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1 The Black-Scholes Theory of Derivative Pricing 1

1.1 Market Model 1

1.1.1　Brownian Motion 2

1.1.2 Stochastic Integrals 3

1.1.3　Risky Asset Price Model 4

1.1.4　It？'s Formula 6

1.1.5　Lognormal Risky Asset Price 8

1.2　Derivative Contracts 8

1.2.1 European Call and Put Options 9

1.2.2　American Options 10

1.2.3 Other Exotic Options 11

1.3 Replicating Strategies 12

1.3.1 Replicating Self-Financing Portfolios 12

1.3.2 The Black-Scholes Partial Differential Equation 13

1.3.3　Pricing to Hedge 15

1.3.4　The Black-Scholes Formula 15

1.4 Risk-Neutral Pricing 18

1.4.1 Equivalent Martingale Measure 19

1.4.2 Self-Financing Portfolios 20

1.4.3　Risk-Neutral Valuation 21

1.4.4 Using the Markov Property 22

1.5　Risk-Neutral Expectations and Partial Differential Equations 23

1.5.1 Infinitesimal Generators and Associated Martingales 24

1.5.2 Conditional Expectations and Parabolic Partial Differential Equations 25

1.5.3 Application to the Black-Scholes Partial Differential Equation 26

1.5.4　American Options and Free Boundary Problems 27

1.5.5　Path-Dependent Derivatives 29

1.6 Complete Market 31

2 Introduction to Stochastic Volatility Models 33

2.1 Implied Volatility and the Smile Curve 34

2.1.1 Interpretation of the Smile Curve 35

2.1.2　What Data to Use 37

2.2 Implied Deterministic Volatility 38

2.2.1　Time-Dependent Volatility 38

2.2.2　Level-Dependent Volatility 39

2.2.3　Short-Time Tight Fit versus Long-Time Rough Fit 39

2.3 Stochastic Volatility Models 40

2.3.1 Mean-Reverting Stochastic Volatility Models 40

2.3.2 Stock-Price Distribution under Stochastic Volatility 42

2.4　Derivative Pricing 42

2.5　Pricing with Equivalent Martingale Measures 46

2.6 Implied Volatility as a Function of Moneyness 48

2.7 Market Price of Volatility Risk and Data 49

2.8 Special Case:Uncorrelated Volatility 51

2.8.1 Hull-White Formula 51

2.8.2 Stochastic Volatility Implies Smile 51

2.8.3 Remark on Correlated Volatility 53

2.9　Summary and Conclusions 55

3　Scales in Mean-Reverting Stochastic Volatility 58

3.1 Scaling in Simple Models 58

3.2　Models of Clustering 59

3.2.1 Example:Markov Chain 60

3.2.2　Example:Another Jump Process 64

3.2.3 Example:Ornstein-Uhlenbeck Process 67

3.2.4 Summary 70

3.3 Convergence to Black-Scholes under Fast Mean-Reverting Volatility 70

3.4　Scales in the Returns Process 71

3.4.1 The Returns Process 72

3.4.2　Returns Process with Jump Volatility 72

3.4.3 Returns Process with OU Volatility 73

3.4.4 S＆P 500 Returns Process 75

4　Tools for Estimating the Rate of Mean Reversion 77

4.1　Model and Data 77

4.1.1 Mean-Reverting Stochastic Volatility 77

4.1.2　Discrete Data 78

4.2 Variogram Analysis 79

4.2.1 Computation of the Variogram 79

4.2.2 Comparison and Sensitivity Analysis with Simulated Data 80

4.2.3　The Day Effect 83

4.3　Spectral Analysis 84

5　Asymptotics for Pricing European Derivatives 87

5.1　Preliminaries 87

5.1.1　The Rescaled Stochastic Volatility Model 88

5.1.2　The Rescaled Pricing Equation 88

5.1.3　The Operator Notation 89

5.2　The Formal Expansion 90

5.2.1　The Diverging Terms 90

5.2.2　Poisson Equations 91

5.2.3　The Zero-Order Term 93

5.2.4 The First Correction 94

5.2.5 Universal Market Group Parameters 96

5.2.6　Probabilistic Interpretation of the Source Term 97

5.2.7　Put-Call Parity 98

5.2.8　The Skew Effect 98

5.3　Implied Volatilities and Calibration 99

5.4　Accuracy of the Approximation 102

5.5　Region of Validity 104

6　Implementation and Stability 108

6.1　Step-by-Step Procedure 108

6.2　Comments about the Method 109

6.3　Dividends 112

6.4　The Second Correction 113

7　Hedging Strategies 115

7.1 Black-Scholes Delta Hedging 115

7.1.1　The Strategy and Its Cost 116

7.1.2　Averaging Effect 117

7.2　Mean Self-Financing Hedging Strategy 119

7.3 Staying Close to the Price 121

8　Application to Exotic Derivatives 124

8.1 European Binary Options 124

8.2　Barrier Options 125

8.3　Asian Options 129

9　Application to American Derivatives 132

9.1 American Problem under Stochastic Volatility 132

9.2 Stochastic Volatility Correction for an American Put 133

9.2.1 Expansions 134

9.2.2　First Approximation 136

9.2.3 The Stochastic Volatility Correction 138

9.2.4 Uncorrelated Volatility 140

9.2.5　Probabilistic Representation 141

9.3　Numerical Computation 141

9.3.1 Solution of the Black-Scholes Problem 141

9.3.2 Computation of the Correction 142

10　Generalizations 145

10.1　Portfolio Optimization under Stochastic Volatility 145

10.1.1 Constant Volatility Merton Problem 145

10.1.2 Stochastic Volatility Merton Problem 147

10.1.3　A Practical Solution 151

10.2　Periodic Day Effect 153

10.3　Other Markovian Volatility Models 156

10.3.1 Markovian Jump Volatility Models 156

10.3.2　Pricing and Asymptotics 158

10.4　Martingale Approach 159

10.4.1　Main Argument 159

10.4.2　Decomposition Result 161

10.4.3　Comparison with the PDE Approach 164

10.5　Non-Markovian Models of Volatility 165

10.5.1 Setting:An Example 165

10.5.2　Asymptotics in the Non-Markovian Case 166

10.6　Multidimensional Models 169

11　Applications to Interest-Rate Models 174

11.1　Bond Pricing in the Vasicek Model 174

11.1.1　Review of the Constant Volatility Vasicek Model 174

11.1.2　Stochastic Volatility Vasicek Models 177

11.2　Bond Option Pricing 182

11.2.1　The Constant Volatility Case 183

11.2.2　Correction for Stochastic Volatility 185

11.2.3 Implications 189

11.3　Asymptotics around the CIR Model 190

11.4　Illustration from Data 192

11.4.1　Variogram Analysis 192

11.4.2　Yield Curve Fitting 193

Bibliography 195

Index 199