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Parisian Time of Reflected Brownian Motion with Drift on Rays and Its Application in Banking

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  • Angelos Dassios

    (Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK)

  • Junyi Zhang

    (Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK)

Abstract

In this paper, we study the Parisian time of a reflected Brownian motion with drift on a finite collection of rays. We derive the Laplace transform of the Parisian time using a recursive method, and provide an exact simulation algorithm to sample from the distribution of the Parisian time. The paper was motivated by the settlement delay in the real-time gross settlement (RTGS) system. Both the central bank and the participating banks in the system are concerned about the liquidity risk, and are interested in the first time that the duration of settlement delay exceeds a predefined limit. We reduce this problem to the calculation of the Parisian time. The Parisian time is also crucial in the pricing of Parisian type options; to this end, we will compare our results to the existing literature.

Suggested Citation

  • Angelos Dassios & Junyi Zhang, 2020. "Parisian Time of Reflected Brownian Motion with Drift on Rays and Its Application in Banking," Risks, MDPI, vol. 8(4), pages 1-14, December.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:4:p:127-:d:454257
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    References listed on IDEAS

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    1. Soramäki, Kimmo & Bech, Morten L. & Arnold, Jeffrey & Glass, Robert J. & Beyeler, Walter E., 2007. "The topology of interbank payment flows," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 317-333.
    2. Jean-Charles Rochet & Jean Tirole, 1996. "Controlling risk in payment systems," Proceedings, Board of Governors of the Federal Reserve System (U.S.), pages 832-869.
    3. Céline Labart & Jérôme Lelong, 2009. "Pricing Parisian options using Laplace transforms," Post-Print hal-00776703, HAL.
    4. Céline Labart & Jérôme Lelong, 2009. "Pricing Double Barrier Parisian Options Using Laplace Transforms," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(01), pages 19-44.
    5. Christopher Becher & Marco Galbiati & Merxe Tudela, 2008. "The timing and funding of CHAPS sterling payments," Economic Policy Review, Federal Reserve Bank of New York, vol. 14(Sep), pages 113-133.
    6. Angelos Dassios & Shanle Wu, 2010. "Perturbed Brownian motion and its application to Parisian option pricing," Finance and Stochastics, Springer, vol. 14(3), pages 473-494, September.
    7. J. Anderluh & J. Weide, 2009. "Double-sided Parisian option pricing," Finance and Stochastics, Springer, vol. 13(2), pages 205-238, April.
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