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Contents
Prefaces.. . . . . . . . . . . . . . . V
Contents.. . . . . . . . . . . . . . . XIII
Notation.. . . . . . . . . . . . . . .XVII
Chapter 1 Modeling Tools for Financial Options . . . . . . . . . . 1
1.1 Options.. . . . . . . . 1
1.2 Model of the Financial Market. . . . . . . 8
1.3 Numerical Methods. . . . . . . . . . . . . . . . . 10
1.4 The Binomial Method. . . . . . . . . . . . . . . 12
1.5 Risk-Neutral Valuation. . . . . . . . . . . . . . 21
1.6 Stochastic Processes. . . . . . . . . . . . . . . . 25
1.6.1 Wiener Process. . . . . . . . . . . . . . . 26
1.6.2 Stochastic Integral. . . . . . . . . . . . 28
1.7 Stochastic Differential Equations. . . . . 31
1.7.1 Itˆo Process. . . . . . . . . . . . . . . . . . 31
1.7.2 Application to the Stock Market33
1.7.3 Risk-Neutral Valuation. . . . . . . . 36
1.7.4 Mean Reversion. . . . . . . . . . . . . . 37
1.7.5 Vector-Valued SDEs. . . . . . . . . . 39
1.8 Itˆo Lemma and Implications. . . . . . . . . 40
1.9 Jump Processes.. 45
Notes and Comments.. 48
Exercises.. . . . . . . . . . . . 52
Chapter 2 Generating Random Numbers with Specified
Distributions.. . . . . . . . . . 61
2.1 Uniform Deviates.62
2.1.1 Linear Congruential Generators62
2.1.2 Quality of Generators. . . . . . . . . 63
2.1.3 Random Vectors and Lattice Structure . . . . . . . . . . . . 64
2.1.4 Fibonacci Generators. . . . . . . . . 67
2.2 Transformed Random Variables. . . . . . . 69
2.2.1 Inversion.. 69
2.2.2 Transformations in IR1. . . . . . . . 70
2.3 Normally Distributed Random Variables . . . . . . . . . . . . . . . . . 72
2.3.1 Method of Box and Muller. . . . . 72
2.3.2 Variant of Marsaglia. . . . . . . . . . 73
2.3.3 Correlated Random Variables. . 75
2.4 Monte Carlo Integration. . . . . . . . . . . . . 77
2.5 Sequences of Numbers with Low Discrepancy . . . . . . . . . . . . . 79
2.5.1 Discrepancy. . . . . . . . . . . . . . . . . . 79
2.5.2 Examples of Low-Discrepancy Sequences . . . . . . . . . . 82
Notes and Comments.. 85
Exercises.. . . . . . . . . . . . 87
Chapter 3 Simulation with Stochastic Differential
Equations.. . . . . . . . . . . . . . 91
3.1 Approximation Error. . . . . . . . . . . . . . . . 92
3.2 Stochastic Taylor Expansion. . . . . . . . . 95
3.3 Examples of Numerical Methods. . . . . . 98
3.4 Intermediate Values. . . . . . . . . . . . . . . . . 102
3.5 Monte Carlo Simulation. . . . . . . . . . . . . 102
3.5.1 Integral Representation. . . . . . . . 103
3.5.2 The Basic Version for European Options . . . . . . . . . . . 104
3.5.3 Bias.. . . . . 107
3.5.4 Variance Reduction. . . . . . . . . . . 108
3.5.5 American Options. . . . . . . . . . . . 111
3.5.6 Further Hints. . . . . . . . . . . . . . . . 116
Notes and Comments.. 117
Exercises.. . . . . . . . . . . . 119
Chapter 4 Standard Methods for Standard Options . . . . . . . 123
4.1 Preparations.. . . . 124
4.2 Foundations of Finite-Difference Methods . . . . . . . . . . . . . . . . 126
4.2.1 Difference Approximation. . . . . . 126
4.2.2 The Grid.. 127
4.2.3 Explicit Method. . . . . . . . . . . . . . 128
4.2.4 Stability.. . 130
4.2.5 An Implicit Method. . . . . . . . . . . 133
4.3 Crank-Nicolson Method. . . . . . . . . . . . . 135
4.4 Boundary Conditions. . . . . . . . . . . . . . . . 138
4.5 American Options as Free Boundary Problems . . . . . . . . . . . 140
4.5.1 Early-Exercise Curve. . . . . . . . . . 141
4.5.2 Free Boundary Problems. . . . . . 143
4.5.3 Black-Scholes Inequality. . . . . . . 146
4.5.4 Obstacle Problems. . . . . . . . . . . . 148
4.5.5 Linear Complementarity for American Put Options . 151
4.6 Computation of American Options. . . . 152
4.6.1 Discretization with Finite Differences . . . . . . . . . . . . . 152
4.6.2 Iterative Solution. . . . . . . . . . . . . 154
4.6.3 An Algorithm for Calculating American Options . . . . 157
4.7 On the Accuracy.161
4.7.1 Elementary Error Control. . . . . 162
4.7.2 Extrapolation. . . . . . . . . . . . . . . . 165
4.8 Analytic Methods.165
4.8.1 Approximation Based on Interpolation . . . . . . . . . . . . 167
4.8.2 Quadratic Approximation. . . . . . 169
4.8.3 Analytic Method of Lines. . . . . . 172
4.8.4 Methods Evaluating Probabilities . . . . . . . . . . . . . . . . . 173
Notes and Comments.. 174
Exercises.. . . . . . . . . . . . 178
Chapter 5 Finite-Element Methods. . . . . 183
5.1 Weighted Residuals. . . . . . . . . . . . . . . . . 184
5.1.1 The Principle of Weighted Residuals . . . . . . . . . . . . . . 184
5.1.2 Examples of Weighting Functions . . . . . . . . . . . . . . . . . 186
5.1.3 Examples of Basis Functions. . . 187
5.2 Galerkin Approach with Hat Functions188
5.2.1 Hat Functions. . . . . . . . . . . . . . . . 189
5.2.2 Assembling. . . . . . . . . . . . . . . . . . 191
5.2.3 A Simple Application. . . . . . . . . 192
5.3 Application to Standard Options. . . . . 194
5.4 Error Estimates.. 198
5.4.1 Classical and Weak Solutions. . 199
5.4.2 Approximation on Finite-Dimensional Subspaces . . . 201
5.4.3 C´ea’s Lemma. . . . . . . . . . . . . . . . 202
Notes and Comments.. 205
Exercises.. . . . . . . . . . . . 206
Chapter 6 Pricing of Exotic Options. . . . 209
6.1 Exotic Options.. . 210
6.2 Options Depending on Several Assets. 211
6.3 Asian Options.. . . 214
6.3.1 The Payoff.214
6.3.2 Modeling in the Black-Scholes Framework . . . . . . . . . 215
6.3.3 Reduction to a One-Dimensional Equation . . . . . . . . . 216
6.3.4 Discrete Monitoring. . . . . . . . . . . 220
6.4 Numerical Aspects. . . . . . . . . . . . . . . . . . 222
6.4.1 Convection-Diffusion Problems. 222
6.4.2 Von Neumann Stability Analysis . . . . . . . . . . . . . . . . . 225
6.5 Upwind Schemes and Other Methods. . 226
6.6 High-Resolution Methods. . . . . . . . . . . . 231
6.6.1 The Lax-Wendroff Method. . . . . 231
6.6.2 Total Variation Diminishing. . . . 232
6.6.3 Numerical Dissipation. . . . . . . . . 233
Notes and Comments.. 235
Exercises.. . . . . . . . . . . . 237
Appendices.. . . . . . . . . . . . 239
A Financial Derivatives. . . . . . . . . . . . . . . . 239
A1 Investment and Risk. . . . . . . . . . 239
A2 Financial Derivatives. . . . . . . . . . 240
A3 Forwards and the No-Arbitrage Principle . . . . . . . . . . 243
A4 The Black-Scholes Equation. . . . 244
A5 Early-Exercise Curve. . . . . . . . . . 249
B Stochastic Tools.. 253
B1 Essentials of Stochastics. . . . . . . 253
B2 Advanced Topics. . . . . . . . . . . . . 257
B3 State-Price Process. . . . . . . . . . . 260
C Numerical Methods. . . . . . . . . . . . . . . . . 265
C1 Basic Numerical Tools. . . . . . . . . 265
C2 Iterative Methods for Ax = b. . . 270
C3 Function Spaces. . . . . . . . . . . . . . 272
D Complementary Material. . . . . . . . . . . . 277
D1 Bounds for Options. . . . . . . . . . . 277
D2 Approximation Formula. . . . . . . 279
D3 Software.. . 281
References.. . . . . . . . . . . . . 283
Index.. . . . . . . . . . . . . . . . . . 293
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