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Optimal Portfolios for Financial Markets with Wishart Volatility

Part of: Mathematical finance Stochastic systems and control

Published online by Cambridge University Press:  30 January 2018

Nicole Bäuerle  and
Zejing Li
Nicole Bäuerle*
Affiliation:
Karlsruhe Institute of Technology
Zejing Li*
Affiliation:
Karlsruhe Institute of Technology
*
∗ Postal address: Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
∗ Postal address: Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
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Abstract

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We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

Keywords

Wishart processportfolio problemCRRA utilitystochastic controlHamilton-Jacobi-Bellman equationmatrix exponential

MSC classification

Primary: 93E20: Optimal stochastic control 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91G10: Portfolio theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1025 - 1043
Copyright
© Applied Probability Trust 

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