Portfolio optimization in a multivariate jump-diffusion model

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Abstract
We demonstrate how the optimal portfolio, with respect to expected power utility, can numerically be derived if the underlying price processes are driven by geometric Lévy processes. According to a result by Nutz [ 26 ], the optimal portfolio weights are constant in this setting, both in the unrestricted case and if budget constraints are included. We make use of the Stochastic Gradient Descent Algorithm to numerically arrive at the optimal solution, even for a high-dimensional asset universe. We then showcase the approach, exploring a multivariate Merton model and a multivariate version of the Kou model, in which we investigate how the jumps and the dependence between them influence the optimal portfolio. The solution is compared to the standard Merton portfolio that is optimal in a multivariate Black–Scholes market. Finally, the presented methodology gives immediate access to the indifference pricing measure for the specific utility of our investor, and we demonstrate how to numerically obtain the corresponding utility-indifference price of non-vanilla derivatives within our framework.

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