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Table of contents




1 Introduction

    Characterizing a usable model – the Black-Scholes equation

    How (in)effective is delta hedging?

    On the way to stochastic volatility

    Chapter's digest


2 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


3 Forward-start options

    Pricing and hedging forward-start options

    Forward-start options in the local volatility model

    Chapter’s digest


4 Stochastic volatility – introduction

    Modeling vanilla option prices

    Modeling the dynamics of the local volatility function

    Modeling implied volatilities of power payoffs

    Chapter’s digest


5 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

6 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

7 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




8 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

9 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

10 What causes equity smiles?
    The distribution of equity returns
    Impact of the distribution of daily returns on derivative prices
    Appendix A – jump-difusion/Lévy models
    Chapter’s digest

11 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

12 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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