SET-VALUED SHORTFALL AND DIVERGENCE RISK MEASURES

Çağin Ararat (), Andreas H. Hamel () and Birgit Rudloff ()
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Çağin Ararat: Department of Industrial Engineering, Bilkent University, Ankara 06800, Turkey
Andreas H. Hamel: School for Economics and Management, Free University Bozen, Bozen-Bolzano 39031, Italy
Birgit Rudloff: Institute for Statistics and Mathematics, Vienna University of Economics and Business, Vienna 1020, Austria

International Journal of Theoretical and Applied Finance (IJTAF), 2017, vol. 20, issue 05, 1-48

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.

Keywords: Optimized certainty equivalent; shortfall risk; divergence; relative entropy; entropic risk measure; average value at risk; set-valued risk measure; multivariate risk; incomplete preference; transaction cost; solvency cone; liquidity risk; infimal convolution; Lagrange duality; set optimization (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (8)

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DOI: 10.1142/S0219024917500261

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