IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1003.4797.html
   My bibliography  Save this paper

Hedging under arbitrage

Author

Listed:
  • 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.

Suggested Citation

  • Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    • Handle: RePEc:arx:papers:1003.4797
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1003.4797
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, July-Dece.
    2. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2009.
    3. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    4. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2007.
    5. Wang, Xingchun, 2021. "Pricing volatility-equity options under the modified constant elasticity of variance model," Finance Research Letters, Elsevier, vol. 38(C).
    6. Kevin John Fergusson, 2018. "Less-Expensive Pricing and Hedging of Extreme-Maturity Interest Rate Derivatives and Equity Index Options Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 3-2018.
    7. Pal, Soumik & Protter, Philip, 2010. "Analysis of continuous strict local martingales via h-transforms," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1424-1443, August.
    8. Paolo Guasoni & Miklós Rásonyi, 2015. "Fragility of arbitrage and bubbles in local martingale diffusion models," Finance and Stochastics, Springer, vol. 19(2), pages 215-231, April.
    9. Martin HERDEGEN & Martin SCHWEIZER, 2015. "Economics-Based Financial Bubbles (and Why They Imply Strict Local Martingales)," Swiss Finance Institute Research Paper Series 15-05, Swiss Finance Institute.
    10. repec:uts:finphd:40 is not listed on IDEAS
    11. Ke Du & Eckhard Platen, 2016. "Benchmarked Risk Minimization," Mathematical Finance, Wiley Blackwell, vol. 26(3), pages 617-637, July.
    12. Shane Miller & Eckhard Platen, 2010. "Real-World Pricing for a Modified Constant Elasticity of Variance Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(2), pages 147-175.
    13. Soumik Pal & Philip Protter, 2007. "Analysis of continuous strict local martingales via h-transforms," Papers 0711.1136, arXiv.org, revised Jun 2010.
    14. Martin Herdegen & Martin Schweizer, 2018. "Semi‐efficient valuations and put‐call parity," Mathematical Finance, Wiley Blackwell, vol. 28(4), pages 1061-1106, October.
    15. Baldeaux Jan & Ignatieva Katja & Platen Eckhard, 2014. "A tractable model for indices approximating the growth optimal portfolio," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(1), pages 1-21, February.
    16. Martin HERDEGEN & Martin SCHWEIZER, 2016. "Economically Consistent Valuations and Put-Call Parity," Swiss Finance Institute Research Paper Series 16-02, Swiss Finance Institute.
    17. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299, arXiv.org, revised Dec 2011.
    18. Eckhard Platen, 2008. "A Unifying Approach to Asset Pricing," Research Paper Series 227, Quantitative Finance Research Centre, University of Technology, Sydney.
    19. Fergusson, Kevin, 2020. "Less-Expensive Valuation And Reserving Of Long-Dated Variable Annuities When Interest Rates And Mortality Rates Are Stochastic," ASTIN Bulletin, Cambridge University Press, vol. 50(2), pages 381-417, May.
    20. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2018. "Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts," Papers 1801.07044, arXiv.org, revised Jan 2018.
    21. Sergio Pulido, 2010. "The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions," Papers 1012.3102, arXiv.org, revised Jan 2014.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1003.4797. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.