: Mario Rometsch
: Quasi-Monte Carlo Methods in Finance. With Application to Optimal Asset Allocation
: Diplomica Verlag GmbH
: 9783836616645
: 1
: CHF 38.00
:
: Wirtschaft
: English
: 138
: kein Kopierschutz/DRM
: PC/MAC/eReader/Tablet
: PDF
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.
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 Itˆo integrals, that we will study now.
Quasi-Monte Carlo Methods in Finance With Application to Optimal Asset Allocation1
Abstract3
Acknowledgment4
Contents5
List of Figures7
Introduction9
1 Monte Carlo and quasi-Monte Carlomethods12
1.1 Numerical integration12
1.2 Evaluation of integrals with Monte Carlo methods13
1.3 Quasi-Monte Carlo methods14
1.3.1 Introduction14
1.3.2 Discrepancy14
1.3.3 The Koksma-Hlawka inequality16
1.4 Classical constructions17
1.4.1 One-dimensional sequences17
1.4.2 Multi-dimensional sequences18
1.5 (t,m,s)-nets and (t,s)-sequences21
1.5.1 Variance reduction21
1.5.2 Nets and sequences22
1.5.3 Two constructions for (t,s)-sequences24
1.6 Digital nets and sequences31
1.7 Lattice rules32
1.8 The curse of dimension revisited33
1.8.1 Padding techniques34
1.8.2 Latin Supercube sampling34
1.9 Time consumption of the various point generators36
1.10 quasi-Monte Carlo methods in Finance37
1.10.1 Example: Arithmetic option37
1.10.2 Path generation38
1.10.3 Sampling size45
1.10.4 Results47
2 Malliavin Calculus51
2.1 Wiener-Itˆo chaos expansion51
2.2 Skorohod integral57
2.3 Differentiation of random variables61
2.4 Examples of Malliavin derivatives75
2.5 The Clark-Ocone formula76
2.6 The generalized Clark-Ocone formula81
2.7 Multivariate Malliavin Calculus89
3 Asset Allocation93
3.1 Problem formulation93
3.1.1 Financial market model93
3.1.2 Wealth process95
3.1.3 Expected utility95
3.1.4 Portfolio problem96
3.1.5 Equivalent static problem97
3.1.6 Optimal portfolio99
3.2 Solution of the portfolio problem105
3.2.1 Optimal portfolio105
3.2.2 Optimal portfolio with constant relative risk aversion (CRRA)105
4 Implementation108
4.1 A single state variable model with explicit solution108
4.2 Simulation-based approach111
4.3 SDE system as multidimensional SDE112
4.4 Error analysis113
4.4.1 Discretisation error114
4.4.2 Conditional expectation approximation error115
4.5 Numerical results117
4.5.1 One year time horizon119
4.5.2 Two year time horizon122
4.5.3 Five year time horizon125
4.5.4 Experiments with a small time horizon128
Conclusion130
Summary131
Bibliography134