Contents
Preface xi
Acknowledgements xv
1 Introduction 1
1.1 Financial Assets 1
1.2 Derivative Securities 3
1.2.1 Options 3
1.2.2 Prices of Options on the S&P 500 Index 5
1.3 Modelling Assumptions 7
1.4 Arbitrage 9
2 Financial Mathematics in Continuous Time 11
2.1 Stochastic Processes and Filtrations 11
2.2 Classes of Processes 13
2.2.1 Markov Processes 13
2.2.2 Martingales 14
2.2.3 Finite- and Infinite-Variation Processes 14
2.3 Characteristic Functions 15
2.4 Stochastic Integrals and SDEs 16
2.5 Financial Mathematics in Continuous Time 17
2.5.1 Equivalent Martingale Measure 17
2.5.2 Pricing Formulas for European Options 19
2.6 Dividends 21
3 The Black–Scholes Model 23
3.1 The Normal Distribution 23
3.2 Brownian Motion 24
3.2.1 Definition 25
3.2.2 Properties 26
3.3 Geometric Brownian Motion 27
viii CONTENTS
3.4 The Black–Scholes Option Pricing Model 28
3.4.1 The Black–Scholes Market Model 29
3.4.2 Market Completeness 30
3.4.3 The Risk-Neutral Setting 30
3.4.4 The Pricing of Options under the Black–Scholes Model 30
4 Imperfections of the Black–Scholes Model 33
4.1 The Non-Gaussian Character 33
4.1.1 Asymmetry and Excess Kurtosis 33
4.1.2 Density Estimation 35
4.1.3 Statistical Testing 36
4.2 Stochastic Volatility 38
4.3 Inconsistency with Market Option Prices 39
5 Lévy Processes and OU Processes 43
5.1 Lévy Processes 44
5.1.1 Definition 44
5.1.2 Properties 45
5.2 OU Processes 47
5.2.1 Self-Decomposability 47
5.2.2 OU Processes 48
5.3 Examples of Lévy Processes 50
5.3.1 The Poisson Process 50
5.3.2 The Compound Poisson Process 51
5.3.3 The Gamma Process 52
5.3.4 The Inverse Gaussian Process 53
5.3.5 The Generalized Inverse Gaussian Process 54
5.3.6 The Tempered Stable Process 56
5.3.7 The Variance Gamma Process 57
5.3.8 The Normal Inverse Gaussian Process 59
5.3.9 The CGMY Process 60
5.3.10 The Meixner Process 62
5.3.11 The Generalized Hyperbolic Process 65
5.4 Adding an Additional Drift Term 67
5.5 Examples of OU Processes 67
5.5.1 The Gamma–OU Process 68
5.5.2 The IG–OU Process 69
5.5.3 Other Examples 70
6 Stock Price Models Driven by Lévy Processes 73
6.1 Statistical Testing 73
6.1.1 Parameter Estimation 73
6.1.2 Statistical Testing 74
CONTENTS ix
6.2 The Lévy Market Model 76
6.2.1 Market Incompleteness 77
6.2.2 The Equivalent Martingale Measure 77
6.2.3 Pricing Formulas for European Options 80
6.3 Calibration of Market Option Prices 82
7 Lévy Models with Stochastic Volatility 85
7.1 The BNS Model 85
7.1.1 The BNS Model with Gamma SV 87
7.1.2 The BNS Model with IG SV 88
7.2 The Stochastic Time Change 88
7.2.1 The Integrated CIR Time Change 89
7.2.2 The IntOU Time Change 90
7.3 The Lévy SV Market Model 91
7.4 Calibration of Market Option Prices 97
7.4.1 Calibration of the BNS Models 97
7.4.2 Calibration of the Lévy SV Models 98
7.5 Conclusion 98
8 Simulation Techniques 101
8.1 Simulation of Basic Processes 101
8.1.1 Simulation of Standard Brownian Motion 101
8.1.2 Simulation of a Poisson Process 102
8.2 Simulation of a Lévy Process 102
8.2.1 The Compound Poisson Approximation 103
8.2.2 On the Choice of the Poisson Processes 105
8.3 Simulation of an OU Process 107
8.4 Simulation of Particular Processes 108
8.4.1 The Gamma Process 108
8.4.2 The VG Process 109
8.4.3 The TS Process 111
8.4.4 The IG Process 111
8.4.5 The NIG Process 113
8.4.6 The Gamma–OU Process 114
8.4.7 The IG–OU Process 115
8.4.8 The CIR Process 117
8.4.9 BNS Model 117
9 Exotic Option Pricing 119
9.1 Barrier and Lookback Options 119
9.1.1 Introduction 119
9.1.2 Black–Scholes Barrier and Lookback Option Prices 121
9.1.3 Lookback and Barrier Options in a Lévy Market 123
x CONTENTS
9.2 Other Exotic Options 125
9.2.1 The Perpetual American Call and Put Option 125
9.2.2 The Perpetual Russian Option 126
9.2.3 Touch-and-Out Options 126
9.3 Exotic Option Pricing by Monte Carlo Simulation 127
9.3.1 Introduction 127
9.3.2 Monte Carlo Pricing 127
9.3.3 Variance Reduction by Control Variates 129
9.3.4 Numerical Results 132
9.3.5 Conclusion 134
10 Interest-Rate Models 135
10.1 General Interest-Rate Theory 135
10.2 The Gaussian HJM Model 138
10.3 The Lévy HJM Model 141
10.4 Bond Option Pricing 142
10.5 Multi-Factor Models 144
Appendix A Special Functions 147
A.1 Bessel Functions 147
A.2 Modified Bessel Functions 148
A.3 The Generalized Hypergeometric Series 149
A.4 Orthogonal Polynomials 149
A.4.1 Hermite polynomials with parameter 149
A.4.2 Meixner–Pollaczek Polynomials 150
Appendix B Lévy Processes 151
B.1 Characteristic Functions 151
B.1.1 Distributions on the Nonnegative Integers 151
B.1.2 Distributions on the Positive Half-Line 151
B.1.3 Distributions on the Real Line 152
B.2 Lévy Triplets 153
B.2.1 γ 153
B.2.2 The Lévy Measure ν(dx) 154
Appendix C S&P 500 Call Option Prices 155
References 157
Index 165