1st Edition

Stochastic Volatility Modeling

By Lorenzo Bergomi Copyright 2016
    522 Pages 88 B/W Illustrations
    by Chapman & Hall

    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.

    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





    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