B-spline Collocation Approach to the Solution of Options Pricing Model

Abstract

In this paper, we construct a numerical method to the solution of the Black-Scholes partial differential equation modeling the European option pricing problem with regard to a single asset. W e use an explicit spline-difference scheme which is based on using a finite difference approximation for the temporal derivative and a cubic B-spline collocation for spatial derivatives. The derived method leads to a tri-diagonal linear system. The stability of this method has been discussed and shown to be unconditionally stable. The computational performance of the proposed scheme is compared with those obtained by using a scheme based on the radial basis function.

Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations

“…The Von-Neumann stability analysis technique is applied to investigate the stability of the proposed method. Such an approach has been used by many researchers like [20][21][22][23][24]. Now assume that the solution to the given problem at the point (𝑥 𝑗 , 𝑡 𝑛 ) is:…”

Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation

“…It is known that this method is applicable to linear scheme. Hence, (1) is linearized by assuming all nonlinear terms equal zero [15].…”

“…The proposed method gives compatible results and better approximation compared to Dehghan and Shokri's method in [14]. Rashidinia et al [15] have presented a cubic B-spline collocation method for solving linear KG equation. The results show the proposed scheme is effective and accurate.…”

Superconvergence Analysis of Finite Element Method for Onlinear Klein-Gordon Equation

“…Up to now, there have been some studies about the Klein-Gordon equation (see [1,[3][4][5][6][7][8][9][10][11][12]). In which, [1] discussed the nonlinear generalized Klein-Gordon equation with dissipative term and boundary damping, and proved the global existence and uniform stabilization.…”

“…[7][8] investigated Klein-Gordon equation by finite difference method and proved the stability and convergence of the difference scheme. [9]developed a stable numerical method based on B-spline collocation method to solve linear Klein-Gordon equation. But, few works can be found about the finite element methods for problem (1)(see [10][11][12]).…”