The Volatility Surface: A Practitioner's GuidePraise 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 |
Contents
| 2 | |
The Heston Model | |
The Implied Volatility Surface | |
The HestonNandi Model | |
Adding Jumps | |
mean and standard deviation With the parameters used to generate these plots the characteristic time | |
Modeling Default Risk | |
buy one put with strike 1 0 and sell two puts with | |
Volatility Surface Asymptotics | |
Dynamics of the Volatility Surface | |
1 Illustration of a cliquet payoff This hypothetical SPX cliquet resets atthemoney every year | |
Exotic Cliquets | |
Volatility Derivatives | |
Postscript | |
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Common terms and phrases
approximate arbitrage assume at-the-money barrier is hit barrier level barrier option bid-offer spread Black-Scholes formula Black-Scholes implied bonds call option struck call spreads capped call Chapter characteristic function cliquet compute convexity adjustment coupon credit spreads dashed line default digital cliquet distribution Dupire dynamics equation European binary European call European options expected fair value FIGURE given graph Heston model Heston parameters Heston-Nandi parameters implied variance skew implied volatility skew implied volatility surface instantaneous variance intuition jump diffusion Lévy process line is stochastic log-strike lognormal lookback lookback option MaxCoupon Mediobanca Merton modeling assumptions Napoleon one-touch option out-of-the-money payoff put-call put-call parity quadratic variation realized volatility replicating reverse cliquet risk-free risk-neutral short expirations short-dated solid line stochastic volatility model stock price strike price SVJ model SVJJ trading valuation variance swap volatility assumptions volatility derivatives volatility of volatility volatility smile volatility swap VXB futures zero