scholarly journals Adaptive Wavelet Precise Integration Method for Nonlinear Black-Scholes Model Based on Variational Iteration Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Huahong Yan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Iteration Method ◽  
Integration Method ◽  
Variational Iteration Method ◽  
Precise Integration ◽  
Adaptive Wavelet ◽  
Precise Integration Method ◽  
Black Scholes Model ◽  
Black Scholes

An adaptive wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for Black-Scholes model is proposed. Black-Scholes model is a very useful tool on pricing options. First, an adaptive wavelet interpolation operator is constructed which can transform the nonlinear partial differential equations into a matrix ordinary differential equations. Next, VIM is developed to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method for the nonlinear differential equations. Third, an adaptive precise integration method (PIM) for the system of ordinary differential equations is constructed, with which the almost exact numerical solution can be obtained. At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical result shows the method's higher numerical stability and precision.

scholarly journals Coupling Technique of Haar Wavelet Transform and Variational Iteration Method for a Nonlinear Option Pricing Model

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1642
Author(s):  
Ruyi Xing ◽  
Meng Liu ◽  
Kexin Meng ◽  
Shuli Mei
Keyword(s):  
Iteration Method ◽  
Exact Analytical Solution ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Haar Wavelet Transform ◽  
Variational Iteration ◽  
Black Scholes Model ◽  
Black Scholes

Compared with the linear Black–Scholes model, nonlinear models are constructed through taking account of more practical factors, such as transaction cost, and so it is difficult to find an exact analytical solution. Combining the Haar wavelet integration method, which can transform the partial differential equation into the system of algebraic equations, the homotopy perturbation method, which can linearize the nonlinear problems, and the variational iteration method, which can solve the large system of algebraic equations efficiently, a novel numerical method for the nonlinear Black–Scholes model is proposed in this paper. Compared with the traditional methods, it has higher efficiency and calculation precision.

scholarly journals Construction of Interval Shannon Wavelet and Its Application in Solving Nonlinear Black-Scholes Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Liwei Liu
Keyword(s):  
Differential Equations ◽  
Boundary Effect ◽  
Nonlinear Partial Differential Equations ◽  
Variational Iteration Method ◽  
Nonlinear Pdes ◽  
Precise Integration ◽  
Black Scholes ◽  
Matrix Differential Equations ◽  
Shannon Wavelet ◽  
Matrix Differential

Interval wavelet numerical method for nonlinear PDEs can improve the calculation precision compared with the common wavelet. A new interval Shannon wavelet is constructed with the general variational principle. Compared with the existing interval wavelet, both the gradient and the smoothness near the boundary of the approximated function are taken into account. Using the new interval Shannon wavelet, a multiscale interpolation wavelet operator was constructed in this paper, which can transform the nonlinear partial differential equations into matrix differential equations; this can be solved by the coupling technique of the wavelet precise integration method (WPIM) and the variational iteration method (VIM). At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical results show that this method can decrease the boundary effect greatly and improve the numerical precision in the whole definition domain compared with Yan’s method.

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