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The Volatility Surface: A Practitioner's Guide

The Volatility Surface: A Practitioner's Guide

Jim Gatheral, Nassim Nicholas Taleb (Foreword by)

ISBN: 978-1-118-04645-6

Mar 2011

208 pages



Praise for The Volatility Surface

""I'm thrilled by the appearance of Jim Gatheral's new book The Volatility Surface. The literature on stochastic volatility is vast, but difficult to penetrate and use. Gatheral's book, by contrast, is accessible and practical. It successfully charts a middle ground between specific examples and general models--achieving remarkable clarity without giving up sophistication, depth, or breadth.""
--Robert V. Kohn, Professor of Mathematics and Chair, Mathematical Finance Committee, Courant Institute of Mathematical Sciences, New York University

""Concise yet comprehensive, equally attentive to both theory and phenomena, this book provides an unsurpassed account of the peculiarities of the implied volatility surface, its consequences for pricing and hedging, and the theories that struggle to explain it.""
--Emanuel Derman, author of My Life as a Quant

""Jim Gatheral is the wiliest practitioner in the business. This very fine book is an outgrowth of the lecture notes prepared for one of the most popular classes at NYU's esteemed Courant Institute. The topics covered are at the forefront of research in mathematical finance and the author's treatment of them is simply the best available in this form.""
--Peter Carr, PhD, head of Quantitative Financial Research, Bloomberg LP Director of the Masters Program in Mathematical Finance, New York University

""Jim Gatheral is an acknowledged master of advanced modeling for derivatives. In The Volatility Surface he reveals the secrets of dealing with the most important but most elusive of financial quantities, volatility.""
--Paul Wilmott, author and mathematician

""As a teacher in the field of mathematical finance, I welcome Jim Gatheral's book as a significant development. Written by a Wall Street practitioner with extensive market and teaching experience, The Volatility Surface gives students access to a level of knowledge on derivatives which was not previously available. I strongly recommend it.""
--Marco Avellaneda, Director, Division of Mathematical Finance Courant Institute, New York University

""Jim Gatheral could not have written a better book.""
--Bruno Dupire, winner of the 2006 Wilmott Cutting Edge Research Award Quantitative Research, Bloomberg LP
List of Figures.

List of Tables.




Chapter 1: Stochastic Volatility and Local Volatility.

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 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.


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.


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 Zero Correlation.

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.





“…I do recommend this book…” (Zentralblatt MATH , Vol. 1118 2007/20)
Erratum: Gatheral / The Volatility Surface Download