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Projection and Contraction Method for the Valuation of American Options

Published online by Cambridge University Press:  06 March 2015

Haiming Song  and
Ran Zhang
Haiming Song
Affiliation:
Department of Mathematics, Jilin University, Changchun, 130012, P. R. China
Ran Zhang*
Affiliation:
Department of Mathematics, Jilin University, Changchun, 130012, P. R. China
*
*Corresponding author. Email addresses: songhm11@mails.jlu.edu.cn (H. Song), zhangran@jlu.edu.cn (R. Zhang)
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Abstract

An efficient numerical method is proposed for the valuation of American options via the Black-Scholes variational inequality. A far field boundary condition is employed to truncate the unbounded domain problem to produce the bounded domain problem with the associated variational inequality, to which our finite element method is applied. We prove that the matrix involved in the finite element method is symmetric and positive definite, and solve the discretized variational inequality by the projection and contraction method. Numerical experiments are conducted that demonstrate the superior performance of our method, in comparison with earlier methods.

Keywords

35A3590A0965K1065M1265M60American optionBlack-Scholes variational inequalityfinite element methodprojectioncontraction method

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Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 48 - 60
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Allegretto, W., Lin, Y. and Yang, H., Finite element error estimates for a nonlocal problem in American option valuation, SIAM J. Num. Anal. 39, 834–857 (2001).CrossRefGoogle Scholar
[2]Amin, K. and Khanna, A., Convergence of American option values from discrete- to continuous time financial models, Math. Finance 4, 289–304 (1994).CrossRefGoogle Scholar
[3]Barone-Adesi, G. and Whaley, R.E., Efficient analytic approximation of American option values, J. Finance 42, 301–320 (1987).CrossRefGoogle Scholar
[4]Badea, L. and Wang, J., A new formulation for the valuation of American options, I: Solution uniqueness, Analysis and Scientific Computing, Park, Eun-Jae and Lee, Jongwoo (eds.), pp. 3–16 (2000).Google Scholar
[5]Badea, L. and Wang, J., A new formulation for the valuation of American options, II: Solution existence, Analysis and Scientific Computing, Park, Eun-Jae and Lee, Jongwoo (eds.), pp. 17–33 (2000).Google Scholar
[6]Boyle, P., Broadie, M. and Glasserman, P., Monte Carlo methods for security pricing, J. Econ. Dyn. Control 21, 1267–1321 (1997).CrossRefGoogle Scholar
[7]Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Pol. Econ. 81, 637–659 (1973).CrossRefGoogle Scholar
[8]Cox, J.C., Ross, S.A. and Rubinstein, M., Option pricing: A simplified approach, J. Fin. Econ. 7, 229–263 (1979).CrossRefGoogle Scholar
[9]Han, H. and Wu, X., A fast numerical method for the BlackĺCScholes equation of American options, SIAM J. Num. Anal. 41, 2081–2095 (2003).CrossRefGoogle Scholar
[10]Holmes, A.D. and Yang, H., A front-fixing finite element method for the valuation of american options, SIAM J. Sc. Comp. 30, 2158–2180 (2008).CrossRefGoogle Scholar
[11]Hull, J., Fundamentals of Futures and Options Markets, Prentice Hall (2007).Google Scholar
[12]He, B., A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Opt. 35, 69–76 (1997).CrossRefGoogle Scholar
[13]Ju, N. and Zhong, R., An approximate formula for pricing American options, J. Derivatives 7, 31–40 (1999).CrossRefGoogle Scholar
[14]Jaillet, P., Lamberton, D. and Lapeyre, B., Variational inequalities and the pricing of American options, Acta Appl. Math. 21, 263–289 (1990).CrossRefGoogle Scholar
[15]Jiang, L., Mathematical Modeling and Methods of Option Pricing, World Scientific Press, Singapore (2005).CrossRefGoogle Scholar
[16]Kim, I.J., The analytic valuation of American puts, Rev. Financial Studies 3, 547–572 (1990).CrossRefGoogle Scholar
[17]Kangro, P. and Nicolaides, R., Far field boundary conditions for Black-Scholes equations, SIAM J. Num. Anal. 38, 1357–1368 (2000).CrossRefGoogle Scholar
[18]Kwok, Y.K., Mathematical Models of Financial Derivatives, 2nd edition, Springer Finance, Berlin Heidelberg (2008).Google Scholar
[19]Longstaff, F.A. and Schwartz, E.S., Valuing American options by simulation: A simple least-squares approach, Rev. Financial Stud. 14, 113–47 (2001).CrossRefGoogle Scholar
[20]Ma, J., Xiang, K. and Jiang, Y., An integral equation method with high-order collocation implementations for pricing American put options, Int. J. Economics and Finance 2, 102–112 (2010).CrossRefGoogle Scholar
[21]Oliver, P., Numerical Simulation of American Options, University of Ulm, Germany (2004).Google Scholar
[22]Rogers, L.C.G., Monte Carlo valuation of American options, Math. Finance 12, 271–286 (2002).CrossRefGoogle Scholar
[23]Song, H., Zhang, R. and Tan, W., Spectral method for the Black-Schles model of American options valuation, J. Math. Stud. 47, 47–64 (2014).CrossRefGoogle Scholar
[24]Wong, W.K. and Xu, K., Refining the quadratic approximation formula for an American option, Int. J. Theor. Appl. Finance 4, 773–781 (2001).CrossRefGoogle Scholar
[25]Wilmott, P., Dewynne, J. and Howison, S., Option Pricing: Mathematical Model and Computation, London: Oxford Financial Press (1995).Google Scholar
[26]Wu, L.X. and Kwok, Y.K., A front-fixing finite difference method for the valuation of American options, J. Financial Eng. 6, 83–97 (1997).Google Scholar
[27]Zhang, R., Song, H. and Luan, N., Weak Galerkin finite element method for valuation of American options, Front. Math. China 9, 455–476 (2014).CrossRefGoogle Scholar
[28]Zhang, T., Numerical methods for American option pricing, Acta Mathematicae Applicatae Sinica 25, 113–122 (2002).Google Scholar
[29]Zhu, S.P., An exact and explicit solution for the valuation of American put options, Quantitative Finance 6, 229–242 (2006).CrossRefGoogle Scholar
[30]Zhang, K., Song, H. and Li, J., Front-fixing FEMs for the pricing of American options based on a PML technique, Applicable Analysis, accepted for publication (2014).Google Scholar