Copula Functions Based On Financial Volatility Models: Theory And Applications

With the developments and innovations of financial markets and statistics methods, how to describe the complex volatility characteristics and dependence structure among financial assets more comprehensively and accurately has become one of the key roles of modern finance theory and applications.This issues is very important for portfolio management, risk management and asset pricing. For example, the variance of of the return of a portfolio depends not only on the variance of the individual assets but also on the dependence structure between the assets in the portfolio. But the traditional multivariate statistics models have some pitfalls in modeling the multivariate distribution. On the one hand, it may result in the serious problem of'dimension disaster'with the dimension of financial assets increase. On the other hand, the traditional multivariate Gaussian distribution and the multivariate student t distribution couldn't describe the volatility characteristics and dependence structure comprehensively and accurately.Copula is the function that links together univariate distributions to form a multivariate distribution function.The most attractive property of the Copula function is that the marginal distributions don't need to be same, and the choice of the Copula functon is not constrained by the marginal distributions. As Copula functions have many excellent statistics properties, it has attached more attention in statistics and econometrics modeling. At the same time, it has also been widely used in financial domains, such as market risk and credit risk modeling, the analysis of dependence structure among financial markets, the pricing and risk management of financial derivatives and structured finance products.In this thesis we try to comprise the financial volatility models with the Copula functions based on the theory of statistics, econometrics and finance. We use the financial volatility models to describe the volatility characteristics of financial assets , and use the Copula functions to describe the dependence structure between them.The structure of the thesis is as follows:The first chapter is the introduction. In this section, we first analyse the backgrounds of the economics and statistics models.We also analyse the main pitfalls of the traditional statistics models. We expound the main values in the theory and applications with the properties of Copula functions. In the last we summarize the main results and innovations of the thesis.The second chapter of the thesis is to summarize and expand the financial volatility models. The Copula functions are based on the financial volatility models. In the thesis we analyse the following models: the ARCH, GARCH, EGARCH and TARCH models, the Stochastic Volatility(SV),SV-HS,SV-JPR and MSSV models. The empirical result shows that both the ARCH family and SV family models could describe the dyanmic, time-varying and volatility clustering characteristics of financial assets, and the financial volatility models based on heavy tailed t distribution and models with leverage effect could fit the data well. At the same time, the Shanghai stock market's volatility has obvious regime state characteristics and MSSV models could describe the cycle well. In the last part of the chapter, we analyse the properties and classfication of the traditional Multivariate GARCH models and point out the main pitfalls of it.The third chapter is to analyse the properties, defination and characteristics of the Copula functions, and how to combine the financial volatility models with Copula functions. We introduce the main types of Copula functions such as Gaussian Copula, t Copula, Gumbel Copula, Clayton Copula, Frank Copula and Joe-Clayton Copula. We then analyse the estimation methods and goodness of tests of Copula functions.In the last, we analyse the two frameworks of combining the traditional financial volatility models with the Copula functions.The forth chapter is to compare the performance of three Copula modeling methods: the static Copula methods, the dynamic Copula methods and the markov switching Copula methods. The most used Copula models are based upon the static methods now. But if the sample period is too long, or the dependence structure among the financial markets has observably changes during the sample, then the traditional static Copula models may have some bias in models building. We analyse the properties, characteristics and estimation methods of dynamic Copula and markov switching Copula methods. The empirical study based on Shanghai A and B stock markets shows that the dynamic Copula and markov switching Copula methods could fit the data well than the static methods. Frow this point we can say that the Copula models that considering the dynamic and regime switching characteristics could describe the complex dependence structure among financial markets well.The fifth chapter is to rebuild the traditional multivariate GARCH models with the Copula functions and the Hoeffding lemma. Based on Lee & Long's(2008) modes, we use the Copula function and Hoeffding lemma to expand their models from Gaussian distribution to t distribution(Comparative model II), and analyse the main new properties of the new model. At the same time, we use the property that the Gaussian Copula with the identity correlation matrix is an independent Copula to build the new multivariate GARCH models(Comparative model III).The empirical study based on Shanghai A and B stock markets and the Hongkong stock market shows that the comparative models could fit the data well than the traditional multivariage GARCH models.The sixth chapter is to apply the Copula function to china financial marksts.In the first part we use the Copula function studied in the forth and fifth chapter to analyse the hedging efficency of Hongkong Hengshen stock index and index futures. The result shows that the markov switching Copula models could fit the data well than other models both in in the sample peroid and the out of sample period. In the second part of the chapter we analyse one credit derivatives- Collobatral Obligation Debt(CDO). We use the Copula functions to estimate the fair value of each tranch of CDO and analyse the factors that are related to the pricing of each tranch. In the last part we summarize the other applications of Copula functions in fiance and statistics domain.The seventh chapter is to summarize the thesis and give some advice for further research.The main innvoations of the thesis are as follows:1. We improve the traditional multivariate GARCH models using the Copula functions and Hoeffding lemma in the fifth chapter. Our models are based upon Lee & Long(2008).We expand their models from Gaussian distributions to t distribution(Comparative model II). We also use the property that the Gaussian Copula with the identity correlation matrix is an independent Copula to build the new multivariate GARCH model(Comparative model III). According to the AIC, BIC and the PIT index, though the Lee & Long's(2008) model could fit the data well than the traditional multivariate GARCH modes based on multivariate Gaussian distribution assumption, it couldn't describe the dependence structure among different financial markets well. At the same time, the Comparative model II could fit both the marginal distribution and dependence structure well. The result shows that it is important to consider the dynamic and markov switching characteristics of dependence structure in Copula modeling. We also find that the NAC methods to construct Archimedean Copula function could do well than the EAC methods.2. We analyse three Copula modeling methods including static, dynamic and markov switching Copula in the forth chapter. In the past, the models to build Copula functions are mostly based on static methods. In this thesis we try to compare it with dynamic and markov switching Copula models. We emphasize on the steps to estimate the markov switching Copula. We also give the methods to calculate the probability integration transformation of the dynamic Copula and markov switching Copula. The empirical study applied to Shanghai A and B share stock markets shows that the dynamic Copula and markov switching Copula mehods could fit the data well than static mehods. From this point we can say that the Copula models that considering the dynamic and regime switching characteristics could describe the complex dependence structure among financial markets well.3. We apply the Copula models to some important domains in China financial markets. In the first part we analyse the hedging efficency of Hongkong Hengshen stock index and index futures.using the Copula models discussed in the forth and fifth chapter. According to the in sample and out of sample results, the markov switching Copula models could fit the data best and reduce the volatility of the portfolio most. We also analyse one special credit derivatives- CDO products. Using six listed companies'data that have issued corporate debt, we build one CDO product and analyse the fair value of each tranch and the factors that may be related to the pricing of CDO. 4. We use the financial volatility models to describe the dynamic and volatility clustering characteristics of financial time series in chapter two. We analyse the traditional ARCH models and SV models and the expansion of them in the two directions: one is the models based upon heavy tailed t distribution, the other is to consider the leverage effect. We also consider the Stochastic Volatility models with regime switching volatility. The empirical study applied to Shanghai stock market shows that the financial volatlity models considering the heavy tailed distribution, leverage effect and regime switching volatility could fit the data well.