Portfolio Optimisation within a Wasserstein Ball

TL;DR

This paper develops a framework for active portfolio management that optimally adjusts strategies within a Wasserstein ball around a benchmark, balancing risk and distributional proximity, with practical simulation methods.

Contribution

It introduces a novel approach combining Wasserstein distance and copula constraints for portfolio optimization, with a characterization of optimal strategies and a simulation-based computation method.

Findings

Optimal strategies improve risk measures while maintaining distributional closeness to the benchmark.

Investors' optimal terminal wealth concentrates in regions reducing risk relative to the benchmark.

The approach applies to various risk measures like Tail Value-at-Risk and distortion risk measures.

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 lie within a Wasserstein ball surrounding a benchmark's -- 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 minimises a distortion risk measure of terminal wealth. In a general (complete) market model, we prove that an optimal dynamic strategy exists and provide its characterisation 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.