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Coherent and convex monetary risk measures for unbounded càdlàg processes

Author

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  • Patrick Cheridito

    ()

  • Freddy Delbaen

    ()

  • Michael Kupper

    ()

Abstract

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set. Copyright Springer-Verlag 2006

Suggested Citation

  • Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2006. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 10(3), pages 427-448, September.
  • Handle: RePEc:spr:finsto:v:10:y:2006:i:3:p:427-448
    DOI: 10.1007/s00780-006-0017-1
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    References listed on IDEAS

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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
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    Citations

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    Cited by:

    1. Homem-de-Mello, Tito & Pagnoncelli, Bernardo K., 2016. "Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective," European Journal of Operational Research, Elsevier, vol. 249(1), pages 188-199.
    2. Antoon Pelsser & Mitja Stadje, 2014. "Time-Consistent And Market-Consistent Evaluations," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 25-65, January.
    3. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    4. Jocelyne Bion-Nadal, 2007. "Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk," Papers math/0703074, arXiv.org.
    5. Engsner, Hampus & Lindholm, Mathias & Lindskog, Filip, 2017. "Insurance valuation: A computable multi-period cost-of-capital approach," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 250-264.
    6. Loisel, Stéphane & Trufin, Julien, 2014. "Properties of a risk measure derived from the expected area in red," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 191-199.
    7. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2014. "A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time," Papers 1409.7028, arXiv.org, revised Sep 2017.
    8. Pospisil, Libor & Vecer, Jan & Xu, Mingxin, 2007. "Tradable measure of risk," MPRA Paper 5059, University Library of Munich, Germany.
    9. Christopher W. Miller & Insoon Yang, 2015. "Optimal Control of Conditional Value-at-Risk in Continuous Time," Papers 1512.05015, arXiv.org, revised Jan 2017.
    10. Masaaki Fukasawa & Mitja Stadje, 2017. "Perfect hedging under endogenous permanent market impacts," Papers 1702.01385, arXiv.org.
    11. repec:hal:wpaper:hal-00870224 is not listed on IDEAS
    12. Alexander S. Cherny, 2009. "Capital Allocation And Risk Contribution With Discrete-Time Coherent Risk," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 13-40.
    13. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.

    More about this item

    Keywords

    Coherent risk measures; Convex monetary risk measures; Coherent utility functionals; Concave monetary utility functionals; Unbounded càdlàg processes; Extension of risk measures; 91B30; 91B16; 60G07; 52A07; 46A55; 46A20; D81; C60; G18;

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • G18 - Financial Economics - - General Financial Markets - - - Government Policy and Regulation

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