Journal of Applied Probability

Optimal portfolios for financial markets with Wishart volatility

Nicole Bäuerle and Zejing Li

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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.

Article information

J. Appl. Probab., Volume 50, Number 4 (2013), 1025-1043.

First available in Project Euclid: 10 January 2014

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91G10: Portfolio theory

Wishart process portfolio problem CRRA utility stochastic control Hamilton-Jacobi-Bellman equation matrix exponential


Bäuerle, Nicole; Li, Zejing. Optimal portfolios for financial markets with Wishart volatility. J. Appl. Probab. 50 (2013), no. 4, 1025--1043. doi:10.1239/jap/1389370097.

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