×

Pricing American-style Parisian down-and-out call options. (English) Zbl 1411.91571

Summary: We propose an integral equation approach for pricing American-style Parisian down-and-out call options under the Black-Scholes framework. For this type of options, the knock-out feature is activated only if the underlying asset price continuously remains below a pre-determined barrier for a sufficiently long period of time. As such, the corresponding pricing problem becomes a three-dimensional (3-D) free boundary problem, instead of a two-dimensional (2-D) one as is the case of “one-touch” barrier options, and this poses a computational challenge. In our approach, we first reduce the 3-D problem to a 2-D one, and then, by applying the Fourier sine transform to the resulting 2-D problem, we can derive a pair of coupled integral equations governing the option price at any given time in terms of (i) the option price at the barrier and (ii) the optimal exercise boundary at that time. This pair of coupled integral equations can be solved using the Newton-Raphson iterative procedure, after which, the option price, the optimal exercise boundary, and the hedging parameters can be obtained in a straightforward manner. A complexity analysis of the method, together with numerical results, show that the proposed approach is robust and significantly more efficient than existing uniform finite difference methods with Crank-Nicolson timestepping, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight into several interesting properties of the option price and the optimal exercise boundary.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Le, N.-T.; Zhu, S.-P.; Lu, X., An integral equation approach for the valuation of american-style down-and-out calls with rebates, Comput. Math. Appl., 71, 2, 544-564 (2016) · Zbl 1443.91331
[2] Chesney, M.; Jeanblanc-Picque, M.; Yor, M., Brownian excursions and Parisian barrier options, Adv. Appl. Probab., 29, 1, pp.165-184 (1997) · Zbl 0882.60042
[3] Dai, M.; Kwok, Y. K., Knock-in American options, J. Futur. Mark., 24, 2, 179-192 (2004)
[4] Gauthier, L., Excursions height-and length-related stopping times, and application to finance, Adv. Appl. Probab., 34, 4, 846-868 (2002) · Zbl 1033.60086
[5] Bernard, C.; Courtois, O. L.; Quittard-Pinon, F., A new procedure for pricing Parisian options, J. Deriv., 12, 4, 45-53 (2005)
[6] Moraux, F., Valuing corporate liabilities when the default threshold is not an absorbing barrier, SSRN Working Paper Series (2002), https://ssrn.com/abstract=314404
[7] Chen, A.; Suchanecki, M., Default risk, bankruptcy procedures and the market value of life insurance liabilities, Insur.: Math. Econ., 40, 2, 231-255 (2007) · Zbl 1141.91494
[8] Dassios, A.; Wu, S., Perturbed Brownian motion and its application to Parisian option pricing, Finance Stoch., 14, 3, 473-494 (2010) · Zbl 1226.91073
[9] Haber, R. J.; Schönbucher, P. J.; Wilmott, P., Pricing Parisian options, J. Deriv., 6, 3, 71-79 (1999)
[10] Zhu, S.-P.; Chen, W.-T., Pricing Parisian and Parasian options analytically, J. Econ. Dyn. Control, 37, 4, 875-896 (2013) · Zbl 1346.91242
[11] Chesney, M.; Gauthier, L., American Parisian options, Finance Stoch., 10, 4, 475-506 (2006) · Zbl 1126.91025
[12] Kwok, Y. K.; Barthez, D., An algorithm for the numerical inversion of Laplace transforms, Inverse Problems, 5, 1089-1095 (1989) · Zbl 0749.65086
[13] Cheng, A. H.-D.; Sidauruk, P.; Abousleiman, Y., Approximate inversion of the Laplace transform, Mathematica J., 4, 2, 76-81 (1994)
[14] Carrier, G. F.; Pearson, C. E., Partial Differential Equations: Theory and Technique (1976), Academic Press: Academic Press (London) LTD · Zbl 0323.35001
[15] Le, N.-T.; Dang, D. M.; Khanh, T.-V., A decomposition approach via Fourier sine transform for valuing American knock-out options with time-dependent rebates, J. Comput. Appl. Math., 317, 652-671 (2017) · Zbl 1386.91164
[16] Kevorkian, J., Partial differential equations, Texts in Applied Mathematics, 35 (2000), Springer-Verlag, New York · Zbl 0937.35001
[17] Duffy, D. J., Finite difference methods in financial engineering, Wiley Finance Series (2006), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester · Zbl 1141.91002
[18] Constanda, C., Solution techniques for elementary partial differential equations (2010), CRC Press: CRC Press Boca Raton, FL · Zbl 1213.35001
[19] Hattori, H., Partial differential equations (2013), World Scientific Publishing, , ISBN 9789814407564 · Zbl 1270.35001
[20] Chiarella, C.; Kucera, A.; Ziogas, A., A Survey of the Integral Representation of American Option Prices, Technical Report (2004), Quantitative Finance Research Centre, University of Technology, Sydney
[21] Kim, I., The analytic valuation of American options, Rev. Financial Stud., 3, 4, 547-572 (1990)
[22] Kythe, P. K.; Schaferkotter, M. R., Handbook of Computational Methods for Integration (2014), Chapman and Hall/CRC, , ISBN 1584884282
[23] Rannacher, R., Finite element solution of diffusion problems with irregular data, Numerische Mathematik, 43, 309-327 (1984) · Zbl 0512.65082
[24] Giles, M. B.; Carter, R., Convergence analysis of Crank-Nicolson and Rannacher time-marching, J. Comput. Finance, 9, 563-576 (2005)
[25] Forsyth, P.; Vetzal, K., Quadratic convergence of a penalty method for valuing American options, SIAM J. Sci. Comput., 23, 2096-2123 (2002) · Zbl 1020.91017
[26] Kwok, Y. K.; Lau, K. W., Pricing algorithms for options with exotic path-dependence, J. Deriv., 9, 1, 28-38 (2001)
[27] Bernard, C.; Le Courtois, O.; Quittard-Pinon, F., A new procedure for pricing Parisian options, J. Deriv., 12, 4, 45-53 (2005)
[28] Cheang, G. H.; Chiarella, C.; Ziogas, A., The representation of american options prices under stochastic volatility and jump-diffusion dynamics, Quantitative Finance, 13, 2, 241-253 (2013) · Zbl 1280.91165
[29] Chiarella, C.; Ziogas, A., American call options under jump-diffusion processes-a Fourier transform approach, Appl. Math. Finance, 16, 1, 37-79 (2009) · Zbl 1169.91340
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.