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# Quasi-Monte Carlo Methods in Finance

€ 43,00

pdf eBook
November 2008

## Beschreibung

### Beschreibung

Portfolio optimization is a widely studied problem in finance dating back to the work of Merton from the 1960s. While many approaches rely on dynamic programming, some recent contributions usemartingale techniques to determine the optimal portfolio allocation.Using the latter approach, we follow a journal article from 2003 and show how optimal portfolio weights can be represented in terms of conditional expectations of the state variables and their Malliavin derivatives.In contrast to other approaches, where Monte Carlo methods are used to compute the weights, here the simulation is carried out using Quasi-Monte Carlo methods in order to improve the efficiency. Despite some previous work on Quasi-Monte Carlo simulation of stochastic differential equations, we find them to dominate plain Monte Carlo methods. However, the theoretical optimal order of convergence is not achieved.With the help of some recent results concerning Monte-Carlo error estimation and backed by some computer experiments on a simple model with explicit solution, we provide a first guess, what could be a way around this difficulties.The book is organized as follows. In the first chapter we provide some general introduction to Quasi-Monte Carlo methods and show at hand of a simple example how these methods can be used to accelerate the plain Monte Carlo sampling approach. In the second part we provide a thourough introduction to Malliavin Calculus and derive some important calculation rules that will be necessary in the third chapter. Right there we will focus on portfolio optimization and and follow a recent journal article of Detemple, Garcia and Rindisbacher from there rather general market model to the optimal portfolio formula. Finally, in the last part we will implement this optimal portfolio by means of a simple model with explicit solution where we find that also their the Quasi-Monte Carlo approach dominates the Monte Carlo method in terms of efficiency and accuracy.

### Inhaltsverzeichnis

1;Quasi-Monte Carlo Methods in Finance With Application to Optimal Asset Allocation;1 1.1;Abstract;3 1.2;Acknowledgment;4 1.3;Contents;5 1.4;List of Figures;7 1.5;Introduction;9 1.6;1 Monte Carlo and quasi-Monte Carlomethods;12 1.6.1;1.1 Numerical integration;12 1.6.2;1.2 Evaluation of integrals with Monte Carlo methods;13 1.6.3;1.3 Quasi-Monte Carlo methods;14 1.6.3.1;1.3.1 Introduction;14 1.6.3.2;1.3.2 Discrepancy;14 1.6.3.3;1.3.3 The Koksma-Hlawka inequality;16 1.6.4;1.4 Classical constructions;17 1.6.4.1;1.4.1 One-dimensional sequences;17 1.6.4.2;1.4.2 Multi-dimensional sequences;18 1.6.5;1.5 (t,m,s)-nets and (t,s)-sequences;21 1.6.5.1;1.5.1 Variance reduction;21 1.6.5.2;1.5.2 Nets and sequences;22 1.6.5.3;1.5.3 Two constructions for (t,s)-sequences;24 1.6.6;1.6 Digital nets and sequences;31 1.6.7;1.7 Lattice rules;32 1.6.8;1.8 The curse of dimension revisited;33 1.6.8.1;1.8.1 Padding techniques;34 1.6.8.2;1.8.2 Latin Supercube sampling;34 1.6.9;1.9 Time consumption of the various point generators;36 1.6.10;1.10 quasi-Monte Carlo methods in Finance;37 1.6.10.1;1.10.1 Example: Arithmetic option;37 1.6.10.2;1.10.2 Path generation;38 1.6.10.3;1.10.3 Sampling size;45 1.6.10.4;1.10.4 Results;47 1.7;2 Malliavin Calculus;51 1.7.1;2.1 Wiener-Ito chaos expansion;51 1.7.2;2.2 Skorohod integral;57 1.7.3;2.3 Differentiation of random variables;61 1.7.4;2.4 Examples of Malliavin derivatives;75 1.7.5;2.5 The Clark-Ocone formula;76 1.7.6;2.6 The generalized Clark-Ocone formula;81 1.7.7;2.7 Multivariate Malliavin Calculus;89 1.8;3 Asset Allocation;93 1.8.1;3.1 Problem formulation;93 1.8.1.1;3.1.1 Financial market model;93 1.8.1.2;3.1.2 Wealth process;95 1.8.1.3;3.1.3 Expected utility;95 1.8.1.4;3.1.4 Portfolio problem;96 1.8.1.5;3.1.5 Equivalent static problem;97 1.8.1.6;3.1.6 Optimal portfolio;99 1.8.2;3.2 Solution of the portfolio problem;105 1.8.2.1;3.2.1 Optimal portfolio;105 1.8.2.2;3.2.2 Optimal portfolio with constant relative risk aversion (CRRA);105 1.9;4 Implementatio
n;108 1.9.1;4.1 A single state variable model with explicit solution;108 1.9.2;4.2 Simulation-based approach;111 1.9.3;4.3 SDE system as multidimensional SDE;112 1.9.4;4.4 Error analysis;113 1.9.4.1;4.4.1 Discretisation error;114 1.9.4.2;4.4.2 Conditional expectation approximation error;115 1.9.5;4.5 Numerical results;117 1.9.5.1;4.5.1 One year time horizon;119 1.9.5.2;4.5.2 Two year time horizon;122 1.9.5.3;4.5.3 Five year time horizon;125 1.9.5.4;4.5.4 Experiments with a small time horizon;128 1.10;Conclusion;130 1.11;Summary;131 1.12;Bibliography;134

### Portrait

Born in 1981, Mario Rometsch studied Mathematics and Economics at ulm university. Being both attracted to Financial mathematics and Numerical Analysis / Computer Science, he chose to write his diploma thesis in Computational Finance, at the point of intersection of both disciplines. Right now, Mario Rometsch is a fellow at the Research Training Group 1100 at ulm university, where he is pursuing his PhD studies with Adaptive Wavelet Methods.

### Leseprobe

Chapter 2 Malliavin CalculusIn this chapter, we will now introduce the theory of stochastic calculus of variations. Wewill follow the lecture notes [Øks97] in this chapter. The intention will be to define theMalliavin derivative, to derive the Clark-Ocone-Haussmann-formula and to familiarizewith these instruments. A more general but also more abstract approach can be foundin the book [Nua06] or in the ebook [ Us04]1.A starting point is the orthogonal expansion of square-integrable, measurable randomvariables in terms of iterated Ito integrals, that we will study now.
EAN: 9783836616645
Untertitel: With Application to Optimal Asset Allocation. Sprache: Englisch. Dateigröße in MByte: 1.
Verlag: Diplomica Verlag
Erscheinungsdatum: November 2008
Seitenanzahl: 148 Seiten
Format: pdf eBook
Kopierschutz: Wasserzeichen
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