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Quasi-Monte Carlo Methods in Finance. With Application to Optimal Asset Allocation

:	Mario Rometsch
:	Quasi-Monte Carlo Methods in Finance. With Application to Optimal Asset Allocation
:	Diplomica Verlag GmbH
:	9783836616645
:	1
:	CHF 38.30
:

:	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 Allocation	1
	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