Data-Driven Optimization of Ambiguous Reward-Risk Ratio Measures

Abstract

We investigate a class of distributionally robust optimization problems that have direct applications in finance. They are semi-infinite programming problems with ambiguous expectation constraints in which fractional functions represent reward-risk ratios. We develop a reformulation and algorithmic data-driven framework based on the Wasserstein metric to model ambiguity and to derive probabilistic guarantee that the ambiguity set contains the true probability distribution. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function, and yields a mathematical programming problem in a finite dimensional constraint space. We design modular bisection algorithms with finite convergence property. We specify new ambiguous portfolio optimization models for the Sharpe, Sortino, Sortino-Satchel, and Omega ratios. The computational study shows the applicability and scalability of the framework to solve large, industry-relevant size problems.