provided by Research Papers in Economics
A MARKOVIAN APPROXIMATED SOLUTION TO A
PORTFOLIO MANAGEMENT PROBLEM *
Jacek B. Krawczyk
Victoria University of Wellington, PO Box 600, Wellington
New Zealand; fax +64-4-4955014
Email: Jacek. KrawczykQvuw. ac. nz ;
http://www.vuw.ac.nz/~jacek
Version β.lb
Abstract-: A portfolio management problem is approximated as a Markov decision
chain. The weak Eulcr scheme is applied to discrétise the time evolution of a portfolio
and an inverse distance method is used to describe the transition probabilities.
The approximating Markov decision problem is solved by value iteration. Numerical
solutions of varying degrees of accuracy to a few specific financial portfolio problems
arc obtained. A sample of a fund manager’s objective functions is analysed to toll
which of them generates an acceptable Valuc-at-Risk.
Keywords: Computational economics; portfolio management; approximating Markov
decision chains; weak Eulcr scheme.
JEL: C8, C63, D92, C87, Gll
AMS: 93E20, 93E25, 90C39, 90C40, 90A09
1. INTRODUCTION
The purpose of this paper is to describe a numeri-
cal method capable of solving a class of stochastic
optimal control problems, which includes portfolio
management. The paper draws the idea of solving
a continuous finite-horizon stochastic optimal con-
trol problem as a Markov decision chain from [7]
and [6]. In this paper, the weakly consistent Eulcr
discretisation scheme, used for the approximation
of the stochastic process, and a scaling in the
policy space have greatly improved the solutions,
relative to those reported previously.
Optimal portfolio management can be modelled
as a stochastic optimal control problem. One can
usually solve a problem of this class by solving the
Hamilton-Jacobi-Bcllnian (HJB) equation. This
is a complex procedure in any case. Often this
equation analytically insoluble and a numerical
* Research supported by VUW GSBGM and IGC. Helpful
comments by my colleagues: Graeme Guthrie, Martin
Lally and Leigh Roberts are gratefully acknowledged; all
remaining errors are mine.