Distributionally robust expected shortfall for convex risks

TL;DR

This paper addresses distributional risk through an optimal-transport framework with quadratic cost, deriving a dual formulation that reduces robust expectation calculations to a one-dimensional optimization once the $\lambda c$-transform is known. For convex, piecewise-linear payoffs, it proves a compact representation $f^{\lambda c}(x)=\max_i\{\langle m_i,x\rangle+c_i+\frac{1}{2\lambda}\|m_i\|^2\}$, enabling a closed-form or semi-closed-form computation of the distributionally robust expected shortfall (ES) under an ambiguity set around a baseline measure. The authors apply this to a risk-neutral call option and to a three-asset portfolio (including a call and a put) to quantify how robustness inflates ES via the ambiguity parameter $\theta$. The results illustrate the practical impact of model uncertainty on option-inclusive portfolios and show that, under quadratic transport cost, robust ES remains computationally tractable for convex piecewise-linear payoffs, supporting risk management under model misspecification.

Abstract

We study distributionally robust expected values under optimal transport distance with a quadratic cost function. In general the duality method, for this computation for the payoff function $f$, requires the computation of the $λc-$transform $f^{λc}$. We show that under the quadratic cost function there exists an intuitive and easily implementable representation of $f^{λc}$, if $f$ is convex and piecewise linear. We apply this to the robust expected shortfall under the risk-neutral measure of an unhedged call option, from the point of view of the writer, as well as that of a portfolio mixing underlying shares with a call and a put option.

Figures (3)

• Figure 1: Distributions to illustrate optimal transport distance under quadratic cost
• Figure 2: A heuristic interpretation of Theorem \ref{['thm:lc']}. Slope matters but vertical intercept does not. In comparison with $f$, payoff $f_1$ is more sensitive to the change of distribution, but $f_2$ not.
• Figure 3: $\lambda c$-transform, call and put payoff with same strike.

Theorems & Definitions (10)

• Theorem 1
• Example 1
• Example 2
• Example 3
• Theorem 2
• proof
• Example 4
• Definition 1
• Lemma 1
• proof