Using a meshless kernel-based method to solve the Black-Scholes variational inequality of American options. (English) Zbl 1393.74215
Summary: 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.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S20 | Finite difference methods applied to problems in solid mechanics |
Keywords:
American options; positive definite kernels; variational inequalities; linear complementarity problemsReferences:
| [1] | Buhmann MD (2003) Radial basis functions: theory and implementations, vol 12., Cambridge Monographs on Applied and Computational MathematicsCambridge University Press, Cambridge · Zbl 1038.41001 |
| [2] | Cottle RW, Pang JS, Stone RE (2009) The Linear Complementarity Problem. Society for Industrial and Applied Mathematics, Classics in Applied Mathematics · Zbl 1131.91333 |
| [3] | Cox John, C; Ross Stephen, A; Mark, Rubenstein, Option pricing: a simplified approach, J Finan Econ, 7, 229-263, (1979) · Zbl 1131.91333 · doi:10.1016/0304-405X(79)90015-1 |
| [4] | Dehghan, M; Tatari, M, Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math Comp Model, 44, 1160-1168, (2006) · Zbl 1137.65408 · doi:10.1016/j.mcm.2006.04.003 |
| [5] | Desmond JH (2002) Nine ways to implement the binomial method for option valuation in MATLAB. SIAM Rev 44(4):661-677 [(electronic) (2003)] · Zbl 1029.91030 |
| [6] | 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 · Zbl 1140.91046 |
| [7] | 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 · Zbl 1092.91524 |
| [8] | Ikonen, Samuli; Toivanen, Jari, Pricing American options using LU decomposition, Appl Math Sci (Ruse), 1, 2529-2551, (2007) · Zbl 1140.91046 |
| [9] | Iske Armin (2004) Multiresolution methods in scattered data modelling, vol 37., Lecture Notes in Computational Science and EngineeringSpringer-Verlag, Berlin · Zbl 1057.65004 |
| [10] | John CH (2006) Options, futures, and other derivatives |
| [11] | José LM, Jorge N, Mikhail S (2008) An algorithm for the fast solution of symmetric linear complementarity problems. Numer Math 111(2):251-266 · Zbl 1157.65389 |
| [12] | 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 · Zbl 0787.65005 |
| [13] | Robert S, Holger W (2006) Kernel techniques: From machine learning to meshless methods. Acta Numerica 15:543-639 · Zbl 1111.65108 |
| [14] | Rüdiger US (2009) 4th edn. Tools for computational finance. Universitext. Springer-Verlag, Berlin · Zbl 1160.91017 |
| [15] | 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 · Zbl 0085.31802 |
| [16] | Sarra, SA, Adaptive radial basis function methods for time dependent partial differential equations, Appl Numer Math, 54, 79-94, (2005) · Zbl 1069.65109 · doi:10.1016/j.apnum.2004.07.004 |
| [17] | Tavella D, Randall C (2000) Pricing Financial Instruments: The Finite Difference Method. Wiley |
| [18] | Topper J (2005) Financial engineering with finite elements. Wiley finance series, Wiley |
| [19] | Wilmott Paul (2007) Paul Wilmott introduces quantitative finance, 2nd edn. Wiley-Interscience, New York · Zbl 1170.91005 |
| [20] | 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 · Zbl 1078.91008 |
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