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The joint distribution of Parisian and hitting times of the Brownian motion with application to Parisian option pricing

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  • Dassios, Angelos
  • Zhang, You You

Abstract

We study the joint law of Parisian time and hitting time of a drifted Brownian motion by using a three-state semi-Markov model, obtained through perturbation. We obtain a martingale, to which we can apply the optional sampling theorem and derive the double Laplace transform. This general result is applied to address problems in option pricing. We introduce a new option related to Parisian options, being triggered when the age of an excursion exceeds a certain time or/and a barrier is hit. We obtain an explicit expression for the Laplace transform of its fair price.

Suggested Citation

  • Dassios, Angelos & Zhang, You You, 2016. "The joint distribution of Parisian and hitting times of the Brownian motion with application to Parisian option pricing," LSE Research Online Documents on Economics 64959, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:64959
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    References listed on IDEAS

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    1. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    2. Marc Chesney & Laurent Gauthier, 2006. "American Parisian options," Finance and Stochastics, Springer, vol. 10(4), pages 475-506, December.
    3. Ning Cai & Nan Chen & Xiangwei Wan, 2010. "Occupation Times of Jump-Diffusion Processes with Double Exponential Jumps and the Pricing of Options," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 412-437, May.
    4. J. Anderluh & J. Weide, 2009. "Double-sided Parisian option pricing," Finance and Stochastics, Springer, vol. 13(2), pages 205-238, April.
    5. Marco Avellaneda & Lixin Wu, 1999. "Pricing Parisian-Style Options With A Lattice Method," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(01), pages 1-16.
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    Cited by:

    1. Dassios, Angelos & Li, Luting, 2020. "Explicit asymptotic on first passage times of diffusion processes," LSE Research Online Documents on Economics 103087, London School of Economics and Political Science, LSE Library.
    2. Angelos Dassios & Luting Li, 2018. "Explicit Asymptotics on First Passage Times of Diffusion Processes," Papers 1806.08161, arXiv.org.
    3. Guglielmo D'Amico & Filippo Petroni, 2020. "A micro-to-macro approach to returns, volumes and waiting times," Papers 2007.06262, arXiv.org.
    4. Gongqiu Zhang & Lingfei Li, 2021. "A General Approach for Parisian Stopping Times under Markov Processes," Papers 2107.06605, arXiv.org.

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    More about this item

    Keywords

    Parisian options; excursion time; three state semi-Markov model; Laplace transform;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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