Quasi-Monte Carlo Methods in Finance
With Application to Optimal Asset Allocation
1. Auflage
Erschienen am 10. November 2008
138 Seiten
978-3-8366-1664-5 (ISBN)
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 use
martingale 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.
Weitere Details
Weitere Ausgaben
Person
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.
Inhalt
1 - Quasi-Monte Carlo Methods in Finance With Application to Optimal Asset Allocation [Seite 1]
1.1 - Abstract [Seite 3]
1.2 - Acknowledgment [Seite 4]
1.3 - Contents [Seite 5]
1.4 - List of Figures [Seite 7]
1.5 - Introduction [Seite 9]
1.6 - 1 Monte Carlo and quasi-Monte Carlomethods [Seite 12]
1.6.1 - 1.1 Numerical integration [Seite 12]
1.6.2 - 1.2 Evaluation of integrals with Monte Carlo methods [Seite 13]
1.6.3 - 1.3 Quasi-Monte Carlo methods [Seite 14]
1.6.3.1 - 1.3.1 Introduction [Seite 14]
1.6.3.2 - 1.3.2 Discrepancy [Seite 14]
1.6.3.3 - 1.3.3 The Koksma-Hlawka inequality [Seite 16]
1.6.4 - 1.4 Classical constructions [Seite 17]
1.6.4.1 - 1.4.1 One-dimensional sequences [Seite 17]
1.6.4.2 - 1.4.2 Multi-dimensional sequences [Seite 18]
1.6.5 - 1.5 (t,m,s)-nets and (t,s)-sequences [Seite 21]
1.6.5.1 - 1.5.1 Variance reduction [Seite 21]
1.6.5.2 - 1.5.2 Nets and sequences [Seite 22]
1.6.5.3 - 1.5.3 Two constructions for (t,s)-sequences [Seite 24]
1.6.6 - 1.6 Digital nets and sequences [Seite 31]
1.6.7 - 1.7 Lattice rules [Seite 32]
1.6.8 - 1.8 The curse of dimension revisited [Seite 33]
1.6.8.1 - 1.8.1 Padding techniques [Seite 34]
1.6.8.2 - 1.8.2 Latin Supercube sampling [Seite 34]
1.6.9 - 1.9 Time consumption of the various point generators [Seite 36]
1.6.10 - 1.10 quasi-Monte Carlo methods in Finance [Seite 37]
1.6.10.1 - 1.10.1 Example: Arithmetic option [Seite 37]
1.6.10.2 - 1.10.2 Path generation [Seite 38]
1.6.10.3 - 1.10.3 Sampling size [Seite 45]
1.6.10.4 - 1.10.4 Results [Seite 47]
1.7 - 2 Malliavin Calculus [Seite 51]
1.7.1 - 2.1 Wiener-It^o chaos expansion [Seite 51]
1.7.2 - 2.2 Skorohod integral [Seite 57]
1.7.3 - 2.3 Differentiation of random variables [Seite 61]
1.7.4 - 2.4 Examples of Malliavin derivatives [Seite 75]
1.7.5 - 2.5 The Clark-Ocone formula [Seite 76]
1.7.6 - 2.6 The generalized Clark-Ocone formula [Seite 81]
1.7.7 - 2.7 Multivariate Malliavin Calculus [Seite 89]
1.8 - 3 Asset Allocation [Seite 93]
1.8.1 - 3.1 Problem formulation [Seite 93]
1.8.1.1 - 3.1.1 Financial market model [Seite 93]
1.8.1.2 - 3.1.2 Wealth process [Seite 95]
1.8.1.3 - 3.1.3 Expected utility [Seite 95]
1.8.1.4 - 3.1.4 Portfolio problem [Seite 96]
1.8.1.5 - 3.1.5 Equivalent static problem [Seite 97]
1.8.1.6 - 3.1.6 Optimal portfolio [Seite 99]
1.8.2 - 3.2 Solution of the portfolio problem [Seite 105]
1.8.2.1 - 3.2.1 Optimal portfolio [Seite 105]
1.8.2.2 - 3.2.2 Optimal portfolio with constant relative risk aversion (CRRA) [Seite 105]
1.9 - 4 Implementation [Seite 108]
1.9.1 - 4.1 A single state variable model with explicit solution [Seite 108]
1.9.2 - 4.2 Simulation-based approach [Seite 111]
1.9.3 - 4.3 SDE system as multidimensional SDE [Seite 112]
1.9.4 - 4.4 Error analysis [Seite 113]
1.9.4.1 - 4.4.1 Discretisation error [Seite 114]
1.9.4.2 - 4.4.2 Conditional expectation approximation error [Seite 115]
1.9.5 - 4.5 Numerical results [Seite 117]
1.9.5.1 - 4.5.1 One year time horizon [Seite 119]
1.9.5.2 - 4.5.2 Two year time horizon [Seite 122]
1.9.5.3 - 4.5.3 Five year time horizon [Seite 125]
1.9.5.4 - 4.5.4 Experiments with a small time horizon [Seite 128]
1.10 - Conclusion [Seite 130]
1.11 - Summary [Seite 131]
1.12 - Bibliography [Seite 134]
1.1 - Abstract [Seite 3]
1.2 - Acknowledgment [Seite 4]
1.3 - Contents [Seite 5]
1.4 - List of Figures [Seite 7]
1.5 - Introduction [Seite 9]
1.6 - 1 Monte Carlo and quasi-Monte Carlomethods [Seite 12]
1.6.1 - 1.1 Numerical integration [Seite 12]
1.6.2 - 1.2 Evaluation of integrals with Monte Carlo methods [Seite 13]
1.6.3 - 1.3 Quasi-Monte Carlo methods [Seite 14]
1.6.3.1 - 1.3.1 Introduction [Seite 14]
1.6.3.2 - 1.3.2 Discrepancy [Seite 14]
1.6.3.3 - 1.3.3 The Koksma-Hlawka inequality [Seite 16]
1.6.4 - 1.4 Classical constructions [Seite 17]
1.6.4.1 - 1.4.1 One-dimensional sequences [Seite 17]
1.6.4.2 - 1.4.2 Multi-dimensional sequences [Seite 18]
1.6.5 - 1.5 (t,m,s)-nets and (t,s)-sequences [Seite 21]
1.6.5.1 - 1.5.1 Variance reduction [Seite 21]
1.6.5.2 - 1.5.2 Nets and sequences [Seite 22]
1.6.5.3 - 1.5.3 Two constructions for (t,s)-sequences [Seite 24]
1.6.6 - 1.6 Digital nets and sequences [Seite 31]
1.6.7 - 1.7 Lattice rules [Seite 32]
1.6.8 - 1.8 The curse of dimension revisited [Seite 33]
1.6.8.1 - 1.8.1 Padding techniques [Seite 34]
1.6.8.2 - 1.8.2 Latin Supercube sampling [Seite 34]
1.6.9 - 1.9 Time consumption of the various point generators [Seite 36]
1.6.10 - 1.10 quasi-Monte Carlo methods in Finance [Seite 37]
1.6.10.1 - 1.10.1 Example: Arithmetic option [Seite 37]
1.6.10.2 - 1.10.2 Path generation [Seite 38]
1.6.10.3 - 1.10.3 Sampling size [Seite 45]
1.6.10.4 - 1.10.4 Results [Seite 47]
1.7 - 2 Malliavin Calculus [Seite 51]
1.7.1 - 2.1 Wiener-It^o chaos expansion [Seite 51]
1.7.2 - 2.2 Skorohod integral [Seite 57]
1.7.3 - 2.3 Differentiation of random variables [Seite 61]
1.7.4 - 2.4 Examples of Malliavin derivatives [Seite 75]
1.7.5 - 2.5 The Clark-Ocone formula [Seite 76]
1.7.6 - 2.6 The generalized Clark-Ocone formula [Seite 81]
1.7.7 - 2.7 Multivariate Malliavin Calculus [Seite 89]
1.8 - 3 Asset Allocation [Seite 93]
1.8.1 - 3.1 Problem formulation [Seite 93]
1.8.1.1 - 3.1.1 Financial market model [Seite 93]
1.8.1.2 - 3.1.2 Wealth process [Seite 95]
1.8.1.3 - 3.1.3 Expected utility [Seite 95]
1.8.1.4 - 3.1.4 Portfolio problem [Seite 96]
1.8.1.5 - 3.1.5 Equivalent static problem [Seite 97]
1.8.1.6 - 3.1.6 Optimal portfolio [Seite 99]
1.8.2 - 3.2 Solution of the portfolio problem [Seite 105]
1.8.2.1 - 3.2.1 Optimal portfolio [Seite 105]
1.8.2.2 - 3.2.2 Optimal portfolio with constant relative risk aversion (CRRA) [Seite 105]
1.9 - 4 Implementation [Seite 108]
1.9.1 - 4.1 A single state variable model with explicit solution [Seite 108]
1.9.2 - 4.2 Simulation-based approach [Seite 111]
1.9.3 - 4.3 SDE system as multidimensional SDE [Seite 112]
1.9.4 - 4.4 Error analysis [Seite 113]
1.9.4.1 - 4.4.1 Discretisation error [Seite 114]
1.9.4.2 - 4.4.2 Conditional expectation approximation error [Seite 115]
1.9.5 - 4.5 Numerical results [Seite 117]
1.9.5.1 - 4.5.1 One year time horizon [Seite 119]
1.9.5.2 - 4.5.2 Two year time horizon [Seite 122]
1.9.5.3 - 4.5.3 Five year time horizon [Seite 125]
1.9.5.4 - 4.5.4 Experiments with a small time horizon [Seite 128]
1.10 - Conclusion [Seite 130]
1.11 - Summary [Seite 131]
1.12 - Bibliography [Seite 134]
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 It^o integrals, that we will study now.
