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.

The Black-Scholes equation is a mathematical model of a financial market containing certain derivative investment instruments (definition). The idea behind the Black-Scholes model is that the price of an option is determined implicitly by the price of the underlying stock. 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. Therefore, many nonlinear Black-Scholes equations are proposed in recent years [

Variational iteration method [

The main purpose of this paper is to construct a modified VIM for nonlinear Black-Scholes model with combining the VIM with WPIM. According to WPIM, the nonlinear differential equation should be transformed to a system of ordinary differential equations with the multiscales wavelet interpolation operator, and then the nonlinear PDEs become a system of nonlinear ordinary differential equations. So solving the matrix differential equation (MDE) is the key in solving nonlinear PDEs with WPIM. In fact, the matrix differential equation (MDE) is a crucial mathematical foundation of the system engineering and the control theory. But most matrix differential equations do not have precise analytical solutions except linear time-invariant system. In this paper, 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 also free from the stiff problem.

Consider the nonlinear matrix differential equations as follows:

According to VIM, we can write down a correction functional as follows:

Using VIM, the stationary conditions of (

As a result, we obtain the following iteration formula:

According to VIM, we can start with an arbitrary initial approximation that satisfies the initial condition. So we take the exact analytical solution of

Substituting (

Substituting (

In most cases, the second-order nonlinear PDEs about the unknown function

In this section, we take the quasi-Shannon wavelet function as the basis function to approximate the solution function of the nonlinear PDEs. The quasi-Shannon function is defined as follows:

To construct the multilevel interpolation wavelet operator, it is necessary to discretize the wavelet function and the solution function

In order to solve (

The calculation of the exponent matrix

In order to test the accuracy of the coupling technique of VIM and WPIM for solving nonlinear PDEs, we will consider the nonlinear Black-Scholes equations which have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor’s preferences, or illiquid markets, which may have an impact on the stock price, the volatility, the drift, and the option price itself.

Consider the Black-Scholes equation:

In (

In order to solve the problem, it is necessary to perform a variable transformation as follows:

Initial condition of Black-Scholes model.

The evolution of the call option price with the development of the parameter

Evolution of the call option price with the parameter

In following, an adaptive interpolation wavelet numerical method is used to solve the nonlinear partial differential equation.

It is well known that the analytical solution of the linear Black-Scholes model for call option price (

The error of the call option price between linear and nonlinear Black-Scholes models is shown in Figure

Error of call option price between the linear and nonlinear Black-Scholes models.

The coupling technique of VIM and WPIM developed in this paper can solve nonlinear partial differential equations successfully. Comparison between the numerical results of the linear and nonlinear Black-Scholes models illustrates that the proposed method is an accurate and efficient method for the nonlinear PDEs. In addition, as the coupling technique of VIM and WPIM for matrix differential equations has the uniform analytical solution, it can be easily used to solve various nonlinear problems.

The author would like to express his warmest gratitude to Professor Shuli Mei, for his instructive suggestions and valuable comments on the writing of this thesis. Without his invaluable help and generous encouragement, the present thesis would not have been accomplished. At the same time, the author is also grateful to the support of the National Natural Science Foundation of China (no. 41171337), the National Key Technology R and D Program of China (no. 2012BAD35B02), and the Project for Improving Scientific Research Level of Beijing Municipal Commission of Education.