The Volatility Surface

A Practitioner's Guide

Jim Gatheral

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781118046456
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

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

CHAPTER 1

Stochastic Volatility and Local Volatility

In this chapter, we begin our exploration of the volatility surface by introducing stochastic volatility—the notion that volatility varies in a random fashion. Local variance is then shown to be a conditional expectation of the instantaneous variance so that various quantities of interest (such as option prices) may sometimes be computed as though future volatility were deterministic rather than stochastic.

STOCHASTIC VOLATILITY

That it might make sense to model volatility as a random variable should be clear to the most casual observer of equity markets. To be convinced, one need only recall the stock market crash of October 1987. Nevertheless, given the success of the Black-Scholes model in parsimoniously describing market options prices, it’s not immediately obvious what the benefits of making such a modeling choice might be.

Stochastic volatility (SV) models are useful because they explain in a self-consistent way why options with different strikes and expirations have different Black-Scholes implied volatilities—that is, the “volatility smile.” Moreover, unlike alternative models that can fit the smile (such as local volatility models, for example), SV models assume realistic dynamics for the underlying. Although SV price processes are sometimes accused of being ad hoc, on the contrary, they can be viewed as arising from Brownian motion subordinated to a random clock. This clock time, often referred to as trading time, may be identified with the volume of trades or the frequency of trading (Clark 1973); the idea is that as trading activity fluctuates, so does volatility.

FIGURE 1.1 SPX daily log returns from December 31, 1984, to December 31, 2004. Note the −22.9% return on October 19, 1987!

From a hedging perspective, traders who use the Black-Scholes model must continuously change the volatility assumption in order to match market prices. Their hedge ratios change accordingly in an uncontrolled way: SV models bring some order into this chaos.

A practical point that is more pertinent to a recurring theme of this book is that the prices of exotic options given by models based on Black-Scholes assumptions can be wildly wrong and dealers in such options are motivated to find models that can take the volatility smile into account when pricing these.

In Figure 1.1, we plot the log returns of SPX over a 15-year period; we see that large moves follow large moves and small moves follow small moves (so-called “volatility clustering”). In Figure 1.2, we plot the frequency distribution of SPX log returns over the 77-year period from 1928 to 2005. We see that this distribution is highly peaked and fat-tailed relative to the normal distribution. The Q-Q plot in Figure 1.3 shows just how extreme the tails of the empirical distribution of returns are relative to the normal distribution. (This plot would be a straight line if the empirical distribution were normal.)

Fat tails and the high central peak are characteristics of mixtures of distributions with different variances. This motivates us to model variance as a random variable. The volatility clustering feature implies that volatility (or variance) is auto-correlated. In the model, this is a consequence of the mean reversion of volatility.²

FIGURE 1.2 Frequency distribution of (77 years of) SPX da...

Table of contents

• Title Page
• Table of Figures
• List of Tables
• Praise
• Copyright Page
• Dedication
• Foreword
• Preface
• Acknowledgments
• CHAPTER 1 - Stochastic Volatility and Local Volatility
• CHAPTER 2 - The Heston Model
• CHAPTER 3 - The Implied Volatility Surface
• CHAPTER 4 - The Heston-Nandi Model
• CHAPTER 5 - Adding Jumps
• CHAPTER 6 - Modeling Default Risk
• CHAPTER 7 - Volatility Surface Asymptotics
• CHAPTER 8 - Dynamics of the Volatility Surface
• CHAPTER 9 - Barrier Options
• CHAPTER 10 - Exotic Cliquets
• CHAPTER 11 - Volatility Derivatives
• Postscript
• Bibliography
• Index