基于COPULA理论的金融风险相依结构模型及应用研究
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摘要
相依性研究是金融风险领域中的一个重要问题,组合投资、资产定价、波动的传导和风险管理等问题都涉及到相依性研究。在建立风险管理模型时仅仅考虑变量间的相关度(degree of dependence)是不够的,还必须考虑到变量的相依结构(dependence structrue)。本文在考虑金融时间序列波动特点的基础上,建立了几个基于Copula理论的模型以研究金融时间序列之间的相依结构,并把Copula模型应用于金融时间序列相依结构的研究分析上。论文的主要内容和创新点如下：
1.结合时间序列的两类相依关系,对于两个一阶平稳马尔科夫时间序列,建立Copula函数相依结构模型研究它们之间的相依结构关系。
根据模型的特点提出了三阶段极大似然估计方法(3SPMLE),把参数的估计问题分步简化,这对估计参数的“维数灾难”问题是一个很好的解决方法。
研究了参数估计的性质,应用概率统计理论研究证明了参数估计的一致性和近似正态性,给出了近似正态性方差矩阵的近似估计计算方法。对两个误设的Copula函数,提出了基于三阶段参数准极大似然比统计量(PPLR),以判断两个误设的Copula函数中哪一个更接近真实的Copula函数。对参数准极大似然比统计量(PPLR)的近似统计性质作了研究。
对模型作了模拟研究,提出了模型的模拟方法。对模型的三阶段极大似然估计方法(3SPMLE),作了Monte-Carlo模拟计算。
对模型作了应用研究,考虑两种相依关系的情况下,研究了三个股票市场相互之间的相依关系。经比较检验,考虑了边缘时间序列短期相依关系的模型要好。
2.研究了股价与交易量之间的相依结构。基于VAR误差修正模型,结合Copula函数理论建立VAR-Copula模型研究股市指数与交易量之间Granger因果关系和相依结构。通过对三个股票市场的实证分析,发现各市场的指数与交易量存在长期的协整关系和由指数到交易量的单向因果关系;指数对数差分与交易量对数差分的相依关系复杂,既有正相依成分也包含负的相依结构,且都表现为上尾高的非对称的相依特征。
使用沪深股市指数和交易量的不同数据,建立ARMA-GARCH-Copula模型研究交易量与股价的同期相依关系、交易量对指数波动的GARCH效应的解释作用；应用模型的标准残差研究沪深股市指数序列的相依结构。结果发现：日交易量对数变化率与日指数之间的同期相依关系强于日交易量对数与日指数间的同期相依关系,日交易量的两种数据序列对日指数波动的GARCH效应存在微弱的解释作用。沪深股市的指数极差之间、指数收益率之间存在很强的正相依性,且有上尾高、下尾低的尾部相依结构特征。
3.基于资产的高阶矩风险和Copula函数理论,综合单个时间序列的高阶矩波动的时变性和Copula函数理论建立研究时间序列之间相依关系的模型Copula-NAGARCHSK-M,研究时间序列之间的相依结构,并从二维模型推广到多维的情形。
综合单个时间序列的高阶矩波动的时变性、非对称性和Copula函数理论建立研究时间序列之间相依关系的Copula-TARCHSK-M模型研究时间序列之间的相依结构,并从二维模型推广到多维的情形。
另外,把Copula函数引入熵理论,并定义了相依结构熵度量随机变量之间的相依结构,将相依结构熵和边缘熵从联合熵中分离出来考虑,有利于随机变量间的相依结构的研究。讨论了二维随机向量在单调变换下相依结构熵的不变性并推广到多维的情形。
Modeling dependence between time series in financial risk management field is of key importance to portfolio diversification, international asset pricing, contagion of volatility and risk management. It is insufficient to only consider the degree of dependence between random variables in establishing risk management models, and we must still consider the structure of dependence of them. In this paper, based on the characteristic of financial time series volatility, several copula-based models are established to study the dependence structure between financial time series, and applied to analyse the dependence structure of some financial time series. The key points and main achievements of this work are listed as follows:
1. A new methodology is proposed based on the conditional probability of Markov chains of order 1 and copula theory to identify the dependence between time series of equity returns. A model for the temporal and contemporaneous dependence of vector time series is established to investigate the dependence between them by combining these two theories.
In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE). The method of parametric estimation is helpful to the issue "dimension averseness".
Based on the 3SPMLE, the properties of parametric estimation, the consistency and asymptotic normality, are studied, and approximate calculations of asymptotic normal variance matrixes are given. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio (PPLR) under the 3SPMLE for stationary Markov vector-type models.
The method of simulation to the model is proposed. Furthermore, a Monte-Carlo simulation is employed to examine the performance of 3SPMLE of model.
Furthermore, we apply the model to study the dependence of equity returns obtained from three major stock markets. The dependence structure will perform well if it takes the temporal dependence of marginal variables into consideration.
2. Based on the VAR error correction model and associated with copula technique, a VAR-Copula model is structured to research the Granger causality relation and the dependence structure between the stock price and the trading volume. The empirical study to three stock markets finds that there is a long-rang co-integration between stock price index and the trading volume and a unilateral Granger causality relationship from stock price to the trading volume, and also finds that the complex dependence relationship between the stock price index logarithmic difference and the trading volume logarithmic difference is positive dependence as well as negative dependence and the asymmetrical dependence structure with higher upper tail to all stock markets.
An ARMA-GARCH-Copula model is proposed to investigate the contemporaneous dependence relationship between the trade-volume and the stock price and to examine the effect of trading volume on GARCH effect of conditional volatility of stock price about the different data of Shanghai and Shenzhen stock markets price indices and the trading volumes. Moreover, the standard residual data oft
he model is employed to research the dependence structure between Shanghai and Shenzhen stock markets. The results show that the contemporaneous dependence between the return(volatility)-volume logarithm and the daily stock price indices is stronger than that between the volume logarithm and the daily stock price indices. The GARCH effect of the conditional volatility of stock price indices which is explained by trading volume is weak. There is very strong positive dependence relationship between Shanghai and Shenzhen stock markets about the extreme differences of stock price indices and the returns of stock price indices, and an asymmetrical dependence structure of the upper tail higher than the lower tail.
3. Based on the higher moment risks of assets and copula, a Copula-NAGARCHSK-M model is established to study the dependence relationships between two time series, and extended to multivariate model from the bivariate.
Integrating the time-varying and the asymmetry of higher moment of univariate time series volatility and copula theory, a dependence structure model, Copula-TARCHSK-M model, is proposed to study the dependence structure between time series, and extended to multivariate model from the bivariate in this paper.
In addition, the copula functions are firstly employed to study the entropy theory and a conception of dependence structure entropy is defined for measuring the dependence structure of random variables. The joint entropy is separated into the dependence structure entropy and the marginal entropy which is useful of investigating the dependence structure of random variables. Moreover, we also discuss the invariability of dependence structure entropy of bivariate variables under monotonous transforming and extend it to multivariable cases.
1.结合时间序列的两类相依关系,对于两个一阶平稳马尔科夫时间序列,建立Copula函数相依结构模型研究它们之间的相依结构关系。
根据模型的特点提出了三阶段极大似然估计方法(3SPMLE),把参数的估计问题分步简化,这对估计参数的“维数灾难”问题是一个很好的解决方法。
研究了参数估计的性质,应用概率统计理论研究证明了参数估计的一致性和近似正态性,给出了近似正态性方差矩阵的近似估计计算方法。对两个误设的Copula函数,提出了基于三阶段参数准极大似然比统计量(PPLR),以判断两个误设的Copula函数中哪一个更接近真实的Copula函数。对参数准极大似然比统计量(PPLR)的近似统计性质作了研究。
对模型作了模拟研究,提出了模型的模拟方法。对模型的三阶段极大似然估计方法(3SPMLE),作了Monte-Carlo模拟计算。
对模型作了应用研究,考虑两种相依关系的情况下,研究了三个股票市场相互之间的相依关系。经比较检验,考虑了边缘时间序列短期相依关系的模型要好。
2.研究了股价与交易量之间的相依结构。基于VAR误差修正模型,结合Copula函数理论建立VAR-Copula模型研究股市指数与交易量之间Granger因果关系和相依结构。通过对三个股票市场的实证分析,发现各市场的指数与交易量存在长期的协整关系和由指数到交易量的单向因果关系;指数对数差分与交易量对数差分的相依关系复杂,既有正相依成分也包含负的相依结构,且都表现为上尾高的非对称的相依特征。
使用沪深股市指数和交易量的不同数据,建立ARMA-GARCH-Copula模型研究交易量与股价的同期相依关系、交易量对指数波动的GARCH效应的解释作用；应用模型的标准残差研究沪深股市指数序列的相依结构。结果发现：日交易量对数变化率与日指数之间的同期相依关系强于日交易量对数与日指数间的同期相依关系,日交易量的两种数据序列对日指数波动的GARCH效应存在微弱的解释作用。沪深股市的指数极差之间、指数收益率之间存在很强的正相依性,且有上尾高、下尾低的尾部相依结构特征。
3.基于资产的高阶矩风险和Copula函数理论,综合单个时间序列的高阶矩波动的时变性和Copula函数理论建立研究时间序列之间相依关系的模型Copula-NAGARCHSK-M,研究时间序列之间的相依结构,并从二维模型推广到多维的情形。
综合单个时间序列的高阶矩波动的时变性、非对称性和Copula函数理论建立研究时间序列之间相依关系的Copula-TARCHSK-M模型研究时间序列之间的相依结构,并从二维模型推广到多维的情形。
另外,把Copula函数引入熵理论,并定义了相依结构熵度量随机变量之间的相依结构,将相依结构熵和边缘熵从联合熵中分离出来考虑,有利于随机变量间的相依结构的研究。讨论了二维随机向量在单调变换下相依结构熵的不变性并推广到多维的情形。
Modeling dependence between time series in financial risk management field is of key importance to portfolio diversification, international asset pricing, contagion of volatility and risk management. It is insufficient to only consider the degree of dependence between random variables in establishing risk management models, and we must still consider the structure of dependence of them. In this paper, based on the characteristic of financial time series volatility, several copula-based models are established to study the dependence structure between financial time series, and applied to analyse the dependence structure of some financial time series. The key points and main achievements of this work are listed as follows:
1. A new methodology is proposed based on the conditional probability of Markov chains of order 1 and copula theory to identify the dependence between time series of equity returns. A model for the temporal and contemporaneous dependence of vector time series is established to investigate the dependence between them by combining these two theories.
In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE). The method of parametric estimation is helpful to the issue "dimension averseness".
Based on the 3SPMLE, the properties of parametric estimation, the consistency and asymptotic normality, are studied, and approximate calculations of asymptotic normal variance matrixes are given. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio (PPLR) under the 3SPMLE for stationary Markov vector-type models.
The method of simulation to the model is proposed. Furthermore, a Monte-Carlo simulation is employed to examine the performance of 3SPMLE of model.
Furthermore, we apply the model to study the dependence of equity returns obtained from three major stock markets. The dependence structure will perform well if it takes the temporal dependence of marginal variables into consideration.
2. Based on the VAR error correction model and associated with copula technique, a VAR-Copula model is structured to research the Granger causality relation and the dependence structure between the stock price and the trading volume. The empirical study to three stock markets finds that there is a long-rang co-integration between stock price index and the trading volume and a unilateral Granger causality relationship from stock price to the trading volume, and also finds that the complex dependence relationship between the stock price index logarithmic difference and the trading volume logarithmic difference is positive dependence as well as negative dependence and the asymmetrical dependence structure with higher upper tail to all stock markets.
An ARMA-GARCH-Copula model is proposed to investigate the contemporaneous dependence relationship between the trade-volume and the stock price and to examine the effect of trading volume on GARCH effect of conditional volatility of stock price about the different data of Shanghai and Shenzhen stock markets price indices and the trading volumes. Moreover, the standard residual data oft
he model is employed to research the dependence structure between Shanghai and Shenzhen stock markets. The results show that the contemporaneous dependence between the return(volatility)-volume logarithm and the daily stock price indices is stronger than that between the volume logarithm and the daily stock price indices. The GARCH effect of the conditional volatility of stock price indices which is explained by trading volume is weak. There is very strong positive dependence relationship between Shanghai and Shenzhen stock markets about the extreme differences of stock price indices and the returns of stock price indices, and an asymmetrical dependence structure of the upper tail higher than the lower tail.
3. Based on the higher moment risks of assets and copula, a Copula-NAGARCHSK-M model is established to study the dependence relationships between two time series, and extended to multivariate model from the bivariate.
Integrating the time-varying and the asymmetry of higher moment of univariate time series volatility and copula theory, a dependence structure model, Copula-TARCHSK-M model, is proposed to study the dependence structure between time series, and extended to multivariate model from the bivariate in this paper.
In addition, the copula functions are firstly employed to study the entropy theory and a conception of dependence structure entropy is defined for measuring the dependence structure of random variables. The joint entropy is separated into the dependence structure entropy and the marginal entropy which is useful of investigating the dependence structure of random variables. Moreover, we also discuss the invariability of dependence structure entropy of bivariate variables under monotonous transforming and extend it to multivariable cases.
引文
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