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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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    References listed on IDEAS

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    1. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. 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.
    4. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    5. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    6. Constantinos Kardaras, 2009. "Finitely additive probabilities and the Fundamental Theorem of Asset Pricing," Papers 0911.5503, arXiv.org.
    7. David Heath & Eckhard Platen, 2002. "Perfect Hedging Of Index Derivatives Under A Minimal Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(07), pages 757-774.
    8. Constantinos Kardaras, 2008. "Balance, growth and diversity of financial markets," Annals of Finance, Springer, vol. 4(3), pages 369-397, July.
    9. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    10. 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.
    11. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. 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.
    13. 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.
    14. Eckhard Platen, 2008. "The Law of Minimal Price," Research Paper Series 215, Quantitative Finance Research Centre, University of Technology, Sydney.
    15. Eckhard Platen & Hardy Hulley, 2008. "Hedging for the Long Run," Research Paper Series 214, Quantitative Finance Research Centre, University of Technology, Sydney.
    16. 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, October.
    17. Constantinos Kardaras, 2008. "Balance, growth and diversity of financial markets," Papers 0803.1858, arXiv.org.
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    Cited by:

    1. Claudio Fontana & Wolfgang J. Runggaldier, 2012. "Diffusion-based models for financial markets without martingale measures," Papers 1209.4449, arXiv.org, revised Feb 2013.
    2. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series 296, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Constantinos Kardaras, 2012. "Valuation and parities for exchange options," Papers 1206.3220, arXiv.org, revised Nov 2014.
    4. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    5. Martin Larsson, 2013. "Non-Equivalent Beliefs and Subjective Equilibrium Bubbles," Papers 1306.5082, arXiv.org.

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