| Quasi-Monte Carlo Methods in Finance With Application to Optimal Asset Allocation | 1 |
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| Abstract | 3 |
| Acknowledgment | 4 |
| Contents | 5 |
| List of Figures | 7 |
| Introduction | 9 |
| 1 Monte Carlo and quasi-Monte Carlomethods | 12 |
| 1.1 Numerical integration | 12 |
| 1.2 Evaluation of integrals with Monte Carlo methods | 13 |
| 1.3 Quasi-Monte Carlo methods | 14 |
| 1.3.1 Introduction | 14 |
| 1.3.2 Discrepancy | 14 |
| 1.3.3 The Koksma-Hlawka inequality | 16 |
| 1.4 Classical constructions | 17 |
| 1.4.1 One-dimensional sequences | 17 |
| 1.4.2 Multi-dimensional sequences | 18 |
| 1.5 (t,m,s)-nets and (t,s)-sequences | 21 |
| 1.5.1 Variance reduction | 21 |
| 1.5.2 Nets and sequences | 22 |
| 1.5.3 Two constructions for (t,s)-sequences | 24 |
| 1.6 Digital nets and sequences | 31 |
| 1.7 Lattice rules | 32 |
| 1.8 The curse of dimension revisited | 33 |
| 1.8.1 Padding techniques | 34 |
| 1.8.2 Latin Supercube sampling | 34 |
| 1.9 Time consumption of the various point generators | 36 |
| 1.10 quasi-Monte Carlo methods in Finance | 37 |
| 1.10.1 Example: Arithmetic option | 37 |
| 1.10.2 Path generation | 38 |
| 1.10.3 Sampling size | 45 |
| 1.10.4 Results | 47 |
| 2 Malliavin Calculus | 51 |
| 2.1 Wiener-Itˆo chaos expansion | 51 |
| 2.2 Skorohod integral | 57 |
| 2.3 Differentiation of random variables | 61 |
| 2.4 Examples of Malliavin derivatives | 75 |
| 2.5 The Clark-Ocone formula | 76 |
| 2.6 The generalized Clark-Ocone formula | 81 |
| 2.7 Multivariate Malliavin Calculus | 89 |
| 3 Asset Allocation | 93 |
| 3.1 Problem formulation | 93 |
| 3.1.1 Financial market model | 93 |
| 3.1.2 Wealth process | 95 |
| 3.1.3 Expected utility | 95 |
| 3.1.4 Portfolio problem | 96 |
| 3.1.5 Equivalent static problem | 97 |
| 3.1.6 Optimal portfolio | 99 |
| 3.2 Solution of the portfolio problem | 105 |
| 3.2.1 Optimal portfolio | 105 |
| 3.2.2 Optimal portfolio with constant relative risk aversion (CRRA) | 105 |
| 4 Implementation | 108 |
| 4.1 A single state variable model with explicit solution | 108 |
| 4.2 Simulation-based approach | 111 |
| 4.3 SDE system as multidimensional SDE | 112 |
| 4.4 Error analysis | 113 |
| 4.4.1 Discretisation error | 114 |
| 4.4.2 Conditional expectation approximation error | 115 |
| 4.5 Numerical results | 117 |
| 4.5.1 One year time horizon | 119 |
| 4.5.2 Two year time horizon | 122 |
| 4.5.3 Five year time horizon | 125 |
| 4.5.4 Experiments with a small time horizon | 128 |
| Conclusion | 130 |
| Summary | 131 |
| Bibliography | 134 |