Time consistency of dynamic risk measures in markets with transaction costs

TL;DR

This paper explores the properties of set-valued dynamic risk measures in markets with transaction costs, introducing a new concept called multi-portfolio time consistency and analyzing its relation to recursive representations.

Contribution

It introduces multi-portfolio time consistency as a new, stronger form of time consistency for set-valued risk measures and establishes its equivalence to recursive and additive properties.

Findings

Multi-portfolio time consistency is stronger than standard time consistency.

Recursive form of risk measures is equivalent to multi-portfolio time consistency.

Set-valued risk measures require new consistency notions in markets with transaction costs.

Abstract

The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measures can be generalized to set-valued risk measures in different ways. The most intuitive generalization is called time consistency. We will show that the equivalence between a recursive form of the risk measure and time consistency, which is a central result in the scalar case, does not hold in the set-valued framework. Instead, we propose an alternative generalization, which we will call multi-portfolio time consistency and show in the main result of the paper that this property is indeed equivalent to the recursive form as well as to an additive property for the acceptance sets. Multi-portfolio time consistency is a stronger property than time consistency. In the scalar case, both notions coincide.