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

Huahong Yan

doi:10.1155/2013/735919

Abstract

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.

Highlights

The Black-Scholes equation is a mathematical model of a financial market containing certain derivative investment instruments
In order to test the accuracy of the coupling technique of VIM and WPIM for solving nonlinear PDEs, we will consider
It is well known that the analytical solution of the linear Black-Scholes model for call option price (C) can be obtained as follows: C = S ⋅ N (d1) − Ke−rTN (d2), (36)

Summary

Introduction

The Black-Scholes equation is a mathematical model of a financial market containing certain derivative investment instruments (definition). The Black-Scholes model is a mathematical model based on the notion that prices of stock follow a stochastic process It is widely employed as a useful approximation, but proper application requires understanding its limitations. There are some numerical algorithms that have been proposed based on the difference method to solve those nonlinear problems, but the precision depends on the time step and the discretization in definition domain [3, 4]. Most matrix differential equations do not have precise analytical solutions except linear time-invariant system. A coupling technique of He’s VIM and WPIM is developed to establish an approximate analytical solution of the matrix differential equations. In contrast to the traditional finite difference approximation, the numerical result obtained with PIM for a set of simultaneous linear time-invariant ODEs approaches the computer precision and is free from the stiff problem

Fundamental Theory of Coupling Technique of VIM and WPIM

Coupling Technique of VIM and WPIM for the Nonlinear Black-Scholes Model

Conclusion