COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

한국수학교육학회지시리즈B:순수및응용수학 / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081

2014, v.21 no.2, pp.129-139

https://doi.org/10.7468/jksmeb.2014.21.2.129

Jeong, Darae (Department of Mathematics, Korea University)
Kim, Sungki (Department of Mathematics, Korea University)
Choi, Yongho (Department of Mathematics, Korea University)
Hwang, Hyeongseok (Department of Financial Engineering, Korea University)
Kim, Junseok (Department of Mathematics, Korea University)

Jeong, Darae, Kim, Sungki, Choi, Yongho, Hwang, Hyeongseok, & Kim, Junseok. (2014). COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION. 한국수학교육학회지시리즈B:순수및응용수학, 21(2), 129-139, https://doi.org/10.7468/jksmeb.2014.21.2.129

복사

Abstract

In this paper, we present a detailed comparison of the performance of the numerical solvers such as the biconjugate gradient stabilized, operator splitting, and multigrid methods for solving the two-dimensional Black-Scholes equation. The equation is discretized by the finite difference method. The computational results demonstrate that the operator splitting method is fastest among these solvers with the same level of accuracy.

keywords: Black-Scholes equation, finite difference method, bi-CGSTAB, operator splitting method, multigrid

참고문헌

H. van der Vorst. (1992). Bi-CGSTAB, A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. & Stat. Comput., 13(2), 631-644. 10.1137/0913035.

Y. Saad & M. Schultz. (1986). GMRES, a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. & Stat. Comput., 7(3), 856-869. 10.1137/0907058.

Y. Saad & H. van der Vorst. (2000). Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math., 123(1), 1-33. 10.1016/S0377-0427(00)00412-X.

U. Trottenberg, C. Oosterlee & A. Schuller. Multigrid.

R.P. Fedorenko. (1962). A relaxation method for solving elliptic difference equations. USSR Computational Math. and Math. Phys., 1(4), 1092-1096. 10.1016/0041-5553(62)90031-9.

Y. Daoud & T. Ozis. (2011). The operator splitting method for Black-Scholes equation. Appl. Math., 2(6), 771-778. 10.4236/am.2011.26103.

D. Duffy. Finite Difference Methods in Financial Engineering, A Partial Differential Equation Approach.

R.P. Fedorenko. (1964). The speed of convergence of one iterative process. USSR Computational Math. and Math. Phys., 4(3), 227-235.

W. Hackbusch. Iterative Solution of Large Linear Systems of Equations.

10.

R. Heynen & H. Kat. (1996). Brick by Brick. Risk Magazine, 9(6), 28-31.

11.

E. Haug. The Complete Guide to Option Pricing Formulas.

12.

S. Ikonen & J. Toivanen. (2004). Operator splitting methods for American option pricing. Appl. Math. Lett., 17(7), 809-814. 10.1016/j.aml.2004.06.010.

13.

D. Jeong. Mathematical model and numerical simulation in computational finance.

14.

P. Lotstedt, J. Persson, L. von Sydow & J. Tysk. (2007). Space-time adaptive finite difference method for European multi-asset options. Comput. Math. Appl., 53(8), 1159-1180. 10.1016/j.camwa.2006.09.014.

15.

MATLAB. Users Guide: Natick.

16.

C.W. Oosterlee. (2003). On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal., 15(2-7), 165-185.

17.

A. Ramage & L. von Sydow. (2011). A multigrid preconditioner for an adaptive Black-Scholes solver. BIT Numer. Math., 51(1), 217-233. 10.1007/s10543-011-0316-6.

18.

C. Reisinger & G. Wittum. (2004). On multigrid for anisotropic equations and variational inequalities "pricing multi-dimensional european and american options". Comput. Vis. Sci., 7(3-4), 189-197.

19.

P. Amstera, C. Averbuj, P. de Napoli & M. Mariani. (2010). A parabolic problem arising in financial mathematics. Nonlinear Anal. Real World Appl., 11, 759-763. 10.1016/j.nonrwa.2009.01.019.

20.

F. Black & M. Scholes. (1973). The pricing of options and corporate liabilities. J. Polit. Econ., 81, 637-659. 10.1086/260062.

21.

R. Chin, T. Manteuffel & J. de Pillis. (1984). ADI as a preconditioning for solving the convection-diffusion equation. SIAM J. Sci. Comput., 5(2), 281-299. 10.1137/0905020.

바로가기메뉴

논문 상세

Vol.21 No.2

COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

Abstract

참고문헌

한국수학교육학회지시리즈B:순수및응용수학