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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2001-10
March 2001

Merton's Portfolio Optimization Problem in a Black & Scholes Market with non-Gaussian Stochastic Volatility of Ornstein-Uhlenbeck Type


Fred E. Benth, Kenneth H. Karlsen, and Kristin Reikvam


We study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock. The goal for the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non-Gaussian Ornstein-Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff-Nielsen and Shephard [3]. Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman-Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.

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This paper has now been published in Math. Finance 13, 215-244 (2003)