A new approach to using the cubic B-spline functions to solve the Black-Scholes equation

Subject Areas : Statistics

Hossein Aminikhah ¹ (Associate Professor, Department of Applied Mathematics and Computer Science, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.)
Seyyed Javad Alavi ² (PhD student, Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.)

Received: 2017-08-05 Accepted : 2019-04-07 Published : 2019-05-01

Keywords: توابع بی-اسپلاین&rlm, , اختیار معامله اروپایی و آمریکایی, معادله بلک-شولز, همگرایی, پایداری, طرح تفاضلی,

Abstract :

Nowadays, options are common financial derivatives. For this reason, by increase of applications for these financial derivatives, the problem of options pricing is one of the most important economic issues. With the development of stochastic models, the need for randomly computational methods caused the generation of a new field called financial engineering. In the financial engineering the presentation of Black-Scholes model in 1973, attracted the attention of economists to the partial differential equations more than past. Therefore, we need a simple and precise solution for this kind of partial differential equations to determine the pricing option contracts. In this article the cubic B-spline collocation method has been used in the form of a difference method to solving Black-Scholes partial differential equation. Using this method as simplicity as finite difference method and does not have complex computation of traditional B-spline collocation method. The use of this method leads to a system of tridiagonal algebraic equations which is suitable for computer programming. The stability and convergence of this method is discussed and numerical results are presented for European and American options.

References:

[1] F. Black, M. Scholes. The pricing of options and corporate liabilities. J. Pol. Econ 81: 637-659(1973).

 

[2] P. Wilmott, J. Dewynne, S. Howison. Option Pricing: Mathematical Models and Computation. Oxford Financial Press. Oxford, UK (1993).

 

[3] Y. Hon, X. Mao. A Radial Basis Function Method For Solving Options Pricing Model. JFE 8: 31-44(1999).

 

[4] S. M. M. Kazemi, M. Dehghan, A. F. Bastani. Asymptotic expansion of solutions to the Black-Scholes equation arising from American option pricing near the expiry. Journal of Computational and Applied Mathematics 311: 11-37(2017).

 

[5] Z. Cen, A. Le. A robust and accurate finite difference method for a generalized Black-Scholes equation. Journal of Computational and Applied Mathematics 235(13): 3728-3733(2011).

 

[6] S. Vahdatia, M. Fardib, M. Ghasemi. Option Pricing Using a Computational Method Based on Reproducing Kernel. Journal of Computational and Applied Mathematics 328: 252-266(2018).

 

[7] Z. Cen, J. Huang, A. Xu, A. Le. Numerical approximation of a time-fractional Black-Scholes equation. Computers & Mathematics with Applications 75(8):2874-2887 (2018).

 

[8] J. Rashidinia, S. Jamalzadeh. Collocation method based on modified cubic B-spline for option pricing models. Math. Commun. 22: 89-102(2017).

[9] M. K. Kadalbajoo, L. P. Tripathi, A. Kumar. A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation. Mathematical and Computer Modelling 55:1483-1505(2012).

 

[10] H. B. Curry, I. J. Schoenberg. On spline distributions and their limits: The polya distribution functions. Bull. Amer. Math. Soc. 53:11-14(1947).

 

[11] E. N. Houstisf, E. A. Vavalis, j. R. Rice. Convergence of O(h4) cubic spline collocation methods for elliptic partial differential equations. siam j. Numer. Anal. 25: 54-74(1988).

 

[12] H. Caglar, N. Caglar. Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations. Applied Mathematics and Computation 201: 597-603(2008).

 

[13] R.C. Mittal, R.K. Jain. Redefined cubic B-splines collocation method for solving convection-diffusion equations. Applied Mathematical Modelling 36(11):5555-5573 (2012).

 

[14] C. C. Wang, J. H. Huang, D. J. Yang. Cubic spline difference method for heat conduction. 569 Int. Commun. Heat Mass Transfer 39(2): 224-230(2012).

 

[15] C. D. Boor. A Practical Guide to Splines. Springer Verlag New York Inc. Applied mathematical sciences 27: (2001).

 

[16] M. Dehghan, M. Abbaszadeh, A. Mohebbi. The numerical solution of nonlinear high dimensional generalized

Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions. Comput. Math. Appl. 68(3): 212-237(2014).

 

[17] Q. Liu, Y. T. Gu, P. Zhuang, F. Liu, Y. F. Nie. An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48(1):1-12(2011).