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The Volatility Smile


The Black-Scholes-Merton option model was the greatest innovation of twentieth century finance, and remains the most widely applied theory in all of finance. Nevertheless, the model is fundamentally at odds with the observed behavior of option markets: a graph of implied volatility against strike will typically display a curve or smile, which the model cannot explain.

Option valuation is not a solved problem, and the past forty years have witnessed an abundance of new ideas and models that try to reconcile theory with markets. Beginning with the principles of financial valuation, The Volatility Smile presents a unique and unified treatment of the Black-Scholes-Merton option model and the more advanced models that have replaced it. Celebrated author, quant, and co-originator of the local volatility model 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 the consequences of different assumptions, readers will learn not only how to handle the volatility smile but how to evaluate and build their own financial models. Key features:

  • The principles of valuation
  • The Black-Scholes-Merton model
  • Hedging strategies and transaction costs
  • The behavior of the volatility smile
  • Static and dynamic replication of standard and exotic options
  • New models: their origin, implementation, and consequences
  • Local volatility
  • Stochastic volatility
  • Jump-diffusion

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.

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