Journal of Convex Analysis 26 (2019), No. 1, 245--267
Copyright Heldermann Verlag 2019
On Representing and Hedging Claims for Coherent Risk Measures
Saul Jacka
Dept. of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom
s.d.jacka@warwick.ac.uk
Seb Armstrong
Dept. of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom
seb.armstrong@gmail.com
Abdelkarem Berkaoui
College of Sciences, Al-Imam Mohammed Ibn Saud Islamic University, P. O. Box 84880, Riyadh 11681, Saudi Arabia
berkaoui@yahoo.fr
[Abstract-pdf]
\def\cF{\mathcal{F}} 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 [{\it The structure of m-stable sets and in particular of the set of risk neutral measures}, in: In Memoriam Paul-Andr{\'e} Meyer, Springer, Berlin et al. (2006) 215--258] for the problem of reserving in a collection of num\'eraires {\bf V}, called {\bf 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: {\it Thinking coherently}, Risk 10 (1997) 68--71; {\it Coherent measures of risk}, Math. Finance 9(3) (1999) 203--228]. We also prove that {\bf V}-m-stability is equivalent to time-consistency when reserving in portfolios of {\bf V}, which is of particular interest to insurers.
Keywords: Coherent risk measures, m-stability, time-consistency, Fatou property, reserving, hedging, representation, pricing mechanism, average value at risk.
MSC: 91B24, 46N10, 91B30, 46E30, 91G80, 60E05, 60G99, 90C48.
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