Set-valued risk measures for conical market models

TL;DR

This paper introduces set-valued risk measures for conical market models, providing primal and dual representations, and shows how scalar risk measures with multiple assets are special cases within this framework.

Contribution

It develops a comprehensive set-valued risk measure framework for conical market models, including primal and dual representations, and unifies scalar and set-valued risk measures.

Findings

Collection of super-hedging initial endowments forms set-valued coherent risk measures.

Scalar risk measures with multiple assets are special cases of the set-valued framework.

Primal and dual representation results are established for these risk measures.

Abstract

Set-valued risk measures on $L^p_d$ with $0 \leq p \leq \infty$ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Scalar risk measures with multiple eligible assets also turn out to be a special case within the set-valued framework.