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Hedging under arbitrage

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  • Johannes Ruf

Abstract

It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuous-time Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a square-integrable market price of risk. A new probability measure is constructed, which takes the place of an equivalent local martingale measure. In order to ensure the existence of the delta hedge, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The recently often discussed phenomenon of "bubbles" is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work.

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  • Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
  • Handle: RePEc:arx:papers:1003.4797
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    File URL: http://arxiv.org/pdf/1003.4797
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    References listed on IDEAS

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    1. Constantinos Kardaras, 2009. "Finitely additive probabilities and the Fundamental Theorem of Asset Pricing," Papers 0911.5503, arXiv.org.
    2. David Heath & Eckhard Platen, 2002. "Consistent pricing and hedging for a modified constant elasticity of variance model," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 459-467.
    3. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
    4. Loewenstein, Mark & Willard, Gregory A., 2000. "Rational Equilibrium Asset-Pricing Bubbles in Continuous Trading Models," Journal of Economic Theory, Elsevier, vol. 91(1), pages 17-58, March.
    5. Eckhard Platen & Hardy Hulley, 2008. "Hedging for the Long Run," Research Paper Series 214, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. S. D. Jacka, 1992. "A Martingale Representation Result and an Application to Incomplete Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 239-250.
    7. Constantinos Kardaras, 2008. "Balance, growth and diversity of financial markets," Annals of Finance, Springer, vol. 4(3), pages 369-397, July.
    8. Robert Fernholz & Ioannis Karatzas & Constantinos Kardaras, 2005. "Diversity and relative arbitrage in equity markets," Finance and Stochastics, Springer, vol. 9(1), pages 1-27, January.
    9. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2009. "Strict Local Martingale Deflators and Pricing American Call-Type Options," Papers 0908.1082, arXiv.org, revised Dec 2009.
    11. Eckhard Platen, 2008. "The Law of Minimal Price," Research Paper Series 215, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Constantinos Kardaras, 2008. "Balance, growth and diversity of financial markets," Papers 0803.1858, arXiv.org.
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    Cited by:

    1. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "On the Hedging of Options On Exploding Exchange Rates," Papers 1202.6188, arXiv.org, revised Nov 2013.

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