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A Visual Classification of Local Martingales

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Abstract

This paper considers the problem of when a local martingale is a martingale or a universally integrable martingale, for the case of time-homogeneous scalar diffusions. Necessary and suffcient conditions of a geometric nature are obtained for answering this question. These results are widely applicable to problems in stochastic finance. For example, in order to apply risk-neutral pricing, one must first check that the chosen density process for an equivalent change of probability measure is in fact a martingale. If not, risk-neutral pricing is infeasible. Furthermore, even if the density process is a martingale, the possibility remains that the discounted price of some security could be a strict local martingale under the equivalent risk-neutral probability measure. In this case, well-known identities for option prices, such as put-call parity, may fail. Using our results, we examine a number of basic asset price models, and identify those that suffer from the above-mentioned difficulties.

Suggested Citation

  • Hardy Hulley & Eckhard Platen, 2008. "A Visual Classification of Local Martingales," Research Paper Series 238, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:238
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp238.pdf
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    Citations

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    Cited by:

    1. Antoine Jacquier & Martin Keller-Ressel, 2015. "Implied volatility in strict local martingale models," Papers 1508.04351, arXiv.org.
    2. Xiaoshan Chen & Yu-Jui Huang & Qingshuo Song & Chao Zhu, 2013. "The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations," Papers 1309.0046, arXiv.org, revised Mar 2017.
    3. Michael Schatz & Didier Sornette, 2017. "Uniform Integrability of a Single Jump Local Martingale with State-Dependent Characteristics," Swiss Finance Institute Research Paper Series 17-21, Swiss Finance Institute.
    4. Sascha Desmettre & Gunther Leobacher & L. C. G. Rogers, 2021. "Change of drift in one-dimensional diffusions," Finance and Stochastics, Springer, vol. 25(2), pages 359-381, April.
    5. Aleksandar Mijatovic & Mikhail Urusov, 2009. "On the Martingale Property of Certain Local Martingales," Papers 0905.3701, arXiv.org, revised Oct 2010.
    6. Sascha Desmettre & Gunther Leobacher & L. C. G. Rogers, 2019. "Change of drift in one-dimensional diffusions," Papers 1910.11904, arXiv.org, revised Dec 2020.

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