Set-valued shortfall and divergence risk measures

This paper is concerned with the utility-based risk of a financial position in a multi-asset market with frictions. Risk is quantified by set-valued risk measures, and market frictions are modeled by conical/convex random solvency regions representing proportional transaction costs or illiquidity effects, and convex random sets representing trading constraints. First, with a general set-valued risk measure, the effect of having trading opportunities on the risk measure is considered, and a corresponding dual representation theorem is given. Then, assuming individual utility functions for the assets, utility-based shortfall and divergence risk measures are defined, which form two classes of set-valued convex risk measures. Minimal penalty functions are computed in terms of the vector versions of the well-known divergence functionals (generalized relative entropy). As special cases, set-valued versions of the entropic risk measure and the average value at risk are obtained. The general results on the effect of market frictions are applied to the utility-based framework and conditions concerning applicability are presented.