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Optimal arbitrage under model uncertainty

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  • Daniel Fernholz
  • Ioannis Karatzas
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    Abstract

    In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.

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    File URL: http://arxiv.org/pdf/1202.2999
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1202.2999.

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    Date of creation: Feb 2012
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    Publication status: Published in Annals of Applied Probability 2011, Vol. 21, No. 6, 2191-2225
    Handle: RePEc:arx:papers:1202.2999

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    Web page: http://arxiv.org/

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    1. RØdiger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
    2. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    3. Tiziano Vargiolu & Silvia Romagnoli, 2000. "Robustness of the Black-Scholes approach in the case of options on several assets," Finance and Stochastics, Springer, vol. 4(3), pages 325-341.
    4. Alexander Schied, 2007. "Optimal investments for risk- and ambiguity-averse preferences: a duality approach," Finance and Stochastics, Springer, vol. 11(1), pages 107-129, January.
    5. Jörg Vorbrink, 2010. "Financial markets with volatility uncertainty," Working Papers 441, Bielefeld University, Center for Mathematical Economics.
    6. Joerg Vorbrink, 2010. "Financial markets with volatility uncertainty," Papers 1012.1535, arXiv.org, revised Dec 2010.
    7. Marcel Nutz, 2010. "Random G-expectations," Papers 1009.2168, arXiv.org, revised Sep 2013.
    8. Nicole El Karoui & Monique Jeanblanc-Picquè & Steven E. Shreve, 1998. "Robustness of the Black and Scholes Formula," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 93-126.
    9. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 1(2), pages 149-177, November.
    10. Gunter H. Meyer, 2006. "The Black Scholes Barenblatt Equation For Options With Uncertain Volatility And Its Application To Static Hedging," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(05), pages 673-703.
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