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Stochastic Volatility Modeling

Lorenzo Bergomi
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Chapman & Hall/CRC Financial Mathematics Series
We do not plan to review this book.

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