This paper deals with the numerical analysis of nonlinear Black-Scholes equation with transaction costs. An unconditionally stable and monotone splitting method, ensuring positive numerical solution and avoiding unstable oscillations, is proposed. This numerical method is based on the LOD-Backward Euler method which allows us to solve the discrete equation explicitly. The numerical results for vanilla call option and for European butterfly spread are provided. It turns out that the proposed scheme is efficient and reliable.
One of the modern financial theory’s biggest successes in terms of both approach and applicability has been the Black-Scholes option pricing model developed by Black and Scholes in 1973 [
In this paper, we are interested in the option pricing model with transaction costs proposed by Barles and Soner [
Many results have been reported for the numerical solutions of linear Black-Scholes equations (see, e.g., [
In this paper, we will consider a splitting method with inhomogeneous boundary conditions. This method proposed here is unconditionally stable, monotone, and positivity preserving. It is also consistent and essentially a “limit” version of the LOD-Backward Euler method (see Chapter IV on splitting methods in [
The paper is organized as follows: we begin by transforming the original equations into nonlinear heat equations and considering the spatial semidiscretization. The splitting scheme will be discussed in Section
We first consider the properties of the function
The solution
if
The function
Combining Lemma
The function
After considering the change of variable
To numerically approximate the solution of (
Obviously, the semidiscrete difference scheme (
For system (
To obtain the stability of the semidiscrete difference scheme (
In this section, we will focus on the time integration methods of system (
Let us set
In this subsection, we construct a splitting time-stepping method based on (
With the splitting (
Then, if the Backward Euler method is used to solve every subproblem, we get
By brief calculation, one can obtain an explicit expression of matrix
For the finite-dimensional discrete system (
In this section, we investigate some properties of the numerical scheme proposed here.
In [
The numerical scheme (
To obtain the spectral norm stability inequality (
We note that
A nice property of the numerical scheme for the pricing equation is positivity preserving, since the value of option is nonnegative.
The numerical scheme (
Since all entries of the matrices
Let us now consider the monotonicity of the numerical scheme (
Scheme (
The following theorem shows the monotonicity of the numerical scheme (
The numerical scheme (
From the proof of Theorem
From (
Now, we consider the error. Let
For splitting
With the previous notation, the local discretization error
It follows from (
We can now immediately conclude that the splitting scheme (
The so-called viscosity solutions are the meaningful solutions in financial applications. This has been shown in [
The splitting Backward Euler scheme (
The convergence of the fully-discrete scheme (
In order to illustrate the stability and convergence properties of our proposed scheme, in this section, we present several numerical experiments in which the vanilla call option and the European butterfly spread are considered. To obtain the numerical solution of nonlinear Black-Scholes equations (
Let us consider a vanilla European call option with
Option pricing of a vanilla European call option for several values of parameter
To further confirm that this scheme is unconditionally stable, monotone, and positivity preserving, we also calculate the numerical solutions, which are presented in Figures
Option pricing of a vanilla European call option for several values of parameter
Option pricing of a vanilla European call option for several values of parameter
To illustrate the convergence of scheme (
Convergence results for the scheme (
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0.001487 |
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From the data presented in Figures
This example comes from [
Option pricing of a European butterfly spread for several values of parameter
Option pricing of a European butterfly spread for several values of parameter
Option pricing of a European butterfly spread for several values of parameter
In [
Convergence results for the scheme (
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From the theoretical analysis given in this paper and the numerical results shown in this section, we come to the following remark: the proposed scheme is efficient and reliable.
In this paper, an unconditionally stable splitting scheme has been proposed to solve the nonlinear option pricing model with transaction costs. This method can be viewed as a “limit” version of LOD-Backward Euler method. This “limit” property that all subproblems are one-dimensional allows us to solve the discrete equation explicitly. As a consequence, this method is computationally efficient. The theoretical analysis carried out in this paper shows that this method is unconditionally stable, monotone, and positivity preserving. We also present several numerical experiments in which the vanilla call option and the European butterfly spread are considered. The theoretical analysis presented and the numerical results shown in this paper confirm that the proposed scheme here is efficient and reliable.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous referees and the editor for the important comments that led to a greatly improved paper. This work was supported by the National Natural Science Foundation of China (Grants nos. 11371074 and 11001033), the Hunan Provincial Natural Science Foundation of China (Grant no. 13JJ1020), the Research Foundation of Education Bureau of Hunan Province, China (Grant no. 13A108), and the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities.