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Set-valued shortfall and divergence risk measures

Author

Listed:
  • c{C}au{g}{i}n Ararat
  • Andreas H. Hamel
  • Birgit Rudloff

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.

Suggested Citation

  • c{C}au{g}{i}n Ararat & Andreas H. Hamel & Birgit Rudloff, 2014. "Set-valued shortfall and divergence risk measures," Papers 1405.4905, arXiv.org, revised Sep 2017.
    • Handle: RePEc:arx:papers:1405.4905
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    References listed on IDEAS

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    Cited by:

    1. c{C}au{g}{i}n Ararat & Zachary Feinstein, 2019. "Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations," Papers 1912.06916, arXiv.org, revised Sep 2020.
    2. Francesca Centrone & Emanuela Rosazza Gianin, 2020. "Capital Allocation For Set-Valued Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-16, February.
    3. Çağın Ararat & Zachary Feinstein, 2021. "Set-valued risk measures as backward stochastic difference inclusions and equations," Finance and Stochastics, Springer, vol. 25(1), pages 43-76, January.
    4. Zachary Feinstein & Birgit Rudloff, 2017. "A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle," Journal of Global Optimization, Springer, vol. 68(1), pages 47-69, May.
    5. c{C}au{g}{i}n Ararat & Birgit Rudloff, 2016. "Dual representations for systemic risk measures," Papers 1607.03430, arXiv.org, revised Jul 2019.
    6. Wang, Wei & Xu, Huifu & Ma, Tiejun, 2023. "Optimal scenario-dependent multivariate shortfall risk measure and its application in risk capital allocation," European Journal of Operational Research, Elsevier, vol. 306(1), pages 322-347.
    7. Zachary Feinstein & Birgit Rudloff, 2015. "A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle," Papers 1508.02367, arXiv.org, revised Jul 2016.
    8. Zachary Feinstein & Birgit Rudloff, 2015. "Multi-portfolio time consistency for set-valued convex and coherent risk measures," Finance and Stochastics, Springer, vol. 19(1), pages 67-107, January.
    9. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561, arXiv.org, revised Jan 2018.
    10. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org.
    11. Andreas H. Hamel & Frank Heyde, 2021. "Set-Valued T -Translative Functions and Their Applications in Finance," Mathematics, MDPI, vol. 9(18), pages 1-33, September.

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