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Superhedging and Dynamic Risk Measures under Volatility Uncertainty

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  • Marcel Nutz
  • H. Mete Soner

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.

Suggested Citation

  • Marcel Nutz & H. Mete Soner, 2010. "Superhedging and Dynamic Risk Measures under Volatility Uncertainty," Papers 1011.2958, arXiv.org, revised Jun 2012.
  • Handle: RePEc:arx:papers:1011.2958
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    Cited by:

    1. Jakv{s}a Cvitani'c & Dylan Possamai & Nizar Touzi, 2014. "Moral Hazard in Dynamic Risk Management," Papers 1406.5852, arXiv.org, revised Mar 2015.
    2. Romain Blanchard & Laurence Carassus, 2017. "Convergence of utility indifference prices to the superreplication price in a multiple-priors framework," Papers 1709.09465, arXiv.org.
    3. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.
    4. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
    5. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    6. Francesca Biagini & Yinglin Zhang, 2017. "Reduced-form framework under model uncertainty," Papers 1707.04475, arXiv.org, revised Mar 2018.
    7. Francesca Biagini & Jacopo Mancin, 2016. "Robust Financial Bubbles," Papers 1602.05471, arXiv.org.
    8. Daniel Bartl & Michael Kupper & Ariel Neufeld, 2017. "Pathwise superhedging on prediction sets," Papers 1711.02764, arXiv.org.
    9. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.
    10. Marcel Nutz, 2014. "Robust Superhedging with Jumps and Diffusion," Papers 1407.1674, arXiv.org, revised Jul 2015.
    11. Luo, Peng & Wang, Falei, 2015. "On the comparison theorem for multi-dimensional G-SDEs," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 38-44.
    12. Drapeau, Samuel & Heyne, Gregor & Kupper, Michael, 2015. "Minimal supersolutions of BSDEs under volatility uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2895-2909.
    13. Nutz, Marcel, 2015. "Robust superhedging with jumps and diffusion," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4543-4555.
    14. Dylan Possamai & Xiaolu Tan & Chao Zhou, 2015. "Stochastic control for a class of nonlinear kernels and applications," Papers 1510.08439, arXiv.org, revised Jul 2017.
    15. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227, arXiv.org, revised Feb 2014.
    16. Peter Bank & Yan Dolinsky & Selim Gokay, 2014. "Super-replication with nonlinear transaction costs and volatility uncertainty," Papers 1411.1229, arXiv.org, revised Jun 2015.
    17. Bei├čner, Patrick, 2016. "Radner Equilibria under Ambiguous Volatility," Center for Mathematical Economics Working Papers 493, Center for Mathematical Economics, Bielefeld University.
    18. Neufeld, Ariel & Nutz, Marcel, 2014. "Measurability of semimartingale characteristics with respect to the probability law," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3819-3845.

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