IDEAS home Printed from https://ideas.repec.org/p/ehl/lserod/64959.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://eprints.lse.ac.uk/64959/
    File Function: Open access version.
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Angelos Dassios & Luting Li, 2018. "Explicit Asymptotics on First Passage Times of Diffusion Processes," Papers 1806.08161, arXiv.org.
    2. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Dassios, Angelos & Lim, Jia Wei, 2013. "Parisian option pricing: a recursive solution for the density of the Parisian stopping time," LSE Research Online Documents on Economics 58985, London School of Economics and Political Science, LSE Library.
    3. C. E. Phelan & D. Marazzina & G. Germano, 2020. "Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities," Quantitative Finance, Taylor & Francis Journals, vol. 20(6), pages 899-918, June.
    4. Carolyn E. Phelan & Daniele Marazzina & Guido Germano, 2021. "Pricing methods for $\alpha$-quantile and perpetual early exercise options based on Spitzer identities," Papers 2106.06030, arXiv.org.
    5. Dassios, Angelos & Qu, Yan & Zhao, Hongbiao, 2018. "Exact simulation for a class of tempered stable," LSE Research Online Documents on Economics 86981, London School of Economics and Political Science, LSE Library.
    6. Chenxu Li, 2016. "Bessel Processes, Stochastic Volatility, And Timer Options," Mathematical Finance, Wiley Blackwell, vol. 26(1), pages 122-148, January.
    7. Richard L. Warr & Cason J. Wight, 2020. "Error Bounds for Cumulative Distribution Functions of Convolutions via the Discrete Fourier Transform," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 881-904, September.
    8. Yera, Yoel G. & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2019. "Fitting procedure for the two-state Batch Markov modulated Poisson process," European Journal of Operational Research, Elsevier, vol. 279(1), pages 79-92.
    9. He, Gang & Wu, Wenqing & Zhang, Yuanyuan, 2018. "Analysis of a multi-component system with failure dependency, N-policy and vacations," Operations Research Perspectives, Elsevier, vol. 5(C), pages 191-198.
    10. Shu, Yin & Feng, Qianmei & Liu, Hao, 2019. "Using degradation-with-jump measures to estimate life characteristics of lithium-ion battery," Reliability Engineering and System Safety, Elsevier, vol. 191(C).
    11. David H Collins & Richard L Warr & Aparna V Huzurbazar, 2013. "An introduction to statistical flowgraph models for engineering systems," Journal of Risk and Reliability, , vol. 227(5), pages 461-470, October.
    12. Joseph Abate & Ward Whitt, 1999. "Computing Laplace Transforms for Numerical Inversion Via Continued Fractions," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 394-405, November.
    13. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2018. "Optimal liquidation under stochastic liquidity," Finance and Stochastics, Springer, vol. 22(1), pages 39-68, January.
    14. John F. Shortle & Martin J. Fischer & Percy H. Brill, 2007. "Waiting-Time Distribution of M/D N /1 Queues Through Numerical Laplace Inversion," INFORMS Journal on Computing, INFORMS, vol. 19(1), pages 112-120, February.
    15. Jeffrey P. Kharoufeh & Natarajan Gautam, 2004. "A fluid queueing model for link travel time moments," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(2), pages 242-257, March.
    16. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    17. Corsaro, Stefania & Kyriakou, Ioannis & Marazzina, Daniele & Marino, Zelda, 2019. "A general framework for pricing Asian options under stochastic volatility on parallel architectures," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1082-1095.
    18. Abdel Belkaid & Frederic Utzet, 2017. "Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 957-971, September.
    19. Steven Kou & Cindy Yu & Haowen Zhong, 2017. "Jumps in Equity Index Returns Before and During the Recent Financial Crisis: A Bayesian Analysis," Management Science, INFORMS, vol. 63(4), pages 988-1010, April.
    20. J. Li & A. Metzler & R. M. Reesor, 2017. "A structural framework for modelling contingent capital," Quantitative Finance, Taylor & Francis Journals, vol. 17(7), pages 1071-1088, July.

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ehl:lserod:64959. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: LSERO Manager (email available below). General contact details of provider: https://edirc.repec.org/data/lsepsuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.