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AN APPROPRIATE APPROACH TO PRICING EUROPEAN-STYLE OPTIONS WITH THE ADOMIAN DECOMPOSITION METHOD

Part of: Equations and systems with constant coefficients

Published online by Cambridge University Press:  26 February 2018

ZIWIE KE Open the ORCID record for ZIWIE KE [Opens in a new window] ,
JOANNA GOARD Open the ORCID record for JOANNA GOARD [Opens in a new window]  and
SONG-PING ZHU Open the ORCID record for SONG-PING ZHU [Opens in a new window]
ZIWIE KE*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email zk990@uowmail.edu.au, joanna@uow.edu.au, spz@uow.edu.au
JOANNA GOARD
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email zk990@uowmail.edu.au, joanna@uow.edu.au, spz@uow.edu.au
SONG-PING ZHU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email zk990@uowmail.edu.au, joanna@uow.edu.au, spz@uow.edu.au
*
✉zk990@uowmail.edu.au
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Abstract

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We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.

Keywords

Adomian decomposition methodEuropean optiondigital optionVasicek model

MSC classification

Primary: 35E15: Initial value problems
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 349 - 369
Copyright
© 2018 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables (Dover Publications, New York, 1964).Google Scholar
Adomian, G., “Convergent series solution of nonlinear equations”, J. Comput. Appl. Math. 11 (1984) 225–230; doi:10.1016/0377-0427(84)90022-0.Google Scholar
Adomian, G., “Nonlinear stochastic dynamical systems in physical problems”, J. Math. Anal. Appl. 111 (1985) 105–113; doi:10.1016/0022-247X(85)90203-3.Google Scholar
Adomian, G., “Review of the decomposition method in applied mathematics”, J. Math. Anal. Appl. 135 (1988) 501–544; doi:10.1016/0022-247X(88)90170-9.Google Scholar
Adomian, G. and Rach, R., “Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations”, Comput. Math. Appl. 19 (1990) 9–12; doi:10.1016/0898-1221(90)90246-G.Google Scholar
Adomian, G. and Rach, R., “A further consideration of partial solutions in the decomposition method”, Comput. Math. Appl. 23 (1992) 51–64; doi:10.1016/0898-1221(92)90080-2.Google Scholar
Al-Dosary, K. I., Al-Jubouri, N. K. and Abdullah, H. K., “On the solution of Abel differential equation by Adomian decomposition method”, Appl. Math. Sci. 2 (2008) 2105–2118; http://www.m-hikari.com/ams/ams-password-2008/ams-password41-44-2008/aldosaryAMS41-44-2008.pdf.Google Scholar
Abdelrazec, A. and Pelinovsky, D., “Convergence of the Adomian decomposition method for initial-value problems”, Numer. Methods Partial Differ. Equ. 27 (2011) 749–766; doi:10.1002/num.20549.Google Scholar
Abudy, M. and Izhakian, Y., “Pricing stock options with stochastic interest rate”, Intl J. Portfolio Anal. Manag. 1 (2013) 250–277; doi:10.1504/IJPAM.2013.054408.Google Scholar
Avila, E., Estrella, A. and Blanco, L., “Solution of the Black–Scholes equation via the Adomian decomposition method”, Intl J. Appl. Math. Res. 2 (2013) 486–494; doi:10.14419/ijamr.v2i4.871.Google Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Political Economy 81 (1973) 637–654; http://www.jstor.org/stable/1831029.Google Scholar
Biazar, J. and Pourabd, M., “A Maple program for computing Adomian polynomials”, Intl Math. Forum 1 (2006) 1919–1924; doi:10.13140/RG.2.1.3473.4565.Google Scholar
Biazar, J. and Shafiof, S. M., “A simple algorithm for calculating Adomian polynomials”, Intl J. Contemp. Math. Sci. 2 (2007) 975–982; http://www.m-hikari.com/ijcms-password2007/17-20-2007/biazarIJCMS17-20-2007-1.pdf.Google Scholar
Bohner, M. and Zheng, Y., “On analytical solutions of the Black–Scholes equation”, Appl. Math. Lett. 22 (2009) 309–313; doi:10.1016/j.aml.2008.04.002.Google Scholar
Bohner, M., Marín, F. and Rodríguez, S., “European call option pricing using the Adomian decomposition method”, Adv. Dyn. Syst. Appl. 9 (2014) 75–85; http://campus.mst.edu/adsa/contents/v9n1p5.pdf.Google Scholar
Cherruault, Y., “Convergence of Adomian’s method”, Kybernetes 18 (1989) 31–38; doi:10.1108/eb005812.Google Scholar
Cherruault, Y. and Adomian, G., “Decomposition methods: a new proof of convergence”, Math. Comput. Model. 18 (1993) 103–106; doi:10.1016/0895-7177(93)90233-O.Google Scholar
El-Wakil, S. A., Abdou, M. A. and Elhanbaly, A., “Adomian decomposition method for solving the diffusion–convection–reaction equations”, Appl. Math. Comput. 177 (2006) 729–736; doi:10.1016/j.amc.2005.09.105.Google Scholar
Fang, H., “European option pricing formula under stochastic interest rate”, Program Appl. Math. 4 (2012) 14–21; doi:10.3968/j.pam.1925252820120401.Z0619.Google Scholar
Fatoorehchi, H. and Abolghasemi, H., “On calculation of Adomian polynomials by MATLAB”, J. Appl. Comput. Sci. Math. 5 (2011) 85–88; https://jacsm.ro/view/?pid=11_13.Google Scholar
González-Gaxiola, O., Ruíz de Chávez, J. and Santiago, J. A., “A nonlinear option pricing model through the Adomian decomposition method”, Intl J. Appl. Comput. Math. 2 (2016) 453–467; doi:10.1007/s40819-015-0070-6.Google Scholar
Hull, J. C., Options, futures, and other derivatives (Prentice Hall, Upper Saddle River, NJ, 2006).Google Scholar
Khuri, S. A., “A new approach to Bratu’s problem”, Appl. Math. Comput. 147 (2004) 131–136; doi:10.1016/S0096-3003(02)00656-2.Google Scholar
Lesnic, D., “The decomposition method for linear, one-dimensional, time-dependent partial differential equations”, Intl J. Math. Math. Sci. 2006 (2006) 1–29; doi:10.1155/IJMMS/2006/42389.Google Scholar
Merton, R. C., “Theory of rational option pricing”, Bell J. Econ. Manag. Sci. 4 (1973) 141–183; doi:10.2307/3003143.Google Scholar
Mavoungou, T. and Cherruault, Y., “Convergence of Adomian’s method and applications to non-linear partial differential equations”, Kybernetes 21 (1992) 13–25; doi:10.1108/eb005942.Google Scholar
Mamon, R. S., “Three ways to solve for bond prices in the Vasicek model”, J. Appl. Math. Decision Sci. 8 (2004) 1–14; doi:10.1155/S117391260400001X.Google Scholar
Panini, R. and Srivastav, R. P., “Option pricing with Mellin transforms”, Math. Comput. Model. 40 (2004) 43–56; doi:10.1016/j.mcm.2004.07.008.Google Scholar
Tatari, M., Dehghan, M. and Razzaghi, M., “Application of the Adomian decomposition method for the Fokker–Planck equation”, Math. Comput. Model. 45 (2007) 639–650; doi:10.1016/j.mcm.2006.07.010.Google Scholar
Vasicek, O., “An equilibrium characterization of the term structure”, J. Financial Econ. 5 (1977) 177–188; doi:10.1016/0304-405X(77)90016-2.Google Scholar
Wilmott, P., Howison, S. and Dewynne, J., The mathematics of financial derivatives: a student introduction (Cambridge University Press, 1995).Google Scholar
Wazwaz, A.-M., “A new method for solving singular initial value problems in the second-order ordinary differential equations”, Appl. Math. Comput. 128 (2002) 45–57; doi:10.1016/S0096-3003(01)00021-2.Google Scholar
Zhu, S.-P. and Lee, J., “On the Adomian decomposition method for solving PDEs”, Commun. Math. Res. 32 (2016) 151–166; doi:10.13447/j.1674-5647.2016.02.08.Google Scholar