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The Volatility Smile
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Table of Contents

Preface xi Acknowledgments xiii About the Authors xv CHAPTER 1 Overview 1 CHAPTER 2 The Principle of Replication 13 CHAPTER 3 Static and Dynamic Replication 37 CHAPTER 4 Variance Swaps: A Lesson in Replication 57 CHAPTER 5 The P&L of Hedged Option Strategies in a Black-Scholes-Merton World 85 CHAPTER 6 The Effect of Discrete Hedging on P&L 105 CHAPTER 7 The Effect of Transaction Costs on P&L 117 CHAPTER 8 The Smile: Stylized Facts and Their Interpretation 131 CHAPTER 9 No-Arbitrage Bounds on the Smile 153 CHAPTER 10 A Survey of Smile Models 163 CHAPTER 11 Implied Distributions and Static Replication 175 CHAPTER 12 Weak Static Replication 203 CHAPTER 13 The Binomial Model and Its Extensions 227 CHAPTER 14 Local Volatility Models 249 CHAPTER 15 Consequences of Local Volatility Models 265 CHAPTER 16 Local Volatility Models: Hedge Ratios and Exotic Option Values 289 CHAPTER 17 Some Final Remarks on Local Volatility Models 303 CHAPTER 18 Patterns of Volatility Change 309 CHAPTER 19 Introducing Stochastic Volatility Models 319 CHAPTER 20 Approximate Solutions to Some Stochastic Volatility Models 337 CHAPTER 21 Stochastic Volatility Models: The Smile for Zero Correlation 353 CHAPTER 22 Stochastic Volatility Models: The Smile with Mean Reversion and Correlation 369 CHAPTER 23 Jump-Diffusion Models of the Smile: Introduction 383 CHAPTER 24 The Full Jump-Diffusion Model 395 Epilogue 417 APPENDIX A Some Useful Derivatives of the Black-Scholes-Merton Model 419 APPENDIX B Backward Ito Integrals 421 APPENDIX C Variance Swap Piecewise-Linear Replication 431 Answers to End-of-Chapter Problems 433 References 497 Index 501

About the Author

EMANUEL DERMAN is a professor at Columbia University, where he directs its financial engineering program. He is the author of My Life as a Quant and Models.Behaving.Badly. MICHAEL B. MILLER is the founder and CEO of Northstar Risk Corp. He is the author of Mathematics and Statistics for Financial Risk Management, Second Edition.

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