A positivity-preserving numerical method for nonlinear Black-Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of the numerical solution. The numerical results for European put option and European butterfly spread are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.

It is widely recognized that the value of a European option can be obtained by solving the linear Black-Scholes equation under quite restrictive assumptions (such as liquid, frictionless, and complete markets) [

In this paper, we are interested in the option pricing model with transaction costs proposed by Barles and Soner [

The existence and uniqueness of the solution of (

There have been rich achievements for the numerical method of linear Black-Scholes equations (e.g., see [

On the other hand, since the value of option is nonnegative, it is very important to make numerical schemes preserve the positivity of solution. Several authors have developed some schemes that guarantee the positivity of solutions for ordinary differential equations [

The rest of the paper is organized as follows. In the next section the original problem (

For the convenience in the numerical processing and the study of the numerical analysis, we are going to transform the problem (

The following two lemmas give the properties of the function

The solution

if

Let

Since the value of an option is nonnegative, it is important that numerical scheme is positivity preserving.

We see that the problem (

Clearly, the approximation used to calculate the second partial derivative of

Scheme (

From property (i) of Lemma

Obviously, numerical scheme (

In order to obtain an efficient scheme, we correct the approximation of the nonlinear coefficients in (

Since the calculation of (

Thus the values of boundary points turn out

The left boundary condition is not needed and in fact must not be prescribed in the case of a parabolic equation with degenerating diffusion term at

In order to study the related properties of numerical scheme (such as monotonicity and stability), we need to know the behaviour of

With the previous notation,

From (

Putting (

On the other hand, the most common option price sensitivities are the first and second derivatives with respect to the price of the underlying asset, that is, “delta” and “gamma”, respectively. These are important features in risk management and are challenging to compute numerically. From the transformation (

For the sake of convenience, we introduce the following definition.

If

The following theorem shows that the nonlinear team

Let

The nonlinear team

Property (i) is proved using the induction principle over index

For

Now, let us assume that property (i) holds true up

On the other hand, from (

Since

Since the value of option is nonnegative, a nice property of the numerical scheme for the pricing equation is positivity-preserving.

Clearly, all the coefficients of the numerical scheme (

The numerical scheme (

For the sake of convenience in the presentation, we introduce the following definition of a monotonicity preserving numerical scheme.

In numerical scheme

The next result shows the monotonicity of the numerical scheme.

The numerical scheme (

Let us write

The monotonicity of the scheme in the internal mesh points has been proved.

In an analogous way, we can verify that

Similarly, we can prove that if

Thus the proof of theorem is complete.

If the payoff

The difference scheme (

From (

The proposed new difference scheme (

The following theorem gives the consistency condition of the difference scheme (

With the previous notation, suppose that the exact solution

Let us write the scheme (

From Theorem

In this section, we implement the positivity-preserving scheme (

A European put option is a contract where the owner of the option has the right to sell an underlying asset

Equations (

The values of European put option under different schemes with the computational parameters

Moreover, Figure

Time evolution profiles of the Greeks of European put option with the computational parameters

Standard scheme for delta

New scheme for delta

Standard scheme for gamma

New scheme for gamma

Next we consider the influence of transaction costs (parameter

Influence of the transaction costs (parameter

Linear and nonlinear cases

Nonlinear-linear (

A butterfly spread is a combination of three-call options with three-strike prices, in which one contract is purchased with two outside strike prices and two contracts are sold at the middle strike price. The payoff function

We choose the following parameters:

Figure

The value of European butterfly spread under different schemes with the computational parameters

Time evolution profiles of the Greeks of European butterfly spread with the computational parameters

Standard scheme for delta

New scheme for delta

Standard scheme for gamma

New scheme for gamma

The last two figures show the value of European butterfly spread with different transaction costs (parameter

Influence of the transaction costs (parameter

Linear and nonlinear cases

Nonlinear-linear (

In this paper, we have extended the numerical method in [

The authors would like to thank the anonymous referees for several suggestions for the improvement of this paper. This work is supported by the Fundamental Research Funds for the Central Universities of China (Grant no. JGK101677).