# Pricing Double Barrier Parisian Options Using Laplace Transforms

## Author

Listed:
• 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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File URL: http://www.worldscientific.com/doi/abs/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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### Keywords

Double barrier option; Parisian option; Laplace transform; numerical inversion; Brownian excursions; Euler summation; option price regularity;

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