Stochastic Volatility Modeling  book cover
1st Edition

Stochastic Volatility Modeling

ISBN 9781482244069
Published January 5, 2016 by Chapman & Hall
88 B/W Illustrations

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Book Description

Packed with insights, Lorenzo Bergomi’s Stochastic Volatility Modeling explains how stochastic volatility is used to address issues arising in the modeling of derivatives, including:

  • Which trading issues do we tackle with stochastic volatility?
  • How do we design models and assess their relevance?
  • How do we tell which models are usable and when does calibration make sense?

This manual covers the practicalities of modeling local volatility, stochastic volatility, local-stochastic volatility, and multi-asset stochastic volatility. In the course of this exploration, the author, Risk’s 2009 Quant of the Year and a leading contributor to volatility modeling, draws on his experience as head quant in Société Générale’s equity derivatives division. Clear and straightforward, the book takes readers through various modeling challenges, all originating in actual trading/hedging issues, with a focus on the practical consequences of modeling choices.

Table of Contents

Characterizing a usable model: the Black-Scholes equation
How (in)effective is delta hedging?
On the way to stochastic volatility
Chapter’s digest

Local Volatility
Introduction: local volatility as a market model
From prices to local volatilities
From implied volatilities to local volatilities
From local volatilities to implied volatilities
The dynamics of the local volatility model
Future skews and volatilities of volatilities
Delta and carry P&L
Digression: using payoff-dependent break-even levels
The vega hedge
Markov-functional models
Appendix A: the uncertain volatility model
Chapter’s digest

Forward-Start Options
Pricing and hedging forward-start options
Forward-start options in the local volatility model
Chapter’s digest

Stochastic Volatility: Introduction
Modeling vanilla option prices
Modeling the dynamics of the local volatility function
Modeling implied volatilities of power payoffs
Chapter’s digest

Variance Swaps
Variance swap forward variances
Relationship of variance swaps to log contracts
Impact of large returns
Impact of strike discreteness
Pricing variance swaps with a PDE
Interest-rate volatility
Weighted variance swaps
Appendix A: timer options
Appendix B: perturbation of the lognormal distribution
Chapter’s digest

An Example of One-Factor Dynamics: The Heston Model
The Heston model
Forward variances in the Heston model
Drift of Vt in first-generation stochastic volatility models
Term structure of volatilities of volatilities in the Heston model
Smile of volatility of volatility
ATMF skew in the Heston model
Chapter’s digest

Forward Variance Models
Pricing equation
A Markov representation
N-factor models
A two-factor model
Calibration: the vanilla smile
Options on realized variance
VIX futures and options
Discrete forward variance models
Chapter’s digest

The Smile of Stochastic Volatility Models
Expansion of the price in volatility of volatility
Expansion of implied volatilities
A representation of European option prices in diffusive models
Short maturities
A family of one-factor models: application to the Heston model
The two-factor model
Forward-start options: future smiles
Impact of the smile of volatility of volatility on the vanilla smile
Appendix A: Monte Carlo algorithms for vanilla smiles
Appendix B: local volatility function of stochastic volatility models
Appendix C: partial resummation of higher orders
Chapter’s digest

Linking Static and Dynamic Properties of Stochastic Volatility Models
The ATMF skew
The Skew Stickiness Ratio (SSR)
Short-maturity limit of the ATMF skew and the SSR
Model-independent range of the SSR
Scaling of ATMF skew and SSR: a classification of models
Type I models: the Heston model
Type II models
Numerical evaluation of the SSR
The SSR for short maturities
Arbitraging the realized short SSR
Chapter’s digest

What Causes Equity Smiles?
The distribution of equity returns
Impact of the distribution of daily returns on derivative prices
Appendix A: jump-diffusion/Lévy models
Chapter’s digest

Multi-Asset Stochastic Volatility
The short ATMF basket skew
Parametrizing multi-asset stochastic volatility models
The ATMF basket skew
The correlation swap
Appendix A: bias/standard deviation of the correlation estimator
Chapter’s digest

Local-Stochastic Volatility Models
Pricing equation and calibration
Usable models
Dynamics of implied volatilities
Numerical examples
Appendix A: alternative schemes for the PDE method
Chapter’s digest




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Lorenzo Bergomi heads the quantitative research group at Société Générale, covering all asset classes. A quant for over 15 years, he is well known for his pioneering work on stochastic volatility modeling, some of which has appeared in the Smile Dynamics series of articles in Risk magazine. He was also the magazine’s 2009 Quant of the Year. Originally trained as an electrical engineer and with a PhD in theoretical physics, he was active as a physicist in the condensed matter theory group at IphT, CEA, before moving to finance.


"With this book, Bergomi has actually offered a precious gift to the whole quant community: his very rich and concrete experience on volatility modelling organized in 500 pages and 12 chapters full of insights; and to the academic community as well: new ideas, points of view, and questions that could well feed their research for years."

- Julien Guyon, Quantitative Finance

"[Stochastic Volatility Modeling] should be read by practitioners, as it is the only one providing a strong quantitative framework to the (Delta and Vega) hedging of Equity derivatives. It should also be read by academics who will benefit from practical insights. It should finally be read by (motivated) students, who will definitely find areas to dig deeper in, both theoretically and numerically […] This book should be seen as a strong case for the need of a deeper understanding of derivatives' modelling (and their risks). Lorenzo Bergomi provides us here with new tools (variance curve models, metrics such as the At-The-Money Forward Skew and the Skew Stickiness Ratio) as well as new results on hedging and P&L computations of actual trading strategies, which have been so far too much overlooked in mathematical finance research. Welcome to the new era of Derivatives Modelling!"

- Antoine Jacquier, Newsletter of the Bachelier Finance Society, November 2017