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

Preface

About the Authors

Chapter 1: Overview

Chapter 2: The Principle of Replication

Chapter 3: Static and Dynamic Replication

Chapter 4: Variance Swaps: A Lesson in Replication

Chapter 5: The P&L of Hedged Option Strategies in a Black–Scholes–Merton World

Chapter 6: The Effect of Discrete Hedging on P&L

Chapter 7: The Effect of Transactions Costs on P&L

Chapter 8: The Smile: Stylized Facts and Their Interpretation

Chapter 9: No–Arbitrage Bounds on the Smile

Chapter 10: A Survey of Smile Models

Chapter 11: Implied Distributions and Static Replication

Chapter 12: Weak Static Replication

Chapter 13: The Binomial Model and Its Extensions

Chapter 14: Local Volatility Models

Chapter 15: Consequences of Local Volatility Models

Chapter 16: Local Volatility Models: Hedge Ratios and Exotic Option Values

Chapter 17: Some Final Remarks on Local Volatility Models

Chapter 18: Patterns of Volatility Change

Chapter 19: Introducing Stochastic Volatility Models

Chapter 20: Approximate Solutions to Some Stochastic Volatility Models

Chapter 21: Stochastic Volatility Models: The Smile for Zero Correlation

Chapter 22: Stochastic Volatility Models: The Smile with Mean Reversion and Correlation

Chapter 23: Jump–Diffusion Models of the Smile: Introduction

Chapter 24: The Full Jump–Diffusion Model

Appendix A: Some Useful Derivatives of the Black–Scholes–Merton Model

Appendix B: Backward Itô Integrals

References

Answers to End–of–Chapter Problems

Index

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.