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

Author

Listed:
  • Angelos Dassios

    (London School of Economics)

  • You You Zhang

    (London School of Economics)

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

  • Angelos Dassios & You You Zhang, 2016. "The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing," Finance and Stochastics, Springer, vol. 20(3), pages 773-804, July.
    • Handle: RePEc:spr:finsto:v:20:y:2016:i:3:d:10.1007_s00780-016-0302-6
    DOI: 10.1007/s00780-016-0302-6
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    References listed on IDEAS

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    1. Marc Chesney & Laurent Gauthier, 2006. "American Parisian options," Finance and Stochastics, Springer, vol. 10(4), pages 475-506, December.
    2. Céline Labart & Jérôme Lelong, 2009. "Pricing Parisian options using Laplace transforms," Post-Print hal-00776703, HAL.
    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. Dassios, Angelos & Wu, Shanle, 2011. "Brownian excursions in a corridor and related Parisian options," LSE Research Online Documents on Economics 32042, London School of Economics and Political Science, LSE Library.
    5. J. Anderluh & J. Weide, 2009. "Double-sided Parisian option pricing," Finance and Stochastics, Springer, vol. 13(2), pages 205-238, April.
    6. 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.
    7. 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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    Citations

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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. Guglielmo D'Amico & Filippo Petroni, 2020. "A micro-to-macro approach to returns, volumes and waiting times," Papers 2007.06262, arXiv.org.
    3. Gongqiu Zhang & Lingfei Li, 2021. "A General Approach for Parisian Stopping Times under Markov Processes," Papers 2107.06605, arXiv.org.
    4. Angelos Dassios & Luting Li, 2018. "Explicit Asymptotics on First Passage Times of Diffusion Processes," Papers 1806.08161, arXiv.org.
    5. Gongqiu Zhang & Lingfei Li, 2023. "A general approach for Parisian stopping times under Markov processes," Finance and Stochastics, Springer, vol. 27(3), pages 769-829, July.

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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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