This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit scheme of the finite difference method.

The theory and methodology of partial differential equation started to become popular to study option pricing problems, after the classical Black-Scholes equation was proposed. The available research results include mainly two aspects: one is to give values of options using more powerful numerical and analytic methods; the other is to derive new pricing models that reflect the actual financial market more closely.

The Black-Scholes equation has been increasingly attracting interest over the last two decades since it provides effectively the values of options. But the classical Black-Scholes equation was established under some strict assumptions. Therefore, some improved models have been proposed to weaken these assumptions, such as stochastic interest model [

In the present work, the option price

The value of European put option is taken as a solution of (

Pricing European put option based on (

Fractional Black-Scholes equation (

Numerical and analytical methods [

Take the change of variable

In order to use finite difference approximation, we start by

As pointed out in [

For spatial derivative, we use the following difference approximation:

Substituting (

From (

If the minimum limit of variable

From the terminal and boundary conditions of the European put option, we can get

In the case of American put option, we perform the above procedure. At the same time, we should check for the possibility of early exercise after computing

In this section, we will analyze the stability and convergence of implicit finite difference scheme (

If

Now, we introduce the grid function in [

Recall the result of [

The solution of (

Substituting the above expression into (

If

For

According to (

The difference scheme (

Suppose that

For completing the proof of convergence, we recall several results below which came from [

Similar to the stability analysis, [

Defining

We can let

Substituting the above expressions into (

There exists a positive constant

From [

According to equalities (

The implicit difference scheme (

Option pricing model based on the time-fractional differential equation (

Consider (

Under the condition

Figure

Solution curve with

Absolute error between numerical solution and analytical solution. dash-dotted line:

European put option pricing model is based on (

European put option. dash-dotted line:

Consider American put option pricing model. The following parameters are selected for the present study:

American put option. dash-dotted line:

Using Examples

In this work, the finite difference method is employed to solve the time-fractional Black-Scholes equation together with the conditions satisfied by the standard put options. Application of the fractional differential equation to the pricing theory of option is in its beginning stage and needs more further work. This fractional model mentioned in this paper can model the price of other financial derivatives like warrant, swaps, and so on. The successful application of the finite difference method proves that this technique is effective and requires less computational work to solve fractional partial differential equation.

The authors would like to express their gratitude to reviewers for the careful reading of the paper and for their constructive comments which greatly improve the quality of this paper. The work is partially supported by the National Natural Science Foundation of China (no. 71171035, no. 71273044, and no. 71271045).