Stochastic Volatility Modeling

Lorenzo Bergomi



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.

Features

• Covers forward-start options, variance swaps, options on realized variance, timer options, VIX futures and options, and daily cliquets
• Includes an in-depth study of the dynamics of the local volatility model, its carry P&L, and its delta
• Surveys the uncertain volatility model and its usage
• Discusses the parametrization of local-stochastic volatility and multi-asset stochastic volatility models
• Characterizes the links between static and dynamics features of stochastic volatility models
• Contains a wealth of unpublished results and insights

Table of Contents

Introduction
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
Conclusion
Dividends
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
Discussion
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
Introduction
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
Conclusion
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
Conclusion
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
Conclusion
Appendix A: bias/standard deviation of the correlation estimator
Chapter’s digest

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

Epilogue

Bibliography

Index