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Superhedging and Dynamic Risk Measures under Volatility Uncertainty

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  • Marcel Nutz
  • H. Mete Soner
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    Abstract

    We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a c\`adl\`ag nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

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    File URL: http://arxiv.org/pdf/1011.2958
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1011.2958.

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    Date of creation: Nov 2010
    Date of revision: Jun 2012
    Publication status: Published in SIAM Journal of Control and Optimization, 50/4, 2065--2089, (2012)
    Handle: RePEc:arx:papers:1011.2958

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    Web page: http://arxiv.org/

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    Cited by:
    1. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227, arXiv.org, revised Feb 2014.
    2. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
    3. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and Duality in Nondominated Discrete-Time Models," Papers 1305.6008, arXiv.org, revised Feb 2014.
    4. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.
    5. Patrick Bei├čner, 2013. "Radner equilibria under ambiguous volatility," Working Papers 493, Bielefeld University, Center for Mathematical Economics.
    6. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.

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