Jim Gatheral is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.

Understanding the volatility surface is a key objective for both practitioners and academics in the field of finance. Implied volatilities evolve randomly and so models of the volatility surface-which is formed from implied volatilities of all strikes and expirations-need to explicitly reflect this randomness in order to accurately price, trade, and manage the risk of derivative products. Author and financial professional Jim Gatheral is intimately familiar with these issues and, in The Volatility Surface, he shares his many years of knowledge and experience to help make sense of it all. Written by a practitionerfor practitioners, The Volatility Surface examines why options are priced as they are and-starting from a powerful representation of implied volatility in terms of a weighted average ofrealized volatilities-explores the implications of various popular models for pricing. The first half of this book focuses on setting up the theoretical framework, while the later chapters are oriented towards practical applications. Informative and accessible, The Volatility Surface: Contains a detailed derivation of the Heston model and explanations of many other popular models such as SVJ, SVJJ, SABR, and CreditGrades Discusses the characteristics of various types of exotic options from the humble barrier option to the super exotic Napoleon Exhaustively covers volatility derivatives with elegant and robust presentations of the latest research Examines performance of exotic cliquet contracts through in-depth case studies of actual bonds that have already matured The purpose of The Volatility Surface is not to just present results, but to provide you with ways of thinking about and solving practical problems that should have many other areas of application. So by the time you finish reading this guide, you'll have a firm understanding of volatility surface modeling as well as a better idea of how you can apply the results of these models to real-world situations. Filled with in-depth insights, expert advice, and real-world examples, The Volatility Surface will get you up to speed on the latest theories underlying options pricing as well as familiarize you with the history and practice of trading in the equity derivatives markets.

Focusing on equity derivatives, Jim Gatheral examines why options are priced as they are and, starting from a powerful representation of implied volatility in terms of a weighted average of realised volatilities, explores the implications of various populat models for pricing.

List of Figures.List of Tables.Foreword.Preface.Acknowledgments.

Chapter 1: Stochastic Volatility and Local Volatility.Stochastic Volatility.Derivation of the Valuation Equation,Local Volatility,History,A Brief Review of Dupire''s Work,Derivation of the Dupire Equation,Local Volatility in Terms of Implied Volatility,Special Case: No Skew,Local Variance as a Conditional Expectation of Instantaneous Variance.

Chapter 2: The Heston Model.The Process.The Heston Solution for European Options.A Digression: The Complex Logarithm in the Integration (2.13).Derivation of the Heston Characteristic Function.Simulation of the Heston Process.Milstein Discretization.Sampling from the Exact Transition Law.Why the Heston Model Is so Popular.

Chapter 3: The Implied Volatility Surface.Getting Implied Volatility from Local Volatilities.Model Calibration.Understanding Implied Volatility.Local Volatility in the Heston Model.Ansatz.Implied Volatility in the Heston Model.The Term Structure of Black-Scholes Implied Volatility in the Heston Model.The Black-Scholes Implied Volatility Skew in the Heston Model.The SPX Implied Volatility Surface.Another Digression: The SVI Parameterization.A Heston Fit to the Data.Final Remarks on SV Models and Fitting the Volatility Surface.

Chapter 4: The Heston-Nandi Model.Local Variance in the Heston-Nandi Model.A Numerical Example.The Heston-Nandi Density.Computation of Local Volatilities.Computation of Implied Volatilities.Discussion of Results.

Chapter 5: Adding Jumps.Why Jumps are Needed.Jump Diffusion.Derivation of the Valuation Equation.Uncertain Jump Size.Characteristic Function Methods.L''evy Processes.Examples of Characteristic Functions for Specific Processes.Computing Option Prices from the Characteristic Function.Proof of (5.6).Computing Implied Volatility.Computing the At-the-Money Volatility Skew.How Jumps Impact the Volatility Skew.Stochastic Volatility Plus Jumps.Stochastic Volatility Plus Jumps in the Underlying Only (SVJ).Some Empirical Fits to the SPX Volatility Surface.Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ).SVJ Fit to the September 15, 2005, SPX Option Data.Why the SVJ Model Wins.

Chapter 6: Modeling Default Risk.Merton''s Model of Default.Intuition.Implications for the Volatility Skew.Capital Structure Arbitrage.Put-Call Parity.The Arbitrage.Local and Implied Volatility in the Jump-to-Ruin Model.The Effect of Default Risk on Option Prices.The CreditGrades Model.Model Setup.Survival Probability.Equity Volatility.Model Calibration.

Chapter 7: Volatility Surface Asymptotics.Short Expirations.The Medvedev-Scaillet Result.The SABR Model.Including Jumps.Corollaries.Long Expirations: Fouque, Papanicolaou, and Sircar.Small Volatility of Volatility: Lewis.Extreme Strikes: Roger Lee.Example: Black-Scholes.Stochastic Volatility Models.Asymptotics in Summary.

Chapter 8: Dynamics of the Volatility Surface.Dynamics of the Volatility Skew under Stochastic Volatility.Dynamics of the Volatility Skew under Local Volatility.Stochastic Implied Volatility Models.Digital Options and Digital Cliquets.Valuing Digital Options.Digital Cliquets.

Chapter 9: Barrier Options.Definitions.Limiting Cases.Limit Orders.European Capped Calls.The Reflection Principle.The Lookback Hedging Argument.One-Touch Options Again.Put-Call Symmetry.QuasiStatic Hedging and Qualitative Valuation.Out-of-the-Money Barrier Options.One-Touch Options.Live-Out Options.Lookback Options.Adjusting for Discrete Monitoring.Discretely Monitored Lookback Options.Parisian Options.Some Applications of Barrier Options.Ladders.Ranges.Conclusion.

Chapter 10: Exotic Cliquets.Locally Capped Globally Floored Cliquet.Valuation under Heston and Local Volatility Assumptions.Performance.Reverse Cliquet.Valuation under Heston and Local Volatility Assumptions.Performance.Napoleon.Valuation under Heston and Local Volatility Assumptions.Performance.Investor Motivation.More on Napoleons.

Chapter 11: Volatility Derivatives.Spanning Generalized Euro