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Pricing Double Barrier Parisian Options Using Laplace Transforms

Author

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  • CÉLINE LABART

    (INRIA Paris-Rocquencourt, MathFi Project, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France)

  • JÉRÔME LELONG

    (INRIA Paris-Rocquencourt, MathFi Project, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France)

Abstract

In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class $\mathcal{C}^\infty$ (see Theorem 5.1). This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times (see Theorem 5.3).

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:01:n:s0219024909005154
    DOI: 10.1142/S0219024909005154
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    References listed on IDEAS

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    1. R. Haber & P. Schonbucher & P.Wilmott, 1999. "An American in Paris," OFRC Working Papers Series 1999mf14, Oxford Financial Research Centre.
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    Cited by:

    1. Zhu, Song-Ping & Chen, Wen-Ting, 2013. "Pricing Parisian and Parasian options analytically," Journal of Economic Dynamics and Control, Elsevier, vol. 37(4), pages 875-896.
    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. 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.
    4. 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.
    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.
    6. Yangyang Zhuang & Pan Tang, 2023. "Pricing of American Parisian option as executive option based on the least‐squares Monte Carlo approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(10), pages 1469-1496, October.

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