Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance (eBook)

• Scientific employee at the Institute for Numerical Simulation at the University of Bonn (July 2004 - January 2009) • Involved in several teaching activities and research projects in the area of computational finance partly in close cooperation with financial institutions • Since January 2009 at head office of Baloise Group working on the introduction of stochastic models for life insurance portfolios

Preface 4
Contents 6
1 Introduction 8
2 Dimension-wise Decompositions 17
2.1 Classical ANOVA Decomposition 19
2.1.1 Effective Dimensions 22
2.1.2 Error Bounds 25
2.2 Anchored-ANOVA Decomposition 27
2.2.1 Effective Dimensions 28
2.2.2 Error Bounds 32
3 Dimension-wise Quadrature 34
3.1 Classical Multivariate Quadrature Methods 34
3.1.1 Monte Carlo 36
3.1.2 Quasi-Monte Carlo 39
3.1.3 Product Methods 43
3.2 Dimension-wise Quadrature Methods 48
3.2.1 Truncation and Discretization 48
3.2.2 Error and Costs 50
3.2.3 A priori Construction 52
3.2.4 Dimension-adaptive Construction 54
4 Sparse Grid Quadrature 56
4.1 Sparse Grid Methods 56
4.1.1 Classical Construction 57
4.1.2 Delayed Basis Sequences 60
4.1.3 Generalised Sparse Grids 64
4.1.4 Dimension-adaptive Sparse Grids 66
4.2 Optimal Sparse Grids in Weighted Spaces 68
4.2.1 Cost-Benefit Ratio 69
4.2.2 Cost Analysis 72
4.2.3 Error Analysis 74
4.2.4 Analysis of Error versus Cost 76
4.3 Relation to Dimension-wise Quadrature 79
5 Dimension Reduction and Smoothing 82
5.1 Dimension Reduction 82
5.1.1 Random Walk, Brownian Bridge, PCA 83
5.1.2 Linear transformations 87
5.2 Domain Decomposition 94
5.2.1 Root Finding 94
5.2.2 Hyperplane Arrangements 95
5.2.3 Conditional Sampling 103
6 Validation and Applications 106
6.1 Interest Rates Derivatives 109
6.1.1 Zero Coupon Bonds 109
6.1.2 Collateralized Mortgage Obligations 114
6.2 Path-dependent Options 119
6.2.1 Asian options 120
6.2.2 Barrier options 125
6.3 Performance-dependent Options 130
6.3.1 Framework and Pricing Formulas 130
6.3.2 Numerical Results 136
6.4 Asset-Liability Management in Life Insurance 141
6.4.1 Model and Integral Representation 141
6.4.2 Numerical Results 150
6.5 Summary and Discussion 154
7 Summary and Conclusions 157
A Discrepancy and Tractability 161
A.1 Reproducing Kernel Hilbert Spaces 161
A.2 Notions of Discrepancy 165
A.3 Tractability Results 171
References 176
Index 183