Distributionally robust Expected Shortfall for convex payoffs

TL;DR

This paper develops a tractable method for computing distributionally robust Expected Shortfall for convex payoffs under quadratic cost perturbations, providing explicit formulas for unhedged call options.

Contribution

It introduces a new representation of the $ ext{c}$-transform for convex payoffs under quadratic cost, enabling explicit formulas for robust Expected Shortfall.

Findings

Derived a tractable representation of the $ ext{c}$-transform for convex payoffs.

Characterized robust Expected Shortfall as a 2D minimization problem.

Obtained a closed-form formula for robust Expected Shortfall of unhedged call options.

Abstract

We study distributionally robust Expected Shortfall when the distribution of the underlying is perturbed by a size quantified with optimal transport distance based on the quadratic cost function. In the dual version of the robust expectation problem, which is part of the robest expected shortfall problem, the computation of the so-called $\lambda c$-transform $f^{\lambda c}$ of payoff $f$ is required. We show that under the quadratic cost function there exists a tractable representation of $f^{\lambda c}$, if $f$ is convex. Furthermore, we show that robust expected shortfall can be characterized as the solution of a 2-dimensional minimization problem. We apply these results to obtain a closed-form formula for robust, with respect to the risk-neutral distribution, Expected Shortfall of an unhedged call option, from the point of view of the writer.