· C ( u , v ) is incremental for all its variables;

· for any u and v ( u , v ∈ [ 0 , 1 ] ) , C ( u , 0 ) = C ( 0 , v ) = 0 , C ( u , 1 ) = u , C ( 1 , v ) = v ;

· For any u 1 , u 2 , v 1 and v 2 , if 0 ≤ u 1 ≤ u 2 ≤ 1 and 0 ≤ v 1 ≤ v 2 ≤ 1 , then C ( u 2 , v 2 ) − C ( u 2 , v 1 ) − C ( u 1 , v 2 ) + C ( u 1 , v 1 ) ≥ 0 ;

· For any u 1 , u 2 , v 1 , v 2 , if u 1 ∈ [ 0 , 1 ] , u 2 ∈ [ 0 , 1 ] , v 1 ∈ [ 0 , 1 ] and v 2 ∈ [ 0 , 1 ] , then | C ( u 2 , v 2 ) − C ( u 1 , v 1 ) | ≤ | u 2 − u 1 | + | v 2 − v 1 | ;

· If U and V are independent and obey the uniform distribution on [0, 1], then C ( u , v ) = u v .

Two-dimensional Gaussian Copula function:

C G a ( u , v , ρ ) = ∫ − ∞ Φ − 1 ( u ) ∫ − ∞ Φ − 1 ( v ) 1 2 π 1 − ρ 2 exp ( − s 2 − 2 ρ s t + t 2 2 ( 1 − ρ 2 ) ) d s d t , (2)

where ρ is a dependent parameter, − 1 ≤ ρ ≤ 1 , Φ − 1 and Φ − 1 are normal distribution function and inverse function respectively.

Two-dimensional t-Copula function:

C G a ( u , v , ρ , k ) = ∫ − ∞ t k − 1 ( u ) ∫ − ∞ t k − 1 ( v ) 1 2 π 1 − ρ 2 exp ( − s 2 − 2 ρ s t + t 2 k ( 1 − ρ 2 ) ) − ( k + 2 ) / 2 d s d t , (3)

where the linear correlation coefficient between variables is − 1 ≤ ρ ≤ 1 , the degree of freedom is k.

Gaussian Copula function and t-Copula function have symmetrical tails, but Gaussian Copula function can’t capture the asymmetric tail correlation between random variables. The t-Copula function has a thick tail, which is very sensitive to the changes of random variables distributed in dimensions, and can better capture the tail correlation changes.

The steps to build copula model are as follows:

1) According to the daily closing price data, the daily return rate of stock is calculated and its statistical characteristics are described.

2) The kernel density estimation method is used to construct the edge distribution, so as to fit the variation characteristics and distribution characteristics of random variables.

3) Catching the dependent structure between random variables by Gaussian Copula function and t-Copula function.

This paper selected the daily closing price data of shares of Midea Group and Gree Electric from July 14, 2007 to October 17, 2019, to study the internal relationship between two stock markets. The data comes from Oriental Fortune net. According to the daily closing price, the daily return of the stock is obtained, and its frequency histogram is given below. It can be seen from Figure 1 and Figure 2 that the distribution of the two-time series is relatively uniform, showing the characteristics of peak and thick tail, but the two-time series are not normal distribution.

Kernel density estimation method is selected for nonparametric estimation. It is

defined as: If the random sample of a single variable x is x 1 , x 2 , ⋯ , x n , the nonparametric density estimation method of the probability density distribution function f ( x ) of x is defined as:

f n ( x ) = 1 n h n ∑ i = 1 n K ( x − X i h n ) , (4)

where K ( ) is kernel function.

Table 1 gives the statistics of daily return, such as mean, variance, skewness and kurtosis. The two-time series are not normal distribution since their kurtosis is large than 3.

This paper chooses Gaussian function as the kernel function since the time series of stock return has the characteristics of peak and thick tail.

It can be seen from Figure 3 and Figure 4 that the empirical distribution function of the stock daily return of Midea Group and Gree Electric is basically consistent with the estimation of the nonparametric kernel distribution, and the nonparametric kernel density estimation can well estimate the overall distribution of the sample.

Figure 5 shows the binary frequency histogram. Obviously, the daily return data of the tail are similar, so the tail of the two series is basically symmetrical. So ( u , v ) has a tail symmetric joint density function. This paper chooses the binary Gaussian Copula function and the binary t-Copula function to describe the correlation of the original sample data. The estimated value of linear correlation parameter ρ of Gaussian Copula function is:

ρ ^ 1 = [ 1.0000 0.0654 0.0654 1.0000 ] .

The estimated value of linear correlation parameter ρ and degree of freedom k of t-Copula function is

ρ ^ 2 = [ 1.0000 0.0701 0.0701 1.0000 ] ,

and the estimated value of degree of freedom k of t-Copula function is k ^ = 2.2496 e + 06 .

Figure 6 and Figure 7 show the density distribution functions and distribution functions of the binary Gaussian Copula function and the binary t-Copula function. It can be seen that density distribution functions of binary Gaussian Copula function and the binary t-Copula function are similar.

In order to verify the goodness of fit of binary Gaussian Copula function and the binary t-Copula function, this paper selected the method of squared Euclidean distance. The smaller the square Euclidean distance is, the better the fitting effect of Copula function is. The square Euclidean distance can be calculated by the following formula:

d 2 = ∑ i = 1 n | C ^ n ( u i , v i ) − C ^ ( u i , v i ) | , (5)

where C ^ n ( u i , v i ) is an empirical Copula function. The fitting effect of binary Gaussian Copula function and the binary t-Copula function are similar since the square Euclidean distances between empirical Copula function and the two Copula functions.