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A Class of Strict Local Martingales

Author

Listed:
  • Martin HERDEGEN

    (ETH Zurich)

  • Sebastian HERRMANN

    (ETH Zurich)

Abstract

Many results in stochastic analysis and mathematical finance involve local martingales. However, specific examples of strict local martingales are rare and analytically often rather unhandy. We study local martingales that follow a given deterministic function up to a random time ? at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local martingales, uniformly integrable martingales that are not in H¹, etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale. Moreover, we characterise all local martingale deflators and all equivalent local martingale measures for a given special semimartingale with respect to the smallest filtration that turns ? into a stopping time. Two new counter-examples show, using direct arguments only, that neither of the no-arbitrage conditions NA and NUPBR implies the other. The structural simplicity of these examples allows to understand the difference between NA and NUPBR on an intuitive level.

Suggested Citation

  • Martin HERDEGEN & Sebastian HERRMANN, 2014. "A Class of Strict Local Martingales," Swiss Finance Institute Research Paper Series 14-18, Swiss Finance Institute, revised Oct 2014.
    • Handle: RePEc:chf:rpseri:rp1418
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    References listed on IDEAS

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    1. Mark Loewenstein & Gregory A. Willard, 2000. "Local martingales, arbitrage, and viability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 16(1), pages 135-161.
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    4. 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.
    5. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    6. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
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    Cited by:

    1. Martin Herdegen & Sebastian Herrmann, 2017. "Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble," Papers 1711.06679, arXiv.org.
    2. Martin Herdegen, 2017. "No-Arbitrage In A Numéraire-Independent Modeling Framework," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 568-603, April.
    3. Yuri Kabanov & Constantinos Kardaras & Shiqi Song, 2016. "No arbitrage of the first kind and local martingale numéraires," Finance and Stochastics, Springer, vol. 20(4), pages 1097-1108, October.

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    More about this item

    Keywords

    Single jump; Strict local martingales; Stochastic integrals; Local martingale deflators; No arbitrage; No unbounded profit with bounded risk;
    All these keywords.

    JEL classification:

    • Y80 - Miscellaneous Categories - - Related Disciplines - - - Related Disciplines

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