Set-Valued Risk Measures

Abstract

Set-valued risk measures are defined on an L^p space over R^d. The results presented are in the dynamic framework, with the image space of the dynamic risk measures in the power set of of a subspace of that L^p space for all times. Primal and dual representations are deduced for closed (conditionally) convex and (conditionally) coherent risk measures. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures is equivalent to a strong time consistency property called multi-portfolio time consistency. Further, multi-portfolio time consistency is shown to be equivalent to an additive property for the acceptance sets. When considering closed convex risk measures, it is possible to prove that multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the closed coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. Additionally, utilizing these equivalent properties, it is possible to generate the multi-portfolio time consistent version of any set-valued risk measure.

Under a finite probability space, we propose an algorithm for calculating multi-portfolio time consistent set-valued risk measures in discrete time. Market models with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. Additionally, we motivate why the proposed procedure can be viewed as a set-valued Bellman's principle. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems.

As examples of dynamic set-valued risk measures, we consider and provide numerical examples of superhedging under proportional and convex transaction costs, the relaxed worst case risk measure, average value at risk (as well as a description of the multi-portfolio time consistent version), and the set-valued entropic risk measure.

Finally, we give an overview of three other methods for defining (dynamic) set-valued risk measures. In particular, we prove under which assumptions results within these approaches coincide, and how properties like the primal and dual representations and time consistency in the different approaches compare to each other.