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Dynamic monetary risk measures for bounded discrete-time processes

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  • Patrick Cheridito
  • Freddy Delbaen
  • Michael Kupper

Abstract

We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time, and we show how this property manifests itself in the corresponding process of acceptance sets. For processes of coherent and convex monetary risk measures admitting a robust representation with sigma-additive linear functionals, we give necessary and sufficient conditions for time-consistency in terms of the representing functionals.

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  • Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2004. "Dynamic monetary risk measures for bounded discrete-time processes," Papers math/0410453, arXiv.org.
    • Handle: RePEc:arx:papers:math/0410453
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    References listed on IDEAS

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    1. Wang, Tan, 2003. "Conditional preferences and updating," Journal of Economic Theory, Elsevier, vol. 108(2), pages 286-321, February.
    2. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    3. Berend Roorda & J. M. Schumacher & Jacob Engwerda, 2005. "Coherent Acceptability Measures In Multiperiod Models," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 589-612, October.
    4. Marco Frittelli, 2000. "Introduction to a theory of value coherent with the no-arbitrage principle," Finance and Stochastics, Springer, vol. 4(3), pages 275-297.
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    6. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    7. Weber, Stefan, 2003. "Distribution-Invariant Dynamic Risk Measures," SFB 373 Discussion Papers 2003,53, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Cited by:

    1. Nicole El Karoui & Claudia Ravanelli, 2007. "Cash Sub-additive Risk Measures and Interest Rate Ambiguity," Papers 0710.4106, arXiv.org.
    2. Castaneda, Pablo, 2006. "Long Term Risk Assessment in a Defined Contribution Pension System," MPRA Paper 3347, University Library of Munich, Germany, revised 30 Apr 2007.
    3. Alexander Schied, 2005. "Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach," SFB 649 Discussion Papers SFB649DP2005-051, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Aug 2006.

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