Dependence For Financial Time Series Based On Copula Theory

Nowadays, the relationships between the financial markets are getting increasingly close and complex because of the rapid development of financial markets, deepening of financial innovation, economic globalization and financial internationalization. How to describe the dependence structure between financial variables accurately is the basis of studying issues such as financial risk management, portfolio and asset pricing. Correlation analysis has drawn more and more attention in the modern financial analysis. That the research of the dependence structure has an important theoretical and practical significance to explore the mechanism of financial system and make financial decisions scientifically. However, most of the traditional ways of correlation analysis couldn't describe the nonlinear and asymmetrical features. The existing multivariate joint distribution function couldn't meet the needs of is multivariate practical analysis neither.The emergence and development of the Copula theory has greatly extended the multivariable correlation analysis to a new stage and become the hotspot of the current financial research. This paper employs Copula theory to study the dependence of financial time series. Firstly, we systematically expound the Copula function and its characteristics, especially for the nested Copula and multiple correlations. Then, we discuss the dependency of indicator derived from the Copula function in depth, and demonstrate that the joint generating function method which describes the tail correlation by combining the coefficient of tail dependence and slowly changing function is better than the common tail correlation coefficient. At last, we introduced the latest dynamic of Copula model. The main work and innovation are as following: This paper discusses the key issue of applying the Copula model: the data fitting and selection of Marginal distribution function. The Copula theory method based on correlation analysis overcomes the limitations of traditional methods, but the choice of marginal distribution function must be correct. This paper mainly discusses the kernel density function, extreme model and fluctuation margin distribution function model.This paper employs POT model to estimate extreme risk in SME stock market for the empirically study. The result shows that the exponential regression model is an effective way to the quantities'threshold selection of small POT model, and it avoids the threshold selection uncertainty of Hill estimator. POT model can better describe the fat-tail and asymmetrical distribution features, so it is prior to normal distribution in estimating the extreme risk. The normal distribution will over-estimate risk within the rise period of stock market, under-estimate risk within the decline period and fail the test.This paper further analyzes how to select the Copula functions. The appropriate selection of Copula functions is another key point to properly apply the Copula theory. After thoroughly discussing the Copula models whose marginal distribution functions are based on kernel density function, extreme value theory and volatility models, we empirically analysed the correlation of stock returns in real estate market and financial industry. The result shows that strong correlation exists between these two markets during the downturn period. We also find out that it is important to take into account Copula functions with varied parameter structure, in order to properly model the correlation. In fact, Copula functions with two parameters usually work better than single-parameter Copula functions. We continue to discuss the dependence with the constant copula models and the time-varying copula models based on the experience distributions. The results indicate that the depence structure of stock returns in real estate market and financial industry is Symmetrized Joe-Clayton copula, and the time-varying copula models are prior to the constant ones in simulating the correlation between the financial time series .Lastly, the capability of Copula models to accurately characterize dependencies between financial time series is studied in depth as well. In order to correctly select the marginal distribution function and henceforth the Copula functions, we employ a non-parametric kernel density function as well as a semi-parametric POT model as the marginal distribution, and then explore the correlation between the Shanghai and Shenzhen Stock markets by the proposed Copula functions. The results show that the tail dependence in downturn is slightly higher than that of the boom. In addition, evidence tells that non-parametric method to fit the tail of the data is sometimes better than the semi-parametric method, while in terms of the overall goodness of fit the semi-parametric methods sometimes beats the non-parametric methods. The different choices of marginal distribution functions result in varied parameter estimates of the Copula models, and consequently different degree of estimated correlations. Therefore, the accuracy of Copula models to describe the dependencies in financial time series, to a large extent, depends on how well the marginal distribution functions fit the data.