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Time consistency of dynamic risk measures in markets with transaction costs

Author

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  • Zachary Feinstein
  • Birgit Rudloff

Abstract

Set-valued dynamic risk measures are defined on with and with an image space in the power set of . Primal and dual representations of dynamic risk measures are deduced. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures as well as an additive property for the acceptance sets is equivalent to a stronger time consistency property called multi-portfolio time consistency.

Suggested Citation

  • Zachary Feinstein & Birgit Rudloff, 2013. "Time consistency of dynamic risk measures in markets with transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 13(9), pages 1473-1489, September.
  • Handle: RePEc:taf:quantf:v:13:y:2013:i:9:p:1473-1489
    DOI: 10.1080/14697688.2013.781668
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    References listed on IDEAS

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    1. Cheridito, Patrick & Stadje, Mitja, 2009. "Time-inconsistency of VaR and time-consistent alternatives," Finance Research Letters, Elsevier, vol. 6(1), pages 40-46, March.
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    5. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
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    7. Bion-Nadal, Jocelyne, 2009. "Time consistent dynamic risk processes," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 633-654, February.
    8. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    9. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702, arXiv.org, revised Jan 2013.
    10. Kovacevic Raimund M., 2012. "Conditional risk and acceptability mappings as Banach-lattice valued mappings," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 1-18, March.
    11. repec:dau:papers:123456789/353 is not listed on IDEAS
    12. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    13. Freddy Delbaen & Shige Peng & Emanuela Rosazza Gianin, 2008. "Representation of the penalty term of dynamic concave utilities," Papers 0802.1121, arXiv.org, revised Dec 2009.
    14. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
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    Citations

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    Cited by:

    1. 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.
    2. Yanhong Chen & Yijun Hu, 2019. "Set-Valued Law Invariant Coherent And Convex Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-18, May.
    3. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    4. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978, arXiv.org, revised Jul 2019.
    5. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Jun 2020.
    6. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561, arXiv.org, revised Jan 2018.
    7. 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.
    8. 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.
    9. Çağin Ararat & Andreas H. Hamel & Birgit Rudloff, 2017. "Set-Valued Shortfall And Divergence Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-48, August.
    10. 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.
    11. 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.
    12. 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.
    13. Fei Sun & Yijun Hu, 2018. "Dynamic risk measures on variable exponent Bochner--Lebesgue spaces," Papers 1806.01166, arXiv.org, revised Jul 2019.
    14. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.

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