Set-valued shortfall and divergence risk measures

Abstract

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.

References in corpus (3)

• Benson type algorithms for linear vector optimization and applications
• Multiportfolio time consistency for set-valued convex and coherent risk measures
• Lagrange Duality in Set Optimization

Cited by in corpus (6)

• Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems
• Multiportfolio time consistency for set-valued convex and coherent risk measures
• Dual representations for systemic risk measures
• A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle
• Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations
• Coherent Risk Measure on $L^0$: NA Condition, Pricing and Dual Representation