We study the fractional Black–Scholes model (FBSM) of option pricing in the fractal transmission system. In this work, we develop a full-discrete numerical scheme to investigate the dynamic behavior of FBSM. The proposed scheme implements a known
The classical option pricing model is proposed by Black and Scholes [
Combining Itô lemma and fractional Taylor expansion of the option price
Jumarie [
There is a lot of work in the modeling and calculation of fractional equations. Yang et al. [
As we all know, as an effective method,
In this work, we will develop an efficient full-discrete scheme to approximate the Black–Scholes model with
We organize the rest of the paper as follows. Section
In this work, we will consider the following time fractional Black–Scholes model:
This is a linear parabolic partial differential equation which has been studied extensively.
We transform the problem to an initial value problem by using the time to mature
In order to solve the above model by numerical method, it is necessary to truncate the original unbounded region into a finite interval. Therefore, we will consider problem (
In fact, one can choose homogeneous or inhomogeneous boundary conditions. It all depends on the actual option price. We have tested it, and it does not make any difference in actual numerical examples.
Here, we will develop the time-discrete method for equation (
The coefficients of formula (
Then, we can obtain the following time-discrete scheme:
First of all, we have the following energy stability results for time-discrete (
The time-discrete scheme (
When
It is easy to verify that the following formula is correct:
Thus,
Giving up some positive terms, we have
Assume the following inequality holds:
Next, we will show
Note the fact that
Thus, we get
This yields (
In this part, we will study the Fourier-spectral method for the time-discrete method (
We have the following estimate [
Then, we can develop the following full-discrete scheme:
We now present the stability results of the fully discrete scheme (
Let
Next, we begin to analyze the error estimates of the full-discrete scheme (
From [
We also define the following error functions:
For
Note that
Therefore, we obtain
In [
For the constructed numerical scheme (
For
Subtracting (
Then, we have
Set
Dropping some positive terms, we find
Assume
Next, we will prove that it holds also for
Let
Thus, we have
Note that
Note that
This ends the proof.
In this section, several numerical examples will be present to confirm the accuracy and applicability of the full-discrete scheme (
First, in order to conduct a time accuracy test, an exact solution will be constructed to evaluate the convergence of the full-discrete scheme (
We consider the following FBSM with
It is easy to verify that the exact solution will be
We set
Temporal convergence orders of various time steps for Example
1.8166 | 1.8263 | 1.8345 | 1.8415 | 1.8475 | |
1.6680 | 1.6746 | 1.6797 | 1.6837 | 1.6869 | |
1.4892 | 1.4925 | 1.4947 | 1.4963 | 1.4974 | |
1.3939 | 1.3960 | 1.3974 | 1.3983 | 1.3989 | |
1.2964 | 1.2979 | 1.2987 | 1.2992 | 1.2995 | |
1.0980 | 1.0990 | 1.0995 | 1.0997 | 1.0999 |
Fix
Error in time direction for different
The
This section is devoted to investigate the dynamic behavior of FBSM equation with different
The dynamic behavior of solution to FBSM equation at
The dynamic behavior of solution to FBSM equation at
The dynamic behavior of solution to FBSM equation at
The influence of various
The influence of various
In this paper, a new full-discrete numerical method is developed to solve the FBSM. An efficient
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
Juan He carried out an efficient numerical approach to time fractional Black–Scholes model. Aiqing Zhang helped to draft the manuscript. All authors read and approved the final manuscript.
The work of Juan He was supported by the China Scholarship Council (no. 202008520027). The work of Aiqing Zhang was supported by the Cultivation Project of Major Scientic Research Projects of Central University of Finance and Economics (no. 14ZZD007).