3 14.6 14.1 13.8 13.5 13.3 13.2

21.5 19.3 18.4 17.3 16.2 15.3 13.3 13.6 12.3 12.5 12.2 13.1

put put put put put put put call call call call call

0.46 0.68 0.97 1.37 2.00 3.00 4.54 5.10 3.06 1.68 0.85 0.39

0.56 0.63 0.94 1.38 2.00 3.00 4.00 4.88 2.50 1.38 0.69 0.38

17.3.2. On smiles, jumps, and incomplete information We develop a simple option pricing model when markets can make sudden jumps in the presence of incomplete information. We build on Derman et al. (1991) modeling of jumps on the underlying asset and combine it with Bellalah (1999) approach to include information costs. The same approach can be extended to allow the estimation of implied information costs from market data. The option value is given by: option = wBS(Su , K, σ, r, b, λs , λc , t) + (1 − w)BS(Sd , K, σ, r, b, λs , λc , t) where BS(S, K, σ, r, b, λs , λc , t) is the formula given in Bellalah (1999). The values used for the underlying asset are: Su = S(1 + u),

Sd = S(1 − d).

The jump up and the jump down are related by: d(1 − w) = wu. The price of a European call with a strike K is: C(S, T ) = Se−(λC −λS )T N (d1 ) − Ke−(r+λC )T N (d2 ) with

d1 = ln

S K

√ 1 2 σ T + r + σ + λS T 2

(17.2)

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and

√ d2 = d1 − σ T

(17.3)

where N (.) is the univariate cumulative normal density function. When λS and λC are set equal to zero, this equation collapses to that in shown Barone–Adesi and Whaley (1987). In this case, the value of a European commodity call is: C(S, T ) = Se((b−r−(λC −λS ))T ) N (d1 ) − Ke−(r+λC )T N (d2 ) with

d1 = ln

S K

√ 1 2 + b + σ + λS T σ T, 2

√ d2 = d1 − σ T .

When λS and λC are equal to zero and b = r, this formula is the same as that in Black and Scholes. For an index option, b = r − δ, where δ is the distribution rate. For a foreign currency option, b = r − r∗, where r∗ is the foreign interest rate. For a commodity, b = r − δ, where δ is the convenience yield. The price of a European put with a strike K is: P (S, T ) = −Se((b−r−(λC −λS ))T ) N (−d1 ) + Ke−(r+λC )T N (−d2 ) with

√ 1 S σ T, + b + σ 2 + λS T d1 = ln K 2

(17.4)

√ d2 = d1 − σ T .

17.3.3. Empirical results in the presence of jumps and incomplete information The smile and the jumps Consider the implied volatilities on a given day for the European-style July S&P index options expiring with a given maturity. Table 17.7 shows the implied volatilities and the deltas of S&P calls and puts using the Black– Scholes (1973) model. The option maturity date is in March 2001, the index level is 1264.74, the risk-less interest rate is 5.81%, and the dividend yield is 1.17%. The calibration has been made using the 1200 and the 1250 put options as they correspond to the most liquid options given the maturity we considered. The results are given in Tables 17.8 and 17.9. In Table 17.8, the results are based on direct application of the Derman et al. (1991)

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Extreme Market Movements, Risk and Asset Management Table 17.7. Implied volatilities and the deltas of S&P calls and puts using the Black–Scholes (1973) model. Strike

Type

σimplied (in %)

∆ (in %)

950 975 1025 1050 1100 1125 1150 1175 1200 1250 1275 1300 1325 1350 1375 1400

put put put put put put put put put put call call call call call call

35.49 34.84 34.72 32.51 31.84 30.38 29.80 29.07 28.21 25.71 24.28 24.57 23.65 22.47 22.02 21.20

−0.37 −0.72 −2.26 −3.68 −8.44 −11.97 −16.31 −21.45 −27.29 −40.45 52.42 45.56 38.96 32.77 27.13 22.10

Table 17.8. Parameter estimates using the Derman et al. (1991) methodology. w (in %)

d (in %)

σdiffusion (in %)

3 4 5 6 7 8 9 10 11 12 13 14 15

58.71 46.87 39.73 34.94 31.49 28.90 26.85 25.19 23.82 22.67 21.67 20.79 20.03

19.73 19.51 19.28 19.04 18.80 18.55 18.29 18.03 17.76 17.47 17.19 16.89 16.58

Table 17.9. Parameter estimates using the Derman et al. (1991) methodology with endogenous w parameter. w

d

σdiffusion

15.51%

19.67%

16.43%

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762 Table 17.10.

Comparison between Black and Scholes and model prices.

Strike

Type

Market price

Black–Scholes price

Model price

Market σ (in %)

Model σ (in %)

950 975 1025 1050 1100 1125 1150 1175 1200 1250 1275 1300 1325 1350 1375 1400

put put put put put put put put put put call call call call call call

1.875 2.625 5.750 6.375 12.000 14.500 18.875 24.000 30.000 44.125 54.000 43.625 32.375 22.500 15.875 10.250

0.10 0.21 0.87 1.59 4.60 7.28 11.05 16.13 22.73 41.07 54.00 42.98 33.70 26.03 19.80 14.84

0.813 1.587 4.512 6.746 12.630 16.194 20.186 24.719 30.012 44.140 54.704 41.540 30.549 21.727 14.930 9.907

35.49 34.84 34.72 32.51 31.84 30.38 29.80 29.07 28.21 25.71 24.28 24.57 23.65 22.47 22.02 21.20

31.20 31.90 32.84 32.98 32.38 31.64 30.66 29.49 28.22 25.71 24.60 23.62 22.78 22.06 21.46 20.96

methodology, i.e., the w parameter value has been explicitly chosen. We give the results for a set of reasonable values, starting from w = 3%, as the algorithm was unable to achieve convergence for values less than this figure. In Table 17.9, the w value is endogenously determined, i.e., we let the algorithm to calculate the parameter values (w, δ, and σ diffusion), which best fit the market prices used for calibration. In the remainder, we decided to restrict ourselves to this approach. Table 17.10 gives a comparison between the market price, the Black and Scholes price, and the model price, and between the model-implied diffusion volatility and the Black and Scholes implicit volatility. The input value of sigma for the Black and Scholes formula has been estimated using the 1250 at-the-money (ATM) put.

Introducing information costs We introduce information costs in the Derman et al. (1991) methodology. We considered information costs both on the option market (λC ) and the underlying asset (λS ) and run simulations for different cost levels (from 1% to 5%). However, due to space considerations, we restrict our presentation in Fig. 17.1 to the most significant results. Information cost levels which give the best fitting (λS = 1%) and λC = 2%) are very close to Merton’s estimates although we use a radically different approach. Thus, we can view the model as

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a possible (and reliable) way to extract information costs using option prices. Recent evidence in 2008 and 2009 seems to confirm the results. 17.4. Implied Volatility and Option Pricing Models: The Model and Simulation Results The “liquidity premium” can be defined as an amount f (S0 , K), which is added to (subtracted from) the classical option payoffs according to the position of the initial underlying asset price with respect to the strike price. The liquidity functions may be written as gc (S0 , K) = K + fc (S0 , K) for a call and gp (S0 , K) = K + fp (S0 , K) for a put option, where fc (S0 , K) and fp (S0 , K) stand for the amount of money subtracted or added to the option price to account for the mispricing, S0 is the observed underlying asset price, and K is the strike price. 17.4.1. The valuation model Assumption The function fc (S0 , K) must satisfy the following assumptions: • if S0 > K, then −K < fc (S0 , K) ≤ 0 and fc (S0 , K) is nondecreasing in K; • if S0 < K, then fc (S0 , K) > 0 and fc (S0 , K) is nondecreasing in K and • fc (S0 , K) is differentiable on R+∗2 and fc (K, K) = 0. The function fp (S0 , K) must satisfy the following assumptions: • if S0 > K, then fp (S0 , K) > 0 and fp (S0 , K) is nonincreasing in K; • if S0 < K then −K < fp (S0 , K) ≤ 0 and fp (S0 , K) is nonincreasing in K; ∂fp (S0 ,K) • ∂K < 1 (which implies that gp (S0 , K) is increasing in K) and • fp (S0 , K) is differentiable on

+∗2

and fp (K, K) = 0.

The functions gc (S0 , K) and gp (S0 , K) satisfy the main properties as in the standard Black and Scholes case: they are positive, both nondecreasing in K and gc (K, K) = gp (K, K) = K. Example Assume that the strike price K is in an interval [aS0 , bS0 ] with 0 < a < 1 < b, then the amount of mispricing can be appreciated with respect to

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the position of the observed underlying asset price S0 with respect to the strike price and the degree of moneyness of the option. For the call option, define, for example, the function fc (S0 , K) as: 2 aS0 with (1−a) if S0 > K, then fc (S0 , K) = −α0 1 − SK0 2 > α0 and 2 if S0 < K, then fc (S0 , K) = β0 1 − SK0 with β0 > 0. Then, we have gc (S0 , K) > 0 and is nondecreasing in K. The parameter α0 affects the prices of in-the-money (ITM) calls and the parameter β0 affects OTM call prices. The parameters α0 and β0 represent the coefficients that give the desired shape of the smile. When these parameters are zero, this situation corresponds to the Black–Scholes case. For the put option, the following function fp (S0 , K) is defined as: 2 If S0 > K, then fp (S0 , K) = γ0 1 − SK0 with γ0 > 0 and sufficiently small ∂f (S ,K) in order to have p ∂K0 < 1 and 2 If S0 < K, then fp (S0 , K) = −δ0 1 − SK0 with δ0 > 0 and small enough ∂f (S ,K) bS0 to have p ∂K0 < 1 and (1− 1 2 > δ0 (with b < 3). ) b

The parameter δ0 affects the prices of ITM puts and the parameter γ0 affects OTM put prices. The call option price in the Black–Scholes model is given by: cBS (K) = e−rT EQ [(ST − K)+ ] where the mathematical expectation EQ is taken with respect to the riskneutral probability Q. The call option price accounting for the liquidity premium is given by: cLP (K) = e−rT EQ [(ST − gc (S0 , K))+ ] which is also: cLP (K) = S0 N (d1 ) − gc (S0 , K)e−rT N (d2 ) with

√ 1 2 d1 (σ0 , gc (S0 , K)) = ln σ T + r+ σ T 2 √ d2 (σ0 , gc (S0 , K)) = d1 − σ T S0 gc (S0 , K)

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When the liquidity premium is accounted for in the valuation of European put options, the put formula is given by: pLP (K) = e−rT EQ [(ST − gp (S0 , K))+ ] which is also: pLP (K) = −S0 N (−d1 ) + gp (S0 , K)e−rT N (−d2 ). The model corrects for both biases with respect to the Black–Scholes model. 17.4.2. Simulation results Using Black and Scholes model and our model with “liquidity premiums”, Tables 17.11 and 17.12 develop simulation results for European call and put options. Table 17.11 presents call prices for different levels of the underlying asset price varying from 95 to 105. For illustrative purposes, the table compares the Black and Scholes call price, cBS with the vertical premium call price, cLP . The spread between the two models is reported as Dif to reflect the amount of overvaluation or undervaluation in the Black and Scholes model. Tables 17.12 shows similar information for the values of European put option prices for the same set of parameters. The option’s time to maturity is one year from the 20th March 1999 to the 19th March 2000. The strike price is 100, the risk-less interest rate is 0.1, and the volatility parameter is 0.2. The vertical premium function is defined for the call using Table 17.11. Simulations of European call prices for BS and our model using: T = 1, r = 0.1, σ = 0.2, K = 100, and α0 = β0 = 0.75. S

CBS

CLP

Dif, 10−3

95 96 97 98 99 100 101 102 103 104 105

9.86275014 10.5069582 11.1703759 11.8523936 12.5523797 13.2696859 14.0036514 14.7536068 15.5188787 16.298793 17.0926787

9.86180506 10.506331 11.1700108 11.8522259 12.5523365 13.2696859 14.0036962 14.7537871 15.5192863 16.2995203 17.0938178

−0.94508 −0.62721 −0.36513 −0.16763 −0.00000 0 0 0.18032 0.40765 0.72727 1.13906

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Table 17.12. Simulations of European put prices for BS and our model using: T = 1, r = 0.1, σ = 0.2, K = 100, and γ0 = δ0 = 0.75. S

PBS

PLP

Dif, 10−3

95 96 97 98 99 100 101 102 103 104 105

5.34649194 4.99070003 4.65411775 4.33613563 4.03612149 3.75342774 3.48739318 3.23734858 3.00262046 2.7825348 2.57642053

5.345741 4.990241 4.653872 4.336032 4.036097 3.753428 3.487415 3.237429 3.002789 2.782811 2.576820

0.751 0.459 0.246 0.104 0.025 0 −0.022 −0.081 −0.168 −0.0277 −0.400

α0 = β0 = 0.75. For the put option, we use γ0 = δ0 = 0.75. The above parameters are used for illustrative purposes, but other parameters can also be used. These parameters can be adjusted to obtain any desired form of a smile. In fact, if we modify these parameters, we change the shape of the smile curve. The simulation results show also that the model prices are very close to Black and Scholes prices for call and put options because of the chosen parameters. The sign of spread between the two models depends on the way it is calculated. It can be either positive or negative according to the degree of parity and to the way the difference is calculated between the two models. The difference reflects the amount of overvaluation or undervaluation reported in empirical tests of the Black and Scholes type models.

17.4.3. Model calibration and the smile effect The Black and Scholes model and market prices are used to estimate the implied volatility. The implicit volatility for short-term and long-term options is calculated using an iterative procedure. The algorithm given by Bellalah and Jacquillat (1995) is used for this purpose. Each day, implied volatilities are aggregated with respect to the degree of the options parity. Hence, we obtain each day, 11 average implied volatilities corresponding to different degrees of parity. To obtain an idea about the index volatility estimates, we construct an implied ratio of volatility. This ratio is defined

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as follows: RvK,t =

σK,t σ0,t

for K = −5, . . . , 5.

By construction, this ratio is equal to one for at the money (ATM) options. For each day, the mean volatility is calculated. Then, the volatilities are aggregated. The ratio of volatilities is constructed by dividing the implied volatility for a trade by the volatility of ATM options. This smile is an evidence against the Black and Scholes model. This smile is different from that reported in Dumas et al. (1998). This result is an evidence that the smile effect is different from one market to another for index options. Summary The financial crisis reveals the failure of markets and institutions in risk management and asset pricing. The main reason for this is that models ignored rare and extreme events as well as the importance of information. The Black–Scholes model is the most simple and successful model in the theory of option pricing. Many authors, however, have tried to relax some of its assumptions in order to explain its well-known biases. These biases are shown to be more pronounced for foreign exchange options than for stock options. The extensions of the B–S model include constant elasticity of variance processes and jump-diffusion processes to explain some of the option biases. This chapter presents recent developments along these lines, especially, the models of Merton (1976) and the Cox and Ross (1976). From a theoretical point of view, these models are rather interesting. They point out the difficulty of pricing assets in an incomplete market. From a practical point of view, the use of these models are very limited given the burden of parameter estimation to implement them. We develop a simple model for the valuation of options in the presence of jumps and information costs. The model is an extension of the models of Derman et al. (1991) and Bellalah (1999). Our model has the potential to explain the smile effect. It is calibrated to market data and allows an implicit estimation of the magnitude of information costs. While our methodology and our model are applied only to index options, they can be used in different option markets. Following the approaches in Black and Scholes (1973), this chapter analyzed and applied an option pricing model with liquidity premiums. The concept of liquidity premiums refer to the amount of mispricing in the Black–Scholes model. This can be defined by a function which depends on the spread

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between the underlying asset price and the option strike price. In fact, it is well known that the mispricing in the Black–Scholes model depends on the degree of parity of the option. These premiums defined with respect to the option parity can be justified on the grounds of the risk-reward trade off as in capital asset pricing models. When “liquidity” premiums are ignored, the model reduces to the Black–Scholes model for valuing European options. Since the model presents an additional parameter with respect to the Black–Scholes model, it may be easily calibrated to market prices and may explain biases observed in Black–Scholes type models. The simulation results for a constant volatility show that the model prices when compared to Black–Scholes prices, do not show a “smile” of volatility. This result is very important. In fact, the smile observed for the Black–Scholes model for different strike prices is a surprising result since the volatility is associated with the stock price rather than the strike price.

Questions 1. 2. 3. 4. 5. 6. 7. 8.

What are the specificities of the jump-diffusion model? What are the specificities of the constant elasticity of variance model? What is the empirical evidence regarding the volatility smiles? Describe briefly the main results in the Merton’s (1976) jump-diffusion model. Describe the main results in the Cox and Ross (1976) constant elasticity of variance model. Describe briefly the main results regarding the pricing and hedging of options in the presence of jumps and information costs. Describe briefly the main results in the models for the valuation of options in the presence of jumps and information costs. Describe briefly the main results regarding implied volatility functions and option pricing models.

References Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, 42 (June), 301–320. Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September), 645–664. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927.

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Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3), 617–635. Bellalah, M and J-L Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bellalah, M, JL Prigent and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Cox, JC and SA Ross (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166. Derman, E, A Bergier and I Kani (1991). Valuing Index Options When Markets Can Jump. Quantitative Research Notes (July). New York: Goldman Sachs. Dumas, B, J Fleming and R Whaley (1998). Implied volatility functions: empirical tests. Journal of Finance, 53, 2059–2106. Knoch, H-J (1992). The pricing of foreign currency options with stochastic volatilities. PhD. dissertation, Yale School of Organization and Management. Melino, A and SM Turnbull (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45, 239–265. Merton, RC (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–144. Rubinstein, M (1994). Implied binomial trees. Journal of Finance, 3, 771–818. Willmott P (1998). Derivatives. New York: John Wiley and Sons.

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Chapter 18 RISK MANAGEMENT DURING ABNORMAL MARKET CONDITIONS: FURTHER GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES, AND INFORMATION COSTS

Chapter Outline This chapter is organized as follows: 1. Section 18.1 presents the Hull and White (1987) model, which is one of the simplest models of option pricing with stochastic volatilities. It develops the main results of the Stein and Stein (1991) model. A generalization of derivative asset pricing models to a context of stochastic volatilities within complete and incomplete markets is presented. In particular, the main results in the models of Heston (1993) and Hoffman et al. (1992) are reviewed. It is important to note that an incomplete market is simply a market in which there is not a unique equivalent martingale measure or risk-neutral probability for the asset price. 2. Section 18.2 develops a framework for option pricing in the presence of a stochastic volatility and information costs. 3. Section 18.3 is devoted to the theory of option pricing biases and in particular the “smile effect”. 4. Section 18.4 presents some empirical evidence regarding option pricing within information uncertainty and stochastic volatility.

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Introduction Volatility is an important parameter in option pricing theory. Black and Scholes (1973) proposed an option-valuation equation under the assumption of a constant volatility in a complete market without frictions. Engle (1982) developed a discrete-time model, to show that the volatility depends on its previous values. Many papers have tried to relax the most stringent assumptions in the Black–Scholes (B–S) theory and in particular, the constant volatility. Much of the known bias reported in empirical studies based on the B–S formula has something to do with this assumption. The biases reported in Rubinstein (1985, 1994) for stock options, Melino and Turnbull (1990, 1991) and Knoch (1992) for options on other underlying assets, seem to be more pronounced for foreign currency options. These biases are not surprising since the model assumes a log-normal distribution for the underlying asset with known mean and variance. The stochastic volatility problem has been examined by several authors. For example, Hull and White (1987), Wiggins (1987), and Johnson and Shanno (1987) studied the general case in which the instantaneous variance of the stock price follows some geometric processes. Scott (1989) and Stein and Stein (1991) used an arithmetic volatility in the study of option pricing. All these models describe (with precision) the effects of the volatility on the options prices. Stein and Stein (1991) and Heston (1993) proposed a dynamic approach for the volatility, which is represented by an Ornstein– Ulhenbeck. It is difficult to find an analytic solution for the stochastic volatility option-pricing problem. Merton (1987) proposed a capital-asset pricing model in the presence of the shadow costs of incomplete information. Bellalah (1990) applied the Merton (1987) model to the valuation of options under incomplete information. Bellalah and Jacquillat (1995) and Bellalah (1999) derived the Black and Scholes (1973) equation in the context of Merton’s (1987) model. They obtained another version of the Black and Scholes equation within information uncertainty. Options can be valued in the context of Merton’s (1987) “Simple model of capital market equilibrium with incomplete information”. We provide the derivation of the partial differential equation for options in the presence of shadow costs of incomplete information and stochastic volatility. We illustrate our approach by specific applications and show the dependancy of the option price on information and stochastic volatility. As in Platen and Schweizer (1992), we show that the investor’s choice of the minimal equivalent martingale measure is not changing, but the process of the price of the asset depends on incomplete information. Bellalah et al. (2001) carried out an empirical

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test of the valuation model of options in the presence of information costs developed by Bellalah (1999). They found that the inclusion of the information costs in the option formula, increases the precision of the option price. Due to these costs and stochastic volatility, they proposed an explanation of the skewness observed in the smile even if there is no skewness in the real distribution. We know how the Black–Scholes (1973) option pricing model has generated non-negligible specification errors with respect to the option-market observed data. Model prices as a function of the strike price are systematically diverging from the observed option prices. This feature in term of implied volatility is often called the “smile effect”, where the so-called smile refers to the U-shaped pattern of implied volatilities across different strike prices. Nevertheless, the smile can be asymmetric. This skewness effect can often be described as the addition of monotonic curve to the standard symmetric smile. The smile becomes sometimes a smirk on a skew, since it appears more or less lopsided: the so-called skewness effect. In general, the implied volatility curve has its minimum for out-of-the-money (OTM). Jackwerth and Rubinstein (1996) have linked the smile shape and the risk-neutral density as one. They concluded that observed smiles translate into a left-skewed highly leptokurtic risk-neutral distribution for the future underlying asset price. Moreover, At-Sahalia and Lo (1998) have pointed out that the presence in option prices of an implied volatility smile, whereby OTM put options are more expensive than at-the-money (ATM) options, directly translate into a negatively skewed option implied risk-neutral density. Indeed, an answer to the question, “How to explain the smile (its shape)?” could be translated to, “how to explain the shape of the risk-neutral density?” It is straightforward to assume that both distributions (after the moments of the second orders) are the same under the risk-neutral and the objective distributions. Renault (1997) pointed out that the asymmetry of the implied volatility curves is best characterized with reference to a benchmark model, which produces a symmetric curve. When the volatility is stochastic as in the Hull and White (1987) model, Renault and Touzi (1996) have shown that the shape of the volatility structure with respect to the moneyness of the option is symmetric when the returns innovations and the volatility are uncorrelated. Based on this model, we show that information costs can produce an asymmetric smile (skew pattern) even if the objective distribution has no skewness. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah et al. (2001a), Bellalah et al. (2001b), Bellalah and Prigent (2001) and Bellalah and Selmi (2001) and so on.

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18.1. Option Pricing in the Presence of a Stochastic Volatility 18.1.1. The Hull and White model Hull and White (1988) considered the following model for the dynamics of the stock price and its volatility, which is defined as a separate stochastic variable: dSt = µ(St , σt , t)St dt + σt St dWt1

(18.1)

dνt = (σt , t)νt dt + δt (σt , t)νt dWt2

(18.2)

where St stands for the stock price at date t, νt = σt2 is the instantaneous variance, and Wt1 and Wt2 are Brownian motions correlated with a correlation coefficient ρ. In this context, the pricing of a call option implies the construction of a risk-less portfolio containing the option, the stock, and a second call option with the same strike price but a different time to maturity. Let c(t, St ) be the value of the first call option which depends on time t and the stock price St . This approach, which was also used by Johnson and Shanno (1987), Scott (1987), and Wiggins (1987), yields a partial differential equation for the option price. However, the solution to this equation is not unique unless the price function for the second call is known. To obtain a unique solution, Hull and White (1987) made the additional assumptions that the two Wiener processes are independent and that the variance has no systematic risk. This convenient assumption implies that the volatility risk does not entail any risk premium. With these additional assumptions, it is possible to obtain a unique option price, computed as the expectation of the discounted terminal pay off under a risk-neutral probability measure. The terminal boundary condition at the maturity date T is c(T, S) = ST − K, if ST ≥ K and c(T, S) = 0 otherwise. Using the additional assumption that the variance ν is not influenced by the stock price and setting T 1 ∗ νt,T = νs ds (T − t) t the value of a European call option when the volatility is uncorrelated with the stock price is: ∞ ∗ 2 ∗ cB−S t, St , νt,T | St , σt2 dF νt,T (18.3) c(t, St , σt ) = 0

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where cB−S is the usual B–S option price corresponding to the variance, ∗ ∗ and F is the conditional distribution of νt,T under the risk-neutral νt,T 2 ∗ probability measure, given St and σt . The value of cB−S (t, St , νt,T ) is given by: ∗ = St N (d1 ) − Ke−rT N (d2 ) cB−S t, St , νt,T S ∗ + r + 12 νt,T (T − t) ln K ∗ (T − t). d1 = , d2 = d1 − νt,T ∗ νt,T (T − t) It is important to note that the moments of F cannot be determined ∗ in general, except in special cases when γ and δ are constants, and νt,T is an integral over log-normal variables. In this very particular situation, using a Taylor’s expansion of Eq. (18.3), Hull and White (1988) provided the following solution: c(t, St , σt2 ) = SN (d1 ) − Ke−rtN (d2 ) 1 1 1 4 h + S (T − t)N (d1 )(d1 d2 − 1) 2σ e − h − 2 − 2 h 2

1 + SN (d1 ) (d1 d2 − 3)(d1 d2 − 1) − (d21 + d22 ) 6 σ6

1 + σ6 3 e3h − (9 + 18h)eh + (8 + 24h + 18h2 + 6h3 ) 8 3h with h = 2 t. 18.1.2. Stein and Stein model Stein and Stein (1991) studied stock price distributions when prices follow diffusion processes with a varying volatility parameter. They obtained interesting results regarding option pricing with stochastic volatilities and the relationship between this parameter and the nature of fat tails in stock price distributions. Their model is more general than that of Hull and White for the following two reasons. First, Hull and White used Taylor’s series expansion to solve explicitly for the option price, about the point where the volatility is non stochastic, i.e., δ = 0. Second, it is not clear that the expansion provides a good approximation to option prices when the value of δ is far from zero. Using Stein and Stein’s notations, let the dynamics of the stock price P , to be represented by the familiar following

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equation: dSt = µP dt + σP dW1

(18.4)

where µ and σ stand for the expected instantaneous return and variance, respectively and W1 is a Wiener process. Let the dynamics of the volatility be governed by an arithmetic Ornstein–Uhlenbeck process, with a tendency to revert back to a long-run average level θ as follows: dσ = −δ(σ − θ)dt + κdW2 .

(18.5)

This process has been used by many researchers in modeling and studying the empirical properties of volatility. Examples include Poterba and Summers (1986), Stein (1989), and Merville and Pieptea (1989) among others. From Eqs. (18.4) and (18.5), it is possible to obtain an explicit closed-form solution for the stock price distribution. Setting P0 equal to one and denoting by S0 (P, t) the time t stock price distribution when the stock price drift µ = 0, the solution given by Stein and Stein (1991) is: 1 −3 ∞ 1 t iη ln(P ) η2 + P 2 e S0 (P, t) = dη. 2π 4 2 η=−∞ When the stock has a non-zero drift, µ different from zero, the solution is: S(P, t) = e−µt S0 (P e−µt , t) where the integral I(.) is calculated using Eqs. (18.3) to (18.5) in Stein and Stein (1991). The European call price is: ∞ −rt [P − K]S(P, t | δ, r, κ, θ)dP. C0 (P, t) = e P =K

Using this model implies the choice of reasonable values for the parameters δ, κ, and θ. Results of the empirical studies by Stein (1989) and Merville and Pieptea (1989) show that reasonable values are θ = 0.3, δ = 16, and κ = 0.4. Simulations of the Stein and Stein (1991) model show that stochastic volatility exerts an upward influence on all option prices. Also, the stochastic volatility is more important for OTM or in-the-money (ITM) options than for ATM options, i.e., the implied volatilities in the context of this model exhibit a U-shape as the strike price is varied. Hence, an ATM option has the lowest implied volatility, and this volatility rises in either direction as the strike price moves.

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18.1.3. The Heston model Heston (1993) used a new technique to derive a closed-form solution for a European call option in a stochastic volatility context. His model allows for arbitrary correlation between the spot asset returns and the volatility. Also, he introduced stochastic interest rates and applied the model to the pricing of bond options and foreign currency options. The dynamics of the spot asset at time t are described by the following diffusion equation: dSt = µSdt + ν(τ )SdWt1 . The volatility dynamics are governed by the Ornstein–Uhlenbeck process used by Stein and Stein (1991). (18.6) d ν(τ ) = −β ν(τ )dt + δdWt2 . Using Ito’s lemma and standard arbitrage arguments, Heston (1993) showed that the price of a European call is given by: c(S, ν, t) = Sp1 − KP (t, T )p2

(18.7)

where p1 and p2 are probabilities which can be calculated using the following formula: iφ ln(k) 1 ∞ 1 e fj (x, ν, T, φ) pj (x, ν, T, ln(K)) = + dφ Re 2 π 0 iφ for j = 1, 2, with fj (x, ν, T, φ) given by Eqs. (18.16) and (18.17) in Stein and Stein (1991). The probabilities in Eq. (18.7) multiplying the asset price S and the strike price K in Eq. (18.6) must be calculated to obtain the option price. Following Merton (1973a) and Ingersoll (1990), Heston (1993) incorporated stochastic interest rates in his option pricing model and applied it to options on bonds and options on foreign currencies. This is possible if one modifies the dynamics of the asset to allow time dependence in the volatility of the spot asset, σs (t): dSt = µs Sdt + σs (t) ν(τ )SdWt1 . This equation is satisfied by discount bond prices in the model of Cox et al. (1985a, b). The dynamics for the bond price P (t, T ), at time t, for a

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maturity date T , are specified by the following equation: dP (t, T ) = µP P (t, T )dt + σP (t) ν(t)P (t, T )dWt2 . It is convenient to note that variances of the spot asset and the bond are given by the same variable ν(t). In this context, the valuation equation is given by Eq. (18.22) in Heston (1993a). The model can also be applied to options on foreign currencies. In fact, when S(t) stands for the dollar price of a foreign currency and the dynamics of the foreign price of a foreign discount bond, F (t, T ), are given by the following equation: dF (t, T ) = µP F (t, T )dt + σP (t) ν(τ )F (t, T )dWt2 then Eq. (18.26) in Heston (1993) can be used for the pricing of European currency options. The model can be used for the valuation of stock options, bond options, and currencyoptions. It is interesting since it links most biases in option prices to the dynamics of the spot price and the distribution of spot returns. With a proper choice of its parameters, the stochastic volatility model seems to be flexible and promising in the description of option prices. However, the main drawback of this model is the number of parameters to be estimated. 18.1.4. The Hoffman, Platen, and Schweizer model Hoffman et al. (HPS) (1992) provided an approach to option pricing which allows the specification of general patterns of volatility behavior. The approach combines the use of a high dimensional Markovian model with stochastic numerical methods in an incomplete market. The assumption of an incomplete market implies that there is no unique equivalent martingale measure or risk-neutral probability for the underlying stock price dynamics. When the volatility is both stochastic and past dependent, the following multi-dimensional process was used by Hoffman et al. (1992): dXti = ai (t, Xt )dt +

n

bij (t, Xt )dWtj

for i = 0 to m and j = 1 to n.

i=1

In this formulation, the risk-less asset is represented by X 0 and the dynamics of the underlying asset (the stock) are given by X 1 . The other components of X can be used to model other assets. For example, they could model the stochastic volatility and its dependence on the past.

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It may be noted that such a model goes back to Merton (1971, 1973, 1990). In this formulation, an option or a contingent claim is a random variable of the form c(XT ). Within this general formulation, there is no unique equivalent martingale measure for the stock price process. This leads to the problem of pricing in incomplete markets. To overcome this problem, and to be able to compute option prices, Hoffman et al. (1992) supposed that there is a “small” investor who should use the minimal equivalent martingale measure. This allows them to decompose the space of all those assets compatible with the given stock is a direct sum of two subspaces: purely traded assets and totally non-tradable assets. The flexibility and generality of the model give rise to stochastic differential equations that do not have explicit solutions. Hoffman et al. used numerical methods to compute option prices and simulate the performance of hedging strategies under various possible scenarios. Such a model is rich from a theoretical point of view but is “poor” from a practical perspective, since it is rather difficult to implement by market participants. The main drawback is the number of parameters to be estimated. The valuation model The pricing of derivative securities in the presence of a random volatility needs the use of two processes: one for the underlying asset and one for the volatility. Consider the following dynamics for the underlying asset: dS = µSdt + σSdW1 and the following process for the volatility: dσ = p(S, σ, t)dt + q(S, σ, t)dW2 . The two processes dW1 and dW2 are Brownian motions with a correlation coefficient ρ. The functions p(S, σ, t) and q(S, σ, t) are specified in a way that fits the dynamics of the volatility over time. Hence, the derivative asset price V (S, σ, t) can be a new source of randomness that cannot be easily hedged away. The pricing of options in this context needs the search for two hedging contracts. The first is the underlying asset. The second can be an option that allows a hedge against volatility risk. Following the same logic as in the original Black–Scholes model (1973), consider a portfolio comprising a long position in the option V , a short position of ∆ units of the underlying asset, and a short position of −∆1 units of an other option

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with value V1 (S, σ, t): pi = V − ∆S − ∆1 V1 .

(18.8)

Over a short interval of time dt, applying Ito’s lemma for the functions S, σ, and t gives the change in the value of this portfolio as: ∂ 2V ∂ 2V 1 1 ∂2V ∂V + σ2 S 2 2 + ρσqS + q 2 2 dt dΠ = ∂t 2 ∂S ∂S∂σ 2 ∂σ 2 ∂V1 1 2 2 ∂ V1 1 2 ∂ 2 V1 ∂ 2 V1 dt + σ S + q − ∆1 + ρσqS ∂t 2 ∂S 2 ∂S∂σ 2 ∂σ 2 ∂V ∂V ∂V1 ∂V1 − ∆1 ∆ dS + − ∆1 dσ. + ∂S ∂S ∂σ ∂σ All the sources of randomness in the portfolio value resulting from dS can be eliminated by setting the quantity before dS equal to zero, or ∂V −∆− ∂S 1 ∆1 ∂V = 0 and also by setting the quantity before dσ equal to zero, or ∂S ∂V1 ∂V − ∆ = 0. 1 ∂σ ∂σ After eliminating the stochastic terms, the terms in dt must yield the deterministic return as in a B–S “hedge” portfolio. Hence, the instantaneous return on the portfolio must be the risk-free rate plus information costs on each asset in the portfolio as shown in Bellalah (1999). This gives: dΠ =

∂2V ∂V 1 1 2 ∂ 2V ∂ 2V dt + σ2 S 2 dt + q dt + ρσSq dt ∂t 2 ∂S 2 ∂S∂σ 2 ∂σ 2 ∂V1 1 2 2 ∂ 2 V1 1 2 ∂ 2 V1 ∂ 2 V1 dt + σ S dt + q dt + ρσSq dt − ∆1 ∂t 2 ∂S 2 ∂S∂σ 2 ∂σ 2

= [(r + λV )V − (r + λS )∆S − (r + λV1 )∆1 V1 ]dt. Isolating the terms in V and V1 gives: ∂V ∂t

=

2

2

2

∂ V + 12 σ 2 S 2 ∂∂SV2 + ρσSq ∂S∂σ + 12 q 2 ∂∂σV2 + (r + λS )S ∂V ∂S − (r + λV )V ∂V ∂σ

∂V1 ∂t

2

2

2

∂ V1 1 + 12 σ 2 S 2 ∂∂SV21 + ρσSq ∂S∂σ + 12 q 2 ∂∂σV21 + (r + λS )S ∂V ∂S − (r + λV1 )V1 ∂V1 ∂σ

.

Since the two options differ by their strikes, payoffs, and maturities, this implies that both sides of the equation given above are independent of the

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contract type. Since both sides are functions of the independent variables S, σ, and t, we have: ∂ 2V 1 1 2 ∂ 2V ∂V ∂2V + σ2 S 2 + q + ρσSq ∂t 2 ∂S 2 ∂S∂σ 2 ∂σ 2 ∂V ∂V + (r + λS )S − (r + λV )V = −(p − δq) , ∂S ∂σ for a function δ(S, σ, t) referred to as the market price for risk or volatility risk. This equation can also be written as ∂2V ∂V ∂2V ∂V 1 1 ∂ 2V + σ 2 S 2 2 + ρσSq + q 2 2 + (r + λS )S ∂t 2 ∂S ∂S∂σ 2 ∂σ ∂S ∂V (18.9) + (p − δq) − (r + λV )V = 0. ∂σ This equation shows two hedge ratios known as the risk-neutral drift rate.

∂V ∂S

and

∂V ∂σ

. The term (p − δq) is

18.1.5. Market price of volatility risk Suppose the investor holds only the option V , which is hedged only by the underlying asset S in the following portfolio Π = V − ∆S. Over a short interval of time dt, the change in the value of this portfolio can be written as: ∂ 2V ∂V 1 1 ∂2V ∂ 2V dΠ = + σ2 S 2 2 + ρσqS + q 2 2 dt ∂t 2 ∂S ∂S∂σ 2 ∂σ ∂V ∂V − ∆ dS + dσ. + ∂S ∂σ In the standard delta hedging, the coefficient of dS is zero and we have: dΠ − [(r + λV )V − (r + λS )∆S]dt ∂2V ∂V ∂ 2V ∂V 1 1 2 ∂ 2V + ρσqS + (r + λS )S = + σ2 S 2 + q ∂t 2 ∂S 2 ∂S∂σ 2 ∂σ 2 ∂S − (r + λV )V ] dt +

∂V ∂V dσ = q (δdt + dW2 ). ∂σ ∂σ

This results from Eqs. (18.8) and (18.9). The term dW2 represents a unit of volatility risk. There are δ units of extra return, given by dt for each unit of volatility risk.

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18.1.6. The market price of risk for traded assets In the B–S analysis, the hedging portfolio is constructed using the option and its underlying tradable asset. Consider the construction of a portfolio as before using two options V and V1 with different characteristics. In this case, if the underlying asset is not a tradable security, the initial portfolio value would be Π = V − ∆1 V1 . Using the same methodology, as before, gives the following equation: ∂ 2V 1 ∂V ∂V + σ2 S 2 − (r + λV )V = 0. + (µ − δS σ)S ∂t 2 ∂S 2 ∂S

(18.10)

If the asset is traded, then V = S must be a solution to Eq. (18.10) Substituting V = S in the Eq. (18.10) gives (µ − δS σ)S − (r + λS )S = 0. The market price of risk for a traded asset in the presence of information S) costs δS = µ−(r+λ . Substituting δS in (21) gives the following equation: σ ∂V ∂ 2V ∂V 1 + σ2 S 2 2 + (r + λS )S − (r + λV )V = 0. ∂t 2 ∂S ∂S This is the B–S equation in the presence of information costs. 18.2. Generalization of Some Models with Stochastic Volatility and Information Costs 18.2.1. Generalization of the Hull and White (1987) model Consider the following model: dBt = (r + λBt )Bt dt dSt = dνt =

(18.11)

µ(St , σt , t)St dt + σt St dWt1 γ(σt , t)νt dt + δ(σt , t)νt dWt2

where St denotes the stock price at time t, νt = σt2 its instantaneous variance, r the risk-less interest rate, which is assumed to be constant, and λBt is the shadow cost related to Bt . W 1 and W 2 are Brownian motions under P , they are independent νt has no systematic risk. This yields a unique option price which can be computed as the (conditional) expectation of the discounted terminal payoff under a risk-neutral probability measure P˜ . Put differently, P˜ is obtained from P by means of a Girsanov transformation

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such that: dBt = (r + λBt )Bt dt ˜1 dSt = (r + λSt )St dt + σt St dW t ˜2 dνt = γ(σt , t)νt dt + δ(σt , t)νt dW t

˜ 1, W ˜ 2 are independent Brownian motions under P˜ . The under P˜ , where W risk-neutral dynamics of the bond and the underlying asset are used in Bellalah (1999). The portfolio value would be Π = V − ∆S − ∆ Bt with ∆ units of the bond. When we apply the methodology of the previous section to this model, Eq. (18.8) gives: 1 ∂V ∂ 2V ∂ 2V 1 ∂2V ∂V + νt St2 2 + ρσt3 St ξ + ξ 2 νt2 2 + (r + λSt )St ∂t 2 ∂St ∂S∂νt 2 ∂νt ∂St + (γ − δξ)νt

∂V ∂V + (r + λBt )Bt − (r + λV )V = 0 ∂νt ∂Bt

˜ 2 independent Brownian motions under the probability ˜ 1, W with W ˜ P (ρ = 0) and λSt is the information cost of the security St . The investor paid the shadow cost λSt , if he/she does not know the asset. Also λBt is the information cost of the bond Bt . We suppose that δ = 0, the option price is then given by:

Bt + ˜ ˜ (ST − K)+ | Ft (ST − K) | Ft = e−(r+λBt )(T −t) E V (t, St ) = E BT To obtain a more specific form for V , we use the additional assumption of the independence of W 1 , W 2 (the instantaneous variance ν is not influenced by the stock price S). Setting: T 1 ν t,T = νs ds. T −t t The conditional distribution of SSTt under P˜ , given ν t,T , is log-normal with parameters (r + λBt )(T − t) and ν T −t . This allows to write V as: ∞ 2 V (t, St , σt ) = uBS (t, St , ν t,T )dF (ν t,T | St , σt2 ) 0

where VBS denotes the usual Black–Scholes (1973) price corresponding to the variance ν t,T and F is the conditional distribution under P˜ of ν t,T given St and σt2 . This is equivalent to writing: ∞ 2 V (t, St , σt ) = VBS (t, St , ν t,T )h(ν t,T | St , σt2 )dν t,T 0

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with VBS (ν) = St N (d1 ) − Xe−(r+λSt )(T −t) N (d2 ) d1 =

log(St /K) + (r + λSt + ν/2)(T − t) , ν(T − t)

d2 = d1 −

ν(T − t).

When µ = 0 and as in Hull and White (1987) we have: V

(S, σt2 )

√ 1 S T − tN (d1 )(d1 d2 − 1) = VBS (ν) + 2 4σ3 4 k 2σ (e − k − 1) 4 × −σ k2 √ 1 S T − tN (d1 )[(d1 d2 − 1)(d1 d2 − 3) − (d21 + d22 )] + 6 8σ5 3k e − (9 + 18k)ek + (8 + 24k + 18k 2 + 6k 3 ) + ··· , × σ6 3k3 with k = ξ 2 (T − t),

1 x2 N (x) = √ e− 2 . 2π

18.2.2. Generalization of Wiggins’s model Under the assumption of the continuous trading, without frictions, in a complete market, Wiggins (1987) used the following dynamics for the asset and the volatility: dSt = µ(St , σt , t)St dt + σt St dWSt dσt = f (σt )dt + θσt dWσt with dWSt , dWσt are processes of Wiener, the correlation coefficient between stock returns and volatility movements is ρdt = dWSt dWσt and (dP/P )(dSt /St ) = 0. The instantaneous rate of return on the hedge portfolio P is: dP/P = wdV /V + (1 − w)dSt /St with w the fraction invested in the contingent claim V and (1 − w) the fraction invested in the stock S. Equation (18.8) is equivalent, in this

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case, to: 1 ∂ 2V ∂2V 1 ∂ 2V ∂V + σt2 St2 2 + ρσt2 θSt + θ2 σt2 ∂t 2 ∂St ∂St ∂σt 2 ∂σt2 + (r + λSt )St

∂V ∂V + (f (σt ) − δθσt ) − (r + λV )V = 0. ∂St ∂σt

As in Wiggins (1987), we can write the following equation: 1 ∂V ∂2V ∂2V 1 ∂2V ∂V + σt2 St2 + ρσt2 θSt + θ2 σt2 + (r + λSt )St 2 ∂t 2 ∂St ∂S∂σt 2 ∂σt2 ∂St ∂V + [f (σt ) − (µ − r − λSt )ρθ + Φ(.)θσt (1 − ρ2 )] − (r + λV )V = 0. ∂σt We conclude that the market price of risk affects the term given by Wiggins (1987), Φ(.) = (µP −r−λP )/σP . This term is the expected excess return per unit risk, or the market price of risk, for the hedge portfolio. It represents the return-to-risk tradeoff required by investors for bearing the volatility risk of the stock. Φ(.) =

δσt − (µ − r − λSt )ρ . σt (1 − ρ2 )

The market price of risk depends on the information cost of the stock and the stochastic volatility. 18.2.3. Generalization of Stein and Stein’s model In this model, the stock price dynamics are given by the following process: dSt = µ(St , σt , t)St dt + σt St dW1 The volatility follows an Ornstein–Uhlenbeck process: dσt = (σt − θ)dt + kdW2 . The Weiner processes dW1 and dW2 are uncorrelated. When Eq. (18.8) is applied in this context, we have: 1 ∂V ∂ 2V 1 ∂ 2V ∂V + σt2 St2 2 + k 2 + (r + λSt )St ∂t 2 ∂St 2 ∂σt2 ∂St + [−(σt − θ) − δk]

∂V − (r + λV )V = 0. ∂σt

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When δ = 0 or a constant, this equation has a solution with the same form as given in Stein and Stein (1991). The solution depends on information costs of V and the underlying asset S. The option price has the following form: ∞ [St − K]H(St, t | , r + λSt , k, θ)dSt V = e−(r+λV )T St =K

with H(St , t) is the price distribution of the underlying asset at the time t with a non-zero drift of St . 18.2.4. Generalization of Heston’s model The underlying asset and the volatility follow the diffusion process: √ dSt = µ(St , σt , t)St dt + νt St dWt1 √ dνt = κ(θ − νt )dt + σt νt dWt2 with ρ the correlation coefficient between dWt1 and dWt2 . In this case, the value of any option V (St , νt , t) must satisfy the following partial differential equation: 2 ∂V 1 ∂ 2V ∂2V ∂V 2 1∂ V + ρσt νt St + νt St2 + (r + λSt )St 2 + σt νt ∂t ∂St∂νt 2 ∂St 2 ∂νt2 ∂St

√ ∂V + [κ(θ − νt ) − δσt νt ] − (r + λV )V = 0 ∂νt

(18.12)

Under the same assumption as in Heston (1993), it is possible to obtain solution to Eq. (18.9). This solution depends on information costs λSt . In fact, a European call with a strike price K and maturing at time T , satisfies the Eq. (18.9) subject to the following boundary conditions: V (S, νt , t) = max(o, S − K) V (0, νt , t) = 0 ∂V (∞, νt , t) = 1 ∂St (r + λSt )St

∂V ∂V ∂V =0 + κ(θ) − (r + λV )V + ∂St ∂νt ∂t V (S, ∞, t) = S.

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By analogy with the Black–Scholes (1973) formula, Heston (1993) gave a solution of the form: V (S, νt , t) = SP 1 − KP (t, T )P2

(18.13)

with P (t, t+τ ) = e−(r+λSt )τ , the price at time t of a unit discount bond that matures at time t + τ . The first term of the right-hand side of the solution V (S, νt , t) is the present value of the underlying asset upon optimal exercise. The second term is the present value of the strike price. Both of these terms must satisfy the Eq. (18.9). It is convenient to write them in terms of the logarithm, x = ln(S). By substituting the solution of Eq. (18.10) in Eq. (18.9), P1 and P2 must satisfy the following equation: ∂ 2V 1 ∂ 2 Pj 1 ∂2V ∂Pj + σt2 νt + νt + ρσt νt St ∂t 2 ∂x ∂St ∂νt 2 ∂νt2 + (r + λSt + uj νt )

√ ∂Pj ∂Pj + (a − bj νt ) =0 ∂x ∂νt

√ for j = 1, 2, where u1 = 1/2, u2 = −1/2, a = κθ, b1 = (κ − ρσt ) νt + δσt , √ and b2 = κ νt + δσt . Following the same resolution method as followed in Heston (1993), we obtain the solution of the characteristic function: fj (x, νt , t; φ) = exp[C(T − t; φ) + D(T − t; φ)νt + ixφ] when C(τ ; φ) = i(r + λSt )φτ + D(τ ; φ) =

a σt2

bj − iρσt φ + d 1 − edτ σt2 1 − gedτ

and g=

bj − iρσt φ + d , bj − iρσt φ − d

(bj − iρσt φ + d)τ − 2 ln

d=

1 − gedτ 1−g

(iρσt φ − bj )2 − σt2 (2iuj φ − φ2 ).

By inverting the characteristic functions fj , we obtain the desired probabilities: −iφ ln K ∗ 1 (e ) fj (x, νt , T ; φ) 1 ∞ Re Pj (x, νt , t; ln K) = + dφ 2 π 0 iφ with fj (x, νt , T ; φ) = eiφx .

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18.2.5. Generalization of Johnson and Shanno’s model We consider the following model: dSt = µSt St dt + σt Stα dWt1 dσt = µσt σt dt + σt σσt dWt2 with dWt1 dWt2 = ρdt. When Eq. (18.8) is applied to this model, we obtain: ∂ 2V ∂2V 1 ∂2V 1 ∂V + σt2 St2α + ρσt3 Stα σσt + σt2 σσ2t 2 ∂t 2 ∂St ∂S∂σt 2 ∂σt2 + (r + λSt )St

∂V ∂V + (µσt − δσσt )σt − (r + λV )V = 0 ∂St ∂σt

Johnson and Shanno (1987), supposed that the risk premium of the µ volatility is zero. Consequently, we have: δ = σσσt . t

18.3. The Volatility Smiles: Some Standard Results 18.3.1. The smile effect in stock options and index options Stein and Stein (1991) presented a model for stock price distributions when the volatility is driven by an Ornstein–Uhlenbeck process (AR1). They applied the results obtained from the model to option pricing with stochastic volatilities and to the analysis of the relationships between stochastic volatilities and the nature of fat tails in stock price distributions. In order to simulate their model, they estimated the necessary parameters from data used in Stein (1989) and Merville and Pieptea (1989) concerning individual stocks and the S&P 100 index. When comparing option prices, they calculated option values using their model and the B–S model based on the implied volatility associated with their model. Their results showed that a stochastic volatility exerts an upward influence on all option prices and that stochastic volatility is more important for away-from-the-money options, that is, implied volatilities corresponding to new option prices, exhibit a U-shape, as the strike price is changed. The implied volatility is the lowest for ATM options and rises as the strike price moves in either direction. For away-from-the-money options, the “mixing distribution”, they used the U-shape in implied volatilities. They found that the overall impact on option prices is economically significant, especially for OTM options. For some parametric values, their model prices were 11.7% more than B–S model’s, and even larger effects are observed with cheaper options.

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18.3.2. The smile effect for bond and currency options In order to explain strike biases in the B–S model, Heston (1993) proposed an option pricing model allowing for arbitrary correlation between volatility and spot asset returns when volatility follows an Ornstein-Uhlenbeck process as in Stein and Stein (1991). His model can be applied to bond options and currency options. When examining the effect of stochastic volatilities on option prices in comparison to the B–S model, he used a B–S model with a volatility parameter that matches the variance of the spot return over the option life. The parameters used correspond roughly to those estimated by Knoch (1992) for Yen and Deutschemark currency options. The model links the biases of the B–S model to the dynamics of the spot price and its distribution. Heston showed that B–S prices are virtually identical to his model prices for ATM options, explaining some of the empirical support to the B–S model. Also correlation between volatility and the spot price is necessary in explaining skewness that affects the pricing of ITM options relative to OTM options. In fact, a positive correlation coefficient results in high variance, which spreads the right tail of the probability density and results in a thin left tail in the distribution. This increases OTM call prices and decreases ITM call prices relative to the B–S model with a comparable volatility. A negative correlation coefficient has the opposite effects, i.e., it decreases OTM call prices relative to ITM call prices with respect to B–S model. Hence, without this correlation, stochastic volatility does not change skewness and affects only the kurtosis. This latter affects prices of near-the-money calls relative to far-from-the-money calls. The model seems to impart not only the strike price bias but also other biases reported when testing B–S model. Heston (1993) presented an option pricing model consistent with empirical biases reported in the B–S formula. The model, based on a log-gamma process, explains in particular, the smile effect with respect to the distribution’s skewness. Model simulations show that option prices are similar to those of B–S for ATM options. However, Heston’s model assigns higher prices to OTM options and lower prices to ITM options when compared to B–S prices. The differences in prices are economically significant. The strike biases are similar to the skewness-related biases in models where stochastic volatility is correlated with stock returns.

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18.3.3. Volatility smiles: Empirical evidence The expected volatility of an underlying asset can be inferred from the option price. Implied volatilities can be taken for different time to maturities and different strike prices. Hence, using option prices, a matrix of implied volatilities with rows ordered by the strike price and columns ordered by time to maturities can be constructed. The rows of the implied volatility matrix may provide information about the term structure of expected future volatility. Poterba and Summers (1986), Stein (1989), Franks and Schwartz (1991), and Heynan et al. (1994) studied the term structure of implied volatilities using only two times to maturity. The time series studies done by Merville and Pieptea (1989) and Day and Lewis (1992) used one implied volatility per day and ignored the term structure effects since time to maturity varies from day-to-day. Stein (1989) used two daily time series on implied volatilities for the S&P 100 index options over the period 1983–1987. Based on the assumption that the volatility is mean reverting, he concluded that long-maturity options tend to overreact to changes in the implied volatility of short maturity options. This conclusion has been disputed by other authors who showed that overreaction depends on the model used to represent changes in the volatility. Xu and Taylor (1994) presented and illustrated methods for estimating this term structure from one row of the implied volatility matrix corresponding to nearest-the-money options. They modeled the term structure of expected volatility and its time series properties and use spot currency options on the British pound, Deutschmark, Japanese Yen, and Swiss Franc quoted against the US dollar in their empirical work. The study concerned the period from January 1985 to November 1989. The main results appeared in Xu and Taylor (1994). When examining the relation between short-term and long-term implied volatilities for the European option exchange index and Philips stock, the principal result in Heynen, Kemma, and Vorst study is that the major determinant for the specification of the term structure of implied volatility relations is the level of the unconditional volatility. This latter study seems correctly specified in the case of the EGARCH(1,1) stock return volatility model. The Xu and Taylor’s study presented the following results. First, implied volatilities vary significantly across maturities. Second, the direction of the term structure of implied volatilities changes (up or down) nearly once every two or three months. Third, the variations in expectations regarding long-term volatility are significant although they are more slowly than those regarding short-term volatility.

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18.4. Empirical Results Regarding Information Costs and Option Pricing Option prices are extracted from a CD-ROM of the SBF-BOURSE DE PARIS. The data concern the long maturity option quotations, the strike prices, the maturity, and the CAC 40 quotations during October 1998. Bellalah et al. (2002) used intra-day data for both quotations. These options are European type options written on the CAC 40 index. Our study is based on both calls and puts. Their maturity is six months. The interest rate is taken from DATASTREAM. We choose the Euro Interbank Offered Rate (EURIBOR) six months as the risk-free interest rate. The maturity chosen is March 1999 and the minimum number of purchasable options is 50. The EURIBOR interest rate is a daily rate. 18.4.1. Information costs and option pricing: The estimation method We use the B–S model to estimate the implied standard deviation (ISD) through the simultaneous equations procedure. This procedure gives the implied values of our parameter. The simultaneous equations procedure estimates the value of the parameter, ISD, which minimizes the following sum of squares: n [COBS j − CBS j (ISD )]2 min ISD

j=1

where: n: the number of bid-ask price quotes sampled on a prior day; COBS : the observed call price and CBS ()ISD : the theoretical B–S call price computed with the ISD parameter. Theoretical B–S prices are obtained with the ISD we computed and which is estimated from one-day lagged price observation. We measure option moneyness relative to the price deviations. Option moneyness is calculated as (Ke−rt − SADJ )/Ke−rt, where SADJ is the dividend-adjusted cash price and Ke−rt , the discounted strike price. Note that negative (positive) moneyness corresponds to ITM (OTM) call options with low (high) strike prices. We see that the B–S model undervalues strongly low OTM calls and overvalues slightly high OTM calls. To compute the information cost-adjusted B–S option price model, we estimate the ISD, the option information cost λC , and the underlying information cost λS ,

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through the simultaneous equations procedure, which gives us the implied values of our parameters. The simultaneous equations procedure estimates the values of the parameters, ISD, λC , and λS , which minimizes the following sum of squares: min

ISD,IλC ,IλS

n

[COBS j − CMj (ISD, IλC , IλS )]2 .

j=1

CMj (ISD , IλC , IλS ) is the call price of the proposed model computed with the ISD parameter which includes the implied information costs parameters, and the cost of carrying the commodity. This call price is calculated for any option in a given current day’s sample. We want to go further into the analysis of the differences between the observed price and the theoretical price that could be computed by the B–S model or the information cost model. We use the mean of absolute forecast error (MAE) and the mean absolute percentage forecast error (MAPE). Table 18.1 gives the results obtained by the MAE and MAPE tests. These tests pertain to both call and put options. We clearly see that the information cost model proposed by Bellalah (1999) performs better than the B–S model. 18.4.2. The asymmetric distortion of the smile The prediction of the B–S model produces systematic deviations from the observed price (“volatility smiles”) corresponds to a “gap” in the formula of option valuation as noticed by Black’s (1976) model prices as a function of the strike prices are diverging from the observed option prices, so that the residuals for low strike prices tend to have different signs from the residuals for high strike prices. Following At Sahalia and Lo (2000), the risk-neutral density is the relevant density only for risk-neutral investors. If investors were not indifferent to risk, then the Table 18.1. Comparison between different models: B–S and Bellalah’s model. Call

Black–Scholes Bellalah

Put

MAE

MAPE

MAE

MAPE

15,1903 4,9669

0,1874 0,0692

27,3895 7,6109

0,4207 0,0725

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corresponding subjective probability could be different. The fundamental relation can be expressed as: subjective probabilities = risk-neutral probabilities risk aversion adjustment coefficient. In the context of the Hull and White (1987) model, Renault and Touzi (1996) proved that observed symmetric smiles represent an evidence against the hypothesis that log-asset returns are homoskedastic. Black (1976) pointed out that “perhaps the most obvious causal relation runs from changes in the value of the firm to stock returns and volatility changes. A drop in the value of the firm will cause a negative return on its stock, and will usually increase the leverage effect of the stock (. . .) that rise in the debt-equity ratio will surely mean a rise in the volatility of the stock”. This is the so-called leverage effect which is usually captured by a negative correlation coefficient between the innovations of the two factors: the returns and its volatility. The coupon-bond option model of Geske (1979), the constant elasticity of variance model of Cox and Ross (1976), and stochastic volatility models in incomplete market (with imperfect negative correlation) of Heston (1993) should reproduce these facts (see Bates (1997)). In this last case, both the objective and the risk-neutral distributions of stock returns are leptokurtic (by the stochastic volatility assumption) and skewed (due to the correlation between stock returns and volatility). However, as pointed out by Ghysels et al. (1996) “it is important to be cautious about tempting associations: stochastic implied volatility and stochastic volatility; asymmetry in stocks and skewness in the smile”. 18.4.3. Asymmetric Smiles and information costs in a stochastic volatility model Based on Renault and Touzi (1996) results, let the data-generating process used in this sub-section be defined on a probability space (Ω, F, P ). The underlying asset price process S is described by: dS = µ(t, S, σ)dt + σdW1 (t) S σ 2 = α(t, σ)dt + βdW2 (t) where W = (W1 , W2 ) is a standard bi-dimensional Brownian motion. We denote by r the instantaneous interest rate supposed to be constant, so that the price of a zero-coupon bond maturing at time T is given by e−r(T −t) . Let C be the price process of a European call option on the underlying

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asset S with strike price K and maturity T . We introduce the variable x = ln(S/Ke−r(T −t)). The call option is said to be ITM, if x > 0, OTM, if x < 0, ATM forward, if x = 0 and ATM if x = r(T − t). Following Hull and White (1987), we impose the assumption of non-systematic volatility risk. The risk-neutral data generating bi-variate process is given by: dS ˜ 1 (t) = µ(t, S, σ)dt + σdW S ˜ 2 (t) σ 2 = α(t, σ)dt + βdW ˜ = (W ˜ 1, W ˜ 2 ) is a standard bi-dimensional Brownian motion under where W ˜ 2 = W2 . The Hull and White (1987) the risk-neutral probability with W formula is given by: T 1 2 BS 2 σu du C(S, σ ) = E C S, T −t t In this case, Renault and Touzi (1996) have shown that the shape of the volatility structure with respect to the moneyness of the option is symmetric when the returns innovations and the volatility are uncorrelated. In the presence of information costs without stochastic volatility, the value of a European call is given by the formula given in Bellalah (1999). It can be said that: (1) The option price is equal to the standard B–S price with a new risk-less rate equal to (r + λS ) multiplied by the discount factor e(−(λC −λS ))τ . (2) In the same way, the option price is equal to the standard Black and Scholes price with a new stock price equal to SeλS τ multiplied by the discount factor e−λC τ . Longstaff (1995) documented the evidence of implicit stock prices greater than by about half percent in mean. Even though Longstaff (1995) still computed B–S implicit stock prices, it is clear that this argument can be extended to Hull and White (1987) pricing: T 1 −λC τ BS λS τ 2 Se , C=e E C σu du . T −t t The main interest of this generalization is to use Renault (1997) results, who formally proved that while the random feature of the volatility implies the existence of a volatility smile, a very small discrepancy between S and the implied S (say 0.1%) may explain a sensible skewness in the smile

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when one computes B–S implicit volatilities with S (ignoring the implied S taken into account by the option market). They show that the shape and the order of magnitude of resulting skewness, is conformable to welldocumented empirical evidence.

Summary Jarrow and Eisenberg (1991) and Stein and Stein (1991) built their models on the assumption that volatility is uncorrelated with the spot asset. They obtained solutions that looked like an average of B–S values over different paths of volatility. The absence of this correlation implies that the model cannot capture important skewness effects. Heston (1993) used a new technique to derive a closed-form solution for a European call option in a stochastic volatility context. His model allows for arbitrary correlation between the spot asset returns and the volatility. Also, he introduced stochastic interest rates and applied the model to the pricing of bond options and foreign currency options. Hoffman et al. (1992) considered a very general diffusion model for asset prices allowing the description of stochastic and past-dependent volatilities. Since their model implied an incomplete market, they were unable to get analytical solutions. They used stochastic numerical methods for the study of option prices and hedging strategies. When the underlying asset dynamics are believed to be stochastic, Hull and White (1987), Stein and Stein (1991), and Heston (1993) among others, proved that the implied volatility may vary with the option’s strike price. Simulations of their models show that a plot of theoretical implied against strike prices display a U-shaped curve, (smile effect). Although there have been several models to explain the strike price bias and the smile effect, only little empirical work has been done by Shastri and Wethyavivom (1987), Fung and Hsieh (1991) and Xu and Taylor (1994). This chapter presents recent developements along these lines, especially, the models of Merton (1976), the Cox and Ross (1976), Hull and White (1988), Stein and Stein (1991), Heston (1993), and Hofman et al. (1992). From a theoretical point of view, these models are rather interesting. They point out the difficulty of pricing assets in an incomplete market. From a practical point of view, the use of these models is very limited given the burden of parameter estimation to implement them. We develop a general context for the valuation of options with stochastic volatility and information costs. The shadow costs are integrated in the investor’s portfolio wealth process in the same vein as in Merton (1987), Bellalah and Jacquillat (1995), and

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Bellalah (1999). We carry out an empirical test for the valuation model of options in the presence of information costs proposed in Bellalah (1999). We examine a way of extending the B–S model in order to take into consideration the biases implied by the nonintegration of information costs in the valuation model. We test the statistical performance of the models used. We give an explanation of the skewness observed in the smile on the basis of our model. We find that taking into account the information costs in the option valuation formula, enhances the accuracy of the valuation.

Questions 1. 2. 3. 4. 5. 6. 7.

How can we proceed to price options in a stochastic economy? What are the basic concepts behind Stein and Stein’s model? What are the basic concepts behind Heston’s model? What are the basic concepts behind HPS’s model? What are the basic concepts behind Heston’s model? What are the theoretical reasons for the existence of the volatility smiles? What is the empirical evidence regarding the volatility smiles?

References At-Sahalia, Y and A Lo (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53, 499–548. At-Sahalia, Y and A Lo (2000). Nonparametric risk management and implied risk-aversion. Journal of Econometrics, 94, 9–51. Bates, DS (1997). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. Review of Financial Studies, 9, 69–107. Bellalah, M (1990). Quatre essais pour l’valuation des options sur indices et sur contrats terme d’indice. Doctorat de l’Universite de Paris-Dauphine. Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September), 645–664. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3), 617–635. Bellalah, M and J-L Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020.

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Bellalah, M and WU Zhen (2002). A model for market closure and international portfolio management within incomplete information. International Journal of Theoretical and Applied Finance, 5(5), 479–495. Bellalah, M, JL Prigent and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Cox, JC and SA Ross (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166. Day, ET and CM Lewis (1992) (in press). Stock market volatility and the information content of stock index options. Journal of Econometrics, 52, 115–128. Engle, RF (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987–1008. Geske, R (1979). A note on an analytical valuation formula for unprotected American call options with known dividends. Journal of Financial Economics, 7, 375–380. Ghysels, E, A Harvey and E Renault (1996). Stochastic volatility. In Statistical Methods in Finance, CR Rao and GS Muddala (eds.), pp. 119–191. North-Holland, Amsterdam: Elsevier. Heston, S (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6, 327–343. Hofmann, N, E Platen and M Schweizer (1992). Option pricing under incompleteness, Mathematical Finance, 2(3), (July), 153–187. Hull, J and A White (1987). The pricing of options on assets with stochastic variables. The Journal of Finance, 42, 281–300. Hull, J and A White (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Advances in Futures and Options Research, 3, 29–61. Jackwerth, JC and M Rubinstein (1996). Recovering probability distributions from contemporaneous security prices. Journal of Finance, 51, 1611–1631. Jarrow, RA and LK Eisenberg (1991). Option pricing with random volatilities in complete markets. Working Paper, Federal Reserve Bank of Atlanta. Johnson, H and D Shanno (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22, 143–151. Knoch, HJ (1992). The pricing of foreign currency options with stochastic volatilities. PhD dissertation, Yale School of Organization and Management. Longstaff, FA (1995). Option pricing and Martingale restriction. Review of Financial Studies, 8, 1091–1124. Melino, A and SM Turnbull (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45, 239–265.

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Merton, RC (1971). Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory, 3, 373–413. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, RC (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–144. Merton, R (1987). An equilibrium market model with incomplete information: Journal of Finance, 42, 483–511. Renault, E (1997). Econometric models of option pricing errors. Working Paper, GREMAQ. Renault, E and N Touzi (1996). Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance, 6, 279–302. Rubinstein, M (1985). Nonparametric tests of alternative option pricing models. Journal of Financial Economics, 40, 455–480. Rubinstein, M (1994). Implied binomial trees. Journal of Finance, 3, 771–818. Scott, L (1987). Option pricing when the variance changes randomly: theory, estimation and an application. Journal of Financial and Quantitative Analysis, 22 (December), 419–437. Stein, J (1989). Overreactions in the option markets. Journal of Finance, 44, 1011–1023. Stein, EM and JC Stein (1991). Stock price distributions with stochastic volatility: an analytic approach. The Review of Financial Studies, 4, 727–752. Wiggins, JB (1987). Option values under stochastic volatility. The Journal of Financial Economics, 19, 351–372.

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Part VII Option Pricing Models and Numerical Analysis

Pricing options with specific features becomes a difficult task in a Black– Scholes world. When there is no analytic solution, we can use numerical methods in the pricing of derivatives. For the sake of clarity, we write two chapters in order to explain the basic tools used in numerical analysis. The analysis can be followed by any student with a level which is equivalent to MBA. Chapter 18 develops the basic concepts and applications of numerical methods in option pricing. Chapter 19 concerns numerical methods and partial differential equations for European and American options and different underlying assets.

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Chapter 19

RISK MANAGEMENT, NUMERICAL METHODS AND OPTION PRICING

Chapter Outline This chapter is organized as follows: 1. In Section 19.1, we introduce numerical analysis and simulation techniques. The implicit and explicit difference numerical schemes are applied to the discretization of the Black and Scholes (1973) partial differential equation (PDE). 2. In Section 19.2, a numerical solution to the valuation of European call options on a non-dividend-paying stock is presented. 3. In Section 19.3, we present a model for the valuation of American options with a composite volatility and stochastic interest rates. 4. In Section 19.4, we introduce simulation techniques and in particular, the Monte–Carlo method. 5. Appendix A presents simple concepts in numerical analysis and the heat transfer equation. 6. Appendix B provides an algorithm for the valuation of a European call. 7. Appendix C provides the algorithm for the valuation of American longterm index options with a composite volatility. 8. Appendix D presents the Monte–Carlo method and the dynamics of asset prices.

Introduction The pricing of derivative assets is usually based upon two methods, which use the same basic arguments. The first method involves the resolution of 801

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a partial differential equation under the appropriate boundary conditions corresponding to the derivative asset’s payoffs. This is often referred to as the Black–Scholes method. The second method uses the martingale method, which was initiated by Harrison and Kreps (1979) and Harrison and Pliska (1981), where the current price of any financial asset is given by its discounted future payoffs under the appropriate probability measure. The probability is often referred to as the risk-neutral probability. Both methods are illustrated in detail for the pricing of European call options. Unfortunately, for most problems in financial economics, and in particular, for the pricing of American options, there is often no closedform solutions and option prices must be approximated. Therefore, financial economists often resort to numerical techniques. A brief presentation of these methods is given in this chapter. Finite difference methods approximate each partial derivative in the above equation by its corresponding formula. There are three main numerical schemes in the approximation of the partial derivative: the implicit difference scheme, the explicit difference scheme, and a weighted average of these schemes known as the Crank– Nicolson numerical scheme. Options on the spot-equity index, futures, and options on futures can be ranked among the most remarkable financial innovations and the securities markets have witnessed. These options and their underlying assets are subject to market imperfections, which interact to make cash and carry arbitrage with equity index futures far from riskless. These imperfections include the difficulty of shorting stocks, the tracking errors, the execution risk, and the fact that the underlying index is not traded (only its individual constituents). These reasons make difficult the implementation of a risk-free portfolio. Since these securities are based on the same underlying stock index, their prices must be related. If their prices do not obey the inter-market relationships, then the relative mispricing, often documented in empirical studies, should be instantaneously corrected given the high degree of sophistication of market participants. To circumvent some difficuties, it is a common place in research and practice to price index options using either European formulas or American-style approximation methods. Under the assumption that the underlying index pays dividends at a constant proportional rate, it seems that index options are severely mispriced. The development of option pricing models with stochastic interest rates and stochastic volatility explains some of the bias observed in empirical tests.

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Risk Management, Numerical Methods and Option Pricing

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803

In this chapter, we “mix” some of the ideas underlying these models with the observed behavior of the volatility and interest rates for the valuation of long-term options. The analysis is based on the results of numerous studies, where it is shown that the variance of a portfolio of assets is simultaneously a function of the variance of the firms cash flows and the interest-rate variance. Hence, it is possible to provide a decomposition of the market portfolio volatility into two components. The first is specific to the basket of stocks. The second is linked to the variance of interest rates. We study the implications of this decomposition on the pricing of index options and index futures options and their values in a model with a composite volatility and stochastic interest rates. In particular, the volatility of interest rates may trigger an early exercise of American stock index options. Using the principal results in the literature, a model is derived for the valuation of American long-term index options and index futures options. The model is a two state, with the values of the underlying index and interest rates as state variables. We develop a stable and convergent numerical scheme for the solutions for American index options and index futures options in this context. 19.1. Numerical Analysis and Simulation Techniques: An Introduction to Finite Difference Methods Several problems in financial economics do not have closed-form solutions. Though, it is always possible to get some simulation results using numerical analysis and simulation techniques. The non existence of analytic solutions to some partial differential equation under their appropriate boundary conditions leads to the choice of an appropriate numerical scheme. The reader can refer to the appendix for a short overview of finite difference methods and their applications to the heat transfer equation. Consider, for example, the discretization of the following PDE using finite difference methods: ∂c ∂c 1 2 2 ∂2c + rS σ S + − rc = 0 2 ∂S 2 ∂S ∂t

(19.1)

The discretization of the price function c(S, t) representing the option price can be done with respect to the state variable S and the time variable t. The method consists in dividing the underlying asset price S into N subintervals of length ∆S and the time variable (T − t) into M sub-intervals of length ∆t. The time step is denoted by ∆t = k and the state step

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as ∆S = h. Since the time variable is finite and corresponds to the option maturity, the state variable must also be finite. Therefore, the highest value of the underlying asset will be denoted by SH

and ∆S =

SH N

and ∆t =

(T − t) . M

Starting from zero, then gives (N + 1) state steps and (M + 1) time steps. This information can be represented on a simple diagram with (N + 1)(M + 1) points. Finite difference methods approximate each partial derivative in Eq. 19.1 by its corresponding formula. There are three main numerical schemes in the approximation of the partial derivative: the implicit difference scheme, the explicit difference scheme, and a weighted average of these schemes known as the Crank–Nicolson numerical scheme. 19.1.1. The implicit difference scheme The approximation of the partial derivatives in the Eq. (19.1) can be done using either a forward difference or a backward difference. As denote by c(i, j), the value of the option c(S, t) at position i and time j, where i refers to the space index and j indicates the time variable. Using a forward difference, the value of the option at each interior point in the grid can be approximated by: 1 ∂c = [c(i + 1, j) − c(i, j)] ∂S h

(19.2)

This partial derivative approximates the option value at node (i, j). The difference between the point (i + 1) and i corresponds to the value of h. Using a backward difference, the value of the option at each interior point in the grid can be approximated as: 1 ∂c = [c(i, j) − c(i − 1, j)] ∂S h

(19.3)

On an average of the above two approximations can be used to obtain: 1 ∂c = [c(i + 1, j) − c(i − 1, j)] ∂S 2h

(19.4)

The difference between the point (i − 1) and (i + 1) corresponds to the value ∂2c of 2h. The term ∂S 2 in the PDE can be approximated at the (i, j) point by the following relation: 1 1 1 ∂2c = [c(i + 1, j) − c(i, j)] − [c(i, j) − c(i − 1, j)] ∂S 2 h h h

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which is equal to: 1 ∂ 2c = 2 [c(i + 1, j) + c(i − 1, j) − 2c(i, j)] ∂S 2 h The partial derivative with respect to time, point (i, j) by the forward difference:

∂c , ∂t

(19.5)

can be approximated at

∂c 1 = [c(i, j + 1) − c(i, j)] ∂t k

(19.6)

The computation of the different partial derivatives is a first step in numerical analysis. The second step requires to replace these partial derivatives by their corresponding formulas in the extended Black–Scholes PDE. Hence, for i = 1 to (N − 1) and j = 0 to (M − 1), the PDE can be written as: rc(i, j) =

1 2 2 2 1 σ i h 2 [c(i + 1, j) + c(i − j, j) + c(i − 1, j) − 2c(i, j)] 2 h 1 + rih [c(i + 1, j) − c(i − 1, j)] 2h 1 + [c(i, j + 1) + c(i, j)] (19.7) k

This system can be re-written as: c(i, j + 1) = ai c(i − 1, j) − c(i − 1, j) + bi c(i, j) + ci c(i + 1, j) (19.8)   1 1   ai = rik − σ2 i2 k   2 2   bi = 1 + σ 2 i2 k + rk     1 1   ci = − rik − σ 2 i2 k 2 2 This system must be solved with the appropriate conditions. For a European call option on a non-dividend paying stock, the terminal or maturity condition is: c(S, T ) = max[0, ST − K] This boundary condition can be approximated by: c(i, M ) = max[0, ih − K] for i = 1, 2, 3, . . . , N .

(19.9)

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Equation (19.9) with j = (M − 1) gives (N − 1) simultaneous equations: ai c(i − 1, M − 1) + bi c(i, M − 1) + ci c(i + 1, M − 1) = c(i, M )

(19.10)

for, i = 1, 2, 3, . . . , N − 1, which can be solved for (N − 1) unknowns: c(1, M − 1), c(2, M − 1), . . . , c(N − 1, M − 1), where the c(i, M ) are known and given by Eq. (19.9). Solving this system provides option prices at time zero as a function of different levels of the underlying asset, c(1, 0), c(2, 0), . . . , c(M − 1, 0). The implicit difference method gives a relationship among three different values of the option at time t + jk, which means the values c(i − 1, j), c(i, j), and c(i + 1, j) and one option value at time t + (j + 1)k or c(i, j + 1). The above simple system can be solved by any method of matrix inversion since it needs the solution of a simple tri-diagonal matrix.

19.1.2. Explicit difference scheme It is possible to solve the same PDE using a simpler numerical scheme: the explicit difference scheme. In this case, we can approximate the different partial derivatives in the PDE by: 1 ∂c = [c(i + 1, j + 1) − c(i − 1, j + 1)] ∂S 2h

(19.11)

for the option partial derivative with respect to the state variable: 1 ∂ 2c = 2 [c(i + 1, j + 1) − c(i − 1, j + 1) − 2c(i, j + 1)] ∂S 2 h

(19.12)

and 1 ∂c = [c(i, j + 1) − c(i, j)] ∂t k

(19.13)

for the option partial time derivative. As before, these partial derivatives must be replaced in the PDE in order to obtain the following system: c(i, j) = a∗i c(i − 1, j + 1) + b∗i c(i, j + 1) + c∗i c(i + 1, j + 1)

(19.14)

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with

807

  1 1 2 2 1  ∗  − rik + σ i k ai =    (1 + rk) 2 2    1 1 + σ 2 i2 k b∗i =  (1 + rk)      1 1 2 2 1   rik + σ i k ci = (1 + rk) 2 2

This system can be solved by inverting a tri-diagonal matrix. The value of an option can be computed by imposing on this system the appropriate boundary conditions. The explicit difference scheme gives a relationship between one value of the option c(i, j) at time t+jk and three values at time t + (j + 1)k,

i.e., c(i − 1, j + 1), c(i, j + 1),

and c(i + 1, j + 1).

19.1.3. An extension to account for information costs Consider, for example, the discretization of the following PDE using finite difference methods: ∂c 1 2 2 ∂ 2c ∂c + (r + λS )S σ S + − (r + λc )c = 0 2 ∂S 2 ∂S ∂t

(19.15)

The discretization of the price function c(S, t) representing the option price can be done with respect to the state variable S and the time variable t. The method consists in dividing the underlying asset price S into N sub-intervals of length ∆S and the time variable (T −t) into M sub-intervals of length ∆t. The analysis is similar to the method proposed above. 19.2. Application to European Options on Non-Dividend Paying Stocks Consider the valuation of a European call on a non-dividend paying stock in the presence of information costs. The call price must satisfy the following PDE:

1 2 2 ∂ 2c ∂c ∂c + rS + − rc = 0 σ S 2 ∂S 2 ∂S ∂t under the following terminal or boundary condition c(S, T ) max[0, ST − K].

=

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An additional condition must be imposed to reflect the fact that, for high values of the underlying asset price, the partial derivative of the call price approaches one, lim

S→∞

∂c = 1. ∂S

19.2.1. The analytic solution The valuation of a call option consists in finding a solution to the following PDE:

∂c ∂c 1 2 2 ∂ 2c + (r + λS )S σ S + − (r + λc )c = 0 2 2 ∂S ∂S ∂t under the following terminal condition which must be satisfied by the call price at its maturity date c(S, t∗ ) = max[0, St∗ − K]. The valuation of a put option consists in finding a solution to the following PDE:

∂p ∂p 1 2 2 ∂ 2p )S + (r + λ σ S + − (r + λp )p = 0 S 2 ∂S 2 ∂S ∂t under the following terminal condition which must be satisfied by the put price at its maturity date p(S, t∗ ) = max[0, K − St∗ ]. The appendix is proposed as an exercise to provide the analytic solution to this problem. 19.2.2. The numerical solution The partial derivative with respect to the stock price can be approximated using a forward difference: 1 ∂c(S, T ) = [c(S + h, T ) − c(S, T )] ∂S h or a backward difference: 1 ∂c(S, T ) = [c(S, T ) − c(S − h, T )] ∂S h At points (i, j), the option partial derivative is approximated by the forward difference: ∂c 1 = [c(i + 1, j) − c(i, j)]. ∂S h

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or a backward difference: 1 ∂c = [c(i, j) − c(i − 1, j)]. ∂S h A better approximation of the option value c(i, j) at point (i, j) considers two steps h in the underlying asset price and divides by two: 1 ∂c = [c(1 + i, j) − c(i − 1, j)] ∂S 2h The term

∂2 c ∂S 2

is approximated by: 1 ∂c(S + h, T ) ∂c(S, T ) ∂ 2c = − ∂S 2 h2 ∂S ∂S

When the partial derivative of

∂c ∂S

is replaced, Eq. (19.16) gives:

1 ∂ 2c = 2 [c(i + 1, j) + c(i − 1, j) − 2c(i, j)]. ∂S 2 h The partial time derivative of

∂c ∂t

(19.16)

(19.17)

can be approximated by:

∂c 1 = [c(i, j) − c(i, j − 1)] ∂t k

(19.18)

The option price can be determined at each instant j as a function of its value at an instant (j − 1). Replacing the partial derivatives by their approximations in the PDE gives: 0=

1 2 2 2 1 σ i h 2 [c(i + 1, j) + c(i − 1, j) − 2c(i, j)] 2 h + rih

1 [c(i + 1, j) − c(i − 1, j)] 2h

1 + [c(i, j) − c(i, j − 1)] − rc(i, j) k

(19.19)

for i = 1 to (N − 1) and j = 0 to (M − 1). Multiplying by −k and gathering the terms in c(i − 1, j), c(i, j), c(i + 1, j), and c(i, j − 1) gives the following tri-diagonal

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system: c(i, j − 1) = ai c(i − 1, j) + bi c(i, j) + ci c(i + 1, j)   1 1   ai = rik − σ2 i2 k    2 2   2 2 bi = 1 + σ i k + rk      1 2 2 1   ci = − rik − σ i k 2 2

(19.20)

The terminal or maturity condition for a European call option on a nondividend paying stock c(S, T ) = max[0, ST − K] is approximated by: c(i, M ) = max[0, ih − K] for i = 1, 2, 3, . . . , N. This corresponds to u(i, 0) = ih − K, when ih is higher than K, otherwise it is equal to zero. For sufficiently higher values of the underlying asset, the condition on the option’s partial derivative can be approximated by: c(n, j) − c(n − 1, j) = h

for j = 0, . . . , M.

The resulting system gives (N − 1) linear equations and (N + 1) unknowns u(i, j). Since the option value at expiration is given, it is possible to generate at each time step, all the time prices by inverting the resulting linear system algorithm. 19.2.3. An application to European calls on non-dividend paying stocks in the presence of information costs Consider the valuation of a European call on a non-dividend paying stock in the presence of information costs. The call price must satisfy the following PDE:

∂c ∂c 1 2 2 ∂ 2c )S + (r + λ σ S + − (r + λc )c = 0 S 2 ∂S 2 ∂S ∂t under the following terminal or boundary condition c(S, T ) = max[0, ST − K]. An additional condition must be imposed to reflect the fact that, for high values of the underlying asset price, the partial derivative of the call price approaches one. The same analysis can be applied.

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19.3. Valuation of American Options with a Composite Volatility The starting points are the common factors in the returns on stocks and bonds. There is a large body of empirical work dealing with the relationships among stock prices, interest rates, and inflation (See for example, Copeland and Stapleton, 1985; Peterson and Peterson, for more details, refer to Bellalah et al., 1998). 19.3.1. The effect of interest rate volatility on the index volatility Fama and French (1993) identified common risk factors in the returns on stocks and bonds. They showed that stock returns have shared variation due to the stock market factors and that they are associated with bond returns through shared variation in bond market factors. It is possible to decompose the variance of the market portfolio into a proportion ϑ corresponding to the variance of cash flows and a proportion (1−ϑ) resulting from variations in interest rates. We briefly review the models by Ramaswamy and Sundaresan RS (1985) and Brenner et al. (1987), and Bellalah (2003). In the RS model, the dynamics of the index price, S, are given by the following equation: dS = [αS − δS]dt + σS SdzS

(19.21)

where: • • • •

α: the instantaneous expected return on S; δ: the dividend yield on the stock index; σS : the instantaneous expected standard deviation of returns on S and dzS : the increment of a Wiener process.

The dynamics of the spot interest rates are given by the familiar square root process, which has correlation ρ with dzS , i.e., cov(dzS , dzr ) = ρdt. In the Brenner et al. model, (1987) the effect of the volatility of interest rates is explicitly taken into account. The dynamics of the index price are given by the following equation: dS = [αS − δS]dt + (σI + νσr )SdzS

(19.22)

where δ stands for the dividend yield on the index price, σI is the specific index volatility, σr is the interest-rate volatility and ν is a coefficient which transmits the effect of interest-rates volatility to the index volatility.

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The dynamics of the interest rate r are given by an Ornstein–Uhlenbeck process where cov(dzS , dzr ) = ρdt. 19.3.2. Valuation of index options with a composite volatility Using the main results in the previous section, the model is based on several assumptions. Using the L-EH is equivalent to assuming that the expected instantaneous return on a coupon-paying bond (independent of its time to maturity) is equal to the spot rate r(t) on a bond maturing instantaneously, i.e., Et [dB/B] = r(t)dt

(19.23)

where Et is the mathematical expectation conditional on all the available information at time t. The dynamics of the short interest rates are described by the familiar square-root process:1 √ dr(t) = κ[µ − r]dt + σr rdzr (19.24) In this expression, κ[µ − r] corresponds to the mean reverting drift pulling the short interest rate towards its long-term value µ, where κ defines the speed of the adjustment. It is convenient to note that the square-root process is more suited than the Ornstein–Uhlenbeck process for some wellknown reasons and because it does not allow for negative interest rates. The choice of this process rather than that used by Brenner et al. (1987) is based on the fact that the critics addressed to arbitrage models as the Vasicek (1977) model do not apply to the CIR inter-temporal general equilibrium term structure model. The dynamics of the stock index or the underlying commodity are described by the following equation: √ (19.25) dS = (αS − δS)dt + (σS + νσr r)SdzS where dzS has correlation ρ with dzr , i.e., cov(dzS , dzr ) = ρdt. This formulation is similar in spirit to that by Brenner et al. (1987). In the above formulation, αS stands for the expected instantaneous relative price change of the underlying commodity asset. When there are no arbitrage opportunities, the assumption of a constant proportional cost of 1 This

process was used in Cox et al. (1985) among others.

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carry implies that F = SebT , where F is the futures price, b = r − δ is the cost of carrying the commodity and T is the time to maturity. For a nondividend-paying asset, b = r and for a dividend-paying asset, b = r − δ. The √ volatility of the underlying index is given by σS + νσr r. It is composed of a specific volatility σS , the interest-rate volatility σr and a coefficient ν, which transmits the effect of interest rates volatility to the index volatility. Let us denote by U (S, r, t), the option value as a function of the two underlying state variables and time. Applying Ito’s lemma and using the standard hedging arguments, it is possible to construct a locally risk-less portfolio with a risk-less bond and the underlying stock index. At equilibrium, the expected rate of return on this portfolio under the L-EH must be the short risk-less interest rate and, under the free boundary formulation, the option price must obey the following PDE in the continuation region: √ √ √ ∂ 2U 1 1 ∂2U (σS + νσr r)2 S 2 2 + ρσr S(σS + νσr r) r 2 ∂S 2 ∂S∂r 2 ∂U ∂U ∂U 1 ∂ U + κ(µ − r) − rU + = 0 (19.26) + rσr2 2 + bS 2 ∂r ∂S ∂r ∂t To find solutions for an American stock index call option with a strike price K and a maturity date T , the PDE shown in Eq. (19.26) must be solved under the following boundary conditions: C(S, r, T ) = (S − K)+ C(S, r, t) = (S − K)+ ∂C(S, r, t) =1 ∂S

(19.27)

The last condition is the usual “smooth fit” principle. An American stock index put option with a strike price K and a maturity date T must satisfy the PDE Eq. (19.26) subject to the following boundary conditions: P (S, r, T ) = (K − S)+ P (S, r, t) = (K − S)+ ∂P (S, r, t) = −1 ∂S

(19.28)

19.3.3. Numerical solutions and simulations The PDE is discretized in a grid with respect to the two space-variables S, r and time t. The Crank–Nicholson scheme is used and some simulations

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are run. The numerical scheme is a θ scheme of the implicit type for which θ = 1/2. It is centered in space and in time. It is unconditionally stable and convergent.

Numerical solutions The time to maturity (T − t) is divided into N time intervals of length k. The option value is calculated at time (s − k) in a recursive way as a function of its value at instant s with t ≤ s ≤ T . The instant s = T corresponds to the option’s maturity date. The underlying index price and the interest rate are divided into M intervals of size h. The state variables are considered within the intervals [0, S∗] and [0, r∗]. Note that the larger the values of M and N , the closer is the numerical solution of the discrete system to the real solution of the PDE. Hence, using: S = ih for 0 ≤ i ≤ M r = jh

for 0 ≤ j ≤ M

t = nk

for 0 ≤ n ≤ N

(19.29)

The option value is represented by a 3-dimensional array, U (S, r, t) = U (ih, jh, nk) = U n (i, j). At each time step, s = T − nk, the first and the second derivatives of S and r and the time derivative, in the PDE (19.26) are approximated using the central differences: U n (i + 1, j) − U n (i − 1, j) ∂U n (i, j) = ∂S 2h ∂U n (i, j) U n (i, j + 1) − U n (i, j − 1) = ∂r 2h U n (i − 1, j) − 2U n(i, j) + U n (i + 1, j) ∂ 2 U n (i, j) = 2 ∂S h2 U n (i, j − 1) − 2U n(i, j) + U n (i, j + 1) ∂ 2 U n (i, j) = 2 ∂r h2 U n (i + 1, j + 1) − U n (i − 1, j + 1) − U n(i + 1, j − 1) + U n (i − 1, j − 1) ∂ 2 U n (i, j) = ∂S∂r 4h2 U n (i, j) − U n − 12 (i, j) ∂U n (i, j) = ∂t k/2

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If we replace the derivatives by their values, the PDE (19.26) can be approximated for each instant s = T − n − 12 k at points S = ih and r = jh by the following system2 : 1 1 1 1 1 1 − AδS2 − Bδr2 − CHr HS − DHS − EHr U n (i, j) 2 2 8 4 4 1 2 1 2 1 1 1 Aδ + Bδ − CHr HS + DHS + EHr + F U n−1 (i, j) = 2 S 2 r 8 4 4 with:

√ (σI + νσr jh)2 (i2 k) , 2G √ ρσr i(σI jh + νσr jh)k , C= hG A=

E=

κ(µ − jh)k , hG

1 G = 1 + jhk, 2

F =

B=

σr2 jk 2hG

D=

(bih)k hG

1 − ( 12 jhk) G

1 ≤ i ≤ M − 1 and 1 ≤ j ≤ M − 13

Until now, the Alternating Direction Implicit Method (ADI) scheme shows no particular problems. The approximation of the boundary condition when the interest rate is zero poses a serious problem and makes the difference between our method and the numerical schemes reported in the 2 For ease of exposition, the following operators H , H , H H , δ 2 , δ 2 , δ 2 δ 2 are r S S r S r S r used with:

HS Hr U n (i, j) = HS [U n (i, j + 1) − U n (i, j − 1)] = U n (i + 1, j + 1) − U n (i − 1, j + 1) − U n (i + 1, j − 1) + U n (i − 1, j − 1) 2 n+1 U (i, j) = U n+1 (i + 1, j) − 2U n+1 (i, j) + U n+1 (i − 1, j) δS

δr2 U n+1 (i, j) = U n+1 (i, j + 1) − 2U n+1 (i, j) + U n+1 (i, j − 1) 2 2 n δr U (i, j) = U n (i + 1, j + 1) − 2U n (i, j + 1) + U n (i − 1, j + 1) − 2U n (i + 1, j) δS

+ 4U n (i, j) − 2U n (i − 1, j) + U n (i + 1, j − 1) − 2U n (i, j − 1) + U n (i − 1, j − 1) 3 It

is possible to show that this system of (M − 1)2 equations may be solved in two steps, given a value of n. For each i, it presents (M − 1) equations with (M + 1) unknowns for 0 ≤ i ≤ M . After calculating the values U n∗ (i + 1, j − 1), the second step implies to solve a system of (M − 1) equations with (M + 1) unknowns for 0 ≤ j ≤ M . This method must be repeated N times for 0 ≤ j ≤ M .

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financial literature. The main difficulty consists in preserving the properties of the numerical scheme, i.e., the tri-diagonal structure of the matrix, the stability, and the speed of convergence. When the interest rate r is zero, i.e., when j = 0, the PDE becomes: ∂U ∂U ∂U 1 2 2 ∂2U σ S + bS + =0 + κµ 2 I ∂S 2 ∂r ∂S ∂t

(19.30)

This PDE is also discretized by the Crank–Nicholson scheme, centered in the space, except for the term ∂U ∂r , which is treated explicitly and “decentered inside the scheme”. Hence, all the terms are of the second order, except this latter term, which is of a first order in time and of a second order in space. The following discretization is used: n 3U (i, 0) − 4U n (i, 1) + U n (i, 2) H∗r U n−1 (i, 0) ∂U = = ∂r 2h 2h which gives:

1 2 1 1 − A δS − B HS U n (i, 0) 2 4 1 1 1 = 1 + A δS2 + B HS + C H∗r U n−1 (i, 0) 2 4 2

with A =

σI2 (i2 k) , 2

B =

(bih)k , h

C =

κµk . h

A detailed algorithm is provided in the appendix for the pricing of American call and put options in this context. The algorithm can also be adapted for the pricing of other derivatives like interest-rate options, warrants, etc. The impact of the composite volatility and stochastic interest rates seem to be significant on American at the money call and put option values. This effect is more important for put options and may trigger an early exercise of index puts. The effect reported here is greater than that in standard models and is less than that in BCS. While the negligible effect in standard models is intuitive, the effect in the BCS model may be due to the process of interest rates chosen, allowing for negative interest rates. Also, the numerical scheme presented here is more efficient than those presented in previous studies. Empirical tests can be conducted in order to test if the parameters are statistically stable

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and if option prices are close to the market prices. For more details, see Bellalah (2003). 19.4. Simulation Methods: Monte–Carlo Method Option prices are computed using the expected value of the terminal payoff. Since, this value is known, it can be simulated. Boyle (1976, 1986, 1988) developed a simulation method using the distribution of the underlying asset at the option’s maturity date. When the process specifying the distribution and the future movements in the underlying asset is known, then it is a simple matter to simulate its values. Each time a simulation is conducted, the computer generates a terminal value of the financial asset. Repeating the simulation 1000 times gives the distribution of terminal asset values. The distribution is used to extract the expected terminal asset value. The simulation can be conducted as follows. Given a random variable with a lax µdx, if we draw n times on a computer the values X1 , X2 , . . . , Xn for a high value of n such that Xn follows the same law µdx, and the sequence (Xn )n≥1 is a sequence of independent random variables, then by the law of large numbers, the derivative price F can be expressed as follows:

N 1 f (Xn ) = f (x)µ(dx) lim N →∞ N n=1 The Monte–Carlo method can be implemented on a computer as follows. Construct a sequence of numbers (Un )n≥1 in order to correspond to a uniform sequence of independent random variables on the interval [0, 1]. Then search for a function to which (u1 , u2 , . . . , up ) corresponds to F (u1 , u2 , . . . , up ) such that the law of the random variable F (u1 , u2 , . . . , up ) is the known law of µ(dx). The sequence of random independent variables (Xn )n≥1 with Xn = F [U(n−1) p+1 , . . . , Unp ] follows the law µ. To generate the sequence (Un )n≥1 with a computer, we use the random function which gives a pseudo-random number between 0 and 1 or an integer in a fixed interval. The function random or Rand(.) is often available on computers. The main drawback of the Monte–Carlo method is that it is cumbersome in the valuation of American options. The method is often used in the valuation of European options.

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19.4.1. Simulation of Gaussian variables A random variable X is a Gaussian variable with a zero mean and a unit variance when its density function is given by: 1 e n(x) = √ 2Πσ

“

x2 2

”

The law of X is known as the standard normal distribution. When a random variable Y = µ + σX is of the Gaussian type, it follows a normal distribution with a parameter µ as the mean and σ 2 as the variance. It is often denoted by: N (µ, σ 2 ). If there are two random uniform independent variables (U1 , U2 ) on the interval [0, 1], then the cos(2πU2 ) −2 ln(U1 ) follows a normal distribution with a zero mean and a unit variance. In order to simulate a Gaussian law with mean µ and a variance σ2 , we just use the change in variables: z = µ + σg. For example, the following instuction may be used in a program written in Turbo Pascal to generate the Gaussian law: Gaussian = µ + σ cos (2πRandom) −2 log (Random) 19.4.2. Relationship between option values and simulation methods The risk-neutral random walk for the underlying asset S is given by: dS = rSdt + σSdz Under the risk-neutral probability, the option value is given by its expected value discounted to the present. The option price can be computed by simulating the risk-neutral random walk for the underlying asset. We start by today’s asset value S0 and continue until maturity. This provides a realization of the underlying price path. We use this realization to compute the option payoff. Then, we make similar realizations and compute the average payoff over all realizations. The present value of this average provides the option value.

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The method can be implemented by generating random numbers from a standardized normal distribution. At each timestep, the asset price is generated using these random increments. For example, we can use the following dynamics: √ δS = rSδt + σS δtφ where φ is taken from a standardized normal distribution. The latest value for S is used to compute δS and hence the next value for S. It is possible for the log-normal random walk to use a simpler method with reference to the equation:

1 d(log S) = r − σ 2 dt + σdz 2 over a short timestep, this can be written as:

√ 1 S(t + δt) = S(t) + δS = S(t) exp r − σ 2 δt + σ δtφ 2 This expression is exact. Simulations are provided in the appendix. Summary This chapter contains the basic material for the pricing of derivative assets in a continuous-time framework. The presentation is made as simple as possible in order to enable uninformed readers to understand the main derivations. First, we present in detail the search for an analytic formula for European call option within the PDE method. Second, we illustrate in detail the martingale method for the derivation of a European call formula. Third, we apply finite difference methods for the valuation of European call options. Fourth, we present a model for the valuation of American options with a composite volatility. The option-pricing literature has been concerned with modeling stochastic interest rates and stochastic volatility without an explicit treatment for long-term stock index options and index futures options. In this chapter, we present a specific model for the valuation of these options since the effect of interest rates on the long run is probably more important than on the short-term options. This model is motivated by the results of empirical and theoretical studies regarding the market index,

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as opposed to a single stock and by the relative mispricing of long-term index options. We extend the classic (ADI) based on the Crank–Nicholson scheme in 2-space dimensions and provide an efficient, stable, and convergent algorithm for the pricing of these options. Some numerical receipts are used to reduce the computation time. The algorithm reduces to solving tri-diagonal systems and may be used to handle other complex problems in financial economics. The solution method is quite general to be applied to any option valuation problem in the presence of 2-state variables. The results indicate the significant effect of interest rates and the volatility on the pricing and the early exercise of long-term index options. Fifth, we develop some simulation methods and in particular, the Monte–Carlo Method.

Questions 1. What is an implicit difference scheme? 2. What is an explicit difference scheme? 3. What is the solution method for the valuation of European calls on nondividend paying stocks? 4. Describe the solution method for the valuation of European calls on non-dividend paying stocks in the presence of information costs. 5. What is a composite volatility? 6. What are the main principles of simulation methods? 7. Describe the Monte–Carlo method.

Appendix A: Simple Concepts in Numerical Analysis The heat transfer equation Some equations of the parabolic type, used in the pricing of derivatives, can be transformed to the heat transfer equation. In its simplest form, this equation can be written as: ∂ 2u ∂u = ∂t ∂S 2

(A.1)

The function u(S, t) depends on time and the underlying asset value S. The option’s pay off is known at the option’s maturity date. The problem is to find the value of the function u(S, t), which satisfies the following system

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of three conditions: ∂2u ∂u for 0 ≤ S ≤ L = ∂t ∂S 2 u(0, t) = u(L, t) = 0 for t ≥ 0 u(S, 0) = u0 (S)

for 0 ≤ S ≤ L

The first condition corresponds to the heat transfer equation for which the underlying asset lies between 0 and a specified value. The second condition is a boundary or a limit condition. It shows that the option value is zero when the underlying asset price is zero. The third condition represents a terminal or a maturity condition. It gives the option price at the maturity date. Some simple numerical schemes for the heat transfer equation The following analysis shows how to find a numerical solution for the heat transfer equation. Consider a function u(S, t) for which the underlying asset S is divided into ih and time corresponds to jk, where h is the state step and k is the time step. The indexes (i, j) vary in the plane (S, t). Assume that the discrete option values u(i, j) are known for all space indexes i at each instant of time j, where j takes the values 0, 1, . . . , j. The discretization of the heat transfer equation needs first the computation of the partial derivatives. The time derivative can be approximated using the finite decentered difference as: (j) u(i, j + 1) − u(i, j) ∂u + o(k) = k ∂t i This approximation is exact to the first order. The derivative with respect to the space variable or the underlying asset can be approximated using the following finite centered difference: 2 (j) u(i, j + 1) − 2u(i, j) + u(i − 1, j) ∂ u = + o(h2 ) (A.2) 2 h ∂S 2 i This gives a numerical scheme, often referred to as the explicit scheme. u(i, j + 1) = u(i, j) +

k [u(i + 1, j) − 2u(i, j) + u(i − 1, j)] h2

(A.3)

When k = o(h), the explicit numerical scheme is accurate to the first order.

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Equation (A.2) can be applied at the level (j + 1) rather than j. In this case, using the following two equations: (j+1) u(i, j + 1) − u(i, j) ∂u = + o(k) (A.4) k ∂t i 2 (j+1) ∂ u u(i + 1, j + 1) − 2u(i, j + 1) + u(i − 1, j + 1) = + o(h2 ) 2 h ∂S 2 i (A.5) gives a totally implicit numerical scheme: u(i, j + 1) = u(i, j) +

k [u(i + 1, j + 1) − 2u(i, j + 1) + u(i − 1, j + 1)] h2 (A.6)

The well-known Crank–Nicolson scheme can be obtained using an average of the numerical schemes in Eqs. (A.3) and (A.6): k [u(i + 1, j) 2h2 − 2u(i, j) + u(i − 1, j) + u(i + 1, j + 1)]

u(i, j + 1) = u(i, j) +

− 2u(i, j + 1) + u(i − 1, j + 1)]

(A.7)

This numerical scheme is also of the implicit type. It is a simple form of a more general scheme, where θ is between 0 and 1. In this case, we have: k {(1 − θ)[u(i + 1, j) − 2u(i, j) + u(i − 1, j)] 2h2 + θ[u(i + 1, j + 1) − 2u(i, j + 1) + u(i − 1, j + 1)]} (A.8)

u(i, j + 1) = u(i, j) +

This general scheme takes different forms according to the values attributed to θ with 0 ≤ θ ≤ 1. When θ = 1, this scheme is of the implicit type. When θ = 0, this scheme is of the explicit type. When θ = 12 , this scheme is of the Crank–Nicolson type. Appendix B: An Algorithm for a European Call For j = 1, . . . , M a(1) = 0,

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b(1) = 1 + rk + σ2 k c(1) = − 12 rk − 12 σ 2 k For i = 2 to n − 1 ai = 12 rik − 12 σ 2 i2 k

bi = 1 + σ2 i2 k + rk ci = − 12 rik − 12 σ 2 i2 k di = u(i, j − 1) end (for i = 2 to n − 1) a(n) = −1,

b(n) = 1,

c(n) = 0,

d(n) = h

This is the procedure for the matrix inversion: For i = 1 to n u(i) = c(i)/[b(i) − u(i − 1)a(i)] g(i) = (d(i) − g(i − 1)a(i)]/[b(i) − u(i − 1)a(i)] end (for i = 1to n). u(n, j) = g(n) For i = n − 1 down to 1 u(i, j)g(i) − u(i)u(i + 1, j) end for End (for j = 1 to m). Appendix C: The Algorithm for the Valuation of American Long-term Index Options with a Composite Volatility Read: K, σI , ν, σr , ρ, d, κ, µ, h, k For i = 0, M For j = 0, M, U (i, j) = 0 if (ih − K) > 0, then U (i, j) = ih − K, End if End for M (For a put (ih − K)is replaced by (K − ih)) For n = 1, nn (time to maturity) For i = 1, M − 1 a(0) = 0, b(0) = 1, c(0) = 0, d(0) = U (i, 0)

823

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For j = 1, M − 1 Use gg = G, f f = F , ee = E, dd = D, cc = C, bb = B, and aa = A with a(j) = 14 ee − 12 bb b(j) = 1 + bb c(j) = − 14 ee − 12 bb

d(j) = 14 cc(U (i + 1, j + 1) − U (i + 1, j − 1) − U (i − 1, j + 1) +U (i − 1, j − 1)) − a(j)U (i, j − 1) − c(j)U (i, j + 1) +(f f − bb − 2aa)U (i, j) + 12 dd + aa U (i + 1, j) + aa − 12 dd U (i − 1, j) End For j = 1, M − 1 a(M ) = 0, b(M ) = 1, c(M ) = 0, d(M ) = ih (For a put, d(M ) becomes: if (K − ih) > 0, then d(M ) = K − ih Else d(M ) = 0 End if). Solve the system (M, a, b, c, d, x) by inverting the matrix. For j = 0, M , U U (i, j) = x(j), End End For i = 1, M − 1. This is the end of the first step. Before beginning the second step, a special treatment is done for j = 0. a(0) = 0, b(0) = 1, c(0) = 0, and d(0) = 1 (For a put, d(0) becomes d(0) = K) For i = 1, M − 1 σI2 (i2 k) 2 (bih)k bb = h cc = κµk h a(i) = − 21 (aa

aa =

− 12 )bb

b(i) = 1 + aa c(i) = − 12 (aa + 12 bb)

d(i) = −a(i)U (i − 1, 0) + (1 − aa − 32 cc)U (i, 0) − c(i)U (i + 1, 0) + 12 cc(−4U (i, 1) + U (i, 2))

End For i = 1, M − 1

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a(M ) = b(M − 1) + 4a(M − 1) b(M ) = c(M − 1) − 3a(M − 1) c(M ) = 0 d(M ) = d(M − 1) − 2ha(M − 1) (For a put, a(M ) = 0, b(M ) = 1, c(M ) = 0, d(M ) = 0) The matrix is inverted again and option values are generated for a nil interest rate: for i = 0, M , U (i, 0) = x(i), End. Now, we can generate option values for a fixed j. This the second step. For j = 1, M − 1 a(0) = 0, b(0) = 1, c(0) = 0, d(0) = 0 (For a put, a(0) = 0, b(0) = 1, c(0) = 0, d(0) = K) . For i = 1, M − 1 jhk 2 √ (σI +νσr jh)2 (i2 k) aa = 2gg dd = (bih)k hgg a(i) = ( 14 dd − 12 )aa

gg = 1 +

b(i) = 1 + aa c(i) = − 41 dd − 12 aa d(i) = U (i, j) + c(i)U (i + 1, j)(b(i) − 1)U (i, j) + a(i)U (i − 1, j) End For i = 1, M − 1 a(M ) = b(M − 1) + 4a(M − 1) b(M ) = c(M − 1) − 3a(M − 1) c(M ) = 0 d(M ) = d(M − 1) − 2ha(M − 1) (For a put, the modification is a(M ) = 0, b(M ) = 1, c(M ) = 0, d(M ) = 0). The matrix must be inverted again for the third time: for i = 0, M, U (i, j) = x(i) End, For i = 1, M − 1, U (i, j) = uu(i, M ) End, End For j = 1, M − 1 Since options are of the American type: For i = 0, M For j = 0, M

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if U (i, j) < (ih − K) then U (i, j) = ih − K End if End for i, End for j, End for n = 1, nn (For a put, the modification is: if U (i, j) < (K − ih) then U (i, j) = K − ih End if). Exercises Exercise 1 Consider the following function F (X(t)) = eX(t) Show that t 1 t X(τ ) eX(τ ) dX(τ ) = eX(t) − 1 − e dτ. 2 0 0 Solution: 1 dF (X(t)) = eX(t) dX(t) + eX(t) dt 2 Since:

t 0

dF = eX(t) − eX(0) = eX(t) − 1

we have, 0

t

eX(τ ) dX(τ ) = eX(t) − 1 −

1 2

0

t

eX(τ ) dτ

Exercise 2 Consider the following function: F (X(t)) = aX 2 (t) + betX(t) 1. Show that:

0

t

X(τ )dX(τ ) =

with a, b > 0

X 2 (t) 1 − t 2 2

2. Show that: t t b t 2 τ X(τ ) τ eX(τ ) dX(τ ) + τ e dτ + b X(τ )eτ X(τ )dτ = etX(t) − 1 b 2 0 0 0

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3. Deduce that: t (2aX(τ ) + bτ eτ X(τ ) )dX(τ ) = b(eX(τ ) − 1) + aX 2 (t) − at − b 0

×

0

t

e

τ X(τ )

τ2 X(τ ) + dτ 2

Solution: 1. Let us denote by G(X(t)) = aX 2 (t). Hence, dG = 2aX(t)dX(t) + adt or,

t 0

dG = a(X 2 (t) − X 2 (0)) = aX 2 (t)

since X 2 (0) = 0. We can compute the quantity,

t 0

dG =

t 0

2aX(τ )dX(τ ) + a

t

0

or,

t 0

dG =

t

0

2aX(τ )dX(τ ) + at

Hence, we have: aX 2 (t) =

t 0

2aX(τ )dX(τ ) + at

⇐⇒ t 2 aX (t) − at = 2a X(τ )dX(τ ) 0

then, 0

t

X(τ )dX(τ ) =

X 2 (t) t − 2 2

dt

827

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2. We denote by H(X(t)) = betX(t) , so:

1 dH = btetX(t) dX(t) + bX(t)etX(t) + bt2 etX(t) dt 2 or,

t 0

dH = b(etX(t) − e0X(0) ) = betX(t) − b

Since we have:

t t t τ2 τ X(τ ) τ X(τ ) X(τ ) + dτ dH = b τe dX(τ ) + b e 2 0 0 0 then,

τ2 dτ eτ X(τ ) X(τ ) + 2 0 0 t t t 2 τ τ X(τ ) e etX(t) − 1 = τ eτ X(τ ) dX(τ ) + eτ X(τ )X(τ )dτ + dτ 0 0 0 2 b(etX(t) − 1) = b

t

τ eτ X(τ ) dX(τ ) + b

t

The computation of the first integral gives: t t τ eτ X(τ ) dX(τ ) = etX(t) − 1 − eτ X(τ )X(τ )dτ + 0

0

t 0

τ 2 τ X(τ ) dτ e 2

Multiplying this equation by b, we deduce the result for the second question. 3. Since F (X(t)) = aX 2 + betX(t) , we have:

t2 tX(t) tX(t) X(t) + dt )dX(t) + a + be dF = (2aX(t) + bte 2 or

t

0

dF = aX(t)2 + b(etX(t) − 1)

and 0

t

dF =

0

t

τ X(τ )

(2aX(τ ) + bτ e

+ 0

t

be

τ X(τ )

t

)dX(τ ) +

τ2 X(τ ) + 2

0

dτ

adt

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Using the answer for the first question, we obtain:

t

0

X(t)dX(τ ) =

t2 X(t)2 − 2 2

Hence,

t 0

2aX(τ )dX(τ ) = aX(t)2 − at

and b

t

0

τ eτ X(τ ) dX(τ ) = b(etX(t) − 1) − b

t

0

τ2 dτ eτ X(τ ) X(τ ) + 2

This gives the desired result: 0

t

(2aX(t) + bτ eτ X(τ ) )dX(τ ) = b(etX(t) − 1) + aX(t)2 − at − b ×

0

t

τ2 dτ eτ X(τ ) X(τ ) + 2

Appendix D: The Monte–Carlo Method and the Dynamics of Asset Prices Consider the following data for the application of the Monte–Carlo method. Initial time: 15/06/2002, maturity date: 15/06/2003. The following parameters are used: Initial asset price at time 0: 200, drift 1%, volatility = 40%, interest rate = 3%, and time step = 0.01. The Table D.1 provide the simulations using the above parameters. We report the results for the first, second, and third simulations, Sim 1 to Sim 3. We calculate the values of the calls and the puts for different strike prices. The underlying asset is simulated using the Monte–Carlo method for a drift of 5%. For each realization, the final stock is provided in the last row. We provided the option payoff, the mean of all the payoffs over all simulations. The present values of the means correspond to the option values. This method is suitable for path-dependent options.

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Table D.1. Asset 200

S mean

Time

Sim 1

Sim 2

Sim 3

Drift 1% Volatility 40% Time-step 0.01

200.00 199.34 198.78

0.00 0.01 0.02

200.00 197.88 189.84

200.00 209.89 214.57

200.00 210.36 207.77

Interest Rate 3%

198.99 198.94 198.71

0.03 0.04 0.05

204.55 208.89 228.25

209.43 215.69 213.42

201.61 200.50 203.37

June 2003

200.08 199.67 199.13 231.38

0.06 0.07 0.08 3.61

223.95 230.12 228.91 134.49

219.91 220.61 212.36 255.98

207.10 212.48 225.35 148.51

Strike 1

290.13

CALL Mean PV

Payoff 55.52 49.82

Strike 2

118.42

PUT Mean PV

Payoff 13.31 11.94

Strike 3

623

CALL Mean PV

Payoff 9.38 8.42

Strike 4

80

PUT Mean PV

Payoff 4 3.59

References Bellalah, M (2003). Valuation of long term options. International Journal of Finance. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654. Boyle, P (1976). Rates of return as random variables. Journal of Risk and Insurance, 43 (December), 694–711. Boyle, P (1986). Option valuation using a three jump process. International Options Journal, 3, 7–12. Boyle, PP (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23 (March) 1–12. Brenner, M, G. Courtadon and M Subrahmanyam (1987). The valuation of stock index options, Solomon Center Working Paper, 414 (June). Cox, J, J Ingersoll and S Ross (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407. Fama, E and KR French (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, 3–56.

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Harrison, JM and DM Kreps (1979). Martingales and arbitrage in multiperiod security markets. Journal of Economic Theory, 20, 381–408. Harrison JM and S Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11, 215–260. Ramaswamy, K and S Sundaresan (1985). The valuation of options on futures contracts. Journal of Finance, 5, 1319–1341. Vasicek, O (1977). A equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188.

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Chapter 20 NUMERICAL METHODS AND PARTIAL DIFFERENTIAL EQUATIONS FOR EUROPEAN AND AMERICAN DERIVATIVES WITH COMPLETE AND INCOMPLETE INFORMATION

Chapter Outline This chapter is organized as follows: 1. Section 20.1 presents a numerical solution to the valuation of an American call option on a dividend-paying stock. 2. Section 20.2 develops a numerical solution for the pricing of an American put option on a dividend-paying stock. 3. Section 20.3 provides some numerical procedures in the presence of information costs. An application is given for an American put option. 4. Section 20.4 presents a numerical solution to the valuation of an American convertible bond (CB) with many embedded call and put options. 5. Section 20.5 shows how to apply two-factor interest rate models in the pricing of bonds within information uncertainty. 6. Section 20.6 is devoted to CB pricing within information uncertainty. 7. Appendix A develops an improved finite difference approach to fitting the initial term structure. It applies the model in Vetzal to the valuation of European and American-style options. 8. Appendix B provides a detailed algorithm for the valuation of American calls when there are several dividends. 9. Appendix C presents a detailed algorithm for the valuation of American puts in the same context. 833

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10. Appendix D gives a detailed algorithm for the valuation of American CBs with several call and put provisions in the presence of dividends and coupon payments.

Introduction Financial economists often resort to numerical methods. They particularly use finite difference methods to solve partial differential equations (PDEs) that must be satisfied by the prices of derivative securities. Indeed, these methods are a powerful tool in option pricing when there are no closed-form solutions. The finite difference method consists in discretizing the PDE and the boundary conditions using either a forward or a backward difference approximation scheme. The resulting system is then solved iteratively. This gives the derivative asset price at each instant of time as a function of different levels of the underlying asset price. It is possible to classify the main approaches in pricing interest-rate contingent claims into two classes. The first specifies the evolution of some smaller number of points on the yield curve and incorporates time-dependent parameters into the model. This allows to meet the consistency with observed initial curve. This approach is based on the work of Black et al. (BDT) (1990), Hull and White (HW), (1990a), Jamshidian (1991), and Black and Karasinski (1991). The second approach is based on the work of Heath et al. (HJM), (1992). This method takes all the term structure as a model input and specifies the dynamics in an arbitrage framework. This method is, by definition, consistent with the current yield curve and need not augment the model with time-dependent parameters. But, this approach is hard to implement for American-style claims. Brennan and Schwartz (1977) presented a numerical solution for the valuation of American put options when there are discrete distributions to the underlying asset. The CB is a security-paying periodic coupon. It is more complex than the warrant and involves a dual option. It gives the right to the bondholder to convert the bond into common stocks and provides the issuing firm the right to call the bond for redemption. Hull and White (1990b, 1993) proposed a technique (explicit finite difference method), which is only applicable to a specific interest-rate model, in order to implement a time-dependent parameter approach. However, explicit finite difference methods are prone to stability problems unless certain conditions regarding the size of the

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Numerical Methods and PDEs for European and American Derivatives

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835

time step are satisfied. Besides, the HW method requires that the model be mean reverting after the transformation of the model to one with constant volatility. However, this condition may not be true for some parametric values and it is difficult to vary time step sizes so as to match the dates of cash flows. It is well known that implicit difference methods are quite flexible and unconditionally stable. When implementing implicit difference methods as shown in Uhrig and Walter (1996), there are no needs in transforming the model to one with a constant volatility. Vetzal (1998) proposed a discretizing strategy for mean-reverting models.

20.1. Valuation of American Calls on Dividend-Paying Stocks 20.1.1. The Schwartz model Schwartz (1977) assumed that the Black–Scholes (1973) equation applies between dividend dates: ∂C ∂C 1 2 2 ∂ 2C + rS σ S + − rC = 0 (20.1) 2 ∂S 2 ∂S ∂t He used the following boundary conditions to solve for the American call value when there are dividends: C(S, 0) = max[0, S − K]

(20.2)

C(0, t) = 0

(20.3)

C(S, T + ) = max[0, S − K, C(S − d, T − )]

(20.4)

where d stands for the dividend amount and T − and T + refer to the instants just before and just after the underlying asset goes ex-dividend. The first condition gives the call payoff at the maturity date. The second condition shows that the call is worthless when the underlying asset price is zero. The third condition indicates that the call value cannot be less than its immediate exercise value when the underlying asset goes exdividend. This condition characterizes the existence of a certain level of the underlying asset, for which the option value (with dividends) is equal to its value upon exercise. This is the critical underlying asset price corresponding to the situation where the option intrinsic value is above C(S, T + ). More generally, this situation needs the use of the following condition on the

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option’s derivative with respect to the underlying asset price: ∂C(S, τ ) =1 lim S→∞ ∂S

(20.5)

This condition must be satisfied for a sufficiently high level of the underlying asset price. 20.1.2. The numerical solution Consider a sub-division of the state variable into h equally spaced units of the underlying asset and the time variable into k units of time, or: Si = ih for i = 0 to n Tj = jk

for j = 0 to m.

The option price u(S, T ) can be written as C(S,T ) = C(Si , Tj ) = C(ih, jk). The partial derivative with respect to time, ∂C ∂t , can be approximated at the point (i, j) by the difference: ∂C [C(i, j) − C(i, j − 1)] = . ∂t k The partial derivative with respect to the asset price can be approximated at the point (i, j) by the difference: ∂C [C(i + 1, j) − C(i − 1, j)] = . ∂S 2h 2 The term ∂∂SC2 can be approximated at the (i, j) point by:

∂ 2C ∂S 2

=

[C(i + 1, j) − 2C(i, j) + C(i − 1, j)] . h2

If we replace these partial derivatives with their values in the B–S PDE, we obtain: ai C(i − 1, j) + bi C(i, j) + ci C(i + 1, j) = C(i, j − 1)

(20.6)

with ai =

1 1 rik − σ2 i2 k, 2 2

bi = 1 + σ2 i2 k + rik,

1 1 and ci = − rik − σ2 i2 k. 2 2

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Boundary condition given in Eq. (20.2) for the American call is approximated by: C(i, 0) = ih − K

if i ≥

C(i, 0) = 0 if i
K − ih, P (i, j − ) = P i − , j + h h d P (i, j − ) = K − ih for P i − , j + < K − ih. h This system can be solved by inverting the matrix to give all possible values of the option price at each instant j as a function of the values at an instant before. Appendix C provides a detailed algorithm corresponding to this model. 20.3. Numerical Procedures in the Presence of Information Costs: Applications Finite difference methods allow the valuation of derivatives by solving the differential equation numerically under the appropriate conditions. 20.3.1. Finite difference methods in the presence of information costs Consider for example, the pricing of an option on a non-dividend paying stock. The B–S differential equation in the presence of information uncertainty is written as: ∂ 2V 1 ∂V ∂V + (r + λS )S + σ2S 2 = (r + λV )V. ∂t ∂S 2 ∂S 2

(20.14)

Using a finite number of equally spaced times between the present time t and the option’s maturity date T , we have: ∆t = (TN−t) , where a total of (N + 1) times are considered from t, t + ∆t, t + 2∆t, . . . , T . In the same way, we consider a finite number of equally spaced stock prices from 0 to M , max where ∆S = SM with stock prices taking the values 0, ∆S, 2∆S, . . . , Smax . This allows a diagramatic representation for the different values of the state variable and time variable. Each point in the grid (i, j) corresponds to time

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t + i∆t and a stock price j∆S. Consider now the valuation of an option Vi,j at time i and position j. Each interior point (i, j) can be approximated by its partial derivative using a forward difference approximation: ∂V Vi,j+1 − Vij = ∂S ∆S

(20.15)

or a backward difference approximation: Vi,j − Vi,j−1 ∂V = . ∂S ∆S

(20.16)

It is also possible to use a symmetrical approximation using two space steps by averaging the two partial derivatives: Vi,j+1 − Vi,j−1 ∂V = . ∂S 2∆S

(20.17)

The time partial derivative can be approximated using a forward difference approximation to make a link between time t + i∆t and t + (i + 1)∆t: Vi+1,j − Vij ∂V = . ∂t ∆t

(20.18)

Using Eq. (20.16), the backward difference at the node (i, j + 1) is given by 2 Vi,j+1 −Vi,j . The term ∂∂ 2VS can be approximated at node (i, j) by: ∆S Vi,j+1 −Vi,j V −V − i,j ∆Si,j−1 ∆S ∂ 2V = ∂ 2S ∆S or Vi,j+1 − Vi,j−1 − 2Vi,j ∂2V = . ∂2S ∆2 S

(20.19)

Since S = j∆S, replacing Eqs. (20.17), (20.18) and (20.19) in Eq. (20.14) gives: Vi+1,j − Vij Vi,j+1 − Vi,j−1 + (r + λS )j∆S ∆t 2∆S Vi,j+1 + Vi,j−1 − 2Vij 1 = (r + λV )Vi,j + σ2 j 2 ∆S 2 2 ∆S 2 for j = 1, 2, 3, . . . , M − 1 and i = 0, 1, . . . , N − 1. This last equation can be re-written as: aj Vi,j−1 + bj Vi,j + cj Vi,j+1 = Vi+1,j

(20.20)

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where aj =

1 1 (r + λS )j∆t − σ2 j 2 ∆t, bj = 1 + σ 2 j 2 ∆t + (r + λV )∆t, 2 2 1 1 cj = − (r + λS )j∆t − σ2 j 2 ∆t. 2 2

This is the general method for the B–S PDE in the presence of information uncertainty. This method corresponds to the implicit finite difference method. The implicit method gives a relationship among three values of the option at time (t + i∆t) and a value at time (t + (i + 1)∆t). The four values are respectively, Vi,j−1 , Vi,j , Vi,j+1 , and Vi+1,j . 20.3.2. An application to the American put using explicit or implicit finite difference methods Consider now the pricing of an American put option on a non-dividend paying stock. Re-call that the put’s payoff at maturity T is max[K −j∆S, 0], where K stands for the strike price. This terminal condition can be approximated by: VN,j = max[K − j∆S, 0] for j = 0, 1, 2, . . . , M.

(20.21)

When the underlying asset price is zero, the put price corresponds to its strike price or: Vi,0 = K

for i = 0, 1, 2, . . . , N.

(20.22)

When the underlying asset price tends to infinity, the put price approaches zero or Vi,M = 0,

for i = 0, 1, 2, . . . , N.

(20.23)

Using Eq. (20.20) with i = (N −1) gives (M −1) simultaneous equations: aj VN −1,j−1 + bj VN −1,j + cj VN −1,j+1 = VN,j

(20.24)

for j varying from 1 to (M −1). Besides, since VN −1,0 = K and VN −1,M = 0, we can solve the (M − 1) Eq. in (20.24) for the (M − 1) unknowns VN −1,1 , . . . , VN −1,M−1 . To account for the possibility of an early exercise, each value of VN −1,j must be compared with the option’s intrinsic value K − j∆S.

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If the option value is less than its intrinsic value, then early exercise is optimal at time (T − 1∆t). In this case, the option value is set equal to (K − j∆S). A similar test is done at all the other nodes. Explicit versus implicit finite difference methods It is possible to approximate the partial derivatives (i, j) on the grid as follows: ∂V Vi+1,j+1 − Vi+1,j−1 = , ∂S 2∆S

∂V ∂S

and

∂2V ∂S 2

at point

Vi+1,j+1 − Vi+1,j−1 − 2Vi+1,j ∂ 2V = . ∂S 2 2∆S 2

In this case, Eq. (20.20) can be written as: Vi,j = a∗j Vi+1,j−1 + b∗j Vi+1,j + c∗j Vi+1,j+1 where a∗j

1 1 2 2 1 = − (r + λS )j∆t + σ j ∆t , 1 + (r + λV )∆t 2 2

b∗j =

1 [1 − σ2 j 2 ∆t], 1 + (r + λV )∆t

c∗j =

1 1 1 (r + λS )j∆t + σ2 j 2 ∆t . 1 + (r + λV )∆t 2 2

and

This approximation refers to the explicit finite difference method. The explicit method gives a relationship between one value of the option at time (t + i∆t) and three values at time (t + (i + 1)∆t). The four values are respectively fi,j , fi+1,j−1 , fi+1,j , and fi+1,j+1 . 20.4. Convertible Bonds 20.4.1. Specific features of CB Following Brennan and Schwartz (1977), we use the following notations: V (t): market value of the firm’s securities; W (V, t): market value of a CB with par value $1000; CP (t): call price at time t at which the bonds may be called for redemption;

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B(V, t): value of an otherwise identical bond with no conversion provision and D(t): dividend payment to the common stocks. Suppose that there are N c CB and N0 shares before conversion. We denote by q(t), the number of shares into which a bond can be converted at time t, I the coupon payments at each payment date, and i = NIc , the periodic coupon payment per bond. Since each CB can be converted into q(t) shares, the conversion value, C(V, t) is given by: C(V, τ ) =

q(t)V (t) = z(t)V (t) [N0 + Nc q(t)]

(20.25)

q(t) with z(t) = [N0 +N . Since an optimal conversion strategy implies that c q(t)] the value of the unconverted bond is at least equal to the conversion value, the following arbitrage condition must be satisfied:

W (V, t) ≥ C(V, t)

(20.26)

The bondholder has the choice at each call date to receive the call price CP (t) or the conversion value, C(V, t). Therefore, the value of the called bond VIC (V, t) must satisfy the following condition: VIC (V, t) = max[CP (t), C(V, t)].

(20.27)

Moreover, at time t = t∗ , when the bond becomes callable (because in practice bonds are not called until after a certain period), its value must satisfy W (V, t∗ ) = C(V, t∗ ) if C(V, t∗ ) ≥ CP (t∗ ) and at any time during the call period, the bond’s value cannot exceed the call price, i.e., W (V, t) ≤ CP (t).

(20.28)

20.4.2. The valuation equations The CB can be valued using the B–S PDE: 1 2 2 ∂ 2W ∂W ∂W + rV σ V + − rW = 0 2 ∂V 2 ∂V ∂t

(20.29)

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under the appropriate boundary conditions. Equation (20.30) shows that the total value of the bonds is less than the firm’s value. This is because the firm’s value is equal to that of its stocks and bonds. Nc W (V, t) ≤ V.

(20.30)

Equation (20.31) indicates that the bond value is zero when the firm is worthless. W (0, t) = 0.

(20.31)

Equation (20.32) shows that the CB value is less than the value of an equivalent straight bond and the maximum number of shares in which it can be converted: W (V, t) ≤ B(V, t) + z ∗ (t)V

(20.32)

where z ∗ (t) is the maximum value of z(τ ) for t ∈ (t, T ). Equation (20.33) illustrates the option offered to the bondholder for conversion. W (V, t) > C(V, t) = z(t)V.

(20.33)

Equation (20.34) indicates the pay off of the CB at the maturity date. W (V, T ) = z(T )V, W (V, T ) = 1000, W (V, T ) =

z(T )V ≥ 1000

1000Nc ≤ V ≤

V , Nc

1000 z(T )

(20.34)

V ≤ 1000Nc.

Equation (20.35) shows the constraint on the call price during the call period. It indicates that the CB price cannot exceed the call price, otherwise the issuer will call back the bonds, W (V, t) ≤ CP (t).

(20.35)

Equation (20.36) is a high-contact condition, which applies for a sufficiently high value of V . ∂W (V, t) = z(t). (20.36) lim S→∞ ∂V

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Equation (20.37) must be applied at each dividend date where the instants just before and just after are denoted respectively by t− and t+ . W (V, t− ) = max[W (V − D, t+ ), z(t− )V ].

(20.37)

Equation (20.38) must be applied at each coupon date when the bond is not currently callable. The instants just before and just after are denoted also by t− and t+ , respectively. W (V, t− ) = W (V − I, t+ ) + i

(20.38)

Equation (20.39) must be applied at each coupon date when the bond is currently callable. W (V, t− ) = min[W (V − I, t+ ) + i, CP (t− )]

(20.39)

Brennan and Schwartz (1977) presented a numerical solution for the valuation of American CBs when there are discrete distributions to the underlying asset and call and put provisions. 20.4.3. The numerical solution Since there is no closed-form solution to the B–S differential equation under the above boundary conditions, numerical methods are useful in solving such problems. Using a time variable τ = T − t, instead of the calendar time t, the discretization of the underlying asset price and the time to maturity is: Vi = ih

for i = 0 to n

τj = jk

for j = 0 to m.

The CB is a solution to the B–S PDE which is approximated by: ai W (i − 1, j) + bi W (i, j) + ci W (i + 1, j) = W (i, j − 1)

(20.40)

with ai =

1 1 rik − σ2 i2 k, 2 2

1 1 ci = − rik − σ2 i2 k 2 2 for i = 1 to n − 1 and j = 1 to m.

bi = 1 + σ2 i2 k + rk,

For a given j, the system given in Eq. (20.40) gives (n − 1) equations with (n + 1) unknowns. (when i varies from 0 to n).

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Equation (20.31) is written as: W (0, j) = 0

for j = 0 to m

The discretization of Eq. (20.34) gives: W (i, 0) = z(0)V = zhi for zhi ≥ P P W (i, 0) = P for P ≤ hi ≤ z W (i, 0) = V = hi for hi ≤ P where P stands for the par value of the CB. Equation (20.36) is approximated by: W (n, j) − W (n − i, j) = z. h

(20.41)

Using the above system and Eq. (20.41), the values of W (i, j) can be determined in a recursive manner from W (i, j − 1) since all the W (i, 0) are given for all values of i. At a call date, Eq. (20.36) is replaced by Eq. (20.35), or W (i, j) ≤ CP (j).

(20.42)

Since the bond can be called before the maturity date, the value of W (i, j) is not defined for a certain i greater than a certain value q, given by q = CP(j) zh . On a dividend date, Eq. (20.37) is approximated to: D D W i − , jD = zih if W i − , jD ≥ zV h h D W (i, jD ) = zih if W i − , jD ≤ zih. h On a coupon date, Eq. (20.38) is approximated to: I W (i, jc ) = W i − , jc + I h

(20.43)

and Eq. (20.39) is approximated to I I W (i, jc ) = W i − , jc + I for W i − , jc + I ≤ CP (jc ) h h I W (i, jc ) = CP (jc ) for W i − , jc + I > CP (jc ) h A detailed algorithm is given in Appendix D.

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20.4.4. Simulations Using the algorithm in Appendix D, the CB price is simulated with the following data: Par value of the bond: 40, semi-annual coupon: 1.1, quaterly dividend: 1.1, firm variance rate: 0.0012 per month, Risk-free rate: 0.0057 per month. The bond is not callable for 5 years; it is callable at 43 (plus accrued interest) for the next 5 years, at 42 for the next 5 years and at 41 for the last 5 years. max = 2.5, (T − t) = 240 months and a time Using 200 iterations, h = V200 step of one month, Table 20.1 provides the prices for different levels of the underlying asset V . 20.5. Two-Factor Interest Rate Models and Bond Pricing within Information Uncertainty We denote at time t by Z(r, l, t; T ) the price of a zero-coupon bond as a function of the spot interest rate r, the long interest rate l, and the maturity date T . The dynamics of the spot rate are specified by: dr = udt + wdX1 , where the parameters u and w can depend on r and t. The dynamics of the long rate are given by dl = pdt + qdX2 , where the parameters p and q can depend on l and t. Consider a portfolio with a long position in Z(r, l, t; T ) and two short positions in two zero-coupon bonds with different maturities T1 and T2 . The initial portfolio value is given by: Π = Z(r, l, t; T ) − ∆1 Z(r, l, t; T1 ) − ∆2 Z(r, l, t; T2 ). The change in this portfolio’s value can be written as: ∂Z ∂Z1 ∂Z2 (H(Z) − ∆1 H(Z1 ) − ∆2 H(Z2 ))dt + − ∆1 − ∆2 dr ∂r ∂r ∂r ∂Z ∂Z1 ∂Z2 − ∆1 − ∆2 dl (20.44) + ∂l ∂l ∂l where: H(Z) =

1 ∂2Z 1 ∂ 2Z ∂Z ∂ 2Z + w2 2 + ρwq + q2 2 ∂t 2 ∂r ∂r∂l 2 ∂l ∂Z ∂Z1 ∂Z2 − ∆1 − ∆2 = 0. ∂r ∂r ∂r

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848 Table 20.1. u(0) u(4) u(8) u(12) u(16) u(20) u(24) u(28) u(32) u(37) u(41) u(45) u(49) u(53) u(57) u(61) u(65) u(69) u(73) u(77) u(81) u(85) u(89) u(93) u(97) u(101) u(106) u(109) u(113) u(117) u(121) u(125) u(129) u(133) u(137) u(141) u(145) u(149) u(152) u(156) u(160) u(164) u(168) u(172) u(176) u(180) u(184) u(188) u(192) u(196)

spi-b708

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0 4.740000 7.990000 11.230000 14.480000 17.940000 22.730000 29.400000 34.520000 36.160000 36.215000 36.215875 36.215875 36.215875 36.215875 36.215876 36.215890 36.216100 36.217900 36.227500 36.260000 36.360000 36.540000 36.790000 37.070000 37.350000 37.740000 38.000000 38.360000 38.690000 38.980000 39.260000 39.570000 39.930000 40.340000 40.760000 41.120000 41.420000 41.640000 41.920000 42.200000 42.490000 42.780000 42.960000 43.990000 44.990000 46.000000 47.000000 48.000000 49.000000

Prices for different levels of the underlying asset V . u(1) u(5) u(9) u(13) u(17) u(21) u(25) u(29) u(33) u(38) u(42) u(46) u(50) u(54) u(58) u(62) u(66) u(70) u(74) u(78) u(82) u(86) u(90) u(94) u(98) u(102) u(107) u(110) u(114) u(118) u(122) u(126) u(130) u(134) u(138) u(142) u(146) u(150) u(153) u(157) u(161) u(165) u(169) u(173) u(177) u(181) u(185) u(189) u(193) u(197)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

1.960000 5.560000 8.800000 12.040000 15.300000 18.950000 24.290000 31.000000 35.190000 36.190000 36.215800 36.215875 36.215875 36.215875 36.215875 36.215878 36.215916 36.216300 36.219100 36.233000 36.280000 36.390000 36.590000 36.860000 37.140000 37.430000 37.830000 38.090000 38.440000 38.760000 39.050000 39.330000 39.650000 40.030000 40.450000 40.860000 41.200000 41.500000 41.710000 41.990000 42.270000 42.560000 42.790000 43.260000 44.250000 45.250000 46.250000 47.250000 48.250000 49.250000

u(2) u(6) u(10) u(14) u(18) u(22) u(26) u(30) u(34) u(39) u(43) u(47) u(51) u(55) u(59) u(63) u(67) u(71) u(75) u(79) u(83) u(87) u(91) u(95) u(99) u(104) u(108) u(111) u(115) u(119) u(123) u(127) u(131) u(135) u(139) u(143) u(147) u(151) u(154) u(158) u(162) u(166) u(170) u(174) u(178) u(182) u(186) u(190) u(194) u(198)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

3.230000 6.370000 9.610000 12.850000 16.140000 20.070000 25.960000 32.420000 35.640000 36.210000 36.215800 36.215875 36.215875 36.215875 36.215760 36.215870 36.215951 36.216600 36.220900 36.240600 36.300000 36.440000 36.660000 36.920000 37.210000 37.580000 37.920000 38.180000 38.530000 38.840000 39.120000 39.410000 39.740000 40.130000 40.560000 40.950000 41.280000 41.570000 41.780000 42.060000 42.340000 42.630000 42.810000 43.490000 44.490000 45.490000 46.500000 47.500000 48.500000 49.500000

u(3) u(7) u(11) u(15) u(19) u(23) u(27) u(31) u(36) u(40) u(44) u(48) u(52) u(56) u(60) u(64) u(68) u(72) u(76) u(80) u(84) u(88) u(92) u(96) u(100) u(105) u(109) u(112) u(116) u(120) u(124) u(128) u(132) u(136) u(140) u(144) u(148) u(152) u(155) u(159) u(163) u(167) u(171) u(175) u(179) u(183) u(187) u(191) u(195) u(199)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

3.910000 7.180000 10.420000 13.660000 17.020000 21.330000 27.690000 33.600000 36.080000 36.214000 36.215870 36.215875 36.215875 36.215875 36.215876 36.215880 36.216000 36.217100 36.223600 36.250000 36.320000 36.480000 36.720000 36.990000 37.280000 37.660000 38.000000 38.270000 38.610000 38.910000 39.190000 39.490000 39.830000 40.240000 40.660000 41.040000 41.350000 41.640000 41.850000 42.130000 42.410000 42.710000 42.850000 43.750000 44.750000 45.750000 46.750000 47.750000 48.750000 49.750000

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It is possible to construct a hedged portfolio using the appropriate ∆1 and ∆2 that allow the coefficients of dr and dl to be zero in Eq. (20.44). This allows the construction of a risk-free portfolio which earns the risk-free rate and the return corresponding to information costs. This leads to the following system of three equations: ∂Z ∂Z1 ∂Z2 − ∆1 − ∆2 =0 ∂l ∂l ∂l H (Z) − ∆1 H (Z1 ) − ∆2 L (Z2 ) = 0 with H (Z) = H(Z) − (r + λZ )Z, where λZ corresponds to the shadow cost of incomplete information corresponding to the bond market. Using a matrix notation, this system can be written as:   H (Z) H (Z1 ) H (Z2 )   M = ∂Z/∂r ∂Z1 /∂r ∂Z2 /∂r ∂Z/∂l

∂Z1 /∂l

∂Z2 /∂l

with the additional condition that det(M) = 0. Using this matrix M, it is possible to write: ∂Z ∂Z + (γl q − p) . ∂r ∂l The standard arbitrage arguments allow the derivation of the following equation for the bond price: H (Z) = (γr w − u)

∂ 2Z ∂Z ∂Z 1 ∂ 2Z 1 ∂2Z + w2 2 + ρwq + q 2 2 + (u − γr w) ∂t 2 ∂r ∂r∂l 2 ∂l ∂r ∂Z − (r + λZ )Z = 0 (20.45) + (p − γl q) ∂l where γr and γl correspond respectively to the market prices of risk. The long-term interest rate refers to the yield on a consol bond, i.e., a fixed coupon-paying bond for which the maturity is infinite. The yield on a consol bond paying a coupon of $1 each year until infinity is given by l = C1o . In this context, the pricing equation becomes: H (Co ) + 1 = (γr w − u)

∂Co ∂Co + (γl q − p) ∂r ∂l

This equation is similar to Eq. (20.45), where the additional term 1 corresponds to the coupon payments. Replacing 1l in this equation gives: p − γl q = l2 − (r + λl )l +

q2 . l2

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This expression of the market price of risk is obtained because the consol bond corresponds to a tradable security. In the presence of two state variables; the spot rate and the consol bond, the valuation equation can be written as: ∂Z q 2 ∂Z 2 − l − (r + λl )l + 2 H (Z) = (γr w − u) ∂r l ∂l or ∂2Z ∂Z ∂Z 1 ∂ 2Z 1 ∂ 2Z + w2 2 + ρwq + q 2 2 + (u − γr w) ∂t 2 ∂r ∂r∂l 2 ∂l ∂r 2 q ∂Z − (r + λZ )Z = 0. + l2 − (r + λl )l + 2 l ∂l 20.6. CBs Pricing within Information Uncertainty 20.6.1. The pricing of CBs This analysis assumes that the coupon is known. Since the CB price is a function of the underlying asset price S and time t, it can be written as V = V (S, t). Consider a portfolio with a long position in the CB and a short position in δ units of the underlying asset. Using hedging arguments, it is possible to show that the change in the portfolio’s value over a short-time interval is: dΠ =

∂V 1 ∂V ∂V ∂ 2V dt + dS + σ 2 S 2 dS. dt − ∂t ∂S 2 ∂S 2 ∂S

Risk can be eliminated from the portfolio by choosing ∆ = ∂V . As before, ∂S the return on the risk-free portfolio must be equal to the risk-less rate plus information costs paid by the investor in both markets. This gives: 1 ∂V ∂V ∂ 2V + σ2 S 2 2 + (rS − d(S, t)) − rV ≤ 0. ∂t 2 ∂S ∂S

(20.46)

The maturity condition corresponds to the value of the CB at this date, which is scaled to one: V (S, T ) = 1. At each coupon date, the following condition must be satisfied: + V (S, t− c ) = V (S, tc ) + C.

Since the bond can be converted into q shares of the issuer, its value must be higher than V ≥ qS. The bond’s value just before maturity is given

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by max(nS, 1). Besides, for high values of the underlying asset, i.e., when S → ∞, the CB value approaches the conversion value, or V (S, t) ∼ nS. When the underlying asset price tends to zero, the CB price corresponds to the present value of the future coupons and principal: V (0, t) = e−(r+λv )(T −t) + Ce−(r+λv )(tc −t) . 20.6.2. Specific call and put features The provision that allows the bond issuer to purchase back the bond at a specified call price corresponds to a call feature. The provision that allows the bond holder to return the bond to the issuing company corresponds to a put feature. In the pricing of callable and puttable corporate CBs, the following two conditions must be imposed V (S, t) ≤ CP and V (S, t) ≥ PP , where CP refers to the call price and PP to the put price. 20.6.3. The pricing of CBs in two-factor models within information uncertainty In the presence of stochastic interest rates, the bond price can be written as a function of the underlying asset price, interest rates, and time as V = V (S, r, t). In general, the following dynamics are used for the underlying asset: dS = µSdt + σSdX1 and interest rates: dr = u(r, t)dt + w(r, t)dX2 where the two Wiener processes are correlated by ρ(r, S, t) with: E[dX1 dX2 ] = ρdt. Using Ito’s lemma, it is possible to show that the dynamics of the CB price are given by: dV =

∂V ∂V ∂V dt + dS + dr ∂t ∂S ∂r 2 2 1 ∂ 2V 2 2∂ V 2∂ V dt. + σ S +w + 2ρσSw 2 ∂S 2 ∂S∂r ∂r2

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In this context, the pricing of the CB needs the construction of a portfolio with a long position in the CB and two short positions in the underlying asset and zero-coupon bonds. The short side justifies the use of two deltas because of the two risks linked to the underlying asset price and the interest rate risk Π = V − ∆1 S − ∆2 Z. The appropriate choice of the hedging ratios allows the elimination of risk from this portfolio with ∂V ∂V ∂Z and ∆2 = . ∆1 = ∂S ∂r ∂r Putting together the terms in T1 and T2 , re-arranging and imposing the condition that the return from a hedged portfolio must be equal to the risk-less rate plus information costs on the corresponding markets gives the following equation: 1 1 2 ∂2V ∂V ∂2V ∂V ∂2V + σ2 S 2 + w + ρσSw + (r + λS )S 2 2 ∂t 2 ∂S ∂S∂r 2 ∂r ∂S ∂V + (u − γw) − (r + λV )V = 0 ∂r The presence of information costs is justified by the costs suffered by investors to implement arbitrage because arbitrage is not costless. In fact, not all investors can implement risk-less arbitrage. This is only possible for investors who are informed about the presence of arbitrage opportunities. When the interest rate is constant, this equation reduces to the extended B–S equation. Summary This chapter introduces the reader to the application of finite difference methods to the pricing of American options. First, the numerical methods are applied to the valuation of American call options when there are several dividends. Second, the numerical methods are applied to the valuation of American puts in the same context. Note that in both cases, there are no analytical solutions in the literature. Third, we apply numerical methods for the valuation of options in the presence of incomplete information. Fourth, the numerical methods are illustrated for the valuation of American CBs when there are several dividend dates, coupon dates, and implicit call and put options. A CB on a stock gives its holder the right to receive coupons at regular intervals and to get the principal at the bond’s maturity date. Besides, the holder has the right to convert his/her bond into a specified

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number of shares. Hence, the bond price must lay between its conversion value and its straight value. The conversion value corresponds to the market price of stock times the conversion ratio. Corporate bonds (CBs) correspond to corporate securities that give the holder the right to forgo coupon/or principal payments and convert to a pre-determined number of shares of common stocks instead. In its simpler form, a CB can be viewed as a hybrid security consisting of a straight bond and a call on the underlying equity. Bond issues contain several optionality features like the possibility of early conversion, callability by the issuer, putability by the holder etc. The pricing of CBs needs in general, a simultaneous pricing of the equity and fixedincome components. Several approaches have been proposed to account for default risk in CB pricing. Practitioners account for credit risk in CBs by introducing an effective credit spread in CB valuation tools. These spreads are simple approximations based on the credit spread of a straight bond conditioned for the hybrid nature of the CB. The Hull and White (1990b) trinomial model uses an explicit method and is prone to stability problems. To circumvent these concerns, Vetzal (1998) developed a simple two-point upstream technique in the presence of an implicit scheme. The method can be introduced after reviewing the Hull and White (1990b) trinomial model and standard finite approaches. In each case, a detailed algorithm is given in the appendix to illustrate the determination of the critical levels of the underlying asset price corresponding to an optimal pre-mature exercise. Fifth, we study the pricing of bonds in a two-factor interest rate model within information uncertainty. Sixth, we study the valuation of CBs within information uncertainty.

Appendix A: A Discretizing Strategy for Mean-Reverting Models A simple two-point upstream technique in the presence of an implicit scheme Vetzal (1998) illustrated the methods using the one-factor Vasicek (1977) model. dr = κ(θ − r)dt + σdB with k: the speed of adjustment, and θ: reversion level.

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When the market price for interest-rate risk is zero, the price of an interest rate-contingent claim u satisfies the following PDE: 1 2 σ urr + κ(θ − r)ur + ut − ru = 0 2

(A.1)

The price of a pure discount bond, u(r, t, T ) maturing in T can be found by solving Eq. (A.1) under the terminal condition u(r, t, T ) = 1. The solution can be found by constructing a grid for values of r over some interval [rmin , rmax ] with an M evenly spaced points r1 , r2 , . . . , rM . The distance between these points is ∆r. If we denote by uni , the value of u at the ith grid point at time step n, then a central weighted explicit scheme can be used to approximate Eq. (20.51) as follows: urr =

n+1 + un+1 un+1 i+1 − 2ui i−1 , (∆r)2

ut =

ur =

n+1 un+1 i+1 − ui−1 , 2∆r

− uni un+1 i ∆t

(A.2)

where ∆t corresponds to the length of the time step between n and n + 1. The ru term in Eq. (A.1) is sometimes denoted by ri uni . Substituting in Eq. (A.1) and re-arranging gives: n+1 + pi,i+1 un+1 uni (1 + r∆t) = pi,i−1 un+1 i−1 + pi,i ui i+1

(A.3)

where pi,i−1 =

1 σ 2 ∆t κ(θ − r)∆t σ 2 ∆t , pi,i = 1 − − , 2 2 (∆r) ∆r (∆r)2 1 σ 2 ∆t κ(θ − r)∆t − pi,i+1 = 2 (∆r)2 ∆r

and (A.4)

When the ps are positive and sum to one, this explicit scheme is stable. However, it is not easy to ensure that the ps are all positive. In fact, if r is very low, the upward drift can be strong enough to lead to negative pi, i−1 . When r is very high, the downward drift can lead to negative probabilities pi, i+1 . This leads to the trinomial approach. When the spatial derivatives ur,r and ur are approximated at time n rather than n + 1, this gives the

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following implicit numerical scheme: 1 σ 2 ∆t κ(θ − r)∆t − Ci,i−1 = − 2 (∆r)2 ∆r where Ci,i = 1 +

σ 2 ∆t + ri ∆t, (∆r)2

Ci,i+1 = −

1 σ 2 ∆t κ(θ − r)∆t . − 2 (∆r)2 ∆r

855

(A.5)

(A.6)

It is well known that the implicit scheme is unconditionally stable since there is no constraint on the Cs to be positive. The system is tridiagonal and can be easily solved. The basic idea in the modified trinomial HW approach can be presented as follows. When the grid point of r closest to its expected value after another time step corresponds to the current grid point, i.e., if the absolute value of the expected change in the interest rate over a small interval of time is less than ∆r 2 , a trinomial lattice can be constructed by solving the following system for the ps: pˆ1,i−1 (−∆r) + pî,i (0) + pî,i+1 (∆r) = κ(θ − r)∆t pˆ1,i−1 (−∆r)2 + pî,i (0)2 + pî,i+1 (∆r)2 = σ2 ∆t + [κ(θ − r)∆t]2 pî,i−1 + pî,i + pî,i+1 = 1

(A.7)

In this system, the first and second lines try to match the first and second moments of the change in the state variable r over the small interval of time. The term k(θ − r)∆t corresponds to the expected mean. The term σ 2 ∆t + [k(θ − r)∆t]2 corresponds to the second moment. The third line shows that the probabilities sum to one. This is a system of three equations and with three unknowns which can be solved by very simple methods. If the term [k(θ − r)∆t]2 is dropped, the solution to Eq. (A.7) is equal to the ps in Eq. (A.4). This means that the second moment is approximated by the variance and the error involved is of order (∆t)2 . For very low values of the interest rate, rl , the branching in the lattice can be modified so that r remains constant, goes up by ∆r or by 2∆r. This truncates the computation domain since r will not be less than rl . Using a set of equations analogous to Eq. (A.7), Vetzal (1998) showed that the probabilities that match the first two moments are given by the following system: pˆ1,1 (0) + pˆ1,2 (∆r) + pˆ1,3 (2∆r) = κ(θ − r)∆t 2

pˆ1,1 (0) + pˆ1,2 (∆r)2 + pˆ1,3 (2∆r)2 = σ 2 ∆t + [κ(θ − r)∆t]2 pˆ1,1 + pˆ1,2 + pˆ1,3 = 1.

(A.8)

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In the same way, when the interest become very high at a level, rM , the lattice can be modified in order to constraint r to that level, or to decrease it by ∆r or 2∆r. In this case, the following system is used at this level to compute the probabilities: pˆM,M−2 (−2∆r) + pˆM,M−1 (−∆r) + pˆM,M (0) = κ(θ − r)∆t pˆM,M−2 (−2∆r)2 + pˆM,M−1 (−∆r)2 + pˆM,M (0)2 = σ2 ∆ + κ(θ − r)∆t pˆM,M−2 + pˆM,M−1 + pˆM,M = 1.

(A.9)

In sum, the Hull and White (1990b, 1993, 1994a,b, 1996) methods determine the risk-neutral probabilities so as to match the first moments of the change in the interest rate over the time interval ∆t. Their methods can account for additional state variables and match initial yield curves and term structures of interest-rate volatilities using time-dependent parameters. However, their approach is based on an explicit method and the advantages of implicit methods are known. Vetzal (1998) considered a grid for the state variable from r1 to rM , uses a standard method for all the interior points from r2 to uM−1 and applied the following approach for the end points. For r1 , two Taylor expansions around u(r1 ) are considered. The first is for u(r1 + ∆r). The second is for u(r1 + 2∆r): u(r1 + ∆r) = u(r1 ) + ur ∆r + urr u(r1 + 2∆r) = u(r1 ) + ur 2∆r + urr

∆r2 2

(2∆r2 ) . 2

(A.10)

Equation (A.11) can be re-written as: urr = ur =

u(r1 + 2∆r) − 2u(r1 + ∆r) + u(r1 ) (∆r)2

−u(r1 + 2∆r) + 4u(r1 + ∆r) − 3u(r1 ) . 2∆r

(A.11)

At the level rM , two second-order expansions are used for u(rM − ∆r) and u(rM − 2∆r) as follows: u(rM − ∆r) = u(rM ) − ur ∆r + urr

(∆r)2 2

u(rM − 2∆r) = u(rM ) − ur 2∆r + urr

(2∆r)2 2

(A.12)

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Equation (A.12) can be re-written as: urr = ur =

u(rM − 2∆r) − 2u(rM − ∆r) + u(rM ) (∆r)2

−u(rM − 2∆r) − 4u(rM − ∆r) + 3u(rM ) . 2∆r

(A.13)

This analysis implies that an explicit method can be used as follows. For the interior points r2 , . . . , rM−1 , the explicit scheme in Eqs. (A.3) and (A.4) can be used. For the minimal level of r, the discrete analogs of the spatial derivatives in Eq. (A.10) at time n + 1 is substituted into the first PDE. This gives the following system: + p1,2 un+1 + p1,3 un+1 un1 (1 + r1 ∆t) = p1,1 un+1 1 2 3

(A.14)

where the probabilities are given by: 1 σ2 ∆t 3k(θ − r1 )∆t p1,1 = 1 + − 2 (∆r)2 ∆r σ 2 ∆t 2k(θ − r1 )∆t + 2 (∆r) ∆r 2 1 σ ∆t 3k(θ − r1 )∆t . = − 2 (∆r)2 ∆r

p1,2 = p1,3

(A.15)

In the same way, at the maximum value of r, the discrete analogs of Eq. (A.12) at time n+1 for the spatial derivatives is substituted in Eq. (A.1) to obtain: n+1 n+1 unM (1 + rM ∆t) = pM,M−2 un+1 M−2 + pM,M−1 uM−1 + pM,M uM

where the probabilities are given by: 1 σ2 ∆t κ(θ − rM )∆t pM,M−2 = , − 2 (∆r)2 ∆r 1 σ2 ∆t 2κ(θ − rM )∆t , pM,M−1 = − − 2 (∆r)2 ∆r 1 σ 2 ∆t 3κ(θ − rM )∆t . pM,M = 1 + − 2 (∆r)2 ∆r

(A.16)

(A.17)

The proposed approach in Vetzal (1998) becomes identical to that of HW when the time interval tends to reach zero. Using Taylor’s series, which allows the construction of an implicit scheme by evaluating the derivatives

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in Eqs. (A.11) and (A.13) at time step n rather than (n + 1). Hence, at the minimum value of the interest rate, we have: c1,1 un1 + c1,2 un2 + c1,3 un3 = un+1 1 where: c1,1

(A.18)

1 σ2 ∆t 3κ(θ − r1 )∆t , = 1 + r1 ∆t − − 2 (∆r)2 ∆r σ 2 ∆t 2κ(θ − r1 )∆t , − 2 (∆r) ∆r 1 σ2 ∆t κ(θ − r1 )∆t =− − 2 (∆r)2 ∆r

c1,2 = c1,3

(A.19)

The main advantage in the discretization in Vetzal (1998) is that this approach does not impose a specific boundary condition at r1 . The classic implicit numerical scheme given by Eqs. (A.5) and (A.6) is used for the interior points and at the maximum value of r: cM,M−2 unM−2 + cM,M−1 unM−1 + cM,M unM = un+1 M where: cM,M−2

(A.20)

1 σ 2 ∆t κ(θ − rM )∆t , =− + 2 (∆r)2 ∆r

σ 2 ∆t 2k(θ − rM )∆t − , (∆r)2 ∆r 1 σ2 ∆t 3κ(θ − rM )∆t . = 1 + rM ∆t − − 2 (∆r)2 ∆r cM,M−1 =

cM,M

This implicit scheme does not require a specific boundary condition at the end point. This implicit scheme is quite flexible with regard to grid construction. The implicit and the Crank–Nicolson versions of the scheme are flexible with respect to the choice of the time. This can be done in such a way to match cash-flow dates exactly near the option’s maturity date. This flexibility does not exist in the HW models. This method shares with the HW, the feature of avoiding the specification of boundary conditions for models with mean reversion, but it can be applied to several situations where the HW method cannot. The simulations conducted by Vetzal (1998) show that the Crank–Nicolson scheme is much more accurate than the HW method. For example, the maximum absolute error for the discount

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function of the final grid values is almost 20 times smaller than the HW method. Fitting the initial term structure Black et al. (1990), Black and Karasinski (1991), Jamshidian (1991), and Hull and White (1993, 1994a,b, 1996) among others proposed the use of a time-dependent parameter in the calibration of interest-rate models to a given term structure. Their models are inappropriate for implicit schemes or the Crank–Nicolson scheme. European-style securities can be valued using an adjustment proposed in Theorem 1 in Dybvig (1989). Valuation of American-style options Consider the following dynamics for the spot risk-free rate: dr = κ(θ − r)dt + σrβ dB

(A.21)

When β = 0, this process corresponds to Vasicek (1977) model. When β = 1, this process corresponds to Courtadon (1982) model. When β = 12 , this process is the Cox et al. (1985) model. In this context, the option price must satisfy the following PDE: 1 2 2β σ r urr + [k(θ − r) − Φ(r)σrβ ]ur + ut − ru = 0 2

(A.22)

where Φ(r) is the market price of interest-rate risk. The pricing of contingent claims is based on the choice of a given Φ(r) or θ as a function of time. Vetzal (1998) used the following PDE for the pricing of contingent claims: 1 2 2β σ r urr + [κ(θ − r) − Φ(r)σrβ ]ur + ut − [r + rb (t)]u = 0 2 −1 n−1 −1 n = A−1 un−2 = A−1 n Bn An Bn u . n−1 Bn−1 An−1 Bn−1 u

(A.23)

In a Crank–Nicolson numerical scheme, the solution un at time n is obtained by solving a set of linear equations of the form: An un−1 = Bn un where A, B are square matrices of size M .

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In the same way: −1 n−1 −1 n un−2 = A−1 = A−1 n−1 Bn−1 An−1 Bn−1 u n−1 Bn−1 An Bn u .

(A.24)

This method is much faster than that proposed in Uhrig and Walter (1996). Numerical results Vetzal (1998) used three one-factor models corresponding to Eqs. (20.21) and (20.73) with respectively, β = 0, β = 12 and β = 1. He also used three EGARCH stochastic volatility extensions of these univariate models as shown in Anderson and Lund (1997) and Vetzal (1997): (1) dr κ(θ − r) σrβ dBt (A.25) = dt + (1) (2) d ln σ α + δ ln σ γ(ρdBt + 1 + ρ2 dBt ) where k, α, θ, δ, γ, and ρ are parameters. Contingent claim prices must satisfy the following PDE: 1 2 2β [σ r urr + 2ργσrβ urv + γ 2 uvv ] + [κ(θ − r]ur 2 + [α + δv]uv + ut − [r + rb (t)]u = 0

(A.26)

where v = ln σ. Hull and White (1993), Uhrig and Walter (1996), and Vetzal (1997) showed that single-factor models give similar values for European bond options, except for deep-out-of-the-money options. For European-style options, the adjustment in Dybvig (1989) can be used. The sample problem involves the valuation of a 5-year option on a 10-year coupon bond using weekly time steps. The Crank–Nicolson method is used. The results seem to be similar for single-factor models. Stochastic volatility models produce somewhat higher values for deep-outof-the-money options. The impact of stochastic volatility depends on the parameter, β. Vetzal (1998) proposed a two-point upstream discretization strategy for mean-reverting models. This method improves on the HW trinomial lattice in several ways. Vetzal (1998) proposed an implicit scheme and also the Crank–Nicolson scheme which offer superior stability and time-step flexibility. He also derived European option values consistent with an initial term structure along the lines of Dybvig (1989) in a PDE context.

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Appendix B: An Algorithm for the American Call with Dividends For each level Si = ih of the underlying asset, the terminal boundary condition is written as: For i = 0 to n, C(i, 0) = ih − K if ih − K < 0, then C(i, 0) = 0, End (for i = 0 to n) When the asset price is zero, the option is worthless. For j = 1 to m, C(0, j) = 0, End. The following system corresponds to the inversion of a tri-diagonal matrix by the Gauss method. For j = 1 to m, a(1) = 0, b(1) = 1 + σ 2 k + rk, c(1) = − 21 rk − 12 σ 2 k, d(1) = C(1, j − 1), For i = 2 to n − 1 a(i) = 12 rik − 12 σ 2 i2 k, b(i) = 1 + σ2 i2 k + rik, c(i) = − 21 rik − 12 σ 2 i2 k, d(i) = C(i, j − 1) End, a(n) = −1, b(n) = 1, c(n) = 0, d(n) = h For i = 1 to n do w(i) = g(i) =

c(i) , (b(i)−w(i−1)a(i)) (d(i)−g(i−1)a(i)) (b(i)−w(i−1)a(i))

End, C(n, j) = g(n). Here, the last elements in the system are calculated. We generate and print all the unknowns at each instant of time using the Gauss method for the inverted matrix. For i = n − 1 down to 1, C(i, j) = g(i) − w(i)u(i + 1, j), End For i = 0 to n write C(i, j).

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A special treatment is done for the determination of the critical asset price corresponding to an optimal early exercise. For i = 0 to n do H(i) = C(i, j) + K − ih end k = 0, For i = 1 to n do if h(i) ≤ 0 then C(i, j) = ih − K, k = k + 1 end if if k = 1 then S ∗ = (i − 1)h + hH(i − 1)/H(i − 1) − H(i) end if End (for i = 1 to n). At a dividend date, the following treatment is done. If j is in J2 then k = int(d/h) where int(.) corresponds to the integer part of a number. For i = 0 to k − 1, v(i) = 0 end for For i = k to n if C(i − k, j) < (ih − K) then v(i) = ih − K else v(i) = C(i − k, j) end if end for For i = 0 to n do C(i, j) = v(i) end for For i = 0 to n write C(i, j) end for, end if End.

Appendix C: The Algorithm for the American Put with Dividends For each level ih of the underlying asset, the terminal boundary condition is written as: For i = 0 to n, P (i, 0) = K − ih if K − ih < 0 then P (i, 0) = 0, End (for i = 0 to n) When the asset price is zero, the put value is equal to the strike price. For j = 1 to m, P (0, j) = K, End. For each time step, the system must be solved using for example, the Gauss method. For j = 1 to m, a(1) = 0, b(1) = 1 + σ 2 k + rk, c(1) = − 12 rk − 12 σ 2 k, and d(1) = P (1, j − 1)

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For i = 2 to n − 1 do a(i) = 12 rik − 12 σ 2 i2 k;

b(i) = 1 + σ 2 i2 k + rk;

c(i) = − 21 rik − 12 σ 2 i2 k, and d(i) = P (i, j − 1). End for (i = 2 to n − 1): a(n) = −1, b(n) = −1, c(n) = 0, d(n) = 0 For i = 1 to n do: w(i) = c(i)/(b(i) − w(i − 1)a(i)), g(i) = (d(i) − g(i − 1)a(i))/(b(i) − w(i − 1)a(i)), End for (i = 1 to n) P (n, j) = g(n). Here, the last elements in the system are calculated. We generate all the unknowns at each instant of time. For i = n − 1 down to 1, P (i, j) = g(i) − w(i)u(i + 1, j), End (for i = n − 1 down to 1). For i = 0 to n write P (i, j) end for. A separate treatment is done for the determination of the critical asset price corresponding to an early exercise. For i = 0 to n do H(i) = P (i, j) − K + ih end for k = 0, For i = 1 to n do if H(i) ≤ 0 then P (i, j) = K − ih, k = k + 1 end if if k = 1 then Sc = (i − 1)h + hH(i − 1)/H(i − 1) − H(i) end if, write j, Sc End (for i = 1 to n). For a dividend date, the following treatment is done. If j is in J1, then k = int(d/h) where int(.) corresponds to the integer part of a number.

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For i = 0 to k − 1, v(i) = 0 end for For i = k to n if P (i − k, j) < (K − ih) then v(i) = K − ih else v(i) = P (i − k, j) end if end for. For i = 0 to n do P (i, j) = v(i) end for For i = 0 to n write P (i, j) end for, end if end for, End. Appendix D: The Algorithm for CBs with Call and Put Prices To run the program, you enter V max, the bond price P , the volatility σ, the interest rate r, the number of months until the maturity date, nm, the maximum number of steps for the underlying asset, nV , the number of periods where z changes, np1, the number of periods where the call price changes, np2, the amounts of dividends, dv, the dates of dividends, dd, the coupon amounts, Ic, the coupon dates, dc, the call price vector, CP (k), and the length of the call period, d2(k). Initialization For i = 1 to np1, enter d1(i) the length of period i in months, z(i), For k = 1 to np2, enter d2(k), CP (k)(CP (0): no call) max . nv = nvv; nvt = nvv; h = V nV

For i = 0 to nV , V (s) = ih

if z(1)v(s) ≥ P then W (i, j) = z(1)Vs else if vs ≥ P then W (i, j) = Vs The procedure to be used to invert the matrix is: w(0) =

c(0) ; b(0)

g(0) =

d(0) . b(0)

For i = 1 to nv do w(i) = c(i)/(b(i) − w(i − 1)a(i)) g(i) = (d(i) − g(i − 1)a(i))/(b(i) − w(i − 1)a(i)) End for (i = 1 to n), W (nv) = g(nv)

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For i = nv − 1 down to 0 do; W (i) = g(i) − w(i)u(i + 1), End for For j = 1 to nm (time until maturity) To search for the value of z(i) corresponding to a given month, the following treatment is done. dt = 0, imp1 = 0, Repeat imp1 = imp1 + 1; dt = dt + d1(imp1)until(j ≤ dt) and zz = z(imp1). To search for the call price CP (j) corresponding to a given month j, the following treatment is done. dt = 0,

imp2 = 0,

Repeat imp2 = imp2 + 1 dt = dt + d2(imp2)until(j ≤ dt) CPP = CP (imp2) If CPP = 0 then for i = 1 to nvt to nvv, W (i) = zhi, nv = nvv nvt = nvv end for (i = 1 to nvt to nvv) Else nvt1 = trunc(CPP/zh) + 1 for i = nvt to nvt1, W (i) = zhi end, nvt = nvt1, nv = nvt, end The equation is discretized as follows: For i = 1 to (nv − 1) do a(i) =

1 1 ri − σ2 i2 , 2 2

End for (i = 1 to nv − 1)

1 1 c(i) = − ri − σ 2 i2 , 2 2 d(i) = W (i, j − 1).

b(i) = 1 + σ2 i2 + r,

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The boundary condition when i = 0 is: a(0) = 0,

b(0) = 1,

c(0) = 0,

d(0) = 0

The boundary condition when i = nV is: a(nV ) = −1b(nV ) = 1,

c(nV ) = 0,

d(nV ) = hzz.

If CPP = 0 (no call) solve the tridiagonal system (a, b, c, d, w, nv) else Repeat nv = nv − 1,

a(nv) = 0,

b(nv) = 1,

c(nv) = 0,

d(nv) = CPP

solve the tri-diagonal system (a, b, c, d, w, nv) until W (nv − 1, j) ≤ Cpp End if (CPP = 0) For the dividend, the following treatment is done: j ) k = trunc( dd if (ddk − j = 0) then:

For i = 0 to nv, uu(i) = W (i, j) End for (i = 0 to nv) For i = 0 to nv, rk = i − dd h , k = trunc(rk) If k > 0, then uu(k) ≥ (k + 1 − rk)uu(k) + (rk − k)uu(k + 1) if uu(k) ≥ (zih) then W (i, j) = uuk else W (i, j) = zih end else W (i, j) = uu(i) end if end for end if The treatment for the coupons is as follows: j k = trunc( dc )

If (dck − j) = 0 for i = 0 to nvuu(i) = W (i, j) end for

for i = 0 to nv

Ic h k = trunc(rk) rk = i −

if k > 0, then uu(k) = (k + 1 − rk)uu(k) + (rk − k)uu(k + 1) if (uuk + Ic ≤ CCP ) or (CPP = 0) then, W (i, j) = uu(k) + Ic else W (i, j) = CPP end

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Else W (i, j) = uu(i) end if end for (i = 0 to nv) End if End for (j = 1 to nm) End. Questions 1. 2. 3. 4. 5.

Why are numerical methods used in asset pricing? What is a finite difference scheme? What is an implicit scheme? What is an explicit scheme? What are the main characteristics of CBs?

Exercises Exercise 1 Consider the extended B–S equation for the pricing of an option with a certain payoff at time T . 1. Can you reduce the extended B–S equation using the following transformations? S = Eex ,

t=T −

2τ , σ2

V (S, t) = Ev(x, τ )

2. Can you reduce the extended B–S equation using the following transformations? v = eα+βτ x u(x, τ );

∀α, β

3. Can you give the new payoff and illustrate the method for the pricing of a European call? Solution 1. The extended B–S equation for the pricing of a European call option can be written as: 1 ∂2V 2 2 ∂V ∂V + − (r + λv )V = 0 σ S + (r + λs )S ∂t 2 ∂S 2 ∂S

(20.47)

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under the following boundary condition: V (0, t) = 0 whenS → ∞ lim V (S, t) = S

S→∞

and the terminal condition: V (S, T ) = max(S − E, 0) For the first transformation, the partial derivative with respect to time gives: ∂V ∂Ev(x, τ ) = ∂t ∂t with τ=

S σ2 (T − t), x = Log 2 E

or ∂V ∂v =E ∂t ∂t

S σ2 σ2 Log , T − t E 2 2

The partial derivative with respect to the underlying asset gives: ∂v ∂x ∂v 1 E ∂v ∂V = = = ∂S ∂x ∂S ∂x S S ∂x and ∂v ∂τ σ 2 ∂v ∂V = =− E ∂t ∂τ ∂t 2 ∂τ The second partial derivative gives: ∂2V E ∂ 2v ∂2v ∂2x E ∂v + = = − ∂S 2 ∂x2 ∂S 2 S 2 ∂x S 2 ∂x2 with substitution in Eq. (20.47) we have: 2 ∂v ∂v ∂ v 1 σ 2 ∂v − + Eσ 2 + (r + λs )E (r + λv )Ev(x, τ ) = 0 − E 2 ∂τ 2 ∂x2 ∂x ∂x ⇐⇒

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2 ∂v 1 σ 2 ∂v 1 2 2∂ v + Eσ − (r + λv )v(x, τ ) = 0 − + − σ + r + λs 2 2 ∂τ 2 ∂x 2 ∂x ⇐⇒ ∂2v 2 2 ∂v ∂v + 2 + (r + λs ) 2 − 1 − 2 (r + λv )v(x, τ ) = 0 − ∂τ ∂x σ ∂x σ ⇐⇒ 2 ∂v ∂v ∂ 2v 2 + (r + λ )1 (r + λv )v(x, τ ) = − s ∂x2 σ2 ∂x σ 2 ∂τ Let 2 r=k σ2 hence,

∂ 2v 2 ∂v 2λs ∂v = − k + 2 λv v(x, τ ) + k+ 2 −1 ∂τ ∂x2 σ ∂x σ

with v(x, τ ) → 0 when x → −∞ because: V (0, t) = 0 and S → 0; Log S → −∞ then x → −∞ and V (S, t) ∼ S when S → ∞ since S = Eex ; v(x, τ ) ∼ ex because: S = Eex → ∞ since ex → ∞ since S = Eex , t = T − t = T, τ = 0:

2τ σ2

we have for

V (S, T ) = Ev(x, 0) = max(Eex − E, 0) = max(E(ex − 1), E, 0) and so: v(x, 0) = max(ex − 1, 0) 2. v(x, τ ) = eαx+βτ u(x, τ )

869

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Using the results in Eq. (20.47) gives: ∂ 2v 2 ∂v 2λs ∂v = − k + 2 λv v + k+ 2 −1 ∂τ ∂x2 σ ∂x σ or ∂u ∂v = βeαx+βτ u(x, τ ) + eαx+βτ ∂τ ∂τ and ∂u ∂v = αeαx+βτ u(x, τ ) + eαx+βτ ∂x ∂x also 2 ∂ 2V 2 αx+βτ αx+βτ ∂u αx+βτ ∂u αx+βτ ∂ u = α e u(x, τ ) + αe . + αe + e ∂x2 ∂x ∂x ∂x2

Hence, we have the following equation: ∂u ∂ 2u 2λs ∂u = + 2α + k + − 1 ∂τ ∂x2 σ2 ∂x 2λs 2λv + α2 + α k + 2 − 1 − k − β − 2 u. σ σ We can eliminate the term

∂u ∂x

by choosing α such that:

2α + k − 1 +

2λs = 0. σ2

It is possible to choose the u term by selecting the value: 2λs 2λv β = α2 + α k + 2 − 1 − k 2 . σ σ We can solve the system of two equations for α and β. 2λs =0 σ2 2λs 2λv α2 + αk − k + α + α 2 2 = β σ σ 2α + k − 1 +

hence, we have: α=

1−k− 2

2λs σ2

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or α= and β=

1 4

λs 1 k − − 2 2 σ2

2 2λs 2 k+1+ 2 + 2 (λs + λv ). σ σ

When λu = λv , we have the following equation: ∂u ∂ 2u = ∂τ ∂x2 with boundary conditions: u(x, τ ) = e−αx−βτ v(x, τ ) and v(x, τ ) → 0

as x → −∞

u(x, τ ) → 0

as x → −∞

and

when (−α < 0) this means α > 0 ⇐⇒ (r + λs )
1, the call (put) will start (α − 1) percent OTM (ITM). In the presence

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of a cost of carry b, the forward-start call formula is: c = Se−(b−r)t [e−(b−r)(T −t) N (d1 ) − αe−r(T −t) N (d2 )] 1 ln α + b + 12 σ 2 (T − t) d1 = , d2 = d1 − σ (T − t). σ (T − t) In the same context, the put formula is: p = Se−(b−r)t [e−(b−r)(T −t) N (d1 ) − αe−r(T −t) N (−d2 )] 1 ln α + b + 12 σ 2 (T − t) , d2 = d1 − σ (T − t) d1 = σ (T − t) Table 21.2 provides some simulations of forward-start options values.

21.3. Pay-Later Options Pay-later options provide a certain insurance against large one-way price movements and are traded on stock indices, foreign currencies, and other commodities. The buyer of pay-later options has the obligation to exercise his/her option when it is ITM and to pay the premium. The exercise takes place regardless of the importance of the difference between the underlying asset price and the strike price, i.e., the amount by which the option is ITM.

Table 21.2. Simulations of forward-start call values. S = 100, t = 11/01/2003, T = 05/06/2003, r = 4%, σ = 20%, and forward-start date = 06/03/2003. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

4.29928 4.34406 4.38885 4.43363 4.47841 4.52320 4.56798 4.61277 4.65755

0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478

0 −0.00000 −0.00000 −0.00000 0 −0.00000 0 0 0

0.18910 0.19107 0.19304 0.19501 0.19698 0.19895 0.20092 0.20289 0.20486

0.02625 0.02653 0.02680 0.02707 0.02735 0.02762 0.02789 0.02817 0.02844

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Risk Management: Exotics and Second-Generation Options

Following Turnbull (1989), we denote by: ST : price of the underlying asset at the option’s maturity date; F : current forward rate; cT : option premium paid at the option’s maturity date and r∗ : foreign interest rate. At the option’s maturity date, the pay-later European call’s pay off is: cpayl (ST , 0, K) = (ST − K − cT )1ST >K . The option pays out ST − K − cT when ST > K, otherwise it has a zero payoff. Applying standard arbitrage arguments, the value of the pay-later European call option is given by: ∗

cpayl (ST , T, K) = Se−r T N (d1 ) − (K + cT )e−rT N (d2 ) S √ ln K + r − r∗ + 12 σ 2 T √ d1 = , d2 = d1 − σ T . σ T The value of the pay-later European put option is given by: ∗

ppayl (S, T, K) = −Se−r T N (−d1 ) + (K − pT )e−rT N (−d2 ) S √ ln K + r − r∗ + 12 σ 2 T √ d1 = , d2 = d1 − σ T σ T Tables 21.3 and 21.4 provide some simulations of pay-later options values. The amount to be paid is denoted by A. Table 21.3. Pay-later call values. S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, and A = 10. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

2.93255 3.26276 3.61441 3.98763 4.38255 4.79919 5.23751 5.69740 6.17871

0.32979 0.35132 0.37301 0.39481 0.41664 0.43843 0.46011 0.48163 0.50292

0.02153 0.02169 0.02180 0.02183 0.02179 0.02168 0.02152 0.02130 0.02103

0.39208 0.40454 0.41593 0.42620 0.43530 0.44319 0.44986 0.45530 0.45950

0.01300 0.01351 0.01400 0.01445 0.01488 0.01528 0.01564 0.01598 0.01627

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Derivatives, Risk Management and Value Table 21.4. Pay-later put values. S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, and A = 10. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

13.67278 13.00300 12.35464 11.72787 11.12279 10.53943 9.97775 9.43764 8.91895

−0.67021 −0.64868 −0.62699 −0.60519 −0.58336 −0.56157 −0.53989 −0.51837 −0.49708

0.02153 0.02169 0.02180 0.02183 0.02179 0.02168 0.02152 0.02130 0.02103

0.39208 0.40454 0.41593 0.42620 0.43530 0.44319 0.44986 0.45530 0.45950

0.00422 0.00473 0.00522 0.00568 0.00611 0.00651 0.00687 0.00720 0.00750

21.4. Simple Chooser Options A chooser is a contingent claim that allows its holder at a certain date, known as the “choice date” to trade this claim for either a call or a put. The claim is a regular chooser when the call and the put have identical strike prices and time to maturity. The claim is a “complex chooser” when the call and the put have different strike prices or time to maturities. Hence, the chooser is neither a call nor a put. Chooser options allow the holder, immediately after a pre-determined elapsed time, to choose whether the option is to be either a call or a put. This is the principal idea on which it is based on a standard chooser. The payoff of the standard chooser is: csimple = max[C ∗ (K, T − t), P ∗ (K, T − t); t], where (T − t) is the time to maturity and t < T . Since the buyer has the right to choose between a call and a put before the chooser’s maturity date, the value of the chooser must lie between the value of a standard option and a straddle. Following Rubinstein (1991), we use the notation: S ∗ : the unknown value of the underlying asset after the elapsed time t; R: 1 plus the risk-less interest rate and d: 1 plus the pay out rate. The value of a standard chooser is: √ Cs = Sd−T N (x) − KR −T N (x − σ T ) − Sd−T N (−y) √ + R−T KN (−y + σ t)

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with:

−T Sd + 12 σ 2 T log KR −T √ , x= σ T

−T

Sd + 12 σ 2 t log KR −T √ y= σ t

When there is a cost of carry b, the value of a standard chooser is: √ Cs = Se(b−r)T N (x ) − Ke−rT N (x − σ T ) − Se(b−r)T N (−y ) √ + e−rT KN (−y + σ t) with: x =

S ln K + b + 12 σ 2 T √ , σ T

y =

S ln K + bT + 12 σ 2 t √ . σ t

Table 21.5 provides simulations of chooser option values and the corresponding Greek letters. 21.5. Complex Choosers A complex chooser option is defined in the same way as the simple chooser except that either the strike prices or (and) the time to maturities for the call and the put are different. A complex chooser implies the choice at a future date t, known as the “choice date”, a call or a put with a strike price K1 or K2 and a time to maturity (T1 − t) or (T2 − t). In this spirit, the complex chooser cannot be assimilated to a package of standard options and is identified to a compound option. The payoff of a complex chooser is Table 21.5. Simple chooser. S = 100, K = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, and choose date = 09/06/2003. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

12.56897 12.65713 12.79734 12.98848 13.22917 13.51787 13.85290 14.23244 14.65458

0.08711 0.13944 0.19063 0.24045 0.28870 0.33525 0.37997 0.42276 0.46355

0.05233 0.05119 0.04982 0.04826 0.04655 0.04472 0.04279 0.04079 0.03875

0.62680 0.62893 0.62848 0.62554 0.62024 0.61272 0.60316 0.59174 0.57866

0.00971 0.01031 0.01087 0.01140 0.01189 0.01235 0.01277 0.01314 0.01348

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given by: ccomplex−chooser = max[C ∗ (K1 , T1 ), P ∗ (K2 , T2 ); t] The formula presented in Rubinstein (1991) for the complex chooser is: √ cc = Sd−T1 N (x, y1 , ρ1 ) − K1 r−T1 N (x − σ t, y1 − σ T1 , ρ1 ) √ − Sd−T2 N (−x, −y2 , ρ2 ) + K2 r−T2 N (−x + σ t, −y2 + σ T2 , ρ2 ) with: ρ1 =

t , T1

ρ2 =

1 ln yi = √ σ Ti

t , T2

Sd−Ti Ki r−Ti

x=

ln

Sd−t Scr r−t

√

σ t

1 + σ2 )Ti 2

1 √ + σ t 2

for, i = 1, 2

where Scr is solution to the following equation: Scr d−(T1 −t) N (z1 ) − K1 r−T1 −t N (z1 − σ (T1 − t)) − Scr d−(T2 −t) N (−z2 ) + K2 r−(T2 −t) N (−z2 + σ (T2 − t)) = 0 and where z1 and z2 are given by: Scr d−(Ti −t) 1 1 2 ln + σ (Ti − t) zi = √ 2 Ki R−(Ti −t) σ Ti − t where N (a, b, ρ) is the bi-variate normal distribution function. Table 21.6 provides simulation results for complex choosers and the corresponding Greek letters.

21.6. Compound Options A compound option is an option whose underlying asset is an option. Since an option may be either a call or a put, we may find four types of compound options: a call on a call, a call on a put, a put on a put, and a put on a call. Following Geske (1979), consider a levered firm for which the debt corresponds to pure discount bonds maturing in T years with a face value K1 . Under the standard assumptions of liquidating the firm in T years,

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Risk Management: Exotics and Second-Generation Options Table 21.6. Complex chooser values. S = 100, t = 11/01/2003, r = 4%, σ = 20%, call strike = 100, and put strike = 70, call maturity = 11/01/2004, put maturity = 09/02/2004, and choose date = 11/04/2000. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

8.56897 8.65713 9.79734 9.98848 10.22917 10.51787 11.85290 12.23244 12.65458

0.45984 0.48814 0.51732 0.54654 0.57512 0.60260 0.62867 0.65318 0.67609

0.02829 0.02919 0.02922 0.02858 0.02748 0.02607 0.02451 0.02291 0.02134

0.48553 0.48073 0.47348 0.46429 0.45367 0.44210 0.42999 0.41765 0.40530

0.01192 0.01252 0.01311 0.01369 0.01421 0.01468 0.01509 0.01543 0.01570

paying off the bondholders and giving the residual value (if any) to stockholders, the bondholders have given the stockholders the option to buy back the assets at the debt maturity date. In this context, a call on the firm’s stock is a compound option, C(S, t) = f (g(V, t), t) where t stands for the current time. If we assume that the return on the firm follows a given diffusion process, then changes in the value of the call can be given as a function of the changes in the firm’s value and time. The valuation of options in this context is standard since a risk-less hedge can be constructed by choosing an appropriate mixture of the firm and call options on the firm’s stock. Hence, changes in the call’s value are expressed as a function of the changes in the firm’s value and time. 21.6.1. The call on a call in the presence of a cost of carry The payoff from a call on a call is: Ccall = max[cB−S (V, K1 , T2 ) − K2 ], where cB−S (V, K1 , T2 ) is the Black–Scholes formula with a strike K1 and a time to maturity T2 . K1 indicates the strike price of the underlying option and K2 is the strike price for the option on the option. Following Geske (1979), the formula for a call on a call is: t1 t1 (b−r)T2 −rT2 N z1 , y1 , N z2 , y2 , Ccall = V e − K1 e T2 T2 − K2 e−rt1 N (y2 )

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with:

√ √ V 1 2 + b + σ v t1 σv t1 , y2 = y1 − σv t1 , y1 = ln Vcr 2 V 1 2 z1 = ln + b + σv T2 σv T2 , z2 = z1 − σv T2 K1 2 where N (A, B, ρ) stands for the bi-variate cumulative normal distribution. In this analysis, t1 is the time to maturity on the option and T2 is the time to maturity on the underlying option.

21.6.2. The put on a call in the presence of a cost of carry The formula for a put on a call corresponds to the following payoff: Pcall = max[K2 − cB−S (V, K1 , T2 )] The solution for a put on a call is given by: Pcall = −K1 e

−rT2

t1 t1 (b−r)T2 −Ve N z2 , −y2 , − N z1 , −y1 , − T2 T2

+ K2 e−rt1 N (−y2 ) where the value Vcr is determined using the following equation: cB−S (Vcr , K1 , T2 − T1 ) = K2 . 21.6.3. The formula for a call on a put in the presence of a cost of carry The formula for a call on a put corresponds to the following payoff: cput = max[pB−S (V, K1 , T2 ) − K2 , 0]. The solution for a call on a put is given by: cput = K1 e

−rT2

t1 t1 (b−r)T2 N −z2 , −y2 , N −z1 , −y1 , −Ve T2 T2

− K2 e−rt1 N (−y2 ).

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21.6.4. The put on a put in the presence of a cost of carry The formula for a put on a put corresponds to the following payoff: Pput = max[K2 − pB−S (V, K1 , T2 ), 0]. The solution for a put on a put is given by: Pput = V e

(b−r)T2

t1 t1 −rT2 N −z1 , y1 , − N −z2 , y2 , − − K1 e T2 T2

+ K2 e−rt1 N (y2 ) where the value Vcr is determined using the following equation: PB−S (Vcr , K1 , T2 − t1 ) = K2 . Tables 21.7 to 21.10 provide simulations of compound option values and the corresponding Greek letters using different parameters.

21.7. Options on the Maximum (Minimum) Stulz (1982) and Johnson and Shanno (1987) derived valuation formulas for options on the maximum and the minimum of two or more risky assets. It is important to note that several exchange-traded futures contracts can be valued using the formula for the option on the minimum. Table 21.7. Simulations of the prices of a call on a call. S = 100, K1 = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, compound date = 10/03/2003, and K2 = 3. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

2.31715 2.80488 3.35064 3.95296 4.60917 5.31551 6.06738 6.85953 7.68638

0.48657 0.54490 0.60177 0.65595 0.70634 0.75207 0.79252 0.82732 0.85637

0.05833 0.05687 0.05418 0.05039 0.04573 0.04045 0.03480 0.02905 0.02344

0.23593 0.25010 0.26101 0.26847 0.27253 0.27339 0.27143 0.26718 0.26120

0.00354 0.00420 0.00492 0.00568 0.00647 0.00728 0.00810 0.00891 0.00970

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Table 21.8. Simulations of the prices of a call on a put. S = 100, K1 = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, compound date = 10/03/2003, and K2 = 3. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

4.74495 4.29053 3.86038 3.45517 3.07550 2.72183 2.39444 2.09342 1.81859

−0.45490 −0.43052 −0.40546 −0.37980 −0.35367 −0.32725 −0.30076 −0.27444 −0.24856

0.02438 0.02506 0.02566 0.02613 0.02642 0.02649 0.02632 0.02588 0.02518

0.32804 0.32180 0.31401 0.30471 0.29397 0.28190 0.26861 0.25425 0.23901

0.00364 0.00364 0.00358 0.00349 0.00336 0.00319 0.00300 0.00278 0.00256

Table 21.9. Simulations of the prices of a put on a put. S = 100, K1 = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, compound date = 10/03/2003, and K2 = 15. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

6.73448 7.15562 7.56401 7.95832 8.33760 8.70122 9.04879 9.38018 9.69543

0.42139 0.40859 0.39446 0.37936 0.36361 0.34749 0.33123 0.31500 0.29896

−0.01279 −0.01413 −0.01510 −0.01575 −0.01612 −0.01627 −0.01623 −0.01604 −0.01574

−0.33921 −0.34947 −0.35728 −0.36279 −0.36621 −0.36771 −0.36752 −0.36582 −0.36280

−0.00631 −0.00657 −0.00679 −0.00699 −0.00716 −0.00731 −0.00742 −0.00751 −0.00757

Table 21.10. Simulations of the prices of a put on a call. S = 100, K1 = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, compound date = 10/03/2003, and K2 = 15. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

0.75775 0.58159 0.44135 0.33123 0.24589 0.18060 0.13128 0.09446 0.06729

−0.17692 −0.14072 −0.11039 −0.08545 −0.06529 −0.04926 −0.03672 −0.02704 −0.01969

0.03620 0.03033 0.02494 0.02016 0.01603 0.01254 0.00967 0.00735 0.00551

0.11211 0.09928 0.08602 0.07303 0.06086 0.04983 0.04013 0.03181 0.02485

−0.00062 −0.00048 −0.00037 −0.00028 −0.00021 −0.00015 −0.00011 −0.00008 −0.00006

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21.7.1. The call on the minimum of two assets Following Stulz (1982), let S1 and S2 stand, respectively for the prices of two risky assets. At maturity, the payoff of a European call on the minimum of these two assets is cmin = max[min(S1 , S2 ) − K, 0]. The price of a European call on the minimum of S1 and S2 , with a maturity date T and a strike price K, denoted by cmin (S1 , S2 , K, T − t) is equal to the value of a self-financing portfolio which has the same value as the option at date T . Stulz (1982) provided the appropriate formulas for the valuation of these options. In the presence of a cost of carry b1 for asset 1 and b2 for asset 2, the formula for the pricing of a call on the minimum of two assets is: cmin (S1 , S2 , K, τ ) = S1 e(b1 −r)τ N (y1 , −d, −ρ1 ) √ + S2 e(b2 −r)τ N (y2 , d − σ τ , −ρ2 ) √ √ − Ke−rτ N (y1 − σ1 τ , y2 − σ2 τ , ρ) with:

+ b1 − b2 + 12 σ22 τ ln SK1 + b1 + 12 σ12 τ √ √ , y1 = d= σ τ σ1 τ ln SK2 + b2 + 12 σ22 τ √ y2 = , σ2 = σ12 + σ22 − 2ρ12 σ1 σ2 σ2 τ ln

S1 S2

ρ1 σ = σ1 − ρσ2 ,

ρ2 σ = σ2 − ρσ1

and where N (α, β, ρ) is the bi-variate cumulative normal distribution, where α, β are the upper limits of integration and ρ is the correlation coefficient. We can write in a compact form, the price of a European call on the minimum of two assets as: cmin (S1 , S2 , K, τ ) = S1 e(b1 −r)τ N (β1 , β2 , ρc ) + S2 e(b2 −r)τ N (α1 , α2 , ρc ) − Ke−rτ N (γ1 , γ2 , ρ12 ). If the strike is zero and there is no cost of carry, the formula for a call on the minimum reduces to: cmin (S1 , S2 , 0, τ ) = S1 − cE (S1 , S2 , 1, τ ) = S1 − N (d11 ) + S2 N (d22 )

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with:

d11 =

ln

S1 S2

+ 12 σ 2 τ √ , σ τ

√ d22 = d11 − σ τ .

where cE (S1 , S2 , 1, τ ) stands for the price of an option to exchange one unit of asset S2 for one unit of asset S1 . This formula is also given in Margrabe (1978). 21.7.2. The call on the maximum of two assets As for ordinary options, it is possible to obtain some parity relationships between options on the minimum, the maximum, the underlying asset, and the interest rate. At maturity, the payoff of a European call on the maximum of two assets is: cmax (S1 , S2 , K, τ ) = max[max(S1 , S2 ) − K, 0]. The value of this option is given by: cmax (S1 , S2 , K, τ ) = C(S1 , K, τ ) − cmin (S1 , S2 , 0, τ ) + C(S2 , K, τ ) where C(S2 , K, τ ) stands for the price of a European call on asset S1 , with a strike price K and a maturity date τ . The value of the call on the maximum of two assets can also be written as: √ cmax (S1 , S2 , K, τ ) = S1 e(b−r)τ N (y1 , d, ρ1 ) + S2 e(b−r)τ N (y2 , −d + σ τ , ρ2 ) √ √ − Ke−rτ [1 − N (−y1 + σ1 τ , −y2 + σ2 τ , ρ)]. 21.7.3. The put on the minimum (maximum) of two assets Let pmin (S1 , S2 , K, τ ) be the price of a European put on the minimum of two assets S1 and S2 . Its price must satisfy the following relationship: pmin (S1 , S2 , K, τ ) = e−rτ K − cmin (S1 , S2 , 0, τ ) + cmin (S1 , S2 , K, τ ). The value of the put on the maximum, pmax (S1 , S2 , K, τ ) is given by Stulz (1982) as: pmax (S1 , S2 , K, τ ) = e−rτ K − cmax (S1 , S2 , 0, τ ) + cmax (S1 , S2 , K, τ ).

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Table 21.11. Simulations of the values of a call on the minimum of two assets. S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, and ρ = 0.8%. S1

Price

Delta1

Gamma 1

Delta 2

Gamma 2

96 97 98 99 100 101 102 103 104

2.58960 2.75440 2.92299 3.09510 3.27041 3.44862 3.62939 3.81242 3.99737

0.16282 0.16677 0.17041 0.17378 0.17683 0.17956 0.18196 0.18405 0.18583

0.00395 0.00365 0.00336 0.00305 0.00273 0.00241 0.00209 0.00177 0.00146

0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297

0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033

Table 21.12. Simulations of the values of a put on the minimum of two assets. S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, and ρ = 0.8%. S1

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

96 97 98 99 100 101 102 103 104

12.14544 11.67764 11.22122 10.77603 10.34191 9.91871 9.50625 9.10437 8.71290

−0.47349 −0.46203 −0.45072 −0.43957 −0.42858 −0.41775 −0.40709 −0.39660 −0.38628

0.01145 0.01131 0.01115 0.01099 0.01083 0.01066 0.01049 0.01032 0.01014

−0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858

0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083

Tables 21.11 and 21.12 provide simulation results for the option price as well as the delta and the gamma with respect to the first asset and the second asset. 21.8. Extendible Options 21.8.1. The valuation context The valuation of these options is realized in the Black–Scholes context. Following Longstaff (1990), the valuation equation that must be satisfied by the option price, V (S, t) is: ∂V ∂V 1 2 2 ∂2V σ S 2 + rS − rV + = 0. 2 ∂ S ∂S ∂t

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Let CE (S, K1 , T1 , K2 , T2 , A) be the current value of an extendible call as a function of the two strike prices K1 and K2 , times to maturity T1 and T2 , the underlying asset S, and the premium A to be paid in the event of extension. At date T1 , the call’s payoff is: CE (S, K1 , T1 , K2 , T2 , A) = max(0, C(S, K2 , T2 − T1 ) − A, S − K1 ) i.e., the option holder can choose among three payoffs: the intrinsic value (S − K1 ), zero, or the difference between the premium A and a standard European call with a strike price K2 and a maturity date (T2 − T1 ). This may also be written as: CE = max{max[0, C(S, K2 , T2 − T1 ) − A], max[0, S − K1 ]}. This payoff function corresponds to a maximum of two risky payoffs: the payoff of a standard call and that of a call on a call. The payoff looks like that of an option on the maximum of two assets. When A is positive, there is some critical value of S at T1 denoted by I1 below which the option is not extended, and another critical value I2 above which the option is again not extended. Hence, an extension occurs when S is in the interval [I1 , I2 ]. At T1 , the value of I1 is a solution to the equation: C(I1 , K2 , T2 − T1 ) = A and I1 must lie between A and A + K2 e−r(T2 −T1 ) . When A = 0, I1 = 0 and when I1 ≥ K1 , the extension privilege is worthless. A sufficient condition for I1 to be less than K1 is: A < K2 − K2 e−r(T2 −T1 ) . At T1 , the value of I2 is given by the solution to the equation: C(I2 , K2 , T2 − T1 ) = I2 − K1 + A. 21.8.2. Extendible calls The value of the extendible call as given by Longstaff is: CE (S, K1 , T1 , K2 , T2 , A) = C(S, K1 , T1 ) + SN2 (γ1 , γ2 , −∞, γ3 , ρ)

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σ 2 T1 , γ2 − σ 2 T1 , −∞, γ3 − σ 2 T2 , ρ) − SN (γ1 , γ4 ) + K1 e−rT1 N (γ1 − σ 2 T1 , γ4 − σ 2 T1 ) − Ae−rT1 N (γ1 − σ 2 T1 , γ2 − σ 2 T1 ) − K2 e−rT2 N (γ1 −

where:

σ2 + r+ T1 σ 2 T1 , 2 σ2 S + r+ T1 σ 2 T1 , γ2 = ln I1 2 σ2 S + r+ T2 σ 2 T2 , γ3 = ln K2 2 σ2 S + r+ T1 σ 2 T1 , γ4 = ln K1 2

γ1 =

ln

S I2

and

ρ=

T1 T2

with: N2 (a, b, c, d, ρ): the cumulative probability of the standard bi-variate normal density with correlation coefficient ρ for the region [a, b]x[c, d]; N (a, b): the cumulative probability of the standard normal density in the region [a, b] and C(S, K1 , T1 ): the value of a standard call option in a Black and Scholes context. In the presence of a cost of carry b, the formula for the extendible call is given by: CE (S, K1 , T1 , K2 , T2 , A) = C(S, K1 , T1 ) + Se(b−r)T2 N2 (γ1 , γ2 , −∞, γ3 , ρ) − K2 e−rT2 N (γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, γ3 − σ 2 T2 , ρ) − Se(b−r)T1 N (γ1 , γ4 ) + K1 e−rT1 N (γ1 − σ 2 T1 , γ4 − σ 2 T1 ) − Ae−rT1 N (γ1 − σ 2 T1 , γ2 − σ 2 T1 )

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where the interest rate r is replaced by the cost of carry b in the following formulas: S σ2 T1 σ 2 T1 , + b+ γ1 = ln I2 2 σ2 S + b+ T1 γ2 = ln σ 2 T1 , I1 2 σ2 S + b+ T2 γ3 = ln σ 2 T2 , K2 2 σ2 S + b+ T1 γ4 = ln σ 2 T1 , K1 2 and

ρ=

T1 T2

with: N2 (a, b, c, d, ρ): the cumulative probability of the standard bi-variate normal density with correlation coefficient ρ for the region [a, b]x[c, d]; N (a, b): the cumulative probability of the standard normal density in the region [a, b] and C(S, K1 , T1 ): the value of a standard call option in the Black and Scholes context. Tables 21.13 to 21.15 provide simulation results for the values of options and the Greek letters. 21.9. Equity-Linked Foreign Exchange Options and Quantos As shown in Garman and Kohlhagen (1983), the Black and Scholes (1973) formula for stock options applies to the valuation of options on currencies where the foreign interest rate replaces the dividend yield. When an investor wants to link a strategy in a foreign stock and a currency, he/she can use at least four different types of options: a foreign equity option struck in foreign currency, a foreign equity option struck in domestic currency, fixed exchange-rate foreign equity options also known as quanto options, or an equity-linked foreign exchange option. These different types of options are analyzed and valued in this section.

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Table 21.13. Simulation results for the values of extendible calls. S = 100, K = 100, t = 11/01/2003, T = 11/01/2004, r = 4%, σ = 20%, extendible maturity date = 11/06/2004, K2 = 110, and additional premium = 4. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

8.22621 8.77647 9.34548 9.93290 10.53836 11.16147 11.80182 12.45896 13.13242

0.54988 0.56873 0.58724 0.60537 0.62311 0.64043 0.65731 0.67370 0.68963

0.01885 0.01851 0.01841 0.01774 0.01732 0.01687 0.01640 0.01592 0.01543

0.50571 0.50771 0.50854 0.50823 0.50681 0.50434 0.50083 0.49638 0.49102

0.01001 0.01024 0.01046 0.01066 0.01086 0.01104 0.01120 0.01136 0.01150

Table 21.14. Simulation results for the values of extendible calls. S = 110, K1 = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, extendible date = 27/12/2003, K2 = 105, and A = 5. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

1.49102 2.63070 4.24971 6.38402 9.03184 12.15924 15.71041 19.61847 23.81495

0.18434 0.27411 0.37487 0.47896 0.57920 0.67006 0.74831 0.81275 0.86383

0.01674 0.01951 0.02080 0.02056 0.01905 0.01673 0.01400 0.01125 0.00873

0.24246 0.32142 0.38670 0.42850 0.44261 0.43046 0.39742 0.35076 0.29766

−0.00473 −0.00628 −0.00757 −0.00839 −0.00867 −0.00844 −0.00779 −0.00687 −0.00583

Table 21.15. Simulation results for the values of extendible calls. S = 110, K1 = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%, extendible date = 27/12/2003, K2 = 105, and A = 5. S

Price

Delta

Gamma

Vega

Theta

88 93 99 104 110 115.50 121 126.50 132

7.61027 10.19395 13.17356 16.52056 20.20043 24.17575 28.40893 32.86391 37.50725

0.43251 0.50656 0.57618 0.63997 0.69713 0.74740 0.79094 0.82812 0.85953

0.01370 0.01304 0.01206 0.01090 0.00965 0.00841 0.00722 0.00612 0.00514

0.43886 0.47311 0.49186 0.49591 0.48709 0.46784 0.44074 0.40832 0.37282

−0.00755 −0.00810 −0.00841 −0.00849 −0.00837 −0.00808 −0.00767 −0.00716 −0.00660

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21.9.1. The foreign equity call struck in foreign currency The payoff of a foreign equity call struck in foreign currency is: C1∗ = X ∗ max[S ∗ − K , 0] where S ∗ is the equity price in the currency of the investor’s country and K is a foreign currency amount. The spot exchange rate expressed in domestic currency of a unit of foreign currency, X ∗ , stands in front of the payoff to show that the latter must be converted into domestic currency. The domestic currency value of this call option is: √ C1 = S X(d)−T N (d1 ) − K X(r∗ )−T N (d1 − σS T ) −T √ √ S (d) 1 d1 = log σS T + σS T ∗ −T K (r ) 2 where σS is the volatility of S . For the continuous compounding of interest ∗ rates, the term (d)−T must be replaced by e−dT , the term (r∗ )−T by e−r T , and the term (r)−T by e−rT . 21.9.2. The foreign equity call struck in domestic currency The payoff of a foreign equity option struck in domestic currency is C2∗ = max[S ∗ X ∗ − K, 0], where K is the domestic currency amount. For the foreign option writer, the payoff is

C2∗ = max[S ∗ − KX ∗ , 0],

where X = 1/X.

X corresponds to the exchange rate quoted at the price of a unit of domestic currency in terms of the foreign currency. This pay-off corresponds to that of an option to exchange one asset (K units of our currency) for another asset (a share of stock). Its value is: √ C2∗ = S (d)−T N (d2 ) − KX (r)−T N (d2 − σS X T ); −T √ √ S (d) 1 d2 = log σS X T + σS X T , and −T KX (r) 2 2 − 2ρ σ σ σ(S X ) = σS2 + σX S X S X

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where ρS X is the correlation coefficient between the rates of return on S and X . The domestic value of this option in the same context is: √ C2 = S X(d)−T N (d2 ) − K(r)−T N (d2 − σS X T ) √ √ S X(d)−T 1 σS X T + σS X T d2 = log −T K(r) 2 2 − 2ρ σ σ = σ . σ(S X) = σS2 + σX S X S X (S X ) As stated before, for the continuous compounding of interest rates, the term ∗ (d)−T must be replaced by e−dT , the term (r∗ )−T by e−r T , and the term (r)−T by e−rT . 21.9.3. Fixed exchange rate foreign equity call The payoff of a fixed exchange rate foreign equity call, known as a quanto is: ¯ max[S ∗ − K , 0] = max[S ∗ X ¯ − K, 0] C3∗ = X ¯ is the rate at which the conversion will be made. It can be written where X in reciprocal units as ¯ ∗ max[S ∗ − K , 0]. C3∗ = XX Reiner (for details, refer to Bellalah et al., 1998) gave the value of this option in foreign currency as: −T √ rd −(ρS X σS σX )T −T ¯ e N (d3 ) − K (r) N (d3 − σS T ) C3 = XX S r∗ −T √ √ S (d) 1 σ σ T σS T + σS T . d3 = log − ρ S X S X ∗ −T K (r ) 2 The domestic value of this option is: −T √ rd ¯ S e−(ρS X σS σX )T N (d3 )a − K (r)−T N (d3 − σS T ) C3 = X r∗ d3 = log

S (d)−T K (r∗ )−T

− ρS X σS σX T

√ √ 1 σS T + σS T . 2

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For the continuous compounding of interest rates, the term (d)−T must be ∗ replaced by e−dT , the term (r∗ )−T by e−r T , and the term (r)−T by e−rT . 21.9.4. An equity-linked foreign exchange call The payoff of an equity-linked foreign exchange call is: C4∗ = S ∗ max[X ∗ − K, 0]. It can also be written as: C4∗ = S ∗ max[1 − KX ∗ − K, 0] = KS ∗ max

1 − X ∗ , 0 . K

The foreign value of this call option is given by: C4∗

−T

= S (d)

d4 = log

N (d4 ) − KS X

X(r∗ )−T K(r)−T

rd r∗

−T

√ e−(ρS X σS σX )T N (d4 − σX T )

+ ρS X σS σX T

√ 1 √ σX T + σX T . 2

The domestic value of this call option is: −T √ rd C4 = S X(d)−T N (d4 ) − K S ∗ e−(ρS X σS σX )T N (d4 − σX T ). r Table 21.16 gives the results for the different models: Black–Scholes (B–S), Garman–Kohlhagen (G–K), foreign equity/foreign strike (FE/FS), foreign equity/domestic strike (FE/DS), fixed-rate foreign-equity, (FR/ FE), equity-linked foreign-exchange, (FL/FE), and equity-linked-foreignexchange (EL/FE). Table 21.16. Type

Asset

Strike

Rate

Distribution

σ

B–S G–K FE/FS FE/DS FL/FE EL/FE

S X SX SX ¯ SX

K K K X K ¯ K X

r r r∗ r r

d r∗ d d

σS σX σS σS X σS σX

SX

KS

rd ρS X σS σX e r∗

rd ρS X σS σX e r∗

d

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Tables 21.17 to 21.20 provide simulations results for the values of foreign currency options in different contexts. We also provide the Greek letters. This allows the reader to make some comments.

21.10. Binary Barrier Options Following Rubinstein and Reiner (1991), we denote by: • • • • •

R: 1 plus the risk-less interest rate r; d: 1 plus the instantaneous payout rate; H: barrier level; S(τ ): price of the underlying asset after elapsed time τ ; St : price of the underlying asset at expiration t;

Table 21.17. Simulations of foreign equity call struck in foreign currency. S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%, and (domestic/ foreign) = 1, 1. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

4.31208 6.15817 8.41102 11.06549 14.10374 17.49871 21.21759 25.22464 29.48371

0.32902 0.40989 0.49121 0.57012 0.64441 0.71261 0.77387 0.82786 0.87471

0.01598 0.01633 0.01606 0.01528 0.01416 0.01282 0.01138 0.00993 0.00854

0.30619 0.35417 0.39151 0.41644 0.42862 0.42886 0.41875 0.40031 0.37569

−0.01255 −0.01453 −0.01607 −0.01710 −0.01760 −0.01761 −0.01719 −0.01642 −0.01540

Table 21.18. Simulations of foreign equity call struck in foreign currency. S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%, and S ∗ = 1. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

3.92008 5.59834 7.64638 10.05954 12.82158 15.90791 19.28872 22.93149 26.80338

0.29911 0.37263 0.44655 0.51829 0.58583 0.64783 0.70352 0.75260 0.79519

0.01453 0.01484 0.01460 0.01389 0.01287 0.01166 0.01035 0.00903 0.00777

0.27835 0.32197 0.35591 0.37858 0.38965 0.38987 0.38068 0.36392 0.34154

−0.01141 −0.01321 −0.01461 −0.01555 −0.01600 −0.01601 −0.01563 −0.01493 −0.01400

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Table 21.19. Simulations of foreign equity call struck in foreign currency. S = 0.95, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, and σ = 30%. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

3.72407 5.31842 7.26406 9.55656 12.18051 15.11252 18.32428 21.78492 25.46321

0.28415 0.35400 0.42422 0.49238 0.55654 0.61544 0.66834 0.71497 0.75543

0.01380 0.01410 0.01387 0.01320 0.01223 0.01108 0.00983 0.00858 0.00738

0.26444 0.30587 0.33812 0.35965 0.37017 0.37038 0.36165 0.34573 0.32446

−0.01084 −0.01255 −0.01388 −0.01477 −0.01520 −0.01521 −0.01484 −0.01418 −0.01330

Table 21.20. Simulations of foreign equity put struck in foreign currency. S ∗ = 0.95, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, and σ = 30%. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

20.84295 17.68730 14.88293 12.42544 10.29938 8.48139 6.94315 5.65379 4.58208

−0.66585 −0.59600 −0.52578 −0.45762 −0.39346 −0.33456 −0.28166 −0.23503 −0.19457

0.01380 0.01410 0.01387 0.01320 0.01223 0.01108 0.00983 0.00858 0.00738

0.26444 0.30587 0.33812 0.35965 0.37017 0.37038 0.36165 0.34573 0.32446

−0.01084 −0.01255 −0.01388 −0.01477 −0.01520 −0.01521 −0.01484 −0.01418 −0.01330

• •

η and φ: binary variables taking the value 1 or −1, and K: the strike price.

21.10.1. Path-independent binary options 21.10.1.1. Standard cash-or-nothing options In their simplest forms, the payoff of a binary call is nothing when the underlying asset terminal price, St , is below the strike price, K and is a predetermined amount, A, if the underlying asset terminal price is above the strike price. The payoff of the binary call is: ccon = 0

if St ≤ K

ccon = A else.

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The payoff of a binary put is nothing when the underlying asset terminal price, St , is above the strike price, K and is a pre-determined amount, A, if the underlying asset terminal price is below the strike price. The payoff of the binary put is: pcon = 0 if St ≥ K pcon = A if else. These options can be valued with respect to classic formulas by Black– Scholes. The values of standard options are given by: √ C = φSd−t N (φx) − φKR −t N (φx − φσ t),

Sd−t log KR −t 1 √ √ + σ t x= 2 σ t with φ = 1 for a call and −1 for a put. The Black–Scholes formula comprises two parts. The first term, d−t φSN (φx), corresponds to the present value of the underlying asset price conditional upon exercising the option. The second term, φKR −t N (φx − √ φσ t), refers to the present value of the strike price times the probability of exercising the √ option. The value of a cash-or-nothing call is ccon = AR −t N (x − σ t). The value of a cash-or-nothing put is: √ pcon = AR −t N (−x + σ t). In the presence of a cost of carry b, the value of a cash-or-nothing call is ccon = Ae−rT N (x ). The value of a cash-or-nothing put is: pcon = Ae−rT N (−x ) where

x =

S log K + b − 12 σ 2 T √ . σ T

21.10.1.2. Cash-or-nothing options with shadow costs We denote by λS and λ, the information costs associated with S and the option. The values of standard options with information costs are

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given by: √ C = e(λS −λ+d)t φSN (φx) − φKe−(λ+r)t N (φx − φσ t) S log K + (r + λS − d)t 1 √ √ x= + σ t 2 σ t with φ = 1 for a call and −1 for a put. The first term in this formula, e(λS −λ+d)t φSN (φx), corresponds to the present value of the underlying asset price conditional upon exercising the √ option. The second term, φKe−(λ+r)t N (φx − φσ t), indicates the present value of the strike price times the probability of exercising the option. The value of a cash-or-nothing call in the presence of shadow costs is given by: √ ccon = Ae(λS −λ)t N (x − σ t). The value of a cash-or-nothing put is: √ pcon = Ae(λS −λ)t N (−x + σ t). 21.10.1.3. Standard asset-or-nothing options These binary options are similar to cash-or-nothing options except that in their payoff, the pre-determined amount is replaced by the terminal asset value. The payoff of the asset-or-nothing call is caon = 0, if St ≤ K caon = St else. The value of this option is given by the present value of the underlying asset price conditional upon exercising the call or: caon = Sd−t N (x). The payoff of the asset-or-nothing put is nothing when the underlying asset terminal price, St , is above the strike price, and is the terminal asset price, St if the underlying asset terminal price is below the strike price paon = 0, if St ≥ K paon = St else. The value of this option is given by the present value of the underlying asset price conditional upon exercising the put or: paon = Sd−t N (−x). In the presence of a cost of carry b, the value of an asset-or-nothing call is caon = Se(b−r)t N (x ). The value of an asset-or-nothing put is: paon = Se(b−r)t N (−x ), where S log K + b + 12 σ 2 t √ . x = σ t

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21.10.1.4. Asset-or-nothing options with shadow costs The value of the asset-or-nothing call is given by the present value of the underlying asset price conditional upon exercising the call or: √ caon = S0 e(λS −λ+d)t N (x − σ t). The value of the asset-or-nothing put is given by the present value of the underlying asset price conditional upon exercising the put or: paon = S0 e(λS −λ−d)t N (−x). Tables 21.21–21.24 provide some simulations of option values and the Greek letters. The reader can make some comments regarding the evolution of these parameters. Table 21.21. Cash or nothing options. S = 100, t = 17/12/2003, T = 18/12/2004, r = 2%, σ = 20%, and barrier = 140. Barrier

Price

Delta

100 101 102 103 104 105 110 120 130 140

0.49005 0.47065 0.45150 0.43262 0.41406 0.39585 0.31099 0.17803 0.09350 0.04577

0.01950 0.01947 0.01940 0.01929 0.01913 0.01893 0.01742 0.01290 0.00829 0.00477

Gamma −0.00002 0.00002 0.00007 0.00012 0.00016 0.00021 0.00039 0.00058 0.00054 0.00041

Vega

Theta

−0.00390 −0.00294 −0.00198 −0.00104 −0.00011 0.00078 0.00474 0.00909 0.00917 0.00708

0.00011 0.00008 0.00005 0.00003 0.00000 −0.00002 −0.00013 −0.00025 −0.00025 −0.00019

Table 21.22. Cash or nothing options. S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 90. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

0.62605 0.64456 0.66247 0.67978 0.69647 0.71253 0.72794 0.74271 0.75683

0.01851 0.01793 0.01732 0.01669 0.01605 0.01541 0.01476 0.01411 0.01346

−0.00059 −0.00061 −0.00063 −0.00064 −0.00065 −0.00065 −0.00065 −0.00065 −0.00064

−0.01027 −0.01097 −0.01160 −0.01216 −0.01265 −0.01307 −0.01341 −0.01369 −0.01390

−0.00019 −0.00021 −0.00023 −0.00025 −0.00027 −0.00029 −0.00030 −0.00032 −0.00033

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Table 21.23. Asset or nothing options. S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, and barrier = 110. S

Price

Delta

Gamma

Vega

Theta

96 97 98 99 100 101 102 103 104

32.04676 34.22618 36.46076 38.74601 41.07730 43.44986 45.85890 48.29956 50.76702

2.17828 2.23380 2.28478 2.33107 2.37257 2.40920 2.44095 2.46781 2.48984

0.05551 0.05098 0.04629 0.04150 0.03664 0.03175 0.02686 0.02202 0.01724

1.09187 1.03383 0.96919 0.89846 0.82219 0.74100 0.65550 0.56637 0.47427

0.04444 0.04330 0.04195 0.04039 0.03864 0.03672 0.03465 0.03243 0.03009

Table 21.24. Asset or nothing options. S = 100, t = 17/12/2003, T = 18/12/2004, r = 2%, σ = 20%, and barrier = 140. Barrier

Price

Delta

Gamma

Vega

Theta

100 101 102 103 104 105 110 120 130 140 150

57.94737 55.99851 54.05440 52.11964 50.19862 48.29551 39.17721 23.92946 13.39969 6.98079 3.42846

2.52913 2.52674 2.51958 2.50773 2.49134 2.47053 2.30761 1.78728 1.21145 0.73818 0.41329

0.02286 0.02768 0.03248 0.03723 0.04189 0.04646 0.06657 0.08763 0.08329 0.06478 0.04380

0.00194 0.09883 0.19597 0.29265 0.38822 0.48204 0.90634 0.40286 0.41011 0.12724 0.77475

−0.00001 −0.00267 −0.00534 −0.00799 −0.01062 −0.01319 −0.02483 −0.03839 −0.03846 −0.03062 −0.02094

In the presence of a cost of carry, equal to the difference between the domestic interest rate r and the foreign interest rate, r∗ , these options can be priced in the same way. Table 21.25 gives the values of these options and the associated Greek letters. 21.10.1.5. Standard gap options Gap options are structured to give the following payoff for a call: cgap = 0

if St ≤ K

cgap = St − A

else.

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Risk Management: Exotics and Second-Generation Options Table 21.25. Digital FX-cash or nothing options. S = 1, t T = 07/02/2004, r = 3%, r ∗ = 4%, σ = 20%, and barrier = 1.5.

=

907 07/02/2003,

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

0.00843 0.00969 0.01110 0.01266 0.01438 0.01629 0.01839 0.02069 0.02320

0.12559 0.14020 0.15593 0.17278 0.19078 0.20992 0.23020 0.25162 0.27415

0.34624 0.31487 0.28279 0.24997 0.21641 0.18211 0.14707 0.11130 0.07482

0.00253 0.00279 0.00306 0.00334 0.00363 0.00393 0.00425 0.00457 0.00489

0.00843 0.00969 0.01110 0.01266 0.01438 0.01629 0.01839 0.02069 0.02320

The “gap” refers to the difference (A−K). Note that the payoff of a gap call corresponds to the difference between the payoffs of an asset-or-nothing call and a cash-or-nothing call. Therefore, its value is given by: √ cgap = Sd−t N (x) − AR−t N (x − σ t). This formula is like that of a standard call except for the cash amount sometimes replacing the strike price. The pay-off of a gap option put is: pgap = 0

if St ≥ K

pgap = St − A else. Note that the payoff of a gap put corresponds to the difference between the payoffs of an asset-or-nothing put and a cash-or-nothing put. Hence, its value is given by: √ pgap = −Sd−t N (−x) + AR−t N (−x + σ t). This formula is like that of a standard put except for the cash amount sometimes replacing the strike price. Gap options can be defined with respect to two different strike prices K1 and K2 . The call’s payoff is zero if S ≤ K1 and is S − K2 if S > K1 . The call’s payoff is zero, if S ≥ K1 and is K2 − S if S < K1 . Using the analysis in Rubinstein and Reiner (1991), the call

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formula is: cgap = Se(b−r)tN (d1 ) − K2 e−rt N (d2 ) √ log KS1 + b + 12 σ 2 t √ , d2 = d1 − σ t. d1 = σ t The put formula is: pgap = K2 e−rtN (−d2 ) − Se(b−r)tN (−d1 ) √ log KS1 + b + 12 σ 2 t √ , d2 = d1 − σ t. d1 = σ t 21.10.1.6. Gap options with shadow costs The value of a gap call is given by:

√ cgap = e(λS −λ−d)t SN (x) − Ae−(λ+r)t N (x − σ t).

The value of the gap put is: √ pgap = −e(λS −λ−d)t SN (−x) + Ae−(λ+r)t N (−x + σ t). 21.10.1.7. Supershares The payoff from a supershare option is 0, if KL > S > KH and otherwise. In this setting, the formula for a supershare option is: csupershare =

Se(b−r)t [N (d1 ) − N (d2 )] KL

log KSL + b + 12 σ 2 t √ d1 = σ t

log KSH + b + 12 σ 2 t √ and d2 = . σ t

S KL

(21.1)

21.11. Lookback Options We use the following notations for the maximum and the minimum over the interval [t1 , t2 ]: Mtt12 = max{Ss /s ∈ [t1 , t2 ]} mtt21 = min{Ss /s ∈ [t1 , t2 ]}. Option prices are computed at time 0 and option contracts are assumed to have been initiated at time T0 ≤ 0.

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21.11.1. Standard lookback options Since the payoff of a standard European lookback call at the maturity date is given by ST − mTT0 , its current value is: √ C = S0 N (d ) − e−rT m0T0 N (d − σ T ) 2 −( 2r2 ) √ σ S σ 2r 0 + e−rT S0 N −d + T − erT N (−d ) 2r m0T0 σ with

1 σ2 S0 + rT + T . ln d = √ m0T0 2 σ T

Since the payoff of a standard lookback put is (MT00 − ST ), its current value is: √ P = −S0 N (−d ) + e−rT MT00 N (−d + σ T ) 2 −( 2r2 ) √ σ S σ 0 S0 − + e−rT N d − (2r/σ) T + erT N (d ) 2r MT00 with:

1 σ2 S0 + rT + ln T . d = √ MT00 2 σ T

Note that these options correspond to the ordinary options as in Black– Scholes formulas plus another term corresponding to the specificities of their payoffs. 21.11.2. Options on extrema 21.11.2.1. On the maximum The payoff of a call on the maximum at maturity T is (MT00 − K)+ . When K ≥ MT00 , the current call price is: √ CM = S0 N (d) − e−rT KN (d − σ T ) 2 −( 2r2 ) σ S0 σ 2r √ −rT rT T + e N (d) +e S0 − N d− 2r K σ S0 1 σ2 ln d= √ + rT + T . K 2 σ T

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When K < MT00 , the call’s value is:

√ CM = e−rT (MT00 − K) + S0 N (d ) − e−rT MT00 N (d − σ T ) 2 −( 2r2 ) σ S0 2r √ −rT σ rT S0 − N d − T + e N (d ) +e 2r MT00 σ

21.11.2.2. On the minimum The payoff of a put on the minimum at the maturity date T is (K − m0T0 )+ . When K < m0T0 , the put’s price is: √ p = −S0 N (−d) + e−rT KN (−d + σ T ) 2 −( 2r2 ) √ σ 2r S σ 0 S0 + e−(rT N −d + T − erT N (−d) 2r K σ When K ≥ m0T0 , the put’s value is:

√ p = e−rT (K − m0T0 ) − S0 N (−d ) + e−rT m0T0 N (−d + σ T ) 2 −( 2r2 ) σ S0 σ 2r √ −rT rT +e S0 − N −d − T − e N (−d ) . 2r m0T0 σ 21.11.3. Limited risk options When m and K are constant, then the payoff of a limited risk call at T is: (ST − K)+ 1(MTT

0

≤m) .

The call’s current value is zero when MT00 > m. When MT00 ≤ m, the call’s current value is:

√ √ Clr = S0 [N (d) − N (dm )] − e−rT K[N (d − σ T ) − N (dm − σ T )] (−2r/σ2 ) √ r√ N 2dm − d − 2 +m T −σ T σ √ 2r 2r S0 −rT +σ e− σ 2 T −e K − N dm − σ m r√ r√ × N 2dm − d − 2 T − N dm − 2 T σ σ

S0 m

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911

with S0 σ2 1 ln + rT + T . dm = √ m 2 σ T The payoff of a limited risk put is (K − ST )+ 1(mTT When m0T0 < m, the put is worthless. When m0T0 ≥ m, the put’s current value is:

0

≥m) .

p = −S0 [N (−d) − N (−dm )] √ √ + e−rT K[N (−d + σ T ) − N (−dm + σ T )] −m

S0 m

(−2r/σ2 )

√ 2r √ T +σ T N −2dm + d + σ

− 2r2 +1 √ σ r√ S0 −rT − N − dm + 2 T +σ T −e K σ m 2r √ 2r √ × N − 2dm + d + T − N − dm + T . σ σ These options are issued in foreign exchange markets and also in stock index markets.

21.11.4. Partial lookback options The payoff of this option is (ST − ηmTT0 )+ with η > 1. The current value of a partial lookback call is: √ ln(η) ln(η) d − √ − ηe−rT m0T0 N d − √ − σ T σ T σ T 2r 2 −( 2 ) σ σ 2r √ S0 ln(η) √ + ηS0 × + e−rT N −d − T 2r m0T0 σ σ T

Cpl = S0 N

−e

rT ( σ2r2 )

η

ln(η) N −d − √ . σ T

The payoff of a partial lookback put is (ηMTT0 − ST )+ with 0 < η < 1. When η = 1, these options become standard lookback options.

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The current value of a partial lookback put is: √ ln(η) ln(η) −rT 0 MT0 N d + √ + σ T − ηe p = −S0 N −d + √ σ T σ T 2 ( 2r2 ) σ ln(η) S0 σ 2r √ −rT ηS0 N d + √ − −e T 2r MT00 σ σ T 2r ln(η) − erT η ( σ2 ) N d + √ σ T When η = 1, these options become standard lookback options. Tables 21.26 and 21.27 provide simulations values of lookback option values and the Greek letters for different parameters. The reader can compare the evolution of the different values. Table 21.26. Simulations values of standard lookback calls. S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, and historical minimum for call values = 80. S

Price

Delta

Gamma

Vega

Theta

80 85 90 95 100 105 110 115 120

12.69868 14.07381 16.47870 19.69548 23.51395 27.75571 32.28247 36.99416 41.82216

0.15888 0.38520 0.56967 0.71022 0.81140 0.88085 0.92664 0.95584 0.97392

0.04822 0.04043 0.03155 0.02317 0.01616 0.01081 0.00697 0.00436 0.00266

0.55516 0.55539 0.50224 0.42028 0.33046 0.24686 0.17669 0.12199 0.08168

−0.01517 −0.01516 −0.01370 −0.01144 −0.00897 −0.00668 −0.00477 −0.00328 −0.00219

Table 21.27. Simulations values of standard lookback calls. S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, and historical minimum = 90. S

Price

Delta

80 85 90 95 100 105 110 115 120

12.69868 13.49235 14.28601 15.59962 17.84962 20.86480 24.47425 28.52559 32.89347

0.15873 0.15873 0.15888 0.36200 0.53253 0.66806 0.77083 0.84569 0.89836

Gamma −0.00000 −0.00000 0.04293 0.03687 0.02989 0.02305 0.01702 0.01213 0.00837

Vega

Theta

0.55516 0.58986 0.62455 0.62826 0.58207 0.50555 0.41656 0.32846 0.24952

−0.01517 −0.01611 −0.01706 −0.01716 −0.01588 −0.01377 −0.01132 −0.00891 −0.00675

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913

Summary The option to exchange one risky asset for another is analyzed and valued. The identification of this option allows the pricing of several financial contracts. This concept is useful in the valuation of complex options such as options on the minimum or the maximum of several assets. Margrabe provided valuation formulas for exchange options giving the right to exchange one risky asset for another. The Black–Scholes–Merton formula appears as a particular case of the Margrabe general formula. Forward-start options provide an answer to the following question: how much can one pay for the opportunity to decide after a known time in the future, known as the “grant date”, to obtain an ATM call with a given time to maturity with no additional cost? Pay-later options provide a certain insurance against large one-way price movements and are traded on stock indices, foreign currencies and other commodities. The buyer of pay-later options has the obligation to exercise his/her option when it is in the money and to pay the premium. A chooser allows its holder at the “choice date” to trade this claim for either a call or a put. The claim is a regular chooser when the call and the put have identical strike prices and time to maturity. The claim is a “complex chooser” when the call and the put have different strike prices or time to maturities. A complex chooser option is defined in the same way as the simple chooser except that either the strike prices or (and) the time to maturities for the call and the put are different. A compound option is an option whose underlying asset is an option. Since an option may be either a call or a put, we may find four types of compound options: a call on a call, a call on a put, a put on a put, and a put on a call. Consider a levered firm for which the debt corresponds to pure discount bonds maturing in some years with a certain face value. Under the standard assumptions of liquidating the firm in some years, paying off the bondholders and giving the residual value (if any) to stockholders, the bondholders have given the stockholders the option to buy back the assets at the debt maturity date. In this context, a call on the firm’s stock is a compound option. If we assume that the return on the firm follows a given diffusion process, then changes in the value of the call can be given as a function of changes in the firm’s value and time. The valuation of options in this context is standard. Changes in the call’s value are expressed as a function of the changes in the firm’s value and time. The chapter also presents a framework for the analysis and valuation of forward-start options, pay-later options, simple chooser and complex chooser options and several other forms of compound options as options

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on the minimum and options on the maximum of two assets, extendible options, equity-linked foreign exchange options and quantos, binary-barrier options and lookback options.

Questions 1. What is an exchange option? 2. What are the main applications of this concept? 3. How information costs affect the pricing procedure? What are the definitions of the following options: forward-start options and pay-later options? 4. What are the definitions of simple chooser options and complex choosers options? 5. What are the payoffs of compound options? 6. What are the payoffs of options on the minimum and options on the maximum of two assets? 7. What are the payoffs of extendible options? 8. What are the payoffs of equity-linked foreign exchange options and quantos? 9. What are the payoffs of binary barrier options? 10. What are the payoffs of lookback options?

References Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13, 1895–1927. Bellalah, M, and J-L Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bellalah, M, Prigent JL and C Villa (2001a). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M, Ma Bellalah and R Portait (2001b). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Briys, E, M Bellalah et al. (1998). Options, Futures and Exotic Derivatives. En collaboration avec E. Briys, et al., John Wiley & Sons. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

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Garman, M and S Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance, 2, 231–237. Geske, R (1979). A note on an analytical valuation formula for unprotected American call options with known dividends. Journal of Financial Economics 7, 375–380. Johnson, H and D Shanno (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22, 143–151. Longstaff, FA (1990). Time varying term premiums and traditional hypothesis about the term structure. Journal of Finance, 45, 1307–1314. Margrabe, W (1978). The value of an option to exchange one asset for another. Journal of finance 33, 177–186. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Rubinstein, M (1991a). Pay now, choose later. Risk, 4(2), February, 44–47. Rubinstein, M (1991b). Double trouble. Risk, 4 (December–January), 53–56. Rubinstein, M (1991c). Options for the undecided. Risk, 4 (April), 70–73. Rubinstein, M (1991d). Somewhere over the Rainbow. Risk, 4(10) (November), 63–66. Rubinstein, M and E Reiner (1991). Breaking down the barriers. Risk, 4(8), 28–35. Stulz, RM (1982). Options on the minimum or maximum of two risky assets: analysis and applications. Journal of Financial Economics, 10, 161–185. Turnbull, SM and LM Wakeman (1991). A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis, 26(3), September, 377–389.

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Chapter 22 VALUE AT RISK, CREDIT RISK, AND CREDIT DERIVATIVES

Chapter Outline This chapter is organized as follows: 1. Section 22.1 presents the definition of the value at risk (VaR) concept, the risk measurement framework and risk metrics. 2. Section 22.2 studies the statistical and probability foundation of the VaR concept. 3. Section 22.3 develops a more advanced approach to the VaR concept. 4. Section 22.4 concerns credit valuation. 5. Section 22.5 studies default and credit-quality migration. 6. Section 22.6 develops credit-quality correlations. 7. Section 22.7 studies portfolio management of credit risk in the Kealhofer, McQuown and Vasicek (KMV) approach. 8. Section 22.8 is devoted to credit derivatives. 9. Section 22.9 is about models developed by rating agencies and proprietary models.

Introduction Measuring the risks for a financial market participant or a financial institution has become the main focus of modern finance theory. The interest in measuring market risk and monitoring the positions is a consequence of securitization and the need to measure performance. Valueat-Risk (VaR) is a measure of the maximum potential change in the value 917

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of a portfolio of a financial institution or financial instruments with a given probability over a pre-specified horizon. In the same vein, it is important to appreciate credit risk of a financial institution with respect to the appropriate tools. Credit valuation appears today as a fundamental subject in research and practice. Credit-risk analysis and credit valuation need some preliminary definitions of the basic concepts. Credit exposure refers to the amount, which is subject to changes in value upon a change in credit quality or a loss in the event of default. The average shortfall corresponds to the expected loss given that a loss occurs or exceeds a given level. The counterparty refers to the partner in a transaction in which each side takes broadly comparable credit risk to the other. Credit scoring refers, in general, to the estimation of the relative likelihood of default for a firm. The current exposure corresponds to the amount it would cost to replace a transaction now if the counterparty defaults. The default probability refers to the likelihood that an obligor will encounter credit distress within a given period. Credit-quality migration refers to the possibility that an obligor with a certain credit rating migrates to any other credit rating by the risk horizon. Credit derivatives are financial instruments which isolate credit risk. They facilitate the trading of credit risk, its transfer, and hedging. There are different categories of credit derivatives: forwards, swaps, options, and some building blocks combining some of these main products. Credit derivatives are often presented in the form of three classes of instruments: total return swaps (total rate of return swaps, loan swaps, or credit swaps), credit-default instruments and credit-spread instruments. Total return swaps are conceived to transfer the credit risk to the counterparty. Credit-default instruments give a certain payoff upon the occurrence of a default event. They are often in the form of a creditdefault swap or default options. Credit-spreads instruments are often in the form of forward or option contracts on credit-sensitive assets. Credit derivatives allow a re-structuring of the risk/return profiles of credits and permit investors to access new markets. The investors have the possibility to structure and to optimize the risk-adjusted performance of their liabilities by diversifying them among several markets and instruments.

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Value at Risk, Credit Risk, and Credit Derivatives

919

22.1. VaR and Riskmetrics: Definitions and Basic Concepts The VaR concept gives an answer to the following question: How much the investor will lose with x% probability over a specified period of time? Riskmetrics is a set of tools, allowing the users to estimate their exposure to market risk under the “VaR framework”. The market risk corresponds to the potential changes in the value of a position as a consequence of the changes in market prices. JP Morgan developed the Riskmetrics of VaR methodologies and published them in a technical document via Internet. The integration of VaR in modern financial management requires that all positions to be market-to-market and needs the estimation of future variability of the market value. VaR is used by dealers, non-financial corporations, institutional investors, bank and securities firm regulators, and securities and exchange commissions. In 1995, the Basle Committee on banking supervision proposed allowing banks to determine their marketrisk capital requirements by implementing the bank’s VaR model. In this case, the committee specifies the parameters to load into the VaR model. Full-valuation models are based on re-valuing the portfolio on different scenarios. These scenarios can be generated using historical simulation, distributions if returns generated from a set of variance–covariance matrixes etc. This process is known as stress testing. These methods account for the whole distribution of returns instead of a single VaR number. However, they are time consuming. The second difference between VaR approaches is how market movements are estimated. Riskmetrics assumes at the beginning conditional normality to estimate market movement. However, this approach was refined and accounts now for higher order moments of the distribution of returns (kurtosis and leptokurtosis). Many practitioners think that the VaR number can be used to aid managers in the understanding of their risk position. In practice, the management of an individual trader’s book position requires more careful considerations of the risk parameter sensitivities than the single VaR number. This is important for the management of the option-price sensitivities or Greek letters. The European Union’s Capital Adequacy Directive, recognizes VaR models as a valid model for the determination of capital requirements for foreign-exchange risks and other market-risk capital requirements.

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22.1.1. The definition of risk Risk is often defined as the degree of uncertainty regarding future returns. This global definition of risk can be extended to define different kinds of risks according to the source of the underlying uncertainty. The operational risk indicates the possible errors in settling transactions or in instructing payments. The credit risk refers to the potential loss resulting from the inability of a firm to fund its illiquid assets. Market risk refers to the deviations of future earnings due to the changes in market conditions.

22.1.2. VaR: Definition Value-at-Risk is a measure of the maximum potential change in the value of a portfolio of a financial institution or financial instruments with a given probability over a pre-specified horizon. The VaR concept gives an answer to the following question: How much the investor will lose with x% probability over a specified period of time? The following two examples are adapted from Riskmetrics. If an investor estimates that there is a 95% chance that the Euro/USD exchange rate will not fall by more than 2% of its current value over the next day, he/she can determine the maximum potential loss on, for example, USD100 million Euro/USD position. Example 1. Consider a USD-based firm which holds an 105 million FX position. The manager wants to calculate the VaR over a one-day horizon. He/she thinks that there is a 5% chance that the loss will be higher than what VaR projected. The exposure to market risk must be determined in a first step. The exposure corresponds to the market value of the position in the investor base currency (the US firm). If the foreign exchange rate is 1.05 /USD, the market value of the position is USD100 million or (105/1.05). The VaR of the position in USD is determined in a second step. The VaR is given by 1.65 times the standard deviation. It is approximately equal to the market value of the position times the estimated volatility. If the estimated volatility is 0.55%, then: FX risk = 100 million (1.65)(0.55%) = 907.500 Dollars.

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Hence, in 95% of the time, the firm will not lose more than 907,500 over the next day. Example 2. Consider now a USD-based firm which holds an 105 million position in a 10-year government bond. The manager wants to calculate the VaR over a one-day horizon. He/she thinks that there is a 5% chance of understating the realized loss. In a first step, the exposure to market risk must be determined. The exposure corresponds to the interest-rate risk on the bond and the FX risk resulting from the Euro. The market value of the position is still USD100 million, which is at risk to the two market risk factors (interest rates, exchange rates etc). The estimation of the interest-rate risk in a second step needs the calculation of the standard deviation on a 10-year European bond in Euro. Suppose this standard deviation is equal to 0.58%. In this case, we have: Interest-rate risk = 100 million Dollars (1.65)(0.58%) = 957,000 Dollars. The estimation of the FX risk is given by: FX risk = 100 million Dollars (1.65)(0.55) = 907,500 Dollars. It is important to note that the total risk of the bond must account for the return on the Euro/USD exchange rate and the return on the 10-year European bond. Assume the correlation is equal to −0.3. The total risk of the position is: 2 VaR = σIrate + σF2 X + 2ρIrate, F X σIrate σF X or VaR =

0.9572 + 0.90752 + 2(−0.3)(0.957)(0.9075).

22.2. Statistical and Probability Foundation of VaR Is the distribution of return constant over time? The time series reveals volatility clustering since periods of large returns are clustered and distinct from periods of small returns, which are also clustered. This shows clearly a change in variances referred to as heteroscedasticity. Researchers investigate alternative modeling methods other than the normal distribution. These models use either unconditional (timeindependent) or conditional distributions of returns (time-dependent).

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The first class corresponds, for example, to the normal distribution, finite-variance symmetric and asymmetric stable paretian distributions etc. The second class of models corresponds to GARCH and stochastic volatility models for example. 22.2.1. Using percentiles or quantiles to measure market risk The percentile or quantile corresponds to a magnitude (the dollar amount at risk) and is given by the following formula for a continuous probability distribution: α f (r)dr; p= −∞

where f (r) corresponds to the probability density function. The fifth percentile is the value such that 95% of the observations lie above it. If t) , then r˜t is normal with mean 0 and a unit we define r˜t as r˜t = (rtσ−µ t variance. Example. Suppose an investor wants to find the 5% percentile of rt under the normal distribution. Since probability (r˜t < −1.65) = 5%, or

(rt − µt ) probability r˜t = < −1.65 = 5%, σt

then probability (rt < −1.65σt + µt ) = 5%. This equation says that there is a 5% probability that an observed return at time t is less than −1.65 times its standard deviation plus its mean. When µt = 0, we obtain the classic result for short-term horizon VaR calculation: Probability (rt < −1.65σt) = 5%. 22.2.2. The choice of the horizon Several models consider a horizon of one day and 95% to 99% confidence interval in the measurement of the amount of risk for the institution. This horizon assumes implicitly that markets and assets are very liquid and

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Value at Risk, Credit Risk, and Credit Derivatives Table 22.1.

VaR of a single asset at the 1% level.

Alpha (in %)

1−Alpha (in %)

Z−alpha

Sigma t/t − 1 (in %)

V (t − 1)

VaR

1 1 1 1 1 1 1

99 99 99 99 99 99 99

2.3263 2.3263 2.3263 2.3263 2.3263 2.3263 2.3263

10 11 12 13 15 20 25

100 100 100 100 100 100 100

23.263 25.59 27.916 30.242 34.895 46.527 58.159

Table 22.2.

VaR of a single asset at the 5% level.

Alpha (in %)

1−Alpha (in %)

Z−alpha

Sigma t/t − 1 (in %)

V (t − 1)

VaR

5 5 5 5 5 5 5

95 95 95 95 95 95 95

1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449

10 11 12 13 15 20 25

100 100 100 100 100 100 100

16.449 18.093 19.738 21.383 24.673 32.897 41.121

allow the different participants to unwind their positions in one day. This approximation is mainly valid for market-trading activities. The correlation between financial prices and aggregation Several VaR models use the historic correlations among the different risk factors with respect to the main results in modern portfolio theory. However, the historic correlations are unstable and other measures can be used. For more details, see Garman (1996b), Hoppe (1998), and Bellalah and Lavielle (2003). The following Tables 22.1 and 22.2 provide the VaR of a single asset at the 1% level. 22.3. A More Advanced Approach to VaR The VaR corresponds to a number indicating the potential change in the future value of a given portfolio. In the process of calculating the VaR, the manager must specify the horizon for the calculation as well as the “degree

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of confidence” chosen. VaR calculations can also be done without resorting to the standard deviation. Consider a manager who wants to compute the VaR of a portfolio over one day with a 5% chance that the actual loss in the portfolios value is higher than the VaR estimate. In this case, the VaR calculation is performed in four steps. The first step determines the current value of the portfolio V0 on a markto-market basis. The second step determines the future value of the portfolio V1 = V0 er˜, where r˜ refers to the portfolio’s return over the horizon. The third step estimates the one-day return on the portfolio, r˜ in a way such that there is a 5% chance that the actual return will be less than r˜, i.e., probability (r < r˜) = 5%. The fourth step determines the portfolios future “worse case” value, V˜1 as: V˜1 = V0 er˜ In this context, the VaR estimate is: V0 − V˜1 . This VaR estimate can also be written as: V0 (1 − er˜). When r˜ is small, er˜ is nearly equal to (1 + r˜). In this case, VaR is nearly V0 r˜. Riskmetrics gives an estimation of r˜. Example: Consider a portfolio with a current mark-to-market value V0 = USD700 million. The determination of the VaR requires first the one-day forecast of the mean. J-P Morgan assumed that this mean is zero over one day. Then, we must calculate the standard deviation of the returns of the portfolio. If the return on the portfolio is distributed conditionally normal, then: assumed that the change in value of the portfolio is approximated by its delta. The other greeks can also be used to appreciate the change in value. The second approach involves creating a large number of possible rate scenarios and revaluing the portfolio under each scenario. VaR is defined in this context as the fifth percentile of the distribution of the value changes. More than one VaR model is used in practice since practitioners have

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selected an approach based on their specific needs. The models implemented differ on the way changes in the values of a portfolio are estimated as a reaction to market movements. They also differ on the way the potential market movements are estimated. Basically, there are two approaches to estimate the change in the portfolios value as a result of market movements: the analytic approach and the simulation approach. Tables 22.3 to 22.5 illustrate the computation of VaR for a portfolio of 21 assets. Non-linear positions correspond, for example, to a portfolio of options. Riskmetrics provides two approaches to compute VaR of non-linear Table 22.3. assets.

Weights, drifts, and volatility for a portfolio with 21

Stocks

Weights (in %)

Drift (in %)

Volatility (in %)

A B C D E F G H I J K L M N O P Q R S T

6.945 6.254 7.537 1.507 3.868 0.987 3.305 0.180 5.770 1.165 0.000 2.518 4.018 5.248 3.103 1.951 3.844 1.902 0 4.702

0.79 1 2.8 0.27 2 2.35 1 1 2 2.34 1 1 1.33 1 0.5 0.59 1 0.5 2.47 2.11

20 21 22 23 24 25 26 27 28 29 20 20 21 22 23 24 25 26 20 20

Risk-free asset

35.198

Table 22.4.

0

0

VaR of the portfolio for an horizon of 292 days (1 year).

Alpha

1−Alpha

Z−alpha

VaR

Horizon time (days)

5%

95%

1.6449

8%

292

T5

T6

T7

T8

T9

T10

T11

T12

T13

T14

T15

T16

T17

T18

T19

T20

T21

1 0.086 0.116 0.114 0.108 0.167 0.064 0.119 0.037 0.108 0.191 0.114 0.094 0.055 0.066 0.072 0.051 0.060 0.189 0.099 0

0.086 1 0.096 0.094 0.089 0.137 0.051 0.098 0.028 0.089 0.161 0.093 0.076 0.041 0.051 0.056 0.038 0.045 0.160 0.080 0

0.116 0.096 1 0.137 0.130 0.2 0.074 0.142 0.040 0.130 0.234 0.136 0.111 0.060 0.075 0.082 0.055 0.066 0.232 0.117 0

0.114 0.094 0.137 1 0.144 0.222 0.082 0.158 0.044 0.144 0.336 0.151 0.123 0.065 0.082 0.090 0.061 0.072 0.259 0.129 0

108 0.089 0.130 0.144 1 0.234 0.087 0.166 0.047 0.152 0.295 0.159 0.130 0.069 0.087 0.095 0.064 0.076 0.273 0.136 0

0.167 0.137 0.2 0.222 0.234 1 0.146 0.280 0.079 0.255 0.493 0.268 0.219 0.118 0.148 0.162 0.110 0.130 0.458 0.230 0

0.064 0.051 0.074 0.082 0.087 0.146 1 0.126 0.041 0.115 0.214 0.121 0.1 0.057 0.070 0.076 0.054 0.063 0.201 0.103 0

0.119 0.098 0.142 0.158 0.166 0.280 0.126 1 0.072 0.226 0.428 0.236 0.194 0.107 0.132 0.145 0.099 0.117 0.403 0.201 0

0.037 0.028 0.040 0.044 0.047 0.079 0.041 0.072 1 0.072 0.126 0.075 0.064 0.041 0.047 0.051 0.039 0.043 0.119 0.064 0

0.108 0.089 0.130 0.144 0.152 0.255 0.115 0.226 0.072 1 0.429 0.238 0.196 0.109 0.134 0.147 0.101 0.119 0.406 0.2 0

0.191 0.161 0.234 0.336 0.295 0.493 0.214 0.428 0.126 0.429 1 0.479 0.391 0.207 0.261 0.288 0.192 0.229 0.831 0.4 0

0.114 0.093 0.136 0.151 0.159 0.268 0.121 0.236 0.075 0.238 0.479 1 0.270 0.150 0.185 0.203 0.139 0.164 0.559 0.268 0

0.094 0.076 0.111 0.123 0.130 0.219 0.1 0.194 0.064 0.196 0.391 0.270 1 0.137 0.167 0.182 0.128 0.149 0.485 0.234 0

0.055 0.041 0.060 0.065 0.069 0.118 0.057 0.107 0.041 0.109 0.207 0.150 0.137 1 0.111 0.120 0.091 0.104 0.270 0.135 0

0.066 0.051 0.075 0.082 0.087 0.148 0.070 0.132 0.047 0.134 0.261 0.185 0.167 0.111 1 0.150 0.111 0.129 0.347 0.171 0

0.072 0.056 0.082 0.090 0.095 0.162 0.076 0.145 0.051 0.147 0.288 0.203 0.182 0.120 0.150 1 0.126 0.150 0.392 0.193 0

0.051 0.038 0.055 0.061 0.064 0.110 0.054 0.099 0.039 0.101 0.192 0.139 0.128 0.091 0.111 0.126 1 0.119 0.276 0.141 0

0.060 0.045 0.066 0.072 0.076 0.130 0.063 0.117 0.043 0.119 0.229 0.164 0.149 0.104 0.129 0.150 0.119 1 0.336 0.170 0

0.189 0.160 0.232 0.259 0.273 0.458 0.201 0.403 0.119 0.406 0.831 0.559 0.485 0.270 0.347 0.392 0.276 0.336 1 0.537 0

0.099 0.080 0.117 0.129 0.136 0.230 0.103 0.201 0.064 0.2 0.4 0.268 0.234 0.135 0.171 0.193 0.141 0.170 0.537 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21

Correlation matrix for the portfolio

T2

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Table 22.5. T1

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positions. The first is an analytical approximation. The second is a structured Monte–Carlo simulation. 22.4. Credit Valuation and the Creditmetrics Approach 22.4.1. The portfolio context of credit The need for a quantitative portfolio approach to credit risk management allows the study of concentration risk. This risk refers to additional risk resulting from higher exposure to one or several correlated obligors. A portfolio credit-risk methodology as the one in Creditmetrics allows to capture simultaneously the benefits of diversification and concentration risks and provides an efficient risk-based capital allocation process. If we consider a bond rated BBB, which matures in n periods, then at the end of the year, the bond stays at BBB, the issuer defaults or it migrates up or down to one of the other categories. Hence, the probabilities that this bond will end up in one of the other categories in a period allow the computation of the bond price under each of the possible rating scenarios. The new present value of the bond can then be calculated from the remaining cash flows under its new ratings. The discount rate is obtained from the forward zero curve, which is different for each rating category. The knowledge of the probabilities or likelihoods for the bond to be in a given rating category and the values of the bond in these categories allow the determination of the distribution of value of the bond in one period. 22.4.2. Different credit risk measures The two measures used in creditmetrics in the appreciation of credit risk are the standard deviation and the percentile level. These measures reflect the potential losses from the same portfolio distribution. The first measure (determination of the standard deviation) needs the computation of the mean value for the portfolio by multiplying the values with the corresponding probabilities and then adding the resulting values. This allows the computation of the standard deviation. The second measure is a specified percentile level which indicates the lowest value that the portfolio will acheive 1% of the time (the first percentile). The likelihood that the actual portfolio value is less than this number is only 1%. The above analysis and the proposed concepts apply to other types of exposures like loans, letters of credit, swaps, and forward contracts.

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22.4.3. Stand alone or single exposure risk calculation The procedure used in creditmetrics for the computation of the credit risk for a single or stand-alone exposure is based on three steps.

The first step: Credit-rating migration This step estimates changes in value due to up (down) grades and default. The likelihood of credit-rating migration is conditioned on the senior unsecured credit rating of the obligor. A transition matrix is conceived using public rating migration data. This allows the determination of the likelihoods of migration to any credit-quality state in a given period.

The second step: Valuation The values at the risk horizon are determined for the credit-quality states. When the credit-quality migration corresponds to a default case, the likely residual value is a function on the seniority class of the debt. When the credit-quality migration is in an other category, the forward zero curves for each rating category are used to re-value the bond’s remaining cash flows.

The third step: Credit-risk estimation Using the likelihoods and values, it is possible to calculate the risk estimate: the standard deviation or the percentile level.

22.4.4. Differing exposure type Creditmetrics determines the credit risk for market-driven instruments like swaps and forward contracts. The value of a swap is given by the difference between two components. The first corresponds to the forward risk-free value of the swap cash flows. This component is the same for all forward credit rating states. The second component corresponds to the loss expected on a swap, resulting from a default net of recoveries by the counterparty on all the cash flows after the risk horizon. The difference between this second component and the first one allows the re-valuation of the swap. The re-valuation of the swap is based on the following formula: value of the swap in a period (with rating R) = risk-free value in one period expected loss in period 1 through maturity (with rating R), where R refers to any

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credit-rating category. The expected loss for each forward non-default credit rating is given by: Expected loss (with rating R) = average exposure (from period 1 to maturity). Probability of default in period 1 through maturity (with rating R) (1 − recovery fraction). The average exposure calculation is time consuming. The probability of default for each rating category between year 1 and the maturity corresponds to the expected loss calculation. It is calculated using a transition matrix. This method allows the computation of the swap value in each of the non-default credit-rating categories. In the case of default during the risk interval, the expected loss in the defaulted state is given by: expected loss (case of default) = expected exposure (in the first period) (1 − recovery fraction). This expression assumes that the risk interval is very short (one year for example). 22.5. Default and Credit-Quality Migration in the Creditmetrics Approach 22.5.1. Default Credit-rating agencies assign an alphabetic or numeric label to rating categories. They defined a default event with reference to missed interest and principal payments. The likelihood of credit distress is defined in Creditmetrics with respect to default rates. They use credit ratings as an indication of the chance of default and credit-rating migration likelihood. They consider that the firm-encountered credit distress even in a context when only the subordinated debt realized a default. The methodology in J.P Morgan assumes that the senior credit rating is the most indicative of encountering credit distress. Filling probabilities of default with a transition matrix Creditmetrics uses historical default studies to obtain transition matrices, which comprise one-year default rates. 22.5.2. Credit-quality migration The value of a firm and its assets changes suggest changes in credit quality. As a firm moves towards bankruptcy, the value of equity falls. Since, the

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credit rating of the firm is given, it is possible to work backward to the “threshold” in asset value that delimits default. In the approach used by JP Morgan, the firm default model uses the default likelihood to place a threshold below which default appears. The methodology uses the rating migration probabilities to define thresholds above which the firm would upgrade or downgrade from its current credit rating. The rating migration probabilities are represented by a transition matrix. This matrix is a square table of probabilities which reflect the likelihood of migrating to an other rating category in a period given the obligator’s present credit rating. 22.5.3. Historical tabulation and recovery rates Creditmetrics uses a technique to model different volatilities of creditquality migration conditioned on the actual credit rating. Hence, each row in the transition matrix allows the description of a volatility of creditrating changes, which is unique to that row’s initial credit rating. Some rating agencies publish tables of cumulative default likelihood over longer holding periods. It is possible to use a cumulative default-rate table to get an implied transition matrix which better replicates the default history. In this spirit, Creditmetrics uses a transition matrix to model credit-rating migrations. This matrix can be constructed by using a least squares fit to the cumulative default rates. The Markov process is used to model the proceess of default. Using the Markov process, it is possible to generate a cumulative default rates matrix from an imputed transition matrix. In the event of default, the estimation of recovery rates is not an easy task. The bond market prices is an efficient way the future realized liquidation values. The examination of the recovery statistics by seniority class shows that the subordinated classes are appreciably different from one another in their recovery realizations. However, there are no statistically significant difference between secured and unsecured senior debt. Table 22.6 provides the one-year transition matrix. Table 22.7 gives the recovery rates for different types of debt. Table 22.8 gives the credit spreads for different ratings and years. 22.6. Credit-Quality Correlations The estimation of default correlations is not an easy task.

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Value at Risk, Credit Risk, and Credit Derivatives Table 22.6.

AAA AA A BBB BB B CCC

Input data 1 year transition matrix.

AAA (in %)

AA (in %)

A (in %)

BBB (in %)

BB (in %)

B (in %)

CCC (in %)

Default (in %)

Sum (in %)

90 0.70 0.08 0.03 0.02 0 0.17

8.05 91.14 2.32 0.29 0.09 0.09 0

0.60 6.30 90 6 0.34 0.11 0.24

0.05 0.40 4 85 5.66 0.30 1

0.20 0.10 0.70 5 84 6.34 1.99

0 0.20 0.30 1.10 6.15 86 8.50

0 0.01 1 0.09 0.80 3.50 70

1.10 1.15 1.60 2.49 2.94 3.66 18.10

100 100 100 100 100 100 100

Table 22.7.

Mean (%)

Standard Deviation (%)

65 45 35 25 15

20 18 16 19 10

Senior secured Senior unsecured Senior subordinated Subordinated Junior subordinated

Table 22.8.

AAA AA A BBB BB B CCC

Recovery rates.

Credit spreads.

1 (in %)

2 (in %)

3 (in %)

5 (in %)

7 (in %)

10 (in %)

20 (in %)

30 (in %)

0.12 0.42 0.52 0.62 0.72 0.82 0.92

0.17 0.47 0.77 1.07 1.37 1.67 1.97

0.22 0.52 0.82 1.12 1.42 1.72 2.02

0.27 0.57 0.87 1.17 1.47 1.77 2.07

0.32 0.62 0.92 1.22 1.52 1.82 2.12

0.37 0.67 0.97 1.27 1.57 1.87 2.17

0.42 0.72 1.02 1.32 1.62 1.92 2.22

0.47 0.77 1.07 1.37 1.67 1.97 2.27

The study of the histories of the rating changes and defaults reveal the existence of correlations in credit-rating changes. Creditmetrics use the following formula to compute the average default correlation from the data: σ2 N (µ−µ − 1 2) σ2 ρ= ∼ (N − 1) (µ − µ2 )

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where: N : number of names covered in the data; µ: average default rate over the years in the study; σ: standard deviation of the default rates observed from year to year. The estimation of default correlations is a first step towards the estimation of the joint likelihood of any possible combination of credit-quality outcomes. When the credit rating system uses n states (AAA, . . . ,default), then between two obligators, there exists n(n) possible joint states for which the likelihoods can be estimated. The estimation of joint credit-quality migration likelihoods can be done using credit ratings time series across several firms. This estimation method assumes that all firms with a given credit rating are identical.

22.7. Portfolio Management of Default Risk in the Kealhofer, McQuown and Vasicek (KMV) Approach Corporate bonds and liabilities are subject to default risk. The default risk is in general less than 0.5% for a typical high-grade borrower. This risk cannot be hedged away. However, it can be shifted and someone must bear it in the end. Portfolio theory and quantitative methods have been used to compute the amount of risk reduction attainable through diversification for a portfolio of equity. This theory is widely used by practitioners who developed techniques for computing the asset attributes, which are fundamental for an actual portfolio management tool. These developments have not been implemented for debt portfolios. KMV developed these methods in its practice with commercial banks in order to measure diversification and to maximize return in debt portfolios.

22.7.1. The model of default risk When a lender acquires a corporate note, it is as if he/she is engaged in two transactions. The first is buying a debt obligation. The second is the sale of a put option to the borrower. In fact, when the firm’s assets are less than the face value of the debt, the borrower “puts” the assets to the lender and uses the cash proceeds to pay the note. Hence, the situation of the lender can be represented by a portfolio with two assets: long a risk-free bond and short a default option. The probability of default can be determined using option pricing theory and mainly the volatility of the firm’s assets. If you

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consider, for example, a firm with a market value of 100 million Dollars and a debt of 50 million Dollars (maturing in one year) in the presence of a given volatility, then it is possible to represent the range of possible asset values and their frequencies in a diagram. The diagram representing the future firm asset value and the frequency distribution will show a default point at 50 million (the left-hand side of the frequency distribution). Hence, the default risk of a company can be derived from the behavior of its asset values and its liabilities. 22.7.2. Asset market value and volatility The firm’s equity behavior can be also derived from the firm’s asset values. The position of stockholders can be viewed as a call on the firm’s asset value, where the strike price is given by the face value of debt obligations. If at the debt’s maturity date, the firm’s value is higher than the amount of liabilities, stockholders exercise their calls by paying off their obligations. Otherwise, they default. When the market value changes, this induces changes in the value of liabilities depending on the degree of seniority. When the asset value falls to a critical level, the probability of default increases and the market value of the liabilities decreases. In general, when the asset value falls, the volatility of equity increases. Option pricing theory can be used to infer the volatility and asset value. 22.8. Credit Derivatives: Definitions and Main Concepts 22.8.1. Forward contracts Forward contracts on bonds can be either cash-settled or physical-settled. The cash flows for this forward agreement commits the buyer to buy a given bond at a specified future date at a pre-determined price specified at contract origination (time t = 0). The agreement can specify this instead of using the price, the bond’s spread over a treasury asset or a benchmark will be used. In this operation, there are two maturity dates: the maturity of the forward agreement and the maturity of the reference bond. The maturity of the forward agreement is, in general, shorter than that of the reference bond. Since, in general, the default risk is borne by the buyer of the forward contract on the spread, he/she will pay at the maturity date the following quantity: (spread in forward agreement − spread at maturity) duration (notional amount). When there is a credit event, the transaction is marked to market and unwound.

September 10, 2009 14:41

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22.8.2. The structure of credit-default instruments Credit-default instruments dissociate the risk of default on credit obligations. They are often presented in the form of either credit-default swaps or credit-default options. In general, when there is a credit event by the reference credit, then according to the terms of a credit-default swap, the bank pays the counterparty an agreed default payment. The counterparty pays a periodic fee and benefits from the protection of the risk of default of the reference credit. This credit derivative structure is based on the replication of the total performance of the underlying credit asset (a reference bond, a loan). The swap is done between an investor and the bank. The investor assumes the risks of the reference bond. The bank passes through all payments of the bond and in return, the investor makes a payment akin to a funding cost. The transaction is based on a notional amount. The current bond price is used to compute the settlement value under the transaction. The investor receives interest payments and pays a money-market interest rate plus or minus a specified margin. 22.8.2.1. Total return swaps In a total return swap, the rate payer makes periodic payments to another party (the total return payer). He/she receives the total return less the principal and interest payments plus or minus the price changes of the reference asset. The total return swap is often used in the swap structures on default risk. The two parties in a total return swap define at origination the initial value P0 of the reference asset and agree on the reference rate. At any time between t = 0 and the maturity date corresponding to the settlement dates, the asset receiver obtains the cash flows from the reference asset. He/she pays a certain amount fixed with reference to the reference rate. At the maturity date of the contract, the value of the reference asset PT is used. If this value is greater than P0 , the asset receiver gets the difference (PT − P0 ), otherwise, he/she pays the difference (P0 − PT ). The credit swap can also be based directly on a spread. In this case, the asset receiver pays at maturity the difference in the spread of the reference asset over a treasury security with a comparable maturity at origination and at maturity. 22.8.2.2. Credit-default swaps This derivative contract allows one party (the protection seller) to receive fixed periodic payments from the protection buyer. The payments are in

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return of making a single contingent payment covering losses with respect to a reference asset following a default or an other specified “credit event”. The main idea behind credit-default swaps is that they strip off the default risk of some reference assets. This risk will be traded separately. By implementing a credit-default swap, the protection seller earns an investment income and the protection buyer hedges the risk of default on the reference asset. This contract allows investors to hedge credits without implementing costly strategies consisting of buying and selling cash securities and loans. The following example is adapted from a sponsor’s statement by Barclays capital. Example: Consider an investor A who gains customized access through a bank B to a corporate bond by selling 3-year default protection to the bank B on the bond. The investor A receives a fixed premium of 120 bp per annum and agrees to make a credit event payment if the borrower defaults on the bond. In the credit event, the swap terminates and the investor A pays the bank B, the notional times the percentage fall from par of the bond. The investor A can also settle the swap by buying the bond from the bank B at par. 22.8.2.3. Basket default swaps An investor can sell default protection on several assets. In a first-to-default basket-default swap, the protection seller assumes the default risk on a basket of bonds by agreeing to compensate market losses on the first asset in the basket to default. 22.8.2.4. Credit-default exchange swap It is possible to swap a default risk on an asset for that of an other asset. In a credit-default exchange swap, both parties act simultaneously as protection buyers and protection sellers. Example: Consider two institutions A and B. A trades the default risk of a loan it holds for that of a complementary loan held by B. A lays off the default risk on loan A in return for assuming default risk on loan B. If a reference credit experiences a credit event then the protection seller must make a credit event payment to the protection buyer. This can terminate the trade. The trade could continue with the protection buyer in the rest of

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transaction paying an agreed rate. Parties do not make periodic payments and swap just the contingent payments when default risks are perfectly matched. 22.8.2.5. Credit-linked notes (CLNs) Credit linked notes are associated with the credit performance of the underlying assets. A principal protected note protects a pre-set portion of principal. A principal-linked structure pays enhanced fixed coupons and can redeem principal at a rate associated with the credit performance of reference assets. Credit-default notes allow investors either to buy or sell default protection on reference credits. 22.9. The Rating Agencies Models and the Proprietary Models 22.9.1. The rating agencies models The models of default risk developed by rating agencies are based on the current rating and the time to maturity of the obligation. These models determine the cumulative risk of default (the total default probability over a given period) and/or the marginal risk of default (the change in default probability over some periods). In a first special report by the Standard and Poor’s website, we find that corporate defaults rose sharply in 1998. Their database contain a vast collection of statistics on default and rating migration behavior on CD-ROM under the trade name CreditP roT M . The data used corresponds to the issuer credit ratings that reflect S&Ps opinion of a company’s overall capacity to pay its obligations. This opinion is based on the obligor’s ability to meet its financial commitments on a timely based. This indicates in general the likelihood of default regarding the firm’s financial obligations. The definition of default corresponds to the first occurrence of a payment default on any financial obligation. Preferred stocks are not considered as financial obligations since a missed preferred stock dividend cannot be equated to a default. The studies of default by S&P are based on groupings called static pools. A static pool is constructed at the first day of each year covered by the study and followed from that point on. All obligors are followed year to year within each pool. All of Standard and Poor’s default studies reveal a well-defined correlation between credit quality and default remoteness. The rating letters in the

September 10, 2009 14:41

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study are: AAA, AA, A, BBB, BB, B, and CCC. In general, the higher the rating, the lower the default probability and vice versa. Besides, the lower an obligor’s original rating, the shorter the time it takes to observe a default. Using default ratios, the study shows that default rates over a one-year horizon exhibit a high degree of volatility. The default patterns seem to share broad similarities across all pools. This result suggests that S&P rating standards are consistent over time. Using transition analysis, each one-year transition matrix shows all rating movements between letter categories from the begining through the end of the year. Rating transition ratios give useful information to investors and credit professionals. The oneyear rating transition ratios by rating category reveal that higher ratings are long lived. The S&P study assumes that the rating transition rates follow a first-order Markov process. This allows to model cumulative default rates over several horizons. Rating transtion matrices are constructed to produce stressed default rates. Multiyear transitions are also constructed for periods of 2 through 15 years using the same methodology as for single-year transitions. This allows to compute average transition matrices whose ratios represent the historical incidence of the ratings. The study also reveals that for example, 10-year transition ratios are less reliable than their one-year counterparts. The analysis reveals an increase in corporate defaults in 1998. In general, recoveries are estimated on the basis of the prices the defaulted securities fatch at some time after the default event. The data corresponds to 533 S&P’s rated straight-debt issues that defaulted in the period 1981–1997. We denote by: Pd : the prices at the end of the default month, (referring to a default event), Pe : prices (just) preceding liquidation or emergence from out-of-court settlement or Chapter 11 re-organization, (referring to an emergence event). The methodology uses the fact that recoveries are based on the ultimate values yielded by the completion of the bankruptcy process. Table 22.9 reproduced from the study of S&P summarizes the main findings. with: Nobs : the number of observations; SA: the simple average; SD: the standard deviation;

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938

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Seniority — ranking

Recoveries shortly after default.

Nobs

SA, Pd

Sd, Pd

W A, Pd

CV , Pd

Senior — secured Senior — unsecured Subordinated Junior — subordinated

65 180 144 144

58.52 49.60 38.29 35.30

22.27 26.51 25.23 22.29

58.11 53.66 36.86 36.41

0.38 0.53 0.66 0.63

Total

533

43.77

25.81

43.93

0.59

W A: the weighted average by issue size in Dollars and CV : the coefficient of variation. Table 22.9 shows that investors who liquidate a position in defaulted securities (shortly after default) expect to recover, on average, about 44 Cents on the Dollar. This means that on average, creditors receive about 40 Cents on the Dollar. Table 22.9 also shows a certain predictable degree of variation across seniority classes. The study shows that average recoveries at default and emergency are higher, the higher the seniority rankings of the issues used. It also reveals that excess returns exhibit much uncertainty at default and emergence and that uncertainty is in general higher, the lower the seniority of the debt used.

22.9.2. The proprietary models The default probability can be calculated as the probability that asset values will be lesser than the value of the claims on the firm’s assets. In this spirit, KMV Corporation developed the expected default frequency model (EDF). This model needs an estimation of the market asset values, the volatility of the assets and the market value of the liabilities. The volatility can be implied from an option pricing model. Using the asset value, the volatility and the cumulative liabilities, it is possible to calculate the default risk of the firm. This model determines the default probability using the distance in volatility between the asset value and the point at which the asset value will be less than the liabilities. The EDF model uses large databases of firms in the computation of historical default frequencies. This approach allows the estimation of expected and unexpected default losses and derives default within a volatility framework. Default risk corresponds to the uncertainty surrounding the firm’s ability to service its obligations. The default probability of a firm depends on the market value of its assets,

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on the risk of the assets, and on the firm’s liabilities. The risk of the assets is given by the standard deviation of the annual percentage change in the asset value. The methodology of KMV Corporation looks for a default point or the asset value at which the firm will default. The firm defaults when its market net worth is zero. A ratio of default risk referred to as distance default, compares the market net worth to the size of a standard deviation move in the asset value: Distance default = (market value assets − default point) . (market value assets − asset volatility) The ratio says that a firm is n-standard deviation away from default. When the probability distribution is known, the default probability can be computed directly. KMV Corporation proposes a model of default probability, Creditmonitor which computes the EDF. Expected default frequency is the probability of default during some coming years. The determination of the defaut probability of a firm is done in three steps. The first step The asset value and its volatility are estimated using the asset market value, the volatility of equity, and the book value of liabilities. Using an option pricing based approach where equity is viewed as a call on the underlying assets with a strike price equal to the book value of the firm’s liabilities, it is possible to estimate the value of the firm and the volatility of its assets. The second step The distance to default is calculated using the asset value and its volatility, and the book value of liabilities. The default probability depends on the current value and the distribution of the asset value, the volatility, the expected rate of growth in the asset value, the horizon, and the level of the default point. If the future distribution of asset values is known, the default probability and the EDF correspond to the likelihood that the final asset value be below the default point. The third step The default probability is computed using historical data on default and bankruptcy frequencies. A frequency table is generated to link the likelihood of default to different levels of distance default.

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Creditmetrics models the process of value changes resulting from changes in credit quality. It gives risk management information which is tailored to the name, industry, and sector concentrations. The details and data are freely available on Internet.

Summary The VaR corresponds to a number indicating the potential change in the future value of a given portfolio. In the process of calculating the VaR, the manager must specify the horizon for the calculation as well as the “degree of confidence” chosen. Value at Risk calculations can also be done without resorting to the standard deviation. Non-linear positions correspond, for example, to a portfolio of options. The VaR of a portfolio of options can be determined using the “greeks”. The basic method uses an option pricing model to obtain the delta. This delta is used to determine the amount of a market factor that must be held to compensate for a change in the underlying asset price. The present value of the delta hedge position in the underlying is included in the determination of the portfolio variance. This method is efficient only for very small changes in the underlying asset price. In fact, the delta is a linear measure only for very small changes in the underlying asset price over very small intervals of time. Since the VaR is concerned with the effects of large changes in the underlying asset price, the linearity may lead to an inappropriate assesement of market-risk measures. The estimation method is improved when the second derivatives of the delta (the gamma) is used in the risk measure. Since the option price function is nonlinear for different prices of the underlying asset, a risk measure including gamma may also lead to an inaccurate measure of market risk for significant changes in the underlying asset price. However, the simultaneous use of delta and gamma can improve risk estimation. A credit derivative can be seen as “any instrument that enables the trading/management of credit risk in isolation from the other types of risk associated with an underlying asset. These instruments may include: creditdefault products, credit-spread products, total-return products, basket products, and credit-linked notes. Banks are the principal actors in the market of credit risk. They represent the buyers, sellers, and intermediaries of credit derivatives. Bankers are the principal buyers of credit protection. However, any firm or institution with a portfolio subject to credit risk can use credit derivatives. This is the case for firms with concentrated portfolios who can use credit derivatives to manage credit lines. Sellers of credit

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protection are motivated by the desire to enhance their risk-adjusted return on capital. Banks searching for higher yields are using credit derivatives written on low-rated assets. Straight credit-default swaps seem to be the most widely used of all credit derivative products. The pricing of credit derivatives and credit instruments depends on the default probability of the reference asset, the expected recovery rate, and the nature of the exposure. The pricing approaches concern individual transactions and portfolios. The pricing of individual transactions or single transactions concerns the loss exposure, the default probability, the recovery rate, and the correlations between these features in order to predict future credit losses. The pricing in a portfolio approach is focused on the correlations between individual exposures, which means that we must account for the joint default probabilities (correlations between default) and the correlation between loss exposure and recovery rates. The proprietary models are based on an option approach to default in which the equity of a levered firm is equivalent to a call on the net asset value of the firm. This approach is initiated by Black and Scholes (1973). This approach considers the position of debtholders as a combination of a long position in a bond plus a short position in a put on the firm’s assets. References Bellalah, M and M Lavielle (2003). A decomposition of empirical distributions with applications to the valuation of derivative assets. Multinational Finance Journal, 83, 1871–1887. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Garman, M (1997). Taking VaR to pieces. Risk, 10(10), October, 70–71. Garman, M (1996a). Making VaR proactive. Financial Engineering Associates, Working Paper. Garman, M (1996b). Making VaR more flexible. Derivatives Strategy, April, 52–53. Garman, M (1996c). Improving on VaR. Risk, 9(5), May, 61–63. Hoppe, R (1998). VaR and the unreal world. Risk, 11, July, 45–50.

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September 10, 2009 14:46

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b708-Index

Index

Black–Scholes, 221, 222, 254, 339, 367–369, 372, 375–377, 379–381, 384, 394, 395, 397–399, 403–406, 408–411, 413, 422–425, 427, 441, 508, 515, 516, 523–526, 536, 538, 541, 544, 545, 552, 556, 557, 560, 562, 563, 567, 583, 588, 594, 596, 597, 609, 611, 642, 654, 655, 660–664, 678, 679, 714, 746, 752, 755, 757–762, 764, 765, 767, 768, 772, 773, 779, 783, 787, 792, 794, 799, 802, 805, 835, 878–880, 887, 893, 900, 903, 909, 913 bond, 3, 5, 22–34, 46–48, 51, 67, 68, 73, 75–77, 87, 90, 91, 102–106, 126, 128, 141, 212, 237, 259, 260, 266, 267, 269–283, 286, 287, 289–291, 293–301, 303, 305–313, 316, 318, 320–323, 328, 340, 341, 353, 354, 374, 376, 382, 393, 395, 397, 404, 407–411, 416, 421, 422, 426, 494, 508, 514, 519–523, 527, 535, 536, 546–551, 555–557, 564, 573, 575, 617, 619, 650, 655, 667–677, 679–692, 694–696, 703–718, 721–727, 731–736, 739, 777, 811–813, 833, 834, 842–847, 849–854, 860, 864, 878, 879, 886, 887, 913, 921, 927, 928, 930, 932–935, 941 bond futures, 27, 29, 667, 668, 670–673, 686, 687 bond option, 3, 22, 26, 46–48, 293, 297, 318, 320, 328, 410, 426, 535,

algorithm, 313, 630, 638, 729, 730, 757, 762, 766, 801, 810, 816, 820, 822, 823, 833, 834, 837, 839, 846, 847, 853, 861, 862, 864 American option, 6, 10, 13, 50, 51, 98, 105, 222, 225, 234, 235, 244, 249, 250, 252, 294, 324, 333, 348, 359, 361, 380, 536, 539, 615–618, 621, 622, 627, 628, 632, 634, 637, 640–642, 645, 655, 656, 799, 801, 802, 811, 817, 819, 852, 880 arbitrage, 10, 14, 20, 40, 46, 49, 60, 67, 72, 73, 76–78, 84, 85, 87, 89, 90, 97–99, 106, 125–127, 224, 237, 298, 300, 321, 340, 342, 343, 367, 368, 375, 382, 383, 393, 396, 406, 407, 416, 420, 426, 493, 494, 496, 515–517, 519, 521, 522, 527, 537, 546, 547, 552, 555, 560, 573, 617, 619, 621, 635, 643, 656, 658–660, 669, 670, 673, 674, 686, 704, 710, 711, 714, 720, 734, 777, 802, 812, 834, 843, 849, 852, 878, 883 asset pricing, 339, 367, 368, 398, 425, 509, 517, 551, 767, 768, 771–773, 777, 779, 793, 795, 867 binary barrier, 877, 879, 901, 914 binomial models, 221–224, 228, 231, 237, 246–249, 293, 294, 298, 305, 310, 318, 320, 321, 327, 329, 331, 334, 345–347, 349–352, 360–362, 404, 407, 494, 617, 622 bivariate normal, 368, 400, 401 943

September 10, 2009 14:46

944

spi-b708

9in x 6in

b708-Index


546, 564, 667–669, 681, 683, 686, 703, 712, 713, 739, 777, 778, 789, 793, 795, 860 bond yield, 294, 672, 687–689, 725 bounds on options, 67, 85, 127 CAPM, 338, 339, 367, 369, 373, 377, 378, 387, 389–391, 398, 517, 634 choosers, 32, 47, 877–879, 884–887, 913, 914 combined strategies, 141, 150 commodity futures, 393, 403, 425, 446, 583, 584, 590, 592, 593, 616, 627, 628, 655, 656 composite volatility, 801, 803, 811, 812, 816, 819, 820 compound options, 33, 320, 615–617, 642, 644, 645, 648, 650, 653, 654, 656, 877–879, 885–887, 889, 913, 914 conditional expectation, 493, 496, 522, 529, 536, 552, 555, 782 condor, 67, 100–102, 127, 142, 163–166, 175 contingent claims, 32, 34, 47, 222, 294, 300, 321, 327, 328, 331, 345, 346, 355, 362, 368, 494, 536–538, 552, 555, 588, 668, 779, 784, 834, 854, 859, 860, 884 credit crunch, 4, 30, 260, 323 credit risk, 30, 259, 293, 406, 680, 853, 917, 918, 920, 927, 928, 940 credit valuation, 917, 918, 927 crude oil, 3, 6–10, 12, 13, 46, 68, 71, 72, 79, 81, 83 currency, 3, 21, 22, 31, 32, 46, 48, 51, 75, 213, 217, 287, 288, 330, 334, 337, 341, 342, 377, 393, 403, 414–418, 424–426, 446, 471, 655, 668–670, 687, 746, 760, 772, 777, 778, 789, 790, 795, 896, 898, 899, 901, 902, 920 currency options, 32, 330, 334, 377, 393, 403, 416, 417, 424–426, 446, 471, 655, 668, 670, 746, 760, 772, 777, 778, 789, 790, 795, 901

delta, 53, 110, 111, 113, 134, 135, 144, 170, 171, 175, 184, 196, 197, 199, 201–204, 213, 334–337, 378, 382, 384, 385, 418, 419, 440–443, 446–461, 464, 465, 467, 468, 478, 515, 525, 526, 670, 721, 754, 755, 757, 760, 761, 781, 852, 878, 880–885, 887, 889, 890, 893, 897, 901, 902, 905–907, 912, 924, 940 delta hedging, 452, 781 derivatives, 4, 20, 22, 34, 36, 46, 56, 67, 69, 70, 92, 170, 171, 174, 176, 221, 222, 271, 275, 294, 316, 320, 327, 329, 334, 338–340, 345, 355, 356, 361, 362, 367, 368, 378, 380, 382, 387, 398, 399, 405, 412, 425, 439–444, 454, 457, 458, 460, 461, 466, 469, 471, 476, 477, 493–495, 501, 505, 506, 509, 510, 514–516, 524–529, 535–538, 544, 546, 551, 552, 555, 558, 563, 565, 566, 572, 574, 575, 579, 583–591, 593, 594, 596–598, 609, 620, 622, 625, 650, 668, 681, 687, 689, 692, 698, 699, 733, 735, 737, 750, 771, 779, 799, 801, 802, 804–806, 808–810, 814–817, 819–821, 833, 834, 836, 838–840, 842, 854, 857, 868, 875, 917, 918, 933, 934, 940, 941 diffusion process, 405, 493, 518, 527, 528, 547, 549, 587, 594, 704, 746–749, 755, 767, 775, 786, 887, 913 discounting factors, 263, 348, 706 distributions, 19, 35, 40, 48, 60, 74, 87, 88, 221, 237, 246, 248, 249, 293, 310, 323, 339, 340, 348, 361, 367, 369, 370, 373, 376, 377, 384, 388, 393, 399, 404, 425, 495–500, 509, 519, 524, 526, 535, 539, 571, 615–622, 627, 634, 640, 642, 643, 645, 652, 655–657, 669, 671–673, 686, 691, 692, 703, 720, 726, 728–732, 739, 746, 748, 750, 760, 772, 773, 775, 776, 778, 783, 786, 788, 789, 793, 817–819, 834, 837,

September 10, 2009 14:46

spi-b708

Index

845, 886, 888, 891, 900, 919, 921, 922, 924, 927, 933, 939 dividends, 19, 20, 69, 75, 76, 111, 115, 116, 120, 166, 167, 175, 213, 221, 238, 246–248, 317, 328, 330, 331, 333, 334, 341, 342, 348, 369, 376, 377, 393, 394, 396, 397, 404–406, 413, 539, 584, 589, 615, 618–622, 627, 641, 642, 645, 655, 791, 802, 833–835, 852, 861, 862, 864 early exercise, 87, 88, 98, 105, 237, 334, 425, 615, 618–620, 622–624, 626–629, 632, 633, 641–643, 655, 803, 816, 820, 841, 842, 862, 863 embedded call, 833 equity option, 3, 17, 48, 221, 222, 249, 329, 361, 362, 403–405, 409, 535, 564, 653, 667, 678, 896, 898 European option, 6, 33, 50, 51, 98, 104, 221, 234, 235, 237, 244, 250, 252, 294, 320, 324, 333, 345, 361, 367–369, 377, 380, 388, 393, 404, 408, 415, 508, 549, 550, 557, 559, 560, 616, 628, 637, 642, 645, 662, 678, 681, 684–686, 704, 717, 751, 768, 774, 777, 778, 790, 791, 793, 795, 799, 807, 817, 860, 879, 880 extendible, 32, 47, 877, 879, 893–895, 897, 914 financial innovations, 3, 4, 32, 34, 35, 46, 48, 802 financial instruments, 3, 4, 22, 24, 31, 32, 34, 35, 46, 48, 49, 51, 90, 268, 393, 404, 407, 440, 680, 681, 918, 920 fixed income, 37, 57, 667, 676, 686, 704 forward start, 47 forward, 3–6, 8, 9, 11, 13, 21, 27, 29, 31, 46–48, 67–69, 71–79, 90, 125–127, 175, 213, 259, 261, 276, 277, 279–281, 287–291, 303–308, 321, 340–342, 356, 357, 367, 368, 386, 387, 389, 390, 398, 412, 414, 415, 417, 418, 424, 426, 493, 507,

9in x 6in

b708-Index

945

518, 527, 553, 554, 557, 559–563, 573, 590, 592, 635, 637, 646, 668, 676–678, 681, 682, 692–694, 703, 704, 706, 707, 715, 716, 719, 721, 722, 724–728, 731–737, 739, 755, 794, 804, 805, 808, 834, 840, 877–883, 913, 914, 918, 927–929, 933 futures, 3–10, 12–14, 16–22, 26–31, 35, 36, 40, 46–49, 56, 60, 67–84, 90, 98, 103, 125–127, 142, 144, 152, 175, 211, 213, 226, 250, 259–266, 268, 270, 271, 276–278, 287, 289, 290, 294, 299, 305, 316, 320, 327, 328, 330, 331, 338–340, 342–344, 347, 361, 362, 367–369, 376, 377, 386–393, 397–399, 403–407, 409, 410, 413–415, 420, 422, 424–426, 446, 473, 496, 500, 521, 536, 537, 573, 583, 584, 590–593, 615–617, 619, 627, 628, 632–636, 638–642, 655, 656, 663, 664, 667, 668, 670–673, 676, 680, 686–690, 692, 704, 705, 707, 711, 712, 715, 716, 719, 724–726, 737–739, 755, 773, 790, 802, 803, 813, 817, 819, 851, 878, 880, 885, 889, 913, 919, 920, 923, 924, 930, 933, 939–941 futures options, 27, 327, 330, 347, 362, 368, 376, 388, 391, 393, 403, 404, 415, 425, 426, 446, 615, 616, 627, 628, 632–634, 656, 663, 664, 667, 670, 671, 673, 686, 738, 803, 819 gamma, 170–172, 174, 175, 334–337, 384, 385, 418, 419, 440–443, 446–451, 454–461, 464, 465, 469, 478, 670, 749, 880–885, 887, 889, 890, 893, 897, 901, 902, 905–907, 912, 940 Girsanov theorem, 535, 537, 541, 545, 564, 565, 574 Greek letters, 142, 170, 334, 384, 418, 419, 442, 466, 880, 885, 886, 889, 896, 901, 905, 906, 912, 919

September 10, 2009 14:46

946

spi-b708

9in x 6in

b708-Index


heat equation, 373, 570 hedge ratio, 170, 171, 334, 378, 440–442, 452, 460, 461, 524, 525, 674, 720, 752, 781 hedging, 4, 10, 20, 24, 32, 37, 40, 49, 57, 60, 67, 68, 70, 78–82, 85, 125, 127, 183, 196, 211, 249, 372, 396, 398, 416, 422–425, 439, 440, 445, 452–454, 464, 465, 493–495, 516–518, 523, 525–527, 538, 673, 721, 731, 745–747, 749, 750, 752, 754, 768, 779, 781, 782, 795, 813, 850, 852, 875, 878, 918 historical volatility, 37, 57, 167–169, 175, 422, 460 implied volatility, 37, 57, 167, 170, 175, 300, 460, 526, 671, 673, 686, 692, 732, 745–747, 754–758, 760, 761, 763, 766–768, 773, 776, 788, 790, 793, 795 incomplete information, 338, 339, 344, 494, 560, 583–585, 588–590, 593, 596, 615, 636, 650, 655, 669, 687, 747, 752, 759, 760, 772, 833, 849, 852 incomplete market, 767 index options, 18, 36, 37, 47, 49, 56, 57, 221, 330, 394, 396, 403, 405–407, 424, 426, 640, 642, 668, 673, 686, 728, 754, 756, 758, 760, 767, 788, 790, 801–803, 812, 820, 823 information costs, 85, 327, 329, 338–341, 343–348, 360–362, 514, 516, 521, 524–526, 540, 544, 560–563, 583–586, 588–591, 593–597, 617, 634–639, 646, 648, 650–654, 656, 661, 673–676, 687, 703, 732–734, 736, 745, 747, 749, 751–753, 759, 762, 763, 767, 768, 771, 773, 780, 782, 783, 785, 786, 791–796, 807, 810, 820, 833, 839, 849, 850, 852, 879, 903, 914

information uncertainty, 338, 342, 346, 348, 521, 535, 544, 583, 591, 617, 634, 645, 650, 655, 656, 664, 667, 676, 771, 772, 833, 839, 841, 847, 850, 851, 853 interest rate trees, 293, 303, 313, 320, 355 interest rates, 777, 782, 791, 793, 795 Itˆ o lemma, 379, 493–495, 501–505, 507, 509–512, 514, 517, 527, 530–532, 553, 554, 561, 562, 575, 576, 579, 584, 587, 590, 591, 594, 651, 663, 674, 695, 699, 777, 780, 813, 851 jump process, 320, 745–747 lattice approach, 221, 222, 249, 250, 293, 320, 327–329, 331, 343, 344, 346, 348, 360–362 lookback, 32, 47, 875, 877, 879, 908, 909, 911, 912, 914 market conditions, 367, 441, 445, 455, 457, 459, 461, 688, 771 market volatility, 141, 407 martingale approach, 535, 538, 539, 554, 555, 557–560, 563, 564, 615 martingale measure, 547, 574, 771, 772, 778, 779 Merton model, 376, 772 monitoring, 34, 171, 418, 425, 439–441, 445, 446, 451, 454, 455, 457, 458, 461, 917 Monte–Carlo simulation, 927 mortgage backed securities, 291 normal distributions, 310, 367, 369, 370, 373, 384, 399, 495, 497–500, 509, 519, 524, 526, 535, 571, 643, 652, 669, 672, 726, 728, 729, 731, 732, 746, 772, 818, 819, 886, 888, 891, 921, 922 numerical analysis, 799, 801, 803, 805, 820

September 10, 2009 14:46

spi-b708

9in x 6in

947

Index

numerical procedure, 221, 328, 353, 361, 833, 839 numerical schemes, 801–804, 806, 814–816, 821, 822, 855, 858, 859 oil markets, 3, 6–9, 12, 46, 71, 78, 125 option, 3–6, 8, 10, 13, 17–22, 24, 26–28, 31–33, 36, 37, 40, 41, 46–53, 55–57, 60, 61, 67, 69, 70, 77, 85–96, 98, 99, 102–107, 110, 111, 113, 114, 116–129, 141–144, 147, 148, 150, 152–154, 156, 157, 159, 160, 166–168, 170, 171, 174–177, 183–188, 194, 196, 202, 208–217, 221–238, 240–246, 248–254, 270, 293, 294, 297, 299–302, 316, 318, 320, 324, 325, 327–334, 338, 339, 343–351, 353, 359–362, 367–383, 386–399, 403–418, 420–426, 439–446, 448, 449, 451–462, 466, 471, 478, 494, 508, 513–518, 523–527, 535, 536, 538–542, 545, 546, 549, 550, 552–554, 556, 557, 559–565, 568, 569, 572, 583, 588, 611, 612, 615–623, 627–651, 653–657, 659–664, 667–671, 673, 676–688, 690–696, 698, 703, 704, 710, 712–717, 719, 720, 728, 731, 732, 737–739, 745–747, 751–760, 762–768, 771–783, 786, 788–796, 799, 801–814, 816–821, 823, 825, 829, 833–842, 844, 852, 853, 858–861, 867, 875, 877–889, 891–896, 898–914, 918, 919, 925, 932–934, 938–941 option combinations, 67, 94 option pricing, 141, 142, 161, 167, 170, 175, 221, 224, 294, 299, 320, 328, 349, 361, 365, 367–371, 375, 391, 397, 398, 403, 412, 416, 425, 439, 458, 491, 494, 517, 518, 523, 527, 538, 564, 572, 613, 616, 617, 627, 642, 653–655, 667, 668, 670, 671, 686, 714, 719, 720, 743, 745, 746, 754, 759, 763, 767, 768,

b708-Index

771–775, 777, 778, 788, 789, 791, 799, 801, 802, 819, 834, 932, 933, 938–940 option strategies, 94, 126, 141–143, 409 options markets, 3, 8, 17–19, 22, 33, 40, 46, 48, 49, 55, 60, 69, 348, 368, 369, 374, 415, 426, 453, 516, 517, 646, 751 parametric approach, 667, 670 partial differential equation, 353, 379, 381, 387, 420, 493–495, 515, 516, 518, 535, 536, 549, 555–558, 561, 563–568, 588, 609, 619, 621, 624, 628, 651, 698, 735, 772, 774, 786, 799, 801–803, 833, 834 path dependent options, 32, 33, 47, 829 pay later, 31, 47, 875, 877–879, 882–884, 913, 914 portfolio insurance, 36, 56, 67, 103, 127, 128, 406, 414, 415, 421 pricing biases, 771 pricing bonds, 259, 266, 328, 573 pricing derivatives, 538, 546, 735, 779 pricing theory, 367, 368, 397, 403, 517, 564, 671, 772, 932, 933 probability, 178, 224, 296, 298, 300, 303, 307, 308, 322, 323, 344, 349–351, 355, 356, 360, 393, 395, 400, 401, 496, 498, 518, 522, 527–529, 535–538, 541, 542, 544–548, 551, 552, 555–558, 563–565, 573–575, 619, 621, 623, 628, 632, 636, 655, 669, 672, 673, 691, 692, 711, 712, 714, 728–731, 738, 746, 750–752, 754–759, 764, 771, 774, 775, 778, 782, 794, 802, 818, 895, 896, 903, 904, 917–922, 924, 929, 932, 933, 936–939, 941 probability foundation, 917, 921 quantos, 33, 877, 896, 914

September 10, 2009 14:46

948

spi-b708

9in x 6in

b708-Index


ratio spread, 102, 103, 127, 174, 404, 460 rating, 24, 31, 917, 918, 927–932, 936, 937 real world, 403, 404, 422, 439, 592 replication, 209, 421, 465, 493, 494, 513, 514, 516, 527, 537, 538, 879, 934 risk management, 34, 37, 57, 67, 141, 403, 493, 535, 615, 667, 721, 767, 771, 801, 877, 927, 940 risk measures, 170, 425, 439–441, 460, 461, 466, 478, 927, 940 risk neutral, 303, 327, 349, 351, 361, 589, 735, 771, 773–775, 778, 781–783, 792–794 risk neutral probability, 303, 771, 774, 775, 778, 782, 793, 794 risk parameters, 417, 418, 461, 919 smile effect, 746, 754, 766, 767, 771, 773, 788, 789, 795 speculation, 14, 49, 67, 78, 84, 125–127, 416 spot assets, 46, 142, 329, 388–391, 398, 399, 557, 583, 584, 590, 593, 635, 656, 663, 664, 777, 778, 789, 795 spot options, 426, 553, 664 stochastic interest rates, 320, 369, 404, 555–558, 562, 667, 668, 678, 681, 686, 777, 795, 801–803, 816, 819, 851 stochastic process, 493, 494, 498, 528, 536, 686, 711, 712, 715, 731, 746 stochastic volatility, 369, 404, 745, 771–778, 782, 785, 788, 789, 793–795, 802, 819, 860, 922 straddles, 67, 69, 94, 95, 126, 127, 142, 150–154, 174, 175, 460, 884 subprime crisis, 4 synthetic positions, 128, 142, 143 Taylor series, 274, 352, 353, 379, 493–495, 501, 509, 513, 518, 520, 523, 524, 529, 530, 775, 857

term structure, 790 theta, 170, 171, 173–175, 334–337, 384, 385, 418, 419, 440, 441, 443, 446–451, 456–461, 464, 465, 473, 475, 478, 670, 729, 730, 811, 882–885, 887, 889, 890, 897, 901, 902, 905–907, 912 trading, 3, 5–7, 9–13, 17, 18, 21, 22, 25–28, 36, 37, 40, 46–48, 51–57, 60, 67–71, 73, 78, 85, 87, 88, 90, 98, 126, 127, 141, 167, 168, 170, 174, 338, 374, 375, 377, 397, 407, 418, 421–423, 425, 457, 524, 678, 720, 756, 758, 784, 918, 923, 940 trading mechanisms, 67 trinomial trees, 293, 313, 314, 316, 317, 320, 328, 353, 361 valuation, 34, 68, 74–76, 85, 107, 117, 120, 177, 212, 213, 221, 222, 228, 231, 234–238, 244–246, 249–253, 287, 293, 294, 305, 312, 316, 318, 320, 321, 324, 325, 327–329, 331, 332, 338–343, 345, 358, 360–362, 367–374, 376, 378, 379, 390, 397–399, 403–405, 407–411, 414–416, 424, 425, 427, 446, 461, 478, 508, 514, 528, 535, 539, 544, 546, 551, 552, 554, 560, 562, 564, 583–585, 592, 593, 596, 615–618, 620, 622, 627, 628, 632, 634, 637–640, 642, 643, 645, 648, 650, 654–657, 659, 667–669, 671, 680, 684, 686, 687, 692, 704, 713, 716, 718, 720, 731, 737, 739, 745–747, 749, 754, 763, 765–768, 772, 773, 778, 779, 792, 795, 796, 801, 803, 807, 808, 810–812, 817, 819, 820, 823, 833–835, 837, 839, 840, 843, 845, 850, 852, 853, 859, 860, 875, 878, 879, 887, 889, 891, 893, 896, 913, 917–919, 927, 928 value at risk, 254, 917, 920, 940 vega, 170–172, 174, 175, 334–337, 384, 385, 418, 419, 441, 444,

September 10, 2009 14:46

spi-b708

Index

446–451, 458–461, 469, 470, 478, 670, 882–885, 887, 889, 890, 897, 901, 902, 905–907, 912 Wiener process, 295, 378, 493–497, 505, 509, 524, 527, 585, 587, 590, 643, 695, 748, 774, 776, 784, 811, 851

9in x 6in

b708-Index

949

yield curve, 259, 260, 275–277, 289, 290, 297, 316, 320, 328, 355, 520, 669, 670, 687, 704–706, 714, 715, 720–722, 725, 731, 736, 737, 739, 834, 856 yield to maturity, 259, 260, 272, 408, 519, 680, 681, 705, 722

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