Volatility Surface: A Practitioner's Guide / Edition 1

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Understanding the volatility surface is a key objective for bothpractitioners and academics in the field of finance. Impliedvolatilities evolve randomly and so models of the volatilitysurface—which is formed from implied volatilities of allstrikes and expirations—need to explicitly reflect thisrandomness in order to accurately price, trade, and manage the riskof derivative products.

Author and financial professional Jim Gatheral is intimatelyfamiliar with these issues and, in The Volatility Surface, heshares his many years of knowledge and experience to help makesense of it all. Written by a practitionerfor practitioners, TheVolatility Surface examines why options are priced as they areand—starting from a powerful representation of impliedvolatility in terms of a weighted average ofrealizedvolatilities—explores the implications of various popularmodels for pricing.

The first half of this book focuses on setting up thetheoretical framework, while the later chapters are orientedtowards practical applications. Informative and accessible, TheVolatility Surface:

  • Contains a detailed derivation of the Heston model andexplanations of many other popular models such as SVJ, SVJJ, SABR,and CreditGrades
  • Discusses the characteristics of various types of exoticoptions from the humble barrier option to the super exoticNapoleon
  • Exhaustively covers volatility derivatives with elegant androbust presentations of the latest research
  • Examines performance of exotic cliquet contracts throughin-depth case studies of actual bonds that have alreadymatured

The purpose of The Volatility Surface is not to just presentresults, but to provide you with ways of thinking about and solvingpractical problems that should have many other areas ofapplication. So by the time you finish reading this guide, you'llhave a firm understanding of volatility surface modeling as well asa better idea of how you can apply the results of these models toreal-world situations.

Filled with in-depth insights, expert advice, and real-worldexamples, The Volatility Surface will get you up to speed on thelatest theories underlying options pricing as well as familiarizeyou with the history and practice of trading in the equityderivatives markets.

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Editorial Reviews

From the Publisher
“…I do recommend this book…” (Zentralblatt MATH , Vol. 1118 2007/20)
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Product Details

  • ISBN-13: 9780471792512
  • Publisher: Wiley
  • Publication date: 9/11/2006
  • Series: Wiley Finance Series, #357
  • Edition number: 1
  • Pages: 208
  • Product dimensions: 6.42 (w) x 9.11 (h) x 0.83 (d)

Meet the Author

JIM GATHERAL is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.Dr. Gatheral obtained a PhD in theoretical physics from Cambridge Universityin 1983. Since then, he has been involved in all of the major derivative product areasas a bookrunner, risk manager, and quantitative analyst in London, Tokyo, and New York. From 1997 to 2005, Dr. Gatheral headed the Equity Quantitative Analytics group at Merrill Lynch. His current research focus is equity market microstructure and algorithmic trading.

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Table of Contents

List of Figures.

List of Tables.




Chapter 1: Stochastic Volatility and LocalVolatility.

Stochastic Volatility.

Derivation of the Valuation Equation,

Local Volatility,


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 InstantaneousVariance.

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.


Implied Volatility in the Heston Model.

The Term Structure of Black-Scholes Implied Volatility in theHeston Model.

The Black-Scholes Implied Volatility Skew in the HestonModel.

The SPX Implied Volatility Surface.

Another Digression: The SVI Parameterization.

A Heston Fit to the Data.

Final Remarks on SV Models and Fitting the VolatilitySurface.

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.

Lèvy 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 andVolatility (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.


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.


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.


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.




Chapter 10: Exotic Cliquets.

Locally Capped Globally Floored Cliquet.

Valuation under Heston and Local Volatility Assumptions.


Reverse Cliquet.

Valuation under Heston and Local Volatility Assumptions.



Valuation under Heston and Local Volatility Assumptions.


Investor Motivation.

More on Napoleons.

Chapter 11: Volatility Derivatives.

Spanning Generalized European Payoffs.

Example: European Options.

Example: Amortizing Options.

The Log Contract.

Variance and Volatility Swaps.

Variance Swaps.

Variance Swaps in the Heston Model.

Dependence on Skew and Curvature.

The Effect of Jumps.

Volatility Swaps.

Convexity Adjustment in the Heston Model.

Valuing Volatility Derivatives.

Fair Value of the Power Payoff.

The Laplace Transform of Quadratic Variation under ZeroCorrelation.

The Fair Value of Volatility under Zero Correlation.

A Simple Lognormal Model.

Options on Volatility: More on Model Independence.

Listed Quadratic-Variation Based Securities.

The VIX Index.

VXB Futures.

Knock-on Benefits.





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