Application of G-Brown Motion in the Stock Price

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1. Introduction

Asset pricing theory is one of the themes of the financial economy. Lèvy and Paras [1] proposed an uncertain volatility model, but the study could not give a dynamic option price. Peng [2] [3] defines G-expectation and G-Brown motion to provide a solution to this problem. Describing the theoretical basis of the option price. Yang and Zhao [4] simulate the G-normal distribution, and study the numerical simulation of G-Brown motion and the simulation of the second variation of G-Brown motion, then the finite difference method is given to solve the G-heat equation. Xu [5] [6] study the European call option price formula and Girsanov theorem under G-expectation. Wang [7] study the G-Jensen inequality under G-expectation. Wang [8] study the comparison theorem and Asian option pricing under G-expectation. Kang [9] study the Brownian motion martingale representation theorem under G-expectation.

The main purpose of this paper is to introduce the price change of the asset driven by G-geometric Brown. First we give the martingale property of discount value under G-framework. We simulate the stock price. We compare the stock price under normal with the stock price under G-. And we give the stock price under different G-. Then we study the G- influence on under G-framework.

2. The G-Martingale Property of Discount Value

Definition 1 [2] : is G-brown motion, is a division on, when, we denote G-quadratic variation process by:

(1)

Definition 2 [2] : A nonlinear expectation is a function satisfying the following properties:

1) Monotonicity: If and then;

2) Preserving of constants:;

3) Sub-additivity,;

4) Positive homogeneity:, ,;

5) Constant translatability:.

Definition 3 [2] : The canonical process B is called a G-Brownian motion under a nonlinear defined on if for each, and for each, , we have

Lemma 1 [2] [G-Itô formula]: for is G-brownian motion, is quadratic variation process of G-brownian miton, is a function about, and, , are continuous function, we have

Lemma 2 [7] [G-Jensen inequality] h is a continuous function defined on R. Then the following two conditions are equivalent:

1) h is a convex function;

2) For, if, we have

Lemma 3 [6] [Girsanov under G-framework]: for, if existing and satisfying:

we have

is a symmetrical martingale under for, .

In this section, we introduce the American call option, give a G-geometric Brownian motion asset. And we prove that the American call price is the same as the European call price.

Considering a stock whose price process is given by

(2)

where the interest rate r and the volatility () are positive and is a G-brownian motion.

Now we compute (2) through G-Itô formula, in [5] the result is:

(3)

where is the stock value at current moment.

Theorem 1: is a nonnegative and convex function, ,. Then the discount value of American option is a G-submartingale.

Proof: is a convex, for and, we have

(4)

. Taking, , and using the fact h(0)=0, we obtain

(5)

for, we have, by (5) and G-expectation property

(6)

According to Lemma 2,

(7)

by Lemma 3 we know that is a G-symmetrical martingale, which implies

(8)

So we conclude that

(9)

and

the is a G-submartingale.

The Inequality (9) implies that the European derivative security price always dominates the intrinsic value of American derivative security. This shows that the option to exercise early is worthless, so the American call option agrees with the price of European option under G-framework.

3. Numerical Simulation

We mainly simulate stock price under G- and

G-. The G- and G- values are simulated in [4]. Yang and Zhao [4] mainly simulate the G-brownian motion by solving a specific HJB equation. Then they give four finite difference methods to solve the HJB equation. Finally they give the numerical algorithms to simulate G-normal distribution, G-brownian motion G-quadratic variation process. The following we give three algorithms.

Algorithm 1 [4] (simulation and):

• For random, calculating approximation;

• For, calculating the difference;

• By calculating density function’s approximation.

by the G-heat equation defining the G-normal distribution and the density function. By Algorithm 1 simulating the and, then we apply these in Algorithm 2 and Algorithm 3.

Algorithm 2 [4] (G-brownian motion numerical simulation):

• For random, using algorithm 1 compute;

• Produing N random numbers in [0,1] obey uniformly distribution ;

• For, calculating;

• By, solving,;

• By, approaching,.

We simulate the values of G-brown motion. By simulating the, we use it in Algorithm 3 to get the.

Algorithm 3 [4] (numerical simulation):

• For random, using algorithm 1 to compute;

• Generating N random numbers in [0,1] for, calculating;

• By, solving,;

• By, approaching,.

The following we simulate the stock price under the G- and G- values.

Example 1: we consider stock price at time t immediately, where interest rate, the volatility,.

Figure 1 denotes the comparison between under G-framework and under classical framework. In Figure 1 we can know that the blue line is simulated by, the red line is simulated by. Figure 2 simulates the price of based on three different G- in Figure 3. Figure 3

is about G- of simulation. In Figure 3, the three lines are respectively under (,), (,), (,). Figure 4 is about

Figure 1. Comparing stock price of simulation between G-expectation framework and classical framework.

Figure 2. Comparing stock price under different G-.

Figure 3. The G- of simulation.

of simulation under classical framework. We can know the G- is different from the according to Figure 3 and Figure 4. And the stock price is a about G-, G-, t function under G-framework. The stock price is a function about and t. That is the main reason to cause the difference. We can know that the G- influence on under G-framework from Figure 5.

The blue line is function. The red line is function . From Figure 6, we can know the G- of simuation values. According to Figure 6 when we replace the with the under G-framework, it has no impact on stock price fluctuations.

Figure 4. The normal of simulation.

Figure 5. The influences on S(t).

Figure 6. The of simulation.

4. Conclusion

This article mainly proves that American call options that do not pay dividends under the G-framework are equal to European call options and simulate the G- image. Comparing stock price images under different, G-. There is a restriction on G-. When is smaller, the G- of simulation shows a downward fluctuation. We need to find the appropriate range of to simulate the stock price.

References

[1] Avellaneda, M., Lèvy, A. and Paras, A. (1995) Pricing and Hedging Derivatives Securities in Markets with Uncertain Volatilities. Applied Mathematical Finance, 2, 73-88. https://doi.org/10.1080/13504869500000005

[2] Peng, S.G. (2007) G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type. In: Benth, F.E. et al., Eds., Stochastic Analysis and Applications. Springer, Berlin, 541-567. https://doi.org/10.1007/978-3-540-70847-6_25

[3] Peng, S.G. (2008) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation. Stochastic Processes and Their Applications, 118, 2223-2253. https://doi.org/10.1016/j.spa.2007.10.015

[4] Yang, J. and Zhao, W.D. (2016) Numerical Simulations for the G-Brownian Motion. Frontiers of Mathematics in China, 11, 1625-1643.

https://doi.org/10.1007/s11464-016-0504-9

[5] Xu, J. and Xu, M.P. (2010) European Call Option Price under G-Framework. Mathematics in Practice and Theory, 4, 41-45.

[6] Xu, J., Shang, H. and Zhang, B. (2011) A Grsanov Type Theorem under G-Framework. Stochastic Analysis and Applications, 29, 386-406.

https://doi.org/10.1080/07362994.2011.548985

[7] Wang, W. (2009) Properties of G-Convex Function under the Framework of G-Expectations. Journal of Shandong University, 44, 43-46.

[8] Wang, Y. (2018) Research of the Comparison Theorem and Asian Option Pricing under the G-Expectation.

[9] Kang, Y. (2012) Brownian Motion Martingale Representation Theorem under the G-Expectation.