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

Published online by Cambridge University Press:  15 February 2017

XIAOPING LU ,
NHAT-TAN LE ,
SONG PING ZHU  and
WENTING CHEN
XIAOPING LU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: xplu@uow.edu.au, tantoankhoa10@gmail.com, spz@uow.edu.au, cwtwx@163.com
NHAT-TAN LE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: xplu@uow.edu.au, tantoankhoa10@gmail.com, spz@uow.edu.au, cwtwx@163.com Department of Fundamental Sciences, MienTrung University of Civil Engineering, 24 Nguyen Du, Tuy Hoa, Phu Yen, Vietnam
SONG PING ZHU*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: xplu@uow.edu.au, tantoankhoa10@gmail.com, spz@uow.edu.au, cwtwx@163.com
WENTING CHEN
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: xplu@uow.edu.au, tantoankhoa10@gmail.com, spz@uow.edu.au, cwtwx@163.com
*
*Corresponding author
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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.

Keywords

Integral equation approachAmerican-style Parisian optionsFourier sine transformoptimal exercise boundaries
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 1 - 29
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Bernard, C., Le Courtois, O. & Quittard-Pinon, F. (2005) A new procedure for pricing Parisian options. J. Derivatives 12 (4), 45–53.Google Scholar
[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), 1–34.Google Scholar
[3] Burdzy, K., Chen, Z.-Q. & Sylvester, J. (2004a) The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 31 (1B), 775–804.Google Scholar
[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), 581–595.Google Scholar
[5] Cheng, A. H.-D., Sidauruk, P. & Abousleiman, Y. (1994) Approximate inversion of the Laplace transform. Math. J. 4 (2), 76–81.Google Scholar
[6] Chesney, M. & Gauthier, L. (2006) American Parisian options. Finance Stoch. 10 (4), 475–506.Google Scholar
[7] Chesney, M., Jeanblanc-Picque, M. & Yor, M. (1997) Brownian excursions and parisian barrier options. Adv. Appl. Probab. 29 (1), 165–184.CrossRefGoogle Scholar
[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.Google Scholar
[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].Google Scholar
[10] Dassios, A. & Wu, S. (2010) Perturbed brownian motion and its application to parisian option pricing. Finance Stoch. 14 (3), 473–494.Google Scholar
[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).CrossRefGoogle Scholar
[12] Evans, J. D., Kuske, R. & Keller, J. (2002) American options on assets with dividends near expiry. Math. Fiance 12 (3), 219–237.Google Scholar
[13] Haber, R. J., Schonbucher, P. J. & Wilmott, P. (1999) Pricing Parisian options. J. Derivatives, 6 (3), 71–79.CrossRefGoogle Scholar
[14] Hattori, H. (2013) Partial Differential Equations, Methods, Applications and Theories, World Scientific Publishing, Singapore.CrossRefGoogle Scholar
[15] Kallast, S. & Kivinukk, A. (2003) Pricing and hedging american options using approximations by kim integral equations. Eur. Finance Rev. 7 (3), 361–383.CrossRefGoogle Scholar
[16] Kevorkian, J. (2000) Partial Differential Equations, Analytical Solution Techniques, Texts in Applied Mathematics, 2nd ed., Vol. 35, Springer-Verlag, New York.Google Scholar
[17] Kim, I. (1990) The analytic valuation of American options. Rev. Financ. Stud. 3 (4), 547–572.Google Scholar
[18] Kwok, Y. K. & Barthez, D. (1989) An algorithm for the numerical inversion of Laplace transforms. Inverse Problems, 5 (6), 1089–1095.Google Scholar
[19] Labart, C. & Lelong, J. (2009) Pricing double parisian options using laplace transforms. Int. J. Theor. Appl. Finance 12 (1), 19–44.Google Scholar
[20] Schröder, M. (2003) Brownian excusions and Parisian barrier options: A note. J. Appl. Probab. 40 (4), 855–864.CrossRefGoogle Scholar
[21] Zhu, S.-P. & Chen, W.-T. (2013) Pricing Parisian and Parasian options analytically. J. Econ. Dyn. Control 37 (4), 875–896.Google Scholar
[22] Zhu, S.-P., Le, N.-T., Chen, W. & Lu, X. (2015) Pricing parisian down-and-in options. Appl. Math. Lett. 43, 19–24.Google Scholar