Energy Risk

ENERGY RISK Valuing and Managing Energy Derivatives SECOND EDITION DRAGANA PILIPOVIC McGraw-Hill New York Chicago Sa...

Author: Dragana Pilipovic

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ENERGY

RISK Valuing and Managing Energy Derivatives SECOND EDITION

DRAGANA PILIPOVIC

McGraw-Hill New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2007 by Dragana Pilipovic. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-159447-7 The material in this eBook also appears in the print version of this title: 0-07-148594-5. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or --otherwise. DOI: 10.1036/0071485945

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C O N T E N T S

PREFACE xiii ACKNOWLEDGMENTS

xv

Chapter 1

Energy Markets: Trading, Modeling, and Hedging

1

1.1. Introduction 1 1.2. Energy Trading 2 1.2.1. Understanding the Fundamentals 2 1.2.2. Liquidity, Volatility, and Intra-Market Correlations 4 1.2.3. Market Deregulation 6 1.3. Energy Modeling 8 1.3.1. Energies Are Still Unique 8 1.3.2. Model Complexity 8 1.3.3. Quants vs. Traders vs. Reality 9 1.4. Energy Hedging and Risk Management 10 1.4.1. Adding Financial Products to the Hedging Mix 10 1.4.2. Risk Management: A Profitable Business Function? 11 1.4.3. Hedging for the Little Guys 12 1.4.4. Assets as Hedges 12 1.4.5. Regulatory Response to “Bad” Stories 13 1.5. Conclusions 14 Chapter 2

What Makes Energies So Different?

17

2.1. Introduction 17 2.1.1. Quantitative and Fundamental Analysis 2.2. What Makes Energies So Different? 19 2.3. Energies Are Harder to Model 20 2.4. Market Response to Cycles and Events 23 2.5. Impact on Supply Drivers 26 2.6. Energies Have a “Split Personality” 28 2.7. Impact of Demand Drivers 28

18

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2.7.1. The Convenience Yield 29 2.7.2. Seasonality 30 2.8. Regulation and Illiquidity 31 2.9. Decentralization of Markets and Expertise 31 2.10. Energies Require More Exotic Contracts 32 2.11. Conclusion 33 Chapter 3

Modeling Principles and Market Behavior

35

3.1. The Modeling Process 35 3.2. The Value of Benchmarks 36 3.2.1. Diffusing Personalized Attachments to Models 36 3.3. The Ideal Modeling Process 38 3.4. The Role of Assumptions: Market Before Theory 38 3.4.1. Typical Assumptions 39 3.4.2. Market Variable vs. Modeling Parameter 41 3.4.3. Testing Assumptions Through Benchmarks 42 3.4.4. Assumptions and Implementation 45 3.5. Contract Terms and Issues 45 3.5.1. Underlying Price or Market 45 3.5.2. Derivative Contract 46 3.5.3. Option Settlement Price 46 3.5.4. Delivery 46 3.5.5. Complexity of Contracts for Delivery 47 3.6. Modeling Terms and Issues 49 3.6.1. Price Returns 49 3.6.2. Elements of a Price Model 49 3.6.3. Convenience Yield 52 3.6.4. Cost of Risk 54 3.7. Quantitative Financial Models Across Markets 55 3.7.1. Lognormal Market 56 3.7.2. Mean-Reverting Market 60 3.8. The Taylor Series and Ito’s Lemma 63 3.8.1. The Taylor Series 63 3.8.2. Ito’s Lemma 64 3.9. Lessons from Money Markets 65 3.9.1. Modeling Price vs. Rate: Defining the Market Drivers 65 3.9.2. Yield vs. Forward Rate Curves 66 3.9.3. Drawbacks of Single-Factor Mean-Reverting Models 68

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3.9.4. 3.9.5.

Drawbacks of Single-Factor Non-Mean-Reverting Models Volatility and Correlation Market Discovery 69

69

Chapter 4

Essential Statistical Tools

71

4.1. Introduction 71 4.2. Time Series and Distribution Analysis 72 4.2.1. Time Series Analysis 72 4.2.2. Distribution Analysis 75 4.3. Other Statistical Tests 81 4.3.1. The Q-Q Plot 81 4.3.2. The Autocorrelation Test 83 4.3.3. Measures of Fit 83 4.4. How Statistics Helps to Understand Reality 85 4.4.1. A Simple Case 85 4.4.2. The Difference Between Price and Return 86 4.4.3. Distinguishing Drift Terms 86 4.5. The Six-Step Model Selection Process 88 4.5.1. Step 1: An Informal Look 89 4.5.2. Step 2: A Shortlist of Possible Models 90 4.5.3. Step 3: Time Series Analysis 90 4.5.4. Step 4: From Underlying Price Models to Distributions 4.5.5. Step 5: Distribution Analysis 92 4.5.6. Step 6: Select the Most Appropriate Model 93 4.6. Relevance to Option Pricing 93 Chapter 5

Spot Price Behavior

95

5.1. Introduction 95 5.2. Looking at the Actual Market Data 96 5.3. A Shortlist of Possible Models 103 5.3.1. The Lognormal Price Model 103 5.3.2. Mean-Reverting Models 105 5.3.3. Cost-Based Models for Electric Utilities 111 5.3.4. Interest Rate Models 111 5.4. Calibrating Parameters Through Time Series Analysis 111 5.4.1. Incorporating Seasonality with Underlying Models 112 5.4.2. Results from Time Series Analysis 113

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5.5. Performing Distribution Analysis 119 5.5.1. Implementation of Distribution Analysis 5.5.2. Results of Distribution Analysis 120 5.6. Analysis Summary 121

119

Chapter 6

The Forward Price Curve

127

6.1. Introduction 127 6.1.1. The Difference Between Forwards and Futures 128 6.2. Reading the Underlying Curve 129 6.3. Seasonality in the Forward Curve 132 6.4. Modeling Concepts Relating Spot, Forwards, and Seasonality 135 6.4.1. S&P 500 136 6.4.2. WTI Crude Oil 136 6.4.3. Seasonal Markets 137 6.5. Linking Spot Price Models to Forward Price Models 143 6.5.1. The Arbitrage-Free Condition 143 6.5.2. Capturing Market Characteristics Within the Model or During Implementation 145 6.5.3. Influence of the Convenience Yield 145 6.6. Modeling the Underlying Forward Price Curve 147 6.6.1. Difference Between Spot and Forward Prices 147 6.6.2. Going from Spot Price Models to Forward Price Models 150 6.6.3. The Risk-Free Portfolio 150 6.6.4. Effect of Dividends 153 6.6.5. Equivalence Between Dividends and the Convenience Yield 155 6.6.6. Adding a Second Factor 156 6.6.7. Seasonality 157 6.7. The Two-Factor Mean-Reverting Model (Pilipovic) 158 6.8. Testing the Spot Price Model on Forward Price Data 162 Chapter 7

Building Marked-to-Market Forward Price Curves: Implementing Forward Price Models 163 7.1. Introduction: What Is a Marked-to-Market Forward Price Curve? 164 7.2. Forward Price Contract Valuation 166 7.2.1. Simple Contract for One-Day Delivery 170 7.2.2. Contract for Delivery Over a Period 173 7.2.3. Bootstrapping and the Problem of Daily Price Discovery 179

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7.3. Fitting the Modeling Needs to Trading Needs 182 7.3.1. Case of Trading Exchange-Traded Products Only 182 7.3.2. Case of Trading OTC 183 7.3.3. Case of Owning Power Production 184 7.4. Building Marked-to-Market Forward Price Curves: Issues to Consider 184 7.4.1. Quote Strips 184 7.4.2. Step-Function Treatment 187 7.4.3. Linear Interpolation 187 7.4.4. Applying Forward-Price Models Based on Spot-Price Analysis 188 7.4.5. Many Degrees of Freedom Within Implementation: Part Art, Part Science 189 7.4.6. From Events to Models 191 7.4.7. Parameter Calibration 192 7.5. Modeling Middle-Term Event Expectations 193 7.6. Modeling Forward Price Seasonality 195 7.6.1. Cosine Seasonality 196 7.6.2. Exponential Seasonality 197 7.6.3. Power-N Model —Flat Seasonality 201 7.6.4. Multiperiod Seasonality Treatment 201 7.7. Special Case of Basis Markets 205 7.8. Noise Versus Events 209 7.9. Markets with Little or No Market Discovery: Off-Peak and Hourly Forward Price Curves 211 7.10. Conclusion 212 Chapter 8

Volatilities

215

8.1. Introduction 215 8.2. Measuring Randomness 216 8.2.1. Standard Deviation and Variance 216 8.2.2. Volatility Defined 217 8.2.3. Comparing Variance and Volatility 218 8.2.4. Variance and Volatility in Spot Price Models 218 8.3. The Stochastic Term 220 8.3.1. Case of Constant Volatility 220 8.3.2. Case of Volatilities with Term Structure 221 8.4. Measuring Historical Volatilities 222 8.4.1. Simple Techniques 222 8.4.2. More Complex Techniques 223

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8.5. Market-Implied Volatilities 224 8.5.1. Option-Implied Volatilities 224 8.5.2. Implied Volatilities from a Series of Options 225 8.5.3. Calibrating Caplet Volatility Term Structure 226 8.5.4. Implied Volatilities from Options on the Average of Price 230 8.5.5. The Volatility Smile 232 8.6. Model-Implied Volatilities 232 8.6.1. The Lognormal Model 233 8.6.2. The Log-of-Price Mean-Reverting Model 234 8.6.3. The Price Mean-Reverting Model 236 8.7. Building the Volatility Matrix 240 8.7.1. Introduction to the Forward Volatility Matrix 241 8.7.2. Discrete Volatilities 242 8.7.3. Tying In Caplet Volatilities 244 8.7.4. Two-Dimensional Approach to Volatility Term Structure 246 8.7.5. Tying In Historical Volatilities 249 8.7.6. Tying In Caplet and Swaption Prices 249 8.8. Implementing the Volatility Matrix 251 Chapter 9

Overview of Option Pricing for Energies

255

9.1. Introduction 255 9.2. Basic Concepts of Option Pricing 256 9.2.1. Parity Value 256 9.2.2. Settlement 258 9.3. Types of Options 258 9.3.1. European Options 259 9.3.2. American Options 259 9.3.3. Asian Options: Options on an Average of Price 259 9.3.4. Swing Options 260 9.4. Effect of Underlying Behavior 261 9.5. Option Pricing Implementation Techniques 263 9.5.1. Closed-Form Solutions 263 9.5.2. Simulations 265 9.5.3. Trees 266 9.5.4. Human Error in Implementation 267 9.6. Choosing the Right Option Pricing Model 267 9.6.1. Three Criteria for Evaluating Option Models 268 9.6.2. Investing in Pricing Model versus Implementation 269 9.6.3. A Model Is Only as Good as Its Implementation 270

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9.7. Option Valuation Process: What Should It Be? 270 9.7.1. Defining Underlying Market Price Behavior 270 9.7.2. Testing Alternative Models 271 9.7.3. Selecting the Most Appropriate Option Model 272 9.8. Did That Option Make Money? 273

Chapter 10

Option Valuation

275

10.1. Introduction 275 10.2. Option Model Implementation 276 10.3. Closed-Form Solutions 276 10.3.1. Pros 276 10.3.2. Cons 277 10.3.3. The Black–Scholes Model 277 10.3.4. The Black Model 279 10.4. Approximations to Closed-Form Solutions 283 10.4.1. Pros 283 10.4.2. Cons 284 10.4.3. The Volatility Smile 284 10.4.4. The Edgeworth Series Expansion 285 10.4.5. Pulling It All Together 288 10.5. The Tree Approach 290 10.5.1. Pros 291 10.5.2. Cons 291 10.5.3. Binomial Trees 292 10.5.4. Trinomial Trees 292 10.5.5. Using a Tree to Value a European-Style Option 293 10.5.6. Using a Tree to Value an American-Style Option 295 10.5.7. Energy-Specific American-Style Options 295 10.6. Monte Carlo Simulations 300 10.7. Conclusions 302 Chapter 11

Valuing Energy Options

303

11.1. Introduction 303 11.2. Daily Settled Options 304 11.2.1. Extending Daily Methodology to Hourly Settled Options 312

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Contents

11.3. Monthly Settled Options 313 11.3.1. Cash-Settled: Look-Back Monthly Settled Average Price Options 314 11.3.2. Monthly-Settled (Look-Forward) Options on Monthly Forwards 317 11.3.3. Incorporating Price Mean Reversion (PMR) into Monthly Settled Options 326 11.3.4. Extending Monthly Methodology to Calendar Year Options 11.4. Optionality in Cheapest-to-Deliver Forward Prices 333 11.5. Types of Energy Swing Options 334 11.6. Demand Swing Contracts 336 11.6.1. Demand Swing Options 336 11.6.2. Demand Swing Forwards 339 11.6.3. Load Behavior 340 11.7. Price Swing Contracts 345 11.7.1. Multiple-Peaker Swing Options 346 11.7.2. Forward Starting Swing 358 11.7.3. Natural Gas Storage 360 11.8. Spread Options 361 11.8.1. Various Approximations to Spread Option Valuation 362 11.8.2. The Tree Approach 370 11.8.3. Crack Spread, Spark Spread, and Basis Spread Options 372 11.8.4. Valuing Power Plants and Transmission Lines 372 11.9. Conclusion 373 Chapter 12

Measuring Risk

375

12.1. Introduction 375 12.2. The Risk/Return Framework 375 12.3. Types of Risk 377 12.3.1. Market Risk 378 12.3.2. Commodity Risk 378 12.3.3. Human Error 378 12.3.4. Model Risk 379 12.4. Definition of a Portfolio 380 12.4.1. Change in Portfolio Value 381 12.4.2. Time Buckets 381 12.5. Measuring Changes in Portfolio Value 12.5.1. Taylor Series 383

383

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12.6. Portfolio Sensitivity: The “Greeks” 385 12.6.1. Delta: Sensitivity to Price Change 385 12.6.2. Vega: Sensitivity to Volatility Change 386 12.6.3. Theta: Sensitivity to Time 388 12.6.4. Rho: Sensitivity to Discounting Rates 391 12.6.5. Gamma: Sensitivity to Changes in Delta 391 12.6.6. Quantity-Specific Risks 394 12.6.7. Sensitivity to Correlation Change 394 12.7. Hedging 395 12.8. Marking-to-Market 396 12.8.1. Information for Marking-to-Market 396 12.8.2. Mark-to-Market Valuation 397 12.8.3. Testing the Mark-to-Market Process 398 Chapter 13

Portfolio Analysis

401

13.1. Introduction 401 13.2. Applications of Portfolio Analysis 402 13.3. Analyzing the Change in Portfolio Value 402 13.4. The Minimum-Variance Method 404 13.4.1. The Hedged Portfolio 405 13.4.2. Per-Deal Hedges 406 13.4.3. Portfolio with Options 410 13.4.4. Lessons from Inadequate Hedging Policies 411 13.5. The Generalized Minimum-Variance Model 417 13.6. Correlations 417 13.7. Value-at-Risk (VAR) Analysis 418 13.7.1. Fixed-Scenario Stress Simulations 420 13.7.2. Monte Carlo Simulations 420 13.7.3. Estimated Variance–Covariance Method 422 13.7.4. Historical “Simulations” 422 13.8. The Special Case of Electricity 423 13.9. The Corporate Utility Function 424 Chapter 14

Risk Management Policies

427

14.1. Introduction 427 14.2. The Case for a Risk-Management Policy 14.2.1. Horror Stories 429

428

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Contents

14.3. Risk-Management Goals and Strategies 430 14.3.1. Speculation 431 14.3.2. Arbitrage 432 14.3.3. Market Maker 433 14.3.4. Treasury 434 14.3.5. Mixed Strategies 434 14.4. Initial Evaluation Checklist 435 14.4.1. Diagnosing and Selecting Trading Strategies 437 14.4.2. Gaps Between Existing and Desired Market Position 14.4.3. Corporate Culture 438 14.5. The “Front/Middle/Back Office” Paradigm 439 14.5.1. Conflicts Between Offices 440 14.5.2. Interoffice Committees 441 14.6. The Energy Team 441 14.6.1. Appropriate Knowledge by Organizational Level and Functions 444 14.6.2. Management Issues 445 14.6.3. Common Management Misconceptions 450 14.7. Implementation of Risk-Management Policies 453 Appendix A: Appendix B: Appendix C:

438

Mathematical and Statistical Notes 455 Models from Interest Rate and Bond Markets 463 Analysis of Markets Published in the First Edition of Energy Risk 467 Glossary of Energy Risk Management Terms 485 Select Bibliography 499 INDEX

503

P R E F A C E

O

ver the many years I have gained experiences in a wide variety of derivative markets: from equities and interest rates to natural gas and electricity. With every new market, I discovered further proof of something that I had only sensed at the very first: markets differ significantly from each other through differences in the types of fundamental price drivers and how they impact the market prices. Each market follows its own unique price behavior: a summer event in the electricity markets is caused by an unexpected temperature spike that typically keeps the prices up for a week or so; a stock price jumps up on news of a take-over and remains at the newly reached levels unless there is further news that the take-over failed. Then why, I ask you, do the pricing experts insist on using the same set of models in markets that are so very different? This question inspired this book. My motivation is to explain why energy markets are so different from the more traditional derivatives markets. My objective is to provide tools capable of handling these differences. Energy risk managers, particularly in the still young power markets, need a comprehensive guide. This book is a practitioner’s book, not an academic one. Energy Risk: Valuing and Managing Energy Derivatives is the product of my years of being a “rocket scientist”—an ex-physicist working in financial markets. I faced all the questions in this book first hand, “on the trading desk” as a quantitative analyst, trader, and consultant. The problems always resembled a double-headed guard dog: first I had to determine a good analytical answer, then I faced the problem of implementation. I approached the problems by establishing benchmarks, setting standards of acceptability, and at the end of the day settled for ideas and technology that got the job done. With the amazing growth of today’s energy markets, particularly in electricity and power, I suspect there are many professionals who now find themselves in a position similar to mine in 1989 when I began trading, natural gas

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Preface

during its dawn of deregulation and again in 1995 when I began modeling electricity: they need practical answers to derivatives and risk management issues. This book is intended as a single-source, desk-top manual for getting reasonable answers to actual modeling and implementation problems surfacing in today’s energy markets. Dragana Pilipovi´c

A C K N O W L E D G M E N T S

I would like to thank many people for their help with this book. First I want to thank John Wengler, Chief Risk Officer of Entergy Services, Inc., who first conceived of this book and then spent many sleepless nights helping me write and edit the first edition. He was also critical in the editing of this new edition, providing many ideas and helpful comments. Thank you, John! I want to thank many professionals in chronological order for their help in developing the concepts and materials for this book: Harvard University’s Deborah Hughes-Hallet for giving me my first job, teaching a course for people scared of math at the college and the wonderful cast of characters attending the Kennedy School of Government summer program; Brown University’s graduate school of physics Professor Augustine Falieros for the joy of applied mathematics and Professor Dave Cutts for giving me the chance to move to Chicago’s Fermilab and understanding why I had to leave physics; Mike Parkinson of the former O’Connor & Associates for giving me my first job in finance and David Weinberger for supporting my research style; Continental Bank’s Ken Cunningham and Philippe Comer for allowing me to form my ideas freely; Linda Rudnick of Harris Bank for providing a safe haven and one of my first consulting contracts; Kay Rigney of the First National Bank of Chicago’s women’s banking unit for invaluable support and advice; Southern Energy Marketing’s Sean Murphy and Jeff Roark for inviting me into the world of electricity; Cinergy Corporation’s Ken Leong and Paul Zhang for helping market-test my theories; the participants in the Chicago, Houston and Aspen seminars that served as the basis for this book; the forward-thinking professionals at Dayton Power & Light, Sonat Marketing and NESI Power Marketing for their special participation in the seminars; Stephen Isaacs of McGraw-Hill for agreeing that the market needed a book like this; Adrian D’Silva of the Federal Reserve Bank of Chicago and his bookshelf; Professor John Bilson of the Illinois Institute of Technology’s Master’s in Financial Markets and Trading for providing a teaching podium; and, last but not least, Rick Dennis of Southern Corp. for suggestions and challenging requests within risk management implementation. Also, thanks to Bob xv Copyright © 2007 by Dragana Pilipovic. Click here for terms of use.

xvi

Acknowledgments

Dylan for his most recent work: Modern Times was my constant companion during the writing of this second edition. A special thanks goes to Entergy Services, Inc., and Francis H. Wang, the Director of Commercial Analytics, for providing invaluable market data for this new edition. Many additional thanks to Francis for also contributing the discussion on Locational Marginal Pricing in Chapter 5. Finally, I would like to thank my family: my children, Sasha and Nevena, for being the wonderful, loving, and positive creatures that they are: your athletic prowess is an incredible motivation in everything I do! My parents, Vera and Nikola, for their labors of love. And my husband, John, for always believing in me.

C H A P T E R

1

Energy Markets: Trading, Modeling, and Hedging Reality is what we take to be true. What we take to be true is what we believe. What we believe is based upon our perceptions. What we perceive depends upon what we look for. What we look for depends upon what we think. What we think depends upon what we perceive. What we perceive determines what we believe. What we believe determines what we take to be true. What we take to be true is our reality . . .

Gary Zukav, The Dancing Wu Li Masters1

1.1.

INTRODUCTION . . . until it starts hurting. As a little girl perhaps I did not know very much about the world at large, but I knew that I did not like going to the dentist. One of my teeth started hurting. I did not like it, I did not enjoy it, but I was going to stand it for as long as I could. I was going to pretend that it was not happening, assume everything was fine—just to avoid the dreaded dentist. In the end, the tooth caught up with me. Once the pain got so bad that I could no longer run out to play, I had to tell my mother. Sure enough, the visit to the dentist was not a pleasant one; the baby tooth was at this point so far gone that it could not be saved, and had to be pulled. The moral of the story is not that you should go to the dentist (although you should!), but rather, that the truth will catch up to you, sooner or later, like it or not. As much as we all have our own realities, our own ways of looking and experiencing the world around us, there are sometimes moments of truth forced upon us. This is a good thing—it is a chance for recalibration of reality, a chance for new growth and new paradigms of thought and experience, much as the process might hurt. In the energy markets there have been many painful lessons since I wrote the first edition of this book in 1997, with some serious moments of truth forced upon us, and there are probably many more awaiting us. But that is what makes the energy business so interesting. 1

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2

Energy Risk

As Brian Hunter, the former trader at Amaranth Advisors, has been quoted to say, Every time you think you know what these markets can do, something else happens.2

With this second edition, we continue exploring energy markets and solutions to valuing energy derivatives and their potential risks. In addition to an all new introduction, there are two new much needed chapters covering forward price curve building and option valuation, plus expanded derivations, explanations, and updates to the original chapters. Many readers over the years had comments and requests for more detailed explanations and I have tried to include these as much as possible. Before we get to the math, let us take some time for a highlevel look at some of the major events through which we have learned and re-learned important risk management lessons during the decade since Energy Risk was first published.

1.2.

ENERGY TRADING 1.2.1.

Understanding the Fundamentals

Energy markets continue to grow and develop as a function of fundamentals. Perhaps some early players underestimated the impact of fundamentals, but by now most energy participants understand first hand just how volatile and eventful these markets can be. The BP Statistical Review of World Energy recap of 2005 energy markets ticks off the kind of fundamentals that continue to drive the market: 2005 was a third consecutive year of rising energy prices. Tight capacity, extreme weather, continued conflict in the Middle East, civil strife elsewhere and growing interest in energy among financial investors led to rising prices . . . World primary energy consumption in 2005 increased by 2.7% . . . World natural gas consumption grew by 2.3% . . . Coal was again the world’s fastest-growing fuel, with global consumption rising by 5% . . .3

Weather is one of the main fundamental price drivers in the energy markets. A heat wave in the summer or a cold spell in the


3

winter can result in sky-rocketing prices, sometimes to magnitudes that are hard to believe. However, even on a day-to-day basis, weather is a dominant player in the market place that has everyone’s attention, as evidenced by the following dispatch in a daily newsletter: In the Midcontinent, the National Weather Service issued a heat advisory for the Oklahoma City area until Thursday evening, with heat indices expected to approach 110 degrees through the end of the week. The oppressive heat continued to test local power generators as they pulled gas supplies from storage and from Western production basins. Natural Gas Pipeline Co. of America’s Midcontinent zone shot up nearly 60 cents, while Natural’s Texok zone added about 50 cents and CenterPoint’s East zone gained more than 45 cents.4

Weather events in the energy markets can easily go both ways. After a price spike due to a large weather event, such as that experienced in the south U.S. markets due to Hurricane Katrina, the market participants become quite weary of the possibility of another such event, particularly given the reality of hurricane seasons and expected long-term hurricane weather patterns. The following magazine excerpt captures how past pain can infuse future expectations: The catastrophic damage to Gulf of Mexico oil facilities wrought by Hurricane Katrina last year leaves the industry extremely jittery as the start of the 2006 hurricane season approaches. So far, most weather predictions do not bode well . . . Worryingly, it is also evident that any new storms are likely to have a bigger impact than in 2005, because the region’s infrastructure is only just recovering from last year.5

These fears were justified during the spring and summer of 2006, despite the fact that prices had been dropping since their post-Katrina highs. But then no serious storms hit the United States, despite the conditions being ripe for a repeat of 2005. In a market with such an event expectation looming large over all the market participants, for the hurricane season to not realize itself is also a huge event, sending prices tumbling down. Although the market participants have to worry about short-term events, trading, and hedging, the long-term market outlook can be just as complex: Consulting company Weed Mackenzie concluded that there is a serious risk of power shortages and extreme price volatility if electricity demand growth is higher than expected during the next five or six years. The record demand peaks of the summer 2006 highlight the

4

Energy Risk

danger of relying on reserve margins that are sufficient for average but not necessarily above-average conditions, according to the company’s report, “A Crisis in the Making?”6

It goes without saying, because we cannot expect the weather to stabilize anytime soon, the need for proper volatility analysis and risk management will continue for years to come!

1.2.2.

Liquidity, Volatility, and Intra-Market Correlations

As a trader in any market will tell you, liquidity issues are a part of a traders’ life. When events hit, even the best-covered markets experience illiquidity, as summarized by a risk manager with a hedge fund: One can never guarantee liquidity in the markets. When events happen, bid–offer spreads widen, volume might decrease. That is just the nature of trading.7

In energy markets, the frequency and magnitude of events can be captured by the high volatility. To make things more complicated, the forward price curves remain imperfectly correlated as the short- and long-term portions of the energy forward price curves tend to be driven by different market factors with usually very little or no relationship. The eventful nature of energy markets, coupled with physical limitations in responding to events, and with relatively limited market participation, can result in what is beginning to appear as a never ending sequence of horror stories for the even highly knowledgeable traders: MotherRock, an energy trading hedge fund led by former Nymex President J. Robert “Bo” Collins, is imploding . . . MotherRock’s troubles stemmed from a series of bad bets on natural gas prices made with leverage, or borrowed money, sources say. Natural gas prices have been volatile in recent months . . . In a May investor update, MotherRock the hedge fund’s “natural gas book was hurt primarily by a loss on volatility spread trading.”8

Understanding the appropriate trading strategy for both the market conditions and the company’s depth of pocket and corporate culture is key to avoiding a market-driven tragedy. When participating in


5

speculative trading in energy markets, the company must have pockets deep enough to cover the types of risk levels the management approves for the traders. In the case of speculative position taking, given excellent traders, a company should expect to see both high profits as well as occasional large losses—this is simply the reality of speculative position taking. One would have thought that investors would eventually take to heart the caveat that appears in most prospectuses: “past results do not guarantee future results,” especially when the promise of success is linked with an individual superstar. The story of Long-Term Capital Management (LTCM), with its array of luminaries, should have proven that hiring the smartest folks does not necessarily guarantee that gambles will always win. And yet, as the Wall Street Journal reported in 2006, we witnessed the same sad story repeated again: . . . Mr. Hunter headed the energy desk for a Connecticut hedge fund called Amaranth Advisors. At the end of August, trading natural gas, he was up roughly $2 billion for the year. Then he lost approximately $5 billion—in about a week. . . . “The cycles that play out in the oil market can take several years, whereas in natural gas, cycles take several months,” Mr. Hunter said in an interview late in July, when his returns were looking rosy . . .9

By all accounts, it appears that Mr. Hunter speculated on the spread between certain months of natural gas delivery. Perhaps this was presented to management and/or investors as an arbitrage strategy based on the idea that positions can be taken to take advantage of market mispricing in such a manner that risks are ideally neutralized but usually minimized. One of the areas in energy trading that require quite a bit of thought both in terms of valuation and hedging are the intra-market correlations: the correlations between forward prices in the same market place but covering different periods of delivery. In these murky correlation waters it is easy to disguise speculation under arbitrage, resulting in potentially miscalculating the market risks and therefore not matching the actual market risks to the depth of the company’s pockets. As the Wall Street Journal noted: Mr. Hunter’s bets ultimately went bad because he misjudged the movement of the difference between prices for different month contracts, known as the spread.10

6

Energy Risk

After the fact it is usually easy to understand why a company might have lost huge amounts of money. Although misjudging the intra-market correlations between the natural gas futures can result in a tragic loss, it is also important to remember that getting these correlations right can earn loads of money as well. However, very rarely do the investors ask for a review of trading strategies when a company makes lots of money: LTCM experienced huge losses on stable correlations that suddenly changed. As in the case of LTCM, and probably Amaranth, when a company makes lots of money, usually the investors indirectly encourage it to invest even more into the strategy. Ultimately, this can be a bad strategy, not because of the strategy itself, but because at some point the company’s pockets may not be able to sustain the magnitudes of risks taken.

1.2.3.

Market Deregulation

It would not be a “decade in review” chapter without revisiting California. One obvious mistake of the California legislators was assuming that they were in a closed system comparable to that of an island, for example, England, a framework that —by the way— the California utilities strongly supported. PG&E valued their plants under the assumption of a closed market. Despite other experts’ voices, cobwebbed within their wishful thinking about the future, and spurred on by the lack of both research and understanding regarding the rest of the U.S. power markets at the time, both the California legislators and the California utilities decided to utterly ignore the existing nature of power markets in the rest of the United States and instead to bury their heads in the sand and pretend that they were just like England. John Wengler summarized the situation at the time: The California experiment with deregulation made two fatal errors early on. First, they looked to England rather than Ohio for inspiration. Prior to liberalization, the British market was far more centralized than California—their solutions simply could not fit our problems . . . California’s other mistake involved promising lower prices rather than price transparency . . .11


7

The volatilities seen in California should not have been perceived as beyond the possible by any of the California utilities prior to deregulation—but in fact, that is exactly what they were. It is funny that the California utilities and the legislators engaged in a legal battle over whose fault it was in the end, when the truth is that both were equally ignorant and irresponsible. Perhaps PG&E did perform mark to market valuations of their plants prior to selling them—perhaps the problem is that they marked to the wrong market! While certainly one could easily argue that the California utilities should not have been forced to sell their plants in order to encourage market competition, it is also true that California utilities did not perform a proper valuation of their plants taking into account the power market price behavior already observed in other parts of the country, and all the potential market states post-deregulation. The local paper summed it up as follows: “There are a lot of smart people at PG&E, but they aren’t exactly creative,” said Harry Snyder, a lawyer for Consumers Union in San Francisco. “So Duke Power, Enron and the other independents came in and ate PG&E’s lunch.” “Those companies paid three times the book value for those PG&E generating plants and PG&E thought they were taking these guys,” he said. “In fact, those independents knew the value of those generating plants, and PG&E sold off way too many of them, so they couldn’t govern their own destiny.”12

You would think that PG&E would have thought twice about why it is that independents were coming in and offering them three times the book value for their plants. PG&E could not possibly have understood the possible energy prices to be seen soon after in California without understanding energy market price behavior outside their own region, just over the Rocky Mountains. Apparently they did not understand price volatility (hence how could they possibly value their assets correctly?) and so—perhaps—it is no surprise that they did not even understand demand volatility. Good intentions, unfortunately, are not enough. Risk management requires both accepting and understanding market price behavior. Sometimes it is good to be the first, but quite often it is much better to be the second, or third, or fourth . . . Eastern European and Asian markets are opening up to power trading, and have the benefit of

8

Energy Risk

learning from mistakes other countries have made in the process of deregulation. Still, it is no easy task: The fledgling electricity markets of central eastern Europe have developed rapidly since liberalization spread to the region at the start of the decade. Some of the incumbent state-run utilities developed sophisticated trading teams from scratch in a remarkably short period of time, and the area has attracted investment from large western European utilities as well . . . The greatest frustrations surround cross-border trading, the lack of transparency around transmission system operators (TSOs), and a prevalence of long-term contracts.13

1.3.

ENERGY MODELING 1.3.1.

Energies Are Still Unique

The “old days” of energy markets saw quite a few agnostics regarding price mean reversion and multifactor energy price modeling. It appears that, over time, the market has more or less accepted the notion that energy markets are indeed different from the financial markets (i.e., interest rate, FX, and stock markets) in some fundamental ways, that indeed the energy markets appear to be driven by more than a single factor (such as spot), and that there is such a thing as mean reversion present in the energy price behavior: Unlike the financial markets, where current and future prices are linked, it is not possible to determine forward electricity prices from present ones. It is also not safe to assume a relationship between forward prices at two adjacent dates, or to rely on price changes between those dates occurring in a predictable manner. To make matters more complex, pricing methods used in the financial markets often break down when applied to the electricity markets . . . . . . Pricing methods must take into account factors such as the meanreversion behavior of electricity prices, price spikes, and non-constant volatility. Modeling future prices via stochastic processes represents one way of including these factors in calculation.14 1.3.2.

Model Complexity

All markets can be quite complex, and even the simpler markets can have extremely complex option valuation problems to solve. In all these


9

cases, modeling begins with discerning between the important market realities and those that can be assumed away or perhaps handled within the model implementation stage. The process of understanding the market realities and simplifying them in order to come up with models that can be feasibly implemented on a trading floor for a valueadded use by traders becomes all the more important the more complex the market behavior. Energy markets perhaps offer the biggest challenge of all. This is perhaps one reason why simulations are so popular in energy markets. Experts today clearly appreciate elegant simplicity, as demonstrated in the following statement by Robert Bothwell in a GARP magazine interview: Lacking intuitive understanding of which aspects of a problem are important and which may be safely ignored, modelers often err on the side of caution and build excessively complex models.15

Another expert, in the same interview, summarizes how simulations and other complex methodologies incorporating numerous degrees of freedom have their limitations: . . . there comes a point when additional complexity begins to reduce rather than enhance a model’s utility. More complex models are slow, and this makes them less useful for real-time decision making. Complexity also increases the risk that the model contains errors. Finally, and most importantly, complexity makes it more difficult to understand why the model produces the results that it does. In other words, it contributes to the black-box syndrome.16

Where does that leave us, because energy markets are unquestionably complex and we will always need complex models? Ultimately, there is no way of getting around the basic problem of understanding which market drivers are the most important and should be included within the modeling process and which can be treated within the implementation stage. Also, once the models are built, we need to make sure that they appeal to both the intuition of the traders, the intellect of the quants, and the proof-hungry skepticisms of true engineers.

1.3.3.

Quants vs. Traders vs. Reality

The walls dividing quants and traders are often quite thick. The different “languages,” the spectrum of response spanning the instinctive and

10

Energy Risk

the intellectual, and perhaps most importantly, the ignorance of both “sides” regarding the value of knowledge on the other side of this wall contribute to building these walls quite thick. The paradox of the situation is that the higher the market complexity, the more transparent these walls must be in order to most realistically model the market. In an article entitled “Quant doublespeak,” Neil Palmer put it this way: If you are on the lookout for obscure and perverse language, then look no further than the theory of option pricing. Not even George Orwell could have devised a more intimidating form of doublespeak. It’s ironic that the principle underlying this terrifying subject—first articulated by Black, Scholes & Merton—is beautifully simple . . . This language has come from finance, and now we’re using it in energy. If you think in terms of a nice easy slogan like “risk-neutral expectation,” then you might just be forgetting about what really lies behind it. In fact, there are some extremely strong assumptions behind the idea of pricing via this method. Continuous trading with no costs is a key requirement. There are many energy markets where this is a remote dream. . . . If you can’t directly compete for attention with the hotshot traders, maybe it pays to be just a little mysterious.17

1.4.

ENERGY HEDGING AND RISK MANAGEMENT 1.4.1.

Adding Financial Products to the Hedging Mix

The decade of energy trading has seen continued use of physical storage as a means of hedging energy exposure, but also an increase in both the availability and use of cash-settled products: . . . Because of high natural gas prices, the summer–winter spread— injecting natural gas into storage when prices are low and withdrawing in winter when prices are high—is not profitable, [Glen] Sudler says . . . With natural gas prices forecasted to remain high, more utilities are buying their own storage facilities, enabling them to swing in and out to meet load demands as needed. The financial markets, says [Keith] Kelly, offer an alternative to physical storage capacity to hedge natural gas prices. Signing long-term contracts and buying storage assets are still mainstays. But the market has more actively traded storage spreads,


11

spark spreads, swing options and basis trades around these physical plays. On the gas side, American Gas Association (AGA) reports local gas distribution companies use financial derivatives to hedge 70% of their physical portfolio, up from 55% just two years ago [2002].18

1.4.2.

Risk Management: A Profitable Business Function?

Risk management should be just risk management. To expect risk management to be a profit function is to disguise other trading strategies under the guise of risk management. This should be a very scary practice for any company. And yet, the idea of risk management adding value outside of reducing risk keeps popping up in the market place every now and then: Utilities and regulators often disagree over the purpose of energy price risk management . . . should utility hedging simply smooth out rates for consumers or actively reduce them?19

The risk management experts, however, know better: “The suggestion that utilities should try to beat the market is just plain wrong,” says a risk manager at another Canadian utility who asked not to be named. “It goes against the purpose of risk management.”20

There is also often a notion of putting a hedge at a right or wrong time encouraging the idea of risk management for profit, resulting in regulators encouraging position taking based on market timing: But a regulator has to monitor hedging programmes continuously to ensure they make sense with regard to costs and rates, says Gerry Gaudreau, secretary to the MPUB (Manitoba Public Utility Board). Moreover, if a utility uses a mechanistic hedging programme at times of unusually high gas prices—such as now—it may be locking in prices that are too high. As a result, the company should use its discretion, he says: a large utility that delivers a lot of gas to its customers should be able to take a relatively educated price view.21

Although it certainly is true that there are times when the hedges are less expensive than other times, it should also be true that the market is pricing all the real market costs in its hedges and who can say that high prices cannot go even higher (or low prices go even lower)?

12

Energy Risk

Therefore, to think that a company is better off costwise to not put on a hedge is to say (1) that the company knows better than the market, and (2) that the company would rather take the “price” of market risk than the price of the hedge. In the case of the first point, if every company knew better than the market, then the market as a whole would converge towards this greater knowledge (making it ultimately impossible for any particular company to know better than the market). The second point is based on whether or not the company has based this fact on the cost analysis of risk or pure conjecture. There is no question that there is such a thing as good hedging vs. bad hedging (as is discussed in Chapter 13 on issues of correlations and proper hedges), it is also true that certain companies do not possess core competencies for speculation. You would be asking for big trouble by asking a utility or a non-trading corporate function that aims at reducing risk to take on speculative views on the market!

1.4.3.

Hedging for the Little Guys

Even the little guys are becoming a part of the traded energy markets. Innovations in price hedging are beginning to reach all the way to the small individual users: Gasoline retailers are exploring ways of enabling customers to personally manage their gasoline price risk, through prepaid cards and price caps. From October this year, Gulf Oil will allow consumers to buy prepaid cards for a fixed amount of gasoline at a prevailing market price—so if the price dips to $2, for example, customers can go online and buy 200 gallons for $200, to be delivered at any time from any Gulf gas station. The firm will also allow holders of its branded credit cards cap their gas prices on any gas purchases made with the credit card, in exchange for a nominal per-gallon fee. Gulf Oil will track what prices its customers are locking in, and then hedge this exposure in the futures market.22

1.4.4.

Assets as Hedges

Although physical assets have always been the necessary and therefore natural hedge for the energy service providers, the energy houses


13

have had to respond to the changing market conditions through risk management via their asset base: Following the demise of Enron, companies that retreated from trading to more asset-based activities, such as generation, are now faced with different market circumstances than a few year ago, when fuel and electricity markets were less volatile. The withdrawal of many companies from trading, combined with considerable M&A activity, has created a shortage of liquidity in many markets. This, in turn, has been partly responsible for increased prices and volatility, and the inability to manage risk through the markets. Some retail suppliers of energy responded several years ago by vertical integration into upstream generation or production activities, which may offset supply risk to some degree but increases the challenge of portfolio management.23

Of course, the financial trading houses conveniently had the capital to buy some distressed generation assets following the post-Enron era. (It reminded me of the movie “It’s a Wonderful Life” when Mr. Potter went about buying bank shares during a panic.) There’s even talk about the financial houses owning their own nuclear power plants. This evolution makes sense because the volatility of power markets carries so much risk that deep pockets and generation may be necessary to stabilize portfolios through the highs and lows.

1.4.5.

Regulatory Response to “Bad” Stories

Huge profits do not occur without huge risks—you can never make as much money in arbitrage as you can in pure bets—but if you know what you are doing, you are taking very little risk of the downside. When companies report large losses, it catches the eye of politicians and they often have an instinctive rather than educated response. The Wall Street Journal reported in 2006: Congress, meanwhile, is jumping in to debate whether hedge funds are to blame for all the volatility.24

Illiquidity contributes to volatility . . . just take a look at the longerterm natural gas futures prices on NYMEX in the late 1980s when the

14

Energy Risk

futures just began trading. Volatility for the longer-term futures was comparable to the short-term futures contracts, in the 60–70% range! Perhaps this is nothing impressive right now, but back then this was a huge volatility driven directly by illiquidity. As soon as the market saw more participants, the volatility of these longer-term natural gas futures dropped down to the 15% range, where, in fact, it remained for quite a few years. The bottom line is, illiquidity adds volatility, not due to the actual price behavior, or market fundamentals, but due to the lack of price discovery (or rather, counterparty discovery). More participants means more liquidity, means less volatility due to price discovery. Energy markets have enough volatility to go around without issues of illiquidity, thank you very much! For Congress to now jump on hedge funds for helping markets reduce illiquidity (for their own gain, of course) is absolutely the wrong response.

1.5.

CONCLUSIONS I wrote the first edition of Energy Risk because my publisher agreed that the fledgling energy market needed its own guidebook. Now that we are ten years down the path, we agree that the guidebook is still needed but with the requisite updates. The balance of this book explores the specifics of modeling and managing the complex task of quantitative and fundamental analysis of the energy derivatives and risk management market. We will follow a progressive path. Chapter 2 introduces the fundamental supply and demand market drivers. ● Chapters 3 and 4 cover the type of modeling principles and skills demanded by the complexities of the energy markets. ● Chapters 5 and 6 describe how to model the underlying price behavior of the spot and forward price markets. The behavioral characteristics of these markets act both as an end to themselves and as valuable inputs for the quantitative analysis covered in the remaining chapters. These chapters were extensively expanded to include some new ideas, such as on distribution analysis, and updated with new market data. ● Chapter 7 is an entirely new chapter. It goes into the details of building marked-to-market forward price curves. At the time ●


15

the first edition was published there was not enough market data to warrant such a chapter, but now it is much needed. ● Chapter 8 explains volatility and introduces a comprehensive method for its modeling. ● Chapters 9 and 10 cover energy option pricing modeling and implementation. ● Chapter 11 is another new chapter discussing the many different types of energy options. Since the publishing of the first edition of this book, traded energy options markets were still in their infancy in comparison to today. ● Finally Chapters 12, 13, and 14 pull together the fundamental and quantitative analysis of market behavior into the context of risk management and portfolio analysis.

ENDNOTES 1. Gary Zukav, The Dancing Wu Li Masters (New York: William Morrow and Company, Inc., 1979) p. 328. 2. “How Giant Bets on Natural Gas Sank Brash Hedge-Fund Trader,” Wall Street Journal, Dow Jones & Company, September 19, 2006. 3. “Quantifying Energy, BP Statistical Review of World Energy,” BP, June 2006. 4. “Heat Drives Power Demand in Midcontinent,” Platt’s Gas Daily, McGraw-Hill Companies, August 10, 2006. 5. Zachary Simecek, “Weathering the Impact of Stormy Price Hikes,” Energy Risk, June 2006. 6. “Consultant Warns that High Demand Growth Could Strain Power Markets and Add Volatility,” Power Markets Week, October 23, 2006. 7. “Hedge Fund Risk: Insights From a Well-Traveled Mind,” interview with Gloria Pilz, Global Association of Risk Professionals, January/February 2006. 8. Matthew Goldstein, Lauren Rae Silva and Melissa Davis, MotherRock Cries Uncle, August 18, 2006, TheStreet.com. 9. “How Giant Bets on Natural Gas Sank Brash Hedge-Fund Trader,” Wall Street Journal, Dow Jones & Company, September 19, 2006. 10. “What Went Wrong At Amaranth,” Wall Street Journal, Dow Jones & Company, September 20, 2006. 11. John Wengler, “Avoid Monday Morning Quarterbacking in California,” Energy Informer, September 2001. 12. Susan Sward and David Lazarus, “How PG&E Missteps Preceded Crisis,” San Francisco Chronicle, January 22, 2001. 13. James Ockenden, “Growing Pains,” Energy Risk, September 2006. 14. Aarzoo Ahah, Riccardo Anacar and Antony Kakoudakis, “The Price Is Right?” Energy Risk, June 2006.

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Energy Risk

15. “Structured Investment Vehicles: Trends, Truths and Myths of Complex Marketplace,” Interview with Robert Bothwell, GARP Risk Review, May/June 2006. 16. “Structured Investment Vehicles: Trends, Truths and Myths of Complex Marketplace,” Interview with Nels Anderson, GARP Risk Review, May/June 2006. 17. Neil Palmer, “Quant Doublespeak,” Energy Risk, April 2005. 18. Catherine Lacoursiere, “Storing up Trouble,” Energy Risk, September 2004. 19. “A Look in the Rear View,” Energy Risk, December 2005. 20. Ibid. 21. Ibid. 22. “Hedging for Drivers,” Energy Risk, June 2006. 23. Colin Cooper, “Optimal Results,” Energy Risk, June 2006. 24. “How Giant Bets on Natural Gas Sank Brash Hedge-Fund Trader,” Wall Street Journal, Dow Jones & Company, September 19, 2006.

C H A P T E R

2

What Makes Energies So Different? America was changing. I had a feeling of destiny and I was riding the changes. New York was as good a place to be as any. My consciousness was beginning to change, too, change and stretch. One thing for sure, if I wanted to compose folk songs I would need some kind of new template, some philosophical identity that wouldn’t burn out. It would have to come on its own from the outside. Without knowing it in so many words, it was beginning to happen.

Bob Dylan1

2.1.

INTRODUCTION Energy markets remain a relatively new world. In dealing with this extraordinary market environment we need all the skills and experience of other, more mature markets, plus some new ways of looking at market behaviors including via volatilities and price distributions. Our learning path should begin with the market, encompass study and research of market variables, in order to ultimately loop back to the market, hopefully with new understanding and knowledge. Throughout this process, our emphasis should be on the managerial and implementation aspects of “quantitative analysis.” Quantitative analysis creates models that reflect market behavior in order to support trading in the actual market. If this book helps a novice build a first forward price curve, or inspires an expert to update a favorite model, then this book, Energy Risk, will have served its purpose. The origin of quantitative analysis is rooted in the concept of “risk” itself. Since the days of the Romans, and perhaps even before then, people have “hedged their bets” against the unknowns of the future by entering into primitive futures and options contracts. Intuition, common sense, and experience probably served as the first quantitative tools for setting prices. (All three remain equally valid tools today!) 17


18

Energy Risk

Energy markets follow the same impulses: energy producers and users alike wish to hedge their exposure to future uncertainty, or to obtain a particular risk/return strategy. Fortunately, in valuing these products, our task will be easier than that of the Romans, thanks to modern mathematics and statistics, and the advent of computers.

2.1.1.

Quantitative and Fundamental Analysis

In addition to quantitative analysis, a second discipline forms the basis of derivatives valuation and risk management: fundamental analysis. Fundamental analysis is an attempt to understand and describe market behavior in terms of the economics of supply and demand. Fundamental analysts attempt to identify, measure, and understand the relationship between the “fundamental price drivers” that cause markets to move up and down.2 Quantitative analysis, on the other hand, attempts to replicate or model market behavior through mathematical models and statistical methodologies. In this book, quantitative analysis plays the leading role, and fundamental analysis contributes to the motivation and the intuition behind the models. The interplay between fundamental and quantitative analysis is very much like the interplay between macroeconomics and microeconomics. Macroeconomics is the study of the forces and causes of economic fluctuations and their relationships. Microeconomics, on the other hand, is the study of the behavior of individual consumers and firms. The two are very much related, as assumptions about the economy depend on the assumptions about the individual players within the economy. A thorough understanding of macroeconomics requires a thorough understanding of microeconomics, and vice versa. Similarly, although fundamental analysts try to understand general price drivers, the quantitative analyst imposes the condition of rational market players who will not allow price arbitrage, resulting in an efficient marketplace. In this sense, fundamental analysis can be likened to macroeconomics, and quantitative analysis can be likened to microeconomics. This book attempts to describe quantitative issues and techniques with very much a fundamental flavor. Every quantitative approach and result is evaluated against the standard of consistency with the fundamental drivers of a marketplace. Therefore, understanding both the quantitative methodologies and the fundamentals of a marketplace is extremely important.


2.2.

19

WHAT MAKES ENERGIES SO DIFFERENT? Energy markets are young maturing markets continuing their transformation by the derivatives and risk management industry. In comparison, the money markets stand as mature markets with relatively few modeling mysteries left to conquer. Bookstores already offer full shelves of excellent introductory and specialized books on fundamental and quantitative analysis for the mature financial markets. Energy markets are slowly catching up. At the time of the first publishing of Energy Risk, there were no other energy market books available. Now, there are a number of excellent energy-specific books. Energies remain very different from money markets (Table 2-1). Fundamental analysis tells us that energy markets respond to underlying price drivers that differ dramatically from interest rates and other well-developed money markets. More importantly, quantitative analysis tells us that the differences in fundamental price drivers can exert a dramatic domino effect as they are applied to pricing and hedging models. The remainder of this chapter will introduce some of the energy market’s fundamental price drivers and cite several examples of

T A B L E

2-1

What Makes Energies Different? Issue

In Money Markets

In Energy Markets

Maturity of market Fundamental price drives Impact of economic cycles Frequency of events Impact of storage and delivery; the convenience yield Correlation between shortand long-term pricing Seasonality

Several decades Few, simple High Low None

Relatively new Many, complex Low High Significant

High

Lower, “split personality”

None

Key to natural gas and electricity Varies from little to very high Lower Decentralized Majority of contracts are relatively complex

Regulation Market activity (“liquidity”) Market centralization Complexity of derivative contracts

Little High Centralized Majority of contracts are relatively simple

20

Energy Risk

fundamental differences between the energy and money markets. Although these examples skim the surface and the individual chapters provide the necessary details for true understanding, we offer these examples in the spirit of market-driven modeling that we hope permeates the entire book.

2.3.

ENERGIES ARE HARDER TO MODEL The interest rate and equity markets are “lucky.” Their fundamental drivers number relatively few and easily translate into quantitative pricing models. For example, the deliverables in money markets consist of “a piece of paper” or its electronic equivalent, which are easily stored and transferred and are insensitive to weather conditions.3 Energy markets paint a more complicated picture. Energies respond to the dynamic interplay between producing and using, transferring and storing, buying and selling, and ultimately “burning” actual physical products. Issues of storage, transport, weather, and technological advances play a major role. In the energy markets, the supply side concerns not only the storage and transfer of the actual commodity, but also how to get the actual commodity out of the ground. The end user truly consumes the asset. Residential users need energy for heating in the winter and cooling in the summer, and industrial users’ own production continually depends on energy to keep the plants running and to avoid the high costs of stopping and restarting them. Each of these energy market participants—be they producers or end users—deals with a different set of fundamental drivers, which in turn affect the behavior of energy markets. These problems lead directly to the need for derivatives contracts. Nothing even approaches these problems in money markets. What makes energies so different is the excessive number of fundamental price drivers, which cause extremely complex price behavior. This complexity frustrates our ability to create simple quantitative models that capture the essence of the market. A hurricane in the Gulf of Mexico will send traders in Toronto into a tailspin. An anticipated technological advance in extracting natural gas could be influencing the forward price curve. How would you go about capturing these kinds of resulting price behaviors into a quantitative model that is also simple enough for quick and efficient everyday use on the trading desk?


21

Figures 2-1 and 2-2 show historical prices for Massachusetts Hub power prices for both the On-Peak and Off-Peak markets (see Chapter 7 for details on contract specifications for both On-Peak and Off-Peak markets). As you can see from the several years of price data in these figures, power prices are not shy in jumping to very high levels during events. Generally speaking, these are upward jumps followed by quick mean reversion back to a more reasonable price level. Also, note that winters and summers tend to be periods of more probable (and serious!) price spikes. Note that the off-peak price history appears quite a bit more volatile, day to day, than even the on-peak power prices, which is counterintuitive; off-peak power is for delivery during the hours of the business day when the demand is less (hence the name “off-peak”) and includes around-the-clock (i.e., all hours of the day) delivery of power on Saturdays, Sundays, and holidays. The large amounts of price volatility we see in Figure 2-2 are a result of the fact that we are “mixing” full days of delivery on weekends and holidays with only a segment of the day for delivery during business days. By separating the two, we obtain Figure 2-3, which shows the price history of off-peak prices only on business days, and is clearly less volatile.

F I G U R E

2-1

Massachusetts Hub On-Peak Power: Sample Price History

22

Energy Risk

F I G U R E

2-2

Massachusetts Hub Off-Peak Power: Sample Price History

F I G U R E

2-3

Massachusetts Hub Off-Peak Power: Sample Price History with Weekends Excluded


F I G U R E

23

2-4

Massachusetts Hub Hourly Power: Sample Price History Averages across Hours of Delivery

Finally, Figure 2-4 shows the historical averages across the few years of sample data of hourly power prices. As you can see from these graphs, not only do power prices exhibit calendar year seasonality, but they also show a strong price term structure across the hours of the day.

2.4.

MARKET RESPONSE TO CYCLES AND EVENTS In the broadest sense, the traditional financial markets demonstrate an almost seamless transition from fundamental to quantitative analysis, but energies do not. The relative impact of economic cycles and frequency of events in the two markets demonstrates this difference. Generally speaking, most economic markets appear to move “up” and “down” around some sort of equilibrium level. This equilibrium level could be a historical interest rate, return on equity, or commodity price. The equilibrium may also be called the “average” or “mean” level. The process of a market returning to its equilibrium level is termed “mean reversion.” Mean reversion will be a recurring theme in this book, because it describes a critical difference between the energy and financial markets. Interest rate markets exhibit relatively weak mean reversion. The actual rate of mean reversion in interest rates appears to be related to

24

Energy Risk

economic cycles, hence fundamental price drivers. The state of the economy as a fundamental driver can be directly translated into financial models through the inclusion of mean reversion. In the case of energies, however, we see stronger mean reversion, and for dramatically different reasons than those that apply to interest rates. The mean reversion in energy commodities appears to be a function of either how quickly the supply side of the market can react to “events” or how quickly the events go away. Droughts, wars, and other news-making events create new and unexpected supply-and-demand imbalances. Mean reversion measures how quickly it takes for these events to dissipate or for supply and demand to return to a balanced state. The Gulf War in the late 1980s and early 1990s, for example, greatly affected crude oil prices. The market forward prices of crude oil contained information on how long it would take the production side to respond to the sudden imbalance between supply and demand. Spot and short-term forward prices spiked, but longer-term futures remained relatively stable. In this case, the mean reversion—as exhibited in forward prices—was tied to how quickly the production side could bring the system back into balance. In another example, summer heat waves over the years have caused electricity prices to jump to multiples of their average price levels. However, in many of the weather-caused events, temperatures spiked only for several days and prices rapidly reverted to equilibrium as the temperatures reverted to their more normal levels. In this case, the mean reversion was related to the dissipation of an event. Figure 2-5 shows quite a few years of natural gas spot price history. Different events tend to have different effects on an energy market. In the case of natural gas, we have observed quite a few eventful situations during this new century. Natural gas prices in the United States appeared to abandon their long-term historical levels for much higher prices under the occasional effects of storage concerns, the longer-term supply problems caused by the Iraq war, seasonal events such as extremely damaging hurricanes, and, perhaps most importantly, the emergence of new agressive players spiking the natural gas prices to a level that could not possibly be understood in the prior years of trading. Natural gas spot prices have always exhibited a high volatility (Figure 2-6). (Many of the natural gas contracts traded are contingent on monthly price averages; the volatility of these monthly-based contracts is smaller, diluted by the averaging effects.) Given the levels of


F I G U R E

25

2-5

Natural Gas: Sample Price History

these historical volatilities you might conclude that this is a market where just about anything could happen! Understanding the possible “anythings” becomes crucial to the risk management of a portfolio in natural gas. F I G U R E

2-6

Natural Gas: Sample Historical Volatility

26

2.5.

Energy Risk

IMPACT OF SUPPLY DRIVERS Energies function with supply drivers that do not exist in money markets: production and storage. Consider the issue of longer-term effects, which have to do with expectations of market production capacity and cost in the long run. Effects of expectations of improvements in the technology of drawing natural gas from the ground will not be seen in the historical data, but—if we are lucky—may be expressed by knowledgeable traders in determining forward prices. Their views would be captured through the levels or yields of long-term forward prices. Similarly, the effects of overcapacity in electricity markets, and how long the overcapacity is expected to last, impact the price over a longer period of time. This “storage limitation” problem creates volatile day-to-day behavior of varying degrees for electricity, natural gas, heating oil, and crude oil. Another consequence of limited storage is that although the spot prices exhibit extremely high volatility, the forward prices show volatilities that decrease significantly as the forward price expirations increase. The latter volatility characteristic has to do with the fact that, in the long run, we expect the supply and demand to be balanced, resulting in long-term forward prices that reflect this relatively stable equilibrium price level. Ultimately, when discussing energy commodities, we are forced to confront the issue of storage capacity. Storage limitations cause energy markets to have much higher spot price volatility than is seen in money markets. Electricity markets represent the extreme case of storage limitation issues. In fact, electricity cannot be readily stored.4 When power plants reach maximum allowable base-load and marginal capacity, there is no more “juice” to go around. While there is no more new electricity to sell, the same unit of electricity may be bought and sold, and hence you may still be able to obtain market price quotes. It should not come as a surprise that such extreme market conditions can cause electricity prices to easily reach levels in multiples of mean price levels. As can be seen from Figures 2-7 and 2-8, power spot prices are even more volatile than natural gas prices, with short-term volatility hitting over 1000% at times! In this sample power market the average on-peak power volatility over the few years of historical data measured 207% and the off-peak prices had an average volatility not too far off at 188%. By comparison—as you can see from Figure 2-9—the average hourly power prices are even more volatile.


F I G U R E

27

2-7

Massachusetts Hub On-Peak Power: Sample Historical Volatility

F I G U R E

2-8

Massachusetts Hub Off-Peak Power: Sample Historical Volatility (Using Weekday Price Returns)

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Energy Risk

F I G U R E

2-9

Massachusetts Hub Hourly Power: Average Sample Historical Volatility (2003–2006)

2.6.

ENERGIES HAVE A “SPLIT PERSONALITY” From the big picture, the issue of storage accounts for energy prices exhibiting a “split personality.” Energy prices are driven both by the short-term conditions of storage and by the long-term conditions of future potential energy supply. Energy forward prices reflect these two drivers, resulting in short-term forward prices with very different behavior from long-term forward prices. Figures 2-10 and 2-11 show a sample historical behavior of the one-month and one-year forward price points of the West Texas Intermediate (WTI) and natural gas forward price curves, respectively. Short-term forward prices reflect the energy currently in storage, and long-term forward prices exhibit the behavior of future potential energy, that is, energy “in the ground,” capturing the energy markets’ “split personality.”5

2.7.

IMPACT OF DEMAND DRIVERS If supply constraints can “shock” the system, demand exerts its own fundamental price drivers. In energies, demand drivers introduce the issues of convenience yield and seasonality that have no parallel in money markets.


F I G U R E

29

2-10

NYMEX WTI Futures’ Prices 1992–1996

2.7.1.

The Convenience Yield

On the industrial user side, the explicit purpose of derivative contracts is to keep plants running. These industrial users drive the market value of convenience yield. Factories seek to minimize their cost of production by avoiding the cost of shutting down and restarting the factory due to high prices or lack of available supply. (In a sense, minimizing

F I G U R E

2-11

NYMEX Natural Gas Futures’ Prices 1992–1996

30

Energy Risk

price risk can be related to this function.) This urgency in maintaining production gives the industrial users an incentive to pay a premium to have the energy necessary to run their plants delivered now, today. This is not because they are being financially inefficient. Quite the contrary; they are factoring in the opportunity cost of having their production stopped while waiting around to get a better deal on energy or waiting for energy to become available. The premium they are willing to pay (or not, depending on the immediate abundance of supply relative to demand) is factored into something called the “convenience yield.” An analogy can be made between the concept of convenience yield and a stock dividend. Consider a shareholder who buys the stock prior to the ex-dividend date. When the dividend is paid, the new shareholder will capture the value of that dividend. But that shareholder would have had to pay a higher price—relative to the price paid post ex-dividend date—which would have included the dividend value. Similarly, the industrial users capture the value of their own production by purchasing energy before they run out of their supply. In doing so, they willingly pay a higher price for this immediate energy supply in order to capture their own, very specific in-house dividend. In the end, the markets will, given specific industrial user demand, reflect a premium of near-term forward prices relative to the longer-term forward prices. To be more specific, the convenience yield is the net benefit minus the cost—other than financing costs—of holding the energy “in your hands.” The benefits include the user-specific value defined above, and the costs include storage.

2.7.2.

Seasonality

On the demand side we have to consider the significant seasonality effects of the residential users. Aggregate residential demand creates seasonality. For example, the United States consumes heating oil mostly during the winter; hence, heating oil prices tend to peak during winter and then drop to their annual lows in the summer months. Electricity, on the contrary, powers air conditioners in the summer months and is used less during winter for heating; its prices tend to reach highest peaks during the summer months, with semi-annual humps during the winter.6 The relative highs of the summer and


31

winter peaks—as clearly exhibited within the electricity forward price curves—are a function of the geographic regions within the United States. These seasonality effects can be seen and measured not only through historical spot price data, but also by observing the forward price markets.

2.8.

REGULATION AND ILLIQUIDITY When modeling energies, we must always remember their relative youth in terms of derivatives and risk management. Natural gas deregulated over a decade ago, and Eastern European and Asian governments are deregulating electricity as this book is being re-written. Even the relatively older markets of heating oil and crude oil took root in the 1980s and continue to evolve in terms of theoretical sophistication and contract complexity and standardization. Although the money markets took decades to evolve, energies are in some ways replicating this evolution in a shorter period. Clearly, lessons from deregulated markets have accelerated the trip up the learning curve. Unfortunately, human character flaws have slowed down the process somewhat. The California crisis and the Enron scandal had wide-felt market effects, dampening energy markets’ development for a period of time.

2.9.

DECENTRALIZATION OF MARKETS AND EXPERTISE When one thinks of financial markets, Wall Street shines at the center. Companies throughout the nation list their stocks on the New York Stock Exchange, and New York also hosts most of the major U.S. banks. Of course, cities outside the Empire State play important roles, but major local and regional banks and financial institutions still turn to Wall Street, Chicago, and other major trading centers to hedge their portfolios. Thus, the financial markets are essentially centralized in terms of location, capital, and expertise. Energy markets, on the other hand, are highly decentralized. To be sure, Houston serves as a mecca, as does Calgary. Energy producers and end users, however, spread from sea to shining sea. To whom does

32

Energy Risk

a Midwestern utility turn to hedge their price risk? If their risks are localized, chances are that their hedges will also be localized. Although many producers and end users may actively use futures contracts in New York and Kansas City, these contracts represent prices at specific delivery points that may behave very differently from the local market being hedged. Decentralization introduces geographic “basis risk,” which is unique to energies. In financial markets, today’s dollar is worth a dollar anywhere in the country. In energy markets, price depends on location. A megawatt of electricity is priced according to delivery point; the same holds true for natural gas. Location is a fundamental driver of price. At the most human level, even the jobs of energy risk managers are more decentralized than in financial markets. Throughout North America, large end users and even moderate sized utilities maintain energy purchasing officers and wholesale analysts at the least, and full trading and risk management staffs at the extreme. Even the people working the energy desks are diverse. These professionals come from a wide variety of backgrounds, including trading, risk management, corporate treasury departments, and even engineering. Not surprisingly, their voices often conflict, sending mixed messages (and occasionally mis-pricing) to the market. These market inefficiencies are being resolved with time, of course, as growth of market understanding (and knowledge transfer, as exemplified by a growing number of energy books) occurs.

2.10.

ENERGIES REQUIRE MORE EXOTIC CONTRACTS The final factor that makes energies so different can be found in the type of financial contracts required by the end users of derivatives. In interest rates, contracts tend to be standardized and relatively easy to model. For the most part, end users of financial derivatives find that relatively simple forwards, swaps, caps, floors, and swaptions suit the majority of their needs. (Not surprisingly, these contracts are made in highly liquid financial markets as compared to energies.) The market even uses the term “vanilla” for these simple contracts; traders immediately term non-vanilla contracts as “exotic.” What makes energy contracts so different is that energy’s typical “vanilla” contract would be


33

considered an “exotic” contract in mature money markets. Due largely to the needs of end users, energy contracts often exhibit a complexity of price averaging and customized characteristics of commodity delivery. The combination of a relatively young derivatives market in development, supporting very sophisticated contracts, presents a terrific challenge to quantitative analysts and risk managers in the energy markets.

2.11.

CONCLUSION Energies differ from nonphysical markets for both fundamental and quantitative reasons. Compared to the traditional markets of interest rates and equities, energies react differently to such fundamental variables as macroeconomic cycles and events. The energy markets suffer from supply-and-demand constraints that dramatically influence both the valuation and management of energy risk. The differences even spread to the company level, where firms that would be considered small by financial market standards must still support trading and risk management operations never seen in like-sized banks. In summary, this chapter introduced energy derivatives and risk management through a comparison of the quantitative differences between energy and money markets. The markets also share many characteristics. The main outcome of these parallel differences and similarities is that the energy markets demonstrate a “split personality.” Energies exhibit some behaviors of traditional financial markets, in particular within long-term price behavior, but at the same time they have their own unique and challenging behavioral intricacies.

ENDNOTES 1. Bob Dylan, Chronicles, Volume One, p. 73. (New York: Simon & Schuster, 2004). 2. Hall & Taylor, pp. 3, 4. Macroeconomics: Theory, Performance, and Policy (New York: W.W. Norton & Company, 1988). 3. However, this was not always so. Remember the gold standard? In the gold standard days, the interest rate markets acted much more like today’s energy markets than like today’s interest rate markets.

34

Energy Risk

4. Water reserves do represent a form of potential electricity storage for hydro plants; several utilities employ off-peak power to pump water up to a reservoir, only to reverse the flow to capture the potential energy during peak periods. 5. “In the ground” is used here as an expression of speech. In the case of electricity, it is not that simplistic. 6. Ironically, most of the residential demand remains in the regulated portion of electricity generation, although this is currently changing.

C H A P T E R

3

Modeling Principles and Market Behavior “Pooh’s found the North Pole,” said Christopher Robin. “Isn’t that lovely?” Pooh looked modestly down. “Is that it?” said Eeyore. “Yes,” said Christopher Robin. “Is that what we were looking for?”

A. A. Milne1

3.1.

THE MODELING PROCESS Modeling market behavior should be approached like any business: with a good amount of common sense. It should not be some mysterious process that Ph.D.s perform in isolation, with no view of the overall trading business goals. The full energy team of managers, traders, quantitative analysts, and engineers should be able to understand the basics of modeling principles and market behavior. This way, modeling can become a well-defined process, with goals and procedures that are discussed, set up, and agreed upon by several key players in a company structure, just like any other business branch of a company. The first step in getting the full energy team to communicate is to define the modeling process, which should include both trader insights about the markets and expert insights about quantifying and valuing the products in that marketplace. In the spirit of developing a standardized language that both traders and valuation experts can use to better define the modeling business goals, this chapter will define the basics of modeling and some common-sense requirements that the modeling process ought to satisfy.

35 Copyright © 2007 by Dragana Pilipovic. Click here for terms of use.

36

3.2.

Energy Risk

THE VALUE OF BENCHMARKS Modeling is often left to itself in its struggle to arrive at pricing models that traders can use. The beginning of the modeling process should consist of an analysis of the available models and their appropriateness for the particular product. Hence, the beginning should be the benchmarking between active market behavior and the modeling choices, resulting in a final choice of a model, given possible implementation constraints. The middle should be the actual development of the chosen model, and the end should be the implementation of the chosen model. The most difficult and also the most important part of this process is the first, the beginning. If the model chosen is not appropriate for the product, given the market in which the product is traded, then the last part of the process, the implementation, is likely to drag on—sometimes for months and even years. Quite often the critical first step of the process, the model benchmarking, is not performed. This can be a very costly mistake. Although the company is paying its valuation experts a good deal of money to finish the long-awaited implementation of the choice model, it is also paying a price for not being able to participate in the trading of the product because the traders cannot yet price it. It is often such poor modeling management (and poor management in general) that results in the traders coming up with their own—however simplistic and maybe even inappropriate—spreadsheets for pricing products. What we are really talking about here is the cost a trading business has to pay for not benchmarking and testing between the models in the laboratory before bringing them out onto the trading desk. If you were to buy a new suit, and you decided to spend a good amount of money on it, surely you would shop around and try on different suits for fit and look? Then why would a company that wants to invest a good amount of money in a modeling methodology not do the same?

3.2.1.

Diffusing Personalized Attachments to Models

I would like to discuss an important issue, which I like to refer to as the “my model, your model” problem. This problem often arises in trading


37

companies that have invested money in research development and there is more than one modeling expert, but each is driven by a separate system of beliefs about modeling. Hence, it would not be surprising to find these valuation experts in what might appear as lethal warfare with no real means of conclusion. Unfortunately, the valuation experts, just like most of us when it comes to something that we know a great deal about and have been working on for years, tend to take the modeling issues very personally. (As hard as I try to be objective, I remain aware of this weakness in myself.) The problem is not that the experts might have different opinions; in fact, this is rather a good thing, as they could probably learn quite a bit from each other. The problem is that they have not agreed on modeling benchmarks and have had no help from the trading or management sides in deciding what benchmarks really ought to be used, given the trading strategies and business goals of the company. Even worse than the “my model, your model” problem is the problem of having a single expert who has a favorite model that the company decides to implement without any benchmarking and testing. The typical story goes as follows: The expert’s favorite model is implemented, but because it might not be appropriate for the market in question, any implementation and new product problems are dealt with using “modeling Band-Aids.” The resulting valuation system very quickly becomes cumbersome if not impossible to use, not to mention that the cost of maintaining it can become quite high. As well as introducing modeling benchmarks, the trading organization must also approach the modeling side with the spirit of always searching for a better understanding of the marketplace and its products. This means that managers, traders, and valuation experts should form a team, which provides a framework for sharing knowledge, and sets the team’s valuation goals, including determining the benchmarks for deciding on methodology routes. With all the above said, I recognize that this book in fact introduces you to one particular view on modeling. However, if the book achieves its purpose, you will not walk away from it thinking about the author’s modeling views. Instead, you will walk away empowered to form your own views and you will encourage others around you to do the same.

38

3.3.

Energy Risk

THE IDEAL MODELING PROCESS The recipe for efficient modeling as applied to a trading operation includes five steps: 1. Establish corporate goals that are within the context of the

2.

3.

4. 5.

risk/return framework and are expressed through the risk management policy (see Chapters 12 and 14). Prioritize the market characteristics, which should be captured by the model. Define the benchmarks that describe the market against which any model will be judged. Select the models to be tested and evaluated against the benchmarks. Perform time series analysis and distribution analysis for comparison. The models should be selected in the order of the evaluation results. Estimate the implementation constraints and costs for each model. Finally, identify the model that best satisfies both the market benchmarks and the implementation requirements.

This process would require the participation of at least the producers (the valuation experts and implementers) and the users (the traders). Ideally, the management also has a representative who adds the necessary degree of management support, understanding, and guidance from a higher level of the trading business goals.

3.4.

THE ROLE OF ASSUMPTIONS: MARKET BEFORE THEORY The goal of quantitative analysis is to develop and implement models that reflect market behavior. The process forces us to make some fundamental assumptions about the marketplace and the products we are trying to model. For example, the famous Black–Scholes differential equation for option prices is based on the fundamental assumption that a hedged portfolio consisting of an option, a stock, and a bond must earn the risk-free rate of return because we have eliminated all the stock price risk by hedging the option with the stock. Expressed in terms of partial differential equations, this fundamental assumption


39

forms the basis of quantitative analysis of option prices.2 One nice feature of making unrealistic assumptions is that we can enforce them to simplify a problem, and then later relax the assumption for a more general, realistic solution. Similarly, if we make the fundamental assumption that electricity prices are related to coal and natural gas prices, we can arrive at a solution for electricity prices by assuming that we can create a risk-free portfolio consisting of electricity, coal, and natural gas. On the other hand, we may assume that electricity prices tend to revert to equilibrium price levels, which are determined by supply-and-demand conditions. In these cases, our different fundamental assumptions would possibly lead us to very different solutions. Fundamental assumptions about the marketplace dramatically influence quantitative models developed and implemented for pricing and risk management purposes. Every quantitative result ought to be consistent with the characteristics the fundamental drivers ultimately give to the behavior of the marketplace. Therefore, understanding the fundamental drivers of the marketplace as well as how these drivers are captured in the behavior of the market is extremely important in arriving at models that reflect market reality. Furthermore, in order to arrive at such models we need not only to understand the fundamental drivers of the marketplace, but also to translate these fundamental drivers into pricing models that are both arbitrage free and practical for implementation onto a trading desk. This is by no means an easy task.

3.4.1.

Typical Assumptions

Some typical assumptions are that the markets are efficient and arbitrage free. In money markets, prices are often assumed to be lognormal. Through such assumptions, we define our version of reality. One person may assume that volatilities are constant, while another may assume the volatilities vary along different points of the forward price curve (i.e., have “term structure”). It is also common to assume continuous hedging. Everyone has probably heard of the Black–Scholes option pricing model.3 While valuing European options on stocks, Black and Scholes assumed that stock prices are lognormal and have constant volatilities.

40

Energy Risk

Hence, the randomness that the stock prices exhibit is assumed to always be of the same magnitude. Although most people agree that stock prices are indeed lognormal, most disagree that the volatilities remain constant. In reality, the randomness of the stock price behavior is not constant, and volatilities do possess term structure. Black and Scholes were forced to make this unrealistic assumption because allowing the volatility to also exhibit a nonconstant behavior made solving for the option price far too difficult. After all, one of the best features of the resulting model is its ease of use. This is a terrific example of how unrealistic assumptions might help to create practical solutions. An important consideration in making assumptions is that they be correctly implemented within models. For example, consider the assumption that price mean reversion exists in interest rates. Although most people believe this to be true, when this assumption is implemented within a single-factor model, the result is a volatility term structure that goes to zero over time. Because most interest rate models are, in fact, single-factor, and because the volatility’s term structure does not in fact go to zero over time, we see a potential conflict.4 The lesson that can be learned from Black–Scholes and similar modeling experiences is that some assumptions that reflect market reality should be relaxed in order to arrive at a workable valuation model. However, when we relax assumptions but recognize them to be true in the real world, we should make sure that the valuation methodology’s implementation captures the assumption—even though the valuation methodology itself does not. So, in the case of Black–Scholes, we can correct for the constant volatility assumption by allowing each option price of different maturity to have a different volatility value. Thus we somewhat capture market reality of the marketplace (at least allowing for marked-to-market option prices), not in the valuation model, but rather in its implementation. If we had just ignored the fact that in reality energy volatilities, intermarket and intramarket correlations are not constant, we could be making a grave mistake, perhaps costing the trading operation a great deal of money. Hence, here is an excellent example of why it is very important to have traders communicating with the valuation experts, particularly when the implementation is very informal. If they do not understand the model assumptions, they may end up using the models blindly and without the appropriate checks on implementation assumptions.


3.4.2.

41

Market Variable vs. Modeling Parameter

The above section on the fundamental assumptions and the modeling process leads us directly to the issue of distinguishing between “market variables” and “modeling parameters” (Table 3-1). A market variable is defined by the marketplace, which exhibits randomness and has a certain set of characteristics associated with its behavior. A modeling parameter is assumed to be either fixed or deterministic—we always know (or hope to know) its value. These distinctions have important implications for the modeling process and risk management. Problems might creep in when we treat a market variable as a modeling parameter for the sake of simplifying the modeling process. During product valuation we are often forced to treat what we know are market variables as modeling parameters for the sake of simplicity or cost reduction in model development and maintenance. Remember the problem of volatility that is treated as a modeling parameter by the Black–Scholes option pricing model, but which is in reality a market variable. The give and take between the cost of model development, which would reflect the true behavior of volatilities and the benefit of having a more realistic valuation model, should determine how volatilities should be treated for valuation purposes. A risk manager wants to ensure that such simplifications, or at least their side effects, are minimized. In the analysis of portfolio risks, in the derivation of optimal hedges, and in the calculations of trading book value-at-risk (VAR) numbers, we want to make sure that

T A B L E

3-1

A Sample Choice of Market Variables and Model Parameters Market Variables

Model Parameters

Spot prices Forward prices Volatilities Correlations Discount rates

Time Mean-reversion rate Equilibrium prices Seasonality factors

42

Energy Risk

we capture all the market variables that are out there, regardless of the types of simplifications we might have made during the valuation process. A lack of understanding of the full market risks can be costly. If a market risk remains unrecognized, then the trading book’s true exposure cannot be discussed, and no attempts can be made to manage the risk. In the event that unknown risks appear, you would truly find yourself caught off guard.

3.4.3.

Testing Assumptions Through Benchmarks

Fundamental assumptions can be tested against modeling benchmarks. If we assume that a market is mean-reverting, we could use the implied market characteristic of the forward price volatilities as modeling benchmarks. In other words, we can ask if a particular model implies the same forward price behavior as that seen in the market. For example, if we assume that a market mean-reverts, resulting in decreasing volatilities of forward prices as we allow their time to expiration to increase, we would expect forward price points that are far out on the forward price curve to be much more stable (i.e., less volatile) than those forward prices in the near-term portion of the forward price curve. The consequence of a mean-reverting assumption can be tested against the market forward price behavior. Figures 3-1 and 3-2 show some sample historical volatilities of the NYMEX futures prices (adjusted for rollovers) for WTI and natural gas (these samples are representative of the markets during the 1990s). In modeling these two markets we might require that our models imply the same types of volatility term structures across the forward prices as seen in Figures 3-1 and 3-2. Modeling benchmarks should be prioritized based on their impact on product valuation. For example, electricity spot prices tend to exhibit extremely volatile day-to-day price returns (see Figure 3-3). With a spot price volatility of 263%, we should expect to see a price range consistent with the sample data “growing” over 6.75 years: roughly 66% of the prices should be in the range between $0 and (very roughly) $450. Yet, over this period of time, the values of the spot prices tend to be mostly within a rather narrow range, roughly between $40 and $75


F I G U R E

43

3-1

NYMEX WTI Rolling Futures’ Historical Volatility: A 1997 Sample

(see Figure 3-4). This is an important reality check, which, if not honored within our assumptions, might cause electricity options to be priced unrealistically high. If a trading operation indeed trades a lot of options, then this is an extremely important market characteristic, which needs to be used as a benchmark when deciding between models.

F I G U R E

3-2

NYMEX Natural Gas Rolling Futures’ Historical Volatility: A 1997 Sample

44

Energy Risk

F I G U R E

3-3

New York Hub A On-Peak Electricity Prices (11/99 Through 08/06 Sample; Spot Price Volatility 263%)

F I G U R E

3-4

New York Hub A On-Peak Electricity Price Distribution (11/99 Through 08/06 Sample)


3.4.4.

45

Assumptions and Implementation

In the previous sections, I have argued that some fundamental assumptions about market behavior ought to be relaxed in order to provide us with a working valuation model—as long as these market characteristics can later be captured through implementation. However, which assumptions should be made and which should be relaxed is a very “personal” decision, to be made only upon a detailed analysis of the costs involved and the benefits to be gained by constraining the model to some fundamental assumptions or relaxing those assumptions. To what extent a model should capture the market realities has to do with how sensitive the particular trading business is to the particular imperfections resulting from relaxing assumptions. For example, a market maker that provides risk management services and primarily makes money off the bid-ask spreads might be wise to invest time and money in a sophisticated portfolio analysis methodology that provides minimum-variance hedges. An arbitraging operation, on the other hand, would want to ensure that their valuation models are extremely sophisticated. How could they expect to capture market arbitrage if they are at the same level of sophistication as the general marketplace? Regardless of what the ultimate valuation/implementation strategy is in capturing the fundamentals of the marketplace, it is always the best practice to capture as much of the market reality as possible. This will ensure that as little money as possible is lost on valuation errors and that as little money as necessary is paid for hedging.

3.5.

CONTRACT TERMS AND ISSUES Derivatives are contracts, so we must define the terms of such contracts as they relate to modeling principles.

3.5.1.

Underlying Price or Market

The underlying price (or underlying market) refers to the spot prices (or market) in the case of an energy market where there is no seasonality.

46

Energy Risk

When we are indeed talking about an energy market with seasonality, the underlying price refers to the spot price with the seasonality factors taken away: St StUnd seasonality factors where:

(3-1)

St the spot price at time t StUnd the underlying spot price at time t

The seasonality effects can sometimes obscure the underlying price processes, that is, the processes of the spot prices stripped of seasonality. We need to strip the effect of seasonality out of price data in order to analyze the underlying price behavior. This nomenclature allows us to talk about modeling seasonality separately from talking about modeling the underlying price processes. (Note: Ultimately, seasonality should be modeled as a stochastic process.) 3.5.2.

Derivative Contract

A derivative is a contract whose value is a function of a spot price. A forward price is a derivative product in the sense that it is a function of the spot price behavior at some future point in time. Similarly, an option on either the spot price or the forward price is a derivative contract. 3.5.3.

Option Settlement Price

The price the option settles on is referred to as the settlement price. If the option is on the spot price, then the settlement price is the spot price. On the other hand, if the option is on a forward price, then the settlement price is the forward price. If the option is an average spot price option, then the settlement price is the average of spot prices over a period of time. I make the above distinctions because in the energy markets the spot price is not often also the option settlement prices. 3.5.4.

Delivery

Delivery is the contractually agreed-upon location and timing of the exchange of the commodity or cash. The actual commodity could be


47

delivered and paid for, or only cash could be exchanged, resulting in cash-settled contracts. 3.5.5.

Complexity of Contracts for Delivery

As stated in this book’s introduction, energy markets differ from money markets in part due to the complexity of the common contracts. Due to the nature of delivery, “vanilla” energy contracts would be considered “exotic” in financial markets. Let us consider the following three modeling issues for energy contracts: Modeling when the underlying market price is a physical commodity ● Valuing contracts for physical delivery ● Valuing cash-settled energy contracts ●

3.5.5.1.

Underlying Market Price Is a Physical Commodity In energy markets, the derivative contracts are typically for delivery of energy: one side pays cash, and the counterparty delivers the energy commodity. This contract requires the specification of location of delivery. Every place of delivery will have its own underlying price, and there are many such locations. Although this problem is great in the magnitude of information, it is really no different from the problem many equity players have to deal with: that of dealing with a huge set of underlying commodities. The large number of underlying price processes in the energy markets offers both good news and bad news. On the good side, the greater the number of different underlying price exposures in a trader’s book, the greater is the effect of diversification. The trader can use the risk diversification to benefit, as long as there is some means of hedging off the systematic (or market) risks,5 very much the way equity traders would use the S&P 500 index to hedge off the systematic risk of their book, which contains a large number of different kinds of stock exposure. The bad news is that the energy trading organizations have to become sophisticated enough to handle such problems and appropriately calculate the minimum-variance hedges. It is often very tempting to dump two different commodity markets into the same commodity bucket without doing the analysis of the

48

Energy Risk

correlation in their behavior. This can be dangerous to the extent that it is the easiest path and also one that is easy for traders to get used to. It allows the traders to sleep at night believing that their books are perfectly hedged purely based on the assumption that because they defined two different market price risks to be perfectly correlated they are off the hook. If this is the case, then it is the management that should spend sleepless nights until they sort out the issue of just how reasonable is the particular assumption of high correlations between the commodities in question. We will devote a good deal more discussion to this issue in the chapters on Risk Management and Portfolio Analysis.

3.5.5.2. Valuing Contracts for Physical Delivery When modeling derivative products that are functions of spot prices of energies for delivery, we have to incorporate all the characteristics of the particular delivery location into the modeling process. In other words, the modeling of the derivative product will be a function of the particular underlying price model for the particular delivery point. A good example here is electricity. The electricity market is extremely local, due to the constraints of production, transmission lines, and even regulation. Hence, even if the same general underlying price model applies to most electricity delivery nodes, we can be sure that the parameters of such a general model would be very different across the delivery nodes. Winter effects are much stronger in the North than in the South; hence winter seasonality will be much more pronounced in the North. Similarly, the Texas region tends to use quite a bit of natural gas in the generation of electricity, but the Northwest region of the United States has a good amount of hydroelectric generation. These different generation methodologies will have an impact on how the two markets tend to react to temperature events.

3.5.5.3. Valuing Cash-Settled Energy Contracts Even when the derivative products are cash-settled, and there is no delivery of energy, as long as the product is linked to the underlying prices of energies that are intended for delivery, the derivative product will have to be modeled as a function of the behavior of the underlying prices of energies for delivery. Even the OTC cash-settled energy derivatives are not able to get away from the fundamental drivers of energies markets.


3.6.

49

MODELING TERMS AND ISSUES Before we get into the details of modeling that will be presented throughout this book, we need to define some modeling terms that will be used.

3.6.1.

Price Returns

The daily price change is simply the difference between today’s price and yesterday’s price, and the daily price return is the daily price change divided by yesterday’s price. In general, a price return over some time period is the percentage price change over that time period. Equations 3-2 and 3-3 show these definitions in mathematical terms: dSt = St + dt − St

dSt St where:

=

St + dt St

−1

(3-2) (3-3)

dS˜t the price change dS˜t /St the price return

Here dt is the time period between price observations S˜t dt and S˜t. In the case of daily price returns, dt would equal 1/365. In the case of weekly observations, dt would be 1/52. Similarly, in the case of monthly price return calculations, dt would be set to 1/12.

3.6.2.

Elements of a Price Model

Every financial model, regardless of whether it is for an interest rate, a price, a log of a price, and so on, starts with the basic assumption of how the market variable being analyzed behaves over a short period of time. The change in the market variable, x, consists of the deterministic (or “drift”) and the stochastic (or “random”) terms: dx˜ deterministic term stochastic term

(3-4)

where: dx˜ the change in the market variable x˜, over time period dt

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Energy Risk

The deterministic term represents the portion of the movement in the market variable x˜, which we expect to see with certainty. The stochastic term represents the portion of the market variable change that is random and cannot be predicted. The deterministic term is also referred to as the drift term, and it is proportional to the time period over which the change in the variable is measured: deterministic term dt

(3-5)

3.6.2.1. Random Variables The stochastic term is proportional to a normally distributed variable, d z˜t. (The little hat, or tilde, above the z˜, denotes that it is a random variable.) This normally distributed variable has a mean of zero, and a variance of dt: stochastic term dz˜t

(3-6)

dz˜t ~ (0, dt)

(3-7)

The randomly distributed variable (aka “random variable”) is a key concept that is used throughout this book. The random variable allows us to capture market movement in our models, and our understanding from mathematics and statistics of random variable properties provides us with many shortcuts to be used in valuation and risk management. We can generalize the definition of the random variable, z˜, from time period t1 to t2, z˜t1, t2, to be normally distributed with a mean of zero and a variance of (t2 t1) zt1, t 2 ∼ ℵ(0, t2 − t1 )

(3-8)

One property of a random variable is that its value is cumulative over time. (We will use this property when solving for forward prices and during portfolio analysis.) Figure 3-5 demonstrates how a random variable “walks” through time. Specifically, the normally distributed variable z˜ has the following characteristic: it is additive. A random variable representing the randomness from time period t1 to t3, let us call it z˜t1,t3, would be equivalent to two random variables, one representing the randomness from time


51

period t1 to t2, z˜t1,t2, and the other representing the randomness from time period t2 to t3, z˜t2,t3: zt1, t 3 = zt1, t 2 + zt 2 , t 3

(3-9)

The mean of zero is preserved, as is the variance being proportional to the time period. Because these are normally distributed variables, the correlation between the random variables during periods that do not overlap is zero: E ⎡⎣ zt1, t 2 , zt 2 , t 3 ⎤⎦ = 0

(3-10)

It follows that the correlation between two random variables with the periods overlapping is not zero. For example, we would have E ⎡⎣ zt1, t 3 , zt1,t 2 ⎤⎦ = E ⎡⎣( zt1, t 2 + zt 2, t 3 ) zt1,t 2 ⎤⎦ = E ⎡⎣(zt1, t 2 )2 ⎤⎦ + E ⎡⎣ zt1, t 2 zt 2, t 3 ⎤⎦ = t2 − t1 (3-11)

For very small changes in time, we have the following: z0 , t + dt = z0 , t + zt , t + dt

(3-12)

We define dz˜t to represent the normally distributed randomness of the process over the time period dt, from time t to time t dt:

dzt ≡ zt , t + dt = z0 , t + dt − z0 , t

F I G U R E

Random Walk

3-5

(3-13)

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Energy Risk

3.6.2.2. Factors In a model, a factor represents a market variable that exhibits some form of random behavior. The Pilipovic Model for the Forward Price Curve is a two-factor model because it assumes that the spot price and the long-term price are both market factors, with their own random behaviors. (See Sections 3.9.3 and 3.9.4 for continued discussion.)

3.6.3.

Convenience Yield

The convenience yield represents the overall benefit minus the cost—with the exception of the financing cost—that a holder of a commodity receives by holding the commodity. The holder of a commodity would reap benefits by using the commodity to generate value that is dependent on the use of the commodity as a fuel, such as for example a factory that needs the fuel to keep running. Because the convenience yield represents the net value of holding the commodity, excluding the financing cost, it can be either positive or negative. It is positive when the benefit of having the fuel on hand outweighs the cost, and it is negative when just the opposite is true. The convenience yield, Cy, although driven by the user’s needs, is in general a measure of the balance between the available supply and the existing demand. If we introduce Lt as the equilibrium price, that is, the price of the commodity when the supply and the demand are in balance, and St as the spot price, both at time t, then the difference between the two represents the measure of the market imbalance. The convenience yield is a function of this imbalance, so we can make a general statement: Cy (St Lt ) K

(3-14)

where K is some constant. (As you will see in Chapter 6, the constant K can be related to the equilibrium price growth rate adjusted for the cost of production and storage as well as the cost of risk.) The contribution to the convenience yield of the market supply and demand imbalance, as captured by the difference between the values of the spot price and equilibrium price, should go to zero over a long period of time, as in the long run we could assume that the prices approach the equilibrium levels: Cy → K as t →

(3-15)


53

The difficulty in modeling the commodity prices is in specifically defining the convenience yield. The measure of the value that the commodity generates for the user by having it on hand as the user needs it—or rather, on the flip side, the value lost to the user in not having the fuel on hand as needed—is user-specific. As there is no standardized formulation of convenience yield in the marketplace, there are huge degrees of freedom in defining it. Typically, it is expressed in terms of a continuously compounded rate. An example of a convenience yield function and how it impacts the derivation of differential equations for commodity derivatives is provided in Chapter 6. The convenience yield helps with understanding the differences between shortand long-term price behavior in energy markets. Short-term markets reflect the fundamentals of the readily available and stored energy; long-term markets reflect the fundamentals of the energy yet to be “dug out of ground” and put into storage. It is really not that surprising that the short-term and long-term products would concentrate on different fundamentals. Convenience yield provides us with a bridge between the short-term and long-term price fundamentals. Some of the short-term fundamental drivers would include supplyside events such as storms, strikes, wars, or other events that might disrupt immediate delivery of the energy. On the demand side, unexpected temperature spikes in the summer and temperature drops in the winter would cause a short-term imbalance between the demand and the immediately available supply. The convenience yield reflects these short-term supply and demand imbalances, as the users are willing to pay a premium for near-term delivery in response to the supply shortage. Another set of fundamentals tends to influence the long-term supply-and-demand sides. Here we are dealing with the expectations of future potential supply and costs of production/storage, and the expectations of future potential demand. New discoveries of energy in the ground and new technologies affect the long-term outlook for energy prices. However, these fundamentals tend to be much more stable and less susceptible to frequent events as compared to the short-term fundamental drivers. It is this divergence between the short-term and long-term market fundamentals that also gives the short-term and long-term energy products a different set of behavioral characteristics. Recall from Chapter 2 how energies exhibit a “split personality”: one that we see in the short term and another that we expect to observe in the long run. Just how “short” the short term is can be measured by how quickly the

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Energy Risk

market tends to revert to the equilibrium levels after an event hits. In the case of crude oil, the short-term markets driven by the short-term behavior fundamentals go out about three to six months relative to today. Typically, event corrections and supply/demand imbalances tend not to vary beyond this time period. For the heating oil and natural gas markets, the short term is even shorter: the short-term fundamentals tend to affect the market out to about three months. Finally, in the electricity markets the short term is truly short-term, with the short-term fundamentals driving the market within only a couple of weeks out. Given the split personality of energy markets, the difficulty in quantitative modeling is in coming up with valuation and risk management models that can capture both the short-term and long-term behavior. The spot prices are driven primarily by the fundamentals of the short-term market factors; however, they are still influenced by the longer-term expectations of equilibrium price levels, slight as this influence might be. Similarly, the longer-term energy products are primarily driven by the effects of the long-term market fundamentals. And yet, they might still feel some small effects of the near-term fundamentals. Most importantly, there is that gray transitional market area where the energy products feel both the short-term and long-term fundamentals, but with different weightings. Any model that we want to test out for consistency with the spot price behavior on a day-to-day basis must also be tested with the spot price behavior over a longer period of time. Similarly, we can use the longer-term energy products, such as the full strip of forward prices, to ensure the consistency between the model and the market reality.

3.6.4.

Cost of Risk

We define the cost of risk, , as the differential between the actual return that an asset pays vs. the risk-free rate, normalized by the asset’s volatility: (µ − τ ) λ = (3-16) σ where: cost of risk rate of return on the asset


55

r risk-free rate

volatility It turns out, in the simple case of a stock price paying no dividends, that the cost of risk is given by

⎛ Et ⎡ ST ⎤ ⎞ ln ⎜ ⎣ ⎦ ⎟ ⎝ Ft ,T ⎠ λ= σ (T − t ) where:

(3-17)

cost of risk S spot price F forward price t time of observation T time of forward price expiration T t time to forward price expiration

volatility

or to put it another way:

Et ⎡⎣ ST ⎤⎦ Ft ,T

= eλσ ( T −t )

(3-18)

The cost of risk is equal to the log of the ratio of the forward price to the expected spot price, Et[ST], normalized by the volatility and the forward price time to expiration, T t.

3.7.

QUANTITATIVE FINANCIAL MODELS ACROSS MARKETS We will use this section to introduce some of the quantitative financial models descriptively rather than mathematically. The next chapter will take you through the details of the models as well as how to quantify them.

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Energy Risk

3.7.1.

Lognormal Market

The lognormal model was the first well-developed quantitative financial model. It has proven to be the most versatile and also the simplest model to be applied in describing price behavior in various markets. We will see, however, that this model does not work as well for energy markets (especially when compared to the mean-reverting models described in the following section). A lognormal price is one that behaves so that the price return— which is the percentage change in the price—is normally distributed. This means that the percentage change in price over some time period is centered on some value—the drift, or the price yield—and its distribution around this drift is symmetric. The price return has as much of a chance of being positive as it does of being negative when the drift is zero. The drift represents the expected price return. Normally distributed price returns translate into lognormally distributed prices. When the price return is normally distributed, the actual prices are guaranteed never to be negative. Figure 3-6 shows the path that the S&P 500 price made from 1992 through 1996. This is an example of a lognormal price path. Because of its simplicity, this model is favored across money markets in general. (Again, its popularity in money markets should not be taken as reason enough for applying it to energy markets.) A graph of the lognormal price distribution (see Figure 3-7) shows that a lognormal price process exhibits a positive skew (i.e., a distribution skewed to the right). In the case when events occur in the marketplace but do not leave lasting effects, the distribution of price returns may be rather wide, F I G U R E

3-6

Sample Path of a Lognormal Price


F I G U R E

57

3-7

Sample Path Lognormal Price Distribution (Drift Rate 13%; Spot Price Volatility 30%)

which reflects the effect of events, and yet the distribution of the prices may not be that wide, which reflects the fact that the events do not have a long-lasting effect. In this case, if we took a stock and an energy that have very similar looking price return distributions, and then compared their price distributions, what we might find is that their price distributions look very different. Figures 3-8 through 3-13 show two paths, Path 1 and Path 2, which have exactly the same price return distributions, but their price distributions look very different. Path 1 is representative of markets where events do not happen often, but when they do happen they tend to have long-lasting effects, such as long-term growth or decline. Path 2 is representative of markets where events happen often, but they do not tend to have long-lasting effects. F I G U R E

3-8

Path 1 Time Series

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Energy Risk

F I G U R E

3-9

Path 1 Return Distribution

Path 2 shows the type of behavior that is mean-reverting in nature. When the price move up tends to be followed by a price move down and vice versa, the short-term price moves might be very large, but the end result is a price range that is fairly narrow. This type of behavior tends to be seen in energy markets: Events tend to be corrected, either through the dissipation of what caused them or through the response of the supply side. Either way, events tend not to have long-lasting effects the way they do in the equity markets. Path 2 behavior has a mean-reverting character, which a simple lognormal model would not capture. Instead, a lognormal model that exhibits similar magnitude of short-term price moves, as in our example above (Path 1), would result in a price range that is much wider F I G U R E

3-10

Path 1 Price Distribution


F I G U R E

59

3-11

Path 2 Time Series

compared to the price range in the example. Hence, although the lognormal model and a mean-reverting model might show very similar magnitudes of daily price moves, when observed over a period of time, the range of actual price levels covered during that time period would be wider for a lognormal model. This is one of the key differences between energies and equities. An event generally has a long-lasting effect on the S&P (remember the stock market crash of 1987?). It should not then be surprising that a lognormal distribution—which works so well in the equity world—does not work well in the physical commodity world. F I G U R E

3-12


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Energy Risk

F I G U R E

3-13


3.7.2.

Mean-Reverting Market

As will be suggested in Chapters 5 and 6, a two-factor mean-reverting model is the most appropriate quantitative model for energy markets.

3.7.2.1. Violin Analogy The effect of mean reversion can perhaps be better understood through the violin string analogy. If we pluck the violin string, the string will revert to its place of equilibrium. We could not possibly measure just how quickly this reversion back to the equilibrium location would happen unless we actually plucked the string. Similarly, the only way to measure mean reversion is when the prices get plucked away from their nonevent levels and we observe them go back to more or less the levels from which they started. If the prices consistently stay at their equilibrium level we have no real means of measuring or deciding on how strong the mean reversion is. We might have to observe a great amount of data, over a long period of time, in order to capture the true behavior.

3.7.2.2. Mean Reversion A mean-reversion process has a drift term that brings the variable being modeled back to some equilibrium level. The end result is that the variable tends to oscillate around this equilibrium. Every time the


61

stochastic term gives the variable a push away from the equilibrium, the deterministic term will act in such a way that the variable will start heading back to the equilibrium. The stronger the mean reversion, the quicker will be the trip of the variable from some extreme point away from the equilibrium and back to it. When a variable, x˜, is mean reverting, it will have a deterministic term defined as Et [dx˜t ] (x¯t x˜t )dt

(3-19)

where: rate of mean reversion x¯ the value around which x tends to oscillate In the above equation, the mean-reverting parameter, , must be positive. If the variable at time t, x˜t, is greater than its equilibrium value at time t, the drift term is negative, resulting in a pull back down toward the equilibrium level. Similarly, if the variable x˜t, is smaller than its equilibrium value at time t, the drift term is positive, pulling x˜t, back up to the higher equilibrium value. Note that the greater the mean-reverting parameter value, , the greater is the pull back to the equilibrium level. Furthermore, for a daily variable change, the change in time, dt, in annualized terms is given by 1/365. If the mean-reverting parameter had a value of 365, the mean reversion would act so quickly as to bring the variable back to its equilibrium within a single day. Although this is just a rough estimate, the value of 365/ gives you an idea of how quickly the variable takes to get back to the equilibrium—in days. Mean reversion can be either in the prices, or the log of prices, or in the yields (or rates). If the mean reversion is in the rates, xt would represent the short-term rate. On the other hand, if mean reversion is in the price, xt would be the price at time t. Finally, if mean reversion is in the log of the price, then xt would be the natural log of the price at time t. 3.7.2.3.

Mean Reversion Expressed as the Inverse of Time When physicists measure the rate of decay of a particular substance, they often use the notion of half-life: the time it would take a given amount of the substance to decay to half its mass. Similarly, we can talk of measuring mean reversion in terms of the expected time it would take the variable to revert to some mean value, given that it is starting at an extreme point away from the mean value.

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Energy Risk

For example, the mean reversion measured from short-term interest rate historical data translates to a period of approximately three years. Hence, on average, it takes roughly three years for interest rates to decline to near their average values, given that they are starting from a point of high extreme, and similarly it takes roughly three years for interest rates to revert to the higher mean levels, given that they are starting from a point of low extreme. As the state of the economy changes relatively slowly, the effect of mean reversion has to be looked at on a timescale of years and even decades. The reversion force remains fairly weak from day to day, week to week, and even month to month. However, it is there nonetheless, and ignoring it would result in being at odds with the market reality. The valuation of long-term products should take this mean-reverting behavior into account. Fundamental economic cycles do not appear to influence energy markets as much as they do interest rates. Instead, the mean reversion in energies appears to be directly related to their event behavior. Either a correction on the supply side, to match the demand side, or the actual dissipation of the event, such as the temperatures reverting to their more average seasonal levels, tends to cause the energy market prices to come back to their typical levels fairly quickly. Mean reversion in the energy markets is extremely strong, particularly when compared to the mean reversion measured in the interest rate markets. In interest rate markets the mean reversion is assumed to directly affect interest rates rather than bond prices. These models are thus referred to as the yield-mean-reverting models. A rate or yield-meanreverting model allows the yields or the rates to mean-revert toward some mean level. By comparison, a price-mean-reverting model allows the prices to revert to some sort of a mean or equilibrium price level. Hence, the basic difference is that a price-mean-reverting model assumes that the mean reversion has a direct effect on the price, whereas a yieldmean-reverting model would have a secondary effect on the actual prices of bonds or assets driven by the underlying rate or yield model. Finally, the log of the price-mean-reverting model assumes that mean reversion affects the log of price directly, and the price itself only indirectly. Although we will be performing benchmarking and quantifying a few models in the next chapter, we will not spend a great deal of time in this book on the actual theoretical derivations of yield- or rate-meanreverting models, as these have been described in quite a bit of detail in many books, articles, and publications.6 Instead we will spend much more time on the model derivations of the other mean-reverting models.


3.8.

63

THE TAYLOR SERIES AND ITO’S LEMMA The “Taylor series” and “Ito’s Lemma” are mathematical relationships that we will use when building differential equations, pricing options, and, perhaps most importantly, performing portfolio analysis. The phrase “Ito’s Lemma” may be justifiably scary to even those of us lucky enough to have technical degrees, but I still feel that a passing knowledge of these concepts is a prerequisite for any risk manager.

3.8.1.

The Taylor Series

Taylor series expansion helps model energy risk and portfolio returns in terms of the market’s discrete building blocks. All of the differential equations that we solve in this book begin with the building of a Taylor series for portfolio returns. The Taylor series allows us to express the change in the value of a function, f, in terms of the changes in the variables determining the value of the function f. Specifically, if f is a function of variable x˜, the Taylor series expansion is expressed as

df = where:

1 ∂2 f 2 ∂f ∂f dx + O( dt ) dx + dt + 2 ∂x 2 ∂t ∂x

(3-20)

f function x˜ variable O(dt) higher order terms in dt

The variable x˜ follows its own process. If it is normally distributed, the process for x˜ is defined by the following: dx˜ adt b ˜

dt

(3-21)

where: a the mean value of x˜ b the annualized standard deviation of x˜ ˜ a normally distributed variable Note that dx˜ has terms of order dt and dt , and (dx˜)2 would have terms of order dt, dt2, and dt3/2. The first important assumption in the above equation is that the stochastic term in the change in the variable x˜, is proportional to the square root of time. The second important assumption is that the

64

Energy Risk

change in time, dt, is assumed to be small. In the Taylor series expansion (Equation 3-20), these two assumptions allow us to retain the terms of order dt and dt, and let all higher order terms in dt go to zero. In the expansion, all the higher order terms in dt are denoted as O(dt). Thus, the assumption is that O(dt) goes to zero. However, we can also relax this assumption, resulting in the inclusion of higher order terms in dt, which we do not assume away as insignificant.

3.8.2.

Ito’s Lemma

Ito’s Lemma represents a specific treatment of the stochastic variable in the Taylor expansion. We assume that as the increment of time dt goes to zero, we can ignore all terms of order higher than dt (such as dt2 or dt3/2). In the derivation of differential equations, this assumption is consistent with the assumption of continuous hedging. Given the assumption that dt goes to zero, and given the character of the normally distributed stochastic variable, ˜, with a mean of zero and a standard deviation of one, the following must hold: E[ ˜ dt ] 0 E[ ˜ 2dt] dt E[ ˜4 dt 2] O(dt 2) → 0

(3-22) (3-23) (3-24)

Becauase the expected value of the stochastic term in dx˜, raised to the fourth power is of order dt2, this term must go to zero, requiring that the stochastic term squared must be a constant, specifically dt:

˜ 2 dt constant

(3-25)

˜ 2 dt → dt

(3-26)

We can now plug the above results into the Taylor series for the function f to obtain the following:

df =

⎛ ∂f 1 ∂2 f 2 ∂f 1 ∂ 2 f 2 ⎞ ∂f ∂f ∂f + + = a b dt dx + dt + b ⎟ dt + b dz ⎜ 2 2 2 ∂x ∂t 2 ∂x ∂t ∂x ∂x ⎝ ∂x ⎠

where: d z˜ ˜ dt

(3-27)


65

We will be using Ito’s Lemma, and thus the assumption of continuous hedging, in deriving the differential equations for forward prices and option prices. We will be using the Taylor series, without the additional constraints of Ito’s Lemma, in performing portfolio analysis.

3.9.

LESSONS FROM MONEY MARKETS The equity markets were the first markets to develop and mature. The interest rate markets followed with the deregulation of the banking industry in the United States. The interest rate markets are now the largest derivative markets in the country, and the crude oil, heating oil, and natural gas markets—although nothing to sneeze at—are still relatively small when compared with the interest rate markets. The electricity markets are currently going through deregulation, and they may prove to be truly a rival in size to the interest rate markets. The electricity markets, as well as other energy markets that have a long way to go toward maturing in terms of trading liquidity and market sophistication, can learn a number of lessons from the growth in the interest rate markets.

3.9.1.

Modeling Price vs. Rate: Defining the Market Drivers

One of the most difficult tasks of financial modeling is to define what exactly ultimately to model in order to provide a basis for all other valuation and risk management calculations. In the example of interest rate markets, do we start by modeling the fundamental market drivers such as inflation and the Federal Reserve moves and then translate these drivers into bond-price models, or do we start by modeling the bond-price behavior? The general market has answered this question by modeling the interest rates; interest rates capture all the fundamental market driver effects and yet directly relate to bond prices. In the interest rate markets, most of the models are applied to short-term interest rates. These interest rate models are then applied to arrive at arbitrage-free pricing of bonds or other derivative products.

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Energy Risk

The underlying process, then, is for the interest rates rather than for the bond prices. In the commodity markets we are faced with the same problem. Do we model the energy generation process or the economics of supply and demand, or do we model the prices for each month of the year, to ultimately value all the energy products? Or maybe we can cut down the number of explanatory variables to just a few, and perhaps model only the spot price, the equilibrium price under no events, and the seasonality factors? Another choice is to model the underlying price yields, very much in the same way as in the interest rate markets, where the rates are modeled and then the prices are modeled as a function of the underlying rate models. This would be consistent with some of the convenience yield models, where instead of modeling the actual prices, forward price yields are used to capture some of the supply-and-demand and storage effects unique to physical commodities. Although in the interest rate markets the trading world truly is driven by the behavior of the interest rates (the Fed moves the discount rates directly), in the physical commodity world, the price yields are, from an intuitive trader view, of secondary importance. Instead, the prices tend to be the direct media within which market factors tend to portray themselves. Of course, this could be very much a discussion of a personal nature unless we have a set of quantitative benchmarks against which we can judge what came first—the chicken or the egg— the commodity prices or their yields. However, if we are indeed to learn from the experience of the interest rate markets, which during their own development of many years had to come up with a set of models that were interest-rate-market specific, we should be looking at new ideas for the physical commodity markets if we are truly to capture all the specifics of the physical commodity behavior, rather than trying to “Band-Aid” the existing models from interest rates and equities to conform to the commodity price behavior. Ultimately, it should be the modeling benchmarks that should help us decide on what model is most appropriate. 3.9.2.

Yield vs. Forward Rate Curves

A topic that has received a good deal of analysis by the interest rate trading houses is not necessarily one that you would see much of in the academic literature on interest rate behavior. Rather, it is more of an


67

issue of implementation and has to do with constructing forward rate or yield curves. A forward rate curve is a curve of short-term rates— such as three-month LIBOR rates—as seen at different points in time in the future. A yield curve, on the other hand, is a discount rate curve, with each point on the curve representing the discount rate from today to the point of discounting. The yield curve represents average rates from today to points along the time axis, whereas the forward rate curve represents the short-term rates at different points in time. The interest rate relationships, required by no-arbitrage conditions, allow the translation of the yield curve to the forward rate curve and vice versa. Because the yield curve is in fact an average-rate curve, it tends to look much smoother and is generally much easier to build. Even when using linear interpolation between the yields, we can obtain a fairly smooth-looking yield curve. So it was that most trading places did exactly this. They built nice, smooth yield curves, typically using linear interpolation between the points, and everybody appeared pretty happy; that is, everybody but the forward rate traders, who often noticed that the forward rates resulting from these generally smooth yield curves tended to take rather nonintuitive jumps into very large or very small (even negative) values at what appeared to be almost random points in the future. The problem was that even the smallest-looking kink in the yield curve, which is an average-rate curve, translated into huge jumps in the forward rate terms. Hence, although linear interpolation might have been a quick and dirty implementation that worked just fine for yield-curve building, it tended to be too crude for implementation in building forward rate curves. We have a similar problem in energies regarding the building of price and volatility curves, particularly in the case of electricity. There is really nothing wrong with building average price and average volatility curves, as long as all the settlement prices of derivative products and all the hedges are also defined as averages. However, this is generally not the case, and all of a sudden, the average price and average volatility curves become problematic. Instead, developing a methodology for the building of discrete price curves and volatility matrices—although requiring a bit more thought in the implementation process than just using linear interpolation—will provide a basis for an extremely flexible and versatile framework for pricing the full spectrum of energy derivative products.

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3.9.3.

Drawbacks of Single-Factor Mean-Reverting Models

Most players in the interest rate markets will agree that the interest rates do exhibit some amount of mean reversion. They might also acknowledge that such a mean-reverting model ought to have at least two drivers, a short-term rate and a long-term mean, or equilibrium rate. The long-term mean rate tends to affect the behavior of the forward rates far out into the future, and the short-term rate defines the behavior of the interest rates in the near future. All the rates in between reflect a mix of the two behaviors, with various weightings on the short- vs. the long-term mean rate. Although most players will agree that this framework might reflect the market reality fairly well, very few have actually implemented a two-factor mean-reverting interest rate model. The reason is typically that the costs tend generally to outweigh the benefits. Even a sophisticated trading operation that has made a large investment in pricing methodologies might shy away from implementing a two-factor option pricing model if that model takes a long time to run for each deal being priced. Instead, a good number of interest rate trading houses have implemented a single-factor interest rate mean-reverting model that assumes that the long-term mean rate remains fixed over time. Of course, this is not representative of reality, and hence the long-term mean rate needs to be recalibrated on a continuous basis in order to ensure that the resulting curves are marked to market. However, the biggest drawback of installing a single-factor meanreverting model is in the case of options pricing: the fact that the longterm rate is fixed results in a model-implied volatility term structure that has the volatilities going to zero as expiration time increases. Hence, in order to get around this point, another model “Band-Aid” needs to be put on. Spot volatilities have to be increased to nonintuitive levels so that the long-term options do not lose all the volatility value— as in the marketplace they certainly do not. It is in such cases that a simpler model, such as Black–Scholes or Black, might have been a better choice, because the model inputs would have retained the flexibility and the intuitiveness that ends up being lost by implementing a half-baked model. Or rather, if you want to be sophisticated in modeling, getting only halfway there may put you in a worse position than not attempting it at all. Quite a few banks had to discover this the hard way, and after they had already spent quite a bit


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of money on the implementation. With this said, let me also say that a good, sophisticated model can be quite valuable, not only for product valuation but also for hedging and risk management precision. And there are also a number of banks that have proven that this approach can be a successful strategy. 3.9.4.

Drawbacks of Single-Factor Non-Mean-Reverting Models

The above section dealt with the downfalls of a single-factor meanreverting model. This section, on the other hand, will deal with the downfalls of not using a mean-reverting model when the market is indeed mean-reverting. Assuming a simple lognormal model for the underlying distribution when the underlying distribution is in fact not lognormal will impact valuation and hedging. The differences between the distributions are particularly obvious when pricing out-of-themoney options, where the tails of the distribution play a very important role. It is no surprise then that if a lognormal model is used to price a far out-of-the-money option, the price can be very different from a mean-reverting model’s price. A general problem that has occurred in the interest rate markets is that a simple Black, and hence lognormal, model was used to price an at-the-money option, and was sufficient when the deal was struck. However, as time went by and the market moved, the option found itself out-of-the-money and therefore extremely sensitive to the lognormal distribution assumption. Most of such tail-effects did generally get treated by the trading groups through the inclusion of volatility strike-structures (which also include other effects, such as that of illiquidity or the smallest price ticks allowed on the exchanges). However, unless the market exists for such out-of-the-money options, it is very difficult to “guesstimate” the effect on volatility without an actual two-factor model implementation. 3.9.5.

Volatility and Correlation Market Discovery

Energy options markets have taken a while to develop, and chances are that certain option products will never achieve the level of liquidity a mark-to-market valuation always craves. The swaptions in interest rate

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markets never reached the liquidity levels of interest rate caps and floors. Even in a mature market there is enough trading to allow a certain amount of market discovery, but not necessarily to paint the full picture. In such situations it is up to the market players to fill in the missing pieces in order to perform the valuation and the hedging. Hence, the lesson to be learned here is that as options markets continue developing in the energy markets, a market maker would be wise to consider volatility estimation methodologies consistent with the traded market option quotes, allowing proper valuation of options in the absence of liquidity. An even greater challenge in the energy options markets consists of obtaining market-implied correlations. Correlations remain the “gold mine” everyone in the market continues their search for: they are critical inputs to valuation, hedging, even bridging across different product volatilities. The chances are that there will never be enough market option information to allow us to market-imply the full intramarket and intermarket correlation matrices.

ENDNOTES 1. A. A. Milne, “Winnie-the-Pooh”, p. 13, Dutton Children’s Books, member of Penguin Putnam, Inc., New York, 1926. 2. Jarrow & Rudd, “Approximate Option Valuation,” p. 104. 3. When the Black–Scholes model first emerged from the academic world, it provided a new paradigm of pricing arbitrage-free options. Once accepted by the marketplace, the model was here to stay. 4. The end result of a single-factor mean-reverting model is that the volatilities of such a process tend to go to zero over a long period of time. Most market players would agree that this is not reflective of the market reality. Yet, some people still use the model by making certain adjustments during implementation. 5. For more information, see Brealy, Myers, and Marcus, Principles of Corporate Finance, Chapter 7 (New York: McGraw-Hill, 1995). 6. For a good introduction to yield-based models, see Hull, Options, Futures and Other Derivative Securities, Chapter 15.

C H A P T E R

4


Throughout this book, you will find us fearlessly editorializing, telling you what you should and shouldn’t do. This prescriptive tone results from a conscious decision on our part, and we hope that you will not find it irritating. We do not claim that our advice is infallible! Rather, we are reacting against a tendency, in the textbook literature of computation, to discuss every possible method that has ever been invented, without ever offering a practical judgment on relative merit. We do, therefore, offer you our practical judgments whenever we can. As you gain experience, you will form your own opinion . . .

Press, Vetterling, Teukolsky, and Flannery1

4.1.

INTRODUCTION To value and manage energy risk, we need to be equipped with the essential quantitative and statistical tools. These tools capture the reality of the market and express its characteristics. In addition, statistics provides the essential benchmarks for testing models and judging between them.2 This chapter will introduce these statistical tools and demonstrate how they can be used. Many books have been written about statistical analysis of various sorts. Here we will go into the details of statistical analysis only to the extent that it is useful in energy modeling.3 Hence, we will view statistical analysis as one particular toolbox with a particular set of tools. We will leave the detailed description of such statistical tools and the discussion of their various uses to other books. The necessity of using statistics offers good news and bad news for energy managers. The good news is that statistics provides terrific valuation tools. The bad news is that even managers need to understand some basic statistics. During the writing of this book, serious thought was given to relegating these methods to an appendix, out of the way, so as not to scare the readers who hated Statistics 101 in college. Instead, let us confront statistical analysis up front, out in the open, because valuing and 71


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managing energy derivatives absolutely depends on time series and distribution analysis. Quantitative experts should be able to explain and defend their models in terms of these tools; managers should possess at least a passing knowledge of the terms and their benefits.

4.2.

TIME SERIES AND DISTRIBUTION ANALYSIS Time series analysis and distribution analysis are two important data analysis methods used throughout this book. They represent two approaches to analyzing data (Table 4-1). Time series analyzes changes in the price from day to day. Distribution analysis, on the other hand, explores price behavior over a period of time. In businesses today, time series analysis occurs much more commonly than distribution analysis; however, distribution analysis is critical in comparing between the suitability of models and in seeing effects of mean reversion and choices of model factors. Time series and distribution analyses each tells a “different part of the story.” Both should be used for proper valuation and risk management.

4.2.1.

Time Series Analysis

Time series analysis is the process of analyzing daily price returns. A very simple type of time series analysis involves taking a data set of prices and calculating the price drift and annualized volatility. T A B L E

4-1

Comparison of Time Series and Distribution Analyses

Purpose Good for

Use in business

Time Series Analysis

Distribution Analysis

Analyzes price change from day to day • Parameter calibration • Event and seasonal calibration

Analyzes price behavior over period of time • Testing, benchmarking, and selecting models • Getting insights about option pricing Uncommon but should be used more

Relatively common


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The most common application of time series analysis is in the calibration of model parameters. The process involves using statistical “fitting” procedures to best match a data set with a model. The object is to estimate model parameter values for the model that “best fit” or explain the data. For complex models, nonlinear statistical estimation may be required. Additionally, the estimated parameters typically will not precisely “match” the data set. The difference between the model predictions and the actual data set results in model residuals: Actual data Model predictions residuals

(4-1)

If the model does a good job, these residuals should essentially be “noise,” the random by-products of a nonbiased difference between the actual data and model predictions. These residuals need to be tested to ensure that they are normally distributed. (See later sections of this chapter for testing model appropriateness.) Price return analysis is performed either through linear or nonlinear regression analysis. Model parameters from a simple model, such as the lognormal model, can be estimated through a linear regression, and more complicated models, such as price mean reversion, might require nonlinear regression analysis. The decision of using linear vs. nonlinear analysis has to do with how the model processes can be translated so that when the regression is performed the residuals can be correctly assumed to be normally distributed. A historical time series of prices will give us some first clues as to the price behavior. When the seasonality factors are very strong—as in the case of electricity markets—they can be identified simply by a quick look at the price time series. By comparison, looking at price returns instead of prices may obscure the effects of seasonality due to the typically very large randomness in the daily price returns of energy markets. By looking at the price time series we might be able to identify the historical seasonality effects. Figure 4-1 shows the time series of spot prices for Massachusetts Hub On-Peak power. We can see both winter and summer seasonality peaks within the historical data, with the fitted exponential seasonality focusing on the first few years of the sample data. Quite clearly, the seasonal behavior exhibited within this price data changes significantly over the full time period of the sample: the winter-specific peaking behavior has been replaced by an October far less quickly dissipating event, fitted in Figure 4-2. We have to be very careful in how we interpret this seasonal calibration as far as it reflects possible future behavior. The past price

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F I G U R E

4-1

Massachusetts Hub On-Peak Electricity Spot Prices: Seasonality Fitted to First Few Years of Data

behavior tells us about some of the possible market states, but not all. This means that the future price behavior may or may not be very similar in terms of magnitudes and calibrated values. There is no question that we have observed a power market with short-term memory when it comes to seasonality. Forward prices over the years have tended to replicate past year’s worth of seasonal behavior, with the memory span expanding as more participants become more aware of the diverse spectrum of possible seasonal states. F I G U R E

4-2

Massachusetts Hub On-Peak Electricity Spot Prices: Seasonality Fitted to Last Year and a Half of Data


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Similarly, if the market tends to experience a good number of events, followed by a strong reversion toward what might be considered the equilibrium price level, then even this may be observed from the time series of price. The general market trend—up, down, or flat—and the general market behavior around some equilibrium level is what typically can be observed through the historical time series of prices.

4.2.2.

Distribution Analysis

Distribution analysis focuses on price behavior over time. It provides meaningful insights into market behavior. The technique helps with

• Creating benchmarks for actual market behavior • Testing models against such benchmarks • Comparing models A price distribution defines the probabilities of prices taking on various values. If we are analyzing actual data, the distribution is defined by the “path” of prices observed over the time period. If we are simulating a model, the distribution shows all the possible values that the spot price might take on over some time period with associated probabilities. We represent distributions visually as histograms or probability graphs.

4.2.2.1. Characteristics of a Distribution One nice feature about distributions is that they demonstrate unique characteristics, which can be used in describing them. Figure 4-3 demonstrates the most important statistical characteristics:

• The “mean” represents the value around which the distribution is centered. • The “standard deviation” suggests the width of the distribution. One standard deviation roughly equals the width of the distribution in which a price will fall 66% of the time the percentage is exact in the case of a normal distribution; two standard deviations roughly represents the price range in which a price would fall 95% of the time; three standard deviations roughly represents the price range for 99% of the time.

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F I G U R E

4-3

Variable X Distribution

• The “skew” reflects whether the prices distribute symmetrically around the mean or are skewed to the left or to the right of the mean. • Finally, “kurtosis” describes the “fatness” of the tails of the distribution. Kurtosis helps us understand the likelihood of extreme events; fat tails suggest higher chances of prices being very high or very low. The concepts of mean, standard deviation, skew, and kurtosis will be used throughout this book, particularly because they represent characteristics that we can visualize. We will not, however, usually employ the concepts in our equations. A related and preferred method is the concept of mathematical “moments,” which are described in the following section.

4.2.2.2. The “Moments” of Truth A distribution can be characterized through mathematical concepts called “moments.” These moments directly relate to mean, standard deviation, skew, and kurtosis of a distribution (see Table 4-2). The energy risk manager should understand and employ moments for three important reasons:

• Moments are relatively easy to calculate. • Moments can be used for correcting modeling errors. • Moments can be very important during option valuation.


T A B L E

77

4-2

Moments Moment

Related to

First, M1

Mean Standard deviation Skew Kurtosis

Second, M2 Third, M3 Fourth, M4

The n-th moment of a distribution for variable x˜ is the expected value of the variable raised to the n-th power. Mathews and Walker4 express this as E[ x n ] = ∫ p( x ) x n dx

(4-2)

where: E[. . .] represents the taking of an expected value x˜ variable p(x) the probability that the variable takes on the value x Although we can calculate as many moments as we wish, the most important moments used in characterizing a distribution are the first four moments. The first moment is the expected value of the variable raised to the first power, or simply put, the mean or average of a distribution. Using spot prices as our variable, we calculate the first moment as M1 = E[ St ] = St

(4-3)

where: M1 the first moment of the distribution S˜t spot price at time t St mean of the distribution of S˜t The second moment is a measurement of the distribution’s width: M 2 = E ⎡⎣( St )2 ⎤⎦

(4-4)

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We can derive the variance of spot price levels by using the first and second moments as follows: Var ( St ) = E ⎡⎣( St − St )2 ⎤⎦ = M 2 − M12

(4-5)

The standard deviation is then given by STD( St ) = Var ( St ) =

M 2 − M12

(4-6)

The third moment is the expected value of the variable raised to the power of three: M 3 = E ⎡⎣( St )3 ⎤⎦

(4-7)

where: M3 the third moment The skew of the distribution is the third moment adjusted for the distribution center. Skew can be expressed as follows: Skew = E ⎡⎣( St − St )3 ⎤⎦ = M 3 − 3 M 2 M 1 + 2 M 13

(4-8)

And finally, the fourth moment is given by M 4 = E ⎡⎣( St )4 ⎤⎦

(4-9)

where: M4 the fourth moment The fourth moment relates to the kurtosis of the distribution as follows: Kurtosis = E ⎡⎣( St − St )4 ⎤⎦ = M 4 − 4 M 3 M1 + 6 M 2 M12 − 3 M14

(4-10)

Note: In the case where the distribution is centered around the mean of zero, the first moment would equal zero. Thus, in this special case, the second moment, M2, equals the variance; the third moment, M3, equals the skew; and the fourth moment, M4, equals the kurtosis.

4.2.2.3. Relating Actual and Model Distributions As suggested, distribution analysis plays a key role in judging the appropriateness of models. We use the distribution moments to relate


79

the actual to the model-generated distributions. Given a spot price model, the model defines what the distribution of the spot prices should look like at some point in time. This model-generated distribution shows all the possible values that the spot price might take at a single instant in time, with associated probabilities. However, the actual historical data that we have to work with represent not a distribution of prices at a single point in time, but rather a path of prices over a period of time. Thus, the actual historical prices reflect a price distribution of a path of prices over time. In order to be able to relate our modelimplied distributions to the historical price distributions, we need to define a set of moments that represent time averages over a period of time t: M nAV =

1 T

∫

T

t

dt M n (t )

(4-11)

where: Mn(t) the n-th moment as observed at time t.

4.2.2.4. Useful Common Distributions Another nice thing about distributions is that we understand very well certain general types of useful and common distributions. When we recognize a type of a distribution, we can then apply what we know about that type of a distribution. Two types of distributions are commonly used in valuation and risk management of derivative products:

• Normal distributions (see Figure 4-4) are commonly used throughout science and business. A normal distribution is perfectly symmetric. The skew is equal to zero, with a kurtosis given by three times the variance squared. If the mean is zero, this can be written as:

(

M 4Normal = 3 M 2Normal

)

2

(4-12)

• Lognormal distributions (see Figure 4-5) are often used in financial models. The lognormal distribution is skewed to the right and its values are always positive.

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F I G U R E

4-4

Normal Distribution

4.2.2.5.

Rate of Growth of the Width of the Distribution: m2/ T Understanding the behavior of a marketplace translates, to a large degree, in understanding the realized forward price values and the behavior of spot and forward price volatilities and correlations. Perhaps the biggest modeling test is whether or not the model predicts the types of volatility term structures we observe in the marketplace. These volatility term structures are directly tied to the way in which the price distributions grow through time. For example, the width of the lognormal price distribution grows differently in time than does the F I G U R E

4-5

Lognormal Distribution


81

width of a price mean-reverting price distribution, as we have already discussed in the previous sections. If we define the normalized second moment as, m2 ≡

M2 ( M1 )2

=

(

Et ⎡⎣ ST 2 ⎤⎦ Et ⎡⎣ ST 2 ⎤⎦

)

2

,

then we have as a measure for the rate of growth of the width of the price distribution, m2 / T. This rate of change is directly related to both the discrete volatility term structure of the underlying price process as well as the effective average volatility of the price distribution over time.

4.3.

OTHER STATISTICAL TESTS In addition to time series and distribution analyses, we can add several statistical tests to our modeling toolbox.

4.3.1.

The Q-Q Plot

A common modeling process includes determining if a data set is normally distributed. For example, consider residuals (between actual and model estimated values) that are assumed to be normally distributed. We need a test to check that they are indeed normally distributed and include no bias. One test for normality is the quantile-to-quantile (Q-Q) test or plot. It also provides a quick visual test. The Q-Q test compares the actual probabilities of the random variable to the expected probabilities if this variable were normally distributed. If the variable is indeed normally distributed, the Q-Q plot looks like a nice diagonal line, indicating that the actual variable probability distribution matches the expected probability distribution for a normally distributed variable. Figure 4-6 shows a Q-Q plot for a normally distributed variable.

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Energy Risk

F I G U R E

4-6

Q-Q Plot for a Simulated Normal Variable

Figure 4-7, on the other hand, shows the Q-Q plot for the case where the random variable is not normally distributed. The actual probability distribution does not match the expected distribution for a normally distributed variable, and we do not get a one-for-one fit from the Q-Q plot. In fact, this kind of “S”-shaped Q-Q plot tells us the following. The rather flat and wide middle section implies that the variable has too many occurrences of values in the middle range, more than it should given that it is supposed to be normally distrib-

F I G U R E

4-7

Q-Q Plot for a Simulated Nonnormal Variable


83

for a normally distributed variable. Figure 4-6 shows a Q-Q plot for a normally distributed variable. Figure 4-7, on the other hand, shows the Q-Q plot for the case where the random variable is not normally distributed. The actual probability distribution does not match the expected distribution for a normally distributed variable, and we do not get a one-for-one fit from the Q-Q plot. In fact, this kind of “S”-shaped Q-Q plot tells us the following. The rather flat and wide middle section implies that the variable has too many occurrences of values in the middle range, more than it should given that it is supposed to be normally distributed. Similarly, the flat ends of the S-shaped graph tell us that the tails of the variable distribution are not what they would be if they indeed were normally distributed. Mathematically, the basic idea is that a normally distributed random variable, with a standard deviation of one, will have the probability of having some value k defined as follows: ⎛ e− ( k 2 / 2 ) ⎞ p( x = k ) = ⎜ ⎟ ⎝ 2π ⎠

(4-13)

Every type of distribution has a specific probability function, which we can use when taking expected values, a process that is very important in valuation and portfolio analysis.

4.3.2.

The Autocorrelation Test

Another test for normality is the autocorrelation test. If indeed a random variable is normally distributed, then the variables will take on values that are uncorrelated. For example, let us say that we go on a ten-step random walk. If our steps are normally distributed, then every step we take will be independent of any of the steps we have already taken. The autocorrelation analysis tests that this is indeed true. The autocorrelation test calculates the various correlations between the steps taken: for adjoining steps, for once-removed steps, for the steps two steps removed, and so on. If indeed the steps are uncorrelated, then all the correlations between the steps will be zero. Figures 4-8 and 4-9 show the plots of two sets of autocorrelations, one for price returns that are normally distributed and one for price returns with mean reversion.

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F I G U R E

4-8

Autocorrelations for Sample Lognormal Price Returns

4.3.3.

Measures of Fit

As described above, the primary application of time series analysis is in the calibration of model parameters using statistical iterations applied to a distribution of actual prices. The optimum parameters are often judged by measures of fit, including the R2 statistic. The “square root of mean-squared error” is another measure of fit directly related to the R2 statistic.

4.3.3.1.

Mean-Squared Error

F I G U R E

4-9

Autocorrelations for Sample PMR Price Returns


85

The mean-squared error is the standard deviation of model residuals. Because we would like our model to predict as much of the actual market data as possible, we would therefore like the mean-squared error to be as small as possible.

4.3.3.2. R-Squared The R2 statistic is a measure that tells us how much of the actual uncertainty in the actual data is captured (or explained) by the model being tested. R2 is measured in percentage terms. If R2 equals 1.0, the model has 100% predictive power. On the other hand, if R2 equals zero, then the model has no predictive value. Specifically, the statistic is given by the following:

⎛ Mean-Squared Error ⎞ R2 = 1 − ⎜ ⎝ Var(actual data) ⎟⎠

(4-14)

And in the case where the mean of the residuals is zero, we have

⎛ Var(model residuals) ⎞ R2 = 1 − ⎜ ⎝ Var(actual data) ⎟⎠

(4-15)

4.4. HOW STATISTICS HELPS TO UNDERSTAND REALITY Now that we have laid some ground rules, let us start with a simple example of how our statistical methods can be used to understand market reality. Then we will proceed to more complex, realistic examples.

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F I G U R E

4-10

Path 1 Time Series

4.4.1.

A Simple Case

Consider an oversimplified example of extreme price mean reversion. Suppose, in a particular market, the price of electricity jumps between $10 and $12 from day to day for a year.5 Using time series analysis of price returns, we calculate a huge annualized daily volatility of over 300%. Does such a large volatility tell us everything we need to know about this price behavior? Definitely not. If we simply assumed that this was a lognormal price process, we would expect prices to range roughly between $0 and $40 roughly 66% of the time, and to be outside this range roughly 34% of the time. But if we also perform distribution analysis, we find that the price distribution over the year remains very narrow. By combining time series and distribution analysis, we can tell the full story of this particular market price behavior. The end result is that a simple lognormal model would not be appropriate for this price behavior.

4.4.2.

The Difference Between Price and Return

Additional examples are provided in Chapter 3 where, for example, we present two price paths that have the same resulting price–return distributions but very different price distributions (see Figures 3-8


F I G U R E

87

4-11


through 3-13). Figures 4-10 through 4-15 show yet another set of two paths. Although Paths 1 and 2 share identical price distributions, their price–return distributions vary significantly. In the above examples, by neglecting to analyze either the price returns or the price distribution, we exclude key information from our analysis.

4.4.3.

F I G U R E

Distinguishing Drift Terms

4-12


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F I G U R E

4-13

Path 2 Time Series

From a modeling point of view, perhaps the biggest reason for using both time series and distribution analyses relates to the need for the use of both methods in order to fully capture the effects of the drift and stochastic elements of a price model. Both drift (deterministic) and stochastic (random) elements contribute to price levels and their returns. The problem is that the stochastic behavior captured by the price return is generally much greater in magnitude than the deterministic behavior. We perform time series analysis of actual market data to estimate model parameters. However, we also need to perform distribution analysis to visibly identify the F I G U R E

4-14



F I G U R E

89

4-15


deterministic behavior over time. This phenomenon also represents one of the most important reasons why using time series analysis cannot enable us to judge between competing models and why distribution analysis is absolutely required.

4.5. THE SIX-STEP MODEL SELECTION PROCESS Our essential statistical tools provide the benchmarks to test the effectiveness of individual models and to compare alternatives. In order to get the “full story,” we will apply different types of reality tests to capture the various behavior characteristics:

• Time series analysis of price returns • Distribution analysis of price levels • Other statistical tests Remember: A good model should be able to capture most of the market characteristics defined by the different types of analysis. Selecting the most appropriate model involves the following six-step model selection process: Step 1: Informally look at the actual market data.

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Step Step Step Step Step

2: Create a shortlist of possible models. 3: Calibrate parameters through time series analysis. 4: Generate distributions from models. 5: Perform distribution analysis. 6: Compare results and select the most appropriate model.

4.5.1.

Step 1: An Informal Look

The very best way to compare models is to begin by analyzing the market data. The first look at the data should involve a descriptive, nonquantitative analysis of the price time series. Ask an experienced staff member to comment on the price spikes or the market turbulence in the data and how these can be related to any fundamental drivers. Such comments can be extremely helpful in getting a first feeling for the price behavior.

4.5.2.

Step 2: A Shortlist of Possible Models

The second step involves creating a shortlist of models that should be considered. The models selected should offer characteristics that fit with what one would expect are the market characteristics based on an informal review of the data or experience in the market. In this book, our short list will include the lognormal price model and mean-reversion models.

4.5.3.

Step 3: Time Series Analysis

The third step of the analysis includes a close look at the historical time series of daily price returns. Analyzing the price returns provides us with estimates for model parameters and seasonality parameters. The analysis of daily price returns yields the model parameter values, which are important in defining the day-to-day behavior of spot prices. Specifically, the expected daily drift in the spot price returns—


91

while generally insignificant compared to the magnitude of the stochastic portion of the daily price return—can still be calibrated from the time series of price returns. The autocorrelation of price returns might point at whether a mean-reverting model might be more appropriate than a non-meanreverting model. Negative autocorrelation is a sign of strong mean reversion in the spot prices. However, performing a rough autocorrelation measurement that assumes a constant drift term will carry a good amount of noise in its estimate. This noise may be overpowered in the case where the mean reversion is very strong. The daily price returns are also valuable sources of information on market volatility, both in noneventful times and during events. Furthermore, they show the effects of events on the marketplace in terms of how long the events tend to stick around and affect the prices. Because the stochastic term is generally so much greater than the deterministic term in the daily spot price returns, chances are that any model fitted to the daily price returns will yield roughly the same R2 values, and they will typically be small. This is all the more true in energies, which have generally much greater daily spot volatility than can be seen in interest rate markets or even equity markets, resulting in a stochastic term that therefore has a greater power over the deterministic term. Using R2 values as a means of benchmarking in the analysis of daily price returns is of no real value when the R2 values are roughly the same across various models. To conclude this third step, the means of testing the spot price model for performance when applied to the daily price returns is not through the model’s forecasting power, but rather through its giving us residuals that are normally distributed. The normality tests, such as the Q-Q plots or the autocorrelation analysis of the residuals, are an indicator of how appropriate the model is to the actual spot price behavior.

4.5.4.

Step 4: From Underlying Price Models to Distributions

The next step is to generate distributions based on the model(s) that are being tested. Every spot price model implies a particular price distribution. Given a model, we can formulate the spot prices as functions of time, the model parameters, and random variables that are normally

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distributed. From this formulation we can either mathematically calculate what the model-implied moments ought to be, or we can simulate the prices and from these simulations estimate the moments. Either way, a model will give us a set of price distribution moments for each point in time. We can integrate these moments over time to obtain the model-implied price distribution moments over a period of time. Although the spot price models specifically provide us with a means of defining the spot price behavior from day to day, their implied distributions provide us with a means of looking at what the models tell us about price behavior over a longer period of time. A good model, which truly captures market reality, will do so both in the short term and in the long run. A good model will capture the day-to-day market behavior characteristics as well as the long-term market price distribution characteristics. When we perform time series analysis and extract the normally distributed residual terms, we should check that these normally distributed residuals are indeed what the model claims they are: normally distributed. As stated earlier, a normally distributed variable will have a well-defined distribution, with well-defined probabilities of the variable reaching certain values. The Q-Q plot and autocorrelation tests are ideal for checking for normality.

F I G U R E

4-16

Comparison of Lognormal and Mean-Reverting Price Distributions 1 Year Out (Spot Vol 100%)


93

4.5.5.

Step 5: Distribution Analysis

Distribution analysis gives us the means of understanding how the fundamental drivers of the marketplace and the financial models ultimately converge to reflect the characteristics of the price behavior. The financial models, through a bit of mathematics or simulations, can be used to tell us how the price distributions ought to look over time, given the particular model assumptions. The fundamentals of the actual market give us the historical price distributions. The comparison of the two tells us how well the models capture the reality. Figure 4-16 shows us two types of price distributions: one is the distribution resulting from a lognormal spot price model, and the other is the distribution resulting from a price mean-reverting spot price

model. In the simulations of these models, both models were given the same daily price return volatility, or randomness, and yet the price mean-reverting distribution has a much narrower width as compared to the lognormal. The above example follows in the footsteps of the two-path examples provided in Chapter 3 (Figures 3-8 through 3-13) and earlier in this chapter (Figures 4-10 through 4-15) to show us the value of distribution analysis. Distribution analysis is the necessary tool in deciding how well the pricing model fits the market reality. It provides us with an almost immediate visual test, and also with a means of translating what might appear as very theoretical and nonintuitive modeling concepts into the concrete reality of price behavior. It is the ultimate benchmarking tool between models.

4.5.6.

Step 6: Select the Most Appropriate Model

Once all the work is done in the first five steps, the energy risk manager must weigh all the evidence and select the most appropriate model. One should employ the types of intuition, statistical tests, and business judgment described through the first four chapters of this book.

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4.6.

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RELEVANCE TO OPTION PRICING 1. William H. Press, William T. Vetterling, Saul A. Teukolsky, Brian P. Flannery, Numerical Recipes in C (Cambridge: Cambridge University Press, 1992) p. 1. 2. Ideally, these statistical tools should always remain simply that: tools. Our objective is to let the market lead us to the appropriate model. In practice, however, the problem of the “mad scientist” often arises: the tools become a means unto themselves. In such a case, we run the danger of having the model tell us what the market ought to be, and not what it is. 3. Useful books on statistics include Anderson, Sweeney, and Williams, Statistics for Business and Economics. Minneapolis: West, 1970; Mathews and Walker, Mathematical Methods of Physics (Glenview, IL: Addison-Wesley, 1996). 4. Mathews and Walker, Mathematical Methods of Physics, pp. 381–82. 5. This would hardly qualify as “random” price behavior. However, for the sake of its educational value, let us treat it as such.

C H A P T E R

5

Spot Price Behavior Imperfect price discovery and the unconventional behavior of energy prices is the first obvious problem any energy markets practitioner will encounter . . . The rapid evolution of the industry and the changing regulatory landscape make historical data irrelevant to the current problems. Another set of problems results from the limited applicability of the stochastic processes used widely in the financial markets to the modeling of the dynamics of energy prices. A combination of seasonality, frequent jumps and dependence of price behavior on the environmental variables, such as weather and the condition of the physical industry infrastructure, creates serious challenges to any trader, quant or risk manager.

Vincent Kaminski1

5.1.

INTRODUCTION All fundamental and quantitative modeling starts with spot price behavior. Supply and demand effects converge in the spot market prices, and all derivative contracts anticipate this convergence. If we can fully understand the market behavior of spot prices, we will possess the means for valuing and managing energy derivatives. In this chapter we will follow the six-step model selection process introduced in Chapter 4 in order to identify the most appropriate model for energy spot prices: Step Step Step Step Step Step

1: Informally look at the actual market data. 2: Create a shortlist of possible models. 3: Calibrate parameters through time series analysis. 4: Generate distributions from models. 5: Perform distribution analysis. 6: Compare results and select the most appropriate model.

As will be seen, the mean-reverting models are the most appropriate for energy spot prices and will serve as the basis for much of the valuation 95 Copyright © 2007 by Dragana Pilipovic. Click here for terms of use.

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and risk management methodologies in this book. We will conclude this chapter with a special paper on “Locational Marginal Pricing” as a methodology for clearing electricity spot markets used by a number of U.S. power markets and generously provided by Francis Wang of Entergy Services. Entergy Services generously provided the data analyzed in this chapter.

5.2.

LOOKING AT THE ACTUAL MARKET DATA We will focus our analysis on natural gas and power markets. All the spot price samples in this chapter snap a good number of years. As you can see from the graph of the natural gas data (Figure 5-1), the natural gas market appears to have entered a new and more volatile stage of price behavior beginning in 2001. The power data do not go as far back in time as the natural gas markets, but there is still plenty of opportunity to show us the price magnitudes these markets are capable of under events. The markets to be analyzed include:

• Henry Hub Natural Gas Spot: Figure 5-1 plots the time series for the natural gas market from March 1991 to August 2006. • Massachusetts Hub On-Peak and Off-Peak Power: Figures 5-2 and 5-3 plot the time series for the on-peak and off-peak spot prices from March 2003 to August 2006. • Massachusetts Hub On-Peak Hourly Power Markets: Figure 5-4 plots the time series for all the on-peak hourly prices between March 2003 and August 2006. • Massachusetts Hub Off-Peak Hourly Power Markets: Figure 5-5 plots the time series for all the off-peak hourly prices from March 2003 to August 2006. • New York A Hub On-Peak and Off-Peak Power Prices: Figures 5-6 and 5-7 plot the time series for the on-peak and off-peak power prices at New York’s delivery hub A from November 1999 to August 2006. • New York C Hub On-Peak and Off-Peak Power Prices: Figures 5-8 and 5-9 plot the time series for the on-peak and off-peak power prices at New York’s delivery hub C (just outside New York City) from November 1999 to August 2006.

Spot Price Behavior

F I G U R E

97

5-1

Time Series of Henry Hub Natural Gas Spot Prices (1991–2006)

F I G U R E

5-2

Time Series of Massachusetts Hub On-Peak Prices (2003–2006)

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Energy Risk

F I G U R E

5-3

Time Series of Massachusetts Hub Off-Peak Prices (2003–2006)

F I G U R E

5-4

Time Series of Massachusetts Hub On-Peak Hourly Power Markets (2003–2006)

Spot Price Behavior

F I G U R E

99

5-5

Time Series of Massachusetts Hub Off-Peak Hourly Power Prices (2003–2006)

F I G U R E

5-6

Time Series of New York A Hub On-Peak Power Prices (1999–2006)

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Energy Risk

F I G U R E

5-7

Time Series of New York A Hub Off-Peak Power Prices (1999–2006)

F I G U R E

5-8

Time Series of New York C Hub On-Peak Power Prices (1999–2006)

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101

F I G U R E

5-9

Time Series of New York C Hub Off-Peak Power Prices (1999–2006)

• New York G Hub On-Peak and Off-Peak Power Prices: Figures 5-10 and 5-11 plot the time series for the on-peak and off-peak power prices at New York’s delivery hub G from November 1999 to August 2006. The graphs of the off-peak power prices above include both business days and weekends—all the days that off-peak power is traded. The weekend off-peak power prices include around-the-clock delivery, whereas the weekday off-peak prices cover only the eight off-peak hours of the day: the weekend off-peak prices do not represent the same hourly periods of the day as the weekday off-peak prices. Therefore, to make the analysis meaningful, the weekend off-peak prices were excluded in the analysis. All of the above are time series of spot prices. However, there are times when it is difficult to obtain a history of spot prices, but a history of exchange traded futures prices is readily available. In the absence of spot prices, first nearby forwards may be used as proxies for spot prices. In this case, the time series analysis becomes a far more complicated matter. In an efficient market, the expected value in the change of the forward price from today to tomorrow is zero (in other words, in an efficient market, the market forward price today is the best representation of what the market expects the forward price to be tomorrow, and the next day, and the next, etc.); thus we would expect to calibrate a zero drift term. However, given the physical nature of energy markets with potentially limited response times to event situations, and given the

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F I G U R E

5-10

Time Series of New York G Hub On-Peak Power Prices (1999–2006)

F I G U R E

5-11

Time Series of New York G Hub Off-Peak Power Prices (1999–2006)

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103

magnitudes of event behaviors common in energy markets, it is not only possible to capture nonzero drift terms, but to even observe mean reversion within time series analysis of first nearby futures (as was observed in the analysis of historical data sets used for the first version of this book). Although the time series analysis of the first nearby forward contract may be questionable, the distribution analysis is not—we should still be able to see the general price distribution characteristics consistent with the underlying spot price model. To add complexity to the situation, the forward prices might actually be monthly discountweighted averages of daily forward prices—the resulting calibrated values may thus be even further diluted by the averaging effect. For market descriptions, analysis, and results published in the original version of Energy Risk, please see Appendix C.

5.3.

A SHORTLIST OF POSSIBLE MODELS The set of models we will discuss are the basic lognormal model and two mean-reverting models seen in the energy markets.

5.3.1.

The Lognormal Price Model

The lognormal model is the most famous model of all, particularly in nonenergy markets. It is extremely simple to use and, as such, provides a good amount of flexibility in its implementation. In a single-factor lognormal model, the change in the price from time t to time t dt, dSt, where dt is very small (dt 1), is given by Equation 5-1: dSt = µ St dt + σ St dzt

(5-1)

where: S spot price t time of observation the drift rate

volatility dz˜ random stochastic variable This change in the price over time dt has two components, the first being the drift, or deterministic term of St dt. The second component

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is the stochastic, or random, contribution to the change in the spot price, Stdz˜t. Both the drift and the stochastic terms are proportional to the spot price level at time t. The greater the price, the greater is both the expected change in the price and the randomness about it. The stochastic term contains the variable dz˜t, which is a normally distributed random variable with a mean of zero, and a standard deviation that grows as the square root of time dt: dz ∼ N (0, dt )

(5-2)

STD( dzt ) = dt

(5-3)

We use the differential Equation 5-1 to solve for the spot price as a function of its model parameters, including the stochastic variable dz˜t, and also to learn about the characteristics of the spot price behavior under the assumption that the spot prices are lognormal. In order to solve for the spot price, we use some tricks of the trade. We start by performing a variable transformation where we define a new variable, xt, to be the natural log of the price: xt ⬅ ln (St )

(5-4)

By applying Ito’s Lemma to the new variable, we find out that it is normally distributed:

⎛ µ −σ2 ⎞ dxt = ⎜ ⎟ dt + σ dzt ⎝ 2 ⎠

(5-5)

This allows us to first solve for the new variable x˜t, and from this solution to derive the solution for the spot price at time T contingent on the spot price at time t: ST = St e

⎛ σ2⎞ ⎜ µ − ⎟ ( T −t )+σ dzt ,T 2⎠ ⎝

(5-6)

t

where: S˜T|t spot price at time T contingent on spot price at time t By taking the expected value of both sides of the above equation, we obtain Equation 5-7, the solution to the expected spot price at time T as observed from time t: Et ⎡⎣ ST ⎤⎦ = St e µ (T −t )

(5-7)

Spot Price Behavior

105

As can be seen from the above derivation, in a lognormal model, the expected spot prices grow exponentially over time, with an expected rate of return given by the Greek character, . Note that the randomness in the price over time is always in the exponential, guaranteeing that the prices will always be positive. If the random variable, z˜, takes on very large negative values, the spot prices approach zero, but are never negative. One of the reasons why the lognormal model is so popular and why so many academics like it is this latter fact that it guarantees that the prices will never be negative. Similarly, the second moment and its time derivative for the lognormal price can be formulated:

)

(

Et ⎡⎣ ST 2 ⎤⎦ = Et ⎡⎣ ST ⎤⎦ eσ 2

2

( T −t )

(5-8)

We have defined in Chapter 4 a normalized second moment as m2 ⬅

M2 ( M1 ) 2

=

Et ⎡⎣ ST 2 ⎤⎦

( E ⎡⎣ S ⎤⎦) t

2

(5-9)

T

We then have for m2 and its rate of change, in the case of a lognormal price process:

m2LN = eσ

2

( T −t )

∂ m2LN = σ 2 m2LN ∂T

(5-10)

(5-11)

The time derivative of the normalized second moment (Equation 5-10) tells us that the width of the normalized distribution will be expanding proportional to the spot price volatility at all times. In this case the Black-equivalent volatility of the price process is always the spot price volatility.

5.3.2.

Mean-Reverting Models

As will be seen from our analysis results, energy markets require mean-reverting models. In fact, the price mean-reverting model turns out to do the best job of capturing the distribution of energy prices. The

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log of price mean-reverting model performs not too badly in capturing the distribution width (i.e., the second distribution moment and standard deviation), but does a poor job of capturing the distribution’s tails (i.e., the fourth distribution moment and kurtosis). Both energy models presented here have the characteristic of giving the changes in the spot prices a negative autocorrelation, a characteristic that is very much a part of the energy markets, particularly electricity. In fact, autocorrelation is a characteristic of a mean-reverting model. Both models presented here are mean-reverting models, one with mean reversion in the log of the price, and the other with mean reversion in the price. 5.3.2.1. Mean Reversion in Log of Price Mean reversion in the natural log of the spot price (following the works of Schwartz and Vasicek) is one of the models used in the energy markets, particularly in electricity. The resulting spot prices are very much like prices derived from other interest rate models. However, it is a bit simpler to use. The nonnegative nature of spot prices is preserved through the modeling of the log of the price rather than the price itself. The mean reversion is therefore applied to the log of the price rather than to the price itself: xt ⬅ ln( St )

(

)

dxt = α b − xt dt + σ dzt

(5-12) (5-13)

where: S spot price t time of observation rate of mean reversion

volatility b long-term equilibrium of x dz˜ random stochastic variable (Note: Equation 5-13 ties back to the Schwartz model by allowing b ⬅ 2/2.) From the above differential equation (5-13), we can solve for the log of the price, x˜T, conditional on time t: xT

t

⎧ xt e−α ( T −t ) + b(1 − e−α ( T −t ) ) ⎫ ⎪ ⎪ =⎨ T ⎬ − α ( T −t ) α ( q−t ) ⎪⎩+σ e ∫q=t dzt ,q e ⎪⎭

(5-14)

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107

and therefore also for the spot price itself:

( )

ST = St t

e− α ( T − t )

( )

⋅ S

(1− e− α ( T − t ) )

⋅ exp ⎛ σ e− α ( T −t ) ∫ dzt ,q eα ( q−t ) ⎞ ⎝ ⎠ q =t T

(5-15)

where: the spot price at time T is contingent on the spot price at time t. From this we can obtain the expected spot price at time T as observed from time t:

( )

Et ⎡ ST ⎤ = St ⎣ t⎦

e− α ( T − t )

( )

. S

(1− e− α ( T − t ) )

⎛σ2 ⎞ x p (1− e−2α ( T −t ) )⎟ ⋅e ⎜ ⎝ 4α ⎠

(5-16)

Similarly, we can just as easily obtain the second moment:

( ) ( )

⎡ Et ⎢ ST ⎣

2

t

⎤ ⎥⎦ = St

2 e− α ( T − t )

( )

⋅ S

2 (1− e− α ( T − t ) )

⎛σ2 ⎞ ⋅ exp ⎜ (1 − e−2α (T −t ) )⎟ ⎝α ⎠

)

(

2 ⎛σ2 ⎞ = Et ⎡ ST ⎤ exp ⎜ (1 − e−2α (T −t ) )⎟ ⎣ t⎦ ⎝ 2α ⎠

(5-17)

As can be seen from the resulting second moment, Equation 5-16, the second moment approaches the square of the first moment times an exponential, the further out in time we go (T >> t), Now we can obtain the rate of the change of the normalized second moment to tell us how the normalized price distribution width will change with time. Equation 5-17 gives us this rate of change:

(

∂ m2LMR ∂T

) =σ e

2 −2α ( T − t )

(5-18)

m2LMR

Equation 5-17 clearly shows that the change in the normalized second moment gets smaller and smaller the further out in time we go, due to the exponential term. The stronger the mean reversion, the quicker this drop off in the magnitude of change occurs. In the case where T is very close to t, that is, T ⬅ t dt and dt > t, the expected spot price and the rate of change of the normalized second moment approach the following values: Et ⎡⎣ ST ⎤⎦ ≅ Se(σ and

(

∂ m2LMR ∂T

2

/ 4α )

)≅0

(5-21)

(5-22)

In fact, the Black-equivalent volatility of this price process over time is given by: LMR σ Black =σ

(1 − e−2α (T −t ) ) 2α (T − t )

(5-23)

As can be seen, the drawback of a single-factor mean-reverting model is that it forces the implied Black-equivalent average volatility of the price distribution to go to zero over a long period of time (as the spot prices approach the immobile long-term mean level). Therefore, caution must be used whenever using a single factor mean-reverting model in valuing longer-term options. Clewlow and Strickland expanded on the mean-reverting model in the log of price to also include a jump process (inspired by Merton’s jump-diffusion model), in order to much better capture the types of price behaviors we see in power prices, not to mention the volatilities that neither the log-of-price mean-reverting model nor Merton’s jump diffusion model can capture individually. By adding a jump-diffusion term they propose what promises to be a rewarding energy spot price model worth the effort of further research and implementation:

(

)

dxt = α b − xt dt + σ dzt + κ dqˆ

(5-24)

where dqˆ is a discrete time process and the natural log of (1 ) is normally distributed (see Clewlow and Strickland’s book Energy Derivatives, Chapter 2). The price process simulated using this model closely resembles the look of an actual energy price time series. When implementing the jump process within forward price and option models, assumptions must be made about expected number of

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109

jumps, their magnitudes, and the times of occurrence. For example, if we have a monthly forward price market quote of $150, should we assume that the daily forwards “average” out—using proper discounting—to $150, or should we assume that the daily forwards “average” out to some smaller price with a certain number of jumps. If we assume the latter, what kinds of average jump sizes are we talking about (which in turn will define the number of jumps) and finally, when exactly do they occur? The answers to these questions can significantly change option valuation for both European- and American-style daily settled options. Ultimately, whether or not we incorporate jumps into our pricing models comes down to both how we look at the world and how we choose to estimate its representation. Although we will not be calibrating jumps within this text, hopefully further market research will be done on the implementation aspects of the mean-reverting jump model. 5.3.2.2. Mean Reversion in Price Another model that has been used in the energy marketplace is a twofactor model, where the first factor is the spot price, and the second factor is a long-term equilibrium price (the Pilipovic Model). The spot price is assumed to mean-revert toward the equilibrium price level, and the equilibrium price level is assumed to be lognormally distributed: dSt = α ( Lt − St ) dt + Stσ dzt

(5-25)

dLt = µ Lt dt + Lt ξ dw t

(5-26)

where: S the spot price L the equilibrium price t time of observation rate of price mean reversion

volatility drift of the long-term equilibrium price volatility in the long-term equilibrium price dz˜ random stochastic variable defining the randomness in the spot price ˜ random stochastic variable defining the randomness in dw the equilibrium price

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We can solve these differential equations to obtain the spot price and its expected value: 1

− (α + σ ST = e 2

2

)( T −t )+σ zt ,T

t

+ α Lt e

St

1 − ( α + σ 2 )( T −t ) 2

T

⋅ ∫ dqe

1 1 ( µ − ξ 2 )( q −t )+σ w t ,q ( α + σ 2 )( q − t )+σ zq ,T 2 2

(5-27)

e

q =t

Et ⎡⎣ ST ⎤⎦ = e−α ( T −t ) St + kLt ( e µ ( T −t ) − e−α ( T −t ) )

(5-28)

= e−α ( T −t ) ( St − kLt ) + kLt e µ ( T −t )

where T some future point in time, i.e., T t, and K⬅

α α+µ

(5-29)

We can approximate K with the value of one in the case where the mean reversion, , is much greater than the equilibrium price rate of return, , giving us Et ⎡⎣ ST ⎤⎦ ≈ e−α ( T −t ) ( St − Lt ) + Lt e µ ( T −t )

(5-30)

If we further assume that >> , , (which turns out to be a good assumption in the energy markets), we have the second moment given by Equation (5-31): Et ⎡⎣( ST )2 ⎤⎦ ≈ e−2α ( T −t )+σ

2

( T −t )

( S t − Lt ) 2

+ 2e−α ( T −t ) ( Lt e µ ( T −t ) )( St − Lt ) + ( Lt e µ ( T −t ) ) 2 eξ

2

( T −t )

(5-31)

From the above, we can easily obtain the rate of change of the normalized second moment. In the long term, as T >> t, we have the spot price approaching the equilibrium price growing at the drift rate, and we have the width of the distribution continue growing proportional to the equilibrium price volatility: Et ⎡⎣ ST ⎤⎦ ≅ Lt e µ (T −t )

m2PMR ≅ eξ

(

∂ m

PMR 2

∂T

2

(5-32)

( T −t )

(5-33)

) ≅ξ m 2

PMR 2

(5-34)

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111

The Black-equivalent volatility in this case becomes the equilibrium price volatility in the long run; the effective volatility of the spot price process never goes to zero.

5.3.3.

Cost-Based Models for Electric Utilities

Historically, electric utilities have used cost-based or structural models to arrive at expected costs in regulated markets. These fundamental cost distribution models for electricity tend to tie in the integrated production cycle for electricity generation in order to arrive at future expected spot prices, for electricity, as well as their distributions. Such models are excellent for understanding the characteristics of electricity cost behavior unique to a particular utility. However, they do not tell the full market price story. First, the cost is not also the market price of electricity. Second, such distribution models cannot satisfy the arbitrage-free requirements of forward prices. Third, and most important, these models do not allow for mark-to-market valuation. The end result of using a cost-based model to value the products in a book is that it gives the energy producers their internal mark-to-cost valuation, and not the mark-to-market valuation. Ideally, a producer has both a cost-based model and a mark-to-market financial model. The different book values resulting from the two approaches define the potential producer-specific arbitrage opportunities.

5.3.4.

Interest Rate Models

Note: Appendix B contains several interest rate and bond models provided for comparison and reference.

5.4.

CALIBRATING PARAMETERS THROUGH TIME SERIES ANALYSIS Next, we will perform the time series analysis of prices and their daily returns. The first step will be for us to perform the time series analysis of price returns in order to obtain seasonality parameters and modelspecific parameters for all three models, the lognormal model and the

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two mean-reverting models. After we perform the time series analysis, we will also calibrate the seasonality factors. Ultimately, this will allow us to perform distribution analysis on the underlying spot prices stripped of seasonality effects. In the case of the lognormal model, as captured by Equation 5-1, the time series analysis will result in estimated values for the price rate of return and the price volatility: and . In the case of the model with mean reversion in the log of the price, as captured by Equations 5-12 and 5-13, the time series analysis will provide us with estimates for the rate of mean reversion, , the log of the price around which mean reversion occurs, b, and the price volatility, . Finally, in the third case where mean reversion is in the price, captured by Equations 5-25 and 5-26, the time series analysis of spot prices will result in the rate of mean reversion, the value of the equilibrium price at the start of the historical data set, the rate of return on the equilibrium price, and the spot price volatility: , L0, , and . Ideally, if our historical price data included simultaneous snapshots through time of both spot prices as well as forward prices, we could calibrate the equilibrium price values from the forward price curves, ultimately resulting in having a time series of both the spot price and the equilibrium price. Unfortunately, if we are trying to fit a two-factor price meanreverting process, and we do not have the corresponding forward price quotes to help us in estimating the long-term equilibrium price on a day-by-day basis, we have to do with just the spot price information. For the sake of simplicity we will assume that the only data we have to work with are the market spot prices. In this case we are forced to reduce the two-factor price mean-reverting model to a single factor, only for the purposes of estimating the historical parameter values but not in forward price curve building or volatility analysis. This means that during this analysis we will assume that the equilibrium prices in the price mean-reverting model do have a rate of return but have zero volatility—that is, they are perfectly stable.

5.4.1.

Incorporating Seasonality with Underlying Models

For all three models we will assume that the spot price is a function of an underlying spot price, StUnd plus seasonal factors: St StUnd seasonality effects

(5-35)

Spot Price Behavior

113

3

St = StUnd + ∑ β n e

− γ n ( rfc ( t − tnC ))2

(5-36)

n=1

where:

St spot price at time t StUnd underlying spot price value i annual seasonality magnitude tnC annual seasonality centering parameter (time of annual peak) n seasonal decay parameter rfc an annually repetitive function; it returns the annualized time to or from the closest annual center, tnC, for that particular seasonal factor

Note that we are allowing for three seasonal factors: summer and winter seasonality, in addition to a third seasonality allowing us to capture any additional repetitive annual event behavior, such as an additional peaking behavior in the summer or winter, or an additional seasonal “hump” in the fall, for example. From the above, we can derive the change in the price over time dt as:

∂ ⎧ 3 − γ ( rfc ( t − tnC ))2 ⎫ dSt = dStUnd + ⎨∑ β n e n ⎬ dt ∂ t ⎩ n=1 ⎭

(5-37)

Thus, the seasonality terms will be defined the same way for all three models. However, the change in the underlying spot price, that is, the spot price stripped of the seasonality effects, will be defined uniquely by each model being tested. The calibration of the model-specific parameters and the seasonality parameters will be performed simultaneously. For each model we will end up calibrating the model-specific parameters, and all the seasonality parameters.

5.4.2.

Results from Time Series Analysis

The time series analysis calibrations for the lognormal, mean-reverting log of price, and price mean-reverting models are provided in Tables 5-1 through 5-3, respectively. Note that the seasonality parameter estimates

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T A B L E

5-1

Parameters from Lognormal Model Market

1

2

3

R2

R2 adj

Nat Gas Mass On Mass Off Mass Hr. 17 Mass Hr. 5 NY A On NY A Off NY C On NY C Off NY G On NY G Off

0.29 11.53 15.73 16.56 8.88 17.78 7.71 86.30 8.02 95.92 9.97

0.06 15.91 15.74 19.98 7.51 1.39 8.58 22.37 2.09 28.43 13.34

0.91 3.56 6.83 19.46 5.71 13.74 0.71 18.48 12.63 11.89 0.36

1% 367% 175% 443% 431% 240% 159% 582% 119% 684% 75%

82% 235% 203% 256% 250% 301% 242% 356% 276% 382% 276%

0.07% 0.83% 1.06% 0.82% 1.62% 0.30% 0.36% 0.92% 0.22% 1.07% 0.15%

0.04% 0.38% 0.62% 0.37% 1.18% 0.07% 0.13% 0.70% 0.01% 0.84% 0.08%

T A B L E

5-2

Parameters from Mean Reversion in Log of Price Model Market

1


0.30 25.28 19.13 34.42 10.15 22.33 8.71 34.04 26.90 60.82 8.23

2 0.02 14.33 14.41 20.14 4.98 21.36 1.10 115.26 6.11 125.44 5.70

3

S¯

R2

R2 adj

0.99 46.96 18.00 143.12 19.00 4.41 4.91 72.51 8.47 77.74 4.57

0.75 175.73 78.46 210.04 106.28 158.92 47.26 235.93 59.12 287.10 56.21

2.79 71.45 49.73 82.29 48.11 54.60 40.96 79.65 44.80 67.02 44.63

83% 235% 199% 261% 242% 299% 237% 364% 269% 397% 269%

0.23% 10.75% 5.67% 12.46% 7.49% 10.06% 3.44% 12.34% 3.53% 16.20% 3.70%

0.11% 10.24% 5.14% 11.95% 6.96% 9.80% 3.17% 12.09% 3.25% 15.95% 3.42%

also change as a function of the model being analyzed, and the differences are significant between the lognormal model and the mean-reverting models, but not that significant between the two mean-reverting models. Also, keep in mind that we had to basically “turn off” the second factor in

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T A B L E

5-3

Parameters from Mean Reversion in Price Model Market

1

2

3


0.34 0.04 1.23 19.38 12.04 53.24 15.43 14.13 17.45 31.39 16.15 153.56 17.53 4.12 18.85 41.51 15.50 4.10 25.94 10.44 10.52 26.36 119.41 86.25 47.75 1.53 7.02 54.75 124.56 79.23 27.47 7.69 11.61

L0

R2

R2 adj

14% 17% 17% 17% 17% 12% 12% 12% 12% 12% 12%

4.49 51.03 25.58 54.87 35.51 52.97 21.84 66.71 28.36 81.28 25.40

1.12 51.10 36.24 57.73 35.71 34.44 25.02 49.34 29.39 41.87 28.11

83% 233% 198% 257% 240% 299% 235% 363% 266% 392% 265%

1.05% 10.91% 6.38% 10.97% 8.66% 10.96% 5.60% 11.22% 6.40% 13.49% 6.16%

0.92% 10.40% 5.85% 10.46% 8.14% 10.70% 5.33% 10.96% 6.13% 13.24% 5.89%

the price mean-reverting model due to the lack of a simultaneous implied equilibrium price observation. However, we will still be able to imply the equilibrium price mean, as well as the equilibrium price volatility (see Section 5.5.2, “Results of Distribution Analysis,” and Table 5-7). This is probably a good place to consider the intuition behind the seasonality factor values. In the case where one seasonality factor is estimated to be significant and positive, and the other factors are roughly zero, the market typically exhibits primarily single annual peaks—either in the winter or the summer. In the case where two seasonality factor are estimated to be significant and positive, while the third factor is roughly zero, the market exhibits typically two annual peaks—in the winter and the summer. However, adding a significant third seasonal factor has been necessary in the building of forward price curves in both power and natural gas markets over the past several years (see Chapter 7 for further discussion). Hence it is no surprise that we find it necessary also in calibrating historical seasonal behavior. For all the markets analyzed here, the first seasonal factor corresponds to a summer seasonality and the third seasonal factor corresponds to the winter seasonality. Hence you see the natural gas results show a good-sized magnitude for winter seasonality across all three models, and a much smaller summer seasonality. In the case of the

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Massachusetts power market, we also see a very strong winter seasonality, corresponding to the geographic location of this power market. The second seasonal factor in the case of natural gas and the Massachusetts hub was allowed to capture any additional seasonal event behaviors in the winter, and we see the magnitudes to be close to zero for natural gas and somewhat strong in the case of the Massachusetts hub. Finally, the New York power markets appear “all over the place” when it comes to seasonal behavior. The one seasonality that jumps out as consistent is the second seasonality factor, which was allowed to capture unusual fall seasonal behaviors. In the case of the New York markets, it appears that this fall seasonality is quite strong. Typically, for both power and natural gas, there are both summer and winter peaks, but one is significantly greater than the other. In the case of electricity, the summer peak tends to be generally greater than the winter peak, as the use of electricity for cooling in the summer tends to be greater than the use of electricity for heat generation in the winter. However, this is most definitely a function of the geographic area, as can be seen from historical analysis of the northeast coast market in the United States. The model-specific parameters vary significantly from market to market. However, we can see some general behaviors here. In the case of all the markets, the spot price volatility across models is roughly the same, regardless of the model being calibrated. This indicates that the drift terms indeed are not nearly as significant as the stochastic terms, resulting in spot price volatilities that are generally indifferent to the type of drift term being calibrated. The R2 values for all three models are given in Table 5-4.2 Also note that the R2 values for the mean-reverting models are far better than for the lognormal model. This tells us that mean-reversion effects are extremely important in predicting the next day price changes. These facts should convince us that mean reversion has to be included in the modeling of spot prices for power and natural gas markets. However, the results also tell us that purely based on next day predictive power, both the log mean-reverting model and the reduced to a single factor price mean-reverting model are extremely similar. The mean-reversion parameter does appear to be fairly diverse in values in the case of the log mean-reverting model for power, but appears to be far less diverse in magnitude in the case of the price mean-reverting model for power.

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117

5-4

Adjusted R2 Summary for “Next Day” Price Change Forecasting Market

Lognormal

Log Mean Reversion

Price Mean Reversion


0.04% 0.38% 0.62% 0.37% 1.18% 0.07% 0.13% 0.70% 0.01% 0.84% 0.08%

0.11% 10.24% 5.14% 11.95% 6.96% 9.80% 3.17% 12.09% 3.25% 15.95% 3.42%

0.92% 10.40% 5.85% 10.46% 8.14% 10.70% 5.33% 10.96% 6.13% 13.24% 5.89%

Finally, we will conclude this section with a brief look at the model residuals, which we expect to be normally distributed. Table 5-4 shows the summary of the next day explanatory power for each market and each model. Tables 5-5 and 5-6 show the “3” Test and the autocorrelation values for the residuals. Table 5-5 shows the fourth moment of the model residuals divided by the second moment squared. As discussed in Chapter 3, this ratio ought to equal exactly 3 in the case of a normally distributed variable. Table 5-5 shows this ratio calculated excluding the moves of ten standard deviations or higher as outliers. With these outliers excluded, note that all three models do a fairly good job with the residuals, with the exception of the natural gas market, which could be better, but is not too bad. For the most part, the ratio is greater than 3, indicating that we really need to incorporate a second factor in the modeling. Similarly, Table 5-6 shows the autocorrelations of model residuals for just a single time lag (roughly one business day). Note that for electricity markets, we see some very strong negative correlations for the lognormal model, indicating strong mean reversion the lognormal model could not incorporate, thus resulting in high negative drift terms.

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T A B L E

5-5

The “3” Test of Moment Residuals Market

Lognormal

Log Mean Reversion



6.32 3.38 3.15 3.12 3.57 3.74 3.04 3.79 3.24 3.94 3.32

5.86 3.01 3.03 2.72 3.36 3.11 2.94 2.94 3.10 3.07 3.17

6.12 3.36 3.12 2.91 3.44 3.14 2.90 3.14 3.03 3.13 3.09

T A B L E

5-6

Autocorrelations Market

Lognormal

Log Mean Reversion



4.23% 6.21% 3.78% 10.73% 11.69% 9.15% 9.66% 10.66% 13.34% 12.41% 16.38%

7.28% 3.47% 0.63% 5.62% 11.27% 4.59% 6.76% 6.38% 12.02% 10.64% 13.79%

6.66% 3.52% 0.76% 3.52% 10.22% 6.16% 5.08% 5.34% 9.23% 6.97% 11.90%

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5.5.

119

PERFORMING DISTRIBUTION ANALYSIS During the time series analysis steps, we capture all the necessary model parameter values. In the distribution analysis we will be testing the models for how well they act over a longer period of time as compared with the actual market. It is this data analysis step that will ultimately give us the answer as to which model is consistent with both short-term and long-term price behavior. Specifically, we will be comparing the historical market spot price distributions to the distributions implied by each of the models we are testing. Although the width of the distribution will be of primary importance, ideally, we also want to look at how well the models capture the skew and the kurtosis—or tails—of the historical price distributions.

5.5.1.

Implementation of Distribution Analysis

We can perform Monte Carlo simulations in order to obtain our modelimplied distributions. The advantage of such simulations is that we get to see what the distributions look like visually, as well as estimate all the distribution moments. However, there is another way to obtain the distribution characteristics, and it does not involve simulations. Instead, we can use mathematics and the probability distributions of random variables to obtain the distribution moments implied by a model. Although this procedure does not give you the visual satisfaction of a plotted distribution, it does give you the moments calculations, which are more precise and a lot quicker to calculate on a computer than the generation of simulations. The drawback is that the procedure requires quite a bit of up-front math work and the results may be complicated. In the case of the lognormal model, the results are actually pretty simple. The moments of a lognormal distribution of spot prices over time are given in the following equations:

( eσ ( T −t ) − 1) M | = 2 (σ (T − t ))

(5-38)

( e3σ ( T −t ) − 1) M | = (3σ 2 (T − t ))

(5-39)

2

T 2 t

2

T 3 t

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( e6σ ( T −t ) − 1) M | = (6σ 2 (T − t )) 2

T 4 t

(5-40)

Calculating these moments for the log of price and price meanreverting models requires quite a bit of time and math muscle, and the results are much more complicated than what you see for the lognormal model. We are now ready to benchmark between models by performing distribution analysis. We do so by calculating model-implied moments and comparing these to the actual market price moments.

5.5.2.

Results of Distribution Analysis

The model-implied distributions second moments can be compared to the actual distributions’ second moments (these data include all the prices, including the outliers). Furthermore, we can use the actual second moment to model-imply the equilibrium price volatility in the case of the price mean-reverting model. As Table 5-7 shows, the reduced T A B L E

5-7

Market

m 2Market

m 2LN

m 2LMR

m 2PMR, 0

m 2PMR, 0

(m 2Market )


1.4453 1.1188 1.1064 1.1625 1.1107 1.1735 1.1567 1.2140 1.1703 1.2089 1.1588

3355.7385 25.6303 0.0000 29.6071 27.5025 3.24E 13 2.79E 17 2.87E 05 4.66E 15 1.98E 05 9.11E 21

1.5974 1.0160 1.0271 1.0165 1.0289 1.0287 1.0616 1.0285 1.0632 1.0279 1.0667

1.3383 1.0281 1.0309 1.0280 1.0296 1.0498 1.0495 1.0520 1.0516 1.0498 1.0495

1.4453 1.1188 1.1064 1.1625 1.1107 1.1735 1.1567 1.2140 1.1703 1.2089 1.1588

8% 20% 19% 24% 19% 16% 15% 18% 16% 18% 15%

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121

to a single-factor price reverting model does a slightly better job of matching the actual market second moment. By model-implying the equilibrium price volatility—also shown in the table—we can match the second moment exactly. Note how consistent the equilibrium price volatility range is across power markets, varying from 15% (for two off-peak markets) to 24% (for the Massachusetts Hub 17th hour power price).

5.6.

ANALYSIS SUMMARY Hopefully, this chapter has presented some case studies of market analysis, and maybe even some useful measures and insights about the energy markets. Clearly, our job here is not done. Although the above analysis certainly could have been done in different ways—looking at more discrete periods of time, looking at other models, and, perhaps most importantly, incorporating the simultaneous analysis of the forward price curve in conjunction with the analysis of the spot price—it did give us some idea as to the basic characteristics of the power and natural gas prices. The conclusions, based on the above results, are twofold: 1. Power and natural gas markets are mean-reverting. 2. Power and natural gas markets are at least two-factor markets. Far more research is needed on two-factor models. Ideally the analysis should incorporate simultaneous snapshots of both spot prices and forward prices for each energy market analyzed, allowing us to perform the simultaneous analysis of the spot price model with the corresponding forward price curve: the wealth of information from this process might well provide a breakthrough in energy price modeling. With this thought, we are ready to move on to the next big section: modeling and marking to market forward prices. This chapter concludes with a discussion of the spot price clearing process in the US markets.

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LOCATIONAL MARGINAL PRICING: AN OVERVIEW BY HUI-FUNG FRANCIS WANG The U.S. electricity spot market began when the Federal Energy Regulatory Commission (FERC) issued Order 888 to establish the Independent System Operators (ISO) to operate the transmission grid and to ensure equal access of the transmission grid by all market participants. In addition to the transmission grid security responsibilities, an ISO also administers the spot market, the Day-Ahead Market (DAM) and the Real time Market (RTM), for the deregulated electricity wholesale markets. Locational Marginal Price (LMP) methodology was first introduced by PJM in 1998 as a methodology to clear the spot electricity energy markets. Since then, New York ISO and ISO New England have adopted the LMP methodology to clear their energy markets beginning in November 1999 and March 2003, respectively. MidWest ISO started its LMP spot energy market in April of 2005. Southwest Power Pool, Electricity Reliability Council of Texas (ERCOT), and California ISO are considering switching from the current single-market clearing energy spot market to the LMP methodology in the near future. LMP is the cost of serving the next MW load at a specific location (buses and/or zones) using the lowest cost generation resources while observing all the transmission constraints. LMP produces a singlemarket clearing price when the transmission system is not congested and it produces the nodal prices when the transmission system is congested. The nodal price is the direct result of LMP methodology to allocate the congestion costs to those locations causing congestion.4 For example, consider two zones in a market area. Zone A has a generation of 1,000 MW and a marginal cost of $25/MWh, and zone B has a generation resource of 500 MW and a marginal cost of $60/MWh. The two zones are connected via a transmission line with a maximum transfer capability of 300 MW. During off-peak hours, the total system load is 450 MW; zone A has a load of 200 MW and zone B has a load of 250 MW. In this case, the ISO dispatches the least-cost Generator A to output 450 MW of which 200 MW services the local load of 200 MW and 250 MW flows across the transmission line to serve the 250 MW load in zone B. Generator B is not dispatched. The transmission line carries 250 MW from zone A to zone B, and so remains under the 300 MW limit. LMP for both zones A and B is $25/MWh. In this example, the LMPs are the same across the zones in the market area when the transmission system is not congested. During peak hours, the total system load increases to 1,000 MW; zone A has a load of 400 MW and zone B has a load of 600 MW. ISO dispatches the least-cost generator

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123

A to output 700 MW, of which 400 MW serves the local load of 400 MW and 300 MW flows across the transmission line to serve zone B. The transmission line is now fully loaded to carry 300 MW. This transmission constraint is binding. Generator B has to be dispatched for the remaining 300 MW. In this case, LMP is $25/MWh for zone A and $60/MWh for zone B. The difference of the LMP between zone A and the LMP at zone B is the congestion cost zone B pays for the congestion it causes during the peak hours. An LMP map from the New York ISO on November 4, 2006, 8:00 am ET is shown in Figure 5-12. It depicts the LMP of the eleven zones in the New York ISO electricity market. New York City and Long Island have higher LMPs than the rest of the State of New York because these two areas rely on importing inexpensive power from neighboring areas before dispatching its own generation resources. An ISO has perfect information required to clear the LMP of a spot electricity market. An ISO requires market participants to submit energy bids, generator status/capability, and bilateral transactions. ISO uses this information, along with the forecasts of load, transmission topology, and regional interchange, to clear the day-ahead LMP. Because

F I G U R E

5-12

LMP: Eleven Zones of the New York ISO Power Market

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market participants do not have all the detailed information, they have to rely on the available information to perform the LMP forecasts. A fundamental LMP analysis starts with selection of LMP market simulation software. There are a couple of commercial graded market simulators implementing the LMP methodology. They vary across speed and ease of use. The analysis begins with preparing the model database to contain the regional generator units’ information, power flow case, fuel prices, and emission prices. Typical generator information includes unit type, size, minimum capacity, maximum capacity, capacity state, heat rate per capacity state, ramp rate, and fuel type. The typical fuel types include natural gas, coal, and fuel oil. The fuel market prices can be obtained from futures markets like NYMEX. Finally, the emission price forecasts are available from publications such as Platts. The power flow case can be found in FERC Form 715 regional transmission owners’ filings. Each ISO publishes transmission-planning reports outlining various transmission constraints and contingencies in the system. ISO also publishes the transmission outage information and transmission limits useful in LMP modeling. Backcasting provides an opportunity for a modeler to organize and fine-tune the input assumptions in order to close the gap between the modeled LMP and the LMP published by the ISO. Backcast starts with collecting the historical fuel prices, generation outage, transmission outage, transmission constraints, emission prices, and control variables. Numerous iterations of model runs are performed during the backcasting process in order to observe the changes in the gap. Once the backcasting result is satisfactory, input assumptions and control variables are locked. The modeler then focuses on updating input assumptions for the forecast period. In the following example, an LMP simulation was performed utilizing the above procedure to test the directional accuracy of the New York Zone A–G DAM price spreads. The test covered the period from May 2004 to August 2004. The simulation results are shown in Figure 5-13. The simulation test captured ~75% of the directional changes of the New York Zone A–G DAM price spreads during the test period. The simulation demonstrates that the fundamental LMP analysis of the electricity spot market is operationally feasible. A long-term LMP study is challenging. It is influenced by fundamental factors such as the boom and burst cycle of generation build, the cyclical nature of fuels, and the available generation technologies. In addition, LMP is also affected by where the future generators are sited and how the transmission systems are to be expanded in the future. Although it is difficult to model the irrational market behavior and to

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125

F I G U R E

5-13

LMP Simulation: New York Zone A–G 60

Spread in $/MWh

NYISO Actual Spread in

_

40 LMP Model Forecast Spread in _

20

Jul 03

Jul 03 Jul 03

Jul 03

Jul 03 Jul 03

Jul 03

Jun 03

Jun 03 Jul 03

Jun 03

Jun 03 Jun 03

Jun 03 Jun 03

Jun 03

Jun 03 Jun 03

Jun 03

Jun 03 Jun 03

Jun 03 Jun 03

May 03

May 03 May 03

May 03

May 03 May 03

May 03

May 03 May 03

May 03

0

Date

predict generation technological innovations, it is reasonable to assume that an LMP modeler can model the rational market behavior based on a sound economic theory in order to project the LMP trend given the current market knowledge of the future.

ENDNOTES 1. Vincent Kaminski, Energy Modelling. London: Risk Books, 2005, p. xiii. 2. Keep in mind that the R2 values here measure how well we can predict the day-to-day price changes and not the price levels. Because the price changes comprise primarily the random term, with the deterministic term being relatively insignificant, the R2 values in Table 5-4 are predictably small. 3. Capturing this tail behavior is particularly important when pricing out-of-the-money options. If we used the log-of-price mean-reverting model to price our options, we would end up assuming fatter tails in the process, giving us option prices biased upward. 4. Prior to the LMP methodology, most of the electricity markets used the single-market price clearing methodology to clear the energy market. If transmission congestion occurred, the system dispatcher re-dispatched the generation resources and the congestion cost was socialized across the users in the market.

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C H A P T E R

6

The Forward Price Curve Historically the majority of work on modeling energy and commodity prices has been focused on the stochastic processes for the spot price and other key variables, such as the convenience yield and interest rates (examples include: Schwartz (1997), Gibson and Schwartz (1990), Hilliard and Reis (1998), Miltersen and Schwartz (1998)). However, this approach has some fundamental disadvantages—firstly the key state variables, such as the convenience yield, are unobservable and secondly the forward price curve is an endogenous function of the model parameters and therefore will not necessarily be consistent with the market observable forward prices.

Les Clewlow and Chris Strickland1

6.1.

INTRODUCTION Forward prices are key inputs to any derivatives pricing and risk management calculation. No matter how sophisticated an option pricing model is, if the forward price curve used as an input to the option pricing calculations is not appropriate, the forward price errors will overshadow any additional value the sophistication of the option pricing model has to offer. A trading operation that invests a good deal of money into product valuation and risk management should budget between valuation projects the same way that the company budgets between businesses and/or investments. One of the projects that should always be on the list of possible valuation projects is forward price curve building methodology development and upgrade. The most common valuation management mistake is to put all the efforts into pricing exotic products, while the forward price curves that affect the valuation of the whole portfolio remain tainted by poor building methodology and/or implementation. The building of marked-to-market forward price curves truly deserves a chapter all to itself (Chapter 7).


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6.1.1.

The Difference Between Forwards and Futures

Before we begin the study of forward prices, we need to distinguish between the futures and the forwards. A forward price contract is an over-the-counter (OTC) agreement between two parties for an exchange at some future point in time of a commodity and its cash value. The cash value is fixed at the time of the contract signing. The individual counterparties will decide how often the value is marked-tomarket and margined. Sophisticated traders will do so daily, while some players will only do so at the end of the contract. A futures contract is a specific type of forward. It is traded at an exchange, and the cash value is marked-to-market on a day-by-day basis. (The largest energy trading center is the New York Mercantile Exchange, which is commonly known as NYMEX.) For example, if we bought a NYMEX futures contract on WTI and agreed on a price of $70 per barrel of crude oil, and the price settled at $69 at the time of the business day’s close, then we would have to pay one dollar per barrel into what is known as the “margin account” at the futures exchange. The next day, if the price closed at $71, we would receive $2 in our margin account from the exchange. This margin account also earns an interest rate. Hence, if the futures prices show a nonzero correlation with the interest rates the margin account earns, then there would be a bias in the futures prices relative to forward prices, because forward prices require no margin accounts and therefore carry no such correlation sensitivity. The futures vs. forwards price bias exists in the bond and interest rate markets, where the futures prices are directly related to interest rates and hence show a good amount of negative correlation to the short-term rates that the futures margin account earns or pays. In these cases a long futures position results in a margin account that earns a smaller interest rate on the profit than it pays on the loss. In these markets the futures prices are smaller than the corresponding forward prices. However, in the energy commodity markets, the correlations of the energy futures prices to the interest rates are typically null, allowing the futures and the forward prices to be valued in a similar fashion: by taking into account the specific delivery and payment dates. If the delivery and payment dates of the specific energy futures contract is identical to the OTC traded forward contract, and there is no possibility of default on either side, then the future price and forward price can be used interchangeably, as both reflect the same value.2


129

In the United States, physical energy markets futures and forwards were originally traded primarily as contracts for physical delivery. However, over the past decade we have seen a slow rise in volume of financial contracts. The financial futures and forwards contracts may in specifications be identical to their physical counterparts, with the cash-only settlement defining the difference. For a physical contract the buyer receives the delivery of the commodity and the seller receives the payment as defined by the fixed contract price. By comparison, under a financial contract the buyer is due the value of the commodity or some prespecified commodity forward/futures contract cash value as defined by contract terms at the time of settlement, and the seller is due the fixed value as defined by the fixed contract price. For a financial contract only the net cash value exchanges hands, and may flow in either direction depending on which is higher at the time of settlement—the settlement price or the fixed contract price. (See Chapter 7 for the details of the forward contract valuation.) NYMEX provides a trading platform for both physical and financial futures.

6.2.

READING THE UNDERLYING CURVE Forward prices are directly tied to the spot price behavior; forward prices are risk-adjusted and net cost-adjusted expectations of the spot prices at forward points in time. Therefore we can use spot price behavior to tell us about forward price behavior, and vice versa. When modeling spot price behavior we want to ensure that the model we choose captures the characteristics of the spot price market both on a day-by-day basis and over a longer period of time. Similarly, we want to ensure that the model that describes the forward price behavior is consistent with the spot price behavior. To understand spot price behavior we begin by observing spot prices through time. Similarly, in order to obtain an understanding of what the forward price model should look like in energy markets, we begin by observing relatively simple forward price curves without seasonality. Figures 6-1 and 6-2 show two sample WTI crude oil forward price curves. The first is the case of a contango, a market in which the forward prices increase with the expiration times. The second case is called backwardation, to reflect that the forward prices actually decrease as the expiration time increases. A particular market may be in contango or backwardation

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F I G U R E

6-1

Crude Oil Forward Price Curve: December 1993

F I G U R E

6-2

Crude Oil Forward Price Curve: October 2001


131

at any given time. Figures 6-1 and 6-2 are simply snapshots of the same commodity market in 1993 and 2001, respectively. As will be seen, other energy markets that have price seasonality can exhibit both of these underlying states, although it may be hard to see this with the seasonality factors laid on top. The contango and backwardation markets are the simplest market states we can find. More typical are more complicated market states, which allow for the short-term and the long-term portions of the forward price curve to independently take on the contango or backwardation states. For example, the forward price curve might exhibit contango in the short-term, that is, for short-term forward price expirations, and backwardation in the medium-term portion of the forward price curve. Figure 6-3 shows one such case for the West Texas Intermediate (WTI) market. Finally, Figure 6-4 shows a slightly more complicated state in 1992, where the contango is in the near-term, followed by backwardation in the middle- and then contango in the longterm portions of the forward price curve. The latter is a case of an event hump, and like all of the previously described market states, it has to do with the expectations of the supply-and-demand imbalances going forward in time. The trading of cross-commodity spread contracts has given

F I G U R E

6-3

Crude Oil Forward Price Curve: June 2006

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F I G U R E

6-4

Crude Oil Forward Price Curve: December 1992

rise to the possibility that some of these “event humps” are directly related to near-term effects of other production-related commodities, particularly seasonal commodities such as natural gas. In the case of WTI, it is easy to see contango and backwardation states in the forward prices, as there are no seasonality effects (other than possibly secondary seasonality effects due to cross-commodity relationships). Because of this, WTI curves serve as good examples of an “underlying” curve, a concept that we will explore further in this chapter.

6.3.

SEASONALITY IN THE FORWARD CURVE Seasonality complicates the forward price curve. The job of detecting the forward price behavior underlying the seasonality factors gets even tougher when we look at energies that exhibit two (or more!) seasonality factors during a single year, thus having two price peaks and two low price periods within a single year.


133

Heating oil and natural gas forward prices provide the next level of complexity. As seen in Figure 6-5, heating oil forward prices exhibit annual seasonality, with the peaks in the winter and the lows in the summer. Heating oil is used primarily for heating in the winter, and this winter demand peak is reflected in the forward prices. Here we see an example of a market where we need to extract the seasonality in order to clearly see the state of the underlying price. In the figure, the underlying curve is shown by the smooth line beneath the seasonal peaks and valleys. Figure 6-5 shows the heating oil market in contango, while Figure 6-6 shows the natural gas market in backwardation. Natural gas is another seasonal market. Like heating oil, it is used for heating in winter, giving it winter seasonal effects. But it also can be used by power generation plants for cooling in the summer, giving it an additional summer seasonal effect. (For a detailed discussion of seasonality modeling, implementation, and ultimately mark-to-market curve building analysis, please see Chapter 7.) The graph of natural gas forward prices in Figure 6-6 shows the evident appearance of a winter peak, while the summer peak appears

F I G U R E

6-5

Heating Oil Forward Price Curve: June 2006

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F I G U R E

6-6

Natural Gas Forward Price Curve: October 2001

very small in comparison. The electricity prices (Figure 6-7 showing the SPP On-Peak market), however, very clearly reflect the summer and the winter peaks, followed by the lows in the spring and the fall. Electricity’s summer peak tends to be dominant, although the magnitude of this dominance over the winter peak varies depending on the region of North America in question. Typically, the use of electricity in the summer for cooling is greater than the use of electricity in the winter for heating, with local weather patterns determining the relative magnitudes. Natural gas, on the other hand, is used like heating oil: for heating in the winter. However, when electric utilities reach the maximum allowed capacity by generating electricity using their first-generation fuel, such as coal, the utilities kick into second-generation production, using natural gas for greater supply capacity. (First-generation fuels earned the “first generation” label by providing the cheapest-to-deliver power. In fact, the utilities hold a cheapest-to-deliver asset-based option by having a diverse set of production plants to use in providing power to their clients.)


F I G U R E

135

6-7

SPP 5 16 Forward Price Curve: August 2005

During very hot summers, natural gas may be used for cooling purposes, giving natural gas prices a rise due to greater demand during the summer, and resulting in forward price curves that recognize this additional seasonality behavior. In fact, if the electric utilities switched entirely to using natural gas for electricity generation, what we could expect to see is a natural gas curve that would have a stronger summer behavior. In general, the greater the volume of trading of crosscommodity spread contracts, the more unified would the energy market price behavior become across commodities.

6.4.

MODELING CONCEPTS RELATING SPOT, FORWARDS, AND SEASONALITY To the extent that the spot price behavior tells us about how the forward prices act, so does the forward price curve tell us about spot price behavior. In this section we will build up our understanding of forward

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price behavior by looking at progressively complicated forward price processes. By following this “evolution,” we will be able to read even the most complex curves for markets like electricity.

6.4.1.

S&P 500

The S&P 500 forward price curves tend to show smooth growth. This is an exponentially increasing curve. Historical correlations between the S&P 500 forward prices and the spot price for the S&P 500 are very high (close to 100%), indicating that a single-factor model would be sufficient for this market. Furthermore, the exponential shape of the curve indicates that a simple single-factor lognormal model would be consistent with this forward price curve.

6.4.2.

WTI Crude Oil

The WTI forward price curves, unlike the S&P 500 forward price curves, can take on a whole variety of different shapes: contango, backwardation, backwardation in the near-term portion of the curve with contango in the back, or event humps followed by contango in the back section of the curve (Figures 6-1 to 6-4). A single-factor lognormal model cannot handle most of these possible market states. The small correlations between the forward prices far out on the curve with the forward prices in the near-term portion of the curve indicate that at least two factors are necessary to explain the behavior of the forward prices. One factor should capture the behavior of the short-term forward prices; another factor should capture the behavior of the longterm forward prices. The long-term section of the forward price curve appears to enter a simple contango state, similar to S&P 500 forwards, implying that the long-term forward prices approach a lognormal longterm price behavior. The short-term forward prices, on the other hand, appear to mean-revert towards the long-term contango state with increasing expiration times. Only a two-factor mean-reverting model is capable of capturing the types of forward price curves seen in WTI markets.


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6.4.3.

Seasonal Markets

There are quite a few energy products that exhibit seasonal behavior. As seen in previous sections, we can break down these products by the type of energy, that is, heating oil, natural gas, power. But within each of these products there are additional price behavior separations due to geographic separations coupled with limited or constrained transportation or transmission volumes. Although in general, the type of energy will define the type of seasonality observed in forward price behavior, the delivery points will further define the specific seasonal magnitude and the speed of seasonal effect dissipation. Chapter 7 provides some sample seasonal models for forward price curve building and a detailed discussion of both mark-to-market curve building and special situations—of which there are many—we have seen in the seasonal energy markets over the past decade.

6.4.3.1. Heating Oil and Other Simple Seasonal Markets Heating oil is an extension of the type of behavior seen in WTI (Figures 6-5 and 6-8). We still need an underlying price model that tends to exhibit both backwardation and contango in the near term and a contango in the long-term portion of the forward price curves. We still have long-term forward prices that are not highly correlated with the shortterm forward prices. However, we also have the additional complication of seasonality effects. In the case of heating oil, only a two-factor meanreverting model with an annual seasonality component added could attempt to capture the different heating oil market states. The seasonal behavior of heating oil has somewhat evolved over the years. It used to be simple enough to be captured by simple cosine functions, as can be observed in Figure 6-8. Note that this graph shows both the actual seasonal forward prices as well as the underlying price model, captured by a two-factor mean-reverting model. As you can see, during the summer of 1995, heating oil forward prices indicated a very small backwardation in the relatively near-term portion of the curve, followed by a contango market. We can compare this to a 2006 heating oil forward price curve (Figure 6-5), where the cosine functions are simply not good enough to capture all the dimensions exhibited by the heating oil seasonality. Additional degrees of freedom may be required to capture the seasonality of even the simplest of seasonal

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F I G U R E

6-8

Heating Oil Forward Price Curve: July 1995

commodities! Exponential seasonality was used in building the forward price curve of Figure 6-5 and is discussed in detail in Chapter 7.

6.4.3.2. Natural Gas Seasonality Figure 6-6 shows natural gas forward prices in 2001 exhibiting strong winter seasonality and some relatively small summer seasonal effects. In the case of natural gas we have to add one more seasonality factor to capture the additional summer seasonality. As you might guess from the underlying price curve also shown in Figure 6-6, it is not a trivial task to distinguish between seasonal behaviors and underlying price behaviors. In the case of the 2001 forward price curve sample, the underlying price behavior was so significant in magnitude that it was clear that the market was predicting a longer-term event dissipation resulting in a curve with strong backwardation. In comparison to the backwardation, the winter and summer seasonality magnitudes were relatively small. However, this is not always so obvious. Take a look at Figure 6-9. This is a more recent natural gas forward price curve incorporating not only a complicated underlying price behavior, but also what has become a fairly typical but rather complicated seasonal behavior.


F I G U R E

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6-9

Natural Gas Forward Price Curve: June 2006

It is fair to say that building forward price curves where there are so many degrees of freedom is part art and part science. Given two traders building forward price curves using the exact same technology, chances are that their resulting forward price interpretations and therefore resulting mark-to-market curves will be different. Although to many involved in the energy marketplace these many degrees of freedom are cumbersome and challenging, they are also the reality of the marketplace and required learning ground to ultimately understand the complexities of energy price behavior. Ultimately, for someone who works on the trading floor or in risk management, the ability to translate market intuitions to product valuation and vice versa is of great value.

6.4.3.3. Seasonality in Power Markets While over the years the natural gas market has behaved more and more like the power markets, the power markets retain the most complex seasonal behavior. The lack of immediate storage when necessary (other than long-term storage in the form of the power plant or water

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storage in the case of hydro plants) causes these markets to be extremely volatile (although natural gas markets are catching up!) in the short term, and often complex in terms of seasonal modeling. In the early years of power trading, the complexity appeared to be limited to capturing seasonality with expected changes in magnitude over the forward years of delivery (as can be seen in Figure 6-10). Such simple states of market existence can still be seen when there are no medium- to long-term event expectations or slow dissipations of a very strong current event. Typically, markets would react to a recently experienced high-price summer, and would build in a higher seasonal magnitude expecting high temperatures for the next summer. The forward prices would show this magnitude to dissipate the following summers as the high temperatures could only stay high for so long (resembling a double-level seasonality: seasonality of seasonality magnitudes). Over the years power market seasonality has exhibited skew in the forward prices, requiring additional degrees of modeling freedom to

F I G U R E

6-10

CINERGY 5 16 Forward Price Curve: October 2001


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capture it. Figure 6-11 shows one such case; note that the seasonal magnitudes also change with forward price expiration. For someone new to building power forward price curves, the volatility of the market poses a significant challenge. The spot prices are extremely volatile, and their volatility drives the near-term forward price volatility. If a model is used to build marked-to-market forward price curves, and the same model parameters implied from market prices at some point in time are used to build a forward price curve at a later point in time, you can expect to see some significant changes in the look of the forward price curve (see an example in Figure 6-12). It is not uncommon for a volatile market to go from a backwardation market to a contango market in a relatively short period of time. By recalibrating the model parameters using forward prices from a later point in time, a proper marked-to-market forward price curve can be built (see Figure 6-13).

F I G U R E

6-11

CINERGY 5 16 Forward Price Curve: August 2005

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F I G U R E

6-12

CINERGY 5 16 Forward Price Curve Built Using Outdated Parameter Values: June 2006

F I G U R E

6-13

CINERGY 5 16 Forward Price Curve With Calibrated Parameters: June 2006


6.5.

143

LINKING SPOT PRICE MODELS TO FORWARD PRICE MODELS By observing market forward prices, we can decide on what we want the forward price model and its implementation to capture and accomplish. The forward price model has to tie back to the spot price model through an arbitrage-free relationship. The spot price model has yet one more test to pass in addition to all the tests we have already discussed in the previous chapters: the test of consistency with the market forward price curves. It is not necessarily an easy step, going from the spot price model to the forward price model. For example, although we may define the forward prices to be proportional to the expected spot prices, they are not—as a rule—equal to the expected spot prices. However, we do have something to lean on in the process of transforming what we know about the spot price behavior into information we can use to define the forward price behavior, and that is the “no arbitrage” condition.

6.5.1.

The Arbitrage-Free Condition

Here’s how it goes: Under the simplest scenario, we should be indifferent between entering into a forward price contract for delivery of the commodity at some time in the future and purchasing the actual commodity now and holding it until that same time in the future. Now, to make the comparison a valid one where we do indeed compare oranges to oranges—rather than oranges to lemons—we need to go through the actual cash-flow analysis in present-value terms. In Section 6.6.1 we will take you through the details of this process. Our forward price model should be “arbitrage-free.” One way to describe the arbitrage-free concept is to assume that we can construct a portfolio of the forward price, the spot price, bonds, and whatever other market products we need to make this portfolio risk-free. A risk-free portfolio must earn the risk-free rate of return. This argument also gives us a mathematical means that we can use to derive the forward price model based on the underlying spot price model assumptions. In this process we end up with a differential equation for forward prices. This differential equation can then be used to derive the forward prices. The particular challenge energy markets face is that the concept of “arbitrage-free” becomes questionable when the supply side is limited in

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volume, constrained by storage or transportation issues (the extreme being the case of power, which cannot be stored directly), or characterized by a small number of producers, or by producers who are either organized, government controlled, or ignorant of market price behaviors and energy contract optionality, or by potential market manipulation, to mention a few. Some examples of such supply side constraints include the price controls experienced in crude oil markets due to OPEC actions in the last century, the Continental Divide as a transmission barrier separating the West Coast USA power market from the Midwest and East Coast markets, or the limited power production capacity coupled with even greater demand causing price spikes above $10,000 per MWh. Similarly, the user side may also taint the behavior of an arbitrage-free market, most commonly through its ignorance of traded markets and contracts but also in their physical limitations in having access to the physical markets. Many users do not have storage or transportation capabilities; their market participation choices may therefore be limited. However, some big users do have the ability to maximize their value by, for example, exercising a contract for full volume where only a part is to be used, with the remainder sold at profit in the marketplace. When a user does not take advantage of such situations when possible, there is potential for market arbitrage. The whole idea of an arbitrage-free market and arbitrage-free pricing is based on the fact that the same type of ultimate benefit of owning an energy commodity can be restructured in at least one other way (thus allowing for hedges and risk-free portfolios). But what happens when there are no other ways to replicate the benefit? One could argue that by building or purchasing, either the whole or part of the commodity production process can be considered the “hedge” or the alternative means of arriving at the ultimate benefit of owning the commodity. I heard of a story while camping on Lake Michigan in Wisconsin of a small island in Wisconsin that decided to build a small power co-op in order to get both cheaper and more reliable power instead of signing contracts with big power producers in the area. Building power plants is certainly a type of alternative path to purchasing power contracts. However, while building power plants as a hedge and a means of power “storage” will guarantee that in the long run the energy market will be arbitrage-free, it still leaves us with the problem of the volatile short-term supply and demand market factors.


6.5.2.

145

Capturing Market Characteristics Within the Model or During Implementation

Early on in the process we have to decide on which of the spot and forward price market characteristics we want to treat within the pure price behavior model and which we want to treat during model implementation. As in any other modeling and implementation project, there is a give-and-take between the modeling and the implementation sophistication. If the model captures all the market realities, then the model implementation should be a fairly well-defined process. However, if the model captures the primary market realities, but leaves some to be dealt with during the implementation stage, then the implementation process is more involved and needs close involvement of the valuation and risk management experts. How much is left to the implementation versus what is captured within the model has to do with the costs and benefits of choosing model sophistication. Capturing all the market realities within the model generally results in solving differential equations for forward prices, which may have closed-form solutions. The cost of arriving at approximations may outweigh the benefits. Similarly, letting the implementation take the burden of capturing some of the market realities might provide the benefit of a very practical and quick methodology development. In my experience, the ideal approach involves capturing the markets’ underlying price behavior within the underlying model while capturing seasonal behavior during the implementation process.

6.5.3.

Influence of the Convenience Yield

The difficulty in the modeling of energy commodity forward prices and solving the forward price differential equation comes in defining the value of convenience yield. The convenience yield is unique to every user of an energy commodity, as it reflects the value it brings to the user of having the energy on hand, as needed, minus the cost that the user would have to pay for storing and maintaining the fuel. If a factory’s production depends on a consistent delivery of an energy, and if there is a cost of stopping and restarting the factory due to lack of energy, as well as a cost to the share price due to the factory appearing

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dysfunctional in the marketplace, then the factory owners may be willing to pay a premium for prompt and consistent delivery of the energy as needed. In other words, the value of having the energy on hand might be very large. The convenience yield, as measured within the market forward price curve, reflects an overall market user’s perception of this convenience yield value. The convenience yield does not directly appear in the modeling of spot prices, although during supply shortages, the spot prices in effect ride up the convenience yield curve—as the spot prices capture the premiums the users are willing to pay to have the commodity on hand. Similarly, the spot prices in effect ride down the convenience yield curve as the events dissipate, resulting in the users being willing to pay less and less of the premium as the supply and demand go back to a balanced state. While the spot prices will exhibit effects of convenience yield during events, the convenience yield is ever present in the market forward prices, and hence it must appear in the modeling of forward prices. It must be incorporated in the differential equation for the forward prices, and hence in the assumptions we make about what a perfectly hedged forward price commodity portfolio should have as its risk-free rate of return. As discussed in previous chapters, in a financial sense, being a commodity holder is just like being a holder of a stock. If the stock pays dividends, then the holder captures this value. On the other hand, a holder of a forward on a stock would not capture this dividend value until the stock is actually delivered. Similarly, the user of an energy commodity who is an owner of a forward on this commodity does not capture its value until the energy is actually delivered. Therefore, just like the risk-free rate is adjusted for the value of the dividend yield in the case of a stock, the risk-free rate must be adjusted for the value of the convenience yield in the case of the energy commodity. The huge ambiguity about just how exactly the convenience yield can be quantified is in part what makes the modeling of energy forward prices difficult. However, there are a couple of pointers we can use. The convenience yield is the value the energy commodity brings to the holder beyond the storage and maintenance costs of holding the commodity. This value is a function of the spot price relative to the equilibrium price and can be positive as well as negative. There may be times when the value the holder obtains from having the commodity readily on hand may be less than the cost of storage, resulting in the cost of having readily available energy outweighing the benefit of holding the energy.


147

This negative convenience yield rate occurs when there is a large abundance of the commodity in the spot market. (Certain areas of Central Europe are a good example of a predominantly negative convenience yield power markets: these areas are saturated with too many functioning nuclear plants.) In this case the user might not see any value in having the energy on hand, and would be better off not having to pay the cost of storage and maintenance; the user would rather buy the energy directly from the spot market than store it. However, on the flip side, the greater the spot price relative to the equilibrium price of the moment, the smaller is the supply relative to the demand, and the greater is the positive convenience yield: the benefits of having readily available energy on hand outweigh the costs of storage. These are the kinds of markets producers like to be in. We can use these intuitions to guide us in ultimately defining the convenience yield in a mathematical form.

6.6.

MODELING THE UNDERLYING FORWARD PRICE CURVE We will start with discussing why it is that the forward prices, as a rule, are not the expected spot prices. This will naturally lead us to the noarbitrage assumption and how to apply this assumption to forward price curve creation.

6.6.1.

Difference Between Spot and Forward Prices

In order to show that the forward prices are not—as a rule—equal to the expected spot prices, we need to go through a cash-flow analysis of two portfolio scenarios. To make this simple, instead of considering energy commodities we will work with a simple stock that pays no dividends and we will apply the no-arbitrage market condition. In the next section we will consider the much more complicated case of an energy commodity. Under the no-arbitrage assumption, we must be indifferent between two scenarios, where one involves the purchase of the forward on the stock and the other involves the purchase of the stock. In the first scenario, we enter into a forward contract. At the time of the forward contract expiration, we pay the cash we agreed in the forward price

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contract and obtain the stock. We immediately sell the stock in the marketplace. Under the second scenario, we borrow the money from the bank and we use it to purchase the stock today and hold it until the same time as the forward expiration. At that point, we sell the stock and we pay the bank what we owe it: the original principal plus the interest. Under both scenarios, the net cash flow at origination time, t, is zero. In the first scenario we simply agree to purchase the stock at future time T for an amount of the forward price, Ft,T. No cash is exchanged at origination time. In the second scenario, we purchase the stock for the market price St at origination time t, and we therefore borrow the principal amount St from the bank. The net cash flow in the second scenario is also zero at time t, as we get the same amount from the bank as we use to purchase the stock. Because we should be indifferent between the two scenarios, the cash flow from the first scenario at expiration time T must be the same as the cash flow from the second scenario at time T. Under the first scenario at expiration we pay out the amount Ft,T for the stock. We get the stock and sell it in the marketplace for ST. Hence, in the first scenario, the cash flow at time T is given by (ST Ft,T). Under the second scenario, we sell the stock and we pay back the bank for the principal amount and the interest. In this case, the cash flow at time T is given by (ST St e r(Tt)), where St er(T t) is both the principal and the interest owed to the bank, the interest compounded continuously at a riskfree rate. Under the no-arbitrage condition the two cash flows must be the same, giving us a solution for the forward price in terms of the spot price, both at origination time t:

Ft ,T = St er ( T − t )

(6-1)

where: Ft,T forward price observed at time t, with expiration at time T St spot price at time t t time of observation T time of expiration r risk-free rate, continuously compounded The forward price is thus given by the stock price at the time of the origination, which is then compounded at the risk-free rate over the forward price expiration time. We can now use Equation 6-1 to relate the forward price to the expected stock price, as seen at the time of forward price contract


149

origination, t. If the stock earns an expected return, , then the expected price of the stock at expiration time T, but as calculated at time t, is given by Et [ ST ] = St eµ ( T −t )

where: the expected compounded

return

on

(6-2) the

stock,

continuously

Given Equation 6-1 for the forward price in terms of the stock price at origination time t, we can express the forward price in terms of the expected stock price at the time of expiration:

Ft ,T = Et [ ST ]e( r − µ )( T −t )

(6-3)

The forward price is proportional to the expected spot price at the time of expiration but not—as a rule—equal to it. The forward price is equal to the expected spot price adjusted for the market cost of risk, where we define the market cost of risk, , as following:3

λ=

(µ − r ) σ

(6-4)

In other words, for a traded asset, the risk-free rate is equal to the expected rate of return on the asset minus the market cost of risk times the volatility for that asset. r = µ − λσ

(6-5)

where: market cost of risk

volatility of the asset price By replacing the r term in Equation 6-3 using Equation 6-5, we generate Equation 6-6. The forward price is equal to the expected spot price adjusted for the market cost of risk:

Ft ,T = Et [ ST ]e− λσ ( T −t )

(6-6)

Although the above arbitrage-free derivation of the forward price in terms of the expected spot price at expiration was applied to the simplest case of a stock price that pays no dividends, we can use the definition of the market cost of risk and the intuitive expectation of the forward price being proportional to the expected stock price to guide us in the analysis of the more complicated forward price markets.

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6.6.2.

Going from Spot Price Models to Forward Price Models

The case above was simple enough to be treated through a simple cashflow analysis. A more general application of the arbitrage-free market condition is the creation of a risk-free portfolio consisting of the commodity product and its hedges. The modeling steps include the creation of the risk-free portfolio consisting of the forward price and all market hedges necessary to make the portfolio risk-free, the incorporation of pricing models for the market hedges, and the derivation of the forward price model given boundary market conditions.

6.6.3.

The Risk-Free Portfolio

We can start with the process of creating a risk-free portfolio for a simple case, and we will build up to the skill level necessary to model forward prices on energy commodities. The risk-free portfolio approach has been used for the derivations of pricing models for options on stocks. Defining differential equations for forward prices is no different. Hence, as we go along, if you are familiar with the option price differential equations you will find that the forward price differential equations really look very similar, only with a different boundary condition at the expiration time. First we will assume that the stock does not pay any dividends, and that the stock follows a simple lognormal model, with a rate of return , and a volatility : dSt = µ St dt + σ St dz

(6-7)

where: mean rate of return

volatility dt the time period over which the change in the price is observed dz˜ random stochastic variable with mean of zero and standard deviation of dt Secondly, we assume that the forward price’s randomness comes purely from the underlying stock price randomness. Hence, a risk-free


151

portfolio can be constructed such that it consists of the forward and some number of shares of the underlying stock: Π t ⬅ Ft ,T + nSt

where:

(6-8)

portfolio value Ft,T forward price expiring at time T n number of stocks in the portfolio S spot price

As the changes in the forward price are due only to the changes in the stock price and the passage of time, for the change in the portfolio value over time dt, we have the following:

Ft ,T = f ( St , t ) d Π t = dFt ,T + ndSt

(6-9)

The above equation specifies the change in the value of the portfolio at time t over some time period dt. The value of the portfolio at any time t is defined as the value of the forward price plus the value of the stocks at time t. The value of the portfolio should not be confused with the initial cash investment in the portfolio. As the forward contracts require no payment from either side entering into the contract at origination, and there is no exchange of payment for delivery of stock until the forward contract expiration, the portfolio cash investment at origination consists only of the money necessary for entering into the stock position of n shares. We still have to figure out just exactly how many shares of the stock we need to hold in the portfolio (hence either buy or sell) in order to make the portfolio risk-free. Because we have yet to define the value of n, we do not know if it is indeed a positive or a negative stock position. If the stock position is positive, it means that we are long the stock, that is, we had to purchase the stock. We would therefore have to borrow money to do so, as our portfolio cash flow at origination would be positive. On the other hand, if it turns out that the value of n is negative, we would be sellers of the stock (short the stock), and hence our investment cash flow would be negative, resulting in our putting the money into the bank at contract origination.

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We know that a risk-free portfolio should earn (or pay) the risk-free rate of return, r: d Πt = rnSt dt

(6-10)

Given a risk-free portfolio investment, we would pay the bank the riskfree rate for borrowing the money for the investment if our overall investment value is positive, and we would receive from the bank the risk-free rate for depositing the money from the investment if our overall investment value is negative. Because the forward contract does not cost any money—as it is an agreement to be settled at the forward expiration date—we only need to worry about the cost of money on the stock position. Hence, for this to be a zero-sum game, our risk-free investment must earn the risk-free rate. Hence we have the above equation. Using Ito’s Lemma (discussed in Chapter 3) for the expansion of the change in the portfolio value over time dt into its subcomponent parts, and by substituting the value of dSt from Equation 6-7 into the above Equations 6-9 and 6-10, we obtain the following differential equation for the option price: 2 ⎞ ⎛ ∂Ft ,T ⎛ ∂Ft ,T ⎞ 1 ∂ Ft ,T 2 2 ∂Ft ,T S n S + + − + + µ St + σ µ r nS dt n ⎟ ⎜ ⎜ ⎟ σ St dzt = 0 t t t ∂t 2 ∂St2 ⎝ ∂St ⎠ ⎠ ⎝ ∂St

(6-11)

Because we want the portfolio to be risk-free, we want to make the stochastic term (the term multiplied by the stochastic variable dz˜t) zero. By doing so we obtain the number of shares of the stock that we need to hold in the portfolio:

n=−

dFt ,T dSt

(6-12)

For the portfolio to be risk-free, it turns out that we need to sell stock, because n is negative (assuming that Ft,T St). Plugging the solution to n back in to the differential equation, we obtain the final equation to be solved for the forward price:

∂F ⎛ ∂F 1 ∂2 F 2 2 ⎞ σ S ⎟ =0 + + r ∂t ⎜⎝ ∂S 2 ∂S 2 ⎠

(6-13)

If we assume that the forward price is linear in S, the solution for the forward price—given the above differential equation, and given the end


153

condition, which requires that at the time of expiration the forward price is equal to the spot price—is given by

Ft ,T = er ( T −t ) St

(6-14)

Hence, we obtain the same result as in the previous section through simple cash flow analysis.

6.6.4.

Effect of Dividends

If we relax the assumption that the stock pays no dividends, the differential equation must be adjusted. In this case, we have two different stock price return formulations, one for the holder of a stock position, and one for a nonholder of a stock position. The nonholder of a stock position does not capture the value of the dividend payments, and hence observes the stock price drop after dividend payments by the exact amount of those payments. If we assume that the dividends, , are paid continuously, then we have: dStnon− holder = St ( µ − δ ) dt + Stσ dzt

(6-15)

On the other hand, a holder of a stock position does capture the dividend values. We will make the assumption that the holder immediately turns around and reinvests the dividends back into the stock. Hence, to such a stock holder, the stock return is given by dStholder = St µ dt + St σ dzt

(6-16)

This distinction is important, because a holder of a forward on a stock is a nonholder of the stock until the forward expiration date. Hence, even though the market value of the forward price will change with the stock price change, it will do so as it would for a nonholder, because we would not see the dividend flow as holders of a forward position. Now we need to answer two questions. How does the payment of the dividend affect what the risk-free portfolio ought to earn over time dt? How does the payment of the dividend affect our cost of money in setting up the portfolio? The solution to this problem is particularly relevant to energy commodities, as the convenience yield value to the

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holder of the commodity acts just like the dividend value to the holder of a dividend-paying stock. As always, the net sum game for a risk-free portfolio where we finance the portfolio through the bank must be zero. On one side we have the difference in the value of the portfolio and on the other we have the cost of money for putting on this portfolio. This has not changed. However, although my stock position does capture the dividend, my forward position does not, hence the change in the market portfolio value is now given by

d Πt =

∂Ft ,T ∂St

dStnon− holder +

2 ∂Ft ,T 1 ∂ Ft ,T ( dStnon− holder )2 + dt + ndStholder (6-17) 2 2 ∂St ∂t

2 ⎛ ∂Ft ,T ⎞ ⎛ ∂Ft ,T ⎞ 1 ∂ Ft ,T 2 2 ∂Ft ,T n S dt n d Πt = ⎜ ( µ − δ ) St + σ S + µ + + + ⎟ ⎜ ⎟ σ St σ dzt t t 2 ∂St2 ∂t ⎝ ∂t ⎠ ⎝ ∂St ⎠

(6-18)

On the financing side, we still have to finance the original investment, which has not changed in value. Furthermore, if we make the portfolio risk-free, then this investment’s financing rate must be the risk-free rate: d Π = rnSt dt

(6-19)

In order to zero out the portfolio risk, the position in the stock must once again be

n=−

∂Ft ,T ∂St

(6-20)

Again applying Ito’s Lemma, we obtain the differential equation for the case where the forward is on a dividend-paying stock:

∂Ft ,T

∂Ft ,T

2 1 ∂ Ft ,T 2 2 σ St = 0 + ( r − ∂) S + ∂t ∂St t 2 ∂St2

(6-21)


155

6.6.5.

Equivalence Between Dividends and the Convenience Yield

So far we have worked with a forward on a dividend-paying stock. How do we then translate the above to a differential equation for the forward price on an energy commodity? In order to perform this transformation, we need to recognize the difference between a stock price and the spot price of an energy commodity. The transformation of the dividendpaying stock price into an energy spot price yields the following transformation of Equation 6-19: dStnon− holder = ( µ − Cy ) St dt + σ St dzt

(6-22)

where the spot price St now refers to the spot price of the commodity, and where the dividend value has been replaced by the convenience yield value of the commodity. Again, applying Ito’s Lemma provides us with a partial differential equation for the forward price:

∂Ft ,T

∂Ft ,T

2 1 ∂ Ft ,T 2 2 σ St = 0 S + + ( r − Cy ) ∂t ∂St t 2 ∂St2

(6-23)

We have to solve Equation 6-23 to obtain the forward price model as a function of spot price and time. When the forward price market is driven by more than a single factor (such as both the spot price and the long-term price) we have to incorporate the additional market drivers into the formulation of the risk-free portfolio. A later section (6.7) will take you through the process of modeling forward prices in the case of a two-factor spot price model. In order to go ahead and actually solve for the forward price as a function of the market drivers, we have to make some assumptions about how to formulate the convenience yield. There is a huge degree of freedom here, as who is to say that one formulation for the convenience yield is better than another? However, we do have some intuitions to guide us in this process. First, setting the convenience yield to a constant would result in a solution to the forward price that would look as follows:

Ft ,T = St e( r −Cy )( T −t )

(6-24)

This assumption results in forward prices that are either in a contango or in backwardation.

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6.6.6.

Adding a Second Factor

The possible forward price market states, where we have backwardation in the front and contango in the back portion of the forward price curve, would simply not be possible under the assumption that the convenience yield is a constant. To have this flexibility, we need to expand the model to include a second factor. If we allow for an introduction of a second factor, the equilibrium price Lt, the above differential equation for the forward price can be generalized to read as follows:

∂F ⎛ ∂F 1 ∂2F 2 2⎞ ⎛ ∂F 1 ∂2F 2 2⎞ ∂2F + + ⎜ ( r − Cy ) + ρξ Lσ S = 0 σ S ⎟ +⎜r ξ L⎟+ ∂t ⎝ ∂S 2 ∂S2 ⎠ ∂ S∂ L ⎠ ⎝ ∂ L 2 ∂ L2

(6-25)

Furthermore, if this equilibrium price is not a traded asset, then the risk-adjusted drift on the equilibrium price4 has to take the place of the risk-free rate, r: ∂F 1 ∂2F 2 2⎞ ∂2F ∂F ⎛ ∂F 1 ∂2F 2 2⎞ ⎛ + ⎜ ( r − Cy ) + S + ( − ) + ρξ Lσ S = 0 σ µ λξ ξ L⎟+ ⎟ ⎜ ∂t ⎝ ∂S 2 ∂S2 ∂ L 2 ∂ L2 ⎠ ⎝ ⎠ ∂ S∂ L

(6-26)

Next, we have to define the convenience yield, Cy. The convenience yield can be either positive or negative, as the relative benefits of holding the energy fuel versus simply purchasing it in the spot market are a function of the general state of the spot market supply and demand. Hence, it would appear reasonable to tie the convenience yield to the relative difference between the current spot price and its equilibrium value as a reflection on the balance between the supply and the demand. Tied to this intuition is also the fact that the convenience yield appears to diminish with increasing forward price expirations. (This would be consistent with a mean-reverting model, where the spot prices approach the equilibrium level prices, resulting in a spot price to equilibrium price spread diminishing, and hence the actual convenience yield value also diminishing.) Putting the above convenience yield characteristics all together would result in the following two examples of possible functional


157

definitions. The first is a convenience yield as a function of the log of the ratio of the equilibrium price to the spot price:5

⎛S ⎞ Cy (t ) = δ (t ) + γ ln ⎜ t ⎟ ⎝ Lt ⎠

(6-27)

Another example of a possible functional definition is very similar to the above, only instead of using the log of the equilibrium to the spot price ratio, it uses the difference between the spot price and the equilibrium price as a percentage of the spot price:

⎛ S − Lt ⎞ Cy (t ) = δ (t ) + γ ⎜ t ⎟ ⎝ St ⎠

(6-28)

We will be using the latter case in the following section, where we go through the steps of solving for the forward price from the differential Equation 6-26. Given the partial differential equation for the forward price, and given the formulation of the convenience yield, we need only one more thing before we can solve for the forward price: the boundary condition. At expiration, the forward price converges to the spot price. Hence, we must have the following condition hold: FT ,T = ST

6.6.7.

(6-29)

Seasonality

Before we proceed toward the final step of solving the differential equation, there is one more issue that we have not yet discussed: seasonality. If you think that the above was complicated, the inclusion of seasonality contributions into the partial differential equation would make the derivation of the forward price model even more difficult. The valuation expert does have the choice of treating seasonality terms either within this level of modeling or within the implementation stage. In the following case study we will leave the seasonality contribution to the implementation stage (and Chapter 7) rather than within the spot price modeling for the sake of simplicity.

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Energy Risk

We will look into the case of a two-factor spot price model where the spot prices mean-revert toward the equilibrium level, which is assumed to be lognormally distributed. This will be the underlying spot price model. This underlying spot price model will lead us to the solution for the underlying forward price model. The seasonality contribution will be added on top of this forward price model to give us the full solution for the forward prices:

Ft ,T = FtUnd + seasonality contribution ,T where:

6.7.

(6-30)

F the market forward price F Und the underlying forward price (stripped of seasonality)

THE TWO-FACTOR MEAN-REVERTING MODEL (PILIPOVIC) Our final step toward the forward price solution is to go from a spot price model to a forward price model. Equation 6-26 defines the differential equation for the forward price in the case of a two-factor model. We need to solve this differential equation for F subject to the boundary constraint FT,T S T

(6-31)

We will assume that the spot prices follow the two-factor price meanreverting model (Pilipovic) introduced in Chapter 5: dSt = α ( Lt − St ) dt + σ St dzt

(6-32)

dLt = µ Lt dt + ξ Lt dw t

(6-33)

where: z˜,w˜ 0 correlation between z and w S spot price L long-term equilibrium price rate of mean reversion

spot price volatility equilibrium price volatility The above Equations 6-32 and 6-33 can be used to solve for the spot price, obtaining the following:6 − (α +1/ 2 σ St = S0 e 0

2

) t +σ z0 ,t

t

( µ −1/ 2 ξ 2 ) x +ξ w 0 , x

+ α L0 ∫ e 0

⋅e

− ( α +1/ 2 σ 2 )( t − x )+σ zx ,t

dx

(6-34)


159

(For the derivation of Equation 6-34, please see Endnote 6.) By taking the expected value of the right-hand side of the above equation, we obtain the expected value of the spot price at time t, conditional on time t 0: E0 [ St ] = S0 e−α t +

α L0 ( e µt − e−α t ) (α + µ )

(6-35)

Following an intuitive expectation that, while the forward prices are not equal to the expected spot prices, we can define the forward price to be proportional to the expected spot price: Ft ,T = β (τ ) ⋅ Et [ ST ] ⎫⎪ ⎧ ⎛ α ⎞ = β (τ ) ⋅ ⎨ St e−ατ + ⎜ Lt ( e µτ − e−ατ ) ⎬ ⎟ ⎝α + µ⎠ ⎪⎩ ⎭

(6-36)

where we have introduced a function of time to define this proportionality:

function of time Tt Because the forward prices are now assumed to be linear in the spot and equilibrium price, the differential equation for the forward price simplifies into the following: (Cyt − rt ) St ∂ S Ft ,T = ( µ − λξ ) Lt ∂ L Ft ,T − ∂τ Ft ,T

(6-37)

By applying the boundary condition on the forward price at the time of its expiration, we obtain the boundary condition for the function :

β (τ = 0) = 1

(6-38)

We can also use Equation 6-28 to define the convenience yield Cy:

Cyt = δ t +

γ ( St − Lt ) St

(6-39)

By importing the forward price as defined by Equation 6-36 and the convenience yield as defined by Equation 6-39 into the differential

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Energy Risk

equation for the forward price, Equation 6-37, we obtain the following relationship: ⎧ ⎡ ⎛ ⎛ α ⎞⎞ ⎤ ⎪β (τ ) ⎢ St δ t + γ − rt − α + Lt ⎜ −γ + ( µ + α − λξ ) ⎜ ⎥ ⎝ α + µ ⎟⎠ ⎟⎠ ⎥⎦ ⎝ ⎢⎣ − ατ ⎪ e ⎨ ⎡ ⎛ α ⎞⎤ ⎪ ⎪ + ∂τ β (τ ) ⎢ St − Lt ⎜⎝ α + µ ⎟⎠ ⎥ ⎣ ⎦ ⎩ ⎫⎪ ⎧ ⎛ α ⎞ ⎡ ⎤ ( ) λ ξβ ( τ ) eµτ ⎨− Lt ⎜ + ∂ β τ ⎦⎬ ⎝ α + µ ⎟⎠ ⎣ τ ⎪⎩ ⎭

(

)

⎫ ⎪ ⎪ ⎬= ⎪ ⎪ ⎭

(6-40)

Allowing the time to approach infinity leaves us with the differential equation for () as the left-hand side of the above equation goes to zero:

β (τ )λξ + ∂τ β (τ ) = 0

(6-41)

Finally, plugging Equation (6-41) back into Equation (6-40), we obtain the following equation to be solved for , , and : St (δ t +γ − rt − α − λξ ) + Lt (α − γ ) = 0

(6-42)

Taking the differential of the above equation relative to spot price and equilibrium price, respectively, while imposing the boundary condition (Equation 6-41) results in the following:

β (τ ) = e− λξτ

(6-43)

γ =α

(6-44)

δ t = λξ + rt

(6-45)

We thus obtain the following formulation for the forward price, as a function of both the spot price and the equilibrium price:

⎛ ⎛ α ⎞ ⎞ − (α +λξ )τ ⎛ α ⎞ ( µ −λξ )τ Ft ,T = ⎜ St − ⎜ L e Le +⎜ ⎝ α + µ ⎟⎠ t ⎟⎠ ⎝ α + µ ⎟⎠ t ⎝

(6-46)


161

Note that in the typical case for energy markets, the drift on the equilibrium price is typically much smaller than the mean-reverting rate, allowing us to make the approximation

⎛ α ⎞ ⎜⎝ α + µ ⎟⎠ ≈ 1

(6-47)

which further simplifies the formulation for the forward price:

Ft ,T ≈ ( St − Lt )e− (α +λξ )τ + Lt e( µ −λξ )τ

(6-48)

If we define risk-adjusted mean reversion, , and risk-adjusted growth rate, , as follows:

α' ≡α + λξ µ' ≡ µ − λξ

(6-49)

we obtain a very simple forward price formulation:

Ft ,T ≈ ( St − Lt )e−α 'τ + Lt eµ 'τ

(6-50)

Note that the risk adjustment term, , is a function of the equilibrium price volatility rather than spot price volatility. This is extremely important in that it suggests that the risk adjustment is much smaller in magnitude relative to what it would be if instead the risk adjustment was a function of spot price volatility. The two terms that define the forward price Ft,T also determine the look of the forward price curve in terms of the near-term and long-term curve backwardation and/or contango. In the case where the spot price is greater than the equilibrium price—giving a positive convenience yield—the near-term portion of the curve is in backwardation. If the spot price was less than the equilibrium price, just the opposite would be true. The near-term portion of the forward price curve would be in contango. If the market cost of risk on the equilibrium price was greater than the equilibrium price drift, then the long-term portion of the curve would be in backwardation. But if the opposite was true, the long-term portion of the curve would show a contango market. Finally, the above formulation can also be generalized to include a long-term equilibrium price convenience yield, which might arise due to long-term costs of production or storage.

162

6.8.

Energy Risk

TESTING THE SPOT PRICE MODEL ON FORWARD PRICE DATA Once the forward price model has been defined, the ultimate test of the model is how well it fits the actual forward price markets. In the case where the forward prices are extremely illiquid and infrequently observed, we are forced to rely to a good extent on the validity of the spot price models and forward price modeling assumptions. In the case where the forward price markets have enough of a history to be used as a test of the forward price theoretical model, we would look for the model parameters to be as stable as possible for a test of forward price model validity.

ENDNOTES 1. Les Clewlow and Chris Strickland, Energy Derivatives Pricing and Risk Management. London: Lacima Publications, 2000, p. 134. 2. See Hull, John C. Options, Futures and Other Derivative Securities. Englewood Cliffs, NJ: Prentice-Hall, 1993, pp. 56–57 and 78–79. 3. Ibid., p. 276. 4. The drift on the equilibrium price, µ, can be generalized to include long-term convenience yield effects, such as the cost of storage. However, if the energy is “in the ground,” it may be argued that the cost of storage is, in fact, zero. 5. Gabillon, Jacques. “Analyzing the Forward Curve.” In Managing Energy Risk, a collection compiled by Financial Engineering, Ltd., London, 1995, p. 36. 6. We can make a transformation of variables, as follows:

(

d e

at + bzt

)

St = e

at + bzt

⎫⎪ ⎞ ⎞ 1 2 ⎪⎧⎛ ⎛ ⎨⎜ ⎜ a + b + bσ − α ⎟ St + α Lt ⎟ dt + (b + σ )St dzt ⎬ 2 ⎠ ⎠ ⎭⎪ ⎩⎪⎝ ⎝

We can choose values for a and b such that the stochastic term and the spot price term on the right-hand side of the equation equal zero:

b ⬅ −σ 1 a ⬅α + σ2 2 Now the differential equation simplifies to

d (e

(α +1/ 2 σ 2 ) t −σ z0 ,t

(α +1/ 2 σ St ) = α Lt e

2

) t −σ z0 ,t

dt

Both sides of the equation can be integrated for time 0 to t, resulting in Equation 6-34. (Note that the following property of the stochastic variable was used in arriving at Equation 6-34: z˜0, t z˜0,x z˜x, t).

C H A P T E R

7

Building Marked-to-Market Forward Price Curves: Implementing Forward Price Models

There is little certain knowledge about future values in finance. Implied values are the rational expectations that make a model fit the market, and provide the best (and sometimes the only) insight into what people expect. During the recent stock market correction, the pre-crash implied volatilities of options with different strikes gave a good indication of the level and variation of post-crash, at-the-money implied volatilities. I expect to see modeling based on implied variables—implied forward rates, volatilities, correlations and credit spreads—continue to grow in applicability and sophistication . . . . . . A good trading model must both match the values of known liquid securities and realistically represent the range of future market variables. Very few models manage this. Academics tend to favour those with a realistic evolution but practitioners who hedge cannot live without well-calibrated models; it is no good valuing an option on a bond with a model that misprices the underlying bond itself. If I were forced to choose, I would prefer to calibrate determinacy first—that is, to get the values of known securities right—and hope for robustness if I get the stochastics a little wrong. Obviously, that’s not perfect. I hope to see progress in building models that are both market calibrated and evolutionarily realistic.

Emanual Derman1


164

7.1.

Energy Risk

INTRODUCTION: WHAT IS A MARKED-TO-MARKET FORWARD PRICE CURVE? Building marked-to-market and marked-to-model forward price curves with daily or hourly granularity is a difficult job in energy markets, even if the organization has agreed upon a perfectly good forward price model. A full-time analyst in an energy trading corporation can easily be appointed the singular job of building energy forward price curves, and he will find himself extremely busy and challenged. In fact, a single person in a trading corporation should never be left alone in such a task. Building forward price curves should be recognized as a process requiring potentially several managerial levels and departments’ cooperation within a corporation. A black-box solution for building forward price curves may work only in a few isolated market situations. So much of building forward price curves in energy markets is based on the art of making assumptions, interpreting market data, and ultimately understanding the market price behavior, so to leave it to a computer program or a single analyst alone is not only unsatisfactory, but potentially dangerous. The process of building forward price curves must involve traders, risk managers, and quantitative analysts on a regular basis. A marked-to-market forward price curve is a string of forward prices with nonoverlapping, continuous delivery periods of equal duration and equivalent specifications, which, when used to price any traded forward price or swap products, provides prices identical to market prices. Therefore, building a marked-to-market forward price curve involves beginning with market price quotes to ultimately loop back to market price quotes. In the case that there are a limited number of market quotes, the model used in building the forward price curves becomes extremely important as it fills in between the available market quotes where there are no available market data. Although the Enron era clearly demonstrated the difference between marking to market versus marking to model, we will continue using the definition of marked-to-market forward price curves when the resulting curves satisfy two constraints: 1. When using the marked-to-market curve to value market-traded products, the resulting valuation yields prices that exactly match the available market quotes. 2. When there are only a few available market quotes, the marked-to-market curve is built in the spirit of attempting to


165

closely arrive at a forward price curve that captures market prices whenever they are available. In other words, the focus should be “What does the market think and how does it behave?” and not “What do I think and how should the market behave based on what I think?” In order to deal with the case where there are limited available quotes true to market behavior requires that the underlying model used in building curves is a best representation of market behavior. In order to build a marked-to-market forward price curve, the level of “granularity” has to be decided so that all the traded market products can indeed be valued using the same marked-to-market forward price curve for a single market. The last thing a trading corporation would want to see is its traders building independent and inconsistent curves based on several different product types, allowing for potential arbitrage situations within its own trading books! Each individual forward price curve should be a common denominator across the trading floor of an organization and thus requires horizontal cooperation across the organization. Similarly, there has to be an agreement between the managerial levels of a trading organization as to which market quotes and products should indeed be used in building forward price curves, which, in turn, are then to be used in pricing and valuing all the products the organization trades or has on the books. Thus, risk managers and traders have to be equally involved, on a daily basis, in discussions of liquidity and validity of available market quotes. Finally, the actual building of forward price curves may be quite a complicated process from both a quantitative and implementation point of view, and should be based on the type of trading in which the organization is involved. An analyst should never be alone in making a multitude of assumptions when building such forward price curves. Traders and risk managers must be involved in order to assure that the theoretical and implementation assumptions of the building process are consistent with both market reality and corporation needs, and that the process benefits and costs are appropriate for the type of products the company trades or strategies it pursues. There is no better way to understand the complexities of building forward price curves than to actually attempt building such curves. With this in mind, we will jump into doing exactly that, and hopefully the above paragraphs will become far more specific as we go through actual examples of forward price curve building.

166

7.2.

Energy Risk

FORWARD PRICE CONTRACT VALUATION In order to be able to build forward price curves, we must begin with the building blocks. We need to understand forward price contracts and their valuation. As with any contract, there are two sides, the buyer and the seller. In the case of energy products, the buyer is the person receiving the energy, and the seller receives the contract-specific predetermined payment for the energy delivered. In the case of cash-settled products, where there is no actual delivery, the buyer receives the cash value of the energy if indeed it were to be delivered, and the seller receives the contract-specific predetermined payment. There are many different types of energy forward price contracts. However, all the forward price contracts are based on the relative valuation of the energy delivered (in actuality or captured as cash value) versus the contract-specific payment for this energy. The required inputs in forward price valuation must therefore be the dates of energy delivery on one side and the contract price and dates of payment of this contract price on the other side. The type of contract delivery is generally defined by the nature of the energy market. For example, in the case of natural gas markets, a typical contract is for delivery over a period of a calendar month, where a delivery is made every single calendar or business day based on contract specifications. An example of such natural gas contract specifications are provided in Figure 7-1. This is an example of a July 2007 contract for natural gas where the delivery is made on calendar days (meaning delivery continues over holidays and weekends). This contract is here defined to end trading (i.e., expire) on the last day of June 2007, the delivery period is “monthly” (monthly tenor), and the quantity to be delivered is the same every single calendar day (weight of 1.0 for every single calendar day of delivery). On the “other side” of the contract we have the payment for the energy defined to take place the fifth day of the month following the last day of delivery, assuming it is a business day, hence August 5th for this example. Note that the delivery in this case is assumed to be uniform throughout the month of July 2007 and therefore there are a total of 31 days of delivery. By comparison, Figure 7-2 shows the date specifications for a natural gas type contract where the delivery occurs only on business days. Notice that delivery of natural gas no longer includes the weekends, and also excludes holidays (July 4th is a Wednesday and is not included


F I G U R E

167

7-1

Forward Price Specifications: Sample Forward Contract for Calendar Day Delivery

as a delivery day). In this example there are only 21 days of delivery. The implementation must thus include a list of holidays. The list of observed holidays may vary by region or by energy exchange; hence, you might have a NERC-specific holiday list as well as a NYMEXspecific holiday list. Holiday assumptions must be defined within the contract and therefore also within valuation. In the case of power there are many different possible contract terms for delivery due to the potential of hourly granularity. The most common traded contract is for On-Peak power, where electricity is delivered during the 16 On-Peak hours of the day and only on business days, thus the contract term 5 16 (Monday through Friday for 16 OnPeak hours). An example of such a contract is the “Remainder of Month” contract shown in Figure 7-3, valued as of August 22, 2006.

168

Energy Risk

F I G U R E

7-2

Forward Price Specifications: Sample Forward Contract for Business Day Delivery

Here we see that delivery is made every single business day for the remainder of the month of August, therefore for a total of 7 days of delivery, and for 16 hours every single day (“Weight” of 16 per delivery day). The Off-Peak contract for power delivers electricity and complements the On-Peak contract by delivering energy on all the hours and days when the On-Peak contract does not deliver. It is also referred to as the 5 8, 2 24 contract, and as you can see from Figure 7-4, it


F I G U R E

169

7-3

Forward Price Specifications: Sample On-Peak Power Forward Contract

covers all the days of the remainder of August (i.e., Monday through Sunday) but the number of hours of delivery are different for business days versus weekends and holidays. On weekends and holidays the delivery occurs all the hours of the day (hence the “Weight” of 24 on those days), while on business days the delivery occurs for the 8 Off-Peak hours of the day (hence the “Weight” of 8 on those days). It is typical to have forward contracts in energy markets covering multiple periods, over two or more months, for example. In this case the seller receives multiple payments, one per period of delivery, as defined by contract terms. Although Figure 7-5 does not show a typical contract (it is an example of a single week of delivery per month, whereas a typical contract would be for a full month of delivery), it shows the level of detail a multiperiod contract requires in terms of date specification. Here we see two periods of delivery, one during the month of July, and the second during the month of August. Each period consists of one calendar week’s worth of on-peak power delivery and one payment. The predetermined contract price defining the value of the payment spans

170

Energy Risk

F I G U R E

7-4

Forward Price Specifications: Sample Off-Peak Power Forward Contract

both periods. However, the value of the payment per period may very well vary due to the different quantities of power delivered per period. In valuing energy contracts, one has to be very careful as to the contract specifications regarding overall quantities per period versus per delivery quantities.2

7.2.1.

Simple Contract for One-Day Delivery

The simplest contract is the contract for a single delivery. In the case of natural gas it would be a contract for a single day of delivery; in the case of power it might be a contract for a single hour of delivery or even for a single day of on-peak delivery, depending on what we consider the building blocks of our trading operation and therefore the building blocks of our forward price curves.


F I G U R E

171

7-5

Forward Price Specifications: Sample Multiple Period Forward Contract

Even though we have a single delivery in the simplest case, we still need to worry about relative discounting of the delivery day versus payment day. At this point we need to introduce some nomenclature. Allow the following definitions: F Tmtm | the marked-to-market price for a contract with the 1,TN t first delivery made on day T1 and the last (N-th) day of delivery on day TN, as observed at time t, Ft,T the daily marked-to-market forward for delivery on day T, as observed at time t. For a single day of delivery (on day T ), the contract marked-to-market price must be a function of the daily forward price as follows: = FTmtm ,T t

Ft ,T dft ,T dft ,T p

(7-1)

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Energy Risk

where dft,T the discount factor for time T as of time t Tp the payment date for the contract (In addition to the above definitions, and as general nomenclature, we will assume that the daily forward price curve is comprised of daily forwards where the payment date and the delivery date are set to be identical; that is, payment for delivery is made on the day of delivery.) In the example shown in Figure 7-6, we have a single day’s worth of on-peak power delivery where the single day’s marked-to-market forward price is here given to us as $66.17 (“Average Forward” is also the single-day’s forward in this case). The discounting factors for the payment date and the delivery date are also provided (0.9978 and 0.9989, respectively) allowing us to plug in these values to obtain the marked-to-market contract value of $66.25 (“Unit Price”). Please note,

F I G U R E

7-6

Forward Price Valuation: Simplest Case of a Single Delivery—Specifications


173

we can assume that this is a marked-to-market contract price to the extent that the daily forward price curve we have used to obtain the single delivery day’s price in valuing this contract is indeed marked-to-market. To price the contract we would follow the above steps. However, to value the contract we need Equation 7-2. If the contract was entered into at a fixed price of FC (C for “contract”), then the present markedto-market value of the contract at time t is given by

(

= q F mtm df − F df p Π t T T ,T t ,T C t ,T t

)

(7-2)

where: qT is the predetermined quantity of the commodity to be delivered at time T. If we purchased the commodity, then the quantity would be positive, we would receive the value of actual commodity on the day of delivery, and in turn we would pay the predetermined contract price on the payment date. The relative present values give us the markedto-market value of this position. If we continued with our pricing example above, and we decided to sell the one-day on-peak power contract at a price of $70.00 and for a quantity of 25 MWh (i.e., 25 MW per hour), then the contract markedto-market would be given by the negative value (as sellers) of 25 MW for 16 hours times the difference of present values of the delivery value and contract payment value, resulting in $1,496.85 (see “transaction value” in Figure 7-7). Note that the delta of the contract is expressed assuming a single dollar move up of the forward price curve. Hence, for a single dollar move of the daily forward price, the contract value would go down by roughly $400, or $1 for each of the 400 MWH (16 hours times 25 MW).

7.2.2.

Contract for Delivery Over a Period

Things get just a bit more complicated when we look at contracts for multiple deliveries within a single period (i.e., with a single payment). Now we need to keep track of quantities to be delivered for every single delivery day, as these might be different.

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Energy Risk

F I G U R E

7-7

Forward Price Valuation: Simplest Case of a Single Day Delivery—Valuation

The general formulation for the marked-to-market forward contract price with multiple deliveries but a single payment is given as follows:

∑ ( q F N

F

mtm T1 ,TN

t

=

n

n=1

t ,Tn

dft ,T

n

)

⎛ N ⎞ ⎜⎝ ∑ qn ⎟⎠ dft ,T p

(7-3)

n=1

where there are N deliveries, each delivery made at time Tn, with n 1. . . N, for a quantity of qn. There is only one payment for all the deliveries at time Tp. If we define Q as the total quantity delivered over the delivery period covered by the payment, then we can rewrite the above as follows:

∑ ( q F N

FTmtm = ,T 1

N

t

n=1

n

t ,Tn

dft ,T

Qdft ,T p

n

)

(7-4)

Our marked-to-market contract price is now a function of every single day’s worth of delivery marked-to-market daily forward prices, as well as the individual delivery quantities for each of the days of delivery. The payment, on the other hand, is simply a function of the discounting factor to the payment date and the overall


175

quantity to be delivered. Figure 7-8 shows an example of a “Remainder of Week” contract for two days worth of delivery of on-peak power. The marked-to-market book value of this contract would then be given by

(

)

= Q F mtm − F df p Π t T ,T C t ,T N

1

(

N

t

)

= ∑ qn Ft ,T dft ,T − QFC dft ,T p n=1

n

n

(7-5)

For the example above, if we were to purchase this “Remainder of Week” contract at the marked-to-market price (assumed in the valuation

F I G U R E

7-8

Forward Price Valuation: Case of Multiple Deliveries but Single Payment—Specifications

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Energy Risk

software used here for sake of demonstration when the entered value for “Transaction Price” is left at zero), we should receive a markedto-market value of zero: the value of energy we are to receive should exactly offset the payment we are to make for the energy. Although our marked-to-market value of the contract would in this case be zero, our risks would not! Figure 7-9 shows the value of this contract across time (in present value terms): the two delivery days exactly offset each other in value, giving us the overall marked-to-market value of zero, but the risks remain. If the market price goes up a dollar, we are to benefit by a dollar amount of almost $800. Of course, this also means that if the market were to go down a dollar, we would lose almost $800. Also, note from Figure 7-9 that the market risks can be expressed in standardized contract terms—in the case of on-peak power, that would be in 5 16 contract terms. Figure 7-10 shows how the marked-to-market present values of this contract would change if we were to purchase this contract at a beneficial price of $60. Although the value of the contract shows a nice positive amount, the risks remain the same, as the contract price does not affect the delta risks of the contract.3 (Note that the Theta as calculated in the above example includes not only the effects of passage of time but also the value “drop-off” due

F I G U R E

7-9

Forward Price Valuation: Case of Multiple Deliveries but Single Payment—Valuation A


F I G U R E

177

7-10

Forward Price Valuation: Case of Multiple Deliveries but Single Payment—Valuation B

to passage of time: the first day of delivery happens to be the next day of trading and includes $2,026.75 worth of energy value to be delivered.) Finally, Figure 7-11 shows what would happen were we to purchase the contract at a disadvantaged price of $65. Risks, once

F I G U R E

7-11

Forward Price Valuation: Case of Multiple Deliveries but Single Payment—Valuation C

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again, remain the same, and we would still benefit if the market were to move up. The general formulation for the marked-to-market forward contract price with multiple deliveries within each payment period and multiple payments M is given as follows:

∑ ∑ (q Nm

M

F

mtm T1,1 ,TM , N

= M

m=1 n=1

m ,n

Ft ,T dft ,T m ,n

m ,n

)

(7-6)

⎛ Nm ⎞ ∑ ⎜ ∑ qm,n ⎟ dft ,Tmp ⎠ m=1 ⎝ n=1 M

t

where for each payment period m there are Nm delivery days, qm,n is the quantity of the commodity delivered on delivery day n (n 1 . . . Nm) of payment period m (m 1 . . . M), Tm,n is the time of the n-th delivery within payment period m, and Tmp is the time of payment for payment period m. Again, if we define the total quantity delivered during payment period m as Qm, Nm

Q m = ∑ qm, n , then we can rewrite the above as follows: n=1

∑ ∑ (q M

FTmtm,T 1,1

M ,N M

=

Nm

m=1 n=1

m ,n

Ft ,T dft ,T m ,n

m ,n

) (7-7)

M

∑Q

t

m=1

m

dft ,T p m

In this most generalized case, we need to incorporate the various per period delivery quantities that are to be paid for at different points in time. Figures 7-12 and 7-13 show the specifications for an on-peak power contract for a typical JAN–FEB multiple period delivery contract. Note that we now have two different payment dates and two different sets of deliveries, each with month-specific number of days of delivery. The marked-to-market book value of this contract would then be given by ⎞ = ⎛⎜ F mtm Π − FC ⎟ ∑ Qm dft ,T p t ⎝ T1,1 ,TM , N M t ⎠ m=1 m M

M

Nm

(

= ∑ ∑ qm,n Ft ,T dft ,T m=1 n=1

m ,n

m ,n

) − F ∑ Q df

(7-8)

M

C

m=1

m

t ,Tmp


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7-12

Forward Price Valuation: General Case of Multiple Deliveries with Multiple Payments—Specifications

Figure 7-14 shows what the value of such a contract would look like across time and overall if we were to sell this JAN–FEB2007 contract at a beneficial price of $70.00 for a quantity of 25 MW per hour.

7.2.3.

Bootstrapping and the Problem of Daily Price Discovery

One of the basic problems of building forward price curves with daily granularity in a market where most of the quotes cover monthly periods of delivery is the problem of implying daily forward prices from the monthly quotes. You will experience the difficulties of solving this problem regardless what forward price model you choose! However, the closer the forward price model comes to expressing real market behaviors, the better.

F I G U R E

7-13

Forward Price Valuation: General Case of Multiple Deliveries with Multiple Payments—Specifications Continued

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F I G U R E

7-14

Forward Price Valuation: General Case of Multiple Deliveries with Multiple Payments—Valuation

In fact, in any market, not just energy markets, you will find that implying more discrete market values from less discrete market quotes gives rise to noncontinuous marked-to-market forward price curves either due to market noise within the quotes or due to model inefficiencies in capturing all market situations. Furthermore, in building more granular forward price curves from market quotes, the noise effect resulting in forward price curve discontinuity tends to grow from quote to quote, all the more so if the forward price model used in building the curve is a poor choice given the market behavior. To see how market noise in market quotes tends to make the more discrete forward price curve noise grow, let us assume a simple single factor lognormal market. Also assume 30-day months, no seasonality, and 0% interest rates. In a perfect world where there is no market noise from quote to quote, the market prices would always perfectly match the exponential model prices and all the forward prices would therefore simply be equal to the current daily price for immediate delivery. However, instead, assume that there is some market noise due to various factors such as supply/demand, bid/ask, illiquidity, and so on, and allow each consecutive market forward price for monthly delivery across the first three months at time t to be $50.05, $49.9, and $50.00, starting with today’s price for immediate delivery at $50.00. (In a perfect world, of course, all the prices should be $50.00.) Using no model, and only using linear interpolation to back out daily forward prices, we can set the price


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at time t to be $50.00 and then imply the 30th day daily forward price such that the simple 30-day average (because of zero interest rates) over the next 30 days gives us the market price of $50.05. When we do so, we obtain a price of $50.10 for the 30th day of delivery daily forward price. Then we continue to the next market quote. We already have a marked-to-market daily forward price for the first 30 days. We can use the second month’s quote to build the curve further out to 60 days. (This is called “bootstrapping.”) We repeat this procedure until we have a daily forward price curve that covers the first 90 days, and with the implied daily forward prices as follows (assuming an exponential market with r 0% for sake of demonstration): St 50.00 mtm F1,30 |t 50.05 mtm| 49.90 F31,60 t mtm F61,90|t 50.00

⇒ Ft,30 ⬇ 50.10 ⇒ Ft,60 ⬇ 49.72 ⇒ Ft,90 ⬇ 50.27

Figure 7-15 graphs the final implied marked-to-market daily forward price curve. Note two things. First, although the market noise in the monthly quotes was only in the 10-cent range, the implied market noise in the daily forward price curve was of the order of 30 cents. Market noise in forwards for delivery over a period of time will always result in at least as great market noise in the daily implied forward

F I G U R E

7-15

Effect of Market “Noise” on Daily Markedto-Market Curve Across Time (in Days) 50.3 50.2 50.1

Theoretical Daily Forward Price Fitted Daily Forward Price Monthly Forward Price

50 49.9 49.8 49.7 49.6 0

50

100

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prices. Especially interesting is the fact that even though the third month’s market forward price is back to the theoretical value of $50.00, the daily implied forward price curve using linear interpolation still exhibits a noise level in the 30-cent range. The second important result to note is that this is not a good-looking daily forward price curve. However, it is still a marked-to-market daily forward price curve. If we used this curve to price the first three months of monthly contracts, we would get back the exact same monthly market prices we used to construct this curve.

7.3.

FITTING THE MODELING NEEDS TO TRADING NEEDS As always, modeling and valuation must be approached as a business venture. There are pros and cons to every model and level of sophistication. One of the dangers of approaching forward price curve building as a layman is the assumption that it should be simple. There is nothing simple about building forward price curves and this is all the more true within energy markets. The above example of dealing with market noise certainly poses some questions. As users we may not like the effect of market noise being amplified as we bootstrap the daily forward price curve from quote to quote. In this case we may opt for a more sophisticated interpolation methodology that might smooth the curve. And this is just a very simple example looking at dealing with a single-factor exponential market. By comparison, we need to deal with multifactor energy markets full of events, event expectations, and seasonal quirky behaviors. The forward price curves are of absolute primary importance in any trading organization. So much attention is usually given to option valuation, and by comparison very little attention goes into the primary inputs to any energy contract valuation: the forward price curve. The treatment of these curves must be viewed first from a very high level: the type of trading and the types of products a corporation is involved with should guide the choice of forward price modeling sophistication.

7.3.1.

Case of Trading Exchange-Traded Products Only

If a trading operation only trades on the exchanges, never takes delivery, never stores energy, and so on—in fact only buys and sells


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exchange-traded monthly contracts—then the needs of that organization in terms of forward price curve building are far simpler than the case of users or producers. As long as a company has only monthly contract exposure and no specific unique daily risk exposures, then this company probably needs only a simple forward price curve of no more than monthly granularity. However, such a simple trading strategy still might require more sophisticated daily forward price methodology for better understanding of the underlying market behavior. This may be of particular importance in trading options on such monthly contracts where understanding monthly contract volatility behavior is very difficult without understanding the discrete daily price behavior and therefore volatility.

7.3.2.

Case of Trading OTC

In the case of over-the-counter forward price contracts, a trading operation that trades only monthly contracts with no discrete daily contingency might get away with building only simplistic monthly forward price curves. However, the moment this trading operation opens its doors to daily-exercise type OTC contracts, the daily implied forward price curves become a must. A typical layman error when first confronted by daily-settlement type contracts is to assume that using the monthly price quote as a flat daily forward price across the month is equivalent to not making any modeling assumptions! Furthermore, any risk manager who observes modeling assumptions being made by any one single person in the company should see the situation as a red flag. Often, modeling assumptions are made in the process of forward price curve building without the recognition that such assumptions are indeed being made. A common phenomenon within a trading operation is to have a single person in charge of forward price curve building and a whole multitude of people who use the curve and do not think twice about the forward price curve inputs and embedded assumptions. Another observable phenomenon is a trading environment with the attitude that forward price curve building should be simple and, if it is not, then modeling assumptions are being made! The reality is that every forward price curve has embedded modeling assumptions,

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whether the users like it or not. The only way to get control over forward price curve assumptions is to understand exactly what they are and then be able to evaluate and change according to need.

7.3.3.

Case of Owning Power Production

Finally, the case of both producers and large users is probably the most complicated of all. The synthetic contracts embedded within production assets or demand needs of a large user are generally of daily if not hourly granularity. In energy markets, producers and users do not have the luxury of not understanding how to build forward price curves with the granularity corresponding to their risk exposures. There are no standardized exchange or OTC contracts that can ever perfectly match the risk exposures and pricing needs of either energy producers or users. There is no easy way out for producers and industrial users: these companies must attempt to develop discrete forward price methodologies as well as forward volumetric curves to capture their supply and demand expectations.

7.4.

BUILDING MARKED-TO-MARKET FORWARD PRICE CURVES: ISSUES TO CONSIDER In building marked-to-market forward price curves we must, as always, begin with the market. What types of market forward quotes are we likely to get on a regular basis that we can use in building our forward price curves? In addition, we need to decide on the type of model and its implementation that we want to use in building these curves. How much market complexity do we want to allow the forward price model to accommodate, and how much of this complexity do we want to take care of within implementation? These are just some of the necessary steps we need to go through in building marked-to-market forward price curves.

7.4.1.

Quote Strips

Generally speaking, each energy market has certain commonly quoted contracts that comprise a daily strip of quotes that can then be used to


185

build marked-to-market forward price curves. An example of such a strip of quotes for a power market is shown in Figure 7-16. In addition to the monthly forward price quotes you see in this strip, there are also some common multiple period quotes, such as the three-month fourth-quarter contract (Q42006), the typical and popular summer contract covering both July and August, as well as January– February and March–April contracts. Often you will see also the F I G U R E

7-16

CINERGY 5 16: Market Quotes

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calendar year contracts that cover 12 months worth of calendar year delivery with monthly payments. In the case of most other energy markets you will see monthly strips together with longer-term swap contracts extending for a number of years. An example of such a monthly strip of data is show in Figure 7-17 for the case of Henry Hub Natural Gas. Be very wary of repetitive monthly market prices as they might indicate that the particular two months where the price quotes are identical are not actually traded separately but rather as a single product! It is a big assumption to say that, for example, JUL2007 and AUG2007 are both at $75, when all you observe in the marketplace is a $75 quote

F I G U R E

7-17

Henry Hub Natural Gas— Sample of Market Quotes


187

for the JUL –AUG2007 contract. This is a clear red flag for a risk manager to deal with. Ultimately, the company might decide that they like the assumption of a flat treatment of the two summer months; however, it is a very different thing to arrive at this assumption through analysis and consensus than to have it assumed without any discussion or even understanding of the assumption that is being made.

7.4.2.

Step-Function Treatment

Both the step function treatment and simplistic linear bootstrapping make assumptions about the marketplace. The step function treatment of the forward price curve is perfectly adequate if all that is traded are monthly contracts, but even in the case of the step function treatment there could be some serious problems with the curve building methodology. Once again, a decision has to be made on the granularity of the forward price curve. The simple reason is that the market strip of quotes might have calendar year quotes that need to be broken down into monthly quotes if indeed you want to properly capture the potential seasonal behavior of the energy in question. Otherwise, a step function treatment of the calendar year quote would result in a flat forward price across the entire calendar year. This is an unlikely scenario, even for a nonseasonal market such as crude oil. This would also be a red flag for a risk manager to deal with. Even if it is decided to use the step function approach, it will probably still be necessary to do some simple calculations in order to convert the available market quotes into forward price curves. You may not need to do even this simple calculation if all you do is trade the standardized monthly contract. However, if you use both exchange traded monthly contracts as well as OTC contracts, you will need to make simple calculation adjustments for the difference in payment and/or delivery dates. In other words, even this most simplistic of all curve building approaches cannot be applied blindly.

7.4.3.

Linear Interpolation

You can indeed build a forward price curve without a specific forward price model. We did it in the linear interpolation example discussed in

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Section 7.2.3 on daily price discovery. Chances are you will find huge magnitudes of discontinuity in building your forward price curves, resulting in an incredibly jagged-looking forward price curve. Would you want to trade daily-settled contracts with such a curve? This is a question only you and your company can answer. And in the process you would be making some serious modeling assumptions with such an approach. The most obvious modeling assumption you would be making is that there is no seasonality. There are also no event expectations, no mean reversion—it is all about randomness. Again, this is a choice only you can make. Simply be aware that you are making these assumptions.

7.4.4.

Applying Forward-Price Models Based on Spot-Price Analysis

The best forward price model to apply is the one that is consistent with the spot price model you believe best represents market reality. By approaching forward price modeling consistently with spot price modeling, you are not only going to be able to build meaningful forward price curves, but you will also find many other rewards in the process. 1. First, you will have a consistent understanding of forward and spot price behavior that will allow you to understand energy volatilities in ways that you would not be able to understand otherwise. This translates to being able to value a multitude of contracts and not just the standardized ones. 2. Second, you will have a means of generating forward price curves in illiquid markets with only a few quotes or maybe even with no quotes, and you will have a means of comparison between model parameters relative to more liquid comparable markets to help you in making adjustments and assumptions about how the forward price curve ought to look. 3. Finally, you will have a means of seeing divergence of the marketplace from model-based expectations. You will have the ability to pin point such divergences and the resulting opportunity to find out why these divergences are taking place: is it an event expectation, is it market noise and if so due to what (bid/ask, liquidity?), or is there something wrong with the


189

market quote? Answering such questions will continuously add to your wealth of knowledge about the marketplace. The process of building marked-to-market forward price curves using a forward price model and its implementation consists of the following steps: 1. Build your theoretical forward price curve based on your model

and its implementation such that this theoretical forward price curve gives you the best possible fit to the existing market quotes. The building of such a theoretical curve will be conditional on the model and implementation parameters you decide you want recalibrated to the current market quotes and market situation. This step will result in the theoretical curve and the recalibrated set of parameters the curve is based on. 2. Build the marked-to-market curve by the bootstrapping methodology using your model and its implementation and giving one of the model inputs an error term such that this error term captures market noise as you bootstrap from quote to quote: Pnmtm P theo ε n

(7-9)

where Ptheo is the value of the input parameter or variable resulting from the theoretical curve calibration, εn is the error term capturing the market noise for the n-th market quote during the bootstrapping process, and Pnmtm is the resulting input parameter or variable used in the marked-to-market curve building for the segment corresponding to the n-th market quote. The end result for this step is the markedto-market curve.

7.4.5.

Many Degrees of Freedom Within Implementation: Part Art, Part Science

Even within the simplest markets the choice of interpolation between quotes gives degrees of freedom and interpretation to the forward price curve builder. These degrees of freedom only grow within the complex energy markets where we need to capture so much information with generally a limited number of market price quotes. In all markets,

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building forward price curves is part art, part science; this is especially so in energy markets. You may choose to not have a model for building forward price curves, and instead you may choose some form of interpolation, linear or a smoothing methodology, perform bootstrapping, and obtain your daily forward price curve from your monthly quotes. In this case the only search you may need to perform will be for the individual daily prices within the bootstrapping methodology, as we did in the linear interpolation example in Section 7.2.3. If instead you decided to use a forward price curve model, then chances are that you will need to additionally fit at least some model parameters in order to obtain the best possible marked-to-market forward price curve. Just capturing seasonality may require a number of parameters, and hence just for seasonality you will have a certain number of degrees of freedom. In all modeling, to have a model with the fewest number of parameters that captures market prices consistently over time is the ultimate goal of any quantitative analyst. Additional unnecessary degrees of freedom not only add to complexity of forward price curve building, but also add to instability of parameter calibration. It would be similar to trying to solve for more variables than there are equations. On the other hand, to have a model that only captures a part of the market price behavior will result in forward price curves that are not consistent with the market prices you observe. This is a situation where additional thought must be given to modeling and/or implementation of the model. One way of solving such a problem is to arrive at a model that captures the additional market behavior. However, this particular solution may not only result in costly model development, but there is no guarantee that such a model—once discovered—will be functional in its implementation on the trading floor. Instead, additional necessary degrees of freedom may be added to the implementation of the model in question. A good example of a model that is widely used—even though most market participants recognize its limitations in capturing market reality—is the Black–Scholes option model. Although most option traders believe that volatilities are not constant, an assumption the Black–Scholes option model applies, most traders still choose to use the Black or Black–Scholes model in trading European-style options, while making adjustments for the reality of volatility behavior outside the model (through implementation), by allowing for two-dimensional volatility


191

treatment (across time of expiration and strike price) outside the option model. In building energy forward price curves we may choose to treat seasonality outside the forward price model and within implementation, for example. Section 7.5 will present some sample treatments of seasonality outside the forward price model and within the implementation.

7.4.6.

From Events to Models

The problem in energy markets is that events have been seen to occur not only in the near-term markets where there is the greatest amount of volatility due to immediate storage and transportation constraints, but also in seasonality across forward years, and even in the middle-term portions of the curve due to longer-term event expectations. When an event occurs anywhere within the forward price curve— that is, short-, middle-, long-term sections of the curve, seasonality or underlying price behavior aspects of the curve—someone who observes the market might see it as an event, would want to understand why there is such an event, but generally would not expect the same type of event to occur again. In general, each event does not warrant a change to the way we look at and model the forward price curve. However, when certain sections of the forward price market continue exhibiting similar events, we need to reevaluate our treatment of the forward price curve. Although such events remain independent of each other, the fact that certain aspects of the forward price curve are more prone to certain behaviors should make us reevaluate our forward price curve methodology. For example, the high volatility and resulting mean reversion of the spot price and the short-term forward prices towards some long-term price equilibrium can be observed to be market responses to usually weather-based events. Weather-based events are independent of each other, but weather is a fundamental factor in energy markets driving at least a portion of energy spot price and nearterm forward price volatility. Weather expectations, coupled with supply-side constraints, similarly, have given rise to volatility in the seasonal months of the forward price curve. This is another type of consistent event driver that will not go away with time. Therefore either our model or our implementation

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Energy Risk

has to capture the potential behaviors in the seasonal months due to this basic event behavior. Similarly, energy markets have the tendency to exhibit events in the middle portion of the curve. Every now and then we will see the market forward price curves build in event expectations a couple of years away. For example, when United States is at war, the near-term portion of the curve is already pricing this in through higher price levels, but the middle portion of the curve is also showing the expectation of this event dissipation perhaps a couple of years away; that is, the expectation of the war ending or the market supply side adjusting. When events in the middle portion of the curve tend to recur and are with us for longer periods of time, we have no choice but to honor this market reality within our models or their implementation. Section 7.6 gives an example of treating such middle-term event expectations within forward price model implementation.

7.4.7.

Parameter Calibration

Once we have defined the manner in which we want to capture market forward prices and build the marked-to-market forward price curve, we are still left with the problem of the potentially difficult and timeconsuming multiparameter calibration resulting in parameter values that give us the smallest fitting error. There are a number of methodologies for performing searches. Chapter 10 of the book Numerical Recipes in C: The Art of Scientific Computing4 is not only a great source of such methodologies, but also provides sample C-code for you to use to implement the search methodology into your valuation system. The SaddlePoint methods, in particular the Method of Steepest Descent,5 are very good in performing multivariable searches, but you still might experience problems with the methodology where the local max/min values are returned in place of global max/min values. Ultimately, a combination of robust simple search engines with more sophisticated search engines might be required depending on the search problem at hand. An example of what a calibration tool might look like is shown in Figure 7-18 for the case of seasonal parameters calibration. Note that as we go from theory to practice we might need to provide the search engine with search boundaries (the max and the min values) as well as some initial values to begin the search.


F I G U R E

193

7-18

Henry Hub Natural Gas—Exponential Seasonality: Average Seasonality

7.5.

MODELING MIDDLE-TERM EVENT EXPECTATIONS You might have noticed from some of the forward price graphs in this chapter that the graphs include some interesting theoretical underlying forward price curves. These are the theoretical curves based on the Pilipovic forward price model to which seasonality treatments are added on. However, these underlying forward price curves also include the additional implementation treatment of the middle-term event expectation. Beginning with the second half of the 1990s we have observed natural gas forward prices implying middle-term events reflecting the market expectation of a price drop due to the expectations of additional deep-well gas storage. Similarly, whenever there is a current event that is expected to dissipate over a longer period of time, we have to deal with both the preexisting short-term event drivers (such as weather and issues of supply constraints) and the superimposed longer-lasting events not expected to dissipate for at least a couple of years. In this case, like it or not, we have to incorporate additional flexibility into our underlying forward price treatment. Specifically, to treat this additional middle-term event behavior, we can add degrees of freedom to the forward price model derived from

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Energy Risk

the two-factor Pilipovic model (Equation 6-48), so that our resulting forward price looks as follows: −α ′ τ −α ′ τ −α ′ τ −α ′ τ Ft ,T ≈ St e MT + LtMT ( e LT − e MT ) + LtLT ( e µ ′τ − e LT ) −α ′ τ −α ′ τ = ( St − LtMT )e MT + ( LtMT − LtLT )e LT + LtLT e µ ′τ

(7-10)

In Equation 7-13 we have added the middle-term equilibrium price in addition to the preexisting long-term equilibrium price, and now have two diversions to the long-term exponential growth of the long-term equilibrium price: the near-term diversion of the difference between the spot and the middle-term equilibrium price that will give the forward price either a dip or a hump in the near term, depending on whether the spot price is greater or less than the middle-term equilibrium price, respectively, and a second term diversion that captures the difference between the middle term and long-term equilibrium price resulting in a middle-term hump or dip in the forward price curve depending on which equilibrium price is greater which also decays and converges towards the long-term lognormal growth of the long-term equilibrium price. Also, note that in the above formulation it is assumed that the middle term mean-reversion parameter is greater in magnitude than the long-term mean-reversion parameter, as it must be by definition. The addition of the middle-term mean-reversion parameter and the equilibrium price variable (effectively the third factor driving the underlying forward price model) constitutes an “implementation” change to the forward price model in order to capture the middle-term event behavior we clearly observe in Figure 7-25, for example. A curve-building tool that implements forward price treatment as captured by Equation 7-10 is shown in Figure 7-19. In this example of the calibration of the natural gas Henry Hub underlying parameter and variable values, we see that the middle-term equilibrium price is significantly higher than the long-term equilibrium price. In fact, the long-term equilibrium price appears to be approaching levels at least somewhat more in line with the historical levels of equilibrium prices for natural gas. Another example of such an application is given in Figure 7-20 but for the case of an on-peak power market—Cinergy 5 16. Note that the relative difference between the Cinergy middle-term equilibrium price and long-term equilibrium price is very small compared to the relative difference in the natural gas example above. Nonetheless, it does appear that


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7-19

Henry Hub Natural Gas—Exponential Seasonality: Underlying Parameters

both natural gas and power markets shown here are reacting to similar market events, although in different magnitudes. In fact, both of these market snapshots were taken on May 31, 2006, and are representative of the combination of the U.S. markets’ reaction to the effects of a war in the Middle East—driving the middle-term equilibrium price up, as well as the more local low summer temperature effects—driving the spot price and therefore the near-term portion of the curve down.

7.6.

MODELING FORWARD PRICE SEASONALITY Chapter 6 suggests capturing seasonality outside the underlying forward price model (Equation 6-30) as one means of dealing with seasonal behavior in the market quotes. This section will provide some examples

F I G U R E

7-20

CINERGY 5 16: Underlying Parameters

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of doing exactly that. In particular, we will look at three examples of seasonality capture: cosine, exponential, and power-N. The three examples will deal with capturing seasonality “outside” the forward price model, that is, within the forward model implementation stage (analogous to allowing volatilities to have a strike structure when implementing the Black option model in valuing European style options). There are, of course, a number of other approaches towards incorporating seasonality into the price behavior. For example, John Putney suggests making the seasonality treatment multiplicative rather than additive (see Chapter 5 of Vincent Kaminski’s book Energy Modeling6).

7.6.1.

Cosine Seasonality

We can attempt to capture seasonal forward price behavior through the simplest of all of seasonal function, the cosine. In general, for seasonal energy markets, most commonly natural gas and power, there is a need for at least two seasonal factors in order to capture both summer and winter seasonality. The cosine function is a relatively simple means of capturing seasonal behavior, requiring relatively few parameters in the process—the magnitude of the seasonal contribution and the time of the year defining the center of the seasonal peak. Specifically, seasonality contribution is defined by cosine (or sine) functions of two periodicities:

1 cos(2 (T t1C )) 2 cos(4 (T t2C ))

(7-11)

The first seasonal term in Equation 7-11 gives an annually repetitive seasonality, with the center at t1C and a magnitude of 1, while the second term is semi-annual and has a center of t2C, with a magnitude of 2. Coupling the above seasonal treatment together with the underlying forward price model, and applying this to the natural gas and power markets, we can obtain forward price curves such as those displayed in Figures 7-21 and 7-23, respectively. Figures 7-22 and 7-24 show the resulting statistics of such a fit for both examples. Although this methodology actually does a pretty good job in capturing seasonal behavior in the natural gas market considering the small number of parameters needed to do so, note that in the case of the Cinergy 5 16 example, there are some serious problems with the cosine seasonality treatment. In particular, the cosine seasonality


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7-21

Henry Hub Natural Gas Cosine Seasonality: Marked-toMarket Daily Curve

defines by default how quickly the seasonal rise and fall of the forward price curve occurs. Another way to put it is that the cosine functions simply do not do a good job of capturing the fast and narrow peaking seasonality of power markets. Better treatment of seasonality, in particular for power markets, is necessary.

7.6.2.

Exponential Seasonality

By comparison, the repetitive exponential functions would give us another degree of freedom that can be used to capture the narrower rise and fall F I G U R E

7-22

Henry Hub Natural Gas Cosine Seasonality: Goodness of Fit

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F I G U R E

7-23

CINERGY 5 16 Cosine Seasonality: Marked-to-Market Daily Curve

of the power seasonal prices. Additionally, exponential functions— although they can be made annually repetitive—have the nice feature of allowing a local treatment. This differ from the cosine functions, which are sinusoidal, resulting in the addition of another seasonal factor affecting the entire curve rather than just the local point of application. Therefore, exponential factors are easily additive and may be used not only to treat seasonality but also unique or repetitive events implied by market forward price quotes. We now can have the seasonality contribution defined by annually repetitive exponential functions,

β1e− γ 1 ( rfc ( T −t1 )) + β2 e− γ 2 ( rfc ( T −t2 )) + β3e− γ 3 ( rfc ( T −t3 )) , C

F I G U R E

2

C

7-24

CINERGY 5 16 Cosine Seasonality: Goodness of Fit

2

C

2

(7-12)


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7-25

Henry Hub Natural Gas Exponential Seasonality: Marked-to-Market Daily Curve

where the function rfc is annually repetitive, and returns the annualized time to or from the closest annual center, tiC, for that particular seasonal factor i. The three seasonal terms in Equation 7-12 give up to three seasonal effects to the forward price curve. Each has a center at tC, a magnitude of , with the width of the seasonal peak defined by the “decay” coefficient . Figures 7-25 to 7-28 show the resulting marked-to-market and theoretical daily forward price curves and goodness-of-fit results based on the same market quotes for natural gas and power as used in the case of the cosine seasonality treatment. The natural gas curve looks smoother, and although we still see some market noise in the early F I G U R E

7-26

Henry Hub Natural Gas Exponential Seasonality: Goodness of Fit

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F I G U R E

7-27

CINERGY 5 16 Exponential Seasonality: Marked-to-Market Daily Curve

spring months of the curve, the fit is better, as you might expect given the additional degrees of freedom and the ability to fit to the seasonality speed of decay. The short-term underlying curve we see here tells us of what price levels the market is capable of dropping down to under a lack of seasonal factors due to a lack of expected weather events. In fact, four months after this curve was built, the prices did drop into the $4.00 range. Of course, this happened due to the unexpected lack of seasonal weather patterns. Just as easily, the prices could have risen due to additional weather events. However, note that the exponential seasonality treatment will result in a different view of the underlying price curve than will the cosine treatment; the exponential seasonality tends to capture the underlying price behavior more realisticly. The power curve shows a marked difference in the fit relative to the cosine treatment. The power curves absolutely require fitting the seasonality speed of decay—which is impossible to do with the cosine treatment—as this decay does change over time and therefore its expectations also change. This probably has to do with the fundamentals of the power market: power prices are more sensitive to weather events than are natural gas prices.


F I G U R E

201

7-28

CINERGY 5 16 Exponential Seasonality: Goodness of Fit

7.6.3.

Power-N Model—Flat Seasonality

Finally, we may be interested in allowing for a “flat” price treatment in the seasonal months, although there is nothing in the market data or the historical data to necessarily justify this treatment. In this case, seasonality contribution is defined by annually repetitive “power-N” functions:

β1

(1 + γ ( fr (T − t 1

C 1

))

N

β2

+

) (1 + γ

2

( fr (T − t2C )) N

)

(7-13)

The two seasonal terms in Equation 7-13 give up to two seasonal effects to the forward price curve. Each has a center at tC, a magnitude of , with the decay of the seasonal peak defined by the “decay” coefficient , and with N a positive even integer defining how “flat” the seasonal peak shape will be (in the examples below, N 12). The Power-N treatment (where N must be a positive, even integer, its value being proportional to the “flat” effect) has the same number of degrees of freedom as the exponential treatment and results in forward price curves with theoretical curve fits comparable to the cosine, but decidedly worse than the exponential treatment (see Figures 7-29 to 7-32, where N is set to 12).

7.6.4.

Multiperiod Seasonality Treatment

If you think the above topics were getting too complicated, then hold on—there is even more complexity coming our way. Temperature patterns tend to show not only annual summer and winter seasonal patterns, but also 10-year and even 100-year cycles. What we are talking about here is the seasonality of seasonality.

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F I G U R E

7-29

Henry Hub Natural Gas Power-N Seasonality: Marked-to-Market Daily Curve

Although certainly we cannot observe 100-year forward price quotes traded, we can observe quotes—if we are so lucky—out for a number of years. From these quotes we can see that the market tends to imply seasonal magnitudes that sometimes tend to decrease, sometimes increase, and sometimes be relatively flat as we go out year by year. We can capture these changes in seasonal magnitudes by allowing multiple periods for calibrating seasonality. Figures 7-33 and 7-34 show the calibrated exponential seasonal magnitudes for natural gas over two periods, the first ranging from May 31, 2006, through October 31, 2007, and the second spanning

F I G U R E

7-30

Henry Hub Natural Gas Power-N Seasonality: Goodness of Fit


F I G U R E

203

7-31

CINERGY 5 16 Power-N Seasonality: Marked-to-Market Daily Curve

November 1, 2007, through the end of the available quotes, and beyond out to the end of December 2012. These calibrated results tell us that the winter seasonality magnitude drops as we go from first to second period, the fall (and not summer!) seasonality remains about the same, and finally the third seasonality is used to capture what appears like an expectation of an event during the month of April in the second period, whereas it appears to be nonexistent during the first period. The graph of the resulting curve is shown in Figure 7-26. Similarly, Figures 7-35 and 7-36 show the calibrated results from the exponential treatment of the power market with the two-period

F I G U R E

7-32

CINERGY 5 16 Power-N Seasonality: Goodness of Fit

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F I G U R E

7-33

Henry Hub Natural Gas Exponential Seasonality: First-Period Seasonality

seasonality, where the first period ranges from May 31, 2006, through May 31, 2007, and the second period extends from June 1, 2007, out past the last quote and through the end of December 2012. In this case, the summer seasonality magnitude increases significantly from first to second period, as does the winter seasonality magnitude. And, like the natural gas market on this day, the third seasonality magnitude can be used to capture an event drop in price (during the month of June) for the second period—although nowhere near the relative magnitude we see in natural gas—and like the case of natural gas, it appears nonexistent in the first period. The curve resulting from the above calibrations is shown in Figure 7-27.

F I G U R E

7-34

Henry Hub Natural Gas Exponential Seasonality: SecondPeriod Seasonality


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205

7-35

CINERGY 5 16: First-Period Seasonality

7.7.

SPECIAL CASE OF BASIS MARKETS We cannot discuss the building of forward price curves without covering the topic of building forward price curves for markets trading as basis to some other, primary market. The specific example here is that of natural gas, where many markets trade as basis spreads to the primary Henry Hub natural gas market. The typical argument you might hear regarding modeling basis markets is whether to model the basis or the actual market given the basis and the primary market. Asking this question is not unlike asking what comes first, the chicken or the egg. However, the basis market

F I G U R E

7-36

CINERGY 516; SecondPeriod Seasonality

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remains defined by the happenings in the actual market and the behavior of the actual market rather than the other way around. Although this is certainly a choice that every trading operation needs to make for itself, the modeling of actual markets versus basis comes with many additional benefits. Modeling the actual market is relatively speaking much easier than modeling the basis. The ultimate result is the consistency of the model chosen from one market situation to another. Although the basis markets can swing wildly, the actual markets will not be nearly as “chaotic” in terms of behavioral changes, as markets respond to different events. Second, by modeling the actual market rather than the basis, we have a parametric means of comparing the market to the primary market in terms of global behaviors such as short-term and middle-term events, relative values of the equilibrium prices, relative behaviors of seasonality, and so on. Finally, the knowledge of the relative behaviors of the market in question versus the primary market can be very useful when we find ourselves in situations of a limited number of basis quotes or illiquid basis quotes. In this case we can use the parameter values to build the forward price curve for the actual market in question and then turn around and obtain the basis spreads to the primary market even though such quotes are not readily available in the marketplace. An example of a strip of basis quotes is provided in Figure 7-37, where the basis quotes are for Columbia Gulf natural gas and the primary market is Henry Hub. The very first step in building a forward price curve for this market is to obtain the actual quotes in place of the basis quotes, as is shown in Figure 7-37. Using the resulting market quotes based on the primary market prices (calculated using the marked-to-market forward price curve for Henry Hub) and the basis quotes, we can build the daily forward price curve first with the parameter values for the Henry Hub market. The resulting graph is shown in Figure 7-38. As you can see from this graph, the actual marked-to-market curve for Columbia Gulf follows the same general shape of the Henry Hub market: the difference between the Henry Hub theoretical curve and the Columbia Gulf marked-to-market curve is not very pronounced. After calibrating the parameter values for Columbia Gulf (see Figures 7-39, 7-40, and 7-41) we can compare these to those of the Henry Hub market (provided previously in Figures 7-19, 7-33, and 7-34).


F I G U R E

7-37

Columbia Gulf—Using Natural Gas Basis Market Quotes

F I G U R E

7-38

Columbia Gulf Forward Price Curve: Built Using Henry Hub Model Parameters

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F I G U R E

7-39

Columbia Gulf: Underlying Parameters

The higher middle-term equilibrium price for Columbia Gulf than for Henry Hub suggests a greater sensitivity of Columbia Gulf to the middle-term event. However, in the long run, Columbia Gulf tends to suggest a slightly smaller longer-term equilibrium price than for Henry Hub. Although the first-period winter seasonality magnitude is similar in magnitude for the two markets, the second-period winter seasonality is significantly smaller for the Columbia Gulf market. Finally, the April event appears far more significant in the Columbia Gulf market, as can also be observed in the graph of the marked-to-market curve based on these calibrated values and shown in Figure 7-42.

F I G U R E

7-40

Columbia Gulf: First-Period Seasonality


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209

7-41

Columbia Gulf: SecondPeriod Seasonality

7.8.

NOISE VERSUS EVENTS When building forward price curves, we have to make decisions regarding the capture of market quotes diverging from the rest of the curve as events or as simply market noise. A great example of such divergence F I G U R E

7-42

Columbia Gulf Natural Gas Forward Price Curve— Incorporating Late Spring Event

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Energy Risk

is given in the example of the Columbia Gulf curve built in the previous section on basis markets. Although we treated the March/ April/May portion of the curve as a recurring event captured with the third seasonality, we could have also assumed that there was no such event, and that that section of the forward price curve was simply exhibiting market noise. We could force the third seasonality to not be used, thus eliminating the capture of this section of the curve as an event, resulting in the graph shown in Figure 7-43. The corresponding seasonality parameters used in the second seasonality period of the curve are provided in Figure 7-44. Clearly, the March/April/May divergence from the rest of the curve persists through several years of the forward price curve and the divergence is significant. However, if we believe that the divergence is superficial and not representative of a repetitive seasonal event, then we should choose this treatment of the forward price curve, despite the lesser quality of the fit to the market quotes. Again, this is another choice that each trading operation has to make for itself.

F I G U R E

7-43

Columbia Gulf Natural Gas Forward Price Curve—Treating Late Spring Event as Market Noise


F I G U R E

211

7-44

Columbia Gulf Natural Gas Forward Price Curve—Treating Late Spring Event as Market Noise—Forcing Spring “Seasonality Effect” to Null

7.9.

MARKETS WITH LITTLE OR NO MARKET DISCOVERY: OFF-PEAK AND HOURLY FORWARD PRICE CURVES If you are a producer, you have residential and industrial user load to support with unique hourly demands. As a producer, you also have a variable production load with exposure to both on-peak and off-peak markets. As a producer you are already involved in what are generally considered illiquid and maybe even nonexistent hourly and off-peak forward markets. Every residential contract or a contract with an industrial user will give you forward price exposure around the clock. Similarly, if you are involved with multiple natural gas markets or pipelines, you probably have to worry about liquidity of the various trading nodes. Although many of the natural gas nodes by now have at least some forward basis contracts traded, chances are that your longer-term forward price exposures may be difficult to hedge due to illiquidity. In the cases where there are only a few or perhaps even no market quotes to use in building the marked-to-market forward price curve, the definition of the marked-to-market curve becomes vague. Still, every attempt should be made to build such a curve in the spirit of consistency with what you do know about the market.

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Energy Risk

Historical analysis in this case becomes quite important as a benchmark and a starting point for forward price curve building. However, historically obtained parameters should never be used blindly, as chances are that there are other market-based looking forward benchmarks that can be helpful in adjusting such historical parameter values to ultimately arrive at a curve that is both true to the nature of behavior of the market in question as well as consistent with related liquid or more liquid traded markets. This is a situation where having a forward price model becomes extremely useful. Parameter values and their historical relationship to the traded or more liquid markets’ calibrated values is a useful benchmark in deciding on what the final forward price curve for the illiquid market should look like. In the case of both hourly power trading as well as the case of exposure to many natural gas delivery nodes, there is the additional problem of managing quite a bit of market and historical data in order to arrive at a methodology and a decision-making process that works efficiently and well. This is not only a difficult forward price curvebuilding problem, but also a potentially very large data management problem. The system design and data relationship design have to be handled careful to ensure the ease of market cross-over and provide market links wherever the curve building or the valuation of products based on such illiquid markets requires it.

7.10.

CONCLUSION Over the years, forward price markets in the various energy markets have continued developing, improving liquidity, and in the process providing the industry participants with more and more information about market forward price behavior. Both the underlying price behavior as well as the seasonal behavior have seen some changes reflecting changing market frameworks both due to events in the world as well as greater understanding of the market participants. This chapter on forward price curve building probably deserves a book of its own. We have covered a number of topics, and hopefully you can walk away and attempt to build some of your own forward price curves. There is no better way of truly understanding the many complexities of building forward price curves than experiencing the challenges of the process for yourself.


213

ENDNOTES 1. “The Future of Modeling,” Interview with Emanuel Derman, Risk, December 1997. 2. For example, in power, the standard is to quote the quantity per hour of delivery, whereas for Natural Gas NYMEX contracts, the standard is to quote a monthly quantity to be spread out evenly for delivery over a month. 3. Note that the Theta of the contract, which tells us how the contract value will change with the passage of time, in this case is equivalent to the value of the first day of delivery. This is due to the fact that as we find ourselves in the middle of the delivery period we will see the contract value begin to incorporate only the remaining delivery days, whereas the already captured value should go into a “Realized MTM” value “bucket.” 4. William H. Press, Saul A. Teukolsky, William T. Vetterinling, Brian P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press, 1992. 5. Jon Mathews, R.L. Walker, Mathematical Methods of Physics. The Benjamin/Cummings Publishing Company, 1970, Section 3–6, pp. 82–90. 6. Vincent Kaminski, Energy Modelling. London: Risk Books, 2005.


C H A P T E R

8

Volatilities Traders use options theory intuitively to understand complex, nonlinear patterns of variation in price in terms of simpler, linear changes in volatility and probability. They do this by regarding a hybrid as a probability-weighted mixture of simpler securities, with the probability depending on the volatility. They think linearly in terms of perceived changes in volatility and probability, and use the model to transform their perceptions into nonlinear changes in price.

Emanuel Derman1

8.1.

INTRODUCTION Volatility is one of the price characteristics that define the behavior of the price process. There are many different types of volatility measures. Spot price volatility tells us about how much randomness there is in the spot price returns over very small time intervals. Option volatility tells us about the randomness of the option’s underlying price over the lifetime of the option. All volatility measures are estimates of the degree to which randomness plays a role in price behavior. When we think about the price of a commodity and how it is going to change from today to tomorrow, there are two very specific things we need to know: What change do we expect to see, and just how wrong may this expectation turn out to be? This relates to the concept that every price change has a deterministic term and a stochastic term, as discussed in Chapter 3. The stochastic term represents the randomness in the price over some time period. The volatility tells us the magnitude of this randomness. Similarly, if we were to plot a histogram of price returns, the width of the distribution would be directly related to the volatility of price returns; the greater the volatility, the greater is the width of the distribution.


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Energy Risk

Because the volatilities tell us about a very important aspect of price behavior, they are used as an important input in the valuation and risk management of a trading book. In option pricing, the width of the price distribution determines the probability of the option expiring in-the-money. If the price distribution is very wide, then the option has a chance of expiring very far in-the-money. Thus, the greater the volatility, the greater is the value of an option. In portfolio analysis and value-at-risk analysis (VAR), volatility is also important in the simulation or estimation of the portfolio value distribution. Volatility itself can be very volatile. In fact, volatility in most energy markets is a function of time, exhibiting a combination of deterministic and random behavior that can exhibit different characteristics in the short term when compared with the long term. In other words, we see “volatility term structure” in energy markets. This, of course, is a fact that many people choose to ignore. Constant or “flat” volatilities are “easier” to handle in modeling. Unfortunately, this simplifying assumption can wreak havoc with valuation and risk management in energy markets. In this chapter we will introduce volatility; explain how to calculate historical, market-implied, and model-implied volatilities; and bring together all the concepts as we develop the kind of discrete volatility matrix required for much of the valuation and risk management necessary in energy markets.

8.2.

MEASURING RANDOMNESS It would probably be very useful to first explain the difference between three terms: variance, standard deviation, and volatility. All three measure the magnitude of randomness in a price process, but each of these measures is expressed within a different framework.

8.2.1.

Standard Deviation and Variance

Let us assume that we have a time series of spot prices, and a corresponding time series of spot price returns, which we will analyze. The

Volatilities

217

standard deviation is a measure of the width of the probability distribution of the price returns: ⎡⎛ dS ⎞ 2 ⎤ ⎛ ⎡ dS ⎤⎞ 2 ⎛ dS ⎞ STD ⎜ ⎟ = E ⎢⎜ ⎟ ⎥ − ⎜ E ⎢ ⎥⎟ ⎝ S ⎠ ⎢ ⎝ S ⎠ ⎥ ⎝ ⎣ S ⎦⎠ ⎣ ⎦

(8-1)

And the variance is simply the standard deviation squared: ⎛ ⎛ dS ⎞ ⎞ Variance = ⎜ STD ⎜ ⎟ ⎟ ⎝ S ⎠⎠ ⎝

2

(8-2)

As such, both the standard deviation and the variance are specific to the time period over which the price returns have been observed. If we take the same spot price process, and look at the standard deviations or variances of the cumulative price returns at different points in time, we will obtain different measures of standard deviation as well as of variance. Therefore, using standard deviations to compare two different distributions would not be meaningful unless we ensured that the two distributions covered the exact same overall time period. And the same is true for the variances.

8.2.2.

Volatility Defined

This problem of comparing apples to oranges motivated the definition of volatilities. Volatility, , is simply the price returns’ standard deviation normalized by time, with time expressed in annual terms:

σ=

⎛ dS ⎞ STD ⎜ ⎟ ⎝ S ⎠

(8-3)

dt

The volatilities give us a very intuitive measure of the magnitude of price randomness. The volatility roughly represents the percentage of the price range within which we can expect to see the prices 66% of the time. For example, if the spot price volatility is 0.1, or 10%, and if the spot price is currently $20, then over the next year we can—very roughly—expect the price to be within the $18 to $22 range 66% of the time.

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Energy Risk

8.2.3.

Comparing Variance and Volatility

Volatility is the annualized standard deviation of price returns. Comparing distribution volatilities instead of standard deviations ensures that we are always comparing apples to apples. There is one more difference between volatilities and standard deviations. The volatility of a price process is always assumed to measure the annualized distribution width of price returns. Standard deviations, on the other hand, are much more general and do not necessarily measure the width of the price return distributions only, but the width of any distribution you choose. Thus we can have standard deviations of price returns or of prices, for example. Hence, although the variance grows with the time period used in obtaining the price return, the volatility measure is always expressed in annualized terms. This normalization of the price return’s variance into volatilities allows us to compare different markets or models through a consistent measure of magnitudes of random behavior, and saves us from the potential mistake of trying to compare apples to oranges. Or, to put it another way, given a normally distributed process with a constant volatility, the variance of the price returns will grow with time. Hence, if we were to compare two such processes, we need to compare either the standard deviations over the exact same time periods, or we can translate the standard deviations into volatilities and compare these standardized measures of randomness magnitudes.

8.2.4.

Variance and Volatility in Spot Price Models

Typically, the spot price models assume that the price returns have a stochastic component that is normally distributed. We will use the lognormal model from Chapter 5’s Equation 5-1—as the simplest with the assumption of flat volatilities—to explain the stochastic behavior characteristics of the spot price: dS = S µ dt + σ Sdz

(8-4)

where: S spot price spot price rate of return

spot price volatility dz˜ normally distributed random variable with a mean of 0 and a variance of dt

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219

The stochastic term’s proportionality to the stock price ensures that the prices always remain positive. If we take the expected value of the price return’s stochastic term squared, we obtain the price return’s variance (applying Ito’s Lemma): ⎡⎛ dS ⎞ 2 ⎤ ⎛ ⎡ dS ⎤⎞ 2 E ⎢⎜ ⎟ ⎥ = ⎜ E ⎢ ⎥⎟ + σ 2 dt ⎢ ⎝ S ⎠ ⎥ ⎝ ⎣ S ⎦⎠ ⎦ ⎣

(8-5)

Please note that the volatility term, 2dt, is proportional to the time period over which the price return is calculated. The volatility is obtained by dividing the variance with time, and then taking the square root of that quantity:

σ=

⎡⎛ dS ⎞ 2 ⎤ ⎛ ⎡ dS ⎤⎞ 2 E ⎢ ⎜ ⎟ ⎥ − ⎜ E ⎢ ⎥⎟ ⎢⎣⎝ S ⎠ ⎥⎦ ⎝ ⎣ S ⎦⎠ dt

(8-6)

If we allow the time period between the price observations to be extremely small (close to zero), we obtain an approximation for the volatility that is purely a function of the price returns squared:

σ≅

⎡⎛ dS ⎞ 2 ⎤ E ⎢⎜ ⎟ ⎥ ⎢⎣⎝ S ⎠ ⎥⎦ dt

(8-7)

By letting dt go to zero, we can omit subtracting the drift term, as it is of order dt, and thus insignificant. One way that this approximation can be tested is through the comparison of volatilities estimated using different models. These volatility estimates tend to be very close to each other, despite the fact that different drift terms are assumed by the different models. This tells us that the drift terms are relatively insignificant as compared to the stochastic terms, and that measuring the volatility purely through the use of Equation 8-7 is a good approximation.

220

8.3.

Energy Risk

THE STOCHASTIC TERM In order to be able to understand how to define volatilities over a period of time, and how to relate the average volatility over a period of time to the shorter-term volatilities that were observed within that period of time, we need to understand the properties of the random variable dz˜ within the stochastic term. Again, using the lognormal model, the stochastic component is the dz˜t term. Recall from Section 3.6.2.1 that the random variable, dz˜t, is normally distributed, with a mean value of zero and a standard deviation of dt: dzt ~ ℵ(0, dt )

(8-8)

This is equivalent to saying that the expected value of the random variable is zero, and the expected value of the random variable squared is exactly dt. Now let us use this knowledge about the random variable dz˜t to define the behavior of the stochastic term in the equation for the price return, dz˜t. The expected value of this stochastic term is zero, because the expected value of the random variable dz˜t is zero. Similarly, the expected value of the stochastic term squared is 2dt. This means that the standard deviation of the price returns is proportional to both the volatility and the square root of the time period between price observations. The greater the time period between observations, the greater is the standard deviation of the price returns. Note that so far we have assumed that the volatility is constant and does not change. Hence the volatility of price returns can remain constant as we allow the time period between the observations to change, and the variance of price returns is proportional to the time period between observations.

8.3.1.

Case of Constant Volatility

Now that we understand how the stochastic term’s variance grows with the time period between observations, what we still need to know is how the randomness of two consecutive time periods relates to the randomness measured as the sum of the two periods. In this case we can use the fact that under the random walk, the overall path the random variable has taken is simply a sum of the individual random steps (see Figure 8-1):

σ z0 ,t = σ z0 ,t + σ zt ,t + … + σ zt N

1

1 2

N −1,t N

(8-9)

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221

F I G U R E

8-1

Sample Random Price Path

where we will assume that the n-th step was made corresponding to time period tn to tn 1, hence each step has a variance of (tn 1 tn). The expected value of the overall path, z˜0,t , is zero, because each of the steps is a random variable with an expected value of zero. Similarly, the variance of the overall path is the sum of the individual steps’ variance, given by N

N −1

σ 2tn = σ 2t1 + σ 2 (t2 − t1 ) + … + σ 2 (t N − t N −1 ) = ∑ σ 2 (tn+1 − tn )

(8-10)

n= 0

This result was derived by incorporating the fact that in a true random walk, the autocorrelation between the steps is zero, hence each step is independent of other steps and shows zero correlation with the other steps.

8.3.2.

Case of Volatilities with Term Structure

The above relationship for the variance of a whole path can be generalized for the case where the volatilities are not the same for each step (i.e., they have term structure). In this case, each step might have a variance of a different magnitude, tn, t . The path’s variance is then given by n 1

σ 02,t n tn = σ t2 ,t t1 + σ t2 ,t (t2 − t1 ) + … + σ t2 0 1

1 2

N −1 , t N

N −1

(t N − t N −1 ) = ∑ σ t2n , t n +1 (tn +1− tn ) n=0

(8-11)

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The volatility of the path is then given by the square root of the variance of the path divided by the time it took to complete the path: N −1

σ 0 ,t =

∑σ n= 0

2 tn,tn +1

(tn+1 − tn )

(8-12)

tN

N

It is important to understand this relationship, as it can be extremely useful in allowing us to go from defining volatilities over short periods of time to average volatilities over longer periods of time.

8.4.

MEASURING HISTORICAL VOLATILITIES We can measure the volatility of historical time series of prices; the values generated are called historical volatilities. Note that these values are average volatilities for the period analyzed.

8.4.1.

Simple Techniques

Volatilities can be observed from historical data of spot or forward prices. Following the pattern of the previous section, we can simply use price returns to obtain the volatility estimates:

(

)

2 σ 2 dt = E ⎡ dS / S ⎤ ⎣⎢ ⎦⎥

(8-13)

or rather, for a data set of N price returns, we have N ⎧ σ = ⎨(1 / N )∑ dSn / S n n=1 ⎩

(

)

1/ 2

2

⎫ / dt ⎬ ⎭

(8-14)

Note that the 1/dt term is the time normalization of the price return variance. In the cases where the prices returns were calculated using calendar daily prices, this term would equal 1/365, for 365 days

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223

in the year. If, on the other hand, the price returns were weekly, then this term would be 1/52, for 52 weeks in the year. When historical price data cover only the business days, and not the weekends, we have a choice of how to annualize the volatilities. One alternative is to ignore the weekend effect and simply let 1/dt equal 1/252 for 252 business days in the year, thus making the assumption that the weekends have no additional impact on the prices. Another choice is to treat the price returns over the weekends by normalizing these returns with 3/365, while all the other price returns are normalized with 1/365, thus treating price returns over the weekends differently from the price returns during the week.

8.4.2.

More Complex Techniques

There are more sophisticated means of obtaining volatility estimates from historical price data. Although the above volatility estimation involves daily price observations, such as close of day or daily settled prices, another alternative is to use not only the closing price, but also the daily high and low prices of the commodity. The estimation process in this case involves correcting for the bias that results in the use of the high and the low prices over the day. However, once the bias is taken care of, the resulting volatility measure carries less of a sampling error than would the simple close-to-close volatility estimate. This technique is primarily limited to analyzing data from futures markets, where the exchanges report highs, lows, and closing prices that are all readily observed by the marketplace and available to the public as historical data. Most of the over-the-counter (OTC) energy markets do not have this kind of luxury in data availability. In these cases, we are lucky if we can just get our hands on the daily settlement price data in order to do simple volatility estimates. Once we have volatility estimates for relatively short time periods, such as on a per-week or per-month basis, we can perform time series analysis of these estimates in order to capture volatility term structure behavior, and in order to be able to provide volatility forecasts. Such volatility forecasts are particularly important in the markets where the options are not very liquid and obtaining market-implied volatility information is practically impossible.

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8.5.

Energy Risk

MARKET-IMPLIED VOLATILITIES This section will discuss volatilities implied by current market prices of an option, a series of options, caplets, and options on average prices. These market-implied volatilities tell us something about the future, whereas historical volatilities only describe the past.

8.5.1.

Option-Implied Volatilities

The option volatility is the volatility we input in order to get the option price. However, we can also go the other way. If we have the option price, we can back out the volatility used in getting the option price. Such option-implied volatility can be very different in value from the historical volatility. Historical volatilities are calculated based purely on the historical underlying market price. As such, the historical volatilities have nothing to do with the traded option prices. The traded option prices, instead, imply what the market thinks the option volatilities should be going forward in time. As with historical average volatilities, we will be backing out a single, average volatility implied by the market option price(s). The volatility implied from the market option prices, called the market-implied volatility, can be very different from the historical volatility. The reason for this difference is that the option-implied volatility looks forward in time, and the historical volatility looks backward through time. The historical volatility uses historical price data in its calculation; thus it is a volatility measure of already-past price behavior. The market-implied option volatility, on the other hand, represents what the market expects the option underlying price uncertainty to be over the time period from today until the option expiration date. Market-implied volatility “looks” forward in time, and as such is a reflection on how volatile the market believes the prices will be over the time period till the option’s expiration. In order to measure the implied volatility from a single option price, the most practical method to use is the simple search method: try many different values of volatility in the option pricing formula. The volatility that provides the desired option price would be considered the modelimplied volatility. If the option price is an actual market quote, then this model-implied volatility also becomes the market-implied volatility.

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225

When we refer to the Black–Scholes equivalent volatility, we simply mean that we have used the Black–Scholes option pricing model in backing out the implied flat or constant volatility, given the option price. Because Black–Scholes is based on a lognormal price process, the Black–Scholes equivalent volatility is also the lognormal equivalent volatility. When we have market prices for European options—which settle on discrete prices, such as a single spot or a single forward price on a specific date—then the option price can be used to imply the volatility of the discrete price. This option-implied volatility represents the average volatility of the price from today to the option’s expiration date.

8.5.2.

Implied Volatilities from a Series of Options

If we have a series of European options with discrete price settlement and with increasing expiration times, we can use these options to back out the rudimentary term structure of such time-averaged volatilities of discrete prices. If you have a series of options that are based on the same settlement price, then you have enough information to begin seeing the underlying volatility term structure specific to that settlement price. For example, if we have three European options, one with an expiration in a month, the second in two months, and the third in three months, and all the options are on the same three-month forward price—which we will assume to be lognormal—then we can use these option prices to back out the term structure of the forward price volatilities over the first three months. In this case, because all three volatilities measure the randomness of the exact same point on the forward price curve, we can use this information to back out the volatility of the forward price for each of the three months. The one-month option price implies the volatility of the forward price over the first month:

σ 0 ,1 = σ 0Option ,1 where:

0,1 the first month’s volatility Option

0,1 the one-month option-implied volatility

(8-15)

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The two-month option volatility is the average volatility of the same forward price but over the first and the second months:

σ

Option 0 ,2

⎛ (σ 02,1 + σ 12,2 ) ⎞ = ⎜ ⎟ 2 ⎝ ⎠

(8-16)

The two-month option-implied volatility is the average of the first month’s volatility and the second month’s volatility, the averaging being carried out as defined by the above equation. As we already have the first month’s volatility and we have the two-month option volatility, we can back out the second month’s volatility of the forward price:

(

σ 1,2 = 2 σ 0Option ,2

)

2

− σ 02,1

(8-17)

Finally, the three-month option volatility is the average of the first, second, and third months’ volatilities: ⎛ (σ 02,1 + σ 12,2 + σ 22,3 ) ⎞ σ 0Option = ⎜ ⎟ ,3 3 ⎝ ⎠

(8-18)

This provides the solution for the third month’s volatility of the forward price:

(

σ 2 ,3 = 3 σ 0Option ,3

)

2

− σ 02,1 − σ 12,2

(8-19)

In the case where we do not have a series of single option prices on the same discrete forward price, as in the above example, but instead have cap or floor prices (still settled on discrete prices), the job of backing out the volatility term structure of the underlying price(s) becomes more difficult. This particular problem is handled in the following section.

8.5.3.

Calibrating Caplet Volatility Term Structure

Option contracts that are commonly called caps and floors in interest rate markets, but translate to energy markets, also provide us with information to calibrate volatility term structure. In interest rate

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227

markets, a cap or a floor is a series of options all priced together, with their prices summed up to give a single cap or floor price. (A cap is a series of call options; a floor is a series of put options.) A cap or a floor has a tenor associated with it. The tenor refers to the expiration date differential between the options within the cap or the floor. For example, a one-year cap with a monthly tenor consists of twelve individual options, referred to as caplets, with the first caplet expiring in the first month, the second in two months, the third in three, and so on, with the last caplet expiring in one year. So, given a series of cap or floor prices instead of individual option prices, the process of backing out volatility term structures, specifically for individual options, becomes more difficult. Exactly the same option structures exist in energy markets, although such option contracts are simply called calls and puts, although generally they consist of a series of calls and puts. For the sake of differentiation between the overall contract “name” and the individual options comprising the overall contract, we will continue referring to these overall contracts as caps and the individual options comprising the caps as caplets within the rest of this chapter.

8.5.3.1. Motivation You may wonder why in the case of caps or floors we need to bother with backing out the individual caplet option volatilities. Why not just back out one cap-specific volatility? And you would be right to make this comment in a world where the volatilities are indeed constant or do not have a strong term structure. Unfortunately, this is not the case in the real world of energies. By backing out the single cap volatility, we would be making the incorrect assumption that the option volatility is constant throughout the cap lifetime. Furthermore, we would find many limitations in the process, including possibly obtaining negative forward variances—a topic that we will get into in the following sections. Volatilities do indeed have a term structure—typically a very obvious and strong term structure in the energy markets—suggesting that the volatilities at which the options of different expirations should be valued will possibly be very different. Ignoring term structure in caplet volatilities would impact not only the pricing of the options, but also the hedging. By not recognizing and backing out this caplet volatility term structure, the trading operation would allow for the existence of arbitrage within its books. Let me

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give you an example here. Suppose that I back out and use only the cap-implied volatilities in the trading and hedging of my cap books. I might use a volatility of 25% for pricing the one-year monthly tenor cap, and a volatility of 15% for pricing the two-year monthly tenor cap. If I did this, my pricing would be inconsistent, not to mention the hedging. Why is this a problem? Well, the one-year cap consists of the first 12 caplet options, the first with an expiration in one month and the last with an expiration in one year. The two-year cap consists of the 24 options, the first with an expiration in one month, and the last with an expiration in two years. Hence, the two-year cap consists of the one-year cap plus the additional 12 options, with expirations past the first year. And yet, by using the two different cap volatilities I would be pricing this one-year cap—on its own—at the 28% volatility, and I would be pricing it—as a part of the two-year cap—at the 22% volatility (Figure 8-2). This is obviously inconsistent pricing, not to mention that my hedging of the first-year cap within the two different deals would also be very different, when in fact it should be the same. The only way of getting around this problem is to indeed back out the caplet, or the individual option volatilities, in order to obtain the true volatility term structure, which will allow consistent pricing and hedging between caps and floors. I should be using the exact same caplet volatility curve to price the two caps in the example (Figure 8-3).

F I G U R E

8-2

Incorrect Treatment of Caplet Volatilities

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229

F I G U R E

8-3

Correct Treatment of Caplet Volatilities

8.5.3.2. Calibration Techniques Suppose that you have the three-month cap, the one-year cap, and the two-year cap, all of monthly tenor. This means that the first cap consists of three caplets or options. The second cap consists of 12 caplets, the first three being identical to the caplets in the first cap. The third cap consists of 24 caplets, the first 12 of which are identical to the caplets in the one-year cap. We are trying to back out the volatilities corresponding to each caplet such that we retain the market price of the three caps while at the same time we use the same volatilities to price the same caplets. The cap prices are then given by CAP1 = C ( F1 , σ 1 , T1 ) + C ( F2 , σ 2 , T2 ) + C ( F3 , σ 3 , T3 )

(8-20)

CAP2 = C ( F1 , σ 1 , T1 ) + C ( F2 , σ 2 , T2 ) + … + C ( F12 , σ 12 , T12 )

(8-21)

CAP3 = C ( F1 , σ 1 , T1 ) + C ( F2 , σ 2 , T2 ) + … + C ( F24 , σ 24 , T24 )

(8-22)

where C(FT, T, T) equals the call option price of the caplet calculated as a function of the forward price FT, using the volatility T, and with the expiration time of T. We try to estimate all the caplet volatilities,

T, such that we preserve the market prices of caps, CAP1, CAP2, and CAP3.2 An example of what such a caplet volatility term structure might look like is given in Figure 8-4. The graph shows the volatilities

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F I G U R E

8-4

Caplet Volatilities with Event Expectation

increasing from the current spot price out to the three-month point. Such a volatility term structure is typical of a market where there is an expectation of an event, such as volatility in the weather, this expectation being reflected in volatilities increasing before they begin declining toward more of the volatility equilibrium levels in the long-term portion of the volatility curve.

8.5.4.

Implied Volatilities from Options on the Average of Price

We discussed the complications of backing out volatilities when the market quotes caps and floors instead of individual options on discrete prices. Here we will discuss an additional complication: that of pricing options where the settlement is based on an average of prices rather than a discrete price. In this case, we need to ensure that our option model can indeed handle the averaging effects, so that we end up with a volatility term structure consistent with discrete price volatilities. If we do not go through this additional trouble, we might have to carry volatility term structures for all kinds of possible averages, such as monthly average of daily prices, quarterly average of daily prices, annual average of weekly prices, and so on. Even more importantly, by

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231

F I G U R E

8-5

Average Price Option Vols for Flat Caplet Volatilities

not translating the volatilities back to the discrete price level we will lose the possibility of linking the volatilities of different types of averages, which will result once again in an inconsistent treatment of the pricing of options on different types of averages. Figures 8-5 and 8-6 show the effects of averaging on option volatilities. The first graph shows the case of flat discrete price volatilities, and the second graph shows the case of discrete price volatilities with term structure. In the energy markets, the OTC options markets tend to trade quite a lot of caps and floors on averages of prices. In trying to back out the option volatility term structures for discrete prices in the energy

F I G U R E

8-6

Average Price Option Vols for a Term Structure of Caplet Volatilities

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markets, we need to worry both about the treatment of caps versus caplets and the treatment of options on discrete versus average price settlements. Obviously, this can be quite a job, and the actual development process requires a good amount of organizational thought.

8.5.5.

The Volatility Smile

Last, but not least, is the problem of capturing the volatility smile. The volatility smile is a phenomenon that shows the lognormal equivalent volatilities for options of the same time to expiration and on the same settlement price to be different across different strikes. When the actual price distribution has fatter tails than does the lognormal model distribution, the out-of-the-money calls and puts tend to show volatilities that increase as the option strikes make the options go further and further out-of-the-money. When graphed across strike prices, such a volatility strike structure ends up looking like a smile, hence the name. In reality, the volatility strike structure does not have to be a smile. It can take on various shapes, depending on what the actual price distribution is, compared to the lognormal price distribution. As you might have already guessed, if the option pricing model is built to incorporate the exact price distribution, the volatility strike structure would be flat, that is, the same volatility would reflect all the options of the same expiration time but of varying strike prices. Unfortunately, it is difficult to come up with such an optionpricing model that incorporates all the price distribution characteristics. Thus the traders are forced to incorporate the strike structure volatility effects in the implementation of the option methodology: the traders maintain the volatilities for various out-of-the-money and in-the-money options.

8.6.

MODEL-IMPLIED VOLATILITIES In this section we will introduce the third set of volatility calculations: the model-implied volatilities. Given model parameters, we can calculate the model-implied volatilities. We will use the three price models

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233

introduced in Chapters 3 through 6 to back out model-specific volatility term structures and correlation matrices:

• the single-factor lognormal • single-factor log-of-price mean-reverting models • the two-factor price mean-reverting model The assumptions we make—through models—about the market spot price behavior affect the behavior of forward prices, and both of these affect the look of volatility term structures and correlation matrices. Given the models for the spot and forward price behavior, we can estimate their model-implied volatilities and correlations. We can use historical data to observe the historical volatilities and correlations. Similarly, we can use the historical data to calibrate the model parameters, and then use these to obtain model-implied volatilities and correlations. The comparison between the historical volatilities and correlations and their model-implied counterparts provides us with yet another means of benchmarking between models. These model-implied volatilities and correlations provide us with another set of tests on model appropriateness, and they also may be used in the case where the options market is highly illiquid, if not almost nonexistent. In such a market, the options on the books still need to be valued regardless of how sporadic and infrequent the option deals are. The model-implied volatilities can be used in such cases to support whatever market information there is and allow the pricing and hedging calculations for the illiquid options.

8.6.1.

The Lognormal Model

The single-factor lognormal model was defined by Equation 5-1, and it is repeated below: dSt = µ St dt + σ St dzt

(8-23)

The volatility is the same as the Black–Scholes implied volatility, as Black–Scholes assumes that Equation 8-23 is the option’s settlementprice behavior. This volatility is assumed to be constant over time, and

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Energy Risk

is exactly the same for the spot price as for all the forward prices on the forward price curve:

σ ts =

F

σ t t ,T =

⎡⎛ dS ⎞ 2 ⎤ Et ⎢⎜ t ⎟ ⎥ ⎢ ⎝ St ⎠ ⎥ ⎦ =σ ⎣ dt

(8-24)

⎡⎛ dF ⎞ 2 ⎤ Et ⎢⎜ t ,T ⎟ ⎥ ⎢⎝ Ft ,T ⎠ ⎥ ⎦ = ⎣ σ dt

(8-25)

The model-implied volatility term structure is flat. In addition, the single-factor lognormal model implies that the forward prices are perfectly correlated with the spot prices and also with each other:

ρS , F t

ρF

t ,T

t ,T 1 , Ft ,T 2

⎡ dS dF ⎤ Et ⎢ t t ,T ⎥ ⎢ St Ft ,T ⎥⎦ =1 = ⎣s F σ t σ t ,T dt

(8-26)

⎡ dF dF ⎤ Et ⎢ t ,T 1 t ,T 2 ⎥ ⎢ Ft ,T 1 Ft ,T 2 ⎥⎦ =1 = ⎣F F σ t ,T 1σ t ,T 2 dt

(8-27)

Because the energy spot and forward price volatilities exhibit strong decaying term structure across the forward prices, and nonperfect correlations, the single-factor lognormal model is not consistent with reality.

8.6.2.

The Log-of-Price Mean-Reverting Model

In the single-factor version of the log-of-price mean-reverting model, the mean reversion affects the volatility term structure by giving it a decreasing effect over time. As discussed to a great extent in the previous chapters, mean reversion has a dampening effect on spot price volatility, estimated from spot distributions over time. The greater the

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235

time, the more we see this dampening effect when compared to what the lognormal model would exhibit with the exact same spot price volatility. When dealing with forward prices—whose behavior is a function of the spot price behavior—the volatility dampening effect is also of importance. In fact, in a single-factor mean-reverting model, the volatility of the forward prices approaches zero as the forward price expiration date is allowed to grow to infinity. Specifically, the volatility of the spot and forward prices in a single-factor mean-reverting model, where the mean reversion is in the log of the spot price, is given by

σ ts =

⎡⎛ dS ⎞ 2 ⎤ Et ⎢⎜ t ⎟ ⎥ ⎢ ⎝ St ⎠ ⎥ ⎣ ⎦ =σ dt

⎡⎛ dF ⎞ 2 ⎤ Et ⎢⎜ t ,T ⎟ ⎥ ⎢⎝ Ft ,T ⎠ ⎥ ⎣ ⎦ = e−α ( T −t )σ dt

F

σ t t ,T =

(8-28)

(8-29)

Note that although the spot price volatility is constant over time, the volatility of the forward price decreases exponentially the greater the forward price time to expiration, T, is. Now the interesting thing about this single-factor mean-reverting model, with the mean reversion in the log of the price, is that although the volatilities are indeed functions of the forward price time to expiration and have a term structure that decreases over time toward zero, the correlations are not a function of the forward price time to expiration. In fact, the correlations remain perfect between all points on the forward price curve:

ρS , F t

ρF

t ,T 2

t ,T 1 , Ft ,T 2

⎡ dS dF Et ⎢ t t ,T ⎢ St Ft ,T = ⎣s F σ t σ t ,T dt

⎤ ⎥ ⎥⎦ =1

⎡ dF dF ⎤ Et ⎢ t ,T 1 t ,T 2 ⎥ ⎢ Ft ,T 1 Ft ,T 2 ⎥⎦ = ⎣F =1 F σ t ,T 1σ t ,T 2 dt

(8-30)

(8-31)

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Energy Risk

The correlations retain the characteristics of a lognormal model with no mean-reversion effects. This is due to the fact that we are working with a single-factor model. The next case, which looks at the two-factor price mean-reverting model, will have a different correlation result.

8.6.3.

The Price Mean-Reverting Model

In the two-factor price mean-reverting model (Pilipovic), we find that the volatilities of the forward prices and the correlations as well are functions of the forward price time to expiration and follow a meanreverting process to the equilibrium price volatility. The stochastic term in the change of the forward price, dFt,T, over some time period, dt, is given by: dFt ,T − E ⎡⎣ dFt ,T ⎤⎦ ≅ e−α ′τ Stσ dzt + ( eµ ′τ − e−α ′τ ) Lt ξ dw t

(8-32)

where: S spot price Ft,T forward price with expiration T observed at time t rate of mean reversion risk-adjusted rate of mean reversion cost of risk time to expiration T t

spot price volatility risk-adjusted drift rate equilibrium price volatility dz˜t random stochastic variable in the spot price return random ˜t stochastic variable in the equilibrium price return dw Assuming that the spot price and the equilibrium price have a correlation of , we have the spot and the forward price model-implied volatilities defined as follows:

σ ts =

⎡⎛ dS ⎞ 2 ⎤ Et ⎢⎜ t ⎟ ⎥ ⎢ ⎝ St ⎠ ⎥ ⎣ ⎦ =σ dt

(8-33)

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237

F

σ t t ,T =

⎡⎛ dF ⎞ 2 ⎤ Et ⎢⎜ t ,T ⎟ ⎥ ⎢⎝ Ft ,T ⎠ ⎥ ⎣ ⎦ dt (8-34)

=

2

⎛ ⎧ α ⎫ µ ´( T − t ) − α ´( T − t ) ⎞ 2 2 e−2α ´( T −t )σ 2 St2 + ⎜ ⎨ −e ⎬ e ⎟ ξ Lt ⎝ ⎩α + µ ⎭ ⎠ 2 Ft T

{

}

While the spot price volatility remains constant over time, the forward price volatility is defined by both the spot price volatility and the long-term equilibrium price volatility. The weighting of the forward price volatility on the spot price volatility decreases, and the weighting on the long-term equilibrium price increases as the forward price expiration time increases. The long-term forward prices have volatilities converging toward the equilibrium price volatility. As the forward price expiration date goes to infinity, its volatility approaches and is almost entirely defined by the long-term equilibrium price volatility. Similarly, for the correlations, we obtain

ρS , F t

ρF

t ,T 1 , Ft ,T 2

t ,T 2

⎡ dS dF ⎤ Et ⎢ t t ,T ⎥ αS ⎢ St Ft ,T ⎥⎦ = ⎣s F = e−α ´( T −t ) F t σ t σ t ,T dt σ t t ,T Ft ,T

⎡ dFt ,T dFt ,T ⎤ 1 2 ⎥ Et ⎢ ⎢⎣ Ft ,T1 Ft ,T2 ⎥⎦ = F F σ t ,T 1σ t ,T 2 dt 2 ⎛ ⎞ ⎛ α ⎞ − α ( T1 − t ) − α ( T2 − t ) 2 2 µ ( T1 − t ) − α ´( T1 − t ) µ ´( T2 − t ) − α ´( T2 − t ) 2 2 − e σ St + ⎜ ( e e )( e − e ) ξ L ⎜e t⎟ ⎜⎝ ⎟⎠ ⎝ α + µ ⎟⎠ = F F σ t ,T1 Ft ,T σ t ,T 2 Ft ,T 1

(8-35)

(8-36)

2

As long as the volatility of the long-term equilibrium price, , is nonzero—that is, the long-term equilibrium price is allowed to exhibit

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Energy Risk

a stochastic behavior—the correlation between the forward prices of different expiration dates, and also the correlation between the spot and forward prices, will be less than one. If the two-factor price mean-reverting model is reduced to a singlefactor model, the above correlations will become one, as is set to zero. Although a single-factor mean-reverting model can capture the decreasing volatility term structure typical of the energy markets, a second factor in the spot price model is necessary to capture the nonperfect correlations between the spot and the forward prices and also between the forward prices of differing expiration times. We can simplify the above expression for volatility and correlation by approximating the spot price with the equilibrium price and assuming that the drift term on the equilibrium price is much smaller than the meanreverting parameter: ⎡ dF0 ,T dF0 ,T ⎤ −α ´( T +T ) 2 1 21 2 ⎥≅e E⎢ σ 2 + (1 − e−α ´T1 )(1 − e−α ´T2 )ξ ⎢⎣ F0 ,T1 Ft ,T21 ⎥⎦ + (e

− α ´T1

(1 − e

− α ´T2

)+e

− α ´T2

(1 − e

− α ´T1

(8-37)

)) ρσξ

where: and where we have allowed the spot and the equilibrium price to exhibit a nonzero correlation, . We can also simplify the correlation formulation:

σ F2 ≅ σ 2 e−2α ´τ + (1 − e−α ´τ )2 ξ 2 + 2 ρσξ e−α ´τ (1 − e−α ´τ )

(8-38)

Figures 8-7 to 8-12 demonstrate that, when compared to historical volatilities in the natural gas and WTI markets, the two-factor price mean-reverting model does well in capturing the behavior of forward prices. The practical implication of the two-factor model is that every forward price point on the curve ends up having its own volatility term structure. The two-factor model forces us to look at volatilities not across a single dimension—time—but across two dimensions—time and forward price. Thus we go from the single-volatility term structure—corresponding to single-factor models—to the two-dimensional matrix of volatilities.

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239

F I G U R E

8-7

Comparison of Model Implied to Historical WTI Volatilities

F I G U R E

8-8

Comparison of Model Implied to Historical Natural Gas Volatilities

F I G U R E

8-9

WTI Historical Correlations

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Energy Risk

F I G U R E

8-10

WTI Model Implied Correlations

F I G U R E

8-11

Natural Gas Historical Correlations

8.7.

BUILDING THE VOLATILITY MATRIX To truly understand the implications of having forward prices that have individual volatility term structures, we will go through the process of building a volatility matrix. Building the volatility matrix depends on all the intuition and techniques introduced so far in this

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241

F I G U R E

8-12

Natural Gas Model Implied Correlations

chapter. The technique represents a comprehensive methodology of viewing volatilities at their most discrete level.3 Any market that exhibits a split personality in the forward prices—as measured by short-term forward prices versus long-term forward prices, with correlation being significantly less than one between the short- and longterm forward prices—needs to be treated with a volatility matrix instead of a single volatility curve. In other words, any market that has a true nature of being driven by two factors (if not more) needs to have a volatility matrix structure rather than a one-dimensional volatility term structure.

8.7.1.

Introduction to the Forward Volatility Matrix

The volatility matrix provides the lowest common denominator volatilities that can be combined in many different ways to capture the pricing of a variety of different types of options in a trading book. These lowest common denominators represent the smallest volatility blocks necessary to provide a consistent volatility framework for the various types of options in the book, and for capturing the specific characteristics of a particular marketplace (see Figure 8-1). Each cell could represent a discrete volatility or the volatility of the particular time bucket.

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Energy Risk

In this volatility matrix framework, we have a means of defining discrete, or very short-term, volatilities for every forward price at any point in time in the future. The volatility matrix has two dimensions: the first is the time dimension, starting with today, and the second is the forward price, starting with the spot price. All the elements in this volatility matrix are thus short-term volatilities, representing the volatilities of forward prices with different expirations and at different points in time, but always over some short time period, dt.

8.7.2.

Discrete Volatilities

We can define the discrete volatility at time t specific to the forward price with expiration time T, Ft,T, as t,Discrete where equals T t, that is, the forward price time to expiration relative to the time of observation t (Figure 8-13). Now, suppose that we can forecast these discrete volatilities at regular and discrete time periods, starting with time t 0, then dt, 2dt, 3dt, and so on. For each of these times we will have a set of discrete volatilities corresponding to all the forward prices. Hence, for time t 0, we will have the current discrete volatilities, across all the forward prices, where is the time to expiration. At time dt, we will have the first discrete period’s discrete volatilities, again across all the forward prices but as observed at time dt, and so on. Now,

F I G U R E

8-13

Discrete Volatility Matrix by Time Buckets

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243

F I G U R E

8-14

Sample Discrete Volatility Matrix

if we let the expiration times of all the forward prices also be defined in discrete time terms, we obtain a matrix of discrete volatilities, with the vertical axis representing the time t, and the horizontal axis representing the forward expiration time—expressed relative to time t. A three-dimensional example of such a volatility matrix is shown in Figure 8-14. Figure 8-15 shows another example of a volatility matrix under event expectations. The volatility forecast for the spot price at some time ndt in the future would then be given by ndt,0, where the first index refers to the time of the volatility forecast, and the second index refers to the point on the forward price curve (in the case of the spot price it is the 0-th point). In Figure 8-13, the spot price discrete volatility term structure is given by the first column. If we take a snapshot of the current discrete volatility for all the points on the forward price curve, these volatilities would be given by 0,ndt. The first index is given the value zero because we are taking the discrete volatility snapshot today, at time t 0, and the second index, ndt, refers to the forward price point on the curve with expiration time of ndt relative to today. Hence, ndt,mdt refers to the volatility at time ndt from today, of the forward price, which expires at time mdt relative to ndt. The volatility term structure of the same forward price point on the forward price

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F I G U R E

8-15

Sample Effect of Event Expectations on Discrete Volatilities

curve would be given by one of the columns in Figure 8-13. On the other hand, if we wanted to find out what the discrete volatilities are at some time t for all the points on the forward price curve, starting with the spot price, we would look at a particular row of the matrix that corresponds to the time of interest.

8.7.3.

Tying In Caplet Volatilities

We are using discrete, or short-term, volatilities to build this matrix, in order to ensure that we have the least-common-denominator volatilities, which can be used in many different ways to ultimately value a diverse set of derivative products. We will go through the steps of how these volatilities relate to the particular option volatilities in the later sections of this chapter. But first it would be insightful to take you through the steps of how these discrete volatilities might equate to Black–Scholes equivalent volatilities in the case of a European option on a forward price. Suppose that at time t 0 we purchase an option on a forward price with the expiration time of 3dt, and the option expires at the same time as the forward price, at time t 3dt. During the first time period, t dt,

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245

the forward price moves down the forward price curve, from the expiration of 3dt at the start of the period to the expiration of 2dt at the end of the period. During the second time period, t 2dt, the forward price keeps moving down the forward price curve, from the expiration of 2dt at the start of the period to the expiration of dt at the end of the period. Finally, during the third and also the last time period, t 3dt, the forward price moves from the point of expiration of dt down the curve until it converges with the spot price at the very end of the third period. In terms of the volatility matrix, the discrete volatilities, which correspond to each of the periods, lie along a “cross-diagonal” line: the discrete volatility corresponding to the first period is found in the first row and third column of the volatility matrix, the discrete volatility corresponding to the second period is in the second row and second column, and the third-period discrete volatility lies in the third row and first column of the discrete volatility matrix. The average volatility across all three periods relates to these discrete volatilities as follows:

σ 0 ,3 =

(σ 12,3 + σ 22,2 + σ 32,1 ) 3

(8-39)

European options with discrete forward price settlement will have average option volatilities that are the average across the cross-diagonal of the discrete volatility matrix. The calculation of these average volatilities can be generalized so that for the option that expires in N months, we have

σ 0,N =

(σ 12, N + σ 22, N −1 + … + σ N2 ,1 ) N

(8-40)

Note that if the volatility of the near-term portion of the forward price curve tends to be much greater than the volatility of the long-term portion of the curve, the forward price discrete volatilities across time—as the forward converges toward the spot price and we come closer and closer to the option expiration—will grow. Figure 8-16 shows a sample path that the discrete volatilities of a forward price might follow as it converges to spot. In volatility markets where the spot price volatility is significantly greater than the long-term forward price volatility, this market characteristic translates into options that capture most of their volatility value close to the expiration. Another way of saying this is that the

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F I G U R E

8-16

Discrete Volatilities Defining the Five-Month Caplet Volatility

option decay is the greatest close to the option expiration. Finally, this fact of increasing discrete volatilities as the forward price converges toward the spot price has a big impact on the hedging calculations. Thus, an option model that uses the average option volatility will do just fine in terms of the price. However, it will do very poorly in terms of providing the appropriate hedges. Instead, an option model that incorporates the discrete volatility term structure will provide both the correct price and the correct hedges.

8.7.4.

Two-Dimensional Approach to Volatility Term Structure

The discrete volatility matrix approach allows us to build in market characteristics that a single-volatility term structure would not allow us to do. (A “single-volatility term structure” is captured by a vector of volatilities rather than a matrix.) As discussed above, the capture of market characteristics through modeling can have a big impact on the appropriate hedging and risk management of options. One such characteristic is that the near-term portion of the forward price curve has very little correlation with the long-term portion of the forward price

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247

curve. It is only intuitive, then, that the near-term portion of the forward price curve has a volatility term structure that can be very different from the volatility term structure of the long-term portion of the forward price curve. Similarly, incorporating the mean-reverting tendencies in the spot price results in the spot price and near-term forward price volatility term structures, which decay over time toward much lower levels. This possibly very strong decay—when the mean reversion is very strong— is not seen in the long-term portion of the forward price curve, which tends to have much flatter volatility term structure. Similarly, expectations of eventful markets might give spot and near-term forward price volatilities a term structure with a “hump.” The existence of such an event hump in the volatility term structure of the long-term forward prices is very unlikely, unless there are events that are expected to affect both the short-term and the long-term market prices. However, even in markets where the mean reversion is relatively small, such as the interest rate markets, there is a market condition under which a single-volatility term structure would yield what should be impossible results. The resulting discrete variances calculated under the single-volatility term structure framework can turn out to be negative. The single-volatility term structure framework is equivalent to the matrix approach where all the rows of the volatility matrix are equivalent to the very first row. In other words, in the single-volatility term structure framework we are making the assumption that the volatilities of individual forward price points are constant over time. Let us work through two examples of single-volatility term structure applications. First we will look at a case where the single-volatility term structure approach yields positive discrete volatilities, and then we will look at a market condition where the resulting discrete volatilities turn out to be mathematically imaginary. If the market quotes only caps and floors, the transformation from the cap volatilities to the caplet volatilities needs to be performed first. Once the caplet volatilities are available, we can proceed to back out the discrete volatilities. Consider the following market situation. Suppose that you have caplet volatilities for the one-month, two-month, and three-month caplets:

1CAPLET 40%

2CAPLET 30%

3CAPLET 25%

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The first month’s discrete volatility is then given by the first month’s caplet volatility:

1 1CAPLET 40% In order to get the second month’s discrete volatility, we need to do a bit of calculation:

σ 2 = 2(σ 2CAPLET )2 − σ 12 = 2(0.3)2 − (0.4)2 = 14.14%

(8-41)

Now that we have both the first and the second months’ discrete volatilities, we can calculate the third month’s discrete volatility:

σ 3 = 3(σ 3CAPLET )2 − σ 22 − σ 12 = 3(0.25)2 − (0.1414)2 − (0.4)2 = 8.66%

(8-42)

Thus, the discrete volatilities that correspond to the caplet volatilities of 40%, 30%, and 25%, are given by 40%, 14.14%, and 8.66%. These discrete volatilities would then be incorporated in the pricing and hedging of options, as is discussed in Chapter 9 on option pricing. Now let us take a look at a slightly different market scenario, which is actually not that different in the caplet volatility values but is very different in the discrete volatility results we obtain. Suppose that the one-month forward price market is just a little more eventful, resulting in the one-month caplet having a volatility of 45% rather than 40%. We now have the one-month discrete volatility given by

σ 1 = σ 1CAPLET = 45%

(8-43)

but now the second month discrete volatility is given by

σ 2 = 2(σ 2CAPLET )2 − σ 12 = 2(0.3)2 − (0.45)2 = −0.0225

(8-44)

an imaginary number! You might ask, how can this be? The problem is that while the volatility of the one-month forward price is currently at 45%, it is not going to remain at that level. Instead, it might drop down to 40% in just a month. Thus, the volatility of that one-month forward price in a month will drop back to the noneventful levels, resulting in a two-month caplet volatility, which is priced assuming the drop from the 45% to the 40% volatility level. The above example forces us to treat discrete volatilities within a two-dimensional matrix rather than within a single-volatility term

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249

structure. This latter market scenario is very common in eventful energy markets, where the caplet volatility drop-off can be quite significant.

8.7.5.

Tying In Historical Volatilities

The historical volatilities represent the historical volatility term structure across the forward price points on the forward price curve, which the term structures of all the forward price points should to approach. In other words, if we built a discrete volatility matrix and let the number of rows of the matrix go to infinity—that is, if we looked at the discrete volatilities across the forward prices at some infinite time in the future—these discrete volatilities ought to converge to the historical average volatilities. For example, if the first month’s forward price currently has an implied volatility of 300%, but historically has an average volatility of 200%, then over time we should see the 300% volatility of the first month’s forward price drop down to 200%. This expectation of the current discrete volatilities approaching the historical volatilities over time should hold for all the forward prices. As we look across the rows of the volatility matrix, we should see the discrete volatilities across rows approach the historical volatility term structure.

8.7.6.

Tying In Caplet and Swaption Prices

We have already gone through an example of how a European option volatility would be translated into the discrete volatilities of the volatility matrix. Specifically, the caplet volatility with an expiration time of Nt would be a function of the discrete volatilities as follows: N

(σ caplet )2 N ∆t = ∑ σ n2, N − n+1∆t

(8-45)

n=1

where: n,N n 1 the discrete volatility corresponding to the n-th row and N n 1-th column of the volatility matrix Figure 8-17 shows the relationship between discrete volatility paths of several forward prices and their corresponding caplet volatilities.

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F I G U R E

8-17

From Discrete Volatilities to Caplet Volatility

In the case of swaptions, we would also need to have a correlation matrix available for relating the discrete volatilities to the swaption volatility. The process is a bit more complicated, but the end results retain consistency between the caplet (and therefore cap) and swaption prices and volatilities. Figure 8-18 shows the discrete volatility paths followed by the two forward prices for the case of a four-month

F I G U R E

8-18

Discrete Volatilities Defining the FourMonth Swaption into a Two-Month Swap

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251

swaption into a two-month swap with monthly tenor. Such a swap is a weighted average of forward prices, with the weights being a function of discounting factors and quantities to be delivered, and the swaption volatility comprising discrete volatilities and correlations across all the forward prices defining the swap price.

8.8.

IMPLEMENTING THE VOLATILITY MATRIX The volatility matrix methodology can be calculated in the following three steps:

• Step 1: Set 1,1 to the first-month caplet volatitity and set T,1

to the historical one-month forward price volatility (Figure 8-19). Fill in the rest of column 1 (Figure 8-20), using a sensible forecasting method.4 In such a way you will have defined the first column of discrete volatilities. • Step 2: Using the cross-diagonal relationship defined by Equation 8-45, calculate 1,2 from the caplet two-month volatility (Figure 8-21). Set T,2 to the historical volatility of

F I G U R E

8-19

Historical and Equilibrium Volatilities as End Conditions

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F I G U R E

8-20

Filling in First Time Bucket

the two-month forward price. Again, interpolate to obtain the second column of discrete volatilities. • Step 3: Repeat the second step for each column of discrete volatilities until all the columns are filled and the full discrete volatility matrix is defined (Figures 8-22 and 8-23).

F I G U R E

8-21

Applying Single Cross-Diagonal Relationship

Volatilities

253

F I G U R E

8-22

Applying Multiple Cross-Diagonal Relationship

F I G U R E

8-23

Abstract of Full Matrix Method Process

ENDNOTES 1. “The Future of Modeling,” Interview with Emanuel Derman, Risk, December 1997. 2. In my professional practice I have implemented this technique trademarked as the Univol methodology. 3. This process will require the use of a statistical search routine and possibly some assumptions about volatility term structure! 4. Possible techniques include ARIMA, VARIMA, ARCH, GARCH, or other volatility forecasting methodology based on historical data.


C H A P T E R

9

Overview of Option Pricing for Energies . . . the fundamental problem of options theory is the valuation of hybrid, nonlinear securities, and options theory is an ingenious but glorified method of interpolation. I don’t mean that as an insult.

Emanuel Derman1

9.1.

INTRODUCTION Options exist all around us, from the financial markets to everyday life. Any insurance you get, whether it is for your car, or health, or house, is a type of option. When you bought your first house, did you know that the mortgage you obtained contained an embedded prepayment option? So options need not scare the energy professional. The secret lies with understanding the option valuation and risk management issues. In money markets, the typical financial options have the advantage of generally being plain vanilla options, with well-understood pricing methodologies and modeling choices. Unfortunately, what is considered exotic in the money markets is usually considered plain vanilla in the energy markets. Issues of proper option pricing for energies do not stop with valuation concerns; the proper option underlying price model guides the hedging and portfolio analysis as well. Certain valuation models can be “fudged” to generate a pretty good option pricing methodology, but the same may not be true for hedging. For illiquid option markets, one must develop a good model to simulate the “option underlying” market. In the ideal case, the model captures actual market behavior. The next two chapters will focus on those aspects of option pricing that are relevant—or peculiar—to energy markets.2 This chapter will introduce the basic concepts and problems that every risk manager should understand. Chapters 10 and 11 will follow with the details, particularly 255


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with regard to the valuation of “exotic” options that traders routinely trade in the energy markets.

9.2.

BASIC CONCEPTS OF OPTION PRICING Every option is a right to do something. As a purchaser of an option I buy the right to do something at some future date based on the terms of the contract. As a holder of a call option I have the right to purchase an asset for some fixed price—determined at the time of the purchase— at some future period in time. The asset is referred to as the option underlying and the fixed price is referred to as the strike price. Similarly, as a holder of a put option, I have the right to sell the option underlying at the strike price. The holder of an option pays for the right the option gives. This payment is the option premium. The right does not have to be exercised. When the right is exercised, then the option is “exercised.” In a typical option contract, when the option is exercised, it also expires. When the right is not exercised, then the option expires upon a predefined date. What determines whether an option will be exercised or not is the relative value of the option underlying price—at the time of the option exercise—to the strike price. Whenever the option underlying price is greater than the option’s strike price, the option is referred to as being in-the-money. On the other hand, when the strike price is greater than the option underlying price, then the option is referred to as being outof-the-money. Finally, when the strike price is the same as the option underlying price, the option is at-the-money.

9.2.1.

Parity Value

The difference between the underlying price and the strike price is referred to as the option’s parity value. At option expiration or at the option exercise, the parity value represents the value of the option. For a call option, this is simply the difference between the option underlying price and the strike price—when the option is in-the-money—and it is zero otherwise (Figure 9-1):


F I G U R E

257

9-1

Call Parity Value

Call Parity Value = max(0,U t − K )

(9-1)

For a put option, this is the difference between the strike price and the option underlying price—when the option is in-the-money—and it is zero otherwise (Figure 9-2): Put Parity Value = max(0, K − U t )

F I G U R E

9-2

Put Parity Value

(9-2)

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Note that we can relate the put parity value to the call parity value as follows: Call Parity Value − Put Parity Value = Ut − K

(9-3)

In other words, given the same type of option and given the same strike price, if the call option is in-the-money, then the put option is outof-the-money, and vice versa. This relationship must always hold, and it is referred to as the put–call parity.

9.2.2.

Settlement

When exercised, an option can be settled either through the delivery of a commodity or the exchange of cash. The exercise of an option “for delivery” requires that the option underlying asset is delivered as defined by the option contract. In turn, the receiving party has to pay the strike price for the delivery. Specifically, a call option holder would pay the strike price to receive the option underlying asset. Similarly, a put option holder would deliver the option underlying asset to receive the strike price. A cash-settled or “financially settled” option, on the other hand, requires no delivery. Instead, while one party pays the strike price, the other party pays the value of the option underlying asset price at the time of option exercise. In this case, a call option holder would pay the strike price to receive the option underlying asset price at the time of the option exercise. Similarly, a put option holder would in theory pay the option underlying asset price at the time of option exercise and receive the strike price. In practice, only the parity value gets delivered. Because the option holder will exercise the option only when this is profitable, the option is never exercised unless it is in-the-money; that is, the option holder would exercise only when she can get positive value out of the exercise.

9.3.

TYPES OF OPTIONS There are European, American, Asian, and swing-type options in energy markets.3 In energy markets, Asian options represent the majority of contracts, in large part due to the market’s need to provide options on averages of prices.


259

9.3.1.

European Options

A European option allows for a single exercise date. On that date, the holder of the option can take advantage of the option right if this is profitable. For a European option the exercise date is also the expiration date of the option.

9.3.2.

American Options

An American-type option allows for more than one date as the possible exercise date. There are American options that allow for a single exercise any day prior to a contract-defined final option expiration date. There are also American options that allow for a single exercise during a particular day in the week or the month until the option expiration. If the option holder exercises her right prior to the final option expiration date, this is referred to as early exercise. This is possible only in the case of the American option. Typically, the American option also expires at the time of early exercise. A holder of an American option will compare the value she would obtain by exercising the option with the market value of the option if she were to hold on to it and not exercise. Hence, the option parity value is compared to the American option’s market value at every instant the holder is allowed to exercise the option. If the parity value is greater than the option market value, then the holder is better off exercising the American option. If this is the case, the holder benefits more by early exercising on that date than by holding on to the option in hopes of a better deal. This is because the American option price at that moment represents all the probabilities of getting a better deal in the future. By comparison, the European option holder will look only at the parity value of the option on the option expiration date to decide whether to exercise or not. If the parity value is positive, she will exercise.

9.3.3.

Asian Options: Options on an Average of Price

A special class of path-dependent European options is commonly seen in energy markets: options on an average of price. These options offer

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average instead of discrete price settlement, and they are also known as average-price or Asian options. There are two different kinds of average-price options: one is cash-settled with the expiration at the end of the averaging period, and the other is an option for delivery at some future time period. The cash-settled average-price options can be seen in WTI and natural gas OTC markets. At the option’s expiration, the option settlement price is calculated as an average over some time period as defined by the option’s contract. Typically, these options are traded as caps and floors. For example, a one-year cash-settled cap on an average of WTI prices with a quarterly tenor would consist of four caplet options, the first with expiration in three months, the second in six months, the third in nine months, and the fourth in one year. Each of these options would settle on the average WTI price calculated over the three months prior to option expiration. Another type of average-price options also very common in energy markets are for delivery of energy over some time period. These types of options expire prior to such delivery. For example, a one-year call option for a one-month delivery of natural gas following the option expiration would settle on the one-month average forward price at the point of expiration. This is still an average-price option, although the average remains a forward price average; that is, its price has not yet settled.

9.3.4.

Swing Options

Swing options can be found in energy contracts that allow the energy quantities delivered or used to swing. There are various types of swing options in the marketplace. However, two distinct groups of swing options exist based on the types of counterparties in question: swing options that are demand-driven, and swing options that are price-driven. The price-driven swing options can be found when the counterparties can both buy and sell the energy in the marketplace. The fact that both sides of the contract can deal energy allows the option holder side to maximize the swing contract value. By contrast, demand swing options tend to be found in contracts where one of the parties can only take or withhold from taking delivery of the energy commodity; hence that party is only set up to purchase the commodity but not to deliver the commodity. The demand-driven swing options are primarily found


261

in the contracts between the dealers and the retail sector of the marketplace. However, the industrial users may also enter into these kinds of demand-driven contracts. The price-driven swing contracts can take on various shapes and forms. The basic swing contract allows the base load of energy delivered to swing a certain amount, with daily and monthly maximum and minimum quantity amounts defined. In addition, the swing option holder may be limited as to how many times the quantity is allowed to swing from the base load. Another spin on this is the forward-strike swing option, where the strike of the option is set at some future date rather than today. Multiple-peaking options, which allow the option holder to purchase the same quantity of energy but only for a fixed number of days over some time period (for example, the option can be exercised five times during the summer for next-day energy delivery) are actually a special subset of swing options, with the base load set to zero. The price-driven swing options can be priced using trees and assuming that there is no arbitrage; that is, the option holder will indeed maximize the option value. By comparison, the demand-driven swing contracts have to incorporate the functional relationship between the prices and the quantity demanded, as there is no means for one of the sides of the contract to deal in the marketplace. (A good example here is the contract between the natural gas providers and residential homes: the homes are set up to take delivery of natural gas, but cannot turn around and participate in the marketplace.) These demand-driven swing options are sometimes allowed to swing without bounds (in theory), and sometimes have the minimum and maximum quantities to be delivered defined. The valuation of these swing options requires knowledge of pricing methodologies as well as some amount of creativity.

9.4.

EFFECT OF UNDERLYING BEHAVIOR We have already introduced the concept of option parity value at the time of option expiration. In order to price an option we need to also know the price behavior and the characteristics of the price distribution (expressed in risk-adjusted terms) to determine the option price. Figure 9-3 shows the price distribution overlaid across the call parity value with a strike of K. In the case of a European option, the option

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F I G U R E

9-3

Call Parity Value and Price Probability Distribution

price is given by the integral of the product of the probability distribution at each value of the underlying price, times the option parity value at that underlying price. The wider the probability distribution, the greater is the probability of having the option expire in-the-money. Similarly, the fatter the option distribution tails—that is, the greater the kurtosis—the greater is the probability of having the option expire far in-the-money. As you can see, the probability distribution, and thus the assumptions about the option settlement price behavior, directly affect option valuation. In order to come up with option pricing models, we need to start by understanding the behavior of the option underlying price, which will ultimately determine whether the option will expire in-the-money and by how much, as well as the underlying market price behavior, which drives the option underlying price behavior. The two can be the same, but in the energy commodities they are usually not, as the most common energy option is on an average of prices. If the option underlying is the spot price, then the underlying market behavior is defined by the behavior of the spot price. In this case, the option underlying price is the same as the underlying market behavior. (This makes the option valuation process quite a bit easier.) However, if the option underlying was some predefined average of spot prices—such as, for example, an average of spot prices over a period of three months—then the option underlying price would be the average price, and the spot price would be the underlying market price driving the behavior of the option underlying average price. In this case, users


263

of the option valuation process need to know how the spot prices behave so that they may know how the average of the spot prices behaves, in order to ultimately value the option. This is a more difficult case of option valuation. In the case where the option underlying is a function of—rather than being equal to—the underlying market price, we have the choice of developing option valuation based on the behavior of the underlying market price or the behavior of the option underlying price. The second is usually the easier, because it is generally easier to model a function of a market price rather than modeling a function of a function. However, if we do decide to go the generally easier way of modeling the option underlying, we still have to ensure that the behavior of the option underlying price remains consistent with the behavior of the market underlying price. Unfortunately, imposing this consistency can be as difficult as developing option valuation based on market underlying price behavior.

9.5.

OPTION PRICING IMPLEMENTATION TECHNIQUES Once the actual option underlying process is well defined, either directly or as a function of the underlying market price process, the valuation expert has to decide on how to implement this process to ultimately arrive at a valuation methodology for the option price, hedge, and risk calculations. There are various choices available to use in the implementation. We will cover some of them here. Table 9-1 summarizes the various implementation methodologies across capturing market characteristics and option settlement characteristics.

9.5.1.

Closed-Form Solutions

A closed-form solution is the solution to a differential equation that expresses the change in option value relative to all the key variables, subject to hedging assumptions and end conditions. Closed-form solutions for option pricing are the ideal in that they provide us with a single equation to use in the pricing and risk calculations of options. The Black–Scholes equation is an example of a closed-form solution.

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T A B L E

9-1

Comparison of Option Pricing Techniques Issue

Closed-Form Solutions

Simulations

Trees

Difficulty of developing American optionality Path-dependent option Multifactor approach Discrete volatility term structure

high no probably not probably not probably not

low no yes yes yes

medium yes no practical up to two factors yes

The closed-form solution is the solution to the differential equation that follows the arbitrage-free argument of a risk-free portfolio earning the risk-free rate of return. Hence, we solve for the closed-form solutions to the option price, just as we solved for the forward price closed form solution in Chapter 6: by finding the solution to the differential equation. A closed-form solution is quick and easy to use, and provides a great amount of implementation flexibility. Unfortunately, the more complicated the underlying market process and the more complicated the type of option, the more difficult— if not impossible—it becomes to arrive at the closed-form solutions to option prices. Hence, solving for closed-form solutions often remains the ultimate and yet also the unattainable option valuation technique. Another choice we have in obtaining the closed-form solution is by taking the expected value of the parity value at the expiration time (Figure 9-3), given the option underlying price distribution, and present valuing this quantity:

((

) (U

Ot ,T = Et ⎡⎢ max −1 ⎣ where:

n

T

)

)

− K , 0 ⎤⎥ e− r ( T −t ) ⎦

(9-4)

n 1 in the case that the option is a call option n 1 in the case that it is a put option UT the option underlying asset price at the time of option expiration T K the option strike price r the risk-free rate


265

The above is a valid valuation process only if two conditions hold: •First, the option must be a European option, as we are assuming that the exercise can only occur at the expiration time T. •Second, the expectation value is taken on a risk-adjusted price distribution. This second condition guarantees that the resulting option valuation will remain arbitrage-free. If we were to solve for the simple European option using Equation 9-4, assuming that the underlying price is lognormal, we would derive the exact same Black–Scholes option price equation as if we were to do it by solving the arbitrage-free differential equation (see Equation 6-13).

9.5.2.

Simulations

Monte Carlo simulations can also be used in option price valuation. This technique simulates either the underlying market prices or the option underlying prices at the time of option expiration. The simulated prices at the option expiration are then used to calculate the expected option parity value at expiration, and then this value is discounted back to obtain the present value of the option. This can be an excellent pricing technique, as all the complexities of multivariable markets can be factored in. With simulations we are not limited to the number of market factors we want to incorporate into describing the price behavior. However, simulations have two drawbacks. The simulations capture the probability distribution through the sheer number of simulation points. The greater the number of simulations, the more precisely do the simulations converge to the underlying price probability distribution. Unless you have a powerful enough computer, do not bother using simulations on a trading desk, where the deals often have to be priced very quickly, or for large portfolio valuation and risk calculations, where the simulations would take a very long time to run through all the deals on the books. This is all the more true when performing mark-to-market calculations. The second drawback to simulations is that they cannot be used for valuing American options. The problem with applying simulations to value American options has to do with the simple question of early

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exercise. A holder of an American option will compare the value he would obtain by exercising the option versus the market value of the option at the time. Hence, the option’s parity value is compared to the option’s market value at every instant the holder is allowed to exercise the option. If we are running the simulations in order to calculate the American option value, we cannot do this parity versus option price comparison within the simulations, given that for each simulated option underlying price we must already have the American option price so that we can decide on early exercise. Although the simulations are no good in the case of American option pricing and in general for day-to-day trading needs, they provide an excellent testing ground for the valuation models considered as candidates for implementation on the trading floor.

9.5.3.

Trees

Another implementation methodology is the building of underlying price trees. Like the simulations, the trees also reflect the option underlying price distribution. Unlike the simulations, which converge toward the underlying market distribution through the simulations of a great many prices, the trees have a calculated probability associated with each node in the tree; hence the trees are more precise than the simulations. In addition, unlike the simulations, the trees do provide a means of pricing American-style options. Because the trees provide us with a tree of price levels with associated probabilities, we can move back and forth in time through the tree, allowing us to calculate the American option prices as needed. We compare the American option price at each node of the tree to the parity value at that node in the tree. Although the trees allow us to price American-style options, something that simulations cannot do, they typically fail in pricing average price path-dependent options, which is something that the simulations usually can manage. A path-dependent option is one that depends on the underlying prices at different points in time during the life of the option. In a tree, this translates to relating nodes at one time step to nodes at another time step. Although this is possible to do, there are so many combinations that one would have to consider that the


267

practicality of the tree approach quickly diminishes as the number of time steps necessary to be combined at the same time increases. This path dependency is something that the simulations are capable of handling through the simulation of underlying paths.

9.5.4.

Human Error in Implementation

As we can see from the above discussion, the option valuation implementation may be limited by the type of the option, by the manner in which the valuation is intended to be used—that is, on the trading floor or for research purposes—and by the complexity of the problem. In addition to these valuation issues, there are also user issues. As much as sophisticated models can add precision to the option price and risk calculations, so can their use be prone to human error in cases where the traders do not understand how to use them. And it is very important for traders to understand how to use the given option pricing models. For example, a trader who does not understand mean reversion in prices should not be using a mean-reverting option pricing model without a proper support network. Because traders have to be able to convert their views on the existing market situation into the specifics of how they affect the option pricing, they need to know exactly how the market state translates into model parameters. This is all the more true in illiquid markets, where there is not enough of an options market to give the traders a solid understanding of how everyone else perceives the state of the market.

9.6.

CHOOSING THE RIGHT OPTION PRICING MODEL Choosing the right option pricing model should be treated in the same way as any business decision-making process. The valuation is an important part of the trading business, particularly under certain trading strategies. In as much as it is a behind-the-scenes aspect of the trading operation, it is still key to a successful trading operation, particularly under the arbitrage-seeking and risk management service provider trading strategies (discussed in detail in Chapter 14).

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9.6.1.

Three Criteria for Evaluating Option Models

In order to make an intelligent decision about which models should be ultimately implemented on the trading desk, each option model needs to be evaluated across three areas:

• the ability to capture market reality • ease of implementation • ease of maintenance Ease of implementation and the ease of maintenance are important to the degree that they match the kind of trader, valuation, and software expertise and support you have or intend to have in your trading group. There is a give-and-take between pricing and hedging precision and the cost necessary to obtain and maintain these at a high-quality level. A huge amount of frustration at all company levels can be avoided by recognizing the reality of the support costs for entering into the more complex option products or option markets up front, and providing such support as necessary. On the other hand, by opting for decreased support, and hence also for low cost of support, the company has to accept the cost of low valuation and hedging precision, which may rear its ugly head in the form of real costs of losses resulting from the inability to properly value the more sophisticated options or even plain vanilla options in more complex markets. In order to evaluate a model for its ability to capture market reality, the particular characteristics of the market reality important to option valuation need to be defined. We have dealt with the building of forward price curves and volatility matrices in the previous chapters. Both of these are inputs to option pricing models. All options have first-order sensitivity to the forward price curves. They also have firstorder sensitivity to volatility matrices. But how about correlations between the forward price points? Are these important to the types of option pricing that your trading operation needs to do? What about the capturing of the tails of the particular market price distribution? How important is this characteristic to the types of options you already have or want to have in your trading books? These are just some sample questions you might want to ask in the process of defining which market characteristics are important to your particular options market.


269

Given a traded options market with at least some option modeling choices, the preceding issues are the ones you need to go through in order to ultimately choose which models best fit your trading needs. But what do you do in the case when the option markets are so illiquid that there is not enough market information to tell you whether the models you are using are precise enough? Unfortunately, in such a case—which, by the way, is more of a norm in the energy options markets—you are forced to use what you have. Spot and forward price historical data may be your only clue for defining volatility matrices. In this case, you need a valuation expert who is as comfortable with data crunching and parameter modeling as with the mathematics of option derivation and implementation techniques.

9.6.2.

Investing in Pricing Model versus Implementation

Given that you probably have a limited budget for option valuation, how much should you invest in the modeling part of the option valuation process and how much in implementation? The answer will generally follow your model choice. The less your option pricing model incorporates the market characteristics, the more you will have to use the implementation of the model to capture the characteristics that the model is missing. For example, a two-factor option pricing model might need only a single volatility curve to define the volatility of the spot price over time, and the model would imply the rest of the volatility matrix, while a simple lognormal model would need the full volatility matrix to define the volatility term structures for each forward price on the forward price curve. In other words, the two-factor model might imply the whole volatility matrix with many fewer parameters than the lognormal model would require. In some cases, you do not have much of a choice in the matter. If no models are indeed available for the type of option you need to price, you might end up using the existing option valuation techniques but with implementation adjustments to take care of the biases that might result through this process. When this is the case, you should perform thorough testing of the approximating model in its application on the more complicated option, to ensure that you know all the drawbacks and biases that might result.

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9.6.3.

A Model Is Only as Good as Its Implementation

Although the more sophisticated models might end up explaining a good amount of market behavior that a simpler model might not, it still requires the proper implementation and support for it to give that added value of providing a more precise valuation and hedging means. Having a valuation group that is heavily concentrated on the model mathematics instead of model implementation often results in sophisticated models that are poorly implemented. In fact, running the option pricing models through benchmarking tests will show you that a simple well-implemented model can be more valuable than a sophisticated but poorly implemented model. By ensuring that a balanced amount of attention is paid to model derivation and model implementation, you can avoid the problem of ending up with an option pricing software that contains bugs. To make things worse, if the traders get used to these bugs, they will perceive an option pricing model that prices correctly as the one that is flawed.

9.7.

OPTION VALUATION PROCESS: WHAT SHOULD IT BE? The option valuation process should follow three steps:

• Define the option underlying market price behavior. Create benchmarks from this behavior for testing alternative models. Make sure forward price and volatility inputs are valid. •Test alternative models against benchmarks. •Select the most appropriate model.

9.7.1.

Defining Underlying Market Price Behavior

The first step to be taken in the option valuation process is to define the underlying market price processes. All option valuation methodologies should be tested against the spot and forward price models that best define the market reality. We can then proceed with defining what types of options need to be valued and how this should be done given the underlying price behavior.


271

The option models ultimately chosen for implementation will need forward price curves and volatility term structures as inputs, if not also additional options-specific parameters. The question of where all these model parameter values come from needs to be answered prior to implementation. The answer should be part of the understood policies and procedures. The parameters should also be realistic. Do not expect to be reestimating some of the parameters on a weekly basis if your valuation group does not have enough people power to be doing so. Whenever possible, give the traders a feeling for parameter stability and confidence levels. This is particularly important in an illiquid market. Unfortunately, in an illiquid market the data may not be readily available nor extensive. In this case, you have no choice but to do the best you can with what you have and try to draw on the information from similar energy markets that are more liquid.

9.7.2.

Testing Alternative Models

Whatever option pricing models are ultimately chosen for implementation, these models ought to be thoroughly tested using simulations and ensuring that the characteristics of the underlying market prices are captured. This test is performed through model benchmarking. If any of the characteristics are not captured, the traders need to be aware of this so that they can ensure a more conservative approach to pricing whenever the missing characteristics do affect the option valuation. Specifically, the models being tested should be ranked by how closely they reflect market reality. Similarly, these same models ought to be ranked by the amount of implementation support necessary and the amount of maintenance support necessary. No option model will be able to capture all the market characteristics. Given the model limitations, identify which parameters can be “fudged” to obtain more realistic market prices under certain market scenarios.4 While this is in general not a good practice to follow, and can be quite dangerous in that it opens the trading business up to fraudulent behavior, chances are that in a developing market all participants will be playing a catch-up game with respect to their valuation and risk management software. However, if managed well, the parameter “fudging” can be a very educational experience for all involved—traders, valuation and risk management experts, and management. However,

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just as risk limits are set for traders to follow, so should parameter limits be set in the cases where there is a good amount of ambiguity regarding the parameter values. A parameter that is going over a limit should trigger a valuation and risk management group discussion and consensus on what needs to be done. We can learn a lesson here from the mortgage markets. Most mortgage pricing models assumed that the prepayment rate was constant, and most users of such models assumed that there will be no variation in the prepayment rate. Everyone was caught by surprise when indeed the prepayment rate increased drastically in the early 1990s, causing huge changes in portfolio values, particularly in the IO/PO books. The prepayment rate was a parameter that should have been tracked for variation, and risk limits should have been placed on this risk, even though the models assumed it to be constant over time. The value of model testing prior to implementation cannot be stressed enough. Ideally, the option model results are practical to use, intuitive to trade on, and stable enough that maintenance does not require a great amount of support. Unfortunately, the energy markets are illiquid and complicated enough that they make the job of arriving at such model results sometimes quite difficult. Recognizing this fact should not discourage you. Quite the opposite, as in any profession, it is the layman who always thinks that the job should be easy to do. Or to put it another way, the more you know the more you become aware of how much more there is to know. So, do not become discouraged. Remember that everyone else is dealing with the same set of issues— whether they are aware of it or not. The ones who are aware have a distinct advantage, as they are more likely to find the solutions that work. After all, how can you come up with a solution to the problem without realizing that there is a problem?

9.7.3.

Selecting the Most Appropriate Option Model

If Steps 1 and 2 are done properly, the managers, traders, and quantitative analysts should have meaningful information with which to evaluate, discuss, and ultimately select the most appropriate model or sets of models.


9.8.

273

DID THAT OPTION MAKE MONEY? One topic worth discussion is, “How do you decide whether an option made money for you or not?” Different option trading strategies will require different types of profit-and-loss analysis. If you bought an option as a treasury or hedging function, in order to minimize risk, you did not enter this contract in order to make money; you entered it in order to reduce the risks of your corporation. In this case, the value that the option brought to you has to be measured in risk terms: How much less was I exposed to the market price swings because of the option I bought as an insurance policy? If you bought an option in the hopes of profit, you would value the deal differently than the hedging strategy. You probably perceived the option to be undervalued. You wanted to capture the spread between the market price and what you perceived its value to be. Ultimate profit or loss analysis is done in present-value terms and includes the hedging you have entered into to offset option risks along the way. Specifically, there are two ways that this spread can be captured. One is a short-term strategy where you buy the option and sell it in the market the moment the market makes the correction. The other is delta hedging—you buy the option and you rehedge the delta continuously. In the process, your hedges capture the true volatility of the option underlying price, providing you—on the average—with a capture of the spread between the market volatility you bought the option at and the true volatility captured through your delta hedges. In this second strategy, not including the hedges in the analysis of the value the option brought to you would leave you with an unrealistic picture of what happened.

ENDNOTES 1. “The Future of Modeling,” interview with Emanuel Derman, Risk, December 1997. 2. My favorite book on options is Option Pricing by Jarrow and Rudd from the Irwin Series in Finance, published by Richard D. Irwin, Inc., in 1983. Unfortunately, this text is out of print. Other favorites include Hull, Options, Futures and Other Derivative Securities, and Cox and Rubenstein, Options Markets. 3. These terms do not really reflect any geographical meaning. “European” and “American” have simply become part of the derivatives lexicon. Once average price

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contracts became popular, traders continued this “continental drift” by referring to options on average prices as “Asian” options. 4. For example, when pricing an averaging (or Asian) option, one could use Black–Scholes by manipulating the volatility term structure and even making additional corrections for skew and kurtosis effects.

C H A P T E R

10

Option Valuation In the real world of traded securities, few of the assumptions of Black, Scholes and Merton are strictly respected. But their view of hybrid nature of a stock option as a probability-weighted mixture of stock and bond captures a core of truth that provides the foundation for the model’s robustness.

Emanuel Derman1

10.1.

INTRODUCTION There is nothing magical about the valuation of options, just a lot of hard work. The task is to pull together all our knowledge about the markets. In Chapter 9 we introduced the basics of option pricing, and in Chapter 8 we introduced you to the basic (and not so basic) volatility calculations and issues. In this chapter, we will draw on many of the concepts and groundwork of those two previous chapters. In Chapter 6, we described an example of arbitrage-free pricing framework for forward prices. We can follow a similar approach for pricing options, with the results being closed-form solutions to option prices. Under this arbitrage-free framework, we follow the cash flow at contract origination and price settlement. At contract origination we pay the option price, whereas we simply agree to the forward contract without any cash exchange. With options we have the choice of exercising or not exercising at expiration time, while with forward prices we have no such choice: we have to exchange cash for delivery. Compared to the forward price solution, the difference in the cash flow at option contract origination causes the differential equation for the option price to include the cost-of-financing term on the option price. The difference in contract settlement causes the boundary condition for the option price at time of expiration to be very different from that of the forward price. In this chapter, we will introduce you to the choices you have in option implementation and we will take you through the valuation issues and modeling processes of some common energy options. The 275


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options market is ever changing, with new contract types being invented every day. Thus, this chapter will detail select contracts as case studies to demonstrate the thinking and the process behind their valuation.

10.2.

OPTION MODEL IMPLEMENTATION Model implementation should not be confused with option model derivation, although with some implementation techniques it is hard to separate the two. Model derivation—in its basic form—is the derivation of the differential equation for the option price. How we get from this differential equation to the option price is what I refer to as the model implementation. There are a number of implementation techniques. We will concentrate on the most common ones: the closed-form solutions (as exemplified by the famous Black–Scholes and Black equations), approximations to the closed-form solutions, and the tree-building methodologies.

10.3.

CLOSED-FORM SOLUTIONS The closed-form implementation methodology involves the solving of the differential equation for the option price to obtain an equation that defines the option price as a function of the market variables and modeling parameters, which played a role in the definition of the option price differential equation. This is a math-intensive procedure, particularly if the differential equation contains more than one market variable.

10.3.1.

Pros

Closed-form solutions for option prices are the ideal implementation methodology. The beauty of closed-form solutions for option prices is that they provide us with a simple equation, which can be easily programmed and implemented on the trading floor. Such equations are easy to use and quick to give us the option value, as well as the risk calculations, when we need them.

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277

10.3.2.

Cons

Unfortunately, the closed-form solutions are typically extremely hard to arrive at. The more complicated the marketplace is, the more complicated is the differential equation for the option price. The more complicated the terms of settlement of the option, the more difficult it becomes to satisfy the boundary condition of the option in solving the differential equation. In the end, in order to arrive at closed-form solutions, we usually need to make many simplifying assumptions about both the market variables and the option settlement character. The end result of these simplifications is that, while providing us with a practical and easy-to-use option pricing methodology, the closed-form solution may not reflect the reality of the market behavior. Examples of such simplification include assuming that the volatilities are constant over time when they are not, assuming that the underlying market price is lognormal when it is mean reverting, and treating the option settlement price as a discrete price when it is actually an average of discrete prices. It is such simplifications that force us to calculate corrections to the closed-form option price implementation. The later section on closed-form solutions with corrections will take you through some sample ways of dealing with such simplifications. Two famous closed-form option pricing models are the Black–Scholes model and the Black model. Both assume that the option settlement prices are lognormal and have constant volatilities. Next, we will take you through the derivations of these models.

10.3.3.

The Black–Scholes Model

The Black–Scholes closed-form solution for option prices is probably the most famous option pricing methodology out there. It is so easy to use that it can be implemented on the trading floor by the traders themselves. The Black–Scholes option valuation assumes that the option settlement price is the spot price at the time of option expiration. It also assumes that the spot prices follow a simple lognormal process with a drift term of and a spot price volatility of : dS = µ Sdt + σ Sdz

(10-1)

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Energy Risk

In this simple world, we make the assumption that an option position can be perfectly hedged with the spot price and that we can use a bank’s services to borrow and lend money at a risk-free rate. This leads us to derive the differential equation for the option price: ∂C 1 ∂ 2C 2 2 ∂C + − rC = 0 σ S + rS 2 ∂S ∂t 2 ∂S

(10-2)

where: C call option price S spot price K strike price r discount (risk-free) rate

spot price volatility Solving this differential equation and imposing the boundary constraint that the option price must equal the option parity value at expiration, we obtain (after quite a bit of mathematics) the closed-form solution for the option price: C BS = Sℵ( d1 ) − Ke− r ( T −t )ℵ( d2 )

d1 =

d2 =

⎛ σ2⎞ ln( S / K ) + ⎜ r + ⎟ (T − t ) 2⎠ ⎝

(10-3)

(10-4)

σ T −t ⎛ σ2⎞ ln( S / K ) + ⎜ r − ⎟ (T − t ) 2⎠ ⎝

σ T −t d2 = d1 − σ T − t

ℵ( x ) = ∫

x

e

−∞

where: T time of option expiration t time of option valuation

−

(10-5)

(10-6)

y2 2

2π

dy

(10-7)

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279

This is the famous Black–Scholes option solution. It is a function of the current spot price, the spot price volatility, the risk-free rate, option’s strike price, and the time to option’s expiration.

10.3.4.

The Black Model

If, instead, the option settles not on the spot price at the time of the option’s expiration, but rather on a forward price, we end up using the forward price to hedge the option and not the spot price. As already shown in Chapter 6, the forward price on a lognormal spot price, as defined above, is given by F = Ser ( T −t )

(10-8)

and the change in the forward price over time dt is then given by dF = ( µ − r ) F dt

(10-9)

As the forward price contract is an agreement that carries no cost of financing, our hedge to the option price requires no borrowing of money from the bank. This changes our option differential equation to look as follows: ∂C 1 ∂ 2C 2 2 + σ F − rC = 0 ∂t 2 ∂F 2

(10-10)

Solving this differential equation for the option price results in the closed-form solution in terms of the forward price rather than the spot price. This is the (also famous) Black option pricing model: C B = Fe− r ( T −t )ℵ( d1 ) − Ke− r ( T −t )ℵ( d2 )

d1 =

d2 =

⎛σ2⎞ ln( F / K ) + ⎜ ⎟ (T − t ) ⎝ 2⎠

σ T −t ⎛ σ2⎞ ln( F / K ) + ⎜ − ⎟ (T − t ) ⎝ 2⎠

σ T −t

(10-11)

(10-12)

(10-13)

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d2 = d1 − σ T − t ℵ( x ) = ∫

x −∞

e

−

(10-14)

y2 2

2π

dy

(10-15)

where: T time of option expiration and forward price expiration t time of option valuation Using probability distributions in a risk-adjusted world leads us to exactly the same option models as derived above and may in the process be a bit more intuitive. In this case, we value the European option as the sum of risk-adjusted values of possible outcomes multiplied by the probability of each possible outcome: Ct E[max (0, F˜T,T K)]dft,T. For a call option, at expiration, the outcome is either zero (the option expires at/out-of-money), or it is the positive difference between the settled/delivered forward price/value less the call’s strike price (known as the “parity value”). The graph of call’s parity value is shown in Figure 10-1. We weigh these outcomes with the probability density

F I G U R E

10-1

Call Option Parity Value Call Parity Value K=$30 55

Call Parity Value

45 35 25 15 5 –5

0

8

15 23 30 38 45 53 60 68 Forward Price at Option Expiration Parity Value

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F I G U R E

10-2

Sample Probability Density

0.006

50

0.005

40

0.004

30

0.003 20

0.002

10

0.001 0

Call Parity Value

Probability Density Function

Probabilities for Two Options With Different Vols F 1 = F 2 = $25, vol 1=100%, vol 2=50%

0 0

8 15 23 30 38 45 53 60 68

Forward Price at Option Expiration Prob 1

Prob 2

Parity Value

function (an example shown in Figure 10-2) in order to arrive at the option’s price. When the probability density is assumed to be for a lognormal price process (the forward prices are therefore assumed lognormal), the Black call option model can be derived. The probability density function is rich with market price information. In fact, it characterizes what we expect to see in the price behavior. We can control the resulting price distribution at two levels: through the parameter inputs used to define the probabilities of outcome, and within the valuation framework or model. For example, we might use Black to value options on foreign exchange futures, but we would also want to imply the volatility strike structure as part of our valuation framework. In other words, although we might feel that Black does not capture all the complexities of the marketplace, we deal with these complexities by controlling the parameters we feed into the model. Figures 10-3 and 10-4 show two different tail behaviors. Figure 10-3 represents the typical lognormal distribution with a typical tail behavior expectation. By comparison, Figure 10-4 shows a fat tail, symbolizing a market expectation that if prices go high they will cluster around

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F I G U R E

10-3

Comparison of Distribution Tails Probabilities for Two Options With Different Vols 0.006

50

0.005

40

0.004

30

0.003 20

0.002

10

0.001 0

Call Parity Value


F 1 = F2 = $25, vol 1 = 100%, vol 2 = 50%

0 0

8 15 23 30 38 45 53 60 68 Forward Price at Option Expiration Prob 1

F I G U R E

Prob 2

Parity Value

10-4

Distribution Tails, Continued

0.006

35

0.005

30 25

0.004

20

0.003

15

0.002

10

0.001

5

0

0 0

8 15 23 30 38 45 53 60 68 Forward Price at Option Expiration Prob 1

Prob 2

Parity Value

Call Parity Value


Tail Effects (Event Expectations) on Far Out-Of-Money Options F1=F2

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283

some high point. If we were to imply Black-equivalent volatility from options priced using the fat tail distribution, we would find that these options exhibit a strike structure where the out-of-money calls show higher volatility than at-the-money calls. Similarly, we can reverse the process, and directly read what market is telling us about price distributions given volatility strike structures.

10.4.

APPROXIMATIONS TO CLOSED-FORM SOLUTIONS Making simplifying assumptions in the derivation of closed-form solutions, such as that the option settlement prices are lognormal with a constant volatility—that is, a flat volatility term structure—and that the settlement price is defined by a single-factor model, when in reality this may not be the case, leads us to come up with approximation and/or correction techniques that allow us to continue using the closedform solutions, such as Black–Scholes or Black. The emphasis of option valuation thus moves from the actual derivation of closed-form solutions in complex markets, to how we implement the simplistic closed-form solution in complex market environments.2

10.4.1.

Pros

This approximation technique may be as simple as adjusting the volatility inputs fed into the closed-form solution, to properly reflect the way markets act. Or they may be as complicated as calculating the higher-order correction terms to the closed-form solution to capture the skew or kurtosis effects that the closed-form solution does not capture. Either way, we still end up with an equation for calculating the option prices, and as such it remains relatively easy to program and use on a trading desk. Such corrections to Black–Scholes or Black option pricing equations allow us to price—fairly easily—all kinds of European-style options, including Asian options on averages of prices, whose settlement prices may be path-dependent.

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10.4.2.

Cons

The potential problem of making adjustments to closed-form solutions is that we have to know when it is OK to do so, and when the corrections simply do not capture all there is to capture. In other words, if the BandAid formed by the corrections covers the wound—that is, takes care of the simplifications made by the closed-form solutions—then we are fine. But if the wound is too big for the Band-Aid and cannot be covered fully, then this methodology is no longer appropriate. Thus, such methodology always needs to be used with caution and with an understanding (by the traders) of its boundaries. The model corrections attempt to allow for correct pricing. However, even if we achieve this, we are still left with potentially incorrect risk calculations. This is probably the greatest drawback of this methodology. The tree-building methodology (see Section 9.5) offers a way to get this right as well.

10.4.3.

The Volatility Smile

The implied volatility term structure and volatility smiles, discussed in great detail in Chapter 6, are the results of using a model that does not capture the full complexity of the marketplace. In other words, these option-implied volatilities are adjusted to capture the true nature of market behavior. In an ideal world, we can incorporate all the market drivers—such as stochastic volatilities, two-factor models for spot and equilibrium price, and stochastic seasonality factors. However, the reality of option implementation, particularly in the case of closed-form solutions, forces us to make simplifying assumptions, such as that the volatilities are constant and prices are lognormal. Although most people in the marketplace have probably come to accept market-implied volatility behavior as something independent of other market variables, the truth is that market-implied volatilities are just as much a function of market behavior as of the models you are using to back out these implied volatilities. Hence, if your models do not capture a certain aspect of market behavior, such as fat distribution tails of the spot prices, then your implied volatilities are bound to show a volatility smile or frown across the option strikes.

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10.4.4.

The Edgeworth Series Expansion

The Edgeworth series expansion offers a very useful technique for estimating the actual price distribution with an approximating price distribution. This technique can be applied to option pricing, resulting in methodologies that can be used to capture the actual market price behavior while still using the assumption of lognormal price behavior with additional price corrections. We begin with our old friend, the Taylor series expansion, only in this case it is applied to the distribution probability function. The result is the Edgeworth series expansion: c2 ∂ 2 a( P ) c3 ∂3a( P ) c4 ∂ 4 a( P ) +ε − + 2 ! ∂P 2 3! ∂P3 4 ! ∂P 4

(10-16)

c2 = STD actual − STD approx

(10-17)

c3 = SKEWactual − SKEWapprox

(10-18)

c4 = KURTOSISactual − KURTOSISapprox + 3 c22

(10-19)

f ( P ) = a( P ) +

where:

P option settlement price f (P) actual probability function of the settlement price a(P) approximating probability function of the settlement price higher order corrections to the approximating distribution STDactual, SKEWactual, KURTOSISactual the standard deviation, skew, and kurtosis of the actual settlement price distribution STDapprox, SKEWapprox, KURTOSISapprox the standard deviation, skew, and kurtosis of the approximating settlement price distribution

Equation 10-16 gives us an expression for the actual probability distribution function, f, in terms of the approximating probability function, a. Note that the correction terms are functions of the actual versus the approximating distributions’ standard deviation, skew, and

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kurtosis. If we make the further assumption that the approximating distribution is lognormal, we can apply the Edgeworth series expansion to option pricing in the case of a European call option, to obtain the following: C=

{∫

}

+∞ K

( P − K ) f ( P ) dP / fvd

⎫ ⎧ +∞( P − K ) a( P ) dP + ⎪⎪ ⎪ ∫K C=⎨ ⎬ / fvd c4 d 3a( P ) c2 da( P ) c3 d 2 a( P ) ⎪ ⎪ | + | − 3! dP 2 P = K 4 ! dP3 P = K ⎭ ⎩ 2 ! dP where:

(10-20)

(10-21)

C price of the call option fvd future value of a dollar at time of option expiration

Finally, if we use the additional trick of requiring our approximating distribution to have the exact same standard deviation as the actual distribution, we have the following simplifications: c2 = M 2 − M 2 where:

approx

=0

(10-22)

M2 actual second moment of the option settlement price M2 approx approximating second moment of the option settlement price

Now the second correction term in the option equation goes to zero: c2 d 2 a( P ) →0 2 ! dP 2

(10-23)

and we have the second moment of the approximating distribution equal to the second moment of the actual distribution: M2 = M2

approx

(10-24)

In the case of lognormal distributions, we can calculate explicitly the moments of the distribution. They are given by M1 E[P]

(10-25)

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287

where the expectation is taken in risk-adjusted terms: M2

2 σ approx τ

approx

= M12 e

2 τ 3σ approx

M3

approx

= M13e

M4

approx

= M14 e

2 τ 6σ approx

(10-26) (10-27) (10-28)

From this we can define the approximating lognormal distribution’s volatility, which allows the second moment of the lognormal distribution to be exactly equal to the second moment of the actual distribution: 2 σ approx τ

M 2 = M12 e

(10-29)

where: approx approximating volatility time to option expiration This gives us (see Hull, Equation 10.9)

σ approx =

=

⎛M ⎞ ln ⎜ 22 ⎟ ⎝ M1 ⎠ (T − t )

(10-30)

⎛ ⎛ STD( P ) ⎞ 2 ⎞ ln ⎜ 1 + ⎜ ⎟ ⎜⎝ ⎝ M1 ⎟⎠ ⎟⎠ (T − t )

With the above equations holding, we are left only with the skew and kurtosis correction terms: c3 d 3a( P ) c4 d 4 a( P ) + 3! dP3 4 ! dP 4

(10-31)

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where c3 and c4 are now given by c3 = ( Eactual [ P3 ] − Eapprox [ P3 ])

(10-32)

c4 = ( Eactual [ P 4 ] − Eapprox [ P 4 ] − 4 ⋅ Eactual [ P] ⋅ c3 )

(10-33)

10.4.5.

Pulling It All Together

We are finally ready to apply the Edgeworth series expansion to the call and put pricing problem. The above requirement of the second moments of the actual and approximating distributions being equal results in the call and put option prices being given by: ⎫⎪ ⎧⎪ c d 2 a( P ) c4 d 3a( P ) Cactual = C BS ( P, σ approx ) + ⎨− 3 | + |P = K ⎬ / fvd (10-34) P=K 2 3 4! dP ⎭⎪ ⎩⎪ 3! dP where: Cactual the call option price CBS the Black–Scholes call option value calculated using the approximating volatility 2 ⎫⎪ c4 d 3a( P ) ⎪⎧ c d a( P ) Pactual = PBS ( P, σ approx ) + ⎨− 3 | + |P = K ⎬ / fvd (10-35) P=K 2 3 4! dP ⎭⎪ ⎩⎪ 3! dP

where: Pactual the put option price PBS the Black–Scholes put option value calculated using the approximating volatility The above also works for the case where the options are on forwards rather than spot prices: 2 ⎫⎪ c4 ∂3a( F ) ⎪⎧ c ∂ a( F ) Cactual = C B ( F , σ approx ) + ⎨− 3 | + |F = K ⎬ / fvd P=K 2 3 4 ! ∂F ⎭⎪ ⎩⎪ 3! ∂F

(10-36)

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289

2 ⎫⎪ c4 ∂3a( F ) ⎪⎧ c ∂ a( F ) Pactual = PB ( F , σ approx ) + ⎨− 3 | + |P = K ⎬ / fvd P=K 2 3 4 ! ∂F ⎭⎪ ⎩⎪ 3! ∂F

(10-37)

where: CB the Black call option value calculated using the approximating volatility PB the Black put option value calculated using the approximating volatility In summary, in order to use this methodology we need to calculate the following:

approx approximating volatility c3 third-order correction term, function of third moment, M3 c4 fourth-order correction term, function of fourth moment, M4

V3 =

d 2 a( P ) |F = K = third-order sensitivity of the approximating distribution dP 2

(10-38)

V4 =

d 3 a( P ) |F = K = fourth-order sensitivity off the approximating distribution dP3

(10-39)

In order to calculate the sensitivities of the approximating distributions, we need the probability function of the lognormal distribution. It is given as:

a( P ) =

2 ⎛ ⎛ ⎛ P ⎞ ⎞ ⎞ 2 ⎜ − ⎜ ln ⎜ ⎟ + σ pτ ⎟⎠ ⎟ ⎜ ⎝ ⎝ E[ P] ⎠ ⎟ exp ⎜ 2 ⎟ 2σ pτ ⎜ ⎟ ⎜⎝ ⎟⎠

P 2πσ 2pτ

(10-40)

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where: a the probability function for a lognormal distribution the option settlement price

p the volatility of the option settlement price, hence the approximating volatility time to option expiration, T t the constant pi (3.14 . . .) We can take derivatives of this probability function to obtain values for V3 and V4.

10.5.

THE TREE APPROACH The last option implementation methodology we will discuss here is the tree-building methodology. The idea is that we build a tree (see Figure 10-5) for the option settlement price that defines the movements, up and down, from node to node, of the option settlement price from now until the time of option expiration. The greater the volatility in the option settlement price, the greater is the up and down jump from node to node. The requirement that there is no arbitrage between current and future value of the

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10-5

Binomial Tree Building

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291

settlement price defines the probabilities of going from one node in the tree to another. Thus equipped, we have the values of the settlement price at each node of the tree, as well as the probabilities associated with being at each node of the tree. In this way, we can solve for the option prices backwards: we know what the parity value of the option is at each node of the tree at the expiration time, and we know what the probabilities are. We can then move backwards in time through the tree to arrive ultimately at the present value of the option’s price.

10.5.1.

Pros

Trees can be relatively easy to build and use in option pricing. Furthermore, they allow us to incorporate the volatility term structure within the tree itself and to price American-style options. Thus, the resulting prices and hedges can both be correct.

10.5.2.

Cons

The main drawback of trees is that the Asian path-dependent options, such as options that settle on an average of prices over some time period, cannot practically be priced using trees. The problem is that at the expiration nodes of the tree there are so many possibilities of calculating the average prices backwards from the tree that it becomes so time-consuming to arrive at the solution that it is impractical to use. A means of dealing with this issue is to combine two methodologies: tree building (for incorporating volatility term structure) and closed-form solutions with corrections (for incorporating the behavior of the average price settlement). In this case, the tree is built to the point where the averaging period starts, and at each node at that point in time, the closed-form solution with corrections is used to calculate the value of the option price. The rest of the procedure is simple movement backwards in time through the tree to finally obtain the present value of the option price.

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10.5.3.

Binomial Trees

The building of binomial trees has been covered by many books on option pricing. We will not spend a great deal of time on this methodology here. However, we will summarize the process. In the simple case of flat volatility term structure, and an option on a forward price, F0, the tree is built so that the moves up and down are given by F0e 冪dt and F0 e 冪dt, where dt is the time step between the nodes. The corresponding probability of an up move is then given by p=

1 − e− σ eσ

∆t

∆t

− e− σ

(10-41)

∆t

The probability of a down move is then 1 p. If we wanted to add the volatility term structure within the binomial tree, we could indeed do so. At every time step, then, the tree would have a different discrete volatility used in deciding the prices along the nodes of the particular time step. The end result is that the tree would in fact imply a form of mean reversion when the volatility term structure is decreasing. (In order to eliminate this mean reversion, you would be forced to go to the trinomial trees.) In this case, the probability of the up move is given by ⎡ e N (σ 0 −σ1 ) ∆t − e−σ1 p=⎢ ⎢⎣ eσ1 ∆t − e−σ1 ∆t

∆t

⎤ ⎥ ⎥⎦

(10-42)

where

0 ⬅ n, the volatility at the n-th time nodes in the tree (10-43)

1 ⬅ n 1, the volatility at the (n 1)-th time nodes in the tree (10-44) When volatility curvature is significant, there is a chance that we may end up with negative probabilities. In that case we would be forced to use trinomial trees.

10.5.4.

Trinomial Trees

Trinomial trees are built just like the binomial trees, only now instead of the up and down move, we have one more degree of freedom: the

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F I G U R E

10-6

Trinomial Tree Building

sideways move (Figure 10-6). In this case, the price moves up and down become F0e 冪3dt and F0e 冪3dt, while the side move is really not a move, that is, F0 stays at F0. Now the probability of the up move becomes p=

(

(1 − q ) 1 − e−σ eσ

− e− σ

3∆t

3∆t

)

(10-45)

3∆t

The parameter q is the probability of the sideways move. Requiring that the second price moment correspond to that of a lognormal price process gives us the value for q: q = 1 − ( eσ

2

∆t

(

− 1) / eσ

3∆t

− e− σ

3∆t

)

2

(10-46)

The probability of the down move is 1 p q.

10.5.5.

Using a Tree to Value a European-Style Option

If we use a simple binomial tree to value a European-style option on an energy forward price, using a market-implied average volatility for the duration of the option, our tree progression would be defined by probabilities and forward price values at each node as follows for a time-step n within the tree with a total of N time steps (N /t, where is the time of option expiration and t is the time period between time steps within the tree):

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Fnm = Fe( 2 m− n )σ cpnm =

∆t

n! p m (1 − p ) n− m m!( n − m)!

(10-47)

m = 0… n, n = 0… N The cumulative probability of outcome Fnm at node m of time step n above, cpnm, is given as a function of the probability of up move, p, where p can be simplified (from Equation 10-42) to the following: p=

1 − e− σ eσ

∆t

∆t

− e− σ

∆t

(10-48)

due to the assumption of an average constant volatility. For a European option, then, the only time step when we can exercise the option is the last time step (n N), and therefore the call and put option values are given by N

C = ∑ cpNm max(( FNm − K ), 0) dfτ m= 0 N

P = ∑ cpNm max(( K − FNm ), 0) dfτ

(10-49)

m= 0

These binomial solutions to call and put prices, of course, converge to the Black option pricing model as we allow the number of time steps, N, to go to infinity, thereby decreasing the time between steps, t, towards zero. Note that if we are to use a binomial approach/tree to value an option given a flat volatility during the lifetime of the option, we must price back through the tree as long as there are potential option exercise periods remaining. However, once we have exhausted all the potential time steps of option exercise, we can value the option backwards through time (and therefore to the present) using the cumulative probabilities. This is a particularly useful technique when dealing with energy options where the time period of potential exercise does not occur for a period of time, but once the exercise period begins it is as discrete as hourly. Purely from a numeric calculation point of view, coming up with methodologies to most efficiently (that is, with least processing time) value such potential multiple exercise options becomes critical.

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295

10.5.6.

Using a Tree to Value an American-Style Option

The American-style option allows early exercise. This means that at every single node of every single time step, we need to make sure that our position is financially optimal. This requires comparing the exercise value of the contract versus the value of the option if we choose not to exercise. At the very last time step, our actions are limited to the simple choice of exercise or not. At this time step N, for each node m, where m 1 . . . N, our option values are given by: C Nm = max(( FNm − K ), 0)

(10-50)

PNm = max(( K − FNm ), 0)

At every other time step n 0. . . N 1 prior to the final exercise, we will be comparing the exercise versus the no-exercise scenario and choosing the one that gives us greatest financial satisfaction:

( (

)( )(

) )

Cnm = max ⎡⎣ max(( Fnm − K ), 0) , ( pCnm++11 + (1 − p )Cnm+1 ) df ∆t ⎤⎦ Pnm = max ⎡⎣ max(( K − Fnm ), 0) , ( pPnm+1+1 + (1 − p ) Pnm+1 ) df ∆t ⎤⎦ 10.5.7.

(10-51)

Energy-Specific American-Style Options

In energy markets, you might find American-style options that allow for early exercise, but only during a particular period prior to expiration. In fact, a typical American-style energy contract is the price swing option, where there are a contract-specified number of swing rights translating into a finite number of multiple-exercise rights with exercise into typically next-day delivery (if not next-hour delivery for power) over a prespecified period of time. These are contingent American-style options, and will be discussed in the next chapter, which is devoted purely to energy market options. However, here we will build the groundwork for valuing the more complex energy swing options by looking at the singleexercise American-style option, where the methodology is adjusted to the forward price term structure. This type of option contract gives rise to the need for daily, if not hourly, forward price curves in the energy markets. Not only are such curves necessary, but they are a must. To be valuing such an option assuming a spot price model could be very dangerous, for two very important reasons:

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1. Using a spot price model to build a tree would give discrepancies between your option valuation and your hedging instruments—forwards or futures. 2. A spot price model could never possibly incorporate all the information the energy market has built into its forward price curves. Although Monte Carlo simulations are indeed available for use incorporating multiple factors, even these would have to be adjusted extensively to assure that, in addition to providing an option valuation methodology, these simulations also converge towards the market-specific forward prices (not to mention the additional problem of using Monte Carlo simulations for valuing American style options!). So from here on, this book will strictly attempt to adhere to the First Principle of trading energy markets: using market-implied and market-calibrated inputs whenever possible. This, in turn, requires that we at the least use Black, instead of Black–Scholes in pricing European-style energy options (assuming that we are comfortable with the lognormal assumption), and using option trees that at the least incorporate the necessary forward price term structure obtained from a marked-to-market forward price curve. The basic idea here is that if we have an American-style option, which when exercised settles into next-day delivery of the commodity, for example, then we need to have a tree that will incorporate every single day’s worth of possible exercise scenarios during the period when we are allowed to exercise the option for next-day delivery. Furthermore, the tree better be built around the daily forward price curve that was built at least in the spirit of marking to market, and whenever market forward price quotes are available is exactly marked-to-market. To build such a tree requires us to think first in terms of building a forest—a tree for each forward price corresponding to a day of potential exercise. However, we only need the time step in such a tree corresponding to the exercise for next-day delivery. In fact, what we will be doing is jumping from tree to tree as we switch our hedges from the current day’s forward to the next day’s forward. If we make the simplifying assumption, for the sake of this example, that the volatilities are flat, the forward price at each node m of each time step n looks as follows: Fnm = Ft ,MTM e ( 2 m− n ) σ t + n∆t

∆t

,

m = 0 … n, n = 0 … N

(10-52)

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297

where F MTM t, t nt is the marked-to-market daily forward price corresponding to the n-th day of early exercise period. For each node we now have the probability of an up move—in a most generalized form—defined as follows: m n

F pnm =

Ft ,MTM t +( n+1) ∆t MTM t ,t + n∆t

F

− Fnm+1

Fnm+1+1 − Fnm+1

,

m = 0 … n, n = 0 … N − 1

(10-53)

We can also calculate the cumulative probabilities going forward through time in this resulting tree and accumulating the probabilities of up and down moves from time step to time step: cpnn++11 = cpnn pnn cpn0+1 = cpn0 (1 − pn0 ) cpnm+1 = cpnm−1 pnm−1 + cpnm (1 − pnm )

(10-54)

m = 1 … n, n = 0 … N − 1 Using this methodology, we can also obtain the most likely exercise date by comparing the time-step-specific sums of cumulative probabilities of nodes where early exercise occurs. When comparing European option valuation methodologies—trees versus closed-form solutions—it has to be kept in mind that values obtained using the tree methodology should converge to the closed-form solutions, but will never be exactly equal whenever we have a limited number of time steps. Generally, the binomial tree will underestimate the volatility value of the option due to the limited number of scenarios, in particular in the tails of the distribution. In fact, for every time step, the second moment of the forward price in a binomial tree is given by E ⎡⎣ F12 ⎤⎦ E ⎡⎣ F1 ⎤⎦

2

= eσ

dt

+ e− σ

dt

−1 (10-55)

= 2 cosh(σ dt ) − 1 The second moment should in fact be e 2 dt; thus we see the difference in the width of the forward price distribution already at a single

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node. This bias grows as the tree grows and can be adjusted by requiring the European options’ closed-form values and tree values to match for identical contracts. The resulting implied tree volatilities will include this bias adjustment. This adjustment will be different across strikes due to tail effects. Examples of most likely early exercise days for American-style option valuation are provided Table 10-1 and Figure 10-7 for the different cases of a flat daily forward price curve, an increasing and decreasing daily forward price curve, and a seasonal forward price curve (with constant volatility of 50%, number of time steps equal to 30, and expiration time of one month). Similarly, we can add volatility term structure; some resulting comparisons are provided in Table 10-2. Figure 10-8 shows the discrete volatility curves used in the calculations. The discrete volatilities were such that the effective average volatility over the lifetime of the option was held at 50% in all cases. In energy markets there are some general rules of thumb when it comes to American-style options. Due to the price mean reversion in energy markets, the discrete volatility of forward prices will increase as forward prices approach expiration. The intensity of this increase can be T A B L E

10-1

Option Value Comparisons for Various Forward Price Term Structures Ex Day/ Most Likely EE

F (1) Black Binomial—Flat Euro* Binomial—Flat Am* Binomial—Inc Euro* Binomial—Inc Euro* Binomial—Inc Am* Binomial—Deac Euro* Binomial—Deac Euro* Binomial—Deac Am* Binomial—Seas Euro* Binomial—Seas Am*

$50.0000 $50.0000 $50.0000 $31.2903 $40.6452 $40.6452 $69.3548 $59.3548 $59.3548 $52.0791 $52.0791

F (T) $50.0000 $50.0000 $50.0000 $50.0000 $59.3548 $59.3548 $50.0000 $40.6452 $40.6452 $50.0000 $50.0000

Fmonth $50.0000 $50.0000 $50.0000 $40.6452 $50.0000 $50.0000 $59.3763 $50.0000 $50.0000 $50.0000 $50.0000

Call $2.8766 $2.8766 $2.8766 $2.8766 $9.8167 $9.8167 $2.8766 $0.2115 $9.3548 $2.8766 $9.9523

*Adjusted for volatility bias relative to Black: Black-implied volatility is 50.42%

Put $2.8766 $2.8766 $2.8766 $2.8766 $0.4619 $9.3548 $2.8766 $9.5763 $9.5763 $2.8766 $10.0404

Call

Put

30 30 30 30 30 30 30 30 1 30 8

30 30 30 30 30 1 30 30 30 30 23

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299

F I G U R E

10-7

Graph of Forward Price Curves Used as Inputs in Calculating Values in Table 10-1 70 65

Forward Price

60 55

Flat F=$50 Inc F, F(T) = $50 Inc F, av F = $50 Dec F, F(T) = $50 Dec F, av F = $50 Seas F, F(T)=av F= $50

50 45 40 35 30 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Day

T A B L E

10-2

Option Value Comparisons for Forward Price and Volatility Term Structures Ex Day/ Most Likely EE

F (1) Black Binomial—Flat F, Inc Vol—Euro Binomial—Flat F, Inc Vol—Am Binomial—Seas F, Inc Vol—Euro Binomial—Seas F, Inc Vol—Am Binomial—Seas F, Seas Vol—Euro Binomial—Seas F, Seas Vol—Am

$50.0000 $50.0000 $50.0000 $52.0791 $52.0791 $52.0791 $52.0791

F (T)

Fmonth

$50.0000 $50.0000 $50.0000 $50.0000 $50.0000 $50.0000 $50.0000

$50.0000 $50.0000 $50.0000 $50.0000 $50.0000 $50.0000 $50.0000

Call $2.8766 $2.8766 $2.8766 $2.8766 $9.9475 $2.8766 $9.9617

Put $2.8766 $2.8766 $2.8766 $2.8766 $10.0129 $2.8766 $10.0437

Call

Put

30 30 30 30 8 30 8

30 30 30 30 23 30 23

quite staggering for power daily forward prices. This reality of the marketplace translates into options with a large “chunk” of their volatility value “sitting at the expiration.” For an American-style option, where the width of trees grows more and more the closer we get to expiration, this volatility value would put a bias toward not exercising early. However, the seasonal effects of the forward prices can be so significant that it overwhelms the volatility value of the option, and makes American calls and puts early exercise due to intense seasonal price decreases and

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F I G U R E

10-8

Graph of Volatilities Used as Inputs to Calculating Values in Table 10-2 70.00% Discrete Volatility (over dt)

65.00% 60.00% 55.00% 50.00% 45.00% 40.00% 35.00% 30.00% 1

6

11

16

21

26

Time Step Discrete Volatility - Flat

Discrete Volatility - Inc

Discrete Volatility - Seas

increases. We see an example of this seasonal effect in Table 10-2 where, despite the increasing volatility case, the call option still optimally exercised early at the seasonal peak and the put option still exercised early at the seasonal low. Of course, if instead we were dealing with Americanstyle options on the same monthly forward price (instead of for next-day delivery—daily forwards/spot) throughout the option lifetime, we would be strongly biased against early exercising due to making sure that we capture this large volatility value at the expiration. The above exercise not only takes us through different types of market environments, but also through the very important practice of comparing models. In valuing a portfolio it is important to recognize the differences in valuation between models and to adjust for these in order to ensure consistency in valuation. At the same time, if market prices are used to imply volatilities, as long as there is consistency between retrieval of implied parameter values and the pricing of contracts, the model bias becomes irrelevant, because it is already accounted for in the implied values.

10.6.

MONTE CARLO SIMULATIONS The computer age continues developing greater processing power and places better and better software tools at our disposal to use in simulating market behaviors, price, and volatility processes. A good review

Option Valuation

301

of the use of Monte Carlo simulations in valuing energy contracts is provided by John Putney in Chapter 5, “Modelling Energy Prices and Derivatives using Monte Carlo Methods,” in Vince Kaminski’s book Energy Modelling. Although computation time is getting faster and faster, the problem of using Monte Carlo simulations for valuing contracts on a fastpaced trading floor remains a practical concern, both in terms of processing time as well as in ensuring a marked-to-market compliance. Although simulations may not be appropriate for a trading floor, they are most certainly a great tool for testing methodologies being considered for the trading floor. However, for valuing assets where the valuation is already filled with long-reaching assumptions potentially covering several decades, and where the valuation processing time is not an important factor, Monte Carlo simulations provide an excellent way of getting an idea about ranges of reasonable values given multiple scenarios and a multitude of possible factors. Examples of some more simple Monte Carlo simulations used to compare two different models—lognormal and price meanreverting—and their resulting price distributions are shown in Figures 10-9 and 10-10.

F I G U R E

10-9

Simulations of Lognormal and Price Mean-Reverting Models (Both With 100% Spot Volatility Texp 0.25; BlackImplied Vol from PMR Distribution 29.85%)

Observations (not normalized)

900 800 700 600 500 400 300 200 100 0 –100 Spot Price

5

10

15

20

25

30

35

40

45

50

55

60

Price LN-Spot At 100% Vol

PMR Spot At 52 Alpha, 100% Spot Vol, 20% Eq. Vol

65

70

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F I G U R E

10-9

Simulations, Continued (Both With 200% Spot Volatility Texp 0.25; Black-Implied Vol from PMR Distribution 50.48%) Observations (not normalized)

700 600 500 400 300 200 100 0 5

10

15

20

25

30

35

40

45

50

55

60

65

70

Price LN-Spot At 200% Vol

10.7.

PMR Spot At 52 Alpha, 200% Spot Vol, 20% Eq Vol

CONCLUSIONS In this chapter, we discussed a few option valuation approaches. Our discussion was limited to closed-form solutions, their approximations, and trees, and even within this small spectrum of possible valuation methodologies we have met with a few challenges in applying such methodologies to the real world. Ultimately, it is the methodology that best weighs practicality of use with maximum possible capture of market reality that generally wins and finds its place on the trading floor.

ENDNOTES 1. “The Future of Modeling,” interview with Emanuel Derman, Risk, December 1997. 2. For excellent example of this process, see Turnbull, Stuart M. and Lee Macdonald Wakeman. “A Quick Algorithm for Pricing European Average Options.” Journal of Financial and Quantitative Analysis 26 (September 1991).

C H A P T E R

11

Valuing Energy Options Options theory works because it aims at relative, rather than absolute, value. A necessary prerequisite is the notion, sometimes scorned by academics, of value calibration: the effort to ensure that the derivative value matches the value of each underlyer under conditions where mixtures become pure and certain. Without that, the relativity of value has no foundation. . . Options theory is rational and causal, based on logic. It is mathematical but the mathematics is secondary. Mathematics is the language used to express dynamics. There are still many traders, even options traders, who have a taste for mathematics without reason—for voodoo number-juggling and patterns and curve fitting and forecasting. I think we will continue to see successful models based on ideas about the real world, as opposed to mathematical-looking formulas alone.

Emanuel Derman1

11.1.

INTRODUCTION To treat many of the real-life energy contracts, in particular the contracts tied to specific but diverse commodity services provided by producers and asked for by users, as simplified financial option contracts may indeed be quite a jump of faith. However, these contracts are indeed being priced and valued regularly. Unifying the intuitions and practices of the energy marketplace with quantitative financial contract valuation should be the big quest of current energy markets. The practitioners who have been involved with the energy services for years, if not decades, are rich with experience and understanding of the fundamentals of the industry, but need to gain some of the quantitative skills in this unification challenge. However, an even bigger step needs to be taken by the quantitative and academic world towards understanding the fundamentals of the energy markets. This step cannot possibly be made without looking at the calibrated or implied market


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parameters grounded in traded market contacts and prices and then learning about and modeling these calibrated parameters and variables. The bridge between the academic and practical world should be through market calibration. This should be the point of departure for both practitioners and academics. This chapter tries to value various energy market option contracts beginning with the available market information. This attempt is flawed; the valuation tools available are too simplistic to deal with the multitude of complexities found in energy contracts. However, as flawed as it might be, it is our point of departure. As any trader knows, model values are benchmarks—incredibly valuable and sometimes greatly predictive benchmarks—but all the same, they remain benchmarks. It is in the spirit of arriving at and understanding some of these benchmarks that this chapter was written. The options discussed in this chapter are specific to the energy marketplace. They include the various European-style options, including daily and monthly settled, monthly cash settled Asian-style options on simple price averages, swaptions, swing options including multiplepeaker swing options and forwards, crack spread options, and their application—albeit extremely simplistic—in valuing energy assets.

11.2.

DAILY SETTLED OPTIONS European-style options with a discrete daily price settlement are typically seen in natural gas and electricity markets. Most commonly, the settlements span a particular month. In this case, the owner of a call option contract has a series of call options that expire every calendar or business day of the month into next day’s (again, calendar or business) delivery of on-peak power or delivery of natural gas. The days of delivery span the month of the option contract and the expiration dates are determined accordingly. An example of such a contract is provided in Figure 11-1 for a typical and popular power option for the months of July through August (JUL-AUG contract). Note that although this is a single contract, it includes a total of 40 options expiring every single business day; if exercised, the settlement is for energy to be delivered the following business day. The nomenclature of the marketplace varies across energies. For power, the quoted unit price is the dollar price per MWh, and the contract


F I G U R E

305

11-1

Sample Power Daily-Settled Daily Option Contract Specifications

quantity quoted is generally assumed to represent the quantity delivered per hour. In power, the overall cost of the on-peak contract can then be calculated by multiplying this unit price with the quantity times the number of days of delivery times 16 (for on-peak hours). In natural gas, the contracts may be either in calendar day terms or business day terms (for delivery) and the unit quote is per MMBtu. The natural gas contract quantity can be expressed either on per day terms or as the overall quantity for the month. Such daily settled options are probably the simplest energy options out there, and yet calling them “simple” is a gross understatement. The difficulty is in making sure that the daily settle options capture the decreasing volatility term structures as expiration times grow (which translates to “riding up the volatility curve” for a forward contract nearing delivery) as well as the seasonal effects. Examples of possible daily forward and volatility term structures are shown in Figures 11-2 and 11-3 for PJM 5 16 power market in the fall of 2006. In this

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Energy Risk

F I G U R E

11-2

PJM 5 16 Daily Forward Price Curve

F I G U R E

11-3

PJM 5 16 Daily (Black-M2) and Monthly (Black) Market-Implied Volatilities


307

case the daily volatilities were model-implied from monthly market option prices (this methodology will be discussed in the later sections of this chapter). Notice the significant increases and decreases in both forwards and daily volatilities in this case during the seasonal months. Daily settled option contract valuation must take such volatility behavior into account in order to capture the market value of these options. Figure 11-4 also shows PJM volatilities, but implied directly from daily settled call options contracts in the winter of 1997. As you can see, the seasonality effects observed almost a decade earlier remain; in comparison, the longer-term volatility levels appear to have grown somewhat over time. Just by observing these graphs it should be apparent that an option methodology that did not incorporate the daily forward and volatility term structures shown in these graphs and instead used flat values for forwards and volatilities would result in a very different set of daily settle option values. An options trader coming from the “stock-world” might believe that there is clear arbitrage when first confronted with the daily option prices in energies: occasionally, the first day’s option value can be observed to be greater than the second day’s option value. This trader might get very excited about taking advantage of a market where the option with the later expiration is valued at less! However, he should beware. First, it is important to remember that these daily

F I G U R E

11-4

Expiration Date

12/23/98

11/23/98

9/23/98

10/23/98

8/23/98

7/23/98

6/23/98

5/23/98

4/23/98

3/23/98

2/23/98

1/23/98

3.00 2.50 2.00 1.50 1.00 0.50 0.00 12/23/97

Volatility (ann.)

Market-Implied Volatilities for Daily Settle

308

Energy Risk

options are all on essentially different underlyings; each daily forward price has its own volatility and forward price. This is not to say that these forward prices are entirely unique. In fact, if we followed the logic of a two-factor mean-reverting model, the further out the expiration time the more correlated should these daily forward prices become. However, as we get close to expiration, the correlations become far more “tricky” to determine and become a function of the current event’s mean reversion in the simplest possible case. Secondly, there is a paradox in the energy markets that the practitioners have to live with due to the strong price mean reversion and extremely high spot volatility: a potentially large portion of the volatility value of the option is captured close to the expiration. As the forward prices converge to spot and ride up that potentially incredibly increasing discrete volatility curve, we are left feeling more comfortable discussing the possible range of values for a forward further out in time than we do once it is close to expiration. In other words, we feel we know less about what will happen tomorrow than we do about what will happen—on average—over a period of time. An example of such daily settled option price valuation is provided in Table 11-1 for the case of three daily-settled options. Black was used to value these options, with the forward prices and volatilities used as inputs also shown in Figures 11-5 and 11-6. Notice how the rise in the volatility the closer the expiration offsets the loss of option value due to time decay in this example. The resulting probability distributions are shown in Figure 11-7. Notice that they are fairly similar, with the time

T A B L E

11-1

Sample Daily Settled Call Option Valuation Days to Expiration 0 1 2 3 4 Payment Date

Date

Texp

Forward Price

Volatility (%)

Discount Factor

31 1 2 3 4 15

0.000000 0.002740 0.005479 0.008219 0.010959 0.041096

40.00 36.32 34.62 34.19 34.56 45.03

299.84 268.32 211.39 185.52 173.28 130.66

1.000000 0.999808 0.999617 0.999425 0.999233 0.997127

Jul Aug Aug Aug Aug Aug

Option Value (per MWh)

1.80 1.81 2.15


309

F I G U R E

11-5

Au g

28

21

14

7

Au g

Au g

Ju 31

F I G U R E

Au g

50 45 40 35 30 25 20 l

Forward Price

Forward Price Term Structure Example

11-6

3 2 1

F I G U R E

g Au

g

28

Au

g Au

21

7

14

Au

Ju

g

l

0

31

Volatility (in %, ann.)

Volatility Term Structure Example

11-7

0.007 0.006 0.005 0.004 0.003 0.002

54.5

50.4

46.3

42.1

38

33.9

29.8

25.6

21.5

17.4

0.001 0

13.2


Sample Daily Settled Underlying Price Distributions (F1 $34.62, F2 $34.19, F3 $34.56, Vol1 268.32%, Vol2 211.39%, Vol3 185.52%, Texp1 1 day, Texp2 2 days, Texp3 3 days)

Forward Prices at Options' Expirations Prob 1

Prob 2

Prob 3

310

Energy Risk

decay of the earlier expiration options being replaced by an increase in the volatility value of the options. At the danger of sounding like a broken record, we are dealing with two factors causing these underlying price distributions for different expiration options to be so similar:

• First, these are options on different forward prices, so if we are to compare them we need to remember this fact. • Second, we are valuing and characterizing these options using the Black-equivalent volatility and option pricing model, and in doing so we are not being entirely “truthful” about the real market behavior of the underlying daily forward prices. These daily forward prices have the tendency to “ride up” the volatility curve as they converge towards highly volatile spot price behavior as the mean-reverting tendency towards the equilibrium price decays close to its expiration. Over time, the daily forward price term structure exhibits the effects of both the extremely volatile spot price behavior as well as the far more stable equilibrium price behavior. Therefore, using either the spot price or the equilibrium price alone to value these options is not appropriate. It would be the equivalent of using the spot price to hedge a long-term forward price position, or, vice versa, to use the longterm forward price to hedge an immediate spot price position. That would be a dangerous hedging strategy indeed! Although some of the behaviors we have observed in energy options markets are certainly due to the fundamentals of the marketplace—with multiple market factors driving some interesting marketimplied volatility behavior—it may also be true that some of diverse magnitudes observed in the volatility term structures are due to market illiquidity. If so, we should expect these volatility term structures to change over time as the market becomes more liquid. Finally, in addition to worrying about implementing the appropriate volatility term structure into the valuation of the daily settled options, it is also critical to implement the volatility strike structure as well. Figure 11-8 shows an example of the rich strike structure across various expiration times for the Cinergy 5 16 power market in the fall of 2000. (This volatility matrix was implied directly from the market quotes for the daily-settled option contracts.) There is no model that could possibly match the strike structure and term structure for


F I G U R E

311

11-8

Volatility Strike and Term Structure for Daily-Settled Power Options (Cinergy 5 16 Market-Implied Daily Vols, September 2000)

these Cinergy volatilities perfectly! The reality is that no matter what model you end up using in your option valuation, you will have to allow for the implementation of both volatility term and strike structure. This means that you will have to build volatility matrices for the daily settled options. This is nothing new for an options trader. The strike and term structure of volatilities exists in all markets, even though they might be quite a bit simpler in comparison to energy markets. So it should be of no surprise that energy markets require at least as much volatility attention. The Cinergy volatility matrix shown here exhibits just about every possible strike structure across the various expiration times. We have smiles, we have smirks, we even have frowns. This is consistent with the energy market’s strike structure across both the types of energies and observed over the years (Figure 11-9 shows a recent PJM 5 16 model-implied daily settle volatility strike structure for November 2006 delivery).

312

Energy Risk

F I G U R E

11-9

PJM Volatility Strike Structure Example (November Option, October 2006)

11.2.1.

Extending Daily Methodology to Hourly Settled Options

Hourly settled options are an important component of the service contract mix that power providers carry. In terms of valuation methodology, going from options on daily power prices to hourly options settling into next hour of delivery is no different than making the transition from monthly settled options to daily settled options as we have discussed above. Although the methodology is no cause for concern, the inputs to option valuation are. Multiply a single daily forward price curve by 24 and the volatility matrices by 24, not to mention the correlation matrices, and you get an idea of how much more information you need to be able to value such options within just a single power market. The additional constraints of ensuring consistency with the market daily forward prices and the market daily volatilities add complexity to the situation, but do not add enough market data to provide for a full spectrum of implied hourly forward prices or volatilities. Instead, the hourly


313

traders, like it or not, must make assumptions about the hourly markets relative to their daily counterparts. It is the task of the valuators and risk managers to ensure that these assumptions are indeed consistent with the available market information.

11.3.

MONTHLY SETTLED OPTIONS There are two general types of monthly settled energy options: the cash-settled Asian monthly options and the more standard European options into next month’s forward price for delivery. Both types of options are traded both OTC as well as on the exchanges. Both options are options on an average of prices, but they are distinctly different types of averages. The cash-settled Asian option settles on a simple arithmetic average of observations of a single underlying forward price, where the observations are made during a calendar month. By comparison, the monthly European options settle into the next calendar month’s forward price, or interchangeably into the delivery of energy for the next calendar month. The cash-settled options are typically seen in crude oil markets, and somewhat in natural gas markets. The European options into monthly calendar delivery are typical for natural gas and power markets, and are the preferred choice for market players with “juice.” Unlike the Asian option, the monthly European option settles into a forward-looking monthly position, where the monthly forward value is in fact a weighted average of the individual, and therefore distinct, daily forward prices over the next month’s worth of delivery. In the case of the Asian option, because we are looking at exactly the same underlying price, but at different points in time, the correlation between the discrete observations of this underlying price is perfect2. However, in the case of the European options into monthly forwards we are looking at a monthly average of distinct daily forward prices that are not generally perfectly correlated, especially as the option gets closer to expiration. In fact, relating the monthly settled European options to the daily settled European options discussed in the previous section of this chapter becomes an exercise in intramarket correlation discovery.

314

Energy Risk

11.3.1.

Cash-Settled: Look-Back Monthly Settled Average Price Options

The cash-settled European-type options, which settle into averages of prices over some period of time, and which expire at the end of the averaging period, are typically seen in WTI and natural gas OTC markets. NYMEX now also provides a trading platform for these Asian options with cash settlement based on the average of the front month crude oil future over the course of the calendar month. In order to value these contracts, we need to include the effects of look-back averaging. Because of the averaging effect, these Asian options are cheaper than their European equivalents—the standard monthly options settling into next month’s delivery or forward contract. The value of these options is contingent on the past-period price averaging, and therefore these options are cheaper both due to the potential diversification effect as well as time-decay effect of averaging. In OTC markets these options are generally traded with monthly tenor, but occasionally quarterly tenor can be observed. In valuing these options we will begin with a very simple case study. We will assume flat volatilities, cash settlement at the end of the averaging period, and a simple two-price average, with the prices assumed to be perfectly correlated. (In fact, the perfect correlation is appropriate in the case where the average is of the single underlying price, such as a single futures contract observed every business day of the calendar month.) These simplifications must be a function of market reality and the option contract specifics; we make them here for the sake of simplicity. Allow the forward price to be observed and averaged over a period of time to be lognormal. Specifically, let us assume that there are only two observations of the same forward price in the average, taken at different times t1 and t2. The forward price at those two observation times is then given as follows: 1

2

( t1 −t0 )+σ z0 ,1

(11-1)

1

2

( t2 −t0 )+σ z0 ,2

(11-2)

− σ F t = F t e 2 1

F

t2

0

− σ = F t e 2 0

where: F˜|t underlying forward price observed at time tn where n t0 t1 t2.


315

For simplicity, the volatility, , is assumed constant, and z˜0,n is a normally distributed stochastic variable with a mean of zero and a variance of (tn–t0). We have the following relationships between the stochastic variables:

ρ( z0 ,1 z1,2 ) = 0

(11-3)

z0 ,2 = z0 ,1 + z1,2

(11-4)

In this case the option settlement price is simply the average of two prices: ⎛ F + F t FA = ⎜ 1 2 ⎜⎝

t2

⎞ ⎟ ⎟⎠

(11-5)

We will now apply the Edgeworth series expansion methodology to calculate the averaging volatility. We begin by calculating the approximating and actual moments of the average price distribution: ⎛ E ⎡ F ⎤ + E ⎡ F ⎤ ⎞ t ⎢ ⎜ t⎢ t⎥ ⎣ t2 ⎦⎥ ⎟ M1 = Et ⎡⎣ FA ⎤⎦ = ⎜ ⎣ 1 ⎦ ⎟ 2 ⎟⎠ ⎝⎜ ⎛ F + Ft ⎞ M1 = ⎜ t ⎟ = Ft ⎝ 2 ⎠

(11-6)

(Note that the first moment is simply the current forward price value.) 2 t2 σ approx

Mapprox = ( M1 )2 e

(11-7)

2

⎡⎛ F + F t ⎢ 2 M 2 = Et ⎡⎣ FA ⎤⎦ = Et ⎢⎜ 1 2 ⎢⎜⎝ ⎣ 2

⎞ t2 ⎟ ⎟⎠

2

⎤ ⎥ ⎥ ⎥ ⎦

⎛ 1⎞ M 2 = ⎜ ⎟ ⎛ Et ⎡( F t )2 ⎤ + 2 Et ⎡( F t )( F t ) ⎤ + Et ⎡( F t )2 ⎤⎞ ⎥⎦ ⎥ ⎥⎦⎠ ⎣⎢ ⎣⎢ 1 1 2 ⎦ 2 ⎝ 2 ⎠ ⎝ ⎢⎣

(11-8)

316

Energy Risk

Because we require that the second moments of the actual and approximating distribution equal each other, we obtain the approximating average price volatility: 2 M approx = M2

2 σ approx t2

e

σ approx

σ 2t1

=e

(11-9)

1 σ 2 ( t −t ) (3 + e 2 1 ) 4

⎛ 3 + eσ 2 ( t2 −t1 ) ⎞ ⎪⎫ 1 ⎧⎪ 2 = ⎨σ t1 + ln ⎜ ⎟⎬ 4 t2 ⎪⎩ ⎝ ⎠ ⎪⎭

(11-10) 0.5

(11-11)

At this point we would value the option price using the Black option model, with the volatility input being the approximating volatility. If we so choose, we could also go on to calculate the higher-order moment corrections for skew and kurtosis, depending on how large these higherorder corrections tend to be for the energy market we are dealing with. (It is quite possible that these higher-order corrections are insignificant and do not need to be calculated. However, there is only one way of finding this out: by calculating the error terms at least once.) The above procedure can be generalized for the case where the average is based on N prices instead of just two. Now, the average is given by 1 FA = N

N

∑ F n=1

(11-12)

tn

where t1 t2 · · · tn

(11-13)

It turns out that the approximating volatility is then given by

σ approx

⎡ 1 ⎛ N n ( m−1)σ 2∆t N −1 ⎞ ⎤ ⎪⎫ 2 1 ⎪⎧ 2 = + ∑ ( N − n)e( n−1)σ ∆t ⎟ ⎥ ⎬ ⎨σ t1 + ln ⎢ 2 ⎜ ∑ ∑ e ⎠ ⎦ ⎪⎭ n=1 t N ⎩⎪ ⎣ N ⎝ n=1 m=1

where ∆t ≡

t N − t0 N

0.5

(11-14)

. The above can be simplified further for t 1:

σ approx

⎧⎪ t ⎛ t − t ⎞ ⎛ 1 1 1 ⎞ ⎫⎪ =σ ⎨ 0 +⎜ N 0⎟⎜ + + 2⎟⎬ ⎪⎩ t N ⎝ t N ⎠ ⎝ 3 2 N 2 N ⎠ ⎪⎭

0.5

(11-15)


317

This simplifies even further for a large number of observations in the average (N 1):

σ approx = σ

1 = 0.5774σ 3

(11-16)

So far we have assumed that the average of prices was based on the same forward price observed at different points in time. Instead, the above formulation can be further generalized for the case of the average of different forward prices with a forward price term structure, for a volatility term structure corresponding to each of the forward prices in the average, and for nonperfect correlations between the prices comprising the average. However, even in the simpler case where indeed the average is based on the same forward price observed at different points in time, resulting in perfect correlations between the forward prices (as it is the same instrument), it is important to add the volatility term structure to the above formulation. In order to compare the European options on crude oil futures traded on NYMEX to their average option counterparts, adding the volatility term structure becomes important. The European options and the Asian options on the same futures expire at different times; thus, not only is there the difference in price between the two options due to the averaging effect in the case of Asian options (driving the Asian options price below the European counterparts), but there is also the additional difference of the volatility term structure where the Asian option expires at the end of the calendar month, whereas the European option settling into the future expires a few days earlier. The latter difference gives an upward boost to the lower Asian option value, as the Asian option observes the underlying price go to full expiration. So, to compare the two option types requires a bit more work than just simply applying the above flat volatility averaging adjustment.

11.3.2.

Monthly Settled (Look-Forward) Options on Monthly Forwards

If the smallest price segment your trading operation trades is monthly (instead of daily), then the problem of monthly settled options on monthly futures becomes fairly simple. It reduces to knowing and

318

Energy Risk

understanding the behavior of the monthly price forward, and does not necessarily require the understanding of the even smaller price segment, such as daily. There are markets where the smallest price segment traded generally is monthly, and therefore in those markets the monthly price becomes the common denominator across all product types. Crude oil markets are one such example. (Figure 11-10 shows market implied volatility strike structure captured in June 2006 from monthly settled Light Sweet Crude Oil Options for delivery on August 2006, January 2007, and June 2007.) Power and natural gas, on the other hand, are markets where the daily prices (if not hourly!) comprise an extensive universe of market products and services. Even in the case of natural gas and power, if all you trade is monthly contracts and options on these monthly contracts, you may indeed be able to get away with maintaining your trading operation by treating the monthly price segment as your most discrete price segment. However, you would be missing out on a whole other level of understanding and valuing energy options by not beginning with the smallest discrete price level traded. In fact, in the case of power and natural gas, trading the monthly settled options as options on a weighted monthly average of daily forwards becomes a correlation

F I G U R E

11-10

A Snapshot of Market-Implied Light Sweet Crude Oil Volatilities (June 2006): Across Strikes and for Several Expiration Periods


F I G U R E

319

11-11

Example of Monthly Settled Option Specifications: Jul–Aug 2007 Contract

play. (An example of the specifications for a monthly settled Jul–Aug 2007 on-peak electricity option is provided in Figures 11-11 and 11-12.) If the energy markets were single-factor lognormal markets, then we would have daily forward price curves where all the daily forward prices would move in unison and with perfect correlation. In this case, all the daily forward prices would have the same volatility, and furthermore the volatility for the monthly weighted average of daily forward prices (weighted by the discount factors) should be identical to the volatility of the daily forward prices. In such a simple marketplace, the F I G U R E

11-12

Example of Monthly Settled Option Specifications: Jul–Aug 2007 Contract, Continued

320

Energy Risk

monthly settled options for monthly delivery should be valued using exactly the same volatility as for the daily settled options into next-day delivery. However, things are nowhere near that simple in energy markets. Figure 11-13 shows the probability distribution for a flat forward price curve, but where the correlations between the forward prices for different deliveries were allowed to experience different correlations. Clearly, the smaller the correlation, the greater is the diversification effect, and the smaller the width of the distribution (i.e., the more stable the resulting average of daily forward prices). Specifically, if we simplify the world and assume all discount factors to be 1.0, the forward price curve to be flat, and the volatilities to be flat, but we allow for intramarket correlations to be nonperfect, we can easily formulate the volatility of the monthly average. First, we define the monthly forward price to be given by Equation 11-17: Ft ,MT ,T = 1

N

1 N

N

∑F n=1

(11-17)

t ,Tn

where Ft,T is the current (time t) market price for the daily forward n price to be delivered at time Tn. (Equation 11-17 has incorporated the

F I G U R E

11-13

Comparison of Probability Distributions Under Different Intramarket Correlations (F flat $25, Vol flat 100%, Texp 0.25) Probability Density Function

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

8

15

23

30

38

45

53

60

Forward Price at Option Expiration Daily & Monthly @ 100% Monthly @ 50% corr

Monthly @ 15% corr Monthly @ 85% corr

68


321

simplification that the discount factors are all set to 1.0. When ultimately performing option valuation, proper discounting must be reapplied.) What we want is the average volatility of this monthly forward price from now (t) until the option’s expiration (To). We need to see how the width of the distribution for this monthly forward price will grow until the option’s expiration time. Specifically, at the time of option expiration, To, the monthly forward price will be given by Equation 11-18: FTM,T ,T o

1

N

t

= =

N

1 N 1 N

∑ F n=1

To ,Tn

N

∑ Ft ,T e n=1

t

(11-18)

1 − σ 2t +σ zn ,t ,T o 2

n

where z˜n;t,T is the normally distributed stochastic variable correspono ding to the n-th daily forward price in the monthly average, with a mean of zero, and a standard deviation of (To t). We have relaxed the assumption of perfect correlations, but allowed for a simplification of a flat correlation matrix, in order to obtain: ⎡

⎤

⎛

⎞

E ⎢⎢ zn; t,T zm;t,T ⎥⎥ = ⎜⎝ δ n ,m (1− ρ )+ ρ ⎟⎠ (To − t ) ⎣

o

o

(11-19)

⎦

where the delta function, n,m, is equal to one when n equals m and zero otherwise, and the intramarket correlation has also been assumed flat. We can then solve for the monthly volatility, using the assumption of a flat forward price curve:

σ

M t ,T1 ,TN

⎧⎪ ⎡ 1 σ 2 ( To − t) ρσ 2 ( To − t ) ⎪ ρσ 2 ( To − t ) ⎤ ⎫ = −e )+e ⎨ln ⎢ ( e ⎥⎬ (To − t ) ⎪⎩ ⎣ N ⎦ ⎭⎪ 1

0.5

(11-20)

In the event of a single observation (N 1), the monthly volatility in Equation 11-20 reduces to the daily volatility—as it should, because the average reduces to the single daily forward price. Also, if the correlation is set to 100%, the monthly volatility again equals the daily volatility—as it should, because the behavior of the average becomes identical to the behavior of the daily forward prices. Finally, in the case of zero correlation and with many observations in the average (N 1), the monthly volatility approaches zero due to the diversification effect. Given the daily forward price volatility, we can now estimate the monthly volatility—conditional on our assumption regarding the intramarket correlation value. Similarly, given the monthly volatility we

322

Energy Risk

could back out the daily forward price volatility—also conditional on the value of the intramarket correlation. Finally, if we are so lucky as to have market quotes for both the monthly settled and the daily settled options, we can use all this market information to imply the intracorrelation value. The above formulation can be extended to incorporate daily forward price and volatility term structures, as well as a correlation matrix. Figure 11-14 shows the market-implied (using Black) monthly volatilities across both strikes and expiration times for natural gas in October of 2000. Notice the complex term and strike structure for this market. Using a flat intramarket correlation of 10%, the volatility relationships formulated here, and the market-implied monthly volatilities from Figure 11-14, we obtain the model-implied daily volatility across expiration times and strikes in October of 2000, as shown in Figure 11-15. A more recent (June 2006) cross-sectional look of the marketimplied monthly volatilities (using Black) across expiration times and for different strikes (relative to at-the-money, in %) for natural gas is shown in Figure 11-16. The same data are also shown across strikes for

F I G U R E

11-14

Example of Market-Implied Monthly Volatility Term and Strike Structure: Natural Gas October 2000 (Marked-to-Market (Black))


F I G U R E

323

11-15

Example of Model-Implied Daily Volatility Term and Strike Structure: Natural Gas October 2000

F I G U R E

11-16

Example of Market-Implied Monthly Volatility Term Structure: Natural Gas June 2000

324

Energy Risk

different expirations in Figure 11-17. Note that across time the seasonal effects remain quite strong, and across strikes we see both smiles and “smirks.” Using the above methodology, but incorporating the daily forward price and monthly volatility term structure, we can model-imply the daily volatilities. Figures 11-18 and 11-19 show the resulting daily volatility term and strike structures conditional on a simplistic 10% intramarket correlation. Given the simplification of a flat correlation matrix, it is no surprise that the daily volatilities retain the monthly shape across time and strikes. However, with a 10% intramarket correlation, the daily volatility levels are much higher compared to their monthly equivalents. A 10% intramarket correlation is perhaps too low, and certainly the assumption of a flat correlation matrix is far too simplistic and not realistic. This is particularly true for the case of longer expiration times in the above example. For these reasons the above model-implied daily volatilities are probably too high, particularly further out in time. Nonetheless, the above exercise provides a sense for what can be done with some market data and some modeling assumptions, and an insight into the range of daily volatility values that might be possible

F I G U R E

11-17

Example of Market-Implied Monthly Volatility Strike Structure: Natural Gas June 2006


F I G U R E

325

11-18

Example of Model-Implied Daily Volatility Term Structure: Natural Gas June 2006

F I G U R E

11-19

Example of Model-Implied Daily Volatility Strike Structure: Natural Gas June 2000

326

Energy Risk

in energy markets. With a well-studied correlation matrix, or even better, with a correlation matrix implied from the marketplace, we can have a complete and unified view and treatment of the daily and monthly option products.

11.3.3.

Incorporating Price Mean Reversion (PMR) into Monthly Settled Options

So far, in this chapter, we have focused to a large extent on implying volatilities from market option prices. But what might we expect to see in the option volatilities if in fact we assumed a mean-reverting spot price? Answering a question such as this lends itself perfectly to the Monte Carlo simulations; simply simulate the PMR world and see how close it comes to reality. In this section we will employ Monte Carlo simulations in order to simulate the effect of the two-factor price meanreverting model on the resulting Black-implied volatilities for the monthly forwards, both through time as well as across strikes. The volatility strike structure will directly tell us how well the model is fitting reality. The market-implied Black volatility shown in the figures so far in this chapter all reflect a market where the actual price distribution diverges from the distribution as expected under a lognormal model. If the Black model—and therefore the lognormal assumption—fit the market perfectly, then there would be no strike structure, or rather, the strike structure would simply be a flat line, and the same volatility would fit all the market option prices. There is no model that will ever perfectly fit a market; you must always include the flexibility of volatility strike structure in your option valuation. However, different models will result in different price distributions, and therefore you will find that some models capture market reality better than others. The value of a “better” model is twofold: first, it might allow you to better value products or value illiquid products and with less implementation labor; second (and perhaps more importantly), a better model will hopefully give you a better understanding of the market behaviors and therefore a better market paradigm from which to perform product valuation and risk management. As we go through the results of the Monte Carlo simulations for the two-factor price mean-reverting model, the question we will continuously ask is how does the resulting price distribution compare


327

with the lognormal price distribution? The answer will be particularly important in the case of distribution tails, as these will in turn determine the volatility strike structure. For example, if the tails of the simulated price distribution are “fatter” than the tails of the corresponding lognormal distribution, then the resulting Black-implied volatility will be greater than for the lognormal model, resulting in a smile for the volatility strike structure. And if the situation is opposite, the result would be a frown. As we simulate the spot and the long-term equilibrium price in the two-factor price mean-reverting model, we can generate the daily forward price curve at each time step and for each simulation path, and in the simplified case of setting all discount factors to 1.0, the resulting monthly forward price is given by Ft ,Ms;T ,T = 1

N

1 N

N

∑F n=1

(11-21)

t ,s ;Tn

where: Ft,s;Tn the daily forward price at time step t, simulation path s, for delivery at time Tn. Implementing the two-factor price mean-reverting model gives us further formulation for the monthly forward price: Ft ,Ms;T ,T = κ α ' e 1

κα ' ≡

1 N

κ µ' ≡

1 N

∆t ≡ where:

− α '( T0 −t )

N

N

∑e

St ,s + (κ µ ' e

− nα ' ∆t

n=1 N

∑ eµ ' ∆t = n=1

=

µ '( T0 −t )

− κ α 'e

− α '( T0 −t )

) Lt ,s ,

e−α ' ∆t (1 − e− Nα ' ∆t ) , N (1 − e−α ' ∆t )

e µ ' ∆t (1 − e N µ ' ∆t ) , N (1 − e µ ' ∆t )

(11-22)

TN − T0 N

t time step between observations (and also simulation steps) St,s simulated spot price at time step t and for simulation path s Lt,s simulated equilibrium price at time step t and simulation path s T0 time one step prior to the delivery of the first forward price in the monthly average.

328

Energy Risk

If we allow the monthly average to consist of 22 price observations (N 22), and we set equal to 35.0, assuming 1.0 (hence ⬇ 1.0), we obtain ⬇ 0.3. If, furthermore, we set the spot volatility to 350%, equilibrium price volatility to 50%, and the starting values for spot and equilibrium price to $30.00, we can go ahead and perform the simulations as a function of option expiration times, T0. First, let us allow the option expiration to occur a month out (T0 1/12). Figure 11-20 shows the growth of the probability distribution for the spot price, equilibrium price, and the average forward price through time (i.e., for each time step) as captured in Black-equivalent volatility terms. Note that the spot price—which we know has a volatility of 350% within the price mean-reverting model—has a quickly declining Black-equivalent volatility; if we allowed the simulations to continue, this spot price Black-equivalent volatility would continue converging towards the equilibrium price volatility level. If the market is indeed price meanreverting, then we should expect to see implied volatilities converge towards equilibrium price volatility over time due to the mean-reversion effects (this is why using spot price volatility levels in “value at risk” calculations across longer periods of time does not make sense.) Second, note that the equilibrium price Black-equivalent volatility remains constant at

F I G U R E

11-20

Monte Carlo Simulation of Spot, Equilibrium, and Monthly Forward Prices in a Two-Factor Price Mean-Reverting Model: Through Time


329

50%; this is as expected considering that we set the equilibrium price to follow a lognormal price process. But perhaps the most interesting is the Black-equivalent volatility path that the monthly forward price follows over time; it begins at the equilibrium price volatility levels—as you would expect under such strong mean reversion and with the option expiration 30 days out (which mean that the last discrete daily price in the average is initially two months out). However, as we get closer to expiration (day 30), we see the average forward price begin converging towards spot price volatility levels. It never actually gets close for two reasons: at day 30 we are still 30 days away from the last price delivery; second, the nonperfect correlations between the discrete daily prices further dampen the spot price volatility through diversification. Similarly, we can take a closer look at the simulated spot price distribution on the 30th day of simulations and analyze it in terms of Black-equivalent volatilities across price levels. Figure 11-21 shows the results. Notice the smile effect we see in the spot price Black-equivalent volatility after 30 days of simulations. The tail of the simulated distribution is particularly fat in the lower price range, but it also thickens out in the higher price ranges. This strike structure is reminiscent of the market-implied volatility strike structure shown in Figure 11-17.

F I G U R E

11-21

Monte Carlo Simulation Continued: Resulting Spot Price Distribution Analysis on Day 30

330

Energy Risk

F I G U R E

11-22

Monte Carlo Simulation Continued: Resulting Forward Price Distribution Analysis on Day 30

Performing the same analysis but for the monthly forward price, we obtain similar Black-equivalent volatilities across forward price levels, but with the tail on the high-price-level portion of the distribution fatter than for the lower prices. Although the out-of-money puts would show some serious thick tail effects in the case of options on spot (or daily delivery), the out-of-money calls on the monthly forwards would show the serious thick tail effects instead. We can perform the same type of simulation analysis but with only five days to go until option expiration. In this case, Figures 11-23 through 11-25 show the resulting Black-equivalent volatilities across time and price levels for the spot and the monthly forward price. There are some huge lessons to learn about the price mean-reverting process from the above simulations. First, the time over which spot prices are captured to provide you with a price distribution plays a huge role in terms of what that price distribution tells you. If the prices were captured over a long period of time, the spot prices will tell you to a large extent about equilibrium price behavior towards which they meanrevert. The shorter the time period of capture, the more we will learn about spot price short-term behavior and hence mean reversion. This is a beautiful result of a mean-reverting process: spot prices hold an incredible amount of information, we just need to know how to look for


F I G U R E

331

11-23

Monte Carlo Simulation of Spot, Equilibrium, and Monthly Forward Prices: Only Five Days Out

F I G U R E

11-24

Monte Carlo Simulation Continued: Resulting Spot Price Distribution Analysis on Day 5 Spot Black-Equivalent Vol (5-Day Expiration) 4.5 4

Volatility

3.5 3 2.5 2 1.5 1 0.5 20

25

30 Spot Price

35

40

332

Energy Risk

F I G U R E

11-25

Monte Carlo Simulation Continued: Resulting Forward Price Distribution Analysis on Day 5

it. Second, you should expect to see some serious strike structure in the market-implied Black volatilities for the monthly forward prices if in fact the market is well represented by a two-factor price mean-reverting process. And we do see some comparisons between the volatility behaviors across price levels in the simulations and the market-implied volatilities of the previous sections. What makes things very difficult in energies is that the reality is not just a two-factor process, but instead at least a four-factor process including the seasonal factors. So, no matter what model we use in valuation, we will absolutely have to calibrate market volatilities if we are indeed to value options mark to market. Still, the value of getting some insight into how the models reflect behaviors, and how market behaviors should influence our models, is critical to progress in valuing energy options.

11.3.4.

Extending Monthly Methodology to Calendar Year Options

Calendar year options are a further extension of the averaging methodology applied in valuing monthly settled European options of the previous section. They are truly swaptions: options settling into a swap. In


333

the case of energy markets, these are options settling into a calendar year forward, common in power markets, where the forward consists of a series of monthly forward prices requiring proper discounting and valuation taking into account all the delivery days and all the payments. All the averaging effects we have discussed in Section 11.3 so far are in effect with an additional magnitude when applied to calendar year options. These options tend to have fairly low Black-equivalent volatilities due to the fact that the option exercise time occurs well before the vast majority of the daily forward prices within the average have a chance to ride up that very steep volatility curve close to delivery time, and due to the averaging diversification effects between the numerous daily forward prices comprising this average. In valuing calendar year options, the two greatest challenges have to do with correlations and volatilities. First, we need to have a very good idea of the correlations between the daily (or monthly if we treat this as a monthly price average) prices comprising the calendar year forward over the time period prior to the option’s expiration time. And second, the volatility to be used for each of the underlying forward prices comprising the calendar year forward is not the same as the volatility used in the monthly settled or daily settled options for the corresponding forwards: the calendar year option expires well before most of the underlying forward prices reach their delivery, hence the appropriate volatility for the underlying forwards corresponds to those forward prices’ behavior from now until calendar year option expiration and not their delivery time. In other words, the volatility of the underlying forward prices will be potentially much lower than the volatility corresponding to their daily or monthly settlement.

11.4.

OPTIONALITY IN CHEAPEST-TO-DELIVER FORWARD PRICES The cheapest-to-deliver forward price contracts are typically seen in natural gas and electricity markets. Within the contract, the party that delivers the energy has the choice of delivering it at one of two delivery points. These forwards carry embedded optionality. The pricing of a cheapest-to-deliver forward can be done through a closed-form solution: Fctd = F1 (1 − N ( h+ctd )) + F2 ( N ( h−ctd ))

(11-23)

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Energy Risk

h+ctd/ − =

⎛F⎞ ⎛1 2 ⎞ ln ⎜ 1 ⎟ + / − ⎜ σ ctd τ⎟ ⎠ ⎝2 ⎝ F2 ⎠

(11-24)

σ ctd τ

2 σ ctd = σ 12 + σ 22 − 2σ 1σ 2 ρ

(11-25)

where: 1 volatility of F1

2 volatility of F2 Note that the price of the cheapest-to-deliver forward is sensitive to the correlation between the prices at the different points of delivery. Options that settle on such cheapest-to-deliver forward prices can also be derived.3 Energy producers carry unique and producer-specific optionality within their books. If they sell a forward contract for delivery of energy at the contract expiration, they have the choice of delivering their own energy—that is, energy that they produced—or simply buying energy in the market and then delivering it. Thus, the producer carries the cheapest-to-deliver optionality, where the valuation involves comparing the cost function to the market price.

11.5.

TYPES OF ENERGY SWING OPTIONS Energy swing options allow variable volume. There are two types of swing options depending both on the contract specifications but also on the participant’s physical constraints: price swing and demand swing options and forwards. In the case of price swing contracts, the participants can (and will) exercise the options whenever it is financially optimal to do so, regardless of their actual need for energy. These are American-style options, with the constraint that, on any day, only one of the swing options may be exercised. Clearly, in order for such an early exercise to be financially optimal, the participant must have a means of both receiving the energy and then turning around to sell it in the marketplace, or the options are cash-settled. Whenever it is optimal to early exercise a swing right, the holder of the option will exercise the full allowed amount, because if it is indeed optimal to exercise, then exercising as


335

much as possible is also optimal. Such swing options can be bundled within a base-load swap (long or short) and with no upfront premium; instead, the premium is embedded within the swap price. Such options can be frequently seen in both natural gas and power markets. In the case of natural gas, these options are also often forward starting; the strike is defined at some point in the future, just prior to the start of the swing period. In electricity these are the multiple-peaker or loadcurtailment options. By comparison, in the case of demand swing contracts the volume is allowed to change continuously as a function of user need for energy. The users are assumed not to have the physical means of turning around to sell the energy back into the marketplace; the user is capable of only taking delivery of energy. In fact, a demand-swing forward or option is continuously exercised as a function of need, and the maximum quantity is defined by physical constraints. Here are some of the specific inputs that swing options include:

• Swing Direction: the option holder has the right to take or make delivery of energy. If the option holder has the right to take, or “call,” for energy, then the option holder has a call swing option or a long swing forward position. Conversely, if the option holder has the right to sell energy, then the option holder has a swing put or short swing forward position. • Swing Rights: the contract defines the number of swing rights, or exercise opportunities. In the case of load-following contracts, these exercise opportunities are continuous. The exercise period defines when the option holder may exercise. • Quantity Limits: the maximum swing quantity must be defined for both the individual exercise as well as for the overall quantity. For demand swing contracts the quantity is defined by physical constraints. • Upfront or Embedded Premium: the swing premiums may be charged up front (with a predefined payment date) or may be embedded within a base-load swap price. A standard base-load swap can be accompanied by a purchase or sale of a swing option, resulting in a forward price greater or smaller than for a forward without swing rights. • Fixed vs. Floating Strike: the fixed strike contracts are exactly that—they define the strike price today. By comparison, the floating strike contracts allow for the strike to be defined at

336

Energy Risk

some future point in time by market prices, and therefore these may be considered forward starting options. Power markets appear to prefer fixed-strike contracts, and natural gas markets tend to prefer the floating strike contracts. • Delivery: the delivery in the case of exercise of a swing right might be for the next day, or it might be for the remainder of the term of the option. The following two sections will go into details of valuing both price and demand swing contracts.

11.6.

DEMAND SWING CONTRACTS Demand swing call options and forwards in power and natural gas markets are held by the user sector, where the option exercise is primarily a function of the need for energy; the quantity used reflects—in simplest terms—the user’s give and take between the level of need for energy and the dislike for the cost of energy. In the following analysis of such contracts, the user is assumed to have only the ability to draw energy and does not have the ability to turn around and sell it in the marketplace. If the user had the ability to turn around and sell the energy to the market, the user would always exercise the maximum possible quantity (given a deep enough marketplace) whenever it is optimal to exercise. For such demandbased contracts, the volatility of the quantity of energy drawn and the correlation of this quantity with the price level of energy constitute key inputs to the valuation. Furthermore, these quantities are specific to the type of user sector in question (for example, industrial vs. residential), if not to the actual user.

11.6.1.

Demand Swing Options

Generally speaking, the users hold demand swing forward contracts with their energy providers in the case of fixed-price, flow upon demand contracts. However, the moment a user has access to more than one provider, and the user is savvy enough to shop around for the best deal, whenever the market price drops below the fixed contract price the user


337

might go to another provider for a better deal. The moment this happens the contract effectively becomes a demand swing call, where the user only draws energy when it is financially beneficial to do so. But keep in mind that this can happen only when three conditions are satisfied: 1. The user has access to more than one service provider. 2. The user is savvy enough to shop for best market price. 3. The available service providers are market efficient and competitive. In order to value the demand swing call option, let us first define the forward price, F, and the quantity, Q, both at the time of option expiration, To, and for delivery at time T (as observed at current time t) as lognormal: − σ ( T −t )+σ F zt ,T o FT ,T = Ft ,T e F o 2

o

F

t

− σ Q ( To −t )+σ Q zt ,T o Q T ,T = Qt ,T e 2

o

Q

(11-26)

t

Given that the quantity and the forward price exhibit a correlation ,4 we can establish the following relationship for their respective stochastic variables: ztF,T = ρ ztQ,T + 1 − ρ 2 εt ,T o

o

(11-27)

o

where E ⎡ ztQ,T ⎤ ⎣ o ⎦t E ⎡( ztQ,T )2 ⎤ ⎣ o ⎦t E ⎡ ztQ,T ztF,T ⎤ ⎣ o o ⎦t E ⎡εt ,T ztF,T ⎤ ⎣ o o ⎦t

= E ⎡ ztF,T ⎤ = E ⎡εt ,T ⎤ = 0 ⎣ o ⎦t ⎣ o ⎦t = E ⎡( ztF,T )2 ⎤ = E ⎡(εt ,T )2 ⎤ = (To − t ) ⎣ o ⎦t ⎣ o ⎦t = ρ (To − t ) =0

The demand swing call option then has the following payoff at expiration time To, as observed at time t: CtDS = EzQ ⎡⎢ Ez F ⎡Q T ,T max(0, FT ,T − K ) ⎤ ⎤⎥ dft ,T = o o ⎣⎢ o t ⎦⎥t ⎦t t ⎣ 1 1 ⎡ ⎡ ⎤⎤ − σ Q2 ( To −t )+σ Q ztQ,T − σ F2 ( To −t )+σ F ztF,T o o EzQ ⎢ Ez F ⎢Qt ,T e 2 max(0, Ft ,T e 2 − K ) ⎥ ⎥ dft ,T o ⎢⎣ ⎣ ⎦t ⎥⎦t

(11-28)

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Energy Risk

Because the quantity exhibits a correlation to the forward price, F, we then have

CtDS

1 ⎡ ⎡ − σ Q2 ( To −t )+σ Q ztQ,T o ⎢ ⎢Qt ,T e 2 ⎢ ⎢ = EzQ ⎢ Eε ⎢ ⎛ 0,, ⎢ ⎢ max ⎜ 1 1 − ρ 2σ 2 ( T −t )+ ρσ F ztQ,T − (1− ρ 2 )σ F2 ( To −t )+ o 2 ⎢ ⎢ ⎜⎝ ( F e 2 F o ) e t ,T ⎣ ⎣

1− ρ 2 σ F εt ,T

o

⎤⎤ ⎥⎥ ⎥⎥ ⎞ ⎥ ⎥ dft ,To ⎟⎥ ⎥ − K ⎟⎠ ⎥ ⎥ ⎦t ⎦ t

(11-29) The expectation value for the stochastic variable corresponds to the problem definition for the Black option model, so we can reduce the above problem to the following expected value: 1 ⎡ ⎤ − σ Q2 ( To − t ) +σ Q ztQ,T o CtDS = EzQ ⎢Qt ,T e 2 CtBLACK ( FBLACK ( ztQ,T ), K , σ BLACK ) ⎥ o ⎣ ⎦t 1 2 ⎡ ⎤ − σ Q ( To − t ) +σ Q ztQ,T o ( FBLACK ( ztQ,T ) N ( h+ ( ztQ,T )) − K N ( h− ( ztQ,T )) ⎥ dft ,T = EzQ ⎢Qt ,T e 2 o o o o ⎣ ⎦t

(11-30) where FBLACK ( ztQ,T ) = Ft ,T e

1 − ρ 2σ F2 ( To − t ) + ρσ F ztQ,T o 2

,

o

σ BLACK = σ F 1 − ρ 2 ,

h+ / − ( ztQ,T ) =

1 ⎛ − ρ 2σ F2 ( To − t ) + ρσ F ztQ,T ⎞ o 2 F e ⎟ ± 1 σ 2 (1 − ρ 2 )(T − t ) ln ⎜⎜ t ,T o ⎟ 2 F K ⎟⎠ ⎜⎝

σ F (1 − ρ 2 )(To − t )

o

N ( x) =

∫

x

−∞

e

−

y2 2

2π

dy

(11-31) ,


339

You might get worried looking at the above remaining problem—but we are in luck: we can apply a useful property for N(x):

∫

+∞

e

−∞

−

x2 2

1 2 ⎛ A + BC ⎞ C eCx N ( A + Bx )dx = e 2 N ⎜ ⎟ ⎝ 1 + B2 ⎠ 2π

(11-32)

to obtain the following closed-form solution for the demand swing call option: CtDS = Qt ,T ( Ft ,T N ( h DS ) − K N ( h DS ) dft ,T +

−

o

(11-33)

where

h+DS/ − =

⎛F ⎞ 1 ln ⎜ t ,T ⎟ ± σ F2 (To − t ) + ρσ F σ Q (To − t ) ⎝ K ⎠ 2

σ F (To − t )

(11-34)

We truly simplified the problem of the draw-as-you-want contract valuation on many different levels. On the contract level we assumed that there were no maximum bounds put on the quantity. Additionally, we assumed that the payment occurs at the same time as the delivery— a very simple adjustment needs to be made to the above discounting treatment to correct for this. Finally, we assumed that the quantity and the forward price were both lognormal. The above derivation becomes quite a bit more strenuous when we start implementing mean-reverting multifactor models in place of lognormal models. Regardless of the ultimate model used, the above demand swing call valuation must use market inputs for volatility matrices and forward prices. Similarly, the demand quantity must be analyzed on a per user sector basis and corresponding volatility and term structures used as inputs. Section 11.6.3. provides some examples of quantity term structures.

11.6.2.

Demand Swing Forwards

Demand swing forward contracts assume that the user will use energy on an as-needed basis, and will not be shopping around for a better price

340

Energy Risk

than the contract price. The problem of valuation is identical to that of the demand swing call options except that the user is always exercising the option regardless of price and primarily based on demand (of course, the user might adjust the demand quantity based on the level of price, such as in the case of the residential sector reducing the usage of electricity or gas in times of high price levels). The demand swing forward price under the simplifying assumptions of lognormal quantity and market forward price as given by Equation 11-26 is provided in Equation 11-35: F tDS = Ft,T e ,T

ρσ F σ Q ( T −t )

(11-35)

The market forward price already captures the overall market demand effects on the price levels. However, each user will have a quantity demand term structure and variability unique to them, and therefore the relationship of the market forward price to their particular demand behavior should determine their particular demand swing forward price position.

11.6.3.

Load Behavior

There are four general user client sectors: wholesale clients, residential users, industrial users, and commercial users. Each user sector load will exhibit behavior specific to that user group, and the same user groups across regions will exhibit similar behaviors with varying degrees of magnitude depending on market factors specific to the geographic region and service area. (These user groups can be broken down even further into smaller client groups, based on typical load size or based on even more detailed type of use.) For example, the residential sector need for power and gas will be driven by the need for cooling and heating, and hence will be very much driven by weather—so we can expect to see seasonal behavior in the residential load. The industrial sector, on the other hand, we would expect to be far more stable and generally independent of weather (of course, this is also a function of the type of industry inhabiting a particular geographic region). And similarly, the commercial sector might be linked to economic cycles and might have its own unique demand term structure and volatility. Figure 11-26 shows a comparison of sample wholesale, residential, commercial, and industrial load across a calendar year.


F I G U R E

341

11-26

Sample Load Through Time

The load values can be modeled just like spot prices. They exhibit mean reversion, seasonality, even mean-reverting volatility behavior. In addition to forward-looking load curves for valuing some of the loadfollowing contracts, we would also need to have both load volatility and correlation to price. Given a sample correlation set (see Table 11-2), and given a sample forward load curve for residential load and its volatility term structure (see Figures 11-27 and 11-28), we can value loadfollowing contracts using the above methodology. A newcomer to the power markets or a simple user might wonder why the residential pricing is different from the OTC quoted forward prices. The following exercise will show the difference in value between the fixed quantity forward and a load-following position. To see the additional value load-following adds to a simple forward, let us begin by valuing a PJM Calendar 2007 Forward for a quantity of a single MWh. Figure 11-29 shows the price for such a standard contract. Using the PJM forward price curve built in October 2006 (see Figure 11-2), we obtain a 2007 Calendar Forward price of $72.90. Now let us see what happens when we keep the fixed quantity but add the demand swing forward price features of flat quantity volatility of 45% and correlation to price of 25%. Now, revaluing the demand swing forward, we obtain a higher price of $78.14, reflecting the additional value

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Energy Risk

T A B L E

11-2

Sample Load to Price Correlations

Wholesale Residential Industrial Commercial

Cinergy

Entergy

PJM

8.07% 6.99% 5.15% 3.33%

13.41% 15.63% 3.29% 5.81%

19.40% 21.34% 2.67% 9.07%

of the positive correlation between the price and the load; when the load goes up, so does the price (Figure 11-30). Finally, let us use the forward load curve, load volatility term structure, and its correlation to the PJM market—all shown above—to value this Calendar 2007 Residential contract. Figure 11-31 shows the results. Note how the residential seasonal behavior has significantly raised the resulting demand swing forward price to $82.44. Figure 11-31 also shows this valuation performed assuming that the producer sold the contract at the standard fixed quantity forward price of $72.90. As you can see from the resulting valuation, the producer would have ended up

F I G U R E

11-27

Sample Forward Load Curve (PJM Residential Load, 4 October 2006)


F I G U R E

343

11-28

Sample Daily Settle Load Volatilities

F I G U R E

11-29

Valuing a Standard PJM Calendar Forward

F I G U R E

11-30

Valuing a Volatile Quantity with Flat Expected Value of 1300 MW

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Energy Risk

F I G U R E

11-31

Incorporating Residential Load Behavior into the Forward Price Valuation

with some serious loss had he not incorporated the added value of load following into the contract valuation! Also note how large the delta risk is for a single dollar move up of the PJM forward price curve. Some of the necessary details in valuing such a contract are shown in Figure 11-32. Note that we valued this contract in daily settled terms. The reality is that these are continuously (or hourly) settled contracts. If you consider the level of information necessary to value such a contract already to be extensive, then imagine at least 24 times that amount for valuing hourly settled contracts. Figure 11-33 shows us the resulting value (or rather, loss, in this case) and market price risks across time. Clearly, the value that power suppliers provide through load following can be quite large—in this simple example it is in the order of billions for a period over a year— and that is assuming that there was no hourly quantity volatility (that


F I G U R E

345

11-32

A Snapshot of Pricing Details for a Demand Swing Forward

is, the quantity was allowed to change on a daily basis rather than even more discretely on an hourly basis). We would expect an even greater value add for an hourly load-following contract. Because the quantity is now allowed to be volatile, we now have new risks to deal with. Figure 11-34 shows the contract sensitivity to volumetric changes per one MWh moves, across time.

11.7.

PRICE SWING CONTRACTS Price swing contracts, when exercised optimally, are contracts where the holder of the option is continuously maximizing the value of the contract position. Price swing options are co-dependent American-style options. F I G U R E

11-33

Demand Swing Forward Market Value and Price Risks Across Time

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Energy Risk

F I G U R E

11-34

Demand Swing Forward Volumetric Risks Across Time

11.7.1.

Multiple-Peaker Swing Options

Generally, for a multiple-peaker on an on-peak power market, each swing corresponds to a unique day in the allowed swing period; in other words, on any day during the swing period, only one of the options may be exercised. (Of course, in power, the contract might be hourly rather than daily.) Expressed in a per MWh price, the value of these options will be somewhere between a pure American-style option (with one swing right)—as the most expensive—and the daily European option with possible exercise every single business day in the period—as the cheapest. So, the end condition for the swing option per unit value should be daily

CN

D

swing

CN

S

american

C1

(11-36)

where the option’s number of exercise rights and exercise periods are defined by: 1 NS ND T american 僆 (T1,TND) swing Tnswing 僆 (T1,TN ); Tnswing Tm(m n); n,m 1 . . . NS D daily T n ; n 1 . . . ND


347

When the number of swing rights drops to one, we are left with a simple American-style option. On the other hand, when the number of swing rights expands to allow for exercise every single day of the swing period, we are dealing with a simple daily settled European option covering the entire swing period. Quite often, the swing rights are purchased or sold bundled up within a base-load swap. A user might agree to a base-load contract, but give the producer a certain fixed number of call-back rights for some maximum quantity during the period of the base-load contract (this is the load curtailment contract). Similarly, a user might agree to a baseload contract but ask for an additional fixed number of swing rights; the additional load has a maximum possible quantity and the user has a fixed number of times to call this additional load over the base-load contract period. In both cases the swap price is adjusted by the swing premium. However, the swing premium will be added or subtracted depending on whether the swing rights are being sold or purchased. The swap price in Equation 7-6 is therefore adjusted for the swing premium as follows:

∑ ∑ (q M

FTswing ,T 1,1

M ,N M

= t

Nm

m=1 n=1

m ,n

Ft ,T dft ,T m ,n

m ,n

) + − SwingPremium

⎛ Nm ⎞ ∑ ⎜ ∑ qm,n ⎟ dft ,Tmp ⎠ m=1 ⎝ n=1 M

(11-37)

In the case of the load curtailment contract, the swing premium is subtracted, because the user is giving up the certainty of the continuous base load, thus bringing down the swap price for the resulting baseload contract. And similarly, in the case of the user wanting additional load beyond the base load for a certain number of days of user’s choosing during the base-load period, the swing premium is added; the user should be willing to pay a swap price greater than he would were this a contract for just the base load. To see some sample values of such swing contracts, let us begin with a simple JAN–FEB forward. Its value without any embedded swing rights is shown in Figure 11-35. Now let us value a 25 MW swing option with five swing call rights, for exercise on any business day (as the example is based on the 5 16 PJM market) during the January through February period of 2007 at a strike price of $150. The unit price is given at $2.26 (Figure 11-36).

348

Energy Risk

F I G U R E

11-35

Example of a 25 MW Base-Load Swap with No Swing Rights

F I G U R E

11-36

Example of a 25 MW Swing Call with Five Swing Rights, Each for 25 MW at a Fee of $150/MWh


F I G U R E

349

11-37

Example of a 25 MW Base-Load Swap Allowing Five Swing Call Backs for Full Load at an Additional Fee of $150/MWh: Valued at Fair Price

Finally, if we allow the JAN–FEB forward contract to have five days worth of full quantity call backs—but at a fee of $150/MWh on the called-back quantity, we would expect this swap price to go down—and it does (as you can see in Figure 11-37); the resulting forward with swing call back rights (at a strike/fee of $150) is now $75.55. If instead there is no additional fee in the case of call-backs, the contract price goes down far more, as this becomes a base-load swap with five swing forwards (that is, the strike price goes to zero), and we are left with a price of $66.25/MWh (which is to be applied towards the full amount of the base-load swap to determine the price the receiver of energy pays) (see Figure 11-38). If we were to purchase this contract but at the same forward price as if there were no swing call-backs, we would stand to lose value, as can be seen in Figure 11-39. Valuing the swing premium will consist of understanding the codependent option tree methodology5. In the simple problem of allowing for only two swing rights, we need to consider the values of these codependent American options within the same tree, such that we exercise one or the other in a manner that maximizes the overall value of our portfolio (where the portfolio consists of the two options). Specifically, at the very last time step in the tree we will still have only one exercise right. Therefore, at that last time step, TN, one of the two

350

Energy Risk

F I G U R E

11-38

Example of a Base-Load Swap Allowing Five Swing Call-Backs for Full Load at No Fee (Load Curtailment at No Price): Valued at Fair Price

options will have a value of zero (because it can no longer be exercised) and the other will have the exercise value: C 1N

T

,m

= max(0,( FN

C N2

T

,m

=0

T

,m

− K )) (11-38)

m = 0… N T F I G U R E

11-39

Example of a Base-Load Swap Allowing Five Swing Call-Backs for Full Load at No Fee (Load Curtailment at No Price): Valued at Base-Load Swap Price with No Load Curtailment (User’s Contract Value)


351

At the second to last time step in the tree, we have two options and two exercise possibilities—at the second to last time step (TN 1), and at the last time step (TN). Therefore, we can use the second to last time step to exercise the second option and save the other one for the last time step: C N1 C

T −1,m

2 N T −1,m

= ( pC N1

T

,m+1

+ (1 − p )C N1

= max(0,( F( N

T −1),m

T

,m

) dfT

N −1 ,TN

− K ))

(11-39)

m = 0… ( N T − 1) Now we step back through the tree. At each node we have to consider the value of exercising one of the options. If in fact the exercise value is greater than one of the options, then we should exercise the option that has holding value less than the exercise value. If the exercise value is greater than both of the option values at that node, then we would want to exercise the one with lesser value in order to end up with the highest value for our portfolio: Cn1−1,m = ( pCn1,m+1 + (1 − p )Cn1,m ) dfT

n −1 ,Tn

C

(

min C

1 n−1,m

,C

2 n−1,m

2 n−1,m

= ( pC

2 n ,m+1

+ (1 − p )C

) = max (( F

( n−1),m

2 n ,m

) dfT

n −1 ,Tn

− K ),min(C

1 n−1,m

,C

2 n−1,m

)

)

(11-40)

n = 1… ( N T − 1); m = 0… ( n − 1) (A note about discounting: for the sake of simplicity we will assume that the payment for the strike occurs on the same day as the delivery of the commodity, and that the delivery occurs immediately after exercise. The discounting will have to be appropriately adjusted when the payment, the delivery, and the exercise all occur on different days.) As an example of valuation for such a simple swing contract with only two swing rights, consider the following: assume the discount factors to equal 1.0, the volatility to be flat at 50%, and the forward prices to be flat at $50. Let us value a call swing option with two swing rights with allowed exercise over the next month (TN 1/12) using only a fourstep binomial tree. (Clearly, none of this is realistic, but carries educational value we can use to build on in order to create realistic valuation.) We can then construct a tree that will look as shown in Figure 11-40. The resulting cumulative probability tree is then given by Figure 11-41. We can go ahead and implement Equations 11-38 through 11-40 in order to value this swing call contract. As you can see from Figure 11-42, we must carry two swing option values throughout the tree, always maximizing the portfolio value where the portfolio consists of the sum of the

352

Energy Risk

F I G U R E

11-40

Forward Tree

F I G U R E

11-41

Cumulative Probability Tree


F I G U R E

353

11-42

Swing Call Tree

two swing options. In this case there is no reason to exercise these options any sooner than we have to. So, in the case of a flat forward price curve we would hold onto these calls until the last two time steps, where we finally exercise the two options, one per time step. Now let us consider a much more interesting example where we allow for forward price term structure. We are now replicating reality much better, as forward price term structure can be quite colorful in the seasonal power and natural gas markets. Figure 11-43 shows the resulting forward price term structure where we allowed the forward prices at the five time nodes in the tree to be $50, $57, $60, $55, and $50, thus mimicking a seasonal behavior. And now the swing contract valuation is also far more interesting (Figure 11-44). Note that there is now quite a bit of early exercise within the tree due to the new forward price term structure. This is really a far in the money call swing option

354

Energy Risk

F I G U R E

11-43

Forward Tree

contract where the most likely exercise will occur at the first and second steps in the tree (roughly corresponding to the end of the first and second weeks of the month, because we assumed a one-month expiration), capturing the highest forward price scenarios as given by the forward price seasonality. For the general case where we are allowed a certain number of swing rights, NS, we can generalize the above approach as given by Equations 11-38 through 11-40. In this general case, for the last time step in the tree we have, just like in the previous simple case of two swing rights, the exercise being possible for only one of the swing options, C 1N

T

,m

= max(0,( FN

C Ns

T

,m

=0

T

,m

− K ))

m = 0… N T ; s = 2… N S

(11-41)


F I G U R E

355

11-44

Swing Call Tree

and for the remaining NS swing options we populate the values of the remaining swing options, while carrying back the ones already populated. So, for s 2. . . NS we have Cnk−1,m = ( pCnk,m+1 + (1 − p )Cnk,m ) dfT

n −1 ,Tn

C

s n−1,m

= max(0,( Fn−1,m − K ))

k = 1… ( s − 1)

(11-42)

n = N T − s + 2; m = 0…(( n − 1) Finally, for the rest of the tree we must continually compare early exercising one of the options in the case that the early exercise value exceeds the value of holding on to any of the swing options without

356

Energy Risk

exercising at that node in the tree. At each node (n 1, m) this is a twostep process. First, we calculate the holding value of each swing option: Cns−1,m = ( pCns,m+1 + (1 − p )Cns,m ) dfT

n −1 ,Tn

(11-43)

s = 1… N S

Then, we compare the early exercise value at that node to see if it is greater than the value of any of the swing option holding values at that node. If indeed there are one or more swing option holding values that are less than the exercise value at that node, we replace the least valuable option with the early exercise value, thus maximizing the overall contract value at that node: min(C 1n 1, m, C 2n 1,m, . . . , C NnS1,m) max((F(n 1),m K ), min(C 1n1, m, C 2n1, m, . . . , C NS

n 1, m

))

(11-44)

For the case of a swing option where the exercise can occur only during a particular portion of the tree, for those time steps where the exercise is not allowed the call values simply need to be carried backwards through the tree without the option of exercise. In fact, this is where having cumulative probabilities for the time step of the first possible early exercise carries huge computational value—particularly if valuing hourly swing options! Because there is no early exercise from present to the first possible early exercise time step, the swing option values at that first possible early exercise time step can be simply “carried” back through time using cumulative probabilities, saving us huge amounts of calculation time were we to, instead, step backwards through the tree at each time step. Finally, the quoted price is generally expressed on a per energy unit basis. Only once we have all the swing option values brought back to the very first node in the tree (T 0) can we calculate the per unit price. You should expect to find that the greater the number of swing rights the smaller is the per unit price, but the greater is the premium, and vice versa. Finally, allowing for maximum and minimum overall quantities has the effect of creating structured products. For example, when buying a swing call contract, a positive minimum quantity forces the exercise of some of these calls, thus transforming the contract from a pure swing call contract to a combination of swing calls and forwards. If


357

instead you are buying a swing call contract but with a maximum quantity less than the overall possible quantity given the number of calls, you in fact have a contract with a reduced number of calls actually purchased. Finally, you could have a straddle swing contract, meaning that the quantity can swing either way. Such contracts can provide the means for valuing storage. An example of a straddle swing contract with only two swing rights for a single MWh (for simplicity) and maximum and minimum overall quantities of zero (hence one of the swing rights has to be a call and the other a put) is shown in Figure 11-45. The resulting values and risks are shown across time buckets in Figure 11-46. In fact, in this example, the price swing contract reduces to a floating time spread due to the overall maximum and minimum quantity constraints. An example of a straddle swing contract with four swing rights for a single MWh (for simplicity) and maximum and minimum quantities of 32 MW (hence two of the swing call rights must be exercised) and the resulting possible exercise dates and risks are shown in Figures 11-47 through 11-49.

F I G U R E

11-45

Example of a Price Swing Forward with Two Swing Forwards (Buy or Sell) but Max/Min Quantity: Each Swing for Total of 16 MWh (as it is a 5 16 market), Required Position at End of Swing Period Net Sum Zero

358

Energy Risk

F I G U R E

11-46

Example of a Price Swing Forward with Two Swing Forwards (Buy or Sell) but Max/Min Quantity: Overall Position Across Time Buckets

11.7.2.

Forward Starting Swing

A forward starting swing contract is typically seen in natural gas markets, where the strike is set at some future point in time. Commonly the strike is set to the monthly forward price covering the swing period. The difficulty in valuing these types of contracts comes down to understanding the discrete daily volatility at the forward point in time.

F I G U R E

11-47

Example of a Price Swing Forward with Four Swing Forwards (Buy or Sell) but Max/Min Quantity: Each Swing for Total of 16 MWh, Required Position at End of Swing Period Purchase 32 MWh


F I G U R E

359

11-48

Example of a Price Swing Forward with Four Swing Forwards but Max/Min Quantity Continued: The Four Swing Details

Forward prices will ride up the volatility curve as they near their expiration or delivery date, increasingly mimicking the spot price behavior. Therefore the volatility to be used in such a forward starting swing contract must be one that recognizes this “close to expiration” behavior. At the same time, we need to be aware of the passage of time, because we are not interested in the short-term volatility observed right now, but rather at the point in time when the strike price will be set. F I G U R E

11-49

Example of a Price Swing Forward with Four Swing Forwards but Max/Min Quantity Continued: Overall Position Across Time Buckets

360

Energy Risk

11.7.3.

Natural Gas Storage

One way of thinking about the natural gas storage problem is to equate it to a floating time-spread problem. However, it is in fact more appropriate to treat the natural gas storage valuation problem as a price swing straddle, with the number of exercise rights corresponding to the number of times gas can be pumped in or out during a particular period, with the strike price set to the cost of pumping in/out, and with the daily, maximum, and minimum quantities being defined both by physical constraints and the storage rental contract. By keeping it this general we can capture the full spectrum of possible optimal “exercise” scenarios, where each exercise corresponds to pumping in or withdrawing gas. Two simplistic examples of such a valuation are shown in Figures 11-50 and 11-51. F I G U R E

11-50

Example of Simplistic Natural Gas Storage


F I G U R E

361

11-51

Future Example of Simplistic Natural Gas Storage

Clearly, this is not an approach incorporating all the real physical issues involved in valuing natural gas storage. For example, the maximum quantity that can be pumped in may not be well established. In fact, if you pump too much into storage you could burst the “bubble” and lose it all!

11.8.

SPREAD OPTIONS Energy markets are filled with both traded and synthetic asset-based cross-commodity and cross-market spread options. Crack spread options take advantage of the natural relationships between energies in the fuel production process. Basis spread options, on the other hand, are common in natural gas markets, where so many of the delivery points trade as basis to the Henry Hub natural gas. And finally, asset valuation, whether it is power plants or transmission lines, is composed of synthetic spread options under many simplifying assumptions, of course. These are generally European-style options, where the settlement terms may vary (daily, monthly, and so on).

362

Energy Risk

11.8.1.

Various Approximations to Spread Option Valuation

Let us begin by looking at the spread options in the most general sense first. Allow the following definitions: FTB,T = FTL,T − kFTS ,T o

t

o

t

o

t

FTL,T = Ft ,LT e

− σ L2 ( To −t )+σ L ztL,T

FTS ,T = Ft ,ST e

− σ S2 ( To −t )+σ S ztS,T o

o

o

t

o

(11-45)

t

where the basis spread forward value at the time of option expiration B To, conditional on value at time t, is F˜ To,T|t, the respective short and long markets for delivery at time T are F˜TL ,T|t and F˜ TS ,T|t, and o o the relative weight (quantity) of the short market to long market is defined by k (this gives us the most flexibility for valuing different types of spread options in energy markets). The spread call option value with a strike of K is then given by: CtSPREAD = Ez S ⎢⎡ Ez L ⎡ max(0, FTB,T − K ) ⎤ ⎤⎥ dft ,T o o ⎣⎢ ⎦⎥t ⎦t t ⎣ 1 1 ⎡ ⎤⎤ ⎡ − σ L2 ( To −t )+σ L ztL,T − σ S2 ( To −t )+σ S ztS,T o o − kFt ,ST e 2 − K ) ⎥ ⎥ dft ,T = Ez S ⎢ Ez L ⎢ max(0, Ft ,LT e 2 o ⎢⎣ ⎦t ⎥⎦t ⎣

(11-46)

If we allow for a correlation, , between the long and the short market commodity, then we have the following relationship between the stochastic, normally distributed variables of the individual markets: ztL,T = ρ ztS,T + 1 − ρ 2 εt ,T

(11-47)

= E ⎡ ztS,T ⎤ = E ⎡εt ,T ⎤ = 0 ⎣ o ⎦t ⎣ o ⎦t 2 S = E ⎡( zt ,T ) ⎤ = E ⎡(εt ,T )2 ⎤ = (To − t ) ⎣ o ⎦t ⎣ o ⎦t

(11-48)

o

o

o

where E ⎡ ztL,T ⎤ ⎣ o ⎦t E ⎡( ztL,T )2 ⎤ ⎣ o ⎦t E ⎡ ztL,T ztS,T ⎤ ⎣ o o ⎦t E ⎡εt ,T ztS,T ⎤ ⎣ o o ⎦t

= ρ (To − t ) =0


363

Now we can use these relationships to further evaluate the call value of the spread:

CtSPREAD = dft ,T

o

⎡ ⎡ ⎛ 0, ⎢ ⎢ ⎜ 1 1 ⎢ ⎢ − ρ 2σ L2 ( To −t )+ ρσ L ztS,T − (1− ρ 2 )σ L2 ( To −t )+ ⎜ o Ez S ⎢ Eε ⎢ max ⎜ ( Ft ,LT e 2 )e 2 ⎢ ⎢ 1 ⎜ − σ 2 ( T −t )+σ S ztS,T ⎢ ⎢ o ⎜⎝ − ( kF S e 2 S o + K) t ,T ⎢⎣ ⎣

⎞⎤ ⎤ ⎟⎥ ⎥ 1− ρ 2 σ Lεt ,T ⎟ ⎥ ⎥ o ⎟⎥ ⎥ ⎟⎥ ⎥ ⎟⎠ ⎥ ⎥ ⎦t ⎥⎦t (11-49)

The expectation value for the stochastic variable corresponds to the problem definition for the Black option model, so we can reduce the above problem to the following expected value: CtSPREAD = Ez S ⎡CtBLACK ( FBLACK ( ztS,T ), K BLACK ( ztS,T ), σ BLACK ) ⎤ o o ⎣ ⎦t (11-50) S S S = Ez S ⎡ FBLACK ( zt ,T ) N ( h+ ( zt ,T )) − K BLACK ( zt ,T ) N ( h− ( ztS,T )) ⎤ dft ,T o o o o o ⎣ ⎦t where FBLACK ( ztS,T ) = Ft ,LT e

1 − ρ 2σ L2 ( To −t )+ ρσ L ztS,T o 2

,

o

K BLACK ( ztS,T ) = kFt ,ST e

1 − σ S2 ( To −t )+σ S ztS,T o 2

o

+ K,

σ BLACK = σ L 1 − ρ 2 ,

h+ / − ( ztS,T ) =

1 ⎛ ⎞ − ρ 2σ L2 ( To −t )+ ρσ L ztS,T L o 2 F e ⎜ ⎟ ± 1 σ 2 (1 − ρ 2 )(T − t ) t ,T ln ⎜ o 1 2 ⎟ 2 L S − σ ( T −t )+σ S zt ,T o ⎜⎝ kF S e 2 S o ⎟⎠ + K t ,T

σ L (1 − ρ 2 )(To − t )

o

Ν ( x) = ∫

(11-51)

x −∞

e

−

y2 2

2π

dy

Clearly, this remains a complicated problem. The above problem simplifies enormously in the special case when the strike is zero. With the strike set to zero we can employ a useful property for N(x)

364

Energy Risk

provided by Equation 11-32 above, to obtain the following closed-form solution for the zero-strike call option on the spread based on Equation 11-50, CtSPREAD

K =0

=0 K =0 K =0 K =0 = CBLACK ( FBLACK , K BLACK , σ BKLACK ) K =0 K =0 = ( FBLACK N ( h+K =0 ) − K BLACK N ( h−K =0 )) dft ,T

(11-52)

o

where, only in this special case, K =0 FBLACK = Ft ,LT , K =0 K BLACK = kFt ,ST , K =0 σ BLACK = σ L2 − 2 ρσ Lσ S + σ S2 ,

h+K/ −=0 =

(11-53)

⎛ F ⎞ 1 K =0 2 ln ⎜ ⎟ ± (σ BLACK ) (To − t ) ⎝ kF ⎠ 2 L t ,T S t ,T

K =0 σ BLACK (To − t )

The problem, of course, is that if we were to trade options on spreads, then we must have a methodology allowing the strike price to be nonzero (positive or negative!). A nonzero strike price forces us to re-examine Equation 11-50. The good news here is that this is a European option, and the “only” remaining step is to integrate the Black call values—as a function of the stochastic variable corresponding to the short market commodity—across all of its values with appropriate probabilities: SPREAD t

C

=

∫

+∞ −∞

dz 2π (To − t )

−

e

z2 2 ( To − t )

⎡⎣ FBLACK ( z ) N ( h+ ( z )) − K BLACK ( z ) N ( h− ( z )) ⎤⎦ dft ,T t o

(11-54) In the absence of the closed-form solution, we will consider several possible routes from here on: this integral can be calculated numerically, this integral can be approximated using the Edgeworth series expansion discussed in Chapter 10, we can apply a Taylor series expansion on the natural log term in h / expression assuming small expiration time, or we can apply an approximation based on the Margrabe formula.5


365

11.8.1.1. Numerical Approximation In the case of numerical integration, we obviously need to define the “infinity” values, such that they are large enough to allow for N(h) to be approximately zero, but small enough to make the numerical computation economical in terms of time. Given these boundaries for the variable z, we can perform the approximating summation based on Equation 11-54: SPREAD t

C

≈

∆z =

z =+ B

∑

z =− B

∆z

−

2π (To − t )

e

z2 2 ( To − t )

⎡⎣ FBLACK ( z ) N ( h+ ( z )) − K BLACK ( z ) N ( h− ( z )) ⎤⎦dft ,T , o

2B N

(11-55)

The problem with the numerical approximation is that the natural log term in the h / expression in Equation 11-55 grows extremely large in the case of negative strike values and values of z such that 1 − σ S2 ( To −t )+σ S ztS,T

S o + K ) gets close to zero. This makes numerithe term ( kFt ,T e 2 cal approximation potentially problematic.

11.8.1.2. Edgeworth Series Approximation In the case of Edgeworth series expansion, we can calculate an approximating volatility replacing the behavior of the short market term with the behavior of the short market term plus the strike: ( kFt ,ST + K )e

1 2 − σ approx ( To −t )+σ approx ztS,T o 2

≈ kFt ,ST e

1 − σ S2 ( To −t )+σ S ztS,T o 2

+K

(11-56)

which gives us the approximating volatility:

σ approx

⎧ ⎛ ( kF S )2 eσ S2 ( To −t ) + 2 kF S K + K 2 ⎞ ⎫ ⎪ ⎪ t ,T t ,T ⎟⎬ = ⎨ln ⎜ 2 S ⎟⎠ ⎪ ( kFt ,T + K ) (To − t ) ⎪ ⎜⎝ ⎭ ⎩ 1

0.5

(11-57)

Now, applying once again the useful derivation of the integral of N(x) (Equation 11-32), we obtain an approximating solution for the call option on the spread: CtSPREAD

K =0

E E E E ≈ CBLACK ( FBLACK , K BLACK , σ BLACK ) E E = ( FBLACK N ( h+E ) − K BLACK N ( h−E )) dft ,T

o

(11-58)

366

Energy Risk

where we now have E FBLACK = Ft ,LT , E K BLACK = kFt ,ST + K , E 2 = σ L2 − 2 ρσ Lσ approx + σ approx σ BLACK ,

h+E/ − =

(11-59)

⎛ FL ⎞ 1 E ln ⎜ S t ,T ⎟ ± (σ BLACK )2 (To − t ) ⎝ kFt ,T + K ⎠ 2 E σ BLACK (To − t )

In the case where the time to expiration is small, we can further approximate the above solution to obtain

σ approx

⎛ kFt ,ST ⎞ ⎯τ⎯ →⎜ S ⎟ σS

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