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In this paper, we study the option price theory of stochastic differential equations under G-Lévy process. By using G-It ô formula and G-expectation property, we give the proof of Black-Scholes equations (Integro-PDE) under G-Lévy process. Finally, we give the simulation of G-Lévy process and the explicit solution of Black-Scholes under G-Lévy process.

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 [

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 [

d S t = a S t d t + b S t d W t + c S t d L t , t ∈ [ 0, T ] , (1)

where a is the interest rate, b is the volatility and c is the jump range of asset price, W t is a G-Brownian motion and L t is a G-Lévy process under the G-framework.

Yang and Zhao [

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 S t by using the new algorithms. Meanwhile, we give a numerical example to verify the result of simulation.

We introduce some notation as follows:

● C b k ( ℝ q ) : the space of functions φ : x ∈ ℝ q → ℝ with uniformly bounded partial derivatives ∂ x k 1 φ for 1 ≤ k 1 ≤ k .

● C: a generic constant depending only on the upper bounds of derivatives of a , b , c 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.

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

| x | = 〈 x , x 〉 1 2 be the Euclidean norm in ℝ q and 〈 x , y 〉 is the scalar product of x , y . If A is a vector or matrix, its transpose is denoted by A^{T}. Next, we will give the definition of Sublinear expectation and G-Lévy process.

Definition 1. [

● monotonicity: E [ X 1 ] ≥ E [ X 2 ] for X 1 ≥ X 2 .

● constant preserving: E [ c ] = c with c ∈ ℝ .

● sub-additivity: E [ X 1 + X 2 ] ≤ E [ X 1 ] + E [ X 2 ] .

● positive homogeneity: E [ λ X 1 ] = λ E [ X 1 ] for λ ≥ 0 .

Therefore, we call the triple ( Ω , ℍ , E ) a sublinear expectation space.

Definition 2. [

● for s ≥ 0 , there exists a Lévy process ( X s f , X s g ) satisfies X s = X s f + X s g .

● process X s f and X s g satisfy the following growth conditions:

lim s ↓ 0 E [ | X s f | 3 ] s − 1 = 0 ; E [ | X s g | ] < C s for all s ≥ 0,

where C is a positive constant.

Lemma 1. [

X t i = X 0 i + ∫ 0 t a s i d s + ∑ j = 1 q ∫ 0 t b s i , j d W s j + ∫ 0 t ∫ E c ( e , s ) L ( d e , d s ) ,

where E ∈ ℝ q \ { 0 } , W s is a G-Brownian motion and L ( d e , d s ) is a G-Lévy process. For h ∈ C b 2 ( ℝ q ) , we deduce

h ( X t ) = h ( X 0 ) + ∑ i = 1 q ∫ 0 t a s i ∂ h ( X s ) ∂ x i d s + 1 2 ∑ i , k = 1 q ∑ j = 1 q ∫ 0 t b s i , j b s k , j ∂ 2 h ( X s ) ∂ x i ∂ x k d 〈 W 〉 s + ∑ i = 1 q ∑ j = 1 q ∫ 0 t b s i , j ∂ h ( X s ) ∂ x i d W s j + ∫ 0 t ∫ E [ h ( X s − + c ( e , s ) ) − h ( X s − ) ] L ( d e , d s ) .

Lemma 2. [

G X [ h ( ⋅ ) ] : = l i m t ↓ 0 E [ h ( X t ) ] t − 1 , (2)

where h ∈ C b 3 ( ℝ q ) . If Equation (2) is true, we have the following Lévy-Khintchine representation

G X [ h ( ⋅ ) ] = sup ( λ , a , b ) ∈ U { 〈 D h ( 0 ) , a 〉 + 1 2 t r [ D 2 h ( 0 ) b b T ] + ∫ E h ( e ) λ ( d e ) } ,

where h ( 0 ) = 0 , E = ℝ q \ { 0 } , U ⊂ V × ℝ q × ℚ , V is a set of all Borel measures of E and ℚ is a set of all positive definite symmetric matrix.

Lemma 3. [

∂ u ∂ t − sup ( λ , a , b ) ∈ U { 〈 D u , a 〉 + 1 2 t r [ D 2 u b b T ] + ∫ E ( u ( x + c ( e ) , t ) − u ( x , t ) ) λ ( d e ) } = 0

where D 2 u is the Hessian matrix of u and a ∈ ℝ q , b ∈ ℝ q × q .

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 u = u ( S t , t ) is the option price and S t is the stock price. For Equation (1), we can obtain the following integral partial differential Equation (Integro-PDE) under G-Lévy process

∂ u ∂ t + sup ( λ , a , b , c ) ∈ U { a S ∂ u ∂ S + b 2 S 2 2 ∂ 2 u ∂ S 2 + ln ( 1 + c ) λ ( E ) S ∂ u ∂ S + ln 2 ( 1 + c ) λ ( E ) ( b 2 S 2 2 ∂ 2 u ∂ S 2 + b 2 S 2 ∂ u ∂ S ) } − a u = 0,

where a , b , c ∈ ℝ , U ⊂ V × ℝ × ℝ × ℝ , V is a set of all Borel measures of E and λ ( E ) = ∫ E λ ( d e ) .

Proof. We define a uniform time partition on time interval [ 0, T ] and 0 = t 0 < t 1 < ⋅ ⋅ ⋅ < t n < ⋅ ⋅ ⋅ < t N = T , Δ t = t n + 1 − t n for 0 ≤ n ≤ N . Let the function u ( S , t ) be sufficiently smooth, Δ 〈 W 〉 n = 〈 W 〉 t n + 1 − 〈 W 〉 t n and Δ W n = W t n + 1 − W t n . Using the G-Itô formula, we can obtain the explicit solution of Equation (1):

S t n + 1 = S a exp { a Δ t − 1 2 b 2 Δ 〈 W 〉 n + b Δ W n + ∫ t n t n + 1 ∫ E [ ln ( 1 + c ) ] L ( d e , d s ) } . (3)

In the G-expectation space, we have the following product rule:

d W t ⋅ d W t = d 〈 W 〉 t , d L t ⋅ d L t = λ ( E ) d t + ( λ ( E ) d t ) 2 , d L t ⋅ d t = 0, d W t ⋅ d L t = 0.

Then, it is well known that the option pricing formula following form

u ( S a , t n ) = 1 r E ( [ u ( S t n + 1 , t n + 1 ) − u ( S n , t n ) ] | S t n = S a ) + 1 r u ( S a , t n ) . (4)

Next, we introduce the Black-Scholes model under G-Lévy process. Using Taylor formula for u ( S t n + 1 , t n + 1 ) , we have

u ( S t n + 1 , t n + 1 ) − u ( S a , t n ) = ∂ u ( S a , t n ) ∂ t Δ t + ∂ u ( S a , t n ) ∂ S ( S t n + 1 − S a ) + 1 2 ∂ 2 u ( S a , t n ) ∂ S 2 ( S t n + 1 − S a ) 2 + O ( Δ t ) 3 2 . (5)

Substituting Equation (3) into (5), we obtain

u ( S t n + 1 , t n + 1 ) − u ( S a , t n ) = ∂ u ( S a , t n ) ∂ t Δ t + ∂ u ( S a , t n ) ∂ S ( S a exp { X n } − S a ) + 1 2 ∂ 2 u ( S a , t n ) ∂ S 2 ( S a exp { X n } − S a ) 2 + O ( Δ t ) 3 2 ,

where X n = a Δ t − 1 2 b 2 Δ 〈 W 〉 n + b Δ W n + ∫ t n t n + 1 ∫ E ln ( 1 + c ) L ( d e , d s ) . Let λ ( E ) = ∫ E λ ( d e ) , it induces from Taylor expansion for exp { X n } that

u ( S t n + 1 , t n + 1 ) − u ( S a , t n ) = [ ∂ u ∂ t + a S a ∂ u ∂ S ] Δ t − S a b 2 2 ∂ u ∂ S Δ 〈 W 〉 n + S a ln ( 1 + c ) ∂ u ∂ S L E + S a b ∂ u ∂ S Δ W n + [ S a ∂ u ∂ S + S a 2 ∂ 2 u ∂ S 2 ] 1 2 ( X n ) 2 + O ( Δ t ) 3 2 = [ ∂ u ∂ t + a S a ∂ u ∂ S ] Δ t − S a b 2 2 ∂ u ∂ S Δ 〈 W 〉 n + S a ln ( 1 + c ) ∂ u ∂ S L E + S a b ∂ u ∂ S Δ W n + [ S a ∂ u ∂ S + S a 2 ∂ 2 u ∂ S 2 ] 1 2 ( b 4 4 ( Δ 〈 W 〉 n ) 2 + b 2 ( Δ W n ) 2 − b 3 Δ W n Δ 〈 W 〉 n + ln 2 ( 1 + c ) λ ( E ) Δ t + ln ( 1 + c ) λ ( E ) ( Δ t ) 2 ) + O ( Δ t ) 3 2 ,

where L E = ∫ t n t n + 1 ∫ E L ( d e , d s ) . Inserting the above result into Equation (4), we can deduce

u = 1 r E [ ( ∂ u ∂ t + a S a ∂ u ∂ S ) Δ t − S a b 2 2 ∂ u ∂ S ( Δ 〈 W 〉 n ) + S a b ∂ u ∂ S ( Δ W n ) + S a ln ( 1 + c ) ∂ u ∂ S L E + [ S a ∂ u ∂ S + S a 2 ∂ 2 u ∂ S 2 ] 1 2 ( b 4 4 ( Δ 〈 W 〉 n ) 2 + b 2 ( Δ W n ) 2 − b 3 Δ W n Δ 〈 W 〉 n + ln 2 ( 1 + c ) λ ( E ) Δ t + O ( Δ t ) 3 2 ) | S t n = S a ] + 1 r u .

It induces from the G-expectation property and the fact u = u ( S t n , t n ) and Δ W n ∼ N ( 0 ; [ σ _ 2 Δ t , σ ¯ 2 Δ t ] ) that we can deduce

( 1 − 1 r ) u = Δ t r ( ∂ u ∂ t + sup ( λ , a , b , c ) ∈ U { a S a ∂ u ∂ S + ( b 2 S a 2 2 ∂ 2 u ∂ S 2 ) + σ ¯ 2 − ( b 2 S a 2 2 ∂ 2 u ∂ S 2 ) − σ _ 2 + ln ( 1 + c ) S a ∂ u ∂ S λ ( E ) + ln 2 ( 1 + c ) ( b 2 S 2 2 ∂ 2 u ∂ S 2 + b 2 S 2 ∂ u ∂ S ) λ ( E ) } + O ( Δ t ) 1 2 ) ,

where r = 1 + a Δ t and a is risk-free rate. Consequently, we obtain the following integro-partial differential equation:

∂ u ∂ t + sup ( λ , a , b , c ) ∈ U { a S ∂ u ∂ S + b 2 S 2 2 ∂ 2 u ∂ S 2 + ln ( 1 + c ) λ ( E ) S ∂ u ∂ S + ln 2 ( 1 + c ) λ ( E ) ( b 2 S 2 2 ∂ 2 u ∂ S 2 + b 2 S 2 ∂ u ∂ S ) } − a u = 0.

The proof is completed. □

In this section, we will give a numerical example for option pricing in stock market. And we study the stock price S t under G-Lévy process. Platen [

Algorithm 1. (The simulation of Poisson jump)

● Setting up the values of intensity λ and the terminal time T.

● Generating random number z i obeying exponential distribution with parameter lambda.

● Then by the formula t k = ∑ i = 1 k z i , we get the occurrence time t k ofn events.

● Plotting a ladder figure for the Poisson jump process.

Assume the intensity λ = 1.5 and the number of jumps are equal to 10. And

Algorithm 2. (The simulation of G-Lévy process)

● Setting up the terminal time T and the intensity functions λ ( t ) , where λ ( t ) ≤ λ with λ is a constant.

● Generating the Poisson jump process random number with intensity λ and obtaining the time of occurrence s 1 , s 2 , ⋯ , s n .

● Generating the uniformly distributed random number x i on ( 0,1 ) . If x 1 ≤ λ ( s i ) / λ , we retain the s i , else we give up the time s i .

● Plotting the time s i which are obtained in the above step and the number of jumps.

Suppose the intensity function λ ( t ) = t 5 and the number of jumps are equal to 25. For λ = 10 and λ = 15 , we simulate the G-Lévy process in

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 [

Example 1. Consider the stock price S t has the following form:

d S t S t = a d t + b d W t + c d L t , t ∈ [ 0 , T ] , (6)

where the initial value S 0 = 0 , the interest rate a and volatility b are positive, W t is a G-brownian motion and L t is a G-Lévy process. Next, we give the explicit solution of Equation (6) on t ∈ [ 0 , 1 ]

S t = S 0 exp { a t − 1 2 b 2 〈 W 〉 t + b W t + ∫ 0 t ∫ E [ ln ( 1 + c ) ] L ( d e , d s ) } .

In this example, we firstly use three different coefficients a 1 = 0.1 , b 1 = 0.3 , c 1 = 0.1 , a 2 = 0.2 , b 2 = 0.2 , c 2 = 0.2 and a 3 = 0.3 , b 3 = 0.1 , c 3 = 0.3 to simulate the stock price S t . And the simulation of S t are given in

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 S t . Let coefficients a = 0.1 , c = 0.1 , we plot the stock price S t with the time t under the different coefficients b = 0.1 , b = 0.2 , b = 0.3 in

Let coefficients a = 0.1 , b = 0.1 , we plot the stock price S t with the time t under the jump intensity coefficients c = 0.5 , c = 1 , c = 5 in

By comparing Figures 3-5, we obtain that coefficients a and b have a great influence on stock price S t than coefficient c. And the stock price S t has a small variety when coefficient c changes.

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 S t , and obtain the coefficients a , b that have a great influence on stock price S t . In the future, we will study the numerical scheme for solving the Integro-PDE. And the numerical scheme is important in financial field.

The authors declare no conflicts of interest regarding the publication of this paper.

Xin, Y.F. and Zheng, H. (2021) Black-Scholes Model under G-Lévy Process. Journal of Applied Mathematics and Physics, 9, 3202-3210. https://doi.org/10.4236/jamp.2021.912209