Abstract
Under the Black–Scholes model, the value of an American option solves a free boundary problem which is equivalent to a variational inequality problem. Using positive definite kernels, we discretize the variational inequality problem in spatial direction and derive a sequence of linear complementarity problems (LCPs) in a finite-dimensional euclidean space. We use special kind of kernels to impose homogeneous boundary conditions and to obtain LCPs with positive definite coefficient matrices to guarantee the existence and uniqueness of the solution. The LCPs are then successfully solved iteratively by the projected SOR algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buhmann MD (2003) Radial basis functions: theory and implementations, vol 12., Cambridge Monographs on Applied and Computational MathematicsCambridge University Press, Cambridge
Cottle RW, Pang JS, Stone RE (2009) The Linear Complementarity Problem. Society for Industrial and Applied Mathematics, Classics in Applied Mathematics
Cox John C, Ross Stephen A, Mark Rubenstein (1979) Option pricing: a simplified approach. J Finan Econ 7:229–263
Dehghan M, Tatari M (2006) Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math Comp Model 44(11–12):1160–1168
Desmond JH (2002) Nine ways to implement the binomial method for option valuation in MATLAB. SIAM Rev 44(4):661–677 [(electronic) (2003)]
Duffy DJ (2006) Finite difference methods in financial engineering. A partial differential equation approach, With 1 CD-ROM. Windows, Macintosh and UNIX. Wiley Finance Series. Wiley, Chichester
Fischer B, Myron S (2012) The pricing of options and corporate liabilities [reprint of J. Polit. Econ. 81 (1973), no. 3, 637–654]. In: Financial risk measurement and management
Ikonen Samuli, Toivanen Jari (2007) Pricing American options using LU decomposition. Appl Math Sci (Ruse) 1(49–52):2529–2551
Iske Armin (2004) Multiresolution methods in scattered data modelling, vol 37., Lecture Notes in Computational Science and EngineeringSpringer-Verlag, Berlin
John CH (2006) Options, futures, and other derivatives
José LM, Jorge N, Mikhail S (2008) An algorithm for the fast solution of symmetric linear complementarity problems. Numer Math 111(2):251–266
Powell MJD (1992) The theory of radial basis function approximation in 1990. In Advances in numerical analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ. Oxford University Press, New York, pp 105–210
Robert S, Holger W (2006) Kernel techniques: From machine learning to meshless methods. Acta Numerica 15:543–639
Rüdiger US (2009) 4th edn. Tools for computational finance. Universitext. Springer-Verlag, Berlin
Salomon B (1959) Lectures on Fourier integrals. With an author’s supplement on monotonic functions, Stieltjes integrals, and harmonic analysis. Translated by Morris Tenenbaum and Harry Pollard. Annals of Mathematics Studies, No. 42. Princeton University Press, Princeton
Sarra SA (2005) Adaptive radial basis function methods for time dependent partial differential equations. Appl Numer Math 54(1):79–94
Tavella D, Randall C (2000) Pricing Financial Instruments: The Finite Difference Method. Wiley
Topper J (2005) Financial engineering with finite elements. Wiley finance series, Wiley
Wilmott Paul (2007) Paul Wilmott introduces quantitative finance, 2nd edn. Wiley-Interscience, New York
Yves A, Olivier P (2005) Computational methods for option pricing, vol. 30 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jorge Zubelli.
Rights and permissions
About this article
Cite this article
Moradipour, M., Yousefi, S.A. Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options. Comp. Appl. Math. 37, 627–639 (2018). https://doi.org/10.1007/s40314-016-0351-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-016-0351-7