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A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle


  • Zachary Feinstein
  • Birgit Rudloff


A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.

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  • Zachary Feinstein & Birgit Rudloff, 2015. "A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle," Papers 1508.02367,, revised Jul 2016.
  • Handle: RePEc:arx:papers:1508.02367

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    References listed on IDEAS

    1. Andreas Löhne & Birgit Rudloff, 2014. "An Algorithm For Calculating The Set Of Superhedging Portfolios In Markets With Transaction Costs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-33.
    2. c{C}au{g}{i}n Ararat & Andreas H. Hamel & Birgit Rudloff, 2014. "Set-valued shortfall and divergence risk measures," Papers 1405.4905,, revised Sep 2017.
    3. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    4. Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
    5. 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.
    6. Matthias Ehrgott & Andreas Löhne & Lizhen Shao, 2012. "A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming," Journal of Global Optimization, Springer, vol. 52(4), pages 757-778, April.
    7. 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.
    8. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    9. Ignacio Cascos & Ilya Molchanov, 2013. "Multivariate risk measures: a constructive approach based on selections," Papers 1301.1496,, revised Jul 2016.
    10. Andreas Hamel & Andreas Löhne & Birgit Rudloff, 2014. "Benson type algorithms for linear vector optimization and applications," Journal of Global Optimization, Springer, vol. 59(4), pages 811-836, August.
    11. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702,, revised Jan 2013.
    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.
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    Cited by:

    1. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561,, revised Jan 2018.

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