The Volatility Surface: A Practitioner's Guide (Wiley Finance) (Hardcover)
by Jim Gatheral (Author), Nassim Nicholas Taleb (Foreword)

Product Details

Hardcover: 208 pages
Publisher: Wiley (August 28, 2006)
Language: English
ISBN-10: 0471792519
ISBN-13: 978-0471792512
CHAPTER 1
Stochastic Volatility and Local Volatility 1
Stochastic Volatility 1
Derivation of the Valuation Equation 4
Local Volatility 7
History 7
A Brief Review of Dupire’s Work 8
Derivation of the Dupire Equation 9
Local Volatility in Terms of Implied Volatility 11
Special Case: No Skew 13
Local Variance as a Conditional Expectation
of Instantaneous Variance 13
CHAPTER 2
The Heston Model 15
The Process 15
The Heston Solution for European Options 16
A Digression: The Complex Logarithm
in the Integration (2.13) 19
Derivation of the Heston Characteristic Function 20
Simulation of the Heston Process 21
Milstein Discretization 22
Sampling from the Exact Transition Law 23
Why the Heston Model Is so Popular 24

CHAPTER 3
The Implied Volatility Surface 25
Getting Implied Volatility from Local Volatilities 25
Model Calibration 25
Understanding Implied Volatility 26
Local Volatility in the Heston Model 31
Ansatz 32
Implied Volatility in the Heston Model 33
The Term Structure of Black-Scholes Implied Volatility
in the Heston Model 34
The Black-Scholes Implied Volatility Skew
in the Heston Model 35
The SPX Implied Volatility Surface 36
Another Digression: The SVI Parameterization 37
A Heston Fit to the Data 40
Final Remarks on SV Models and Fitting
the Volatility Surface 42
CHAPTER 4
The Heston-Nandi Model 43
Local Variance in the Heston-Nandi Model 43
A Numerical Example 44
The Heston-Nandi Density 45
Computation of Local Volatilities 45
Computation of Implied Volatilities 46
Discussion of Results 49
CHAPTER 5
Adding Jumps 50
Why Jumps are Needed 50
Jump Diffusion 52
Derivation of the Valuation Equation 52
Uncertain Jump Size 54
Characteristic Function Methods 56
L´evy Processes 56
Examples of Characteristic Functions
for Specific Processes 57
Computing Option Prices from the
Characteristic Function 58
Proof of (5.6) 58

Editorial Reviews

Book Description
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

From the Inside Flap
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.

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