Cash Sub-additive Risk Measures and Interest Rate Ambiguity

TL;DR

This paper introduces cash sub-additive risk measures that better model interest rate ambiguity and default risk, extending traditional risk measures by relaxing the cash additivity axiom and providing new theoretical representations.

Contribution

It proposes a novel class of risk measures called cash sub-additive, addressing interest rate ambiguity and default risk, with new representations and dynamic formulations using BSDEs.

Findings

Cash sub-additive risk measures model interest rate ambiguity.

Representations involve penalty functions on sub-linear probability measures.

Dynamic measures are characterized via BSDEs with state-dependent generators.

Abstract

A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, nonfinancial and insurance positions. The debated cash additive axiom is relaxed into the cash sub additive axiom to preserve the original difference between the numeraire of the current reserve amounts and future positions. Consequently, cash sub-additive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented and in such contexts cash additive risk measures cannot be used. Several representations of the cash sub-additive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sub-linear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash sub-additive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash sub-additive risk measure.