Multi-currency reserving for coherent risk measures

TL;DR

This paper develops a framework for dynamic multi-currency reserving under coherent risk measures, establishing conditions for time-consistent portfolios and linking the problem to a generalized asset pricing theorem.

Contribution

It introduces optional m-stability for multi-currency reserving and connects it to a fundamental asset pricing theorem in a market with transaction costs.

Findings

Reserving portfolios are time-consistent if optional m-stability holds.

The problem is equivalent to dynamic trading in baskets of currencies.

A version of the Fundamental Theorem of Asset Pricing is proved in this context.

Abstract

We examine the problem of dynamic reserving for risk in multiple currencies under a general coherent risk measure. The reserver requires to hedge risk in a time-consistent manner by trading in baskets of currencies. We show that reserving portfolios in multiple currencies $\mathbf{V}$ are time-consistent when (and only when) a generalisation of Delbaen's m-stability condition \cite{D06}, termed optional $\V$-m-stability, holds. We prove a version of the Fundamental Theorem of Asset Pricing in this context. We show that this problem is equivalent to dynamic trading across baskets of currencies (rather than just pairwise trades) in a market with proportional transaction costs and with a frictionless final period.