Bounds on Choquet Risk Measures in Finite Product Spaces with Ambiguous Marginals

TL;DR

This paper extends optimal transport theory to nonadditive measures, providing bounds for Choquet risk measures in finite spaces with ambiguous marginals, and explores their properties and applications in credit risk.

Contribution

It generalizes optimal transport to capacities, characterizes solutions explicitly, and connects to linear programming, with applications to counterparty credit risk.

Findings

Explicit solutions for finite marginal spaces.

Optimal transport for capacities is a linear program.

Numerical examples demonstrate the approach and applications.

Abstract

We investigate the problem of finding upper and lower bounds for a Choquet risk measure of a nonlinear function of two risk factors, when the marginal distributions of the risk factors are ambiguous and represented by nonadditive measures on the marginal spaces and the joint nonadditive distribution on the product space is unknown. We treat this problem as a generalization of the optimal transport problem to the setting of nonadditive measures. We provide explicit characterizations of the optimal solutions for finite marginal spaces, and we investigate some of their properties. We further discuss the connections with linear programming, showing that the optimal transport problems for capacities are linear programs, and we also characterize their duals explicitly. Finally, we investigate a series of numerical examples, including a comparison with the classical optimal transport problem, and applications to counterparty credit risk.