Browse > Article
http://dx.doi.org/10.12941/jksiam.2021.25.093

AN EFFICIENT HYBRID NUMERICAL METHOD FOR THE TWO-ASSET BLACK-SCHOLES PDE  

DELPASAND, R. (DEPARTMENT OF APPLIED MATHEMATICS AND MAHANI, MATHEMATICAL RESEARCH CENTER FACULTY OF MATHEMATICS AND COMPUTER SHAHID BAHONAR UNIVERSITY OF KERMAN)
HOSSEINI, M.M. (DEPARTMENT OF APPLIED MATHEMATICS AND MAHANI, MATHEMATICAL RESEARCH CENTER FACULTY OF MATHEMATICS AND COMPUTER SHAHID BAHONAR UNIVERSITY OF KERMAN)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.25, no.3, 2021 , pp. 93-106 More about this Journal
Abstract
In this paper, an efficient hybrid numerical method for solving two-asset option pricing problem is presented based on the Crank-Nicolson and the radial basis function methods. For this purpose, the two-asset Black-Scholes partial differential equation is considered. Also, the convergence of the proposed method are proved and implementation of the proposed hybrid method is specifically studied on Exchange and Call on maximum Rainbow options. In addition, this method is compared to the explicit finite difference method as the benchmark and the results show that the proposed method can achieve a noticeably higher accuracy than the benchmark method at a similar computational time. Furthermore, the stability of the proposed hybrid method is numerically proved by considering the effect of the time step size to the computational accuracy in solving these problems.
Keywords
Two-asset option pricing; Black-Scholes equation; Radial basis functions; convergency;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 J. Persson, L. Von Sydow, Pricing European multi-asset options using a space-time adoptive F. D-method, Comput. Visualization. Sci, 10 (4) (2007) 173-183.   DOI
2 D. Joeng, J. Kim, I. -S Wee, An accurate and efficient numerical method for Black-Scholes equations, Commun. Korean. Soc, 24 (4) (2009) 617-628.   DOI
3 O. Pattersson, E. Larsson,G. Marcusson, J. Persson, Improved radial basis function methods for multi-dimensional option pricing, J. Comput. Appl. Math, 222 (1) (2008) 82-93.   DOI
4 J. A. Rad, K. Parand, L. V. Ballestra, Pricing European and American options by radial basis point interpolation, Appl. Math. Comput, 251 (2015) 363-377.   DOI
5 V. Shcherbakov. E. Larsson, Radial basis function partition for unity methods for pricing vanilla basket options, Comput. Math. Appl, 71 (1) (2016) 185-200.   DOI
6 Sh. Zhang, Radial basis functions method for valuing options: A multinomial tree approach, J. Comput. Appl. Math, 319 (2017) 97-107.   DOI
7 Y. Chen, H. Yu, X. Meng, X. Xie, M. Hou and J. Chevalier , Numerical solving of the generalized Black-Scholes differential equations using Lequerre neural networks, Digit. Signal. Process, 112 (2021) 103003.   DOI
8 P. Roul, V. M. K. Prasad Goura, A new higher order compact finite difference method for generalised Black-Scholes partial differential equations: European call option, J. Comput. Appl. Math, 363 (2020) 464-484.   DOI
9 M. Zaka Ullah, An RBF-FD sparse scheme to simulate high dimensional Black-Scholes partial differential equations, Comput. Math. Appl, (2019) DOI: http://doi.org/10.1016/j.Camwa.2019.07.011.   DOI
10 M. Buhmann, N. Oyn, Spectral convergence of multiquadratic interpolation, Proc. Edinb. Math. Soc, 36 (2) (1993) 319-333.   DOI
11 X. Liu, Y. Cao, C. Ma, L. Shen, Wavelet-based option pricing: An emprical study, Eur. J. Oper. Res., 272 (2019) 1132-1142.   DOI
12 F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Political. Econ, 81 (3) (1973) 637-659.   DOI
13 K. in't Hout,R. Volk, Numerical solution of a two-asset option valuation PDE by ADI finite difference discretization, AID. Conf. Proc, 1648 (2015) 850054.
14 L. V. Ballestra, G. Pacelli, Pricing European and American options with two stocastic factors: a highly efficient radial basis function approach, J. Econom. Dynam. Control, 37 (6) (2013) 1142-1167.   DOI
15 W. R. Madych, Miscellaneaus error bounds for multiquadratic and related interpolants, Comput. Math. Appl, 24 (12) (1992) 121-130.   DOI
16 W. Margrabe, The value of an option to exchnge one-asset for another, J. Finac, 33 (1978) 177-186.   DOI
17 C.H. Tsai, J. Kolibal, M. Li, The golden section search algorithm for finding a good shape parameter for meshless colocation methods, Eng. Anal. Bound. Elem., 34 (2010) 738-746.   DOI
18 H. Johnson, Options on the minimum or the maximum of several assets, J. Finac. Quant. Anal, 22 (1987) 277-283.   DOI
19 M. Rubinste, Somewhere over the rainbow, Risk Magazine, 4 (1995) 63-66.
20 R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci, 4 (1) (1973) 141-183.   DOI
21 R. Stulz, Options in the minimum or the maximum of two risky assets, J. Finac. Econ, 10 (1982) 161-185.   DOI
22 H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005.
23 Y. Hon, X. Mao, Aradial basis function method for for solving option pricing model, Finac. Eng, 8 (1) (1999) 31-49.