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Finite Volume Method for Pricing European and American Options under Jump-Diffusion Models

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  02 May 2017

Xiao-Ting Gan ,
Jun-Feng Yin  and
Yun-Xiang Guo
Xiao-Ting Gan*
Affiliation:
School of Mathematical Science, Tongji University, Shanghai 200092, PR China School of Mathematics and Statistics, Chuxiong Normal University, Chuxiong 675000, Yunnan Province, PR China
Jun-Feng Yin*
Affiliation:
School of Mathematical Science, Tongji University, Shanghai 200092, PR China
Yun-Xiang Guo*
Affiliation:
School of Mathematical Science, Tongji University, Shanghai 200092, PR China
*
*Corresponding author. Email addresses:[email protected] (X.-T. Gan), [email protected] (J.-F. Yin), [email protected] (Y.-X. Guo)
*Corresponding author. Email addresses:[email protected] (X.-T. Gan), [email protected] (J.-F. Yin), [email protected] (Y.-X. Guo)
*Corresponding author. Email addresses:[email protected] (X.-T. Gan), [email protected] (J.-F. Yin), [email protected] (Y.-X. Guo)
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Abstract

A class of finite volume methods is developed for pricing either European or American options under jump-diffusion models based on a linear finite element space. An easy to implement linear interpolation technique is derived to evaluate the integral term involved, and numerical analyses show that the full discrete system matrices are M-matrices. For European option pricing, the resulting dense linear systems are solved by the generalised minimal residual (GMRES) method; while for American options the resulting linear complementarity problems (LCP) are solved using the modulus-based successive overrelaxation (MSOR) method, where the H+-matrix property of the system matrix guarantees convergence. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of these methods.

Keywords

Finite volume methodoption pricingjump-diffusion modelslinear complementarity problemsGMRES methodmodulus-based successive overrelaxation method

MSC classification

Secondary: 65M08: Finite volume methods 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N40: Method of lines
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 227 - 247
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Amin, K., Jump diffusion option valuation in discrete time, J. Financial 48, 1863–1883 (1993).Google Scholar
[2] Andersen, L. and Andreasen, J., Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res. 4, 231–262 (2000).Google Scholar
[3] Borici, A. and Luthi, H.J., Fast solution of complementarity formulations in American put option pricing, J. Comp. Fin. 9, 63–81 (2005).Google Scholar
[4] Bai, Z.Z., Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear. Algebra. Appl. 17, 917–933 (2010).Google Scholar
[5] Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ. 81, 637–654 (1973).Google Scholar
[6] Cryer, C. W., The solution of a quadratic programming using systematic overrelaxation, SIAM J. Control 9, 385–392 (1971).Google Scholar
[7] Forsyth, P.A. and Vetzal, K.R., Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput. 23, 2095–2122 (2002).Google Scholar
[8] Li, R.H., Chen, Z.Y. and Wu, W., Generalized Difference Methods for Differential Equations (Numerical Analysis of Finite Volume Methods), Marcel Dekker, NewYork (2000).Google Scholar
[9] Li, Y., Lin, J. and Yang, M., Finite volume element methods: An overview on recent developments, Int. J. Numer. Anal. Model, Series B 4, 14–34 (2013).Google Scholar
[10] d’Halluin, Y., Forsyth, P.A. and Labahn, G.A, A penalty method for American options with jump diffusion processes, Numer. Math. 97, 321–352 (2004).Google Scholar
[11] d’Halluin, Y., Forsyth, P.A. and Vetzal, K.R., Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal. 25, 87–112 (2005).Google Scholar
[12] Salmi, S. and Toivanen, J., An iterative method for pricing American options under jump-diffusion model, Appl. Numer. Math. 61, 821–831 (2011).Google Scholar
[13] Toivanen, J., Numerical valuation of European and American options under Kou's jump-diffusion model, SIAM J. Sci. Comput. 30, 1949–1970 (2008).Google Scholar
[14] Toivanen, J., A high-order front-tracking finite difference method for pricing American options under jump-diffusion models, J. Comp. Fin. 13, 61–79 (2010).Google Scholar
[15] Zhang, X.L., Numerical analysis of American option pricing in a jump diffusion model, Math. Oper. Res. 22, 668–690 (1997).Google Scholar
[16] Zhang, K. and Wang, S., A computational scheme for options under jump diffusion processes, Int. J. Numer. Anal. Mod. 6, 110–123 (2009).Google Scholar
[17] Zhang, K. and Wang, S., Pricing options under jump diffusion processes with fitted finite volume method, Appl. Math. Comput. 201, 398–413 (2008).Google Scholar
[18] Zheng, N. and Yin, J.F., Modulus-based successive overrelaxation method for pricing American options, J. Appl. Math. Informatics 31, 769–784 (2013).CrossRefGoogle Scholar
[19] Zhang, K., Yang, X.Q. and Teo, K.L., Augmented Lagrangian method applied to American option pricing, Automatica 42, 1407–1416 (2006).Google Scholar
[20] Zhang, K., Wang, S., Yang, X.Q. and Teo, K.L., A power penalty approach to numerical solutions of two-asset American options, Numer. Math: Theory, Methods and Applications 2, 202–223 (2009).Google Scholar