Stochastic Volatility Modeling: 1st Edition (Hardback) book cover

Stochastic Volatility Modeling

1st Edition

By Lorenzo Bergomi

Chapman and Hall/CRC

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pub: 2016-01-05
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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.


"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

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




About the Author

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.

About the Series

Chapman and Hall/CRC Financial Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General