This book addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These problems are important to investors ranging from large trading institutions to pension funds. The authors present mathematical and statistical tools that exploit the "bursty" nature of market volatility. The mathematics is introduced through examples and illustrated with simulations, and the approach described is validated and tested on market data.

Introduction

1 The Black-Scholes Theory of Derivative Pricing

 1.1 Market Model

1.1.1 Brownian Motion

1.1.2 Stochastic Integrals

1.1.3 Risky Asset Price Model

1.1.4 Ito's Formula

1.1.5 Lognormal Risky Asset Price

 1.2 Derivative Contracts

1.2.1 European Call and Put Options

1.2.2 American Options

1.2.3 Other Exotic Options

 1.3 Replicating Strategies

1.3.1 Replicating Self-Financing Portfolios

1.3.2 The Black-Scholes Partial Differential Equation

1.3.3 Pricing to Hedge

1.3.4 The Black-Scholes Formula

 1.4 Risk-Neutral Pricing

1.4.1 Equivalent Martingale Measure

1.4.2 Self-Financing Portfolios

1.4.3 Risk-Neutral Valuation

1.4.4 Using the Markov Property

 1.5 Risk-Neutral Expectations and Partial Differential Equations

1.5.1 Infinitesimal Generators and Associated Martingales

1.5.2 Conditional Expectations and Parabolic Partial Differential Equations

1.5.3 Application to the Black-Scholes Partial Differential Equation

1.5.4 American Options and Free Boundary Problems

1.5.5 Path-Dependent Derivatives

 1.6 Complete Market

2 Introduction to Stochastic Volatility Models

 2.1 Implied Volatility and the Smile Curve

2.1.1 Interpretation of the Smile Curve

2.1.2 What Data to Use

 2.2 Implied Deterministic Volatility

2.2.1 Time-Dependent Volatility

2.2.2 Level-Dependent Volatility

2.2.3 Short-Time Tight Fit versus Long-Time Rough Fit

 2.3 Stochastic Volatility Models

2.3.1 Mean-Reverting Stochastic Volatility Models

2.3.2 Stock-Price Distribution under Stochastic Volatility

 2.4 Derivative Pricing

 2.5 Pricing with Equivalent Martingale Measures

 2.6 Implied Volatility as a Function of Moneyness

 2.7 Market Price of Volatility Risk and Data

 2.8 Special Case: Uncorrelated Volatility

2.8.1 Hull-White Formula

2.8.2 Stochastic Volatility Implies Smile

2.8.3 Remark on Correlated Volatility

 2.9 Summary and Conclusions

3 Scales in Mean-Reverting Stochastic Volatility

 3.1 Scaling in Simple Models

 3.2 Models of Clustering

3.2.1 Example: Markov Chain

3.2.2 Example: Another Jump Process

3.2.3 Example: Ornstein-Uhlenbeck Process

3.2.4 Summary

 3.3 Convergence to Black-Scholes under Fast Mean-Reverting Volatility

 3.4 Scales in the Returns Process

3.4.1 The Returns Process

3.4.2 Returns Process with Jump Volatility

3.4.3 Returns Process with OU Volatility

3.4.4 S&P 500 Returns Process

4 Tools for Estimating the Rate of Mean Reversion

 4.1 Model and Data

4.1.1 Mean-Reverting Stochastic Volatility

4.1.2 Discrete Data

 4.2 Variogram Analysis

4.2.1 Computation of the Variogram

4.2.2 Comparison and Sensitivity Analysis with Simulated Data

4.2.3 The Day Effect

 4.3 Spectral Analysis

5 Asymptotics for Pricing European Derivatives

 5.1 Preliminaries

5.1.l The Rescaled Stochastic Volatility Model

5.1.2 The Rescaled Pricing Equation

5.1.3 The Operator Notation

 5.2 The Formal Expansion

5.2.1 The Diverging Terms

5.2.2 Poisson Equations

5.2.3 The Zero-Order Term

5.2.4 The First Correction

5.2.5 Universal Market Group Parameters

5.2.6 Probabilistic Interpretation of the Source Term

5.2.7 Put-Call Parity

5.2.8 The Skew Effect

 5.3 Implied Volatilities and Calibration

 5.4 Accuracy of the Approximation

 5.5 Region of Validity

6 Implementation and Stability

 6.1 Step-by-Step Procedure

 6.2 Comments about the Method

 6.3 Dividends

 6.4 The Second Correction

7 Hedging Strategies

 7.1 Black-Scholes Delta Hedging

7.1.1 The Strategy and Its Cost

7.1.2 Averaging Effect

 7.2 Mean Self-Financing Hedging Strategy

 7.3 Staying Close to the Price

8 Application to Exotic Derivatives

 8.1 European Binary Options

 8.2 Barrier Options

 8.3 Asian Options

9 Application to American Derivatives

 9.1 American Problem under Stochastic Volatility

 9.2 Stochastic Volatility Correction for an American Put

9.2.1 Expansions

9.2.2 First Approximation

9.2.3 The Stochastic Volatility Correction

9.2.4 Uncorrelated Volatility

9.2.5 Probabilistic Representation

 9.3 Numerical Computation

9.3.1 Solution of the Black-Scholes Problem

9.3.2 Computation of the Correction

10 Generalizations

 10.1 Portfolio Optimization under Stochastic Volatility

10.1.1 Constant Volatility Merton Problem

10.1.2 Stochastic Volatility Merton Problem

10.1.3 A Practical Solution

 10.2 Periodic Day Effect

 10.3 Other Markovian Volatility Models

10.3.1 Markovian Jump Volatility Models

10.3.2 Pricing and Asymptotics

 10.4 Martingale Approach

10.4.1 Main Argument

10.4.2 Decomposition Result

10.4.3 Comparison with the PDE Approach

 10.5 Non-Markovian Models of Volatility

10.5.1 Setting: An Example

10.5.2 Asymptotics in the Non-Markovian Case

 10.6 Multidimensional Models

11 Applications to Interest-Rate Models

 11.1 Bond Pricing in the Vasicek Model

11.1.1 Review of the Constant Volatility Vasicek Model

11.1.2 Stochastic Volatility Vasicek Models

 11.2 Bond Option Pricing

11.2.1 The Constant Volatility Case

11.2.2 Correction for Stochastic Volatility

11.2.3 Implications

 11.3 Asymptotics around the CIR Model

 11.4 Illustration from Data

11.4.1 Variogram Analysis

11.4.2 Yield Curve Fitting

Bibliography

Index