Scalar multivariate risk measures with a single eligible asset

TL;DR

This paper investigates scalar risk measures in markets with transaction costs, providing dual representations and a weaker form of time consistency that aligns with market frictions, especially in superhedging contexts.

Contribution

It introduces a weaker notion of time consistency for scalar risk measures in markets with transaction costs and establishes its equivalence to a backward recursion under fixed pricing.

Findings

Dual representation of scalar risk measures with market frictions

Weaker time consistency concept compatible with market models

Equivalence between time consistency and backward recursion

Abstract

In this paper we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs, as in Jouini and Kallal (1995), Roux and Zastawniak (2016), and Loehne and Rudloff (2014), that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.