Stochastic Implied Volatility: A Factor-Based Model

Lecture Notes in Economics and Mathematical Systems Founding Editors:

M. Beckmann H. P. Kiinzi Managing Editors: Prof. Dr. G. Fandel

Fachbereich Wirtschaftswissenschaften Femuniversitat Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut far Mathematische Wirtschaftsforschung (IMW)

Universitat Bielefeld Universitatsstr. 25, 33615 Bielefeld, Germany Editorial Board:

A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Karsten, U. Schittko

545

Reinhold Hafner

Stochastic Implied Volatility A Factor-Based Model

4) Springer

Author Dr. Reinhold Hafner risklab germany GmbH Nymphenburger StraBe 112-116 80636 München Germany

Library of Congress Control Number: 2004109369 ISSN 0075-8442 ISBN 3-540-22183-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

42/3130Di 5 4 3 2 1 0

Ftir meine Eltern

Preface

This monograph is based on my Ph.D. thesis, which was accepted in January 2004 by the faculty of economics at the University of Augsburg. It is a great pleasure to thank my supervisor, Prof. Dr. Manfred Steiner, for his scientific guidance and support throughout my Ph.D. studies. I would also like to express my thanks to Prof. Dr. Glinter Bamberg for his comments and suggestions. To my colleagues at the department of Finance and Banking at the University of Augsburg, I express my thanks for their kind support and their helpful comments over the past years. In particular, I would like to thank Dr. Bernhard Brunner for many interesting discussions and also for the careful revision of this manuscript. At risklab germany GmbH, Munich, I would first of all like to thank Dr. Gerhard Scheuenstuhl and Prof. Dr. Rudi Zagst for creating an ideal environment for research. I would also like to express my thanks to my colleagues. It has been most enjoyable to work with them. In particular, I would like to thank Dr. Bernd Schmid. Our joint projects on stochastic implied volatility models greatly influenced this work. I am also indebted to Anja Fischer for valuable contributions during her internship and Didier Vermeiren (from Octanti Associates) for carefully reading the manuscript. Further, I am extremely grateful to Prof. Dr. Martin Wallmeier for his continuous support and advice, his thorough revision of the manuscript, as well as for many fruitful discussions. The results of our joint projects on the estimation and explanation of implied volatility structures entered this work. Most of all, I want to thank my girlfriend Heike for endless patience, encouragement, and support, and also my mother Lieselotte and my brother Jurgen for being there all the times.

Mering, May 2004

Reinhold Hafner

Contents

1

Introduction 1.1 Motivation and Objectives 1.2 Structure of the Work

1

1 5

2

Continuous-time Financial Markets 2.1 The Financial Market 2.1.1 Assets and Trading Strategies 2.1.2 Absence of Arbitrage and Martingale Measures 2.2 Risk-Neutral Pricing of Contingent Claims 2.2.1 Contingent Claims 2.2.2 Risk-Neutral Valuation Formula 2.2.3 Attainability and Market Completeness

9 10 10 13 15 15 18 20

3

Implied Volatility 3.1 The Black-Scholes Model 3.1.1 The Financial Market Model 3.1.2 Pricing and Hedging of Contingent Claims 3.1.3 The Black-Scholes Option Pricing Formula 3.1.4 The Greeks 3.2 The Concept of Implied Volatility 3.2.1 Definition 3.2.2 Calculation 3.2.3 Interpretation 3.3 Features of Implied Volatility 3.3.1 Volatility Smiles 3.3.2 Volatility Term Structures 3.3.3 Volatility Surfaces 3.4 Modelling Implied Volatility 3.4.1 Overview 3.4.2 Implied Volatility as an Endogenous Variable 3.4.3 Implied Volatility as an Exogenous Variable

23 24 24 25 27 29 32 32 34 35 38 38 39 41 43 43 45 51

X

Contents

3.4.4

Comparison of Approaches

56 59 60 60 61 63 63 68 68 70

4

The General Stochastic Implied Volatility Model 4.1 The Financial Market Model 4.1.1 Model Specification 4.1.2 Movements of the Volatility Surface 4.2 Risk-Neutral Implied Volatility Dynamics 4.2.1 Change of Measure and Drift Restriction 4.2.2 Interpretation of Terms in the Risk-Neutral Drift 4.2.3 Existence and Uniqueness of the Risk-Neutral Measure 4.3 Pricing and Hedging of Contingent Claims

5

Properties of DAX Implied Volatilities 73 73 5.1 The DAX Option 73 5.1.1 Contract Specifications 5.1.2 Previous Studies 75 76 5.2 Data 5.2.1 Raw Data and Data Preparation 76 78 5.2.2 Correcting for Taxes and Dividends 82 5.2.3 Liquidity Aspects 5.3 Structure of DAX Implied Volatilities 83 5.3.1 Estimation of the DAX Volatility Surface 83 92 5.3.2 Empirical Results 5.3.3 Identification and Selection of Volatility Risk Factors 99 102 5.4 Dynamics of DAX Implied Volatilities 5.4.1 Time-Series Properties of DAX Volatility Risk Factors . 102 5.4.2 Relating Volatility Risk Factors to Index Returns and other Market Variables 109 113 5.5 Summary of Empirical Observations

6

A Four-Factor Model for DAX Implied Volatilities 6.1 The Model under the Objective Measure 6.1.1 Model Specification 6.1.2 Model Estimation 6.1.3 Model Testing 6.2 The Model under the Risk-Neutral Measure 6.2.1 Risk-Neutral Stock Price and Volatility Dynamics 6.2.2 The Market Price of Risk Process 6.2.3 Pricing and Hedging of Contingent Claims 6.2.4 Model Calibration 6.3 Model Review and Conclusion

115 115 115 118 124 131 131 133 137 140 144

Contents

XI

7

Model Applications 7.1 Pricing and Hedging of Exotic Derivatives 7.1.1 Product Overview 7.1.2 Exotic Equity Index Derivatives 7.1.3 Volatility Derivatives Value at Risk for Option Portfolios 7.2 7.2.1 The Value at Risk Concept 7.2.2 Computing VaR for Option Portfolios 7.2.3 A Case Study 7.2.4 Beyond VaR: Expected Shortfall Trading Volatility 7.3 7.3.1 Definition and Motivation 7.3.2 Volatility Trade Design 7.3.3 Profitability of DAX Volatility Trading Strategies

145 145 145 147 153 158 158 160 162 167 170 170 171 178

8

Summary and Conclusion

187

Some Mathematical Preliminaries

A.1 Probability Theory A.2 Continuous-time Stochastic Processes

193 193 194

Pricing of a Variance Swap via Static Replication

201

A

B

List of Abbreviations

205

List of Symbols

207

References

215

Index

225

1 Introduction

1.1 Motivation and Objectives Financial derivatives or contingent claims are specialized contracts whose intention is to transfer risk from those who are exposed to risk to those who are willing to bear risk for a price. Derivatives are heavily used by different groups of market participants, including financial institutions, fund managers (most notably hedge funds), and corporations. While speculators intend to benefit from the derivative's leverage to make large profits, hedgers want to insure their positions against adverse price movements in the derivative's underlying asset, and arbitrageurs are willing to exploit price inefficiencies between the derivative and the underlying asset. During the last two decades the market for financial derivatives has experienced rapid growth. From 2000 to 2002 alone, global exchange-traded derivatives volume nearly doubled, to reach almost 6 billion contracts traded in 2002. With a market share of approximately 50%, equity index derivatives are thereby the most actively traded contracts.' Huge volumes of derivatives are also traded over the counter (OTC). In addition to standard products, the OTC market offers a wide variety of different contracts, including so-called exotic derivatives. Exotic derivatives were developed as advancements to standard derivative products with specific characteristics tailored to particular investors' needs. The latest development in this area are volatility derivatives. These contracts, written on realized or implied volatility, provide direct exposure to volatility without inducing additional exposure to the underlying asset. The increasing use and complexity of derivatives raises the need for a framework that enables for the accurate and consistent pricing and hedging, risk management, and trading of a wide range of derivative products, including all kinds of exotic derivatives. The first important attempt in this direction was the Black-Scholes option pricing model, developed by Black/Scholes (1973), formalized and extended in the same year by Merton (1973). It builds a corSee FIA (2003).

2

1 Introduction

nerstone in the theory of modern finance, and has led to many insights into the valuation of derivative securities. In 1997, the importance of the model was recognized when Myron Scholes and Robert Merton received the Nobel Price for Economics. The Black-Scholes model provides a unique "fair price" for a (European) option that is traded on a frictionless market and whose underlying asset exhibits lognormally distributed prices. Under the model's assumptions, an option's return stream can be perfectly replicated by continuously rebalancing a self-financing portfolio involving stocks2 and risk-free bonds. In the absence of arbitrage, the price of an option equals the initial value of the portfolio that exactly matches the option's payoff. The Black-Scholes model is often applied as a starting point for valuing options. However, the empirical investigation of the Black-Scholes model revealed statistically significant and economically relevant deviations between market prices and model prices. A convenient way of illustrating these discrepancies is to express the option price in terms of its implied volatility, i.e. as a number that, when plugged into the Black-Scholes formula for the volatility parameter, results in a model price equal to the market price. If the Black-Scholes model holds exactly, then all options on the same underlying asset should provide the same implied volatility. Yet, as is well known, on many markets, Black-Scholes implied volatilities tend to differ across exercise prices and times to maturity. The relationship between implied volatilities and exercise prices is commonly referred to as the "volatility smile" and the relationship between implied volatilities and times to maturity as the "volatility term structure". Volatility surfaces combine the volatility smile with the term structure of volatility. The existence of volatility surfaces implies that the implied volatility of an option is not necessarily equal to the expected volatility of the underlying asset's rate of return. It rather also reflects determinants of the option's value that are neglected in the Black-Scholes formula. The obvious shortcomings of the Black-Scholes model have led to the development of a considerable literature on alternative option pricing models, which attempt to identify and model the financial mechanisms that give rise to volatility surfaces, in particular to smiles. One strand of the literature concentrates on the nature of the underlying asset price process which was assumed to be a geometric Brownian motion with constant volatility in the BlackScholes framework. Here the main focus is on models which assume that the volatility of the underlying asset varies over time, either deterministically or stochastically. Derman/Kani (1994b), Derman/Kani (1994a), Dupire (1994), and Rubinstein (1994) were the first to model volatility as a deterministic function of time and stock price, known as local volatility. The unknown volatility function can be fitted to observed option prices to obtain an implied price process for the underlying asset. In an empirical study Dumas et al. (1998) 2 We use the term "stock" as a general expression for the underlying asset of a derivative security, although it could as well be an equity index, an exchange rate, or the price of a commodity.

1.1 Motivation and Objectives

3

conclude that, as fax as S&P 500 options are concerned, local volatility models are unreliable and not really useful for valuation and risk management. The stochastic volatility approach was motivated by empirical studies on the time series behavior of (realized) volatilities. They suggest that volatility should be viewed as a random process. Specifications for a stochastic volatility process have been proposed by a number of authors, including Hull/White (1987), Wiggins (1987), Scott (1987), Stein/Stein (1991), and Heston (1993). A problem of stochastic volatility models is that unrealistically high parameters are required in order to generate volatility smiles that are consistent with those observed in option prices with short times to maturity. 3 A third explanation for implied volatility patterns that is related to the asset price process are jumps. 4 When jumps occur, the price process is no longer continuous. Jumps have proved to be particularly useful for modelling the crash risk, which has attained considerable attention since the stock market crash of October 1987. In the attempt to correctly reproduce the empirically observed implied volatility patterns, neither (one-factor) stochastic volatility models nor simple jump-diffusion models are successful. Furthermore, both types of models are incomplete. Consequently, the requirement of no-arbitrage is no longer sufficient to determine a unique preference-free price of the contingent claim. Another problem of models based on the underlying asset to describe the dynamic behavior of option prices is that infinitesimal quantities such as the local or stochastic volatility or the jump intensity, are not directly observable but have to be filtered out either from pricing data on the underlying asset or "calibrated" to options data. In the first case, the quantity obtained is model-dependent and in the second case it is the solution to a non-trivial optimization problem. A second strand of the literature identifies market frictions as another possible explanation for the smile pattern. Transaction costs, illiquidity, and other trading restrictions imply that a single arbitrage-free option price no longer exists. Longstaff (1995) and Figlewski (1989a) examined the effects of transaction costs and found that they could be a major element in the divergences of implied volatilities across strike prices. Yet, Constantinides (1996) points out that transaction costs cannot fully explain the extent of the volatility smile. McMillan (1996) argues that the crash of 1987 lessened the supply of put option sellers, whereas at the same time fund managers showed a higher demand for out-of-the-money puts. Because hedging the risk exposure of written out-of-the-money puts turned out to be expensive, higher prices for out-of-the-money puts were charged. This could also partly explain the observed strike pattern of implied volatility. It is generally acknowledged that the above influences are interrelated, and no single explanation completely captures all empirical biases in implied volatilities. The increasing liquidity in the market for standard options, especially in the area of equity index options considered here, has had two major conse3 See,

e.g., Andersen et al. (1999), p. 3, and Das/Sundaram (1999), p. 5.

4 See,

e.g., Bates (1996a), Trautmann/Beinert (1999).

4

1 Introduction

quences: 5 First, there is no more need to theoretically price standard options. The market's liquidity ensures fair prices. Second, hedging of standard options becomes less important as positions can be unwound quickly. These developments and the inability of the models described above to accurately reflect the dynamic behavior of option prices or their implied volatilities have brought up a second modelling approach. In directly taking as primitive the implied volatility (surface), this approach is usually referred to as a "market-based" approach. 6 Market-based models have the advantage to be automatically fitted to market option prices. In difference to fundamental quantities such as an (unobservable) instantaneous volatility or a jump intensity, implied volatilities are highly regarded and continuously monitored by market participants. A market scenario described in terms of implied volatilities is therefore easy to understand for a practitioner. Due to the noticeable standard deviation found in time series of implied volatilities, deterministic implied volatility models do not seem to be appropriate. A natural step of generalization is to let implied volatilities move stochastically. In contrast to (traditional) stochastic volatility models, where the instantaneous volatility of the stock return is modelled, stochastic implied volatility models focus on the (stochastic) dynamics of either a single implied volatility (e.g., Lyons (1997)), the term structure of volatility (e.g., Schônbucher (1999)), or the whole volatility surface (e.g., Albanese et al. (1998) and Ledoit et al. (2002)). A major advantage of stochastic implied volatility models is their completeness. While the approaches of Schtinbucher (1999) and Ledoit et al. (2002) dealt with the problem of stochastic implied volatility from a theoretical perspective, Rosenberg (2000), Cont/Fonseca (2002), Goncalves/Guidolin (2003), among others, focus on the empirical aspects of the problem. For example, Cont/Fonseca (2002), using S&P 500 and FTSE 100 option data, suggest a factor-based stochastic implied volatility model where the abstract risk factors driving the volatility surface are obtained from a Karhunen Loève decomposition. A natural application of this model is the simulation of implied volatility surfaces under the real-world measure, for the purpose of risk management. However, the models are not intended to determine the consistent volatility drifts needed for risk-neutral pricing of exotic derivatives. "How best to introduce the ideas from these models into a no-arbitrage theory remains an open question" . 7 It has mainly been this question that motivated this work. The overall goal of this work is to provide a stochastic implied volatility model that allows for the integrated and consistent pricing and hedging, risk management, and trading of equity index derivatives as well as derivatives on the index volatility. As we assume that the evolution of the volatility surface See SchtInbucher (1999). approach is similar to the approach of Heath-Jarrow-Morton (HJM) in the field of interest rates. See Heath et al. (1992). 5

6 This 7 See

Lee (2002), p. 25.

1.2 Structure of the Work

5

is driven by a small number of (fundamental or economic) risk factors, the model is termed "factor-based". Specifically, in the first, theoretical part of this work, we aim at developing a unifying theory for the analysis of contingent claims under both the real-world measure and the risk-neutral measure in a stochastic implied volatility environment. Based on the theory developed, the objective of the second, empirical part is to specify, estimate, and test a factor-based stochastic implied volatility model for DAX implied volatilities. 8 In the final part of this work, we will present potential applications of the model.

1.2 Structure of the Work This work is organized as follows (see Figure 1.1). In Chapter 2, we discuss the principles of continuous-time financial markets in a rather general framework, which will also serve as a reference in the later chapters. Special emphasis is put on the valuation of contingent claims. We comment on the class of results — referred to as a fundamental theorem of asset pricing — which says, roughly, that the absence of arbitrage opportunities is equivalent to the existence of an equivalent martingale measure. The main result of this chapter is the risk-neutral valuation formula, which states that the arbitrage price of any attainable contingent claim is the expectation of the discounted claim under the equivalent martingale measure. Chapter 3, devoted to implied volatility, starts with a description of the Black-Scholes model. We present the model's assumptions, derive the BlackScholes formula for European-style stock options, and state the main option's sensitivities, better known as the Greeks. We then introduce the implied volatility concept and describe some well-known patterns in the behavior of implied volatility as the strike price and the maturity date of the option change, namely the volatility smile, the term structure of volatility, and the volatility surface as the combination of the two. In the remainder of this chapter, we discuss the various approaches to value options in the presence of implied volatility structures. We highlight their individual strengths and weaknesses and explain the difficulties induced by them. In Chapter 4, we develop a general mathematical model of a financial market in continuous time where in addition to the usual underlying securities stock and risk-free bond, a collection of standard European options is traded. The prices of the standard options are given in terms of their implied Black-Scholes volatilities, whose joint evolution is driven by a small number of risk factors. The chapter starts with a description of the financial market model under the real-world or objective probability measure. Then, we derive necessary and sufficient conditions that have to be imposed on the drift 8 The DAX option is one of the most heavily traded equity index options in the world.

6

1 Introduction Continuous-time financial markets (Chapter 2) special case

Black-Scholes model (3.1) leads to Implied volatility surface (3.2 & 3.3) is

endogenous (3.4.2) 21 Market frictions Stock price is not a geometric Brownian motion

exogenous (3.4.3) volatility surface is

stochastic

deterministic

General factor-based stochastic implied volatility model (Chapter 4)

Properties of DAX implied volatilities (Chapter 5) examine

Structure (5.3)

Dynamics (5.4)

Four-factor model for DAX implied volatilities (Chapter 6)

11,

Objective measure (6.1)

Value at risk (7.2)

Risk-neutral measure (6.2)

Volatility trading (7.3)

Figure 1.1.

1.

Pricing and hedging of exotic derivatives (7.1)

Structure of the work

coefficients of the options' implied volatilities in order to ensure discounted call prices to be martingales under the risk-neutral measure. We also discuss existence and uniqueness of the risk-neutral measure. Finally, we show how to price and hedge a general stock price dependent contingent claim. The goal of Chapters 5 and 6 is the specification, estimation and testing of a factor-based stochastic implied volatility model for DAX implied volatilities.

1.2 Structure of the Work

7

Chapter 5 concentrates on the identification of the main properties of DAX implied volatilities both in a cross-sectional ("structure") and a time-series ("dynamics") setting. Our database contains all reported transactions of options and futures on the DAX, traded on the DTB/Eurex over the sample period from January 1995 to December 2002. To the best of our knowledge, this database is one of the largest databases, at least in Europe, that has ever been used in such a study. In preparing the data, we have carefully accounted for potential biases such as tax effects or non-simultaneous options and underlying prices. Based on the empirical results of Chapter 5, Chapter 6 proposes a fourfactor model for the stochastic evolution of the DAX volatility surface. We begin with the specification of the model under the real-world measure, we then show how to estimate the model parameters from historical data, and finally we perform various in- and out-of-sample tests to assess the quality of the model. In the second part of this chapter, we derive the risk-neutral dynamics of the DAX index, the DAX volatility surface, and the instantaneous DAX volatility. We discuss the issues of existence and uniqueness of a martingale measure and show how to price and hedge contingent claims. Finally, we calibrate the model to market data. Chapter 7 presents applications of the factor-based stochastic implied volatility model in the fields of pricing and hedging, risk management, and trading. In particular, we consider the pricing and hedging of selected exotic derivatives, including derivatives on index volatility, then apply the model to calculate the value at risk and expected shortfall for an option portfolio, and finally discuss volatility trading. Here, we describe several ways of how to trade volatility, discuss the advantages and disadvantages of each strategy, and empirically test some of these strategies on their ability to generate abnormal trading profits. The work concludes with a short summary of the main results and suggestions for further research (Chapter 8). For readers who are not familiar with stochastic analysis, we give a short introduction to this subject in Appendix A. Finally, Appendix B gives a proof on the pricing of a variance swap via the method of static replication.

2

Continuous-time Financial Markets

Understanding a theory means to solve a certain problem.

(.

.)

understanding it as an attempt

Sir Karl Popper

This chapter discusses the principles of continuous-time financial markets in a rather general framework, which will also serve as a reference in the later chapters. Special emphasis is put on the valuation of contingent claims. Following the path-breaking work of Harrison/Kreps (1979) and Harrison/Pliska (1981), we start in Section 1 by developing a rigorous mathematical model of a financial market in continuous time. In distinction to Harrison/Pliska (1981), who model the evolution of asset prices by a possibly discontinuous, semimartingale process, we restrict ourselves to continuous processes of the Ito type. The agents' activities in the market are modelled by trading strategies. A particularly important class of trading strategies in the context of contingent claim valuation is the class of self-financing trading strategies. It is described in some detail. In Section 2, we first introduce the concept of an arbitrage opportunity and then comment on the class of results — referred to as a fundamental theorem of asset pricing — which says, roughly, that the absence of arbitrage opportunities is equivalent to the existence of an equivalent martingale measure. The last section focuses on the arbitrage pricing of contingent claims. We present the risk-neutral valuation formula, which states that the arbitrage price of an attainable contingent claim is the expectation of the discounted claim under the equivalent martingale measure, discuss the attainability of contingent claims and introduce the notion of a complete market.

10

2 Continuous-time Financial Markets

2.1 The Financial Market 2.1.1 Assets and Trading Strategies We start with a frictionless security market, where investors are allowed to trade continuously up to some fixed terminal time horizon T'> 0. 1 A security market is called frictionless, if there are no transaction costs or taxes, no bidask spreads, no margin requirements, no restrictions on short sales, and all assets are perfectly divisible. 2 The uncertainty in the financial market is characterized by the complete probability space (S2, F, P) where f2 is the state space, .7. is the a-algebra representing measurable events, and IP is the objective or real-world probability measure. 3 Information evolves over the trading interval [0, TX] according to the Brownian filtration F = {,Ft : t E [0, , generated by a (p + 1)-dimensional standard Brownian motion W =- {Wt : t E [0, T*]} , Wt = (Wo,t, • • . , W,4'. 4 The a-algebra .Ft represents the information available at time t. Throughout this work, we assume that F satisfies the usual conditions: IF is complete, i.e. To contains all P-null sets of F and IF is right-continuous. 3 Moreover, we assume that the a-field ..7-0 is trivial, i.e. F. = {0, n} , and that FT' The market consists of d + 1 (stochastic) primary traded assets (stocks, bonds, options, etc.), whose (spot) price processes are given by stochastic processes Zo, Zd. We assume that Z = {Z t : t E [0, T 3 ]} , Zt =follows a positive (d + 1)-dimensional Itô process 6 with re-Z4', spect to W on the filtered probability space (St, F, P, F). 7 The movement of the security prices relative to each other will be important to study, so it is convenient to normalize the price vector Z. We set Zt Dt

Zr = — = with Zr t = Zi, t /Dt, the numéraire.

j

,

,

Vt E [0, TX]

,

(2.1)

= 0, ... d. The (one-dimensional) process D is called

'This section follows closely the presentations given in Bingham/Kiesel (1998), Chapter 6, Korn/Korn (1999), Chapter 6, Musiela/Rutkowski (1997), Chapter 10, and Hasrison/Pliska (1981). 2 See, e.g., Bingham/Kiesel (1998), p. 7. 3 Appendix A provides a brief account to fundamental concepts of probability theory and the theory of continuous-time stochastic processes, so far as they are used in this text. For more information, the reader is advised to consult the references cited there. 4 x' denotes the transpose of the vector x. 'By definition, the natural filtration of a Brownian motion a (W, : s < t) is right- and left-continuous but not complete. However, if we extend -Fr by the a-algebra containing all IF-null sets of .7., we obtain a filtration with the desired property. See Lamberton/Lapeyre (1996), p. 30. 6 Note that this specification does not allow for jumps in the price processes. 7 We shall henceforth denote a probability space (SI, F , P) endowed with a filtration F a filtered probability space (0, T,P,F) .

yr, =

2.1 The Financial Market

11

Definition 2.1 (Numéraire). A numéraire D = {Dt : t E [0, T1 } is a price process which is P-a.s. positive for each t E [0, T*] . Mostly, the money market account or a zero-coupon bond is used as

numéraire. This explains why Z* is usually called the discounted price process. From now on, we assume that Zo is the numéraire, i.e. D = Zo• The market participants' activities over time are described by trading strategies or portfolio strategies. Definition 2.2 (Trading Strategy). Let us fix a time horizon T < T* . Then, a trading strategy or portfolio strategy over the time interval [0, 7 ] is an R d+1 -valued progressively measurable process 0 = {g5t : t E [0, T1 } , ,ffit = (0o,t , 0147 • • • 1 Old,t) such that the stochastic integrals fip t

08dZ8

fo 08 dZ:

and

exist. The portfolio holding 0i ,t denotes the number of units of asset i held at time t.8 Since we have assumed that the market is frictionless, 0i,t may be any positive or negative value. The value and the gains process associated with a trading strategy are introduced next. Definition 2.3 (Value and Gains Process). Let 0 be a trading strategy over the time interval [0, 7 ] . 1. The value of the portfolio 0 at time t is given by d

Vt (0) = Ot • Zt

E i=o

Vt E [O, T] ,

(2.2)

The process V(0) is called the value process or wealth process of the trading strategy 0 with initial value or wealth Vo (çb). 2. The gains process G(0) is defined by d

Gt (0) =

t

f .9 c1Zt s , °

f 0 8C1Z8 = E

Vt E [0, .

(2.3)

8 1n a more general framework, where asset prices follow a continuous-time semimartingale, the trading strategy has to be predictable. Intuitively, this means that the number of assets held at time t are determined on the basis of information available before time t but not t itself. In our setting, however, it can be shown that it is actually enough to require that 4 is progressively measurable. See also Musiela/Rutkowski (1997), p. 230.

12

2 Continuous-time Financial Markets

The value G(0) represents the gains or losses accumulated up to and including time t. Thereby, we implicitly assume that the securities do not generate any cash payments such as dividends. With Z0 as numéraire, we define the discounted value process Vt*(0) as

Vt (4))

Vt* (0) =

d

= ,t

E ot,tzzt,

Vt E [0, 7],

(2.4)

.

(2.5)

and the discounted gains process as d

G; (0)

=

E i=1 0

t

Vt

E [0,7]

Note that G;(0) does not depend on the numéraire. A trading strategy, where all changes in the value of the portfolio are due to capital gains, as opposed to withdrawals of cash or injections of new funds, is called self-financing. Definition 2.4 (Self-financing Trading Strategy). A trading strategy 0 is called self financing over the time interval [0, T] if the value process V(0) satisfies -

Vt E [0,

,

(2.6)

or equivalently: d

dVt(0)

=

E f ,=0 0

t Vt E [0, 7 ]

.

(2.7)

Our goal is to be very flexible with respect to the chosen numéraire. The next result underscores this. Theorem 2.5 (Numéraire Invariance Theorem). Self financing trading strategies remain self financing after a numéraire change. -

-

Proof. This can easily be shown using It6's product rule. For a formal proof, see Bingham/Kiesel (1998), p. 173. Using the Numéraire Invariance Theorem we can restate the self-financing condition (2.6) in terms of the discounted processes: A trading strategy 0 is self-financing if and only if

vt*(sb) = 11 (0) +

(

(2.8)

0),

and, of course, 14 (0) > 0 if and only if Vt*(0) > 0 for all t E [0, 7 ] . This result shows that a self-financing trading strategy is completely characterized by its initial value Vo* (q5) and the components 0 1 , ,ç5 Therefore, any trading strategy can be uniquely extended to a self-financing strategy q5 with initial value V0*(0) = y by setting .

2.1 The Financial Market d

00,t =

Ef i _1

t

o

13

d

— i=1

0, t z:, t ,

vt c [0,

.

(2.9)

In real life finance, there is usually a limit of how much loss an investor is willing to tolerate, or in other words, there is a lower boundary on the portfolio value. The class of tame trading strategies incorporates this restriction as the following definition shows. Definition 2.6 (Tame Strategy). A self-financing trading strategy 0 over the time interval [0,2 ] is called tame if 9 V * () > 0,

IF'-a.s., Vt E [0,T] .

The class of tame strategies over [0, T] is denoted by TT. Correspondingly, T = UT 0) > O.

(2.10)

We say that a market is T-arbitrage-free (arbitrage-free) if there are no arbitrage opportunities in T T (T). An arbitrage opportunity would allow investors to make limitless profits without being exposed to the risk of incurring a loss. In order for the continuous-time market model to be reasonable from an economic point of view, it should be free of arbitrage opportunities. Unfortunately, it is very difficult to check directly if a model has any arbitrage opportunities. However, there is an important necessary and sufficient condition for the model to be consistent with the absence of arbitrage. This condition involves the concept of equivalent martingale measures. Definition 2.8 (Equivalent Martingale Measure). We say that a probability measure Q defined on the measurable space (SI, .F) is a (strong) equivalent martingale measure or a risk-neutral measure if:

/. Q is equivalent to IP, 2. the discounted price process Z* is a Q-martingale.

9 This condition can be weakened: In fact, it is enough to demand that Vt (0) is P- as. lower bounded. See, e.g., Musiela/Rutkowski (1997), p. 235.

14

2 Continuous-time Financial Markets

The set of equivalent martingale measures is denoted by P. A useful criterion in practical applications to test whether a given equivalent measure belongs to P is the observation, that the drift rates relative to the numéraire of all given primary securities under the measure in question must be the same. In particular, if the numéraire asset Z0 is the money market account B with price process Bt = ert, , Vt E [0, where r is the constant risk-free interest rate, the discounted asset prices Z* have an expected instantaneous rate of return of r under the measure Q. This is exactly the return a risk-neutral investor demands from an investment. For this reason, an equivalent martingale measure is alternatively called a "riskneutral measure". In the following we shall use the two terms interchangeably. In our model setting, all equivalent martingale measures QEP can be obtained using Girsanov's theorem (or the Cameron-Martin-Girsanov theorem). Theorem 2.9 (Girsanov's Theorem). Given the standard Brownian mo, tkp,t r, tion W defined on ( ft, P, F), let 71) = : t E [0, T*]1 , Ibt = OP

be a (p +1)-dimensional progressively measurable process satisfying the condition fot tivls : Rd -- R, then H is called path-independent, otherwise H is called path-dependent. The function (I) is called the payoff or contract function. 12 One can think of a contingent claim as part of a contract that a buyer and seller make at current time t E [0, T*], in which the seller promises to pay the buyer the amount H at future time T> t. Note that this definition does not cover American-style derivatives (i.e. contingent claims which can be exercised at any time t up to the maturity date T) because the main focus of this work is on equity index derivatives and almost all exchange-traded equity index derivatives are of European-type. Derivative securities might be classified into three different groups: Forwards and futures, options and swaps. During this text we will mainly focus on options, although the techniques developed may be readily applied to forwards, futures and swaps as well. We shall now describe the basic features of these types of derivatives, as far as they are needed for the further understanding. 13 Therefore, let us consider the case d = 1 with the asset Zi being a non-dividend paying stock S." The stock price at the current date t (expiration date T) is denoted by St (ST). A forward contract is a particularly simple derivative. It is an agreement to buy or sell the underlying stock S at a certain future date T for a certain price K. A forward contract is traded in the over-the-counter (OTC) market — usually between two large financial agents (banks, institutional investors, etc.). The agent who agrees to buy the stock is said to have a long position, the other agent assumes a short position. The maturity date T is sometimes called delivery date and the prespecified price K is referred to as delivery price. The forward price F, (T) is the delivery price which would make the contract have zero value at time t, t < T. It is important to distinguish between the forward price and the delivery price. The two are the same when the contract is first entered into but are likely to be different at later times. Since a forward contract is settled at maturity and a party in a long position is obliged to buy an asset worth ST at maturity for K, it is clear that the payoff from the long position (from the short position, respectively) corresponds to the time T contingent claim H (—H, respectively), where H = (I) (ST) = ST — K.

(2.12)

"In the literature, as, e.g., in Pliska (2000), P. 112, it is sometimes distinguished between contingent claims and derivative securities. The term "contingent claim" is then reserved for a European-style derivative, whereas the notion "derivative security" also includes American-style derivatives. "See Hull (2000), pp. 1-10 and Bingham/Kiesel (1998), pp. 2-4. 14 We will often use the term "stock" as a general expression for the underlying asset of a derivative security.

2.2 Risk-Neutral Pricing of Contingent Claims

17

A futures contract is conceptually equal to a forward contract. However, unlike forward contracts, futures contracts are normally traded on an exchange. To make trading possible, the exchange specifies certain standardized features of the contract. The given price is now called the futures price and is paid via a sequence of installments over the contract's life. These payments reset the value of the futures contract after each trading interval - usually a day; the contract is marked to market. In general, the futures price is different from the forward price. However, it can be shown that if Zo is a deterministic process, e.g., in the case that Zo is the money market account and interest rates are a deterministic function of time, then the futures price equals the forward price. 15 Since this is the only relevant case in this work, we no longer distinguish between forward and futures prices. An option is a financial instrument giving the right but not the obligation to make a specified transaction at (or by) a specified future date T at a specified price K. Call options give the right to buy, and put options the right to sell the underlying asset S. The price K in the contract is known as strike price or exercise price and the date T as the option's maturity or expiration. The simplest call and put options are called standard or plain vanilla options. A standard European call option is formally equivalent to the claim H whose payoff at time T is contingent on the stock price ST, and equals H = (I) (ST) = max {ST — K; 0} .

(2.13)

Similarly, a standard European put option is a contingent claim of the form H

= '(ST) = max {K

—

ST;0} .

(2.14)

At a given instant t before or at expiry, we say that a standard call option is in - the - money (ITM) and out- of- the money (OTM) if St > K and St < K, respectively. Conversely, a put option is in-the-money (out-of-the-money) if St < K (S t > K). An option is said to be at- the - money (ATM) if St = K. In addition, the terms near-the-money for options close to being at-the-money and deep out-of-the-money (in-the-money) for options that are far off being at-the-money are frequently used. Instead of the spot price St , the forward or futures price Ft (T) is sometimes used in the above definitions. In addition to standard options, modern options markets offer a wide variety of different contracts, including so-called exotic options. Types include: Asian options, whose payoff depends on the average price over a period, lookback options whose payoff depends on the minimum or maximum price over a specified period, and barrier options, whose payoff depends on some price level being attained or not. Finally, a swap is an agreement whereby two parties undertake to exchange, at known dates in the future, various financial assets (or cash flows) according to a prearranged formula. A swap can always be decomposed into a basket of 15 See,

Hull (2000), pp. 60-62 and 85-86, and Musiela/Rutkowski (1997), p. 86.

18

2 Continuous-time Financial Markets

forwards and/or options. Therefore, to value a swap it suffices to be able to value the individual components.

2.2.2 Risk- Neutral Valuation Formula Assuming that there exists at least one equivalent martingale measure for the market model, we now approach the problem of pricing derivatives. The problem of interest is to determine the time t value of the payoff H. In other words: what is the fair price of claim H at time t that the buyer should pay the seller in order to satisfy both parties. One might suppose that the value of a contingent claim would depend on the risk preferences and utility functions of the buyer and seller, but in a many cases this is not so. It turns out that by the arguments of arbitrage pricing theory there is often a unique, "correct", value of the claim at time t, a value that does not depend on investors' risk preferences. In the remainder of this section, we are going to derive this value. Let us therefore fix an arbitrary equivalent martingale measure Q*EP and an arbitrary maturity date T < T*. Definition 2.13 (Admissible and Attainable Claims). 1. A tame trading strategy 0 E TT is called Qs-admissible for time T if V*(0) is a (2*-martingale. We denote the class of Q* -admissible trading strategies by TT(Q * )• 2. A contingent claim H with maturity date T is called (Q* )-attainable if it admits at least one trading strategy 0 E TT(Q* ) such that

VT(0) = H,

P- a.s.

We call such a trading strategy 0 a replicating strategy for H.16 Now we are ready to present the central idea of arbitrage pricing theory: If a T-contingent claim H is (Q* )-attainable, H can be replicated by a trading strategy 0 E TT(Q*). This means that holding the portfolio and holding the contingent claim are equivalent from an economic point of view. In the absence of arbitrage opportunities, the value process V(0) of 0 replicating H and the arbitrage price processII(H) = Mt (H) : t E [0, T] } of H must therefore satisfy

H(H) = Vt (0).

(2.15)

16 Remark: Strictly speaking, it is not necessary to demand a Q*-admissible strategy to be tame. See Bingham/Kiesel (1998), p. 177. Moreover, we want to emphasize that the definition of an attainable claim depends on the class of trading strategies, but not on the numéraire. It can easily be shown that an attainable claim in one numéraire remains attainable in any other numéraire and the replicating strategies are the same.

2.2 Risk-Neutral Pricing of Contingent Claims

19

Otherwise, an astute investor would buy (sell) the replicating strategy and sell (buy) the contingent claim to make a riskless profit of Illt (H) — Vt(0)1 > O. Naturally, the questions arise what will happen if there exists more than one replicating strategy for H, and what the relation of the arbitrage price process to the equivalent martingale measure is. The following central theorem is the key to answering these questions. Theorem 2.14 (Risk- Neutral Valuation Formula). In the standard market model the arbitrage price process II(H) = fli t (H) : t E [0, 7 ] } of any Q* attainable claim with maturity date T is given by the risk-neutral valuation formula

rit(H) =

H ZO,t 15()* [

4,0,T

Ft]

/

Vt e [0, 7] .

(2.16)

Proof. Since H is (Q*)-attainable, there exists a trading strategy 0 e TT (Qs) with value process V(0) which replicates H, i.e. VT (0) = H. As the discounted value process V*(0) is a Qs-martingale we can write

Vt* (0) = EQ. [ Tr; (95)1 Ft ] , and by the definition of the discounted value process

Vt(0) _ E r VT(0) Ft] Zo,t Q. L _0,T z

.

Using lit (H) = Vt (0), which holds by the absence of arbitrage, the assertion follows. 0 Taking a different perspective, the risk-neutral valuation formula (2.16) was first derived by Ross (1976) and Cox/Ross (1976). Based on the insight of Black/Scholes (1973) that the fair price of an option does not depend on the risk preferences of the individual investors, they assumed investors to be risk-neutral. Under this assumption, (2.16) follows directly. 17 The first mathematically rigorous proof of the risk-neutral valuation formula stems from

Harrison/Pliska (1981). Coming back to the question of uniqueness of the replicating strategy, it is apparent from the risk-neutral valuation formula (2.16) that if the replicating strategy would not be unique, it would be possible to construct an arbitrage strategy. Hence, the replicating strategy is unique, up to indistinguishability of stochastic processes. An apparent drawback of Definition 2.13 is the dependence of the class of admissible strategies on the choice of the martingale measure.' To circumvent 17 Thus the valuation is preference free in the sense that it is valid independent of the specific form of the agents' preferences, as long as they prefer more wealth to less wealth. 18 Note, that the martingale property of the discounted value process is in general not invariant to an equivalent change of a martingale measure. See Musiela/Rutkowski (1997), p. 235, footnote 5. -

2 Continuous-time Financial Markets

20

this problem we generalize our definition of an admissible trading strategy and an attainable claim to: Definition 2.15. A tame trading strategy 0 E TT is called admissible for time T if it is Q-admissible for some Q EP, and H is called attainable, if there exists an admissible strategy 0 replicating it. The next result shows that the arbitrage price process is unique. Theorem 2.16. For admissible trading strategies 01 E TT(Q1) and 02 E TT(Q2) replicating H, we have

H

lit(H) = Zo,tEQ1[ zo,T .Ft] = Zo,t 1E4:22

H

Vt E [0, 7].

Ft]

Proof. See Musiela/Rutkowski (1997), pp. 235-236. 2.2.3 Attainability and Market Completeness

If we wish to price a contingent claim H it is sufficient to find an equivalent martingale measure, but for hedging purposes we are more interested in the replicating trading strategy. The following useful result shows that the existence of a replicating strategy for H is equivalent to the existence of a stochastic integral representation of the claim. Theorem 2.17. Let Q EP be any equivalent martingale measure and H a contingent claim with maturity date T. If the Q-martingale defined by

r

Ht(H) (H) = Li ,t

H I

FAQ

1

Vt E [0, 7 ] ,

admits an integral representation of the form d

t

11; (H) = v +E f

Vt E [0,7'],

(2.17)

i=1 °

with 0i,t progressively measurable and such that fct, 0i,t d4"3 exists for i = 1, . , d, then H is attainable. Proof. See Bingham/Kiesel (1998), p. 180. From relation (2.17) in the above theorem, we see that attainability is closely related to martingale representation theorems. These exist in various degrees of generality. In our model setting, the most important version states that a square-integrable Brownian martingale, i.e. a martingale with respect to a Brownian filtration, can be represented in terms of an It8 integra1. 19 19 See,

e.g., Lamberton/Lapeyre (1996), P. 67.

2.2 Risk-Neutral Pricing of Contingent Claims

21

Provided that an equivalent martingale measure exists, we can use the risk-neutral valuation principle to derive a unique price for every attainable contingent claim. The problem, of course, is that the claim might not be attainable. In this case, it is not clear what the "correct" price should be. In particular there is no reason why it should equal its arbitrage price. We therefore need a convenient method for checking if a contingent claim is indeed attainable. One method was illustrated in theorem 2.17. But there exists an alternative, even more powerful, method. It builds upon the definition of a complete market: Definition 2.18 (Complete Market). A financial market model is called complete if every contingent claim H with arbitrary maturity date T < T* is attainable. Otherwise, the model is said to be incomplete.

The method establishes a connection between market completeness and the number of existing equivalent martingale measures, namely: Theorem 2.19. A financial market model is complete if and only if P consists of exactly one equivalent martingale measure.

Proof. See Harrison/Plislca (1981), p. 241 ff. This extremely powerful theorem tells us that we only have to prove the uniqueness of the equivalent martingale measure to infer the attainability of any contingent claim. To summarize, if the model is complete we know how to price all the contingent claims. On the other hand, if the model is incomplete, we know how to price some of the contingent claims, namely, all the attainable ones. But we do not know how to value the claims that are not attainable. It turns out, however, that we can identify an interval within which a fair, reasonable value for the claim must fall. To derive a unique value, we have to make assumptions on the risk preferences of the agents in the economy. This implies that the valuation of contingent claims is no longer preference-free. Of course this is not desirable from a modelling point of view, explaining partly the popularity of complete-market models, where the price of the contingent claim is unaffected by the agents' preferences. 2° Before we close, let us state an extremely useful rule of thumb, which helps us to judge whether a given model is respectively arbitrage-free or not and/or complete or not. Without proof, we formulate what Bji5rk (1998), p. 106, calls a "meta-theorem": Theorem 2.20 (Meta theorem). Let d +1 be the number of stochastic primary traded assets in a financial market model, and let p+1 denote the number of random sources. In general, we then have the following relations: -

20 See Pliska (2000). For a comprehensive treatment of questions on pricing and hedging in incomplete markets see, e.g., Carr et al. (2001).

22

2 Continuous-time Financial Markets

1. The model is arbitrage-free if and only if d < p. 2. The model is complete if and only if d > p. 3. The model is arbitrage-free and complete if and only if d = p.

3 Implied Volatility

A smiley implied volatility is the wrong number to put in the wrong formula to obtain the right price. Riccardo Rebonato

As a special case of the standard market model, the Black-Scholes option pricing model builds a cornerstone in the theory of modern finance and was the starting point of a whole branch of research concerning the pricing and hedging of contingent claims. In this text, the Black-Scholes model functions as a reference model to all other models. In Section 1, we describe the Black-Scholes assumptions about the financial market, derive the BlackScholes formula for European-style stock options, and state the main option's sensitivities, commonly referred to as the Greeks. Given the market price of an option, the volatility implied by this price can be determined by inverting the Black-Scholes option pricing formula. In Section 2, we give a formal definition of this so-called implied volatility, show how to compute it and discuss possible interpretations under different sets of assumptions. If the Black-Scholes model holds exactly, all options on the same underlying asset should provide the same implied volatility. Yet, empirical implied volatilities differ systematically across strike prices and maturity dates such that the misspecified model produces the correct market prices. Section 3 describes the most important of these implied volatility patterns: the volatility smile, the volatility term structure, and the volatility surface as the combination of the two. In the last section, we discuss the various approaches to price options in the presence of volatility surfaces and assess their individual strengths and weaknesses.

24

3 Implied Volatility

3.1 The Black-Scholes Model 3.1.1 The Financial Market Model Black/Scholes (1973) assume a frictionless security market with continuous trading up to some fixed terminal time horizon T* > 0 (Assumption 1). The uncertainty in the financial market under the objective probability measure is characterized by the complete filtered probability space (SI, F I?, 1F). The filtration is generated by a one-dimensional (i.e. p = 0) standard Brownian motion W = {Wt : t E [0, T* ] 1 which satisfies the usual conditions. The primary traded securities are the money market account and a non-dividend paying stock (i.e. d = 0). ,

Assumption 2. The price process of the money market account is given by the ordinary differential equation (ODE): dB t= rBt dt,

Vt E [0, 71,

(3.1)

where B o = 1 and the continuously compounded interest rate r is supposed to be constant and nonnegative. The solution of (3.1) is Bt = ers ,

Vt E [0, T*I .

(3.2)

In the Black-Scholes financial market the money market account is taken as the numéraire. Assumption 3. The stock price follows a geometric Brownian motion (GBM). Specifically, the dynamics of the stock price process S= {St : t E [0,71} is described by the linear stochastic differential equation (SDE):

dSt = PStdt + vSt dWt ,

Vt E [0,

,

(3.3)

where A E R, y > 0 are deterministic constants and So > 0 is the initial stock price. The coefficient is a constant appreciation rate of the stock price and the coefficient y, referred to as the (stock price) volatility, is interpreted as a measure of uncertainty about future stock price movements.' As volatility increases, the chance that the stock will perform very well or very poor increases. Related definitions stem from Taleb (1997), p. 88 and Natenberg (1994), p. 51. The former states: „Volatility is best defined as the amount of variability in the returns of a particular asset" and the latter describes volatility as a "measure for the speed of the market". More formally, the volatility of a 'See, e.g., Hull (2000), p. 241.

3.1 The Black-Scholes Model

25

stock is often defined as the annualized standard deviation of its continuously compounded returns . 2 Using formula, it is elementary to check that the stock price process given by S St = So exp

–

1

t + yWt ,

Vt E [0, T*I ,

(3.4)

is indeed a solution of (3.3), starting from So at time O. Since '1St and vSt are Lipschitz continuous for all t E [0,11, the solution S is unique, according to a general result of It45.3 It has the following properties: S is Yradapted, S has continuous trajectories, and S is a Markov process. It is apparent from

(3.4) that the continuously compounded (t – u)- period stock returns ln (P-) are normally distributed with mean (.1 – 10) (t – u) and variance y2 (t – u) under the objective measure P for any dates u R is the Black-Scholes put option pricing formula: PBS (t7

S)

= Ke — r(T—t) N(—d2(t,$))— sN(— di (t,$)).

(3.23)

The functions di (t, s) and d2(t, s) are defined as in (3.19). Since in typical situations it is not difficult to find a proper form of the put-call parity, we shall usually restrict ourselves to the case of a call option. Originally, the Black-Scholes formula was derived by introducing a continuously rebalanced risk-free portfolio containing an option and underlying stocks. In the absence of arbitrage, the instantaneous return from such a portfolio needs to be equal to the risk-free rate. 5 This property leads to a partial differential equation, which is then solved for the price process of the option. More specifically, if H = (I) (ST) is a path-independent contingent claim, then the arbitrage price process is also given by H t (H) = f (t, St ), where f solves the (parabolic) Black-Scholes partial differential equation (PDE):6

af(t,$)

Of(t,$) 2 2 (92 f(t, rs as + 2 v s 0.92

S)

r f (t, s) = 0, V(t, s) E (0, T) x (0, oo) (3.24)

with terminal condition

f (T, s) = (1)(s).

(3.25)

In particular, solving (3.24)-(3.25) for (Ks) = max {s — K; 0} yields the BlackScholes call option pricing formula CBs(t, s). Theoretically, the connection between the risk-neutral valuation formula (3.10) and the Black-Scholes PDE for claims of the form H = (I) (ST), i.e. for path-independent claims, is established by the Feynman -Ka e stochastic representation formula, which basically expresses the solution of a parabolic PDE as the expected value of a certain functional of a Brownian motion. 7 3.1.4 The Greeks To assess the risk of an option's position, we will now examine the impact of the option's underlying (risk) factors on its price. The Black-Scholes option 6 1t appears, however, that the risk-free portfolio does not meet the formal definition of a self-financing trading strategy. See Musiela/Rutkowski (1997), p. 109. 6 The Black-Scholes PDE is a so-called Cauchy problem. For further details, see Musiela/Rutkowski (1997), pp. 124-129. 7 See Bj6rk (1998), pp. 58-60.

30

3 Implied Volatility

values of standard call and put options depend on the current time, the underlying stock price, the volatility, the interest rate, the maturity date, and the strike price. The sensitivities of the option price with respect to changes in the first four factors are commonly referred to as the Greeks. Each Greek measures a different dimension of the risk in an option position. At a first glance, the sensitivity of the option price to fluctuations in the model parameters volatility and interest rate seems self-contradictory, since a model parameter is by definition a given constant, and thus cannot change within a given model. In fact, these Greeks measure the sensitivity of the option price with respect to misspecifications of the model parameters. Denoting by Cgs = CBs(t,S, K,T,r,v) and PBS = PBS (t, K,T, r, v) the Black-Scholes pricing function of a standard European call option and a standard European put option, respectively, we can determine the Greeks by taking partial derivatives. The most common Greeks (or more precisely Greek functions) for call options are OCns

as

N(d 1 ) > 0

("Delta"),

(3.26)

82 CBS = 0 ("Gamma"), 052 t acBs s > 0 ("Vega"), Ov sn(di)v OCBs r K e - r(T-t) N (d2) < O ("Theta"), at 21,/7 t OCBs = K(T - t)e - r(T-t) N (d2) < 0 ("Rho"), Or where d1 = d1 (t, s, K, T, r, y) -

ln(*)

+ (r + ly2 ) y

(T - t)

t

d2 = d2(t,s,K,T,r, y) =d1 - v and t E [0, 7]. Similarly, in the case of a put option we get: OPBS

Os 492 PB s

= N(C11) — 1 G

aPBs

("Delta"),

n(di)_ 02 C

8.52 sv OP.6.3 =

av

0

s-Vn(di 7 )

sn(cli)v

0 sa2c

(3.27) ("Gamma"),

> 0 ( "Vega" ),

rice_ r(T-t) iv,„.

(_d2)

("Theta"),

- 2 °Pas = K(T - t)e - T (T-t) (N(d2 ) - 1) < 0. ("Rho"). Or Ot

3.1 The Black-Scholes Model

31

The delta of an option is defined as the rate of change of the option price with respect to the price of the underlying stock, when all else remains the same. At any time t, it gives the number of shares 01, , in the replicating portfolio for the option. Therefore, the option's delta is also called "hedge ratio". 'When the stock price changes, the delta changes, too. This effect is captured by the option's gamma, the second partial derivative of the option price function with respect to the stock price. The vega (also known as lambda) of an option measures the rate of change of the option price compared with the change in the underlying's volatility. Similar statements hold for the Greeks theta and rho. Theta is also sometimes referred to as the (deterministic) time decay of the option. In contrast to the theta of a standard call option which is always positive, the theta of a put option may become negative. Yet, in practice, such a case hardly ever occurs. Compared with the other Greeks, rho is small in magnitude, and is therefore often neglected. When we relax the assumption of a constant volatility in the later chapters, two further Greeks will become important. Following the nomenclature of Taleb (1997), p. 200, we define:

92 CBS

at,2

= .97 Nr'tn(di)

I92 CBS

d2

°say

y

di d2

=—n(c/i) —

a2 PBS at,2

a2 PBS

("DVegaDVol" ),

( "DDeltaDVol" ).

asav

DVegaDVol (DDeltaDVol) corresponds to the change in vega (delta) resulting from a change in volatility. For convenience, we introduce the following standard notation for the Black-Scholes call option Greeks: Definition 3.5. We define OCBs

&CBS

03

OCBs

acBS

as2

ABS

OV

OBS =

at ,

and VBS

a2 CBS

a2 CBS

(91) 2

080V

The Black-Scholes PDE (3.24)-(3.25) can be used to obtain the relation between the Greeks. In the case of a standard European call option, we have: 1 2

et ± rSt 6e + _ v 2 s2,-,t t

rCt

(3.28)

where we have assumed a priori that the arbitrage price of the call option at time t equals Ci = CBs(t, St). The Greeks at time t are given by (it = 8Bs(t,St), et = OB s(t,St ), and rt = rBB(t, St ). This relation also holds for a portfolio, when the portfolio value can be expressed as a function of time and stock price only. The portfolio can thus consist of a position in the

32

3 Implied Volatility

underlying stock itself, as well as positions in various forwards, futures and (path-independent) options written on the underlying stock. The portfolio Greeks are then obtained as the weighted sums of the Greeks of the portfolio components. In general, a portfolio which is insensitive with respect to small changes in one of the risk factors is said to be neutral or hedged with respect to this factor. Formally, this means that the corresponding Greek equals zero. For example, a portfolio is called delta-neutral or delta-hedged if the portfolio delta is zero. If we consider a stock with delta 1 and a call option on this stock with delta 0.5, delta neutrality can be achieved, among other things, by buying two call options and selling one stock. A portfolio which has a positive (negative) sensitivity with respect to small changes in one of the risk factors is said to be long (short) with respect to this factor. For example, a long call is said to be delta long or long delta.

3.2 The Concept of Implied Volatility 3.2.1 Definition The Black-Scholes formula relates the price of an option to the current time, the underlying stock price, the volatility of the stock, the interest rate, the maturity date, and the strike price. All parameters other than the stock's volatility can be observed directly in the market. Given that these parameters are known, the pricing formula relates the option price to the volatility of the underlying stock. If one may observe the market price of the option, then the volatility implied by the market price can be determined by inverting the option pricing formula. This volatility is known as the implied volatility. 8 Definition 3.6 (Implied Volatility). Let Ct (K,T) be the market price of a standard European call option with strike price K > 0 and maturity date T at time t E [0, T). The (Black-Scholes) implied volatility crt (K,T) is then defined as the value of the volatility parameter which equates the market price of the option with the price given by the Black-Scholes formula (3.18):9

Ct (K,T)= CBs(t,S t ,K,T,r,cr t (K,T)).

(3.29)

In fact, this definition can be somewhat misleading these days, since option traders often quote implied volatility directly, and then calculate the option's market price implied by this volatility quote. 1 ° 8 See Mayhew (1995), p. 8. 8 For ease of notation, we will use the symbol at(K,T) for both, the implied volatility of an option with fixed strike K and fixed maturity T and for the implied volatility function with respect to strike price and maturity. 1° In the OTC options markets it is very common to quote option prices in terms of their implied volatilities. On the other hand, for exchange-traded options typically the option's price is quoted.

3.2 The Concept of Implied Volatility

33

Discussions involving implied volatilities will typically also incorporate two other notions of volatility with very different meanings — instantaneous and realized volatility. Whereas implied volatility is derived from the market price of an option, the concepts of instantaneous and realized volatility are based on the price process of the underlying stock. The term instantaneous or actual volatility refers precisely to the volatility that appears in the SDE describing the evolution of the underlying asset. In general, it cannot be observed directly. Consider, for example, the BlackScholes model with stock price dynamics: dSt = p,St dt + vStdWt• Here, the constant y is the instantaneous volatility of the stock price. In a more general setting, the instantaneous volatility might also vary over time, either deterministically or stochastically. If there is no chance of confusion, we will simply refer to the instantaneous volatility as the "volatility". Historical data may also be used to estimate the volatility parameter, which can then be used to compute the theoretical option values. The most natural approach uses an estimate of the standard deviation based upon the ex-post continuously compounded stock returns measured over a specific sample period in the past. This estimate is usually called realized or historical volatility: 11 Definition 3.7 (Realized Volatility). Let us assume that we can observe the stock price process S under the objective measure P at N* + 1 equidistant points in time to, t 1 , , tN, where At denotes the length of the sampling t,,-1, n = 1, . . . , N* . The stock price at time tt, is interval, i.e. At = tn denoted by S. Then the realized volatility (or historical volatility) I', (N) of the stock at current time tti for the last N periods, N < n < N*, is defined as the annualized sample standard deviation of the continuously compounded st stock returns Rtn = ln ( —n— stn _ i :

\I At (N —1)

(Rt

i — Rtn(N))2 , — +1

(3.30)

where R tn (N) is the N-period sample mean at time tn Rin (N)

= — E Rtn-i+1 • i=1

Under the assumptions of the Black-Scholes model, realized volatility (and the sample mean) is independent of n, i.e. Ytn (N) = Y for all n, N < n < N*. b ee e.g., Hull (2000), p. 242. For an in-depth discussion of realized volatility, "See, see Figlewski (1997).

34

3 Implied Volatility

Statistically, 'Y it is a consistent estimator of the constant volatility parameter y. 12 Its standard error can be shown to be approximately i'')/./V. A word on terminology: Let us assume that the current date is to. Then the implied volatility at (K,T) at time t is more precisely called the spot implied volatility, if t = to, the past or historical implied volatility, if t < to, and the future implied volatility, if t > to . The same terminology applies to all other types of volatility used throughout, except for realized (or historical) volatility, which is, by definition, related to the past. 3.2.2 Calculation

Since the Black-Scholes call pricing formula CBS (y) = CBS(t, St, K ,T, r, y) is a continuous - indeed, differentiable - increasing function of y, with boundaries" liMCBS(V) = v-40

- Ke - r(T- t)

t

o

if St > K e- r(T-t)

if st < Ke -r(T-t)

(3.31)

liM CBS(V) = St)

V—.00

the inverse function exists. 14 Therefore, a unique positive implied volatility a-t (K,T) can always be found. Unfortunately, there is no closed-form solution for implied volatility, although the Black-Scholes formula is given in analytic form. Instead one has to use numerical methods to find the implied volatility. Perhaps the most commonly used method in this context is the Newton-Raphson iteration procedure. If the transcendental equation

Ct(K,T) = CBS (V) is to be solved for y, and vk is the current approximation, then the next approximation V k+1 is given by the Newton-Raphson formula CBS (Vk) — Ct(K,T) Vki-1 = Vk

8C25(v) v lv—vk

(3.32)

of the Black-Scholes formula with respect to where the derivative acBs(v) av the volatility parameter - the option's Black-Scholes vega ABs(v) - is always positive. The iteration step (3.32) is repeated until convergence to the desired 12 Notice that realized volatility is a biased estimator of true volatility y, but the bias disappears asymptotically. On the other hand, realized variance 1D2 is an unbiased estimator of true variance y 2 . For a definition of consistency and unbiasedness, see Kmenta (1997), pp. 156-169. "See Cox/Rubinstein (1985), p. 216. 14 See KOnigsberger (1995), p. 30.

3.2 The Concept of Implied Volatility

35

accuracy is achieved (typically only a couple of iterations). 15 Alternatively, implied volatility can be computed using the simple approximation formula of Corrado/Miller (1996). It provides the implied volatility of an option in closed-form across a wide range of underlying prices. In the absence of arbitrage and market frictions, the implied volatility of a European put option and a European call option with the same strike price and the same maturity date must coincide if the underlying stock pays no dividends. 16 To see this, suppose that, for a particular value of y, Cgs (t, St, K,T,r,v) and PR s(t, St, K,T,r,v) are the time t Black-Scholes values of European call and put options with strike price K and maturity date T. Because put-call parity is satisfied for the Black-Scholes model we must have Cgs (t, St, K,T,r,v)

—

PB s(t, St , K,T,r,v) = St

—

Ke — r(T—t) .

(3.33)

Since it also holds for the market prices Ct (K,T) and Pt (K,T):

Ct (K,T)

—

Pt (K,T)= St

—

Ke — r(T—t) ,

(3.34)

we get by subtracting (3.34) from (3.33)

CBs(t,S t ,K,T, r, y)

—

Ct (K,T)= PBs(t,St ,K,T,r,v)— P t (K,T).

(3.35)

Equation (3.35) shows that the absolute pricing error when the BlackScholes model is used to price a European call option with strike price K and maturity date T must be equal to the absolute pricing error when it is used to price the corresponding European put option. Let us now suppose that the implied volatility of the call option is at (K,T). This means that the left-hand side of equation (3.35) becomes zero when y = at (K,T) in the Black-Scholes formula. From equation (3.35), it then follows that Ps(K, T) = PBS(t, St, K,T,r,at (K,T)). This argument shows that in the case of a standard European option it should not matter whether one uses a call price or a put price to back out implied volatility. 3.2.3 Interpretation The interpretation of implied volatility depends on whether the Black-Scholes model provides a good description of reality or not. Let us first consider the case where the Black Scholes model holds. If it holds exactly, the volatility implied in an option's market price is widely regarded as the (subjective) -

15 Another frequently used method to derive the implied volatility is the bisection method. For background information on the Newton-Raphson and the bisection method, see Press et al. (1992), Sections 9.1 and 9.4. "See Carr (2000), p. 9 and Hull (2000), pp. 435-436.

3 Implied Volatility

36

market's expectation of the future constant volatility y of the stock's continuously compounded returns. 17 Relaxing the assumption of a constant volatility and assuming instead that volatility is time-dependent, i.e. the stock's actual volatility is a deterministic function y : [0, T1 R + of time only, it can be shown that the Black-Scholes formula is still valid when using the average volatility D(t,T) over the remaining life of the option T — t, i.e.

I

D(t,T) — \ I T 1 t ) T V 2 (u du

(3.36)

in place of y. 1 8 In that case, implied volatility is interpreted to be the market's expectation of the future average volatility £3(t, T).' 9 Now, if capital markets are informationally efficient 20 , implied volatility should be an efficient forecast of future (average) volatility. This means that implied volatility should subsume all information in the market information set, especially the historical price record of the stock, that can be used to predict the future volatility of the underlying stock. 21 On the other hand, if the Black-Scholes model does not hold, the logical foundation for the belief that implied volatility is the efficient market's forecast of the future (average) volatility may be rather weak. For example, if the actual volatility is a deterministic function of time and stock price or if it is stochastic, implied volatility can generally not be interpreted as a time-averaged volatility — not even as a true volatility measure. 22 Although implied volatility may, in fact, have little to do with the stock's actual volatility, it is nevertheless a highly regarded number by option traders and other market participants, but "See Christensen/Prabhala (1998), Figlewski (1997), and Mayhew (1995), among others. result was first shown by Merton (1973). "Note that the term "average volatility" can be somewhat misleading, since the quantity defined in (3.36) does not in general coincide with the value fT v(u)du, as one would normally expect from the name. However, this term is so common that we will use is here, too. "Financial economists have a strong belief that capital markets are infor?nationally efficient, at least in their weak form. In an efficient market, prices fully and instantaneously reflect all available relevant information. The concept of market efficiency was mainly developed by Fama (1970). For further details, see Copeland/Weston (1988), pp. 330-356. 21 More formally, implied volatility is called an efficient forecast of future volatility, if the forecast error is a white noise process that is uncorrelated with any variable in the market's information set. See also Christensen/Prabhala (1998). 22 0ne exception is the stochastic volatility model of Hull/White (1987). Assuming that actual volatility follows a GBM and volatility risk is unpriced in the market, Hull/White (1987) have shown that the option price is the Black-Scholes price integrated over the distribution of the average volatility. For a thorough discussion of the interpretation of implied volatility as a future average volatility under different models for the stock price, see Lee (2002). 18 This

3.2 The Concept of Implied Volatility

37

for quite different reasons. 23 For them, implied volatility is a measure of an option's price or as Lee (2002) points out "... a language in which to express an option price", one that controls for option-specific characteristics such as the strike price, the maturity, etc. The translation of option prices into implied volatilities brings the advantage of eliminating a substantial amount of nonlinearity. 24 Implied volatility is also interpreted as an implicit parameter which embodies all deviations from the Black-Scholes assumptions. If the BlackScholes assumptions are wrong and implied volatility cannot be interpreted as a volatility forecast, it is worth stressing that there is nothing special about using the Black-Scholes formula in the definition of implied volatility. The Black-Scholes formula is then just a convenient and well-known mapping from option prices to implied volatilities. Other functions with similar properties as the Black-Scholes formula will work just as wel1. 25 The question whether implied volatility efficiently predicts future volatility has been the subject of a vast number of empirical studies. 26 In these studies, typically a weighted average of implied volatilities, rather than the implied volatility of a single option, is used as a point estimate for the future volatility. 27 Early papers, e.g., Latane/Rendleman (1976), examine the informational content of implied volatilities in a cross-sectional setting. These papers essentially document that stocks with higher implied volatilities also have higher ex-post realized volatility. The time series literature on the predictive power of implied volatilities has produced mixed results. On the one hand, Canina/Figlewski (1993), for example, find for S&P 100 index options that implied volatility has little or no correlation at all with future return volatility. On the other hand, Christensen/Prabhala (1998), using the same data set, come to the conclusion that implied volatility is a useful source of information to predict future volatility, although it fails to be efficient. In particular, they find that historical volatility has no incremental explanatory power over implied volatility in some of their specifications. Although there is still a lot of discussion, the general conclusion to be drawn from this large body of research is that the informational content of implied volatility goes beyond that of past return data. 28 However, implied volatility fails to be an efficient forecast of future volatility. Moreover, it tends to be biased. this context, the term "implied volatility" is somewhat misleading. Rosenberg (2000), pp. 51-52. 25 In principle, every bijective function of the option price with respect to volatility is suitable. See also Ledoit et al. (2002). 26 For a systematic overview on tests of the Black-Scholes model, see Bates (1996b). See also Mayhew (1995). 27 The reasoning behind this approach is as follows: If the Black-Scholes model is correct, market microstructure effects such as price discreteness and nonsynchronous trading, causing implied volatilities to differ across options, represent noise, and noise can be reduced by using more observations. For an in-depth discussion of the different weighting schemes, see Bolek (1999), pp. 134-143. 28 See Mayhew (1995), p. 13. 23 In

24 See

38

3 Implied Volatility

If the Black-Scholes model holds exactly all options on the same underlying asset should provide the same implied volatility. Yet, as is well-known, empirical implied volatilities differ systematically across strike prices and across maturity dates such that the misspecified model produces the correct market prices29 or as Rebonato (1999), p. 78, puts it: "...implied volatility is the wrong number to put in the wrong formula to obtain the right price." This finding delivers the most striking evidence against the Black-Scholes model and the interpretation of implied volatility as an efficient forecast of future volatility. 30

3.3 Features of Implied Volatility In this section we describe some well-known patterns in the behavior of implied volatility as the strike price and the maturity date of the option change. 31 explanations for the existence of these patterns are given in the nextPosible section. 3.3.1 Volatility Smiles The most often quoted phenomenon testifying to the limitations of the BlackScholes model is the smile effect: that implied volatilities vary with the strike price of the option contract. Formally, we define the volatility smile as follows: Definition 3.8 (Volatility Smile). For any fixed maturity date T, T < T*, the function crt(K,-) of implied volatility against strike price K, K > 0, is called the (implied) volatility smile or just smile (for maturity T) at date t E [0, T). Henceforth, we shall use the term "volatility smile" to denote both the implied volatility function at (K, -) with respect to K and its graphical representation. Before the 1987 crash, implied volatilities in equity options markets were, in general, nearly symmetric around the prevailing underlying price, with ITM and OTM options having higher implied volatilities than ATM options. This corresponds with a "U-shaped" form - a smile shape - in the plot of implied volatility against strike (left graph of Figure 3.1). A skew or sneer pattern, however, is more indicative of the pattern since the crash - at least for longer term index options - with implied volatilities decreasing monotonically as the strike price rises (right graph of Figure 3.1). The skew curve tells us that there is a premium charged for OTM puts and ITM calls above their "See, e.g., Mayhew (1995), p. 14. 30 The

Black-Scholes model has been tested in a number of empirical studies. For a summary of the main results, see Hull (2000), pp. 448-450. 31 In the following, see Alexander (2001a), Chapter 2, Dumas et al. (1998), and Rebonato (1999), pp. 83-87.

3.3 Features of Implied Volatility

39

Black-Scholes price computed with the ATM implied volatility. Conversely, for OTM calls and ITM puts there is a premium received. In this work, if not stated otherwise, the term "smile" is used as a general expression for the shape of the implied volatility pattern across exercise prices. It covers a literal smile as well as skew.

// /

OTM puts ITM calls

......____

ATM

Implied volatility

Implied volatility

ATM

OTM puts FTM calls

OTM calls fl-M puts

OTM calls ITM puts

Strike price

Strike price

Figure 3.1. Ideal types of volatility smiles. Left graph: literal smile; right graph: skew

Other qualitative features of volatility smiles for equity index options are that smiles are not constant but vary over time and that smiles are more pronounced for short-dated options when expressed in the strike price K ("flattening-out effect") •32 3.3.2 Volatility Term Structures Next, we consider the relationship between implied volatility and maturity date for a fixed strike option. Definition 3.9 (Volatility Term Structure). For any fixed strike price K, K > 0, the function ci t e ,T) of implied volatility against maturity T, T < Ta, is called the term structure of (implied) volatility or (implied) volatility term structure (for strike K) at date t E [0, T). Again, the notion "term structure of volatility" stands for the function at (., T) as well as for its plot. The strike price K is usually chosen to be the ATM strike. Unless otherwise stated, we will follow this convention. In analogy to the terminology used in interest rate markets, the term structure of volatility is called normal, if implied volatilities for options with longer maturities are higher than those for options with shorter maturities. 3 2 See, e.g., Dumas et al. (1998). For a detailed analysis of the empirical properties of the strike profile of DAX option implied volatilities, see Section 5.3.

3 Implied Volatility

40

Implied volatility

Implied volatility

Conversely, we speak of an inverse shape, if short-dated options have higher implied volatilities than longer-dated options. The term structure is called flat, if the plot of implied volatility against maturity is a horizontal line. The three basic shapes are illustrated in Figure 3.2.

Maturity date

Maturity date

Maturity date

Figure 3.2. Basic shapes of the volatility term structure. Upper left graph: normal shape; upper right graph: inverse shape; lower graph: flat shape The concept of forward implied volatility is introduced next: 33 Definition 3.10 (Forward Implied Volatility). Given the term structura of volatility at(., T) (for some K) at time t, where cr?(.,T2)(T2 — t) > 01(.,To (Ti - t), the forward implied volatility Grr (T1 , T2) between dates T1 and T2 t d), the model is not complete. This requires the specification of the market price of volatility risk. Heston (1993) assumes that this is proportional to the variance of the stock price. Call prices can then be readily obtained in closed-form using Fourier transformation techniques. This is an appealing feature of the model, which explains part of its popularity. Different correlation coefficients p will result in different probability distributions and smile patterns. If volatility is uncorrelated with the stock price, i.e. p = 0, a true smile occurs whose degree depends on the other parameters of the model, especially the volatility of volatility tv. A negative correlation 44

Empirical analyses tend to support stochastic implied volatilities and therefore stochastic option prices. See, Skiadopoulos et al. (1999), Hafner/Wallmeier (2001) and Cont/Fonseca (2002), among others. 45 See Schwert (1990b), Engle (1982), and Ebens (1999), among many others.

48

3 Implied Volatility

coefficient spreads the left tail of the distribution and thus produces a skew pattern. One possible reason for the increase in volatility when the stock price falls refers to the leverage effect. A lower stock price brings about a higher leverage ratio producing an increase in stock return volatility, and vice versa. Yet, in a study on the S&P 100 index, Figlewski/Wang (2000) find a strong leverage effect associated only with falling stock prices. They conclude that the variations of volatility have little direct connection to firm leverage. On the other hand, a positive correlation between stock price and volatility increases the probability of high returns and thus results in a positive skew pattern. In a market with stochastic volatility, the existence of smiles can also be explained by simple (model-free) no-arbitrage arguments. Let us consider an investor who buys an option with positive DVegaDVol (any OTM option, either high- or low strike) 46 If the investor now hedges the outright vega by selling an ATM option (which has roughly zero DVegaDVol), he ends up with a position which gains whatever direction volatility will move. To prevent arbitrage opportunities, thus, the OTM option must have an implied volatility which is above the ATM implied volatility. This explains the (literal) smile profile. The explanation of the skew pattern involves the Greek "DDeltaDVol". Yet, the argumentation is quite similar. A problem of stochastic volatility models is that unrealistically high parameters are required in order to generate volatility smiles that are consistent with those observed in option prices with short times to maturity. 47 This is not the case for long times to expiration. A further problem with stochastic volatility models results from their incompleteness. Since one has to specify and calibrate the market price of risk (process), the model depends on the risk preferences of the investors. This gives rise to specification error, which may, for instance, result in wrong hedging numbers. Moreover, there exists no hedging strategy eliminating all market risk related with volatility.

Jumps A further explanation for implied volatility patterns refers to jumps in the asset price process. 48 When jumps occur, the price process is no longer continuous. Jumps have proved to be particularly useful for modelling the crash risk, which has attained considerable attention since the stock market crash of October 1987. It is often argued that the increased sensitivity of market participants to the crash risk - sometimes called "crash-o-phobia" - has contributed to the skew pattern in S&P options prevailing since 1987. In this context, Bates (1991) has interpreted the high implied volatility of OTM puts prevailing in the forefront of the 1987 crash as an insurance premium against jump risk. 46 In the constant volatility Black-Scholes market DVegaDVol would be zero, by definition. 47 See, e.g., Andersen et al. (1999), p. 3 and Das/Sundaram (1999), p. 5. 48 see, e.g., Bates (1996a), Trautmann/Beinert (1999).

3.4 Modelling Implied Volatility

49

Merton (1976) was the first to study the impact of jumps in the stock price process on the pricing of options. The model of Merton combines a continuous diffusion process with a discontinuous jump component. The evolution of the stock price under the objective measure P is given by the following SDE

dS t

= (II. — Am) dt + vdWt + (I — 1) dIVi t (A),

(3.45)

where the symbols p and y have their usual meaning, g(A) denotes a Poisson process" with intensity A, / is the time-independent random amplitude of the jump, and m = EP [I — 1] is the average jump size. It is assumed that there is no correlation between the Brownian motion W and the Poisson process N(A), and no correlation between the size of the jump I and the occurrence of the jump, represented by N(A). Since the jump component cannot be hedged, there exists no unique replication strategy for contingent claims in this model. The model is incomplete. To avoid the specification of the market price of jump risk, Merton (1976) assumes that the jump component of the stock's return represents nonsystematic risk (i.e. risk not priced in the economy). This means that holding a portfolio consisting of a long position in a call option and a short position in S stocks must earn the riskless rate r over a time step dt. This argument leads to a PDE, which can then be solved for the call price. For the special case that the logarithm of / is normally distributed, there exists a closed-form solution for the call price. 50 In all other cases, numerical procedures have to be applied. The jump-diffusion model gives rise to fatter left and right tails than BlackScholes and can therefore explain the smile effect. Whereas the effect of stochastic volatility increases with longer time to maturity, the impact of jumps diminishes. This is due to the fact that in long time periods positive and negative jumps compensate each other. Therefore, jumps seem especially suitable for modelling the steep implied volatility smile for short maturities. On the other hand, stochastic volatility models are particularly qualified for the modelling of the relatively flat implied volatility smile for long maturities. It seems therefore natural to combine the two approaches. Two examples for stochastic volatility models with jumps are Chernov et al. (1999) and Jiang (1999). Although adding jumps to the stock price process undoubtedly captures a real phenomenon that is missing from the Black-Scholes model, jump-diffusion models are rarely used in practice. Basically, there are three main reasons for this: first, it is hard to find a solution for the pricing PDE, since the governing equation is no longer a diffusion equation, but a difference-diffusion equation; 49 For a discussion of the Poisson process, see, e.g., Brzezniak/Zastawniak (1998), Section 6.2. 50 For an easy to understand derivation of the call price formula, see Wilmott (1998), pp. 329-330.

50

3 Implied Volatility

second, the model parameters are difficult to estimate; and finally risk-free hedging is not possible, since the model is not complete. 51 Alternative Distributions for the Innovation Term A further possible explanation for the smile pattern is that prices move continuously but not according to a geometric Brownian motion. The true underlying distribution may thus be characterized by fat tails and skewness, even if the volatility is constant and jumps do not occur. For example, Eberlein et al. (1998) propose to describe the terminal stock price using a hyperbolic distribution. Analyzing five German stocks, the authors find that the hyperbolic model accurately fits the empirically observed returns and is also able to reproduce the observable smile pattern. Another interesting approach comesfrom Madan et al. (1998). They propose a three parameter generalization of Brownian motion as a model for the dynamics of the logarithm of the stock price. The new process, termed the variance gamma process, is obtained by evaluating Brownian motion at a random time change given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time. They provide control over the skewness and kurtosis of the return. Within an empirical study for S&P 500 options they demonstrate that their model can reproduce the observable implied volatility patterns. A recent development, with promising first results, concerns the use of a so-called fractional Brownian motion in the SDE of the stock price instead of the standard Brownian motion. For more details on the use of fractional Brownian motion in finance and especially in option pricing, the reader is referred to Sottinen (2001). Market Frictions Market frictions are another possible explanation for the smile pattern. Transaction costs, illiquidity and other trading restrictions imply that a single arbitrage-free option price no longer exists. Instead, there is a band of feasible prices. Since arbitrage is no longer sufficient to derive a definite option price, Longstaff (1995) proposes an "unrestricted Black-Scholes model", which does not impose the martingale restriction. In his study of S&P 100 index options the suggested specification is able to neutralize the pricing bias with respect to the strike profile. Longstaff concludes "that transaction costs and liquidity effects play a major role in the valuation of index options" (p. 1093). Similarly, Figlewslci (1989a) (and also Figlewski (1989b)) examined the effects of transaction costs by simulating a large number of price paths and found that they could be a major element in the divergences of implied volatilities across strike prices. Yet, Constantinides (1996) points out that transaction costs cannot fully explain the extent of the volatility smile. 51

See Wilmott (1998).

3.4 Modelling Implied Volatility

51

McMillan (1996) argues that the crash of 1987 lessened the supply of put option sellers, whereas at the same time fund managers showed a higher demand for out-of-the-money puts. Because hedging the risk exposure of written out-of-the-money puts turned out to be expensive, higher prices for out-ofthe-money puts were charged. This could partly explain the observed skew pattern. 52 Tax effects and the capital requirements associated with holding out-ofmoney options in options books are two other cases that could cause a smile in implied volatility. 3.4.3 Implied Volatility as an Exogenous Variable Option Pricing using the Risk-Neutral Density Implied risk-neutral density functions derived from cross-sections of observed standard option prices, or equivalently, the volatility smile, have gained considerable attention during the last years. 53 To introduce the concept of a risk-neutral density, let us consider an arbitrage-free and frictionless market where the money market account evolves according to Bt = exp (rt) for all t E [0, 71. Due to the risk-neutral valuation formula (Theorem 2.14), the time t price of a European path-independent T-contingent claim H = T E (t, T 4], on the stock S is given by

H(H) t(H) = e -r(T-t) EQ [HI .Ft1= e -r(T-t) f

oo

(s) qsT (s)ds,

(3.46)

with qs,,,(s) denoting the risk-neutral density (RND) of ST conditional on the information set .Ft at time t. Setting (I) (Sr) = max {ST — K; 0} yields the price Ct of a standard European call option with strike price K and maturity

T: t = e-r(T-t)

oc

max {.5 — K; qs,,,(s)ds.

(3.47)

Given the market prices Ct (K, T) of a continuum of European call options on the stock S with the same time to maturity T and strike prices K ranging from zero to infinity, we can apply the fundamental result of Breeden/Litzenberger (1978) to fully recover the RND in an easy and unique way.54 They have shown that the discounted RND is equal to the second See also Cochrane/Sag-Requejo (1996). the following, see Brunner/Hafner (2003). 34 Since Breeden/Litzenberger (1978) build their work upon the state-preference theory of Arrow (1964) and Debreu (1959), they speak of a "state-price density". We refrain from using this expression for two reasons: First, it is ambiguous. The term state-price density is also referred to the Radon-Nikodym derivative c/Q/cilP (see, e.g., Plislca (2000), p. 28). Second, the term RND is more popular in the recent literature. 52

53 In

3 Implied Volatility

52

derivative of the European call price function (3.47) with respect to the strike price K. qs, (8)

=

a2c,(K, T) aK2

(3.48) K=s

For any fixed maturity T, T> t, the relation between the RND and the volatility smile is obtained by successive application of (3.48) and the definition of implied volatility (3.29): (K2, T) (s ) .= er(T—t) &cat K

(3.49) K=s

= e r(T t) a2 CBS(t, St K,T,r,cr t (K,T))1 —

I K=s After applying the chain rule for derivatives, we get: 1 gsT (s ) = n(d2(s)) [ scr t (s,T)

2 2d1(s) acrt(K,T)I + c• t (s,T) alf I K=s

—t acrt(IC' T) 1 ) 2 T + scli(s)da(s)V 0,(s, arc K=s

s

(3.50)

maatK (K2" K=sj

for all s > 0, where ln (49t ) -

di(s)

(r

(cr t (s , T)) 2 ) (T — t)

at(s,T) ■/77--t-

, d2 (s)= cli (s)—cr t (s,T).

(3.51) and n(x) is the standard normal density function. For equation (3.50) to be properly defined, the implied volatility function crt (K,T) is required to be twice differentiable in K. If o-t (K, T) is given in closed form, so is the RND qs,.(s). As the above discussion has shown, for a given maturity T three functions contain essentially the same information: the RND qsT (s), the call price function Ct(K,T) with respect to strike, and the volatility smile o-t (K,T). This is also illustrated in Figure 3.5. In practice, option contracts are only available for a discrete set of strike prices within a relatively small range around the at-the-money (ATM) strike price. Therefore, all of the methods for estimating RND functions boil down to the completion of the call price function (or implied volatility function or RND function, respectively) by interpolating between available strike prices and extrapolating outside their range. RNDs have found various applications in finance: Central banks, among others, use the RND to assess the market participants' expectations about underlying asset prices in the future. 55 Ait-Sahalia/Lo (2000) and Jacicwerth 55

ers.

See Bahra (1997), Cooper/Talbot (1999), and Levin/Watt (1998), among oth-

3.4 Modelling Implied Volatility

53

(t, St, K,T,r,Ct(K,T))

CBS(t, St ,

K ,T, r, t (K ,T))

Figure 3.5. Relationship between call price function, volatility smile, and RND

(2000) compare the RND with the objective density to retrieve the investors'

risk preferences. Bates (1996a) uses the RND to estimate the parameters of the underlying stochastic process which generates this RND. In the field of option pricing, the implied RND allows to price illiquid path-independent exotic derivatives in a consistent way. Given qs, (s), we simply have to evaluate (3.46) to obtain the price of a derivative H ---- CST)• The main advantage of the RND approach is that we do not have to specify the stock price process. The continuum of option prices is sufficient to uniquely derive the RND qs, (s) which allows for the pricing of all path-independent options as, e.g., digital or power options. On the other hand, because we do not explicitly model the stock price evolution, we cannot price options where this information is needed. In particular, we are not able to value pathdependent options such as barrier or lookback options. For the same reason, the RND concept cannot be employed for hedging purposes and the analysis of (option) trading strategies. For applications of the RND concept in the area of risk management, the RND has to be transformed into the corresponding objective density. This affords additional assumptions on the risk preferences of the agents in the economy. Deterministic Implied Volatility Models The empirical observation that the implied volatility surface evolves with time and stock price has led practitioners to develop simple rules to estimate its evolution. Following the study of Derman (1999) on S&P 500 options, we distinguish between three different rules, where, in turn, each rule is associated

54

3 Implied Volatility

with a different market regime. 56 The first rule, the so-called sticky-strike rule , postulates that, when the underlying index moves, the implied volatility of an option with a particular strike and a particular maturity remains unchanged from time t to t + At, where At is a small time interval (e.g., a day) - hence the term "sticky-strike". Mathematically, the sticky-strike rule can be expressed as: V(K,T). (3.52) at+At (K) T) = at (K,T), If the market behaves according to the sticky-strike rule, the absolute implied volatility surface has no dependence on the index level. This implies that the ATM implied volatility decreases as the index increases, and vice versa. The sticky-strike rule is typical for situations where future underlying moves are likely to be constrained to a trading range, without a significant change in current volatility. In a trending market, Derman (1999) suggests to use the sticky-moneyness rule (also called sticky-delta rule). This rule stipulates that, when viewed in (M, r) remains constant from relative terms (M, r), the volatility surface time t to t ± At. Mathematically:

a,

at-I-At

(MI T) = at(m, 7),

v(m, T),

(3.53)

with moneyness defined as M = tF,‘-. According to this rule, the ATM (i.e. M = 1) implied volatility is constant over time. However, for a fixed strike option, the implied volatility increases as the index level increases." In a third market environment, characterized by jumpy markets, implied volatilities are supposed to follow the sticky-implied-tree model. As the name suggests, this rule is related to the implied tree approach, presented in Section 3.4.2. It says that the implied volatility of a fixed strike option will decrease when the index goes up, and increase when the index falls. The ATM implied volatility decreases roughly twice as rapidly as the index level increases. For more details on these rules, the reader is referred to Derman (1999) and Alexander (2001a). The above rules are in fact (simple) deterministic laws of motion for the volatility surface: given today's prices and given the current market regime, there is no uncertainty about the implied volatility surface tomorrow. These rules can be extended to models where a t (K, T) or (M, r) evolve in a more general deterministic way. In a detailed study, Balland (2002) shows that in fact the only arbitrage-free models in which at (K, T) is deterministic are Black-Scholes models with at most time-dependent volatility. As Cont/Fonseca (2002) point out, these simple rules are not verified in practice. They show that the implied volatility surface of S&P 500 options has "a noticeable standard deviation that cannot be neglected when considering either hedging or risk management of portfolios of options". Even the 56 See

also Cont/Fonseca (2002). that this is only true in the case of a volatility skew.

57 Note

3.4 Modelling Implied Volatility

55

ATM implied volatility, assumed to be constant or slowly fluctuating in the sticky-moneyness model, exhibits significant variation. These results are also supported by other empirical studies. 88 Stochastic Implied Volatility Models Due to the apparent shortcomings of deterministic implied volatility models, a

natural step of generalization is to let implied volatilities move stochastically. In contrast to (traditional) stochastic volatility models, where the instantaneous volatility of the stock is modelled, stochastic implied volatility models focus on the (stochastic) dynamics of either a single implied volatility, the volatility smile, the term structure of volatility or the whole volatility surface. Lyons (1997) was the first to model the stochastic evolution of a single implied volatility. Amerio et al. (2001) also models a single implied volatility to price derivatives on implied volatility. Following the path-breaking work of Heath, Jarrow, and Morton (HJM) 59 in the area of interest rates, SchtSnbucher (1999) models the evolution of the term structure of volatility in a continuoustime setting. He derives a no-arbitrage condition on the drifts of the options' implied volatilities which is similar to the drift condition derived by HJM for the instantaneous forward rates and furthermore he analyzes the restrictions that have to be imposed to ensure regularity of the option price at expiry. Albanese et al. (1998), extending the work of Scht5nbucher (1999), consider the arbitrage-free evolution of the volatility surface at (K,T) in a generalized 10 setup. Ledoit et al. (2002) independently arrive at a similar model as Albanese et al. (1998), however, their focus is on the modelling of the relative volatility surface Eit (M, 7-) with moneyness defined as M = St/K. They show that in a stochastic implied volatility model the instantaneous volatility v t of the stock can no longer evolve independently. In particular, they prove that the implied volatility of at-the-money options converges to the instantaneous volatility of the underlying asset as the time to maturity approaches zero. Because the number of random sources is smaller than or equal to the number of risky assets, stochastic implied volatility models are in general completemarket models. While the above approaches dealt with the problem of stochastic implied volatility from a theoretical perspective, Rosenberg (2000), Cont (2001), and Cont/Fonseca (2002), among others, focus on the empirical aspects of the problem. Rosenberg (2000) proposes a "dynamic implied volatility function model" to describe the (discrete) evolution of the implied volatility surface of S&P 500 futures options. Thereby he separately estimates the time-invariant implied volatility function with respect to moneyness and time to maturity and the stochastic process of the only state variable, the ATM volatility, that 58 See Alexander (2001b) for FTSE 100 options and Hafner/Wallmeier (2001) for DAX options. "See Heath et al. (1992).

3 Implied Volatility

56

together drive changes in the individual implied volatilities. Based on an empirical study of time series of implied volatilities of S&P 500 and FTSE 100 index options, Cont/Fonseca (2002) suggest a factor-based stochastic implied volatility model. The abstract risk factors driving the volatility surface are obtained from a Karhunen Lc:16\re decomposition. This model extends the stickymoneyness model used by practitioners, while matching some salient features of volatility surfaces. A natural application of the statistical models of Rosenberg (2000) and Cont/Fonseca (2002) is the simulation of implied volatility surfaces under the real-world measure, for the purpose of risk management. However, the models are not intended to determine the consistent volatility drifts needed for risk-neutral pricing of exotic derivatives. 60

3.4.4 Comparison of Approaches In the attempt to correctly reproduce the current volatility surface, neither (one-factor) stochastic volatility models nor simple jump-diffusion models are successful. 61 Local volatility models, on the other hand, are able to exactly reproduce the current volatility surface; however, they perform poorly in grasping its future evolution. Implied volatility models match the current market prices of standard options, by definition. Whether they also match the future option prices depends on the chosen model. Models based on the underlying asset to describe the dynamic behavior of option prices or their implied volatilities, except of local volatility models, are in general incomplete. Consequently, the requirement of no-arbitrage is no longer sufficient to determine a unique price of the contingent claim. Instead, we have several risk-neutral measures, and several market prices of risk. To derive a unique value, we have to make assumptions on the risk preferences of the agents in the economy. This implies that the valuation of contingent claims is no longer preference-free. In contrast, stochastic implied volatility models are complete-market models and thus independent of investors' risk preferences. This implies that there exists a unique replicating strategy for each contingent claim and thus a unique arbitrage price. The fact that we can invoke the risk-neutral valuation principle for valuation purposes in models where implied volatility is endogenous does not imply that any modelling approach that produces the same final price distributions and is consistent with risk-neutrality will lead to equivalent results. For example, hedging numbers may in fact be quite different. It is therefore essential, to identify the financial mechanism, which causes the smile. Since it is generally acknowledged that the financial mechanisms that are responsible for volatility smiles are interrelated, and no single explanation completely captures all empirical biases in implied volatilities, this can be very difficult. 62 60 See

Lee (2002). the following, see also Cont/Fonseca (2002). See Rebonato (1999), p. 93 and Hafner/Wallmeier (2001).

61 In 62

3.4 Modelling Implied Volatility

57

Implied volatility models, on the other hand, do not have these problems as they do not attempt to explain the volatility surface but use it as an input. Another advantage of market-based modelling of implied volatilities is that implied volatilities are directly observable and independent of any modelling assumptions on the processes involved. By contrast, quantities such as the local or stochastic volatility or the jump intensity, are not directly observable and have to be filtered out either from pricing data on the underlying asset using an econometric model or "calibrated" to options data. 63 In the first case, the quantity obtained is model-dependent and in the second case it is the solution to a non-trivial optimization problem. On the other hand, implied volatility models are automatically calibrated to market option prices. In contrast to fundamental quantities such as an (unobservable) instantaneous volatility or a jump intensity, implied volatilities are highly regarded and continuously monitored by market participants. A market scenario described in terms of implied volatilities is therefore easier to understand for a practitioner than the same scenario (re)expressed in terms of these fundamental factors. These arguments motivate the direct modelling of the implied volatility surface. Due to the inability of deterministic implied volatility models to capture the empirically observable fluctuations of the implied volatility surface, stochastic implied volatility models seem favorable. The main disadvantage of these models is their complexity, because, in addition to the stock price process, the (stochastic) joint dynamics of the implied volatilities of all strikes and all maturities have to be modelled in such a way that they are consistent with no-arbitrage. Fortunately, shifts in the level of implied volatility exhibit high correlation across strikes and maturities. This suggests that their joint evolution can be accurately described by a small number of risk factors.

63 Calibration usually means determining the parameters of a financial model such that model prices best possibly fit observed market prices. This is achieved by minimizing a prespecified error function, for example, the sum of squared errors.

4

The General Stochastic Implied Volatility Model

There is nothing more practical than a good theory.

Leonid Ilich Brezhnev

In this chapter we develop a rigorous mathematical model of a financial market in continuous time where in addition to the usual underlying securities stock and money market account, a collection of standard European options is traded. The prices of the standard options are given in terms of their implied volatilities. These, in turn, are described by risk factors, which are stochastic. To meet the objective of providing a framework that is applicable to the pricing and hedging, the risk managing, and the trading of contingent claims, consistent modelling under both the objective measure and the risk-neutral measure is required. In this respect, the model presented here differs from other models proposed in the literature which apply either the objective measure (e.g., Cont/Fonseca (2002) and Rosenberg (2000)) or the risk-neutral measure (e.g., Schtinbucher (1999) and Albanese et al. (1998)), but do not consider both simultaneously. The chapter is organized as follows. In Section 1, we describe the financial market model under the objective probability measure II). In Section 2, we derive necessary and sufficient conditions that have to be imposed on the drift coefficients of the options' implied volatilities in order to ensure discounted call prices to be martingales under the risk-neutral measure Q. In this context, we also discuss existence and uniqueness of a risk-neutral measure. Finally, we show in Section 3 how to price and hedge a general stock price dependent contingent claim.

4 The General Stochastic Implied Volatility Model

60

4.1 The Financial Market Model 4.1.1 Model Specification We consider a frictionless security marketi where investors are allowed to trade continuously up to some fixed finite planning horizon T* (Assumption 1). 2 The uncertainty in the financial market is characterized by the complete probability space (SI, .7',P) where SI is the state space, Y. is the a-algebra representing measurable events, and P is the objective probability measure. Information evolves over the trading interval [0, 71 according to the augmented, right continuous, complete filtration IF = {Ft : t E [0, T*11 generated by a p 1-dimensional standard Brownian motion W = {Wt : t E [0, T*]} , Wi,t, • • • 7 Wp,t) i ) initialized at zero. We assume that the or-field Wt=(o, contains all the P-null sets of .7 -, and that Y .T. = F. trivial and Yo is The primary traded securities are a non-dividend paying stock3 , a money market account and a continuum of standard European call options on the stock. Under the objective probability measure P, we make the following assumptions on the evolution of the money market account and the stock: Assumption 2: The price process of the money market account is given by the SDE: dB t= rBt dt, Vt E [0, 71, (4.1) where Bo =1 and the interest rate r is supposed to be constanti and nonneg-

ative. Assumption 3: The evolution of the stock price is governed by

dSt Stiltdt + StvtdWo,t,

Vt

[0, T *

]

7

(4.2)

with initial non-random stock price So > O. The drift process {A t : t E [0, T*1} is real-valued, progressively measurable and satisfies fot ISuA u l du < co P-a.s. for all t E [0, Tt] . The volatility process {v t : t E [O, T1 } is supposed to be nonnegative, progressively measurable with ft:t) Siiv 2u du < oo P-a.s. for all t E [0, At time t the continuum of European call option prices Ct (K, T) with strikes K > 0 and maturities T> t can be represented by the volatility surface 1

Note, however, that some market frictions are already reflected by market option

prices.

the following, see also Hafner/Schmid (2003). results can be easily extended to the case of a (even stochastic) dividend paying stock. 4 Our approach can be generalized to allow for stochastic interest rates by simply attaching a HJM-type model. 2 In

3 All

4.1 The Financial Market Model

61

t (I( T). Because the volatility surface can be more easily parameterized and estimated as a function in moneyness M and time to maturity r = T t, we rather consider the continuum of call prices dt * (M, T) which is represented by the relative volatility surface at (M, 7). Recall from Section 3.3.3 that there is a one-to-one correspondence between at (K,T) and ai m 7-) of the form —

at (m,T) = cit (m-i(m),t + T) ,

(4.3)

with m 1 (M) being the inverse function of m with respect to K. We demand the moneyness function and the corresponding moneyness to be valid according to the following definition: Definition 4.1 (Valid Moneyness). We call M. a valid moneyness function and M defined as M = Mt = m(t, St , K ,T,r)

a valid moneyness for our financial market model if m has the following properties: 1. m(t, s, K ,T,r) E C 2 ([0,T*] x R++ x R++ x (t, T*] x R+ ) , r) 0. Including the variable moneyness cubed as in Tompkins (2000) does not eliminate the bias. To account for the asymmetry of the strike pattern of implied volatilities Hafner/Wallmeier (2001) propose a quadratic spline function with the two segments M < 0 and M> 0:

•, /3) = fl +132M + 03M2 + /34 (D M2) ,

(5.12)

where the dummy variable D takes the value 0 for M < 0, and the value 1, otherwise. Obviously, (5.12) is only once differentiable in M at the threshold M = 0. However, this presents a violation of Assumption 4 of the general factor-based stochastic implied volatility model, which requires the function g 0 to be twice differentiable in M.29 Furthermore, the inclusion of the variable D M2 increases the correlations between the explanatory variables and the higher the correlations become, the less precise our parameter estimates will be.39 For these reasons, we refrain from using (5.12), but model the volatility smile by the quadratic function(5.11). Graphical analyses of the term structure of volatility "4(•, T, /3) reveal for most days in the sample period a pattern that can be very well approximated by a square root function or a logarithmic function. This is also supported by simple regression analyses. Because the first derivative of the square root function Irr evaluated at r = 0 is zero, the modelling of the volatility term structure by a square root function would lead to problems with regard to the factor-based stochastic implied volatility model. In fact, one can show that the implied volatility processes (expressed in terms of K and T) do not exist when the volatility term structure is represented by a square root function. On the other hand, if we model the volatility term structure by the function "See Shimko (1993) and Ripper/Ganzel (1997). problem can be solved by using a polynomial of higher degree. See, e.g., Brunner/Hafner (2003). "This problem is known as multicollinearity. See Greene (1993), pp. 266-273, for details. 29 This

88

5 Properties of DAX Implied Volatilities

ln(1 + r), no problem occurs, while at the same time, roughly the same fit to the data is achieved. Combining the findings from the analysis of the volatility smile and the volatility term structure, we posit three alternative functional forms for the volatility surface: 31 Model 1: Model Model

-4(M, r 0 ) = 01 ± 02M 133M 2 04 111(1 T), 2: (M T 0) = + 02M ± 133M2 ± 04 ln(1 + r) + 05 M ln(1 + 7), (M,T, 0) = 01 ± 132M ± 03M 2 + 04 ln(1 + r) + 05 M ln(1 + 7) 3: ,

-

+ )36M2 1n(1 + In Model 1, the volatility smile and the volatility term structure are supposed to be independent. Model 2 allows the slope of the volatility smile to vary with time to maturity, and Model 3 additionally accounts for a varying curvature of the volatility smile across different times to maturity. A preliminary statistical analysis reveals that Model 1 and Model 2 are not flexible enough to capture the typical shape of the DAX volatility surface. There is unexplained structure left in the residuals. In particular, the first two models cannot reproduce the "flattening-out" effect that is commonly observed. This effect refers to the fact that the volatility smile becomes flatter when the time to maturity increases. 32 Model 3, on the other hand, is expected to capture most of the variations in implied volatilities attributable to variations in the degree of moneyness and variations in time to maturity. Regression Model and Estimation Method The full specification of Model 3 is given by an,3 = /3 i,n + 02,nMn,3 + /33,n4.2y,3 + /34, n ln(1 +

(5.13)

+05,nAln,i In(1 + Tn,i) + 06,n ML ln(1 rn,i) + en,i, where n E {0,...,N and j = 1, , J. Although the model is expected to capture most of the cross-sectional variations in implied volatilities, it might yet not be well specified as the variables M and M ln(1 +r), on the one hand, and M2 and M2 ln(l+r), on the other hand, and hence also [32 and 06, and 03 and 06, are highly correlated. When we hypothesize that these relationships are stable over time, the model can be simplified by imposing the following restrictions on the regression coefficients: }

05,n = g102,n)

n= 0,...,N,

(5.14)

and 31 Similar parametrizations have been suggested by Dumas et al. (1998) and Ane/Geman (1999). 3 2 See also Das/Sundararn (1999).

5.3 Structure of DAX Implied Volatilities 06,n =

22i33,n,

n = 0,

, N,

89

(5.15)

where 21 and 22 are known constants. Substituting (5.14) and (5.15) into (5.13) leads to the following unrestricted regression model: an,j = /1,n I32iti ,nnj ( 1 + el ln

(1 + rn,i ))

(5.16)

+03, n721,i 11/ ( 1 + 22 in (1 + rn,j)) + )34,nln ( + Tn,j) En,j • In practice, 21 and 22 are not known but have to be estimated. Consequently, it is not possible to apply the least squares method directly to (5.16). Instead, we propose a two step estimation procedure.33 In the first step, we estimate the original regression model (5.13) for all days n = 0, , N. From the obtained time series of regression coefficients ;3i ,n (i = 2, 3, 5, 6), we then estimate 21 and 22 by fitting the auxiliary regression models -

g:42,n +

En",

n -= 0,

, N,

(5.17)

&,n = 02;33,n +

En

n = 0 , • • • , N,

(5.18)

and with €( 1 ) and e(2) being random disturbances. The final estimates of the regression coefficients (31 , , 04 are obtained in the second step by applying the least squares method to model (5.16) where the unknown constants 21 and 22 . , i.e. to: are replaced by their estimates "e,i and ".ô2 ln (1 + r)) = 01,n + 02,n Mn,3 (1 04,n ln (1 + r,) j +03,7114,3( 1 i321n (1 +

(5.19) Enj.

The implied volatility of deep ITM calls and puts is very sensitive to changes in the index level. Since small errors in determining the appropriate index level are unavoidable, the disturbance variance of regression models (5.16) and (5.19) is supposed to increase as options go deeper ITM. Residual scatterplots support this presumption. Using the White-test, the null hypothesis of homoskedasticity was rejected in about 70% of all regressions. To account for the heteroskedasticity of the disturbances we apply a weighted least squares estimation (WLS) assuming that the disturbance variance is proportional to the positive ratio of the option's delta and vega. 34 This ratio indicates how an increase in the index level by one (marginal) point affects the implied volatility of an option, if its price does not change 33 This estimation procedure is similar to the two-step Cochrane-Orcutt method that is sometimes used to estimate regression models where the error terms are autocorrelated. For a detailed description of the Cochrane-Orcutt method, see Kmenta (1997), pp. 314-315. delta and vega are computed using the implied volatility of the corresponding option. The delta of puts is multiplied by —1 to obtain a positive ratio. "

The

90

5 Properties of DAX Implied Volatilities

In view of the large number of intraday transactions it is not astonishing that some extreme deviations occur representing "off-market" implied volatilities. They can, for example, be due to a faulty and unintentional input by a market participant. In this case, the trade can be annulled if certain conditions are fulfilled. To exclude such unusual events we discard all observations corresponding to large errors of more than four standard deviations of the regression residuals where the standard deviation is computed as the square root of the weighted average squared residuals. We then repeat the estimation on the basis of the reduced sample until no further observations are discarded. This procedure is known as applying the "4-sigma-rule" or "trimmed regression".35 We examined the impact of this exclusion of outliers and found it to be negligible in an but very few cases. On the one hand, the precondition of stationarity is best achieved by selecting data from a short time window. On the other hand, however, if the environment does not change dramatically, a larger database may improve the precision of the regression estimates. Our analysis of this trade-off argues in favor of the second view. When selecting a two-hour interval of 2:00 to 4:00 p.m., for example, the average coefficient of determination drastically decreases compared to regressions based on all transactions of one day. In addition, imposing this restriction often strongly reduces the range of strikes and maturities for which call and put prices are available. Therefore, we do not restrict the time window. Certainly, new pieces of information and large intraday variations in the underlying index level may alter the shape of the volatility surface. But scatterplots of moneyness, time to maturity and implied volatilities suggest that intraday the volatility surface is roughly constant. On some very few days, however, the volatility surface experiences a parallel shift within the day. 36 A large percentage of all traded DAX options in the period from 1995 to 2002 features a degree of moneyness between —0.25 and 0.20 (see Figures 5.3 and 5.4) and a time to maturity below 180 days. We discard all observations outside this range in order to eliminate potential problems with extreme degrees of moneyness or time to maturity. As options with fewer than 5 days to maturity have relatively little or no time premium and hence the estimation of volatility is extremely sensitive to measurement errors, we also exclude them. To always ensure a good fit to the data, we eliminate all days from the sample where the adjusted coefficient of determination is lower than 60%. In total, these are 29. In general, plotting the residuals did not reveal any remaining violations of the assumptions of the chosen regression model.

Kmenta (1997), p. 219 and Sachs (1972), p. 265. "This happened, for instance, at the four most extreme market decreases caused by the Asian and Russian crisis and the September 11th Terrorists Attacks (October 28, 1997, August 21, 1998, October 1, 1998, September 11, 2001). 35 See

5.3 Structure of DAX Implied Volatilities

91

General No-Arbitrage Relations Let us consider the market for standard options at an arbitrary time t. If there are no arbitrage opportunities, then standard options satisfy four general arbitrage relations: 37 1. Hedge relation: for any maturity T, T > t, the value of a call is never greater than the stock price and never less than its intrinsic value: St > Ct (K,T)> max {St

—

K e —r(T—t) ;0} ,

VK > 0.

(5.20)

2. Bull spread relation: for any maturity T, T > t, the value of a vertical bull call spread is nonpositive or, respectively, the call price function with respect to strike is monotonically decreasing. The slope of the call price function is never less than -1:

—1 < -

OCt (K,T)

at(

0.

(5.21)

-

3. Butterfly spread relation: for any maturity T, T > t, the value of a butterfly spread is nonnegative or, respectively, the call price function with respect to strike is convex: 82 Ct (K, T)

?°'

VK > 0.

(5.22)

4. Calendar spread relation: for any strike price K > 0, the value of a calendar spread is nonnegative or, respectively, the call price function with respect to time to maturity is monotonically increasing:

OCt (K,T) tn,

0,

VT > 0.

(5.23)

To check these conditions, we consider for each day n E {0, , NI an equally spaced grid of 200 options exhibiting degrees of moneyness between —0.15 and 0.10 and times to maturity between 5 and 120 days. The call option prices are calculated on the basis of the estimated volatility surfaces, and the derivatives in equations (5.21), (5.22) and (5.23) are computed numerically. The moneyness boundaries .114i, = —0.15 and Mu = 0.10 and time to maturity boundaries TL = 5/365 and Tu = 120/365 were chosen such that the number of observations outside these intervals always suffices to ensure an accurate estimate of the implied volatilities within and at the boundaries. Since on many days option trades with a degree of moneyness greater than 0.10 or lower than —0.15 or a time to maturity of more than 120 days, respectively, do not occur, we were not able to enlarge the chosen boundaries. If any of the four no-arbitrage conditions is violated on a specific day, this day is excluded from 37 See,

e.g., Brunner/Hafner (2003) and Carr (2001).

92

5 Properties of DAX Implied Volatilities

the sample. In the overall sample, this happens on 42 days.39 The remaining

days are (re)numbered from 0 to N, with N now being equal to 1938. 39 5.3.2 Empirical Results Goodness of Fit For each day n E {0,...,N } , N = 1938, we estimate a regression of implied volatility on moneyness and time to maturity following the two-step procedure described above. As the result of the first regression we obtain time series of the daily coefficient estimates for the parameters 02, 03, 05, and )36 . These are used to estimate the model constants pl and 02 . We get: -P i = —1.6977 and -P.2 = —3.3768. The corresponding R2 values of 90.45% (Model 5.17) and 95.61% (Model 5.18) support the assumption of an almost deterministic, linear relationship between 02 and 05 , on the one hand, and between 05 and 06 , on the other hand. Repeating the estimation of pi and p2 , using different subsamples of the data, the estimates turn out to be quite stable. This suggests that the relations between )32 and 05 and between 05 and #6 are approximately time-invariant. Based on the parameter estimates -P i = —1.6977 and i*52 = —3.3768, we run regression (5.19). Across the 1939 days in the sample the average adjusted R2 value is 92.44% and the median adjusted R2 value amounts to 94.58%. 49 For comparison, the average adjusted R2 value obtained under the original regression model (5.16), using the same sample, is 93.00%. The loss in accuracy of 0.70% seems acceptable, when contrasted with the increase in model parsimony. All in all, the high R2 values suggest that our regression model captures most of the variation in implied volatilities attributable to moneyness and time to maturity. To further assess the quality of our model, the mean absolute error of the regression, i.e. the mean of the absolute deviations of the reported implied volatilities from the model's theoretical values, is computed each day. For almost all days in the sample, we find this measure to be well within the average bid-ask spread. 41 38 Of the four no-arbitrage relations, the butterfly spread relation is violated most frequently. 39 This results from 2010 — 29 — 42 — 1 = 1938. "It should be noted that in the case of a WLS regression model there exists no single generally accepted definition of R2 . The reported values are based on the non-weighted WLS regression residuals. The meaning of this R2 is not exactly the same as in an ordinary least-squares regression (OLS) regression. For more details, see, e.g., Greene (1993), p. 399. 41 As our database does not contain information on bid and ask prices, we use the average bid-ask spread of all liquid option contracts quoted on December 31, 2002 as a proxy for the bid-ask spread in the whole sample period. We find this value to be roughly 0.3 volatility points.

5.3 Structure of DAX Implied Volatilities

93

In a final analysis, we compute the estimated ATM volatility of DAX options with a time to expiration of 45 calendar days:

ATM„ = LZ(0, 45/365, ;3„),

n = 0, . . . , N,

(5.24)

where 73, denotes the estimated parameter vector on day n, and compare this variable with the German volatility index VDAX. This index represents the implied volatility of ATM DAX options with a remaining lifetime of 45 days. It is constructed as follows: for each DAX option's maturity traded at a given point in time, the Eurex calculates a volatility subindex based on the implied volatilities of the two calls and puts with strikes nearest to the DAX forward price for that maturity. The VDAX is then determined by linear interpolation between the two subindices representing times to maturity next to 45 days.42 Figure 5.5 shows that the ATM variable and the VDAX are almost identical although the estimation methods differ. The median of the difference between ATM and VDAX amounts to —0.0013. The strong correspondence between the two indices manifests itself in an almost perfect positive correlation of 0.9951 within the sample period."

Il 11 III 11111 1 IIIITT 1 1l 03 014 01 0 2 03 04 01 02 03 04 01 0 2 03 04 01 012 03 CH 01

11 11 111 1111111 Iuhlhhlhhlhh l

01

ca

1995

1996

1997

1996

1999

1111111111111,11111111111111

oa

03 04 01 02 03 04 01

2000

2001

oa

li

03 04 01

2002 2003

Date

Figure 5.5. VDAX and ATM on a daily basis over the sample period 1995-2002

Deutsche BtSrse (1999). largest difference between VDAX and ATM was observed on October 4, 2002 with 7.90 percentage points. A close examination of this day's data supports the correctness of ATM. 42 See

43 The

5 Properties of DAX Implied Volatilities

94

Average Parameter Estimates Table 5.1 reports the mean and the standard deviation of the daily coefficient estimates for each parameter, as well as the t-statistic for the mean. The (sample) standard deviation is calculated as

=

— N E

- ai)

2 ,

N

(i n=0

n=0

where

is the parameter estimate of parameter i on day n and

the mean of

denotes

kn. The t-value of 73i is then given by ---16i/sOi-

Table 5.1. Mean, standard deviation and t-value of the daily parameter estimates over the period January 1995 to December 2002

so , (t-value)

02 902 (t-value)

=

..

/33 s-43 (t-value)

134

so, (t-value)

1.4594 0.2361 —0.4966 0.0166 0.1031 (100.84) 0.1427 (-153.23) 1.0369 (61.98) 0.1397 (5.23)

Figure 5.6 gives a graphical representation of these results. It shows a

plot of the average estimated volatility surface, i.e. the function 4 - (M, 7- , 0), for different degrees of moneyness M E [ML, Mu] and times to maturity T E [TL,TU] For the interpretation of the regression results, it is convenient to recall the regression function:

-4(m, 7", 0) = 01 ± 02M (1 + ei ln (1 ± r)) +03M2 (1 + 22 ln (1 + 7-)) + 04 ln (1 + 7) .

(5.25)

The parameter 0 1 is common to all implied volatilities constituting the volatility surface. It may therefore be interpreted as the general level of volatility in the market. It should be closely related to the volatility of the underlying index. During the sample period, the average estimated value of 01 , i.e. 01 , amounts to 23.61%. The shape of the volatility smile is determined by the parameters 02 and 03 as the differentiation of the function ( M,- r , )3) with respect to moneyness shows:

ô-4(111 ' T ' fi) — /32( 1 + ln (1 + -7- )) + 203 M(1 + p2 ln (1 + r)), (5.26) 8M 492-4(m,T, /3 ) = 2,8 3 (1 ± 0 2 ln + 71)am-2

5.3 Structure of DAX Implied Volatilities

95

.0A5

Figure 5.6. Average estimated volatility surface --4(M, T, /3) for the sample period 1995-2002

Given el the parameter /32 reflects the common part of the slope of the volatility smile. As expected, its average estimated value of —0.4966 is negative. The curvature of the smile is represented by the parameter /33 . Since (1 + -02 ln (1 + T)) is positive for all T E [TL TU1 and the daily parameter estimates of 03 are mostly positive, the volatility smile is typically convex. The degree of convexity is however often small. On average, the curvature of the smile amounts to 1.4594. The minimum of the smile is almost always located at degrees of moneyness clearly above zero. Given 02 and /33, the estimates = —1.6977 and "02 = —3.3768 suggest that, in general, the volatility smile is steeper and more convex for shorter-term options. These features of the average volatility smile are also apparent from Figure 5.7, which plots the ,

function g'(M,•,#) for three different times to maturity. With regard to the smile patterns introduced in Section 3.3.1, the skew pattern is the predominant pattern in our sample." However, on some days,

44 Note, however, that a true skew, i.e. a linear function of implied volatility versus moneyness, hardly ever occurs.

96

5 Properties of DAX Implied Volatilities

Figure 5.7. Average estimated volatility smiles for 20, 60, and 100 days to expiration. Sample period: 1995-2002

for instance on October 6, 1995, a nearly literal smile pattern can be observed (see Figure 5.8). The slope of the volatility terra structure of ATM options is represented by the parameter /34 . Its mean estimated value of 0.0166 indicates that, on average, the implied volatilities of shorter-term ATM options are lower than those of longer-term ATM options. This implies that the average volatility term structure features a "normal shape". A detailed investigation of the reveals that the ATM volatility term structure exhibits coefficient estimates on 1316 days a normal shape (-4 4 > 0) and on 623 days ( 34 , 0.24 :12 0.22 -

ra.

E 0.20 -

0.18 -

0.16 -

0.14 -0.20

-0.04 1

-0.12 1

0.04

0.12

0.20

Log simple moneyness

Figure 5.8. Estimated volatility smiles for 20, 60, and 100 days to expiration on October 6, 1995

days in the sample and report them in Table 5.2. As the values show, the -1 correlations are generally quite low, except for the correlation between 4 and -al . This may serve as evidence that not only the coefficient estimates fluctuate, but also the volatility surface itself features considerable variation. Table 5.2. Correlation coefficients between the daily parameter estimates in the period 1995-2002 2

73 3

;3

4

—0.3149 —0.3161 —0.6758 0.4102 0.1191 0.1076 733 ;3

1

;3

2

Risk-Neutral Densities The estimated coefficients of the implied volatility functions can also be used to deduce the shape of the risk neutral density (RND) at the option expiration -

98

5 Properties of DAX Implied Volatilities

dates.45 However, as the function g(M, 0) is only known for degrees of moneyness ranging from ML to Mu, it has to be extrapolated beyond this range to fully recover the RND, using the Breeden/Litzenberger (1978) theorem. One such extrapolation function was proposed by Brunner/Hafner (2003). Given :-

the estimated volatility smile within the range of observable strike prices, and hence the middle part of the corresponding RND, the basic idea of the Brunner/Hafner (2003) method is to complete the RND by attaching nonnegative functions to the lower tail and to the upper tail such that the complete RND is consistent with the absence of arbitrage. As specific choices for the tail functions, Brunner/Hafner (2003) consider mixtures of two lognormal density functions. The implementation of their method involves only straightforward numerical procedures. Moreover, the method is robust, accurate, and fast. 46 For illustration purposes, we use the estimated volatility surface on October 31, 2001 and apply the extrapolation method of Brunner/Hafner (2003) to the volatility smiles of 20, 60, and 100 days to expiration. The current DAX index level is 4589.70. This implies the three RNDs shown in Figure 5.9.

0 .0010 -

- 20 days 60 days - - 100 days

0.0008 -

0.0006 -

0.0004 -

0.0002 -

0.0000 -

5500

6000

Figure 5.9. RNDs for 20, 60 and 100 days to expiration on October 31, 2001. DAX level: 4589.70. Range of observable strike prices: roughly 3440-5500 4 5 For an in-depth discussion of RND estimation methods, see Bahra (1997), Cont (1997), and Jackwerth (1999). 'For further details, see Bruruier/Hafner (2003).

5.3 Structure of DAX Implied Volatilities

99

The smooth continuation of the RNDs outside the range of observable strikes (here from approximately 3440 to 5500) is apparent. The wider variances for 60 and 100 days to expiration reflect the greater probability of large price moves over a longer time period. All distributions are skewed to the left, exactly the opposite of the right-skewness implied by the Black-Scholes assumption of lognormally distributed asset prices. The degree of skewness tends to be independent of time to maturity; on the other hand the degree of kurtosis obviously depends on time to maturity: the shorter the time to maturity the higher the kurtosis and vice versa. Compared with the kurtosis of the lognormal distribution, all distributions exhibit excess kurtosis, i.e. they are leptokurtic. The negative skewness and the excess kurtosis reflect the same deviations from the Black-Scholes world as are also observable from the pattern of implied volatilities. The above findings are not only true for the particular day considered, but are typical for the whole sample. 47

5.3.3 Identification and Selection of Volatility Risk Factors Original versus Abstract Risk Factors The discussion so far has shown that the DAX volatility surface evolves randomly over time. However, as the volatility surface forms a highly correlated complex multivariate system, it is difficult to model. To reduce complexity, we search for a smaller set of abstract risk factors which represents, in the best possible way, the set of original risk factors, i.e. the implied volatilities for different degrees of moneyness and times to maturity. In contrast to the original risk factors, abstract risk factors are not directly observable in the market, but are usually created by transforming the original risk factors in some manner. The set of possible transformations is limited to invertible functions, otherwise the original risk factors cannot be recovered. However, the recovery process may be approximate. In the following, we state some desirable properties of abstract risk factors: • • •

Abstract risk factors should be accurately estimable. The set of abstract risk factors should be parsimonious. Abstract risk factors should be easy to interpret.

Especially the last property, i.e. easy interpretability, is often crucial for a model to be accepted in practice. Fundamental versus Statistical Factors as Abstract Risk Factors Given our regression model, the volatility surface on day n is completely described by the four regression coefficients 0 (i =- 1, , 4) and the timeinvariant parameters ei and e2 . Since the coefficient estimates have proven to 47 Bliss/Panigirtzoglou

(2002) come to similar results for the FTSE 100 index.

100

5 Properties of DAX Implied Volatilities

be highly accurate, the regression coefficients are the most natural . candidates for being used as abstract risk factors. To investigate the issue of parsimony, we run a principal component analysis on the correlation matrix displayed in Table 5.2. Figure 5.10 shows a screeplot of the variances explained by the principal components. As can be seen, the first three principal components explain 92.90% of total variance. This value goes down to 85% when looked at different subsarnples, implying that the fourth factor still explains a substantial part of total variance. Consequently, any further reduction of the number of risk factors would lead to a significant loss in accuracy. The last section has shown that the regression parameters are easy to interpret. The parameter /31 represents the (overall) level of implied volatility, /32 and 03 stand respectively for the (overall) slope and the curvature of the implied volatility smile, and 04 represents the slope of the (ATM) term structure of volatility. As these parameters can be thought of to capture systematic risks an option's investor is facing, and are therefore directly linked to economic activity, they are commonly called fundamental risk factors.

0.501 2.0

1.5 0.782

0.929 0.5 1

0.0 Comp.1

Comp.2

Comp.3

Comp.4

Figure 5.10. Screeplot of variances explained by the principal components. Basis: correlation matrix of the time series

,i34 . Sample period: 1995-2002

5.3 Structure of DAX Implied Volatilities

101

As opposed to fundamental risk factors, one could alternatively use statistical methods such as factor analysis or principal component analysis to derive a set of statistical risk factors that characterize the dynamics of implied volatilities. From the typically nonparametrically estimated implied volatility surface, time series of implied volatilities for different times to maturity and degrees of moneyness are constructed by evaluating the implied volatility function at the respective grid points. Then, on the basis of these time series, a principal component analysis is performed. For instance, Skiadopoulos et al. (1999) analyze the volatility surface of S&P 500 for the years 1992-1995. Depending on the criterion used for factor selection, they find that at least two and at most six factors are necessary to capture the dynamics of S&P 500 implied volatilities. Cont (2001) and Cont/Fonseca (2002) also examine the dynamics of the S&P 500 volatility surface. Applying a Karhunen-Loève decomposition to the daily log-variations of the implied volatility, they report that the first three principal components account for more than 95% of the daily variance. As already mentioned, Fengler et al. (2000) perform a common principal component analysis based on the closing prices of DAX options during the year 1999. They conclude that three factors are sufficient to capture 95% of the daily variations in implied DAX volatilities. The main advantage of the statistical approach is the orthogonality of the obtained factors, i.e. the factors have a correlation of zero among each other. On the other hand, the factors are usually difficult to interpret. Moreover they are not unique, because a factor rotation can yield a different set of factors with the same degree of explanation. Mainly due to their better interpretability, we decide for the regression coefficient estimates )3 1 , ... 034 or transformations of them to serve as our abstract implied volatility risk factors or just volatility risk factors Y1 . • • , Y4. Concretely, we define ,

— hi 01,n)

Yi,n = fii n, i = 2,3,4, ,

(5.27)

for n = 0, , N. The variable 4 - 1 was normalized by taking the natural logarithm, because it represents the level of volatility in the market and as such it has to be positive in any economically meaningful model. Note that the log transformation is not appropriate for the variable 03, although 03 mainly assumes positive values. The reason is that in order to be consistent with no-arbitrage the smile needs not to be convex in moneyness (and strike), but can also be concave."

48 5ee

also Carr (2001).

102

5 Properties of DAX Implied Volatilities

5.4 Dynamics of DAX Implied Volatilities 5.4.1 Time-Series Properties of DAX Volatility Risk Factors Outline of the Analysis Having identified the set of risk factors characterizing the volatility surface, this section is concerned with the question of finding what process is most appropriate for each factor. 49 For that purpose, we individually examine the historical time series of the volatility risk factors {Yi,„ : n = 0, , NI, i = 1, , 4, and determine their main statistical properties so that we can afterwards propose models that are suitable to capture most of the historical features. 50 To reduce complexity, we restrict our search for models to the class of autoregressive integrated moving average (ARIMA) models. The correlation structure defining the relationships between the volatility risk factors and the DAX index will be studied in Section 5.4.2. 51 Our procedure for analyzing the data partly follows the model identification stage of the Box/Jenkings (1976) approach to time series analysis and involves the following four steps: 1. 2. 3. 4.

Graphical inspection of the data. Identification of nonstationarity: testing for unit roots. Analysis of the marginal distributions. Determination of model order.

Graphical Inspection of the Data We start our analysis with a graphical inspection of the data. Figure 5.11 has four panels containing plots of the time series of the volatility factors Y1, Y4 for the sample period January 1995 to December 2002. During the sample period, the values of the estimated volatility level -4 1 lie between 9.16% (August 10, 1996) and 63.13% (July 24, 2002). The values of Y1 , defined as the natural logarithm of OD therefore range between —2.39 and —0.46. As can be seen from the Y1-graph, the volatility levels in the years 1995 and 1996 are distinctly lower than they are in the period from July 49 In the general factor-based stochastic implied volatility model, the price process of the underlying asset is assumed to follow a GBM with stochastic volatility, and the volatility process is implicitly defined in terms of implied volatility. Consequently, the price process of the underlying is almost determined, and therefore not considered here. 5 0 A similar strategy for the specification of multivariate risk factor models is chosen by Algorithmics in their Mark to Future framework for scenario generation. See Reynolds (2001). 51 The relationships between the regression coefficient estimates ;3 1 , , ..;34, which are, except for Y1 equal to the volatility risk factors, have been analyzed before. See also Table 5.2. -

,

-

5.4 Dynamics of DAX Implied Volatilities Y1

103

Y2

c»

Figure 5.11. DAX volatility risk factors Y1,

, Y4 on a daily basis over the period

1995-2002

1997 to December 2002. To formally test on this, we compare the population mean /41, ) for the period January 1995 to December 1996 with the population mean py(21) for the period July 1997 to December 2002 by running a Welch modified two - sample t test. 52 More specifically, we test the null hypothesis Ho : /19) > pr against the alternative hypothesis H1 : 41) < 42). Since -

the sample mean YiP) = —1.3384 for the second period is significantly higher = —2.0691 for the first period, the null hypothesis than the sample mean is clearly rejected. Economically, the reason for this substantial increase in volatility in the first half of 1997 can be seen in the beginning of the Asian crisis. Reaching its peak in October 1997, the Asian crisis was followed by the Russian crisis just one year later in 1998 letting volatility levels — expressed in terms of 731 — climb to more than 60% (see top left graph in Figure 5.11). Subsequently, such high levels of volatility have only been reached during the WTC terrorists attacks on September 11, 2001 and during the stock market's turmoil in fall 2002. 32 This

test assumes normality. For more details see, e.g., Casella/Berger (2002).

104

5 Properties of DAX Implied Volatilities

As Figure 5.11 shows, the time series of the risk factors Y2, Y3, and Y4 tend to be mean-reverting, i.e. they tend to fluctuate around a long-run mean. Astonishingly, the structural break occurring in the Y1 time series does not appear here. Consistent with our previous findings, the slope of the volatility smile Y2 is always negative. On the other hand, the curvature of the volatility smile as represented by the factor Y3 is mostly, but not always, positive. Precisely, on 74 days Y3 assumes a slightly negative value. The Y4 time series alternates between positive and negative values. In 2001 and 2002 negative values clearly dominate. Correspondingly the volatility term structure changes between normal and inverse shapes. In comparison to the Y2 and Y4 series, the Y3 series looks quite erratic. The extreme points in the graphs of the Y2 and Y4 time series match quite well with the extreme points in the graph of the Y1 series, although there seems to be a time lag in some cases. Exceptionally high levels of volatility Y1 usually come along with a distinctly downward-sloping smile and a pronounced inverse term structure of volatility. An inverse term structure generally expresses the expectation of the investors of a quick end of the crisis.

Distributional Properties Table 5.3 presents descriptive statistics on the distribution of the volatility risk factors Y1 , . . . Y4. ,

Table 5.3. Summary statistics for the volatility risk factor time series Yi.,.. • ,Y4. Sample period: 1995-2002

Minimum 25% Quantile Mean Median

75% Quantile Maximum Standard deviation Skewness Excess Kurtosis

Y1

Y2

Y3

Y4

- 2.3896 -1.7962 -1.5297 -1.5289 -1.2872 -0.4599 0.4109 0.1912 -0.3655

-0.9276 -0.5881 -0.4966 -0.4992 -0.4012 -0.1549 0.1427 -0.0893 -0.2537

-0.7275 0.7127 1.4594 1.3057 2.0012 5.5113 1.0369 0.8447 0.6381

-0.7291 -0.0235 0.0166 0.0404 0.0954 0.5704 0.1397 -1.6864 4.7809

The distributions of Y1 and Y2 appear to be close to a normal distribution, as the skewness and excess kurtosis values are around zero and the median closely matches the mean in both cases." Performing a Jarque-Bera test and a Kolmogorov-Smirnov goodness-of-fit test, however, the null hypothesis that 53 The normal distribution exhibits a skewness of 0 and an excess kurtosis (defined as kurtosis minus 3) of 0.

5.4 Dynamics of DAX Implied Volatilities

105

Y1 follows a normal distribution is rejected at the 1% significance level in both tests. 54 In the case of Y2 normality cannot be rejected at the 5% level using the Kolmogorov-Smirnov test, but is rejected at the 1% level, using the JarqueBera test. The skewness and excess kurtosis values for the factors Y3 and Y4 indicate a nonnormal distribution — both empirical distributions exhibit a skewness that is distinctly different from zero and excess kurtosis. Figure 5.12 illustrates these findings graphically. The empirical densities (solid line) are computed as a smoothed function of the histogram using a normal kerne1. 55 Superimposed on the empirical density is a normal distribution having the same mean and the same variance as that estimated from the sample (dotted line).

2

4

6

Y3

Figure 5.12. Distributional properties of Y2 and Y3. Solid line: Kernel estimate of the density function; dotted line: Normal density having the same mean and the same variance as that estimated from the sample

54 For details on the Jarque-Bera test, see, e.g. Kmenta (1997), p. 265-266. More information on the Kolmogorov-Smirnov goodness-of-fit test can be found in Casella/Berger (2002). 5 5 See Silverman (1986), Chapter 3.

5 Properties of DAX Implied Volatilities

106

Although the risk factors Y3 and Y4 are not normally distributed, in both cases the normal distribution (and in part also the lognormal distribution) still represents the "best" approximation to the empirical distribution from a broad spectrum of potential continuous distributions. More specifically, computing the Chi-Square and the Kolmogorov-Smirnov goodness-of-fit measures for the beta, the chi-square, the exponential, the gamma, the inverse Gaussian, the normal, the lognormal, the Student's t, and the Weibull distribution, we find the normal distribution (and for Y3 also the lognormal distribution), to be the one where these measures are lowest. 56 Stationarity and Serial Correlation At a first glance, there is no evidence of a deterministic time trend, a seasonal component or a unit root in any of the four series. To further investigate the key issue of stationarity, we plot the sample autocorrelation function (sample ACF) for each of the series over the sample period 1995-2002 in Figure 5.13. Y1

Y2

6

ci

a

u.

a 6 O

5

10

15

20

25

30

0

10

Lag

15

20

25

30

20

25

30

Lag

Y3

Y4

6

o

0 } is a nondecreasing family of sub a-algebras of ÇF CF for 0 < s < t < co. ,

,

2 1t is called complete, if 1- contains all subsets G of Il with IF-outer measure zero. Any probability space can be made complete. See Taylor (1996).

A.2 Continuous-time Stochastic Processes

195

We assume that the filtration IF also satisfies the "usual conditions":

1. Fo contains all P-null sets of .F, i.e. if A E .F and P (A) = 0, then for any t, A 2. IF is right continuous, i.e. Ft = ns>t.Fs• Condition 1 ensures that if X = Y P-a.s. 3 for random variables X and Y, and Y is Ft-measurable, then X is also Ft -measurable.' In a financial model, the g-algebra Ft represents the information available at time t, and the filtration IF represents the information flow evolving with time. We say that a stochastic process X is adapted to the filtration F, or equivalently, Fradapted, if for any t > 0, Xt E Ft, i.e. Xt is Ft-measurable. Thus Xt is known when Ft is known. We can build a filtration Fx generated by a process X and we write Tic = (X, : s < t) . We call it the natural filtration. Obviously, a process is adapted to its natural filtration. In general the natural filtration does not fulfill condition 1. However, if we extend Fi" by the a-algebra containing all P-null sets of F, we obtain a filtration with the desired property. In this work, we only deal with filtrations satisfying the usual conditions. The class of progressively measurable processes is a slight enlargement of the class of adapted processes. We define: Definition A.5 (Progressively Measurable Process). A (n-dimensional) process X = {Xt : t > 0} on some probability space (Q, F, P) is called progressively measurable with respect to a filtration IF if, for any t > 0, the map (s, (w) from [0, t] x Q —■ Rn is measurable on the product a-algebra

B ([0 , t]) ®F. If all paths of a stochastic process X are right-continuous, then the process X is also progressively measurable. 5 Two classes of stochastic processes are fundamental to continuous-time financial models: martingales and Markov processes. Definition A.6 (Martingale). A stochastic process X = {X t : t > 0} on some probability space (Q, F P) is a martingale relative to a filtration F and relative to a probability measure F if ,

1. X is Ft -adapted, and E [iXtl] < co for all t > 0; 2. E [Xt IF8] = X. for 0 < s < t. The second property is called the martingale property. In a financial context, saying that the price process X = {Xt : t > 0 } of an asset is a martingale implies that, at each time s, the best estimate (in the least square means: P (X (w) = Y (co)) = 1 for all co G O. Lamberton/Lapeyre (1996), p. 30. 3 For a proof, see Korn/Korn (1999), p. 37. 3 This 4 See

196

A Some Mathematical Preliminaries

sense) of Xt is given by X,. The martingale concept is sometimes too restrictive and one needs to generalize it: A process is said to be a local martingale X = {X t : t > 0} if there exists an increasing sequence of stopping times T k -> 00 such that each stopped process X(k) = (X(t A rk) is a martingale. —4 10, cc] is called a stopping time with respect to a A random variable T : every t > 0 the event {7- < t} belongs to the cy-algebra filtration IF, if for martingale is a martingale. Trivially, each local Definition A.7 (Markov Process). A stochastic process X = {X t : t> 0} on some probability space (f ,P) is a Markov process, if for each t, each set A E a (X, : s > t) (the "future") and B E a (X8 : s < t) (the "past"), the Markov property

P (Al X t

,

P ( Xt)

holds. The future conditional distribution of a Markov process, occasionally called "no-memory" process, does only depend on the present, not on the past equivalently, past and future are conditionally independent given the present. If capital markets are informationally efficient 6 , no investor can earn excess returns by developing trading rules based on historical price information. This implies that asset prices have to follow Markov processes. A particularly important example of stochastic process in finance is the (standard) Brownian motion. It belongs to the classes of Markov and martingale processes' and will underlie most continuous-time financial models. Definition A.8. A stochastic process W = {W t : t > 0} is a (one-dimensional) standard Brownian motion on some probability space (S2, .T, P) , if 1. Wo = 0, P 2. W has continuous paths: W(t) is a continuous function oft for all w E St; 3. W has independent increments: Wt+u — Wt is independent of .F tw for all u > 0; 4. w has stationary increments: the law of Wt+, — Wt only depends on u; 5. W has Gaussian increments: Wt-Fu — Wt is normally distributed with mean 0 and variance u, Wt+, — Wt N (0, u) . Standard Brownian motion is also termed Wiener process. 8 . Standard Brownian motion in n dimensions (multi-dimensional standard Brownian mo6 For an extensive discussion on capital market efficieny, see Copeland/Weston (1988), p. 330 ff. 'It also belongs to the large class of Lévy processes, characterized by stationary independent increments. Another important Levy process which is not considered here in detail is the Poisson process. In financial models, it is used to incorporate jumps, e.g, in the price of a stock. For a modern textbook reference on Lévy process see Bertoin (1996). 8 Although the two processes are defined differently, they are equivalent according to a theorem of Levy. See Neftci (1996), p. 149.

A.2 Continuous-time Stochastic Processes

197

tion) is defined by W = (W1 , W2 , , W,)', where W1, , Wn are independent standard Brownian motions in one dimension. To describe the evolution of asset prices in continuous time we introduce the concept of Ito processes.9 Definition A.9 (Ito Process). Let W = (W 1 ,W2, . . , Wm )' be standard mdimensional Brownian motion defined on a filtered probability space (fl, .F, P, IF) . Then,

1. X = {Xt : t > 0} is a mal valued Ito process, if for any t > 0, it has the unique representation

X t = X0 + f as ds f 14,dW s ,

t = Xo + f as ds + Ef b3,3dw,, 8 ,

3=1 0

: t > 0} with where X 0 is To -measurable and a = {as : t > O}, b = bt = (bi,t , , bm,t) are progressively measurable processes satisfying I as I ds < oo and f

o

sds < oc P-a.s. Vt > 0, j = 1, . . . , m.

, X n ) is a n-dimensional Ito process, if X 1 ,. , X n are realvalued Ito processes in one dimension.

2. X = (X 1 ,

Sometimes, a one-dimensional Ito process is defined with respect to a onedimensional standard Brownian motion, i.e. m = 1. 10 The expression f; bs dW3 is usually referred to as the stochastic integral. If X is an Ito process, it is often written in the shorter differential form

dX t = at dt + bt dWt ,

(A.1)

although only the integral notation is meaningful from a mathematical point of view. If at = a(t, X t ) and bt = b(t, Xt ) are deterministic functions of t and Xt for all t > 0, equation A.1 is called a stochastic differential equation (SDE). The solution of an SDE is called a diffusion. In stochastic analysis, the equivalent to the chain rule in ordinary calculus is the Itô formula, also referred to as Itô's Lemma.' Theorem A.10 (1-dimensional Itt5 Formula.). Let X be an Ito process given by

dX t = at dt + btdWt 9 See Korn/Korn (1999), pp. 48-49. "See, e.g., Oksendal (1998), p. 44. II See Oksendal (1998), Chapter 4.

198

A Some Mathematical Preliminaries

and f(t,x) E C2 ([O, co) x R) (i.e. f is twice continuously differentiable on 10, co) x R). Then Vt E [0, oo), Yt = f(t,xt), is again an ItO process, and

a

a at

f 1 a2 f f dYi = —(t, X i )dt — (t, X i )dX i + – — (t, X i )d < X, X >t, Ox 2Ox2 where d < X, X > i = (dXi) 2 , the quadratic variation of X, is computed according to the rules

dt dt = dt • dWi = dWi • dt = 0, dWi •dWt = dt. Proof See Korn/Korn (1999), pp. 51-59. When f is a function of several It6 processes, the general Ito formula may be applied. Theorem A.11 (General Itel Formula.). Let X = , X n ) be an n, f,n (t,x)) be a C2 map from dimensional ItO process and f (t,x) = [0, oo) x Rn into Rm. Then the process

Yt = f (t, Xt),

Vt E [0, 00 ),

is again an Ito process, whose component number k,

Ykt,

is given by

a

k ,t = dy fkt (t,x t )dt

+ Ê aOxfki (t, Xi)dXi,t i=1

1 r Nv ti=1 j=1

a2 fk OXiaXj

(t,Xi)d<Xi,Xi>t,

where d < Xi, Xi >i= dXi,idXj, t , the quadratic covariation (or crossvariation) of Xi and Xi, is computed according to the rules

dt, if i • dWi,t • dWi,t { 0, otherwise. Proof See Korn/Korn (1999), p. 59. In this text, we will also need a generalization of the It6 formula: the generalized It6-Venttsel formula." 12 See

Venttsel (1965) and Brace et al. (2002), Appendix A.

A.2 Continuous-time Stochastic Processes

199

Theorem A.12 (Generalized Itto-Venttsel Formula).

Let W = (W1,W2,... ,W, n )' be an m-dimensional standard Brownian motion. Suppose G(t,u) is twice differentiable with respect to the parameter u and satisfies the SDE dG(t,u) = A i (t,u)dt + Bi(t,u)dWt• If ti t satisfies the SDE

dut = A2 (t,ut)dt + B2 (t, ut)dWt, then an SDE for G(t, ut ) is

dG(t,ut ) = Ai(t,u t )dt + Bi(t,ut )dWt

1 , ., _ ,

aG , t ,)aut , + --kt,u01012 a2G +—kt,u kt,u t .)i12 - a,t Ou 2 Ou2 0B1 +--67--1 (t,ut )B2 (t,ut )dt. Proof. See Venttsel (1965).

0

Pricing of a Variance Swap via Static Replication

This appendix contains the proof of equation (7.21) on p. 156 in the main text. 1 Theorem B.1. Suppose the interest rate r is constant and suppose further the stock price process S = {St : t E [0,7 ] 1 is a diffusion with progressively mea: t E [0,7 ] 1 and progressively measurable volatilsurable drift process p = t : t E [0, T]}: ity process v = {v dSt = StA t dt + StvtdWt,

Vt E [0,7] ,

(B.1)

where W = {Wt : t E [0, T]} is a one-dimensional standard Brownian motion. Then, provided that a continuum of standard European options with maturity date T and strike prices ranging from zero to infinity exist on the stock S, a T-maturity variance swap can be statically replicated and its fair price at time t =- 0 is given by (B.2)

KVARS = BQ [WT] 2

= —T erT o(f0

Fo(T) ---P (K,T)dK + K

1 .2 Co(K ,T)dIC) . Fool n

Here, Co (K, T) and Po (K ,T) , respectively, denote the current market price of a put and a call option of strike K and maturity T, Fo (T) = SoerT is the stock's T -maturity forward price, and wg- , is the realized continuously sampled variance over the interval [0, T] : WT =-

1 f — T

v 2 dt. t

(B.3)

Proof. By the application of Itel's lemma to ln(St), we obtain: 'See also Carr/Madan (2002) and Demeterfi et al. (1999b), for a similar proof.

202

B Pricing of a Variance Swap via Static Replication

1

d (ln (St)) = (ktt - -2 v0 dt +vtdi'Vt,

Vt E [0,T].

Subtracting equation (B.4) from equation (B.1), we obtain 1 dSt — - d (ln (S t )) = .14dt,

Vt E [0,7].

St

(B.5)

Integrating equation (B.5) from 0 to T yields:

foT dsStt ln (ST )

so

=

1

0 qdt.

Dividing (B.6) by T, we finally obtain the continuously sampled variance over [0,7 ] : dS t (ST) ] 2 (B.7) WT = f qdt = --

[fT

Tj o

T o

St

0)

.

Identity (B.7) dictates the replicating strategy for variance. The first term in represents the payoff of a trading brackets on the right-hand side, LT d '—5g strategy which involves the continuous rebalancing of a stock position that is always instantaneously long 1/St units of stock worth 1 dollar. The second

t,

term, - ln (k) 0 , represents a static short position in a contract that pays off the continuously compounded stock return over the contract's lifetime [0, T]. This contract is known as the log contract.2 To obtain the fair delivery price of variance Kvjuis , we compute the riskneutral expectation of the right-hand side of (B.7):

115_ [ f T dSt ln ST)] h st so )

(B.8)

KVARS = BQ [WT] = T

(EQ [foT d,'Stt]

EQ [ln (It)]

.

The first expectation is easily evaluated to

EQ

[fT Jo

fT st

Jo

[dSt i st

f

rdt = rT.

o

(B.9)

The second expectation represents the fair value of the log contract. Unfortunately, the log contract is not a market-traded security. To make the replicating strategy viable in practice, we need to duplicate the log contract with traded standard options. For convenience, let us write the log payoff as: 2 The

log contract was first discussed by Neuberger (1994).

B Pricing of a Variance Swap via Static Replication

ln

— ST (

So

) = ln ( T ST ) + ln

.

GI) , o

203

(B.10)

where S* > 0 is an arbitrary stock price. Since the second term on the righthand side, ln (e), is constant and known at time 0, only the first term, ln (k) , has to be replicated. Assuming that a complete collection of standard European options with strikes K > 0 and maturity T E (t, T*] is traded, it

--) s can be rewritten as:3 can be shown that the payoff ln (s ln

(ST ) ST

— S*

f S. max {K — ST;0} dK 0 K2

S*

S* —

(B.11)

00 1 — max {Sr — K;0} dK. s. K 2

The first term on the right-hand side of equation (B.11) can be interpreted as a long position in (1/S* ) forward contracts struck at S. The second term arises from a short position in (1/K2 ) put options struck at K, for all strikes less than S. Similarly, the last term arises from a short position in (1/K2 ) call options struck at K, for all strikes greater than S. All contracts mature at time T. Taking the risk-neutral expectation of (B.10), where In (k) is given by

(B.11), yields ° B[

( ST : T0

)]

=1

3

(S

_erT

°

S*) — erT f

erT —

S.

1 —2 Po(K,T)dK (B.12)

0 K ' 1 — C0(K T)dK + ln fs. K2 '

( S* ) So

.

Substituting (B.12) and (B.9) into (B.8), we finally obtain the fair value of future variance:

KVARS

2

( so erT — s*) — ln ( S*)

(rT —

+err f

-5T1

S. — K2

_FerT

(B.13)

Po(K T)dK '

oo

Is.,Co(K,T)dif) .

If we choose the arbitrary parameter S* , defining the boundary between call and put strikes, to be the current forward price F0(T) = soerT, equation (B.13) can be further simplified to: 3 See Demeterfi et al. (1999b), p. 18. More general, it can be shown that any twice differentiable payoff can be statically replicated in such a way. For a proof, see Carr/Madan (2002).

204

B Pricing of a Variance Swap via Static Replication

2 err (f F°(T) 1 °.° — Po(K T)dK + f —Co(K T)dK KvARS = — T K2 ' 0 Fo(T) K2 (B.14) This proves the claim.

List of Abbreviations

a.s. ACF ADF AR ARIMA ATM BIS BS CBOE CEV DAX DTB e.g. etc. FIA FTSE GARCH GBM HJM i.e. i.i.d. MAPE ML MLE ITM NFLVR ODE OLS OTC OTM

almost surely autocorrelation function augmented Dickey-Fuller (test) autoregressive (process) autoregressive integrated moving average (process) at-the-money Basle Committee on Banking Supervision Black-Scholes Chicaco Board Options Exchange constant elasticity of variance model Deutscher Aktienindex Deutsche Terminboerse for example (lat: exempli gratia) etcetera (lat: et cetera) Futures Industry Association Financial Times Stock Exchange generalized autoregressive conditional heteroscedasticity geometric Brownian motion Heath-Jarrow-Morton model that is (lat: id est) independent and identically distributed mean absolute percentage error maximum likelihood maximum likelihood estimator or estimation in-the-money no free lunch with vanishing risk ordinary differential equation ordinary least sqaures over the counter (market) out-of-the money

206

List of Abbreviations

OU PACF PCD PDE PP

QMLE RND SD

SDE SW S&P

SOFFEX VaR VAR

VDAX WLS

Ornstein-Uhlenbeck (process) partial autorcorrelation function proportion of correct direction partial differential equation Phillips-Perron (test) quasi maximum likelihood estimation risk-neutral density standard deviation stochastic differential equation factor-based stochastic implied volatility model (specified for DAX options) Standard and Poors Swiss Options and Financial Futures Exchange Value at Risk vector autoregressive (process) DAX volatility index weighted least squares

List of Symbols

Roman case: a, ATM

Bt cx (a) C2

CBS()

Ct (K,T)

b-t(M,T)

CONCt CONPt CONCBs•)

d c/10,d2(-)

D Dt DW

speed of mean reversion of risk factor process Y, ATM volatility of DAX options with a time to maturity of 45 calendar days on day n fixed cash payout of a cash-or-nothing option money market account process value of the money market account at time t mean reversion level of risk factor process Y, a-quantile of distribution (or random variable) X set of twice continuously differentiable functions arbitrage price of a standard European call option at time Black-Scholes call option pricing formula market price at time t of a standard European call option with strike price K and maturity date T; also: call price as a function of K and T market price at time t of a European call option with moneyness M and time to maturity r; also: call price as a function of M and T arbitrage price of a cash-or-nothing call option at time t arbitrage price of a cash-or-nothing put option at time t Black-Scholes price function of a cash-or-nothing call option (d+1) is the number of primary traded assets in the financial market auxiliary functions in the Black-Scholes formula price process of numéraire asset price of numéraire asset at time t Durbin-Watson test statistic

208

List of Symbols

DIV DOC, AD/Vt ,T

ES F,(T) F,1 (TF)

Fx (x)

X.) (G* ( 0)) G(0)

(G;(0)) G(ç5) h H HL, kd

km KVARS

KvoLs m(.) m-10

M = Mt ML Mu MAPE(K) n(x)

N* N(x) N (p ,a2 )

R(A)

gross dividend arbitrage price of a discrete down-and-out call option time T terminal value of the difference dividend incurred between dates t and T the Euler number 2.71... expected shortfall price of a futures or forward contract (on a stock) with maturity date T at time t price of a DAX futures contract with maturity date TF on day n at minute 1 distribution function of X volatility surface function regression function or approximate DAX volatility surface function (discounted) gains process of portfolio 0 (discounted) gains of portfolio accumulated up to and including time t forecasting period in days contingent claim or derivative security half-life of process i time-independent random amplitude of a jump corporate income tax rate for distributed gains marginal investor's tax rate strike price or exercise price of an option delivery price for variance; fair value of variance delivery price for volatility; fair value of volatility number of trading minutes per day moneyness function inverse function of m(.) with respect to the strike price moneyness of an option (at time t) lower and upper moneyness boundary mean absolute percentage error with respect to the parameter vector IC probability density function of a standard normal random variable sample size terminal time horizon measured in units of At cumulative distribution function of a standard normal random variable cumulative distribution function of a normal random variable with mean ti and variance cr2 Poisson process with intensity A

List of Symbols 9)

Pt PBS•)

Pt(K,T)

P&L, P&L; PCD 4.57. (s) r(t) R, R; Rt, R(N) p2 R "act ,) "

RE (S*) S (Sr) St (

n,l) Sn,1

209

cumulative distribution function of a multivariate normal random variable with mean vector it and covariance matrix S2 (p+1) is the number of random sources in the financial market model arbitrage price of a standard European put option at time Black-Scholes put option pricing formula market price at time t of a standard European put option with strike price K and maturity date T; also: put price as a function of K and T one-week profit or loss of strategy s one-week profit or loss of strategy s net of transaction costs proportion of correct direction risk-neutral density for ST constant risk-free interest rate time-dependent risk-free interest rate continuously compounded one-day DAX return on day n one-week return of strategy s one-week return of strategy s net of transaction costs continuously compounded stock return in period i N-period sample mean of continuously compounded stock returns at time t (adjusted) coefficient of determination rebate (discounted) stock price process (discounted) stock price at time t (adjusted) futures-implied DAX level on day n at minute

(DAX) index level on day n; also: underlying price at time t a standard error SE time trading date corresponding to trading day n ta maturity date or expiration date maturity date of a futures contract TF maturity date of an option To terminal time horizon T* barrier level initial value Vo (q5) (discounted) value process of the portfolio 0 (V 3 (95)) 17(0) (discounted) value of the portfolio at time t (Vt3 (95)) Vt (0) value at risk VaR VARS t arbitrage value of a variance swap at time t Sn

List of Symbols

210

VOL St WT

W (W*) Wt (Wt* ) ( 1474t)

w Wi

X Yi YO,n Yi tn n+h

(Z*) Z (Zt) Zt (Zn (ZZt ) Zi,t

arbitrage value of a volatility swap at time t continuously sampled variance over the time interval [0,7] standard Brownian motion (one- or multidimensional) under P (Q) standard Brownian motion (one- or multidimensional) at time t under P (Q) standard Brownian motion i at time t under P (Q) vector Brownian motion i-th component of a vector Brownian motion a general random variable or general stochastic process i-th implied volatility risk factor (process) natural logarithm of stock price Sn value of the i-th volatility risk factor (i = 1. 4) on day n optimal h-days-ahead forecast of volatility risk factor i given the information on day n value of the i-th volatility risk factor at time t (discounted) asset price vector process (Z0, .••, Zd)' (discounted) asset prices (Zo,t, •••, Zd,t)' at time t (discounted) price process of asset i (discounted) price of asset i at time t

Greek case:

fin i,n

rBs rt Sas () St 50 1, E

fi,n

7g(K,T) qt (K, T)

drift rate of Yi at time t vector of regression coefficients on day n regression coefficient i on day n in the DAX volatility surface regression volatility of volatility risk factor Yi variance rate of volatility risk factor Y, with respect to Brownian motion Wi at time t covariance matrix Black-Scholes gamma (function) gamma of a standard European option at time t Black-Scholes delta (function) delta of a standard European option at time t delta of portfolio 0 (vector) Gaussian white noise process; random disturbance value of Gaussian white noise process i on day n random disturbance risk-neutral drift rate of crt (K,T) at time t real-world drift rate of crt (K,T) at time t

List of Symbols

Fit(M)r)

0 efts (.)

et (K ,T) tjt(M,r)

X

ABS (•)

At IL

en

(II* (H)) II(H)

211

real-world drift rate of -6-" t (M,T) at time t parameter vector Black-Scholes theta (function) theta of a standard European option at time t real-world variance rate (vector) of at (K,T) at time t real-world variance rate (vector) of at (m,r) at time t vector of parameters that together determine the market price of risk process speed of mean reversion in the Heston model intensity of Poisson process Black-Scholes vega (function) vega of a standard European option at time t constant instantaneous rate of return (process) from the stock instantaneous rate of return from the stock over [t, t +dt] vector of time-dependent model constants in VAR(1) process model constant in AR(1) process of correlation matrix volatility of volatility in the Heston model (discounted) arbitrage price process of contingent claim

H (11;(11 )) Ilt(H) fi

ei,e2 at(K .) at( . , T) at(K,T)

at tf,r) (

an(m, r) (Ti T2)

En TL, TU

(discounted) arbitrage price of contingent claim H at time t Monte carlo estimate of II correlation coefficient instantaneous correlation between Wi and vvi surface parameters volatility smile volatility term structure time t implied volatility of a standard European option with strike price K and maturity date T; also: (absolute) implied volatility surface at time t time t implied volatility of a standard European option with moneyness M and time to maturity T; also: (relative) implied volatility surface at time t implied volatility of a DAX option with moneyness M and time to maturity T on day n; also: (relative) implied DAX volatility surface on day n forward implied volatility (curve) conditional covariance matrix in VAR(1) model time to maturity, defined as T — t lower and upper time to maturity boundary instantaneous stock volatility (process)

212

List of Symbols

vt v(t) v(t,St) v2 00 0(t, T)

ût (N) 14(N)

T

(TT)

TT(Q*) (ki,t

Sht (Pi 4'(')

,b

1

Ijt

WErs

instantaneous stock volatility at time t time-dependent but deterministic instantaneous volatility local volatility long-run variance in the Heston model average volatility over [0, T] realized or historical stock price volatility over the last N periods at time t realized or historical stock price variance over the last N periods at time t class of all tame strategies (over [0, T]) class of (r-admissible strategies number of units of asset i held at time t vector of portfolio holdings {0 04 ,014 , ..., 0,14 ) at time t trading strategy or portfolio process vector of autoregressive coefficients in VAR(1) model autoregressive coefficient in AR(1) process of Yi payoff or contract function market price of risk process market price of risk vector {0 04 ,0, 4 ,...,07, 4 ) i at time t market price of risk of random source i at time t Black-Scholes DDeltaDVol (function) DDeltaDVol of a standard European option at time t state sample space

Other symbols: the Borel a-algebra unconditional expectation of X with respect to the measure Q conditional expectation of X with respect to the measure

g

Radon-Nikod3'rm density I at time t Ft or-algebra or-algebra representing information at time t the a-algebra generated by the process {X, : s < t} filtration constant elasticity parameter in the CEV model likelihood function nominal value of a variance or volatility swap set of natural numbers set of equivalent martingale measures objective or real-world probability measure -

List of Symbols Q Qs, Qi, Q2

R (Rn) R+ (R7) R++ (R7+ ) Vss() Vt

NIQ [X] VQ[XITt] Z 0

PO

PC / X) t --X,

lA

213

equivalent martingale measure or risk-neutral measure particular equivalent martingale measures set of real numbers (n-dimensional) set of nonnegative real numbers (n-dimensional) set of positive real numbers (n-dimensional) Black-Scholes DVegaDVol (function) DVegaDVol of a standard European option at time t unconditional variance of X with respect to the measure Q conditional variance of X with respect to the measure Q set of integers empty set the absolute value of X; also: the determinant of the matrix X the quadratic variation of X the quadratic covariation or cross-variation of X and Y distributed as approximately equal to the transpose of the vector x indicator function returning 1 if the set A is non-empty and 0 otherwise

Further comments: •

•

A function f(x) is said to be increasing (nondecreasing) on an interval I if f(b) > f (a) (f(b) ? 1(a)) for all b > a, where a,b E I. Conversely, a function f(x) is said to be decreasing (nonincreasing) on an interval I if 1(b) < f (a) (1(b) _< 1(a)) for all b> a with a, b E I. A quantity x is said to be positive (nonnegative) if x > 0 (x > 0) and negative (nonpositive) if x < 0 (x < 0).

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-

-

-

-

-

Index

4-sigma rule, 90 American-style, 16 AR, 108, 116 arbitrage free, 13 opportunity, 13 price, 18, 26 pricing theory, 18

ARIMA, 102 augmented Dickey-Fuller test, 107 autoregressive process univariate, see AR vector, see VAR barrier option delta, 153 down-and-out, 151 pricing, 152 types, 151 bid-ask costs, 181 binomial model, see tree methods

Black-Scholes call pricing formula, 28 market, 24 model, 1, 24 partial differential equation, 29, 67,

173 put pricing formula, 29 sensitivities, see Greeks Box-Jenkins approach, 102 Box-Pierce statistic, 125 Brownian motion deviation from, 43

fractional, 50 geometric, 24 multi-dimensional, 117, 197 standard, 196 vector, 117 butterfly spread, 176 calibration, 57, 140 cash-or-nothing option, see digital option CEV model, 46 Cholesky decomposition, 117 conditional expectation, 194 contingent claim, see derivative security contract function, see payoff function convexity adjustment, 158 crash risk, 48 cross-variation, 198 DAX adjusted index level, 79 correction for taxes and dividends, 78 future, 74, 78 index, 73 option, 74-76, 82 volatility index, see VDAX DAX implied volatility model applications, 145 calibration, 140 correlated form, 115 diagnostic checking, see diagnostic checking

discretization, 118 estimation method, 120

226

Index

hedging, 139 market price of risk, 133 out-of-sample test, 128 parameter estimates, 122 parameter stability, 126 pricing, 137 real-world dynamics, 116 review, 144 risk-neutral dynamics, 131 risk-neutral measure, 137 simulation, see Monte Carlo specification, 115 uncorrelated form, 117 volatility of volatility factor, 116 DAX volatility risk factors

ACF, 106 distributional properties, 104 forecasts of, 129 level, 94 model order, 108

PACF, 108 relation to market variables, 109 smile curvature, 95 smile slope, 95

stationarity, 107 term structure slope, 96 DAX volatility surface choice of moneyness, 86 empirical results, 92 outlier detection, 90 overall estimation procedure, 84 regression model, 89 risk factors, see DAX volatility risk factors shape, 94 stochastic model, see DAX implied volatility model two-step estimation, 89 delta-gamma method, 167 delta-normal method description, 160 drawbacks, 166 derivative security, 15 attainable, 18, 20 forward, see forward contract future, see futures contract Option, see option swap, see swap third-generation, 147

diagnostic checking, 124 difference dividend, 78 diffusion, 155, 197 digital option delta, 150 pricing, 148 types, 147 distribution fat-tailed, 44

leptokurtic, 99 normal, see normal distribution down-and-out option, see barrier option drift condition, 66, 133 Euler method, 118, 138 exotic derivative Asian option, 17 barrier option, see barrier option digital option, see digital option forward-start option, 172 log contract, 202 lookback option, 17 overview, 145 path-dependent, 146 path-independent, 146 variance swap, see variance swap volatility swap, see volatility swap expectation, 194 expected shortfall, 168

Feynman-Kac formula, 29 filtration, 194 financial market

Black-Scholes, 24 complete, 21, 55 general implied volatility model, 60 incomplete, 3, 21 finite difference approach, 139 flattening-out effect, 39, 88 forecast efficient, 36, 129 unbiased, 129 forward contract, 16, 26 futures contract, 17 gains process, 11

GARCH, 179 general implied volatility model assumptions, 60

Index drift condition, see drift condition hedging, 72 instantaneous volatility, 70 interpretation of drift terms, 68 market price of risk, 63 pricing, 72 real-world dynamics, 62 risk-neutral dynamics, 63 risk-neutral measure, 68 Girsanov's theorem, 14 Greeks, 30, 65, 139, 173, 175

DDeltaDVol, 31 delta, 31, 139, 150, 153

DVegaDVol, 31 gamma, 31, 139 theta, 31

vega, 31 half-life, 123 Heston model, 3, 47 Hyperbolic model, 50 implicit method, 80 implied tree, 46 implied volatility as primitive, 4, 45 calculation, 34 concept, 2, 32 deterministic, 53 forward, 40 general model, see general implied volatility model interpretation, 35 risk factors, see DAX volatility risk factors skew, 38 smile, 2, 38, 87 stochastic, 55 surface, 55 surface of DAX options, see DAX volatility surface term structure, 39, 55, 87 indicator function, 168 informational efficiency, 179, 185, 196 Ito formula, 197 process, 197 It&Venttsel formula, 64, 198

227

Jensen inequality, 158 jumps, 3, 44, 48 keep-strike strategy, 178, 182 leverage effect, 48, 109 liquidity, 172 Ljung-Box statistic, 125 mark to market, 17 market frictions, 3, 43, 50 market maker, 179, 181 market price of risk, 14, 63, 133 market risk, 158 market-based approach, 4, 45 Markov process, 25, 196 martingale, 195 local, 196 property, 195 representation theorem, 20 martingale measure, 13, 15, 26, 68, 137 maximum likelihood estimation, 120 likelihood function, 120 quasi, 120 mean absolute percentage error, 141 mean reversion level, 116 speed, 116 Merton model, 49 meta-theorem, 21

moneyness concept, 42 fixed-strike, 42 log simple, 86

Natenberg, 85 simple, 86 standardized, 85 valid, 61, 86 Monte Carlo estimate, 139 simulation, 137 simulation for value at risk, 162, 167 simulation path, 142 standard error, 139

Newton-Raphson procedure, 34 NFLVR condition, 15 no-arbitrage condition, see drift condition

228

Index

relations, 91 nonparametric methods, 84 normal distribution

multivariate, 120 univariate, 28 Novikov condition, 14, 63 numéraire, 11 option at-the-money (ATM), 17, 86 contract, 17 exotic, see exotic option exposures, 175 in-the-money (ITM), 17, 86 maturity, 17 out-of-the-money (OTM), 17, 86 profit or loss, 174 standard, 17, 27, 28 strategies, 176 strike price, 17 Ornstein-Uhlenbeck process, 116 parameter vector, 119 parsimonious, 99 payoff function, 16 Philips-Perron test, 107 pin risk, 148 Poisson process, 49 portfolio delta, 160

delta-neutral, 32 strategy, see trading strategy principal component analysis, 100, 101 probability measure, 193 progressively measurable, 195 proportion of correct direction, 130 put-call parity, 28, 78 quadratic variation, 198 random variable, 194 rebate, 152 risk factors abstract, 4, 99 fundamental, 100 of DAX volatility surface, see DAX volatility risk factors original, 99 statistical, 101

risk-neutral density, 51 density of DAX options, 97 measure, see martingale measure valuation formula, 19, 26, 72 rolling window, 128 sigma-algebra, 193 smile effect, see implied volatility smile models comparison, 56 overview, 43 square-root process, 47 stepwise regression, 136 sticky-implied tree rule, 54 sticky-moneyness rule, 54 sticky-strike rule, 54 stochastic differential equation, 197 stochastic process, 194 straddle, 163, 176 swap, 17 tax effects, 51 tax system

Halbeinktinfte-Verfahren, 80 KtSrperschaftssteueranrechnungsVerfahren, 80 trading strategy, 11 admissible, 18, 20 replicating, 18 self-financing, 12 tame, 13 transaction costs, 50, 181 tree methods, 137, 152 trimmed regression, see 4-sigma rule value at risk computation methods, 160 concept, 159 delta-normal method, see deltanormal method full-valuation method, 162 with DAX implied volatility model,

162 value process, 11 VAR, 119, 125 variance fair value of, 155, 203 realized, 155 swap, see variance swap

Index total, 155 variance gamma process, 50 variance swap description, 154 pricing, 155, 201 replicating strategy, 202

VDAX, 93 vertical spread, 176 volatility actual, 33 average, 36 clustering, 107 derivative, 171 derivatives, 146 deterministic, 2, 3, 45 fair value of, 158 historical, 33 implied, see implied volatility instantaneous, 33 local, 2, 45 of stock price, 24

229

realized, 33 smile, see implied volatility stochastic, 47 surface, see implied volatility swap, see volatility swap term structure, see implied volatility time-varying, 44 trading, see volatility trade volatility swap description, 157 pricing, 158 volatility trade definition, 170 empirical analysis, 180 first-order, 178 second-order, 178 trading instruments, 171 trading objects, 171 weighted least squares, 89

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Vol. 519: A. Klose, G. Speranza, L. N. Van Wassenhove (Eds.), Quantitative Approaches to Distribution Logistics and Supply Chain Management. XIII, 421 pages. 2002. Vol. 520: B. Glaser, Efficiency versus Sustainability in Dynamic Decision Making. IX, 252 pages. 2002. Vol. 521: R. Cowan, N. Jonard (Eds.), Heterogenous Agents, Interactions and Economic Performance. XIV, 339 pages. 2003.

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Vol. 532: K. Marti, Y. Ermoliev, G. Pflug (Eds.), Dynamic Stochastic Optimization. VIII, 336 pages. 2004. Vol. 533: G. Dudek. Collaborative Planning in Supply Chains. X, 234 pages. 2004. Vol. 534: M. Runlcel, Environmental and Resource Policy for Consumer Durables. X, 197 pages. 2004. Vol. 535: X. Gandibleux, M. Sevaux, K. Sorensen. V. T'kindt (Eds.), Metaheuristics for Multiobjective Optimisation. IX, 249 pages. 2004. Vol. 536: R. Briiggenumn, Model Reduction Methods for Vector Autoregressive Processes. X, 218 pages. 2004. Vol. 537: A. Esser, Pricing in (In)Complete Markets. XI, 122 pages, 2004. Vol. 538: S. Kokot, The Econometrics of Sequential Trade Models. XI, 193 pages. 2004. Vol. 539: N. Hautsch, Modelling Irregularly Spaced Financial Data. XII. 291 pages. 2004. Vol. 540: H. Kraft, Optimal Portfolios with Stochastic Interest Rates and Defaultable Assets. X, 173 pages. 2004. Vol. 541: G.-Y. Chen, Vector Optimization (planned). Vol. 542: J. Lingens, Union Wage Bargaining and Economic Growth. XIR. 199 pages. 2004. Vol. 543: C. Benkert, Default Risk in Bond and Credit Derivatives Markets. IX, 135 pages. 2004. Vol. 545: R. Hafner, Stochastic Implied Volatility. XI, 229 pages. 2004. Vol. 546: D. Quadt, Lot-Sizing and Scheduling for Flexible Flow Lines. XVIII, 227 pages. 2004.