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1 The Black-Scholes Theory of Derivative Pricing1

1.1 Market Model1

1.1.1　Brownian Motion2

1.1.2 Stochastic Integrals3

1.1.3　Risky Asset Price Model4

1.1.4　It？'s Formula6

1.1.5　Lognormal Risky Asset Price8

1.2　Derivative Contracts8

1.2.1 European Call and Put Options9

1.2.2　American Options10

1.2.3 Other Exotic Options11

1.3 Replicating Strategies12

1.3.1 Replicating Self-Financing Portfolios12

1.3.2 The Black-Scholes Partial Differential Equation13

1.3.3　Pricing to Hedge15

1.3.4　The Black-Scholes Formula15

1.4 Risk-Neutral Pricing18

1.4.1 Equivalent Martingale Measure19

1.4.2 Self-Financing Portfolios20

1.4.3　Risk-Neutral Valuation21

1.4.4 Using the Markov Property22

1.5　Risk-Neutral Expectations and Partial Differential Equations23

1.5.1 Infinitesimal Generators and Associated Martingales24

1.5.2 Conditional Expectations and Parabolic Partial Differential Equations25

1.5.3 Application to the Black-Scholes Partial Differential Equation26

1.5.4　American Options and Free Boundary Problems27

1.5.5　Path-Dependent Derivatives29

1.6 Complete Market31

2 Introduction to Stochastic Volatility Models33

2.1 Implied Volatility and the Smile Curve34

2.1.1 Interpretation of the Smile Curve35

2.1.2　What Data to Use37

2.2 Implied Deterministic Volatility38

2.2.1　Time-Dependent Volatility38

2.2.2　Level-Dependent Volatility39

2.2.3　Short-Time Tight Fit versus Long-Time Rough Fit39

2.3 Stochastic Volatility Models40

2.3.1 Mean-Reverting Stochastic Volatility Models40

2.3.2 Stock-Price Distribution under Stochastic Volatility42

2.4　Derivative Pricing42

2.5　Pricing with Equivalent Martingale Measures46

2.6 Implied Volatility as a Function of Moneyness48

2.7 Market Price of Volatility Risk and Data49

2.8 Special Case:Uncorrelated Volatility51

2.8.1 Hull-White Formula51

2.8.2 Stochastic Volatility Implies Smile51

2.8.3 Remark on Correlated Volatility53

2.9　Summary and Conclusions55

3　Scales in Mean-Reverting Stochastic Volatility58

3.1 Scaling in Simple Models58

3.2　Models of Clustering59

3.2.1 Example:Markov Chain60

3.2.2　Example:Another Jump Process64

3.2.3 Example:Ornstein-Uhlenbeck Process67

3.2.4 Summary70

3.3 Convergence to Black-Scholes under Fast Mean-Reverting Volatility70

3.4　Scales in the Returns Process71

3.4.1 The Returns Process72

3.4.2　Returns Process with Jump Volatility72

3.4.3 Returns Process with OU Volatility73

3.4.4 S＆P 500 Returns Process75

4　Tools for Estimating the Rate of Mean Reversion77

4.1　Model and Data77

4.1.1 Mean-Reverting Stochastic Volatility77

4.1.2　Discrete Data78

4.2 Variogram Analysis79

4.2.1 Computation of the Variogram79

4.2.2 Comparison and Sensitivity Analysis with Simulated Data80

4.2.3　The Day Effect83

4.3　Spectral Analysis84

5　Asymptotics for Pricing European Derivatives87

5.1　Preliminaries87

5.1.1　The Rescaled Stochastic Volatility Model88

5.1.2　The Rescaled Pricing Equation88

5.1.3　The Operator Notation89

5.2　The Formal Expansion90

5.2.1　The Diverging Terms90

5.2.2　Poisson Equations91

5.2.3　The Zero-Order Term93

5.2.4 The First Correction94

5.2.5 Universal Market Group Parameters96

5.2.6　Probabilistic Interpretation of the Source Term97

5.2.7　Put-Call Parity98

5.2.8　The Skew Effect98

5.3　Implied Volatilities and Calibration99

5.4　Accuracy of the Approximation102

5.5　Region of Validity104

6　Implementation and Stability108

6.1　Step-by-Step Procedure108

6.2　Comments about the Method109

6.3　Dividends112

6.4　The Second Correction113

7　Hedging Strategies115

7.1 Black-Scholes Delta Hedging115

7.1.1　The Strategy and Its Cost116

7.1.2　Averaging Effect117

7.2　Mean Self-Financing Hedging Strategy119

7.3 Staying Close to the Price121

8　Application to Exotic Derivatives124

8.1 European Binary Options124

8.2　Barrier Options125

8.3　Asian Options129

9　Application to American Derivatives132

9.1 American Problem under Stochastic Volatility132

9.2 Stochastic Volatility Correction for an American Put133

9.2.1 Expansions134

9.2.2　First Approximation136

9.2.3 The Stochastic Volatility Correction138

9.2.4 Uncorrelated Volatility140

9.2.5　Probabilistic Representation141

9.3　Numerical Computation141

9.3.1 Solution of the Black-Scholes Problem141

9.3.2 Computation of the Correction142

10　Generalizations145

10.1　Portfolio Optimization under Stochastic Volatility145

10.1.1 Constant Volatility Merton Problem145

10.1.2 Stochastic Volatility Merton Problem147

10.1.3　A Practical Solution151

10.2　Periodic Day Effect153

10.3　Other Markovian Volatility Models156

10.3.1 Markovian Jump Volatility Models156

10.3.2　Pricing and Asymptotics158

10.4　Martingale Approach159

10.4.1　Main Argument159

10.4.2　Decomposition Result161

10.4.3　Comparison with the PDE Approach164

10.5　Non-Markovian Models of Volatility165

10.5.1 Setting:An Example165

10.5.2　Asymptotics in the Non-Markovian Case166

10.6　Multidimensional Models169

11　Applications to Interest-Rate Models174

11.1　Bond Pricing in the Vasicek Model174

11.1.1　Review of the Constant Volatility Vasicek Model174

11.1.2　Stochastic Volatility Vasicek Models177

11.2　Bond Option Pricing182

11.2.1　The Constant Volatility Case183

11.2.2　Correction for Stochastic Volatility185

11.2.3 Implications189

11.3　Asymptotics around the CIR Model190

11.4　Illustration from Data192

11.4.1　Variogram Analysis192

11.4.2　Yield Curve Fitting193

Bibliography195

Index199