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Diffusion-based models for financial markets without martingale measures

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  • Claudio Fontana
  • Wolfgang J. Runggaldier
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    Abstract

    We consider a general class of diffusion-based models and show that, even in the absence of an Equivalent Local Martingale Measure, the financial market may still be viable, in the sense that strong forms of arbitrage are excluded and portfolio optimisation problems can be meaningfully solved. Relying partly on the recent literature, we provide necessary and sufficient conditions for market viability in terms of the market price of risk process and martingale deflators. Regardless of the existence of a martingale measure, we show that the financial market may still be complete and contingent claims can be valued under the original (real-world) probability measure, provided we use as numeraire the Growth-Optimal Portfolio.

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    File URL: http://arxiv.org/pdf/1209.4449
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1209.4449.

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    Date of creation: Sep 2012
    Date of revision: Feb 2013
    Publication status: Published in Risk Measures and Attitudes (F. Biagini, A. Richter and H. Schlesinger, eds.), Springer, EAA Series, pages 45-81 (2013)
    Handle: RePEc:arx:papers:1209.4449

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    1. Eckhard Platen, 2009. "A Benchmark Approach to Investing and Pricing," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 253, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Schweizer, Martin, 1992. "Martingale densities for general asset prices," Journal of Mathematical Economics, Elsevier, vol. 21(4), pages 363-378.
    3. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Ball, Clifford A. & Torous, Walter N., 1983. "Bond Price Dynamics and Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 18(04), pages 517-531, December.
    5. Eckhard Platen & Wolfgang Runggaldier, 2004. "A Benchmark Approach to Filtering in Finance," Asia-Pacific Financial Markets, Springer, Springer, vol. 11(1), pages 79-105, March.
    6. Leonard Maclean & Edward Thorp & William Ziemba, 2010. "Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 10(7), pages 681-687.
    7. Eckhard Platen, 2004. "A Benchmark Approach to Finance," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 138, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Nicola Bruti-Liberati & Christina Nikitopoulos-Sklibosios & Eckhard Platen, 2010. "Real-world jump-diffusion term structure models," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 10(1), pages 23-37.
    9. Eckhard Platen & Wolfgang Runggaldier, 2007. "A Benchmark Approach to Portfolio Optimization under Partial Information," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 191, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Robert Jarrow & Dilip B. Madan, 1999. "Hedging contingent claims on semimartingales," Finance and Stochastics, Springer, Springer, vol. 3(1), pages 111-134.
    11. Eva Strasser, 2005. "Characterization of arbitrage-free markets," Papers math/0503473, arXiv.org.
    12. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, Springer, vol. 5(3), pages 327-341.
    13. Hardy Hulley, 2010. "The Economic Plausibility of Strict Local Martingales in Financial Modelling," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 279, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 280, Quantitative Finance Research Centre, University of Technology, Sydney.
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