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On optimal arbitrage

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  • Daniel Fernholz
  • Ioannis Karatzas
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    Abstract

    In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.

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    File URL: http://arxiv.org/pdf/1010.4987
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1010.4987.

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    Date of creation: Oct 2010
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    Publication status: Published in Annals of Applied Probability 2010, Vol. 20, No. 4, 1179-1204
    Handle: RePEc:arx:papers:1010.4987

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    Web page: http://arxiv.org/

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    References

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    1. Adrian Banner & Daniel Fernholz, 2008. "Short-term relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, Springer, vol. 4(4), pages 445-454, October.
    2. Eckhard Platen & Hardy Hulley, 2008. "Hedging for the Long Run," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 214, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Loewenstein, Mark & Willard, Gregory A., 2000. "Rational Equilibrium Asset-Pricing Bubbles in Continuous Trading Models," Journal of Economic Theory, Elsevier, Elsevier, vol. 91(1), pages 17-58, March.
    4. Gourieroux, Christian & Jasiak, Joann, 2006. "Multivariate Jacobi process with application to smooth transitions," Journal of Econometrics, Elsevier, Elsevier, vol. 131(1-2), pages 475-505.
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    Cited by:
    1. Hugonnier, Julien, 2012. "Rational asset pricing bubbles and portfolio constraints," Journal of Economic Theory, Elsevier, Elsevier, vol. 147(6), pages 2260-2302.
    2. Erhan Bayraktar & Yu-Jui Huang, 2011. "Robust maximization of asymptotic growth under covariance uncertainty," Papers 1107.2988, arXiv.org, revised Sep 2013.
    3. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2012. "Strict local martingale deflators and valuing American call-type options," Finance and Stochastics, Springer, Springer, vol. 16(2), pages 275-291, April.
    4. Huy N. Chau & Peter Tankov, 2013. "Market models with optimal arbitrage," Papers 1312.4979, arXiv.org.
    5. Jarrow, Robert & Protter, Philip, 2012. "Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory," Finance Research Letters, Elsevier, Elsevier, vol. 9(2), pages 58-62.
    6. Constantinos Kardaras & Scott Robertson, 2012. "Robust maximization of asymptotic growth," LSE Research Online Documents on Economics, London School of Economics and Political Science, LSE Library 44994, London School of Economics and Political Science, LSE Library.
    7. Aleksandar Mijatović & Mikhail Urusov, 2012. "Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models," Finance and Stochastics, Springer, Springer, vol. 16(2), pages 225-247, April.
    8. Martin Larsson, 2013. "Non-Equivalent Beliefs and Subjective Equilibrium Bubbles," Papers 1306.5082, arXiv.org.
    9. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "On the Hedging of Options On Exploding Exchange Rates," Papers 1202.6188, arXiv.org, revised Nov 2013.
    10. Ruf, Johannes, 2013. "A new proof for the conditions of Novikov and Kazamaki," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 123(2), pages 404-421.
    11. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, Springer, vol. 18(1), pages 115-144, January.
    12. Johannes Ruf & Wolfgang Runggaldier, 2013. "A Systematic Approach to Constructing Market Models With Arbitrage," Papers 1309.1988, arXiv.org, revised Dec 2013.
    13. Ting-Kam Leonard Wong, 2014. "Optimization of relative arbitrage," Papers 1407.8300, arXiv.org, revised Aug 2014.

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