Distributionally robust Expected Shortfall for convex payoffs

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.