Measuring risk with multiple eligible assets

TL;DR

This paper analyzes the properties of risk measures based on multiple eligible assets, focusing on their nondegeneracy, finiteness, and continuity, and introduces an alternative dual representation approach.

Contribution

It provides new insights into the mathematical properties of multi-asset risk measures and offers a novel dual representation method using external set characterizations.

Findings

Risk measures are nondegenerate iff the pricing functional admits a positive supporting extension.

Finiteness and continuity depend on the interplay between the acceptance set and eligible portfolios.

Applications include set-valued risk measures, superhedging, and risk sharing.

Abstract

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.