Skip to main content

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

  • XIAOPING LU (a1), NHAT-TAN LE (a1) (a2), SONG PING ZHU (a1) and WENTING CHEN (a1)

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
[1] Bernard C., Le Courtois O. & Quittard-Pinon F. (2005) A new procedure for pricing Parisian options. J. Derivatives 12 (4), 4553.
[2] Burdzy K., Chen Z.-Q. & Sylvester J. (2003) The heat equation and reflected Brownian motion in time-dependent domains. II. Singularities of solutions. J. Funct. Anal.l 204 (1), 134.
[3] Burdzy K., Chen Z.-Q. & Sylvester J. (2004a) The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 31 (1B), 775804.
[4] Burdzy K., Chen Z.-Q. & Sylvester J. (2004b) The heat equation in time dependent domains with insulated boundaries. J. Math. Anal. Appl. 294 (2), 581595.
[5] Cheng A. H.-D., Sidauruk, P. & Abousleiman Y. (1994) Approximate inversion of the Laplace transform. Math. J. 4 (2), 7681.
[6] Chesney M. & Gauthier L. (2006) American Parisian options. Finance Stoch. 10 (4), 475506.
[7] Chesney M., Jeanblanc-Picque M. & Yor M. (1997) Brownian excursions and parisian barrier options. Adv. Appl. Probab. 29 (1), 165184.
[8] Chiarella C., Kucera A. & Ziogas A. (2004) A Survey of the Integral Representation of American Option Prices. Technical Report, Quantitative Finance Research Centre, University of Technology, Sydney.
[9] Constanda C. (2010) Solution Techniques for Elementary Partial Differential Equations, CRC Press, Boca Raton, FL. With a foreword by Peter Schiavone, Second edition [of 1910694].
[10] Dassios A. & Wu S. (2010) Perturbed brownian motion and its application to parisian option pricing. Finance Stoch. 14 (3), 473494.
[11] Duffy D. J. (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] Evans J. D., Kuske R. & Keller J. (2002) American options on assets with dividends near expiry. Math. Fiance 12 (3), 219237.
[13] Haber R. J., Schonbucher P. J. & Wilmott P. (1999) Pricing Parisian options. J. Derivatives, 6 (3), 7179.
[14] Hattori H. (2013) Partial Differential Equations, Methods, Applications and Theories, World Scientific Publishing, Singapore.
[15] Kallast S. & Kivinukk A. (2003) Pricing and hedging american options using approximations by kim integral equations. Eur. Finance Rev. 7 (3), 361383.
[16] Kevorkian J. (2000) Partial Differential Equations, Analytical Solution Techniques, Texts in Applied Mathematics, 2nd ed., Vol. 35, Springer-Verlag, New York.
[17] Kim I. (1990) The analytic valuation of American options. Rev. Financ. Stud. 3 (4), 547–572.
[18] Kwok Y. K. & Barthez D. (1989) An algorithm for the numerical inversion of Laplace transforms. Inverse Problems, 5 (6), 10891095.
[19] Labart C. & Lelong J. (2009) Pricing double parisian options using laplace transforms. Int. J. Theor. Appl. Finance 12 (1), 1944.
[20] Schröder M. (2003) Brownian excusions and Parisian barrier options: A note. J. Appl. Probab. 40 (4), 855864.
[21] Zhu S.-P. & Chen W.-T. (2013) Pricing Parisian and Parasian options analytically. J. Econ. Dyn. Control 37 (4), 875896.
[22] Zhu S.-P., Le N.-T., Chen W. & Lu X. (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
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 35 *
Loading metrics...

Abstract views

Total abstract views: 232 *
Loading metrics...

* Views captured on Cambridge Core between 15th February 2017 - 23rd November 2017. This data will be updated every 24 hours.