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 Fractional Black-Scholes model with regularized Prabhakar derivative

2017 ◽  
Vol 102 (116) ◽  
pp. 121-132 ◽  
Author(s):  
Shiva Eshaghi ◽  
Alireza Ansari ◽  
Reza Ghaziani ◽  
Mohammadreza Darani
Keyword(s):  
Analytical Solutions ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
European Options ◽  
Variational Iteration ◽  
Black Scholes Model ◽  
Approximate Analytical Solutions ◽  
Black Scholes ◽  
Fractional Type

We introduce a fractional type Black-Scholes model in European options including the regularized Prabhakar derivative. We apply the reconstruction of variational iteration method to get the approximate analytical solutions for some models of generalized fractional Black-Scholes equations in terms of the generalized Mittag-Leffler functions.

scholarly journals Convergence of Variational Iteration Method for Fractional Delay Integrodifferential-Algebraic Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Yayun Fu ◽  
Hongliang Liu ◽  
Aiguo Xiao
Keyword(s):  
Iteration Method ◽  
Time Lag ◽  
Variational Iteration Method ◽  
Good Choice ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Variational Iteration ◽  
Fractional Delay ◽  
Restricted Variation ◽  
Algebraic Constraint

Fractional order delay integrodifferential-algebraic equations are often used for many practical modeling problems in science and engineering, which have time lag, memory, constraint limit, and so forth. These yield some difficulties in numerical computation. The iterative methods are good choice. In the present paper, we construct variational iteration method for solving them by using the appropriate restricted variation. This overcomes the difficulties caused by limitations of large storage amount and algebraic constraint and extends the previous conclusions.

A NEWLY MODIFIED VARIATIONAL ITERATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350029
Author(s):  
R. YULITA MOLLIQ ◽  
M. S. M. NOORANI
Keyword(s):  
Iteration Method ◽  
Nonlinear Differential Equations ◽  
Method Comparison ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Accurate Method ◽  
Convergence Region ◽  
Variational Iteration ◽  
Approximate Analytical Solutions ◽  
Modified Variational Iteration Method

This paper presents a new reliable modification of the variational iteration method (MoVIM). An enlarged interval of convergence region of series solutions is obtained by inserting a nonzero auxiliary parameter (ℏ) into the correction functional of variational iteration method. Approximate analytical solutions for some examples of nonlinear problems are obtained using variational iteration method. Comparison with the exact solution, Runge–Kutta method 4, and also another modified variational iteration method has shown that MoVIM is an accurate method for solving nonlinear problems.

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 Variational Iteration Method for Initial and Boundary Value Problems Using He's Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-28 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ahmet Yildirim ◽  
M. M. Hosseini
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
User Friendly

This paper outlines a detailed study of the coupling of He's polynomials with correction functional of variational iteration method (VIM) for solving various initial and boundary value problems. The elegant coupling gives rise to the modified versions of VIM which is very efficient in solving nonlinear problems of diversified nature. It is observed that the variational iteration method using He's polynomials (VIMHP) is very efficient, easier to implements, and more user friendly. Several examples are given to reconfirm the efficiency of the proposed VIMHP.

scholarly journals On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Index-3

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Melike Karta ◽  
Ercan Çelik
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Variational Iteration

Numerical solution of differential-algebraic equations with Hessenberg index-3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.

The Variational Iteration Method and the Variational Homotopy Perturbation Method for Solving the KdV-Burgers Equation and the Sharma-Tasso-Olver Equation

2010 ◽  
Vol 65 (1-2) ◽  
pp. 25-33 ◽  
Author(s):  
Elsayed M. E. Zayed ◽  
Hanan M. Abdel Rahman
Keyword(s):  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Alternative Methods ◽  
Variational Iteration ◽  
Homotopy Perturbation

AbstractIn this article, two powerful analytical methods called the variational iteration method (VIM) and the variational homotopy perturbation method (VHPM) are introduced to obtain the exact and the numerical solutions of the (2+1)-dimensional Korteweg-de Vries-Burgers (KdVB) equation and the (1+1)-dimensional Sharma-Tasso-Olver equation. The main objective of the present article is to propose alternative methods of solutions, which avoid linearization and physical unrealistic assumptions. The results show that these methods are very efficient, convenient and can be applied to a large class of nonlinear problems.

scholarly journals Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Iteration Method ◽  
Fractional Derivatives ◽  
Financial Models ◽  
Gordon Equation ◽  
Burgers Equation ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Liouville Derivative ◽  
Black Scholes ◽  
Klein Gordon

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

A general numerical algorithm for nonlinear differential equations by the variational iteration method

2020 ◽  
Vol 30 (11) ◽  
pp. 4797-4810 ◽  
Author(s):  
Ji-Huan He ◽  
Habibolla Latifizadeh
Keyword(s):  
Differential Equations ◽  
Numerical Algorithm ◽  
Iteration Method ◽  
Solution Process ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Content Type ◽  
Variational Iteration ◽  
Laplace Transform Technique

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

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