On Representing and Hedging Claims for Coherent Risk Measures

We provide a dual characterisation of the weak$^*$-closure of a finite sum of cones in $L^\infty$ adapted to a discrete time filtration $\cF_t$: the $t^{th}$ cone in the sum contains bounded random variables that are $\cF_t$-measurable. Hence we obtain a generalisation of F. Delbaen's m-stability condition [The structure of m-stable sets and in particular of the set of risk neutral measures, in: In Memoriam Paul-Andr{é} Meyer, Springer, Berlin et al. (2006) 215–258] for the problem of reserving in a collection of numéraires V, called V-m-stability, provided these cones arise from acceptance sets of a dynamic coherent measure of risk [see P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath: Thinking coherently, Risk 10 (1997) 68–71; Coherent measures of risk, Math. Finance 9(3) (1999) 203–228]. We also prove that V-m-stability is equivalent to time-consistency when reserving in portfolios of V, which is of particular interest to insurers.