Holtz Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

1. Auflage 2010
ISBN: 978-3-642-16004-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 77, 192 Seiten, eBook

Reihe: Lecture Notes in Computational Science and Engineering

This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

Zielgruppe

Research

Autoren/Hrsg.

Holtz, Markus

Inhaltsverzeichnis

1;Preface;4
2;Contents;6
3;1 Introduction;8
4;2 Dimension-wise Decompositions;17
4.1;2.1 Classical ANOVA Decomposition;19
4.1.1;2.1.1 Effective Dimensions;22
4.1.2;2.1.2 Error Bounds;25
4.2;2.2 Anchored-ANOVA Decomposition;27
4.2.1;2.2.1 Effective Dimensions;28
4.2.2;2.2.2 Error Bounds;32
5;3 Dimension-wise Quadrature;34
5.1;3.1 Classical Multivariate Quadrature Methods;34
5.1.1;3.1.1 Monte Carlo;36
5.1.2;3.1.2 Quasi-Monte Carlo;39
5.1.3;3.1.3 Product Methods;43
5.2;3.2 Dimension-wise Quadrature Methods;48
5.2.1;3.2.1 Truncation and Discretization;48
5.2.2;3.2.2 Error and Costs;50
5.2.3;3.2.3 A priori Construction;52
5.2.4;3.2.4 Dimension-adaptive Construction;54
6;4 Sparse Grid Quadrature;56
6.1;4.1 Sparse Grid Methods;56
6.1.1;4.1.1 Classical Construction;57
6.1.2;4.1.2 Delayed Basis Sequences;60
6.1.3;4.1.3 Generalised Sparse Grids;64
6.1.4;4.1.4 Dimension-adaptive Sparse Grids;66
6.2;4.2 Optimal Sparse Grids in Weighted Spaces;68
6.2.1;4.2.1 Cost-Benefit Ratio;69
6.2.2;4.2.2 Cost Analysis;72
6.2.3;4.2.3 Error Analysis;74
6.2.4;4.2.4 Analysis of Error versus Cost;76
6.3;4.3 Relation to Dimension-wise Quadrature;79
7;5 Dimension Reduction and Smoothing;82
7.1;5.1 Dimension Reduction;82
7.1.1;5.1.1 Random Walk, Brownian Bridge, PCA;83
7.1.2;5.1.2 Linear transformations;87
7.2;5.2 Domain Decomposition;94
7.2.1;5.2.1 Root Finding;94
7.2.2;5.2.2 Hyperplane Arrangements;95
7.2.3;5.2.3 Conditional Sampling;103
8;6 Validation and Applications;106
8.1;6.1 Interest Rates Derivatives;109
8.1.1;6.1.1 Zero Coupon Bonds;109
8.1.2;6.1.2 Collateralized Mortgage Obligations;114
8.2;6.2 Path-dependent Options;119
8.2.1;6.2.1 Asian options;120
8.2.2;6.2.2 Barrier options;125
8.3;6.3 Performance-dependent Options;130
8.3.1;6.3.1 Framework and Pricing Formulas;130
8.3.2;6.3.2 Numerical Results;136
8.4;6.4 Asset-Liability Management in Life Insurance;141
8.4.1;6.4.1 Model and Integral Representation;141
8.4.2;6.4.2 Numerical Results;150
8.5;6.5 Summary and Discussion;154
9;7 Summary and Conclusions;157
10;A Discrepancy and Tractability;161
10.1;A.1 Reproducing Kernel Hilbert Spaces;161
10.2;A.2 Notions of Discrepancy;165
10.3;A.3 Tractability Results;171
10.4;References;176
11;Index;183

Über Autor(innen)

• 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