The Volatility SmileThe Black-Scholes-Merton option model was the greatest innovation of 20th century finance, and remains the most widely applied theory in all of finance. Despite this success, the model is fundamentally at odds with the observed behavior of option markets: a graph of implied volatilities against strike will typically display a curve or skew, which practitioners refer to as the smile, and which the model cannot explain. Option valuation is not a solved problem, and the past forty years have witnessed an abundance of new models that try to reconcile theory with markets.The Volatility Smile presents a unified treatment of the Black-Scholes-Merton model and the more advanced models that have replaced it. It is also a book about the principles of financial valuation and how to apply them. Celebrated author and quant Emanuel Derman and Michael B. Miller explain not just the mathematics but the ideas behind the models. By examining the foundations, the implementation, and the pros and cons of various models, and by carefully exploring their derivations and their assumptions, readers will learn not only how to handle the volatility smile but how to evaluate and build their own financial models.Topics covered include:* The principles of valuation* Static and dynamic replication* The Black-Scholes-Merton model* Hedging strategies* Transaction costs* The behavior of the volatility smile* Implied distributions* Local volatility models* Stochastic volatility models* Jump-diffusion modelsThe first half of the book, Chapters 1 through 13, can serve as a standalone textbook for a course on option valuation and the Black-Scholes-Merton model, presenting the principles of financial modeling, several derivations of the model, and a detailed discussion of how it is used in practice. The second half focuses on the behavior of the volatility smile, and, in conjunction with the first half, can be used for as the basis for a more advanced course.

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