Home
• Aa
• Aa

# Pricing American-style Parisian up-and-out call options

Abstract

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

Corresponding author
*Corresponding author
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] C. Bernard , O. Le Courtois & F. Quittard-Pinon (2005) A new procedure for pricing Parisian options. J. Derivatives 12 (4), 4553.

[2] K. Burdzy , Z.-Q. Chen & J. Sylvester (2003) The heat equation and reflected Brownian motion in time-dependent domains. II. Singularities of solutions. J. Funct. Anal.l 204 (1), 134.

[4] K. Burdzy , Z.-Q. Chen & J. Sylvester (2004b) The heat equation in time dependent domains with insulated boundaries. J. Math. Anal. Appl. 294 (2), 581595.

[6] M. Chesney & L. Gauthier (2006) American Parisian options. Finance Stoch. 10 (4), 475506.

[7] M. Chesney , M. Jeanblanc-Picque & M. Yor (1997) Brownian excursions and parisian barrier options. Adv. Appl. Probab. 29 (1), 165184.

[10] A. Dassios & S. Wu (2010) Perturbed brownian motion and its application to parisian option pricing. Finance Stoch. 14 (3), 473494.

[11] D. J. Duffy (2006) Finite Dfference Methods in Financial Engineering, Wiley Finance Series. John Wiley & Sons Ltd., Chichester. A partial differential equation approach, With 1 CD-ROM (Windows, Macintosh and UNIX).

[12] J. D. Evans , R. Kuske & J. Keller (2002) American options on assets with dividends near expiry. Math. Fiance 12 (3), 219237.

[13] R. J. Haber , P. J. Schonbucher & P. Wilmott (1999) Pricing Parisian options. J. Derivatives, 6 (3), 7179.

[15] S. Kallast & A. Kivinukk (2003) Pricing and hedging american options using approximations by kim integral equations. Eur. Finance Rev. 7 (3), 361383.

[17] I. Kim (1990) The analytic valuation of American options. Rev. Financ. Stud. 3 (4), 547–572.

[18] Y. K. Kwok & D. Barthez (1989) An algorithm for the numerical inversion of Laplace transforms. Inverse Problems, 5 (6), 10891095.

[19] C. Labart & J. Lelong (2009) Pricing double parisian options using laplace transforms. Int. J. Theor. Appl. Finance 12 (1), 1944.

[20] M. Schröder (2003) Brownian excusions and Parisian barrier options: A note. J. Appl. Probab. 40 (4), 855864.

[21] S. Zhu -P. & W.-T. Chen (2013) Pricing Parisian and Parasian options analytically. J. Econ. Dyn. Control 37 (4), 875896.

[22] S.-P. Zhu , N.-T. Le , W. Chen & X. Lu (2015) Pricing parisian down-and-in options. Appl. Math. Lett. 43, 1924.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
• ISSN: 0956-7925
• EISSN: 1469-4425
• URL: /core/journals/european-journal-of-applied-mathematics
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 21 *