1. Introduction
Nowadays, many studies are interested in stochastic differential equations (SDEs). And SDEs have been widely applied to economics and finance fields, such as option pricing in stock market see [1]. In the 1970s, Black and Scholes propose the famous option pricing model and promote the development of stocks, bonds, currencies, products. Subsequently, the famous Black-Scholes formula was paid more attention by many scholars. In 1976, Merton [2] proposed the logarithmic jump-diffusion models in stock price, which was described as a combination of Brownian motion and compound Poisson process.
Although option pricing formula has developed for a long time, there are many uncertainty problems in stock market. Many scholars have been studied the uncertainty problem. For example, Peng [3] [4] proposed the sublinear expectation space to solve the uncertainty problem. In particular, the G-expectation space plays an important role in solving them. Then the G-Brownian motion, G-Itô formula and G-center limit theorem are proposed for us in G-expectation framework. In this paper, we consider the following stock price
such that:
(1)
where a is the interest rate, b is the volatility and c is the jump range of asset price,
is a G-Brownian motion and
is a G-Lévy process under the G-framework.
Yang and Zhao [5] introduce the simulation of G-Brownian and G-normal distribution under G-expectation and Chai studies the option pricing for stochastic differential equation under G-framework. Although G-Brownian motion solved many financial issues, some financial models that depend on the Lévy processes remain unresolved. Therefore, Peng and Hu [6] studied the G-Lévy process, which is the generalization of G-Brownian motion. And Krzysztof [7] introduced G-Itô formula and G-martingale representation for G-Lévy process.
In this paper, we study Black-Scholes model under G-Lévy process and prove the Integro-PDE by using G-Itô formula, option pricing formula and G-expectation property. Then we simulate the G-Lévy process and the stock price
by using the new algorithms. Meanwhile, we give a numerical example to verify the result of simulation.
We introduce some notation as follows:
●
: the space of functions
with uniformly bounded partial derivatives
for
.
● C: a generic constant depending only on the upper bounds of derivatives of
and h, and C can be different from line to line.
The outline of the paper is as follows. In Section 2, we introduce some necessary notations and theorems, such as the G-Lévy process and G-Itô formula. In Section 3, we propose a new theorem that gives the proof of Black-Scholes equations (Integro-PDE) under G-Lévy process. Finally, some numerical simulations for G-Lévy process and stock price are given in Section 4.
2. Preliminaries
In this section, we will introduce some basic knowledge and notation that is the focus of this paper. Throughout this paper, we will give the definition of G-Lévy process. Unless otherwise specified, we use the following notations. Let
be the Euclidean norm in
and
is the scalar product of
. If A is a vector or matrix, its transpose is denoted by AT. Next, we will give the definition of Sublinear expectation and G-Lévy process.
Definition 1. [6] (Sublinear expectation) Let
is a linear space and
, we give the definition of sublinear expectation
● monotonicity:
for
.
● constant preserving:
with
.
● sub-additivity:
.
● positive homogeneity:
for
.
Therefore, we call the triple
a sublinear expectation space.
Definition 2. [6] (G-Lévy process) Assume
is a Lévy process,
is a generalized G-Brownian motion and
is of finite variation. We say the X is a G-Lévy process if satisfy the following conditions:
● for
, there exists a Lévy process
satisfies
.
● process
and
satisfy the following growth conditions:
where C is a positive constant.
Lemma 1. [7] (G-Itô formula) For
,
is the k-th component of
and it satisfies the following form:
where
,
is a G-Brownian motion and
is a G-Lévy process. For
, we deduce
Lemma 2. [6] (Lévy-Khintchine representation) Assume X is a G-Lévy process in
, we have the following form
(2)
where
. If Equation (2) is true, we have the following Lévy-Khintchine representation
where
,
,
,
is a set of all Borel measures of
and
is a set of all positive definite symmetric matrix.
Lemma 3. [6] (Integro-PDE) Assume X is a G-Lévy process and the functions
, and by using the Lemma 2 (Lévy-Khintchine representation), we have the following Integro-PDE:
where
is the Hessian matrix of u and
,
.
3. Black-Scholes Equations under G-Lévy Process
In this section, we will give the Black-Scholes equations under G-Lévy process, and prove the Integro-PDE by combining the G-Itô formula and the option pricing formula.
Theorem 1. (Black-Scholes equations) Assume
is the option price and
is the stock price. For Equation (1), we can obtain the following integral partial differential Equation (Integro-PDE) under G-Lévy process
where
,
,
is a set of all Borel measures of
and
.
Proof. We define a uniform time partition on time interval
and
,
for
. Let the function
be sufficiently smooth,
and
. Using the G-Itô formula, we can obtain the explicit solution of Equation (1):
(3)
In the G-expectation space, we have the following product rule:
Then, it is well known that the option pricing formula following form
(4)
Next, we introduce the Black-Scholes model under G-Lévy process. Using Taylor formula for
, we have
(5)
Substituting Equation (3) into (5), we obtain
where
. Let
, it induces from Taylor expansion for
that
where
. Inserting the above result into Equation (4), we can deduce
It induces from the G-expectation property and the fact
and
that we can deduce
where
and a is risk-free rate. Consequently, we obtain the following integro-partial differential equation:
The proof is completed. □
4. Numerical Experiment
In this section, we will give a numerical example for option pricing in stock market. And we study the stock price
under G-Lévy process. Platen [1] introduces the application jump process in stock market of financial field. The simulation of G-Brown under G-framework see [5] [8] [9]. Next, we firstly study the simulation of Poisson jump process and G-Lévy process.
Algorithm 1. (The simulation of Poisson jump)
● Setting up the values of intensity
and the terminal time T.
● Generating random number
obeying exponential distribution with parameter lambda.
● Then by the formula
, we get the occurrence time
ofn events.
● Plotting a ladder figure for the Poisson jump process.
Assume the intensity
and the number of jumps are equal to 10. And Figure 1 shows that the simulation of Poisson jump process.
Figure 1. The plots of time and number of jumps for Poisson jump process.
Algorithm 2. (The simulation of G-Lévy process)
● Setting up the terminal time T and the intensity functions
, where
with
is a constant.
● Generating the Poisson jump process random number with intensity
and obtaining the time of occurrence
.
● Generating the uniformly distributed random number
on
. If
, we retain the
, else we give up the time
.
● Plotting the time
which are obtained in the above step and the number of jumps.
Suppose the intensity function
and the number of jumps are equal to 25. For
and
, we simulate the G-Lévy process in Figure 2.
Next, we will introduce the Black-Scholes formula with jump under the G-Lévy process, and it is the generation of classical Black-Scholes formula. In [6], Peng and Hu use the option pricing formula under the G-Lévy process. There we will give the following examples.
Example 1. Consider the stock price
has the following form:
(6)
where the initial value
, the interest rate a and volatility b are positive,
is a G-brownian motion and
is a G-Lévy process. Next, we give the explicit solution of Equation (6) on
In this example, we firstly use three different coefficients
,
,
,
,
,
and
,
,
to simulate the stock price
. And the simulation of
are given in Figure 3 with three different coefficients.
Because the interest rate a, volatility b and jump intensity c are variable, we study the influence of volatility b and jump intensity c on stock price
. Let coefficients
,
, we plot the stock price
with the time t under the different coefficients
,
,
in Figure 4. And we obtain the stock price
will decrease with the increase of the volatility b.
Figure 2. The simulation of G-Lévy process.
Figure 3. The simulation of stock price
with three different coefficients.
Figure 4. Stock price
with three different coefficients
,
,
.
Figure 5. Stock price
with different coefficients
,
,
.
Let coefficients
,
, we plot the stock price
with the time t under the jump intensity coefficients
,
,
in Figure 5.
By comparing Figures 3-5, we obtain that coefficients a and b have a great influence on stock price
than coefficient c. And the stock price
has a small variety when coefficient c changes.
5. Conclusion
In this paper, by using G-Itô formula and G-expectation property, we prove the Integro-PDE under G-Lévy process. Then we study the influence of coefficients on stock price
, and obtain the coefficients
that have a great influence on stock price
. In the future, we will study the numerical scheme for solving the Integro-PDE. And the numerical scheme is important in financial field.