On representing claims for coherent risk measures

TL;DR

This paper explores the representation of claims for coherent risk measures, introducing generalized concepts of time-consistency and m-stability, and establishing their equivalence and relation to markets with proportional transaction costs.

Contribution

It generalizes the notions of time-consistency and m-stability for coherent risk measures and proves their equivalence, linking them to markets with proportional transaction costs.

Findings

Equivalence of m-stability and time-consistency for coherent risk measures.

Any coherent risk measure can be represented by a market with proportional transaction costs.

Markets with proportional transaction costs correspond to coherent risk measures under certain conditions.

Abstract

We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (weak and strong) time-consistency with respect to a portfolio of assets, generalizing the one defined by Delbaen. In a similar way we extend the notion of m-stability, by introducing weak and strong versions. We then prove that the two concepts of m-stability and time-consistency are still equivalent, thus giving necessary and sufficient conditions for a coherent risk measure to be represented by a market with proportional transaction costs. We go on to deduce that, under a separability assumption, any coherent risk measure is strongly time-consistent with respect to a suitably chosen countable portfolio, and show the converse: that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, essentially establishing the equivalence of the two concepts.