Multivariate risk measures: a constructive approach based on selections

TL;DR

This paper introduces a new set-valued risk measure for multivariate portfolios that accounts for exchange rules and transaction costs, extending classical models with a selection-based approach.

Contribution

It proposes a novel selection-based definition of set-valued risk measures that are coherent, convex, and law invariant, with dual representations and approximation methods.

Findings

The selection risk measure is coherent and convex.

It has a dual representation facilitating analysis.

Connections to existing set-valued risk measures are established.

Abstract

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel, Heyde and Rudloff (2013).