Set-valued shortfall and divergence risk measures

Çagin Ararat, Andreas Hamel, Birgit Rudloff

Publication: Scientific journalJournal articlepeer-review


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.

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Original languageEnglish
Pages (from-to)1750026
JournalInternational Journal of Theoretical and Applied Finance
Issue number5
Publication statusPublished - 2017

Austrian Classification of Fields of Science and Technology (ÖFOS)

  • 101024 Probability theory
  • 101007 Financial mathematics
  • 502009 Corporate finance

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