Cash Sub-additive Risk Measures and Interest Rate Ambiguity

A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, non-financial 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. Several practical examples are presented and in such contexts cash additive risk measures cannot be used. Several dual 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. Some examples of dynamic cash sub-additive risk measures are provided via BSDEs. In contrast to the dynamic cash additive risk measures, the dynamic cash sub-additive risk measures are recursive.