Portfolio Optimization within a Wasserstein Ball

Abstract

We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy’s risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lies within a Wasserstein ball surrounding a benchmark’s terminal wealth—being distributionally close—and that have a specified dependence/copula tying state-by-state outcomes to it. The investor then chooses the alternative strategy that minimizes a distortion risk measure of terminal wealth. In a general complete market model, we prove that an optimal dynamic strategy exists and provide its characterization through the notion of isotonic projections. We further propose a simulation approach to calculate the optimal strategy’s terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different copula and risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped, and lower- and upper-tail distortion risk measures as examples. We find that investors’ optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving the benchmark’s structure.