Merton’s problem for an investor with a benchmark in a Barndorff-Nielsen and Shephard market

Abstract

To try to outperform an externally given benchmark with known weights is the most common equity mandate in the financial industry. For quantitative investors, this task is predominantly approached by optimizing their portfolios consecutively over short time horizons with one-period models. We seek in this paper to provide a theoretical justification to this practice when the underlying market is of Barndorff-Nielsen and Shephard type. This is done by verifying that an investor who seeks to maximize her expected terminal exponential utility of wealth in excess of her benchmark will in fact use an optimal portfolio equivalent to the one-period Markowitz mean-variance problem in continuum under the corresponding Black-Scholes market. Further, we can represent the solution to the optimization problem as in Feynman-Kac form. Hence, the problem, and its solution, is analogous to Merton’s classical portfolio problem, with the main difference that Merton maximizes expected utility of terminal wealth, not wealth in excess of a benchmark.