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Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles

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  • Beatrice Acciaio
  • Hans Föllmer
  • Irina Penner

Abstract

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito et al. (Electron. J. Probab. 11(3):57–106, 2006 ). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalised probability measures on the optional σ-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular, we show how “bubbles” may appear in the dynamic penalisation, and how they cause a breakdown of asymptotic safety of the risk assessment procedure. Copyright Springer-Verlag 2012

Suggested Citation

  • Beatrice Acciaio & Hans Föllmer & Irina Penner, 2012. "Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles," Finance and Stochastics, Springer, vol. 16(4), pages 669-709, October.
    • Handle: RePEc:spr:finsto:v:16:y:2012:i:4:p:669-709
    DOI: 10.1007/s00780-012-0176-1
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Saul Jacka & Seb Armstrong & Abdelkarem Berkaoui, 2017. "On representing and hedging claims for coherent risk measures," Papers 1703.03638, arXiv.org, revised Feb 2018.
    2. Gabriela Kov'av{c}ov'a & Birgit Rudloff & Igor Cialenco, 2020. "Acceptability maximization," Papers 2012.11972, arXiv.org.
    3. 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.
    4. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Papers 1304.4853, arXiv.org.
    5. Yanhong Chen & Zachary Feinstein, 2022. "Set-valued dynamic risk measures for processes and for vectors," Finance and Stochastics, Springer, vol. 26(3), pages 505-533, July.
    6. E. Kromer & L. Overbeck & K. Zilch, 2019. "Dynamic systemic risk measures for bounded discrete time processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 77-108, August.
    7. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2016. "A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective," Papers 1603.09030, arXiv.org, revised Jan 2017.
    8. Drapeau, Samuel & Jamneshan, Asgar, 2016. "Conditional preference orders and their numerical representations," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 106-118.
    9. Aloisio Araujo & Alain Chateauneuf & José Heleno Faro & Bruno Holanda, 2019. "Updating pricing rules," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(2), pages 335-361, September.
    10. Fei Sun & Jingchao Li & Jieming Zhou, 2018. "Dynamic risk measures with fluctuation of market volatility under Bochne-Lebesgue space," Papers 1806.01166, arXiv.org, revised Mar 2024.
    11. Irina Penner & Anthony Réveillac, 2014. "Risk measures for processes and BSDEs," Post-Print hal-00814702, HAL.
    12. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2018. "A Unified Approach to Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 204-221, February.
    13. Daniel Bartl, 2016. "Conditional nonlinear expectations," Papers 1612.09103, arXiv.org, revised Mar 2019.
    14. Marlon Moresco & M'elina Mailhot & Silvana M. Pesenti, 2023. "Uncertainty Propagation and Dynamic Robust Risk Measures," Papers 2308.12856, arXiv.org, revised Feb 2024.
    15. Teemu Pennanen & Ari-Pekka Perkkio, 2016. "Convex duality in optimal investment and contingent claim valuation in illiquid markets," Papers 1603.02867, arXiv.org.
    16. Irina Penner & Anthony Réveillac, 2013. "Risk measures for processes and BSDEs," Working Papers hal-00814702, HAL.
    17. Irina Penner & Anthony Réveillac, 2015. "Risk measures for processes and BSDEs," Finance and Stochastics, Springer, vol. 19(1), pages 23-66, January.
    18. Elisa Mastrogiacomo & Emanuela Rosazza Gianin, 2019. "Time-consistency of risk measures: how strong is such a property?," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 287-317, June.
    19. Eisele Karl-Theodor & Kupper Michael, 2016. "Asymptotically stable dynamic risk assessments," Statistics & Risk Modeling, De Gruyter, vol. 33(1-2), pages 41-50, September.
    20. Samuel Drapeau & Michael Kupper, 2013. "Risk Preferences and Their Robust Representation," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 28-62, February.

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    More about this item

    Keywords

    Dynamic convex risk measures; Cash flows; Discounting ambiguity; Model ambiguity; Robust representation; Time consistency; Dynamic penalisation; Asymptotic safety; Bubbles; Cash subadditivity; 60G35; 91B30; 91B16; D81;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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