The Volatility Smile
The 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 models
The 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.
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 Itoˆ 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.
