分位数回归方法及其在金融市场风险价值预测中的应用
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摘要
金融创新和金融全球化使得现代金融机构的经营活动暴露在更多市场风险之下。在2007年,由美国房地产引发的次贷危机,使华尔街五大投行悉数倒闭,并迅速演变成全球性的金融危机,致使全球经济经历了自上世纪30年代以来最为严重的衰退。事实上,除去当前这次次贷危机,金融市场从来都不是风平浪静。从历史上几次金融危机的发生和变化可以看出金融风险与金融业发展相伴而生。与此同时,金融业的监管也经历了从自由走向初步管制,从初步管制到严格的全面管制；再由严格的全面管制再次走向自由发展的演变历程。毋庸置疑,金融自由化极大地推动了金融业和金融市场的发展。但是,在金融自由化的同时,如果没有同步加强金融监管或者在金融创新的同时,缺乏相应的体制创新特别是监管创新,这都无疑加大了金融风险。随着金融衍生工具出现和不断创新,其所带来的风险品种也得到了快速增长,这些风险给风险管理增添了诸多的困难。于是,人们迫切要求加强金融监管。中国金融市场不断改革,加速对外开放,在给我们带来巨大机遇的同时,也使得我国金融机构暴露在更多的风险之中。因此,提高市场风险管理水平和加强风险控制,对我国金融市场稳定和发展至关重要。
风险管理和控制的关键是风险度量。伴随着金融市场的发展,如何准确对市场风险进行测量受到了高度重视。在众多风险度量模型中,VaR作为一个重要的风险测量工具在各金融机构中获得了广泛应用和推广并且被认为是国际金融风险度量的标准。因此,大力开发和应用VaR方法,对金融风险的防范以及保证金融风险管理的有效性与合理性具有相当重要的意义。尽管VaR的概念比较简单,但是如何计算却存在着各种不同观点和方法。大多数方法的核心都在于估计金融头寸未来收益的统计分布或概率密度函数。本文尝试采用分位数回归方法来计算VaR。与传统方法不同的是,该方法直接对模型进行估计并且无需考虑模型的具体分布形式。除此之外,分位数方法是一种稳健的回归方法,这对于研究具有厚尾分布的金融数据来说尤为有效。通过模拟和实证分析,试图比较分位数回归与其它方法在计算VaR上的优越性和局限性,从而为以后的实际风险度量提供备选方法。
全文共分六个部分,具体结构和内容如下：
第一章为绪论。该章阐述了本文研究的背景和意义,介绍了VaR方法的产生和发展背景,并说明了该方法需要进一步完善的地方。同时,还对VaR技术在国内外的研究进展和实际应用做出了总结和文献综述。最后,给出了本文的研究思路和研究方法以及研究的创新点,其中详细介绍了分位数回归方法,说明了该方法在VaR估计中的优势和长处。
第二章从金融风险的定义、基本特征和分类等相关概念出发,讨论了金融市场风险在众多金融风险中占据着特殊的重要地位。其次,对金融市场风险进行了阐述,其中主要包含了金融市场风险影响、金融市场风险管理以及金融市场风险常用的度量方法。然后,对于金融市场风险中用于描述金融市场特征的一些基本统计量给出了介绍,这里涉及到金融资产收益率的计算和相关统计量的性质。在本章结尾,对中国股票市场的发展和现状做一个简要概述,并以上证和深证综合指数为例,分析了它们收益率分布的特征和性质。
第三章介绍了VaR理论与方法以及其在研究金融市场风险中的应用。大家知道VaR方法是研究和讨论现代市场风险测量的主流方法,从VaR的定义出发对其进行具体介绍。使用VaR度量风险的难点就是如何选择合适的估计方法来计算VaR值。本章依照先理论后实证的顺序,对当前VaR估计技术进行叙述。首先介绍了VaR计算的基本原理与过程,这里包含了估计VaR所使用的参数方法和非参数方法。接着,从易变性和精确性两方面说明如何对VaR进行事后检验,并做出了实证分析。之后,重点分析了当前流行的基于波动性的VaR建模方法并探讨了处理金融数据厚尾和不对称现象的常用方法。在结尾处,使用深证综合指数数据对几种波动方法进行比较说明,实证结果表明使用这些方法计算出的VaR值比较有效。
第四章应用局部多项式非参分位数方法来计算VaR值。这一章主要包含两个方面：一方面,先从非参数模型开始,说明了非参数建模的主要优势是可以避免模型设定误差问题；接着,给出了非参数模型的局部多项式估计步骤并对其中涉及到的变量选择问题进行探讨。随后,使用局部多项式非参分位数估计方法对非参数模型进行估计。期间,也介绍了该估计方法的统计性质和所涉及到的变量选择方法。进一步,使用蒙特卡洛模拟检验了局部多项式非参分位数估计的有限样本性质,证明了局部多项式分位数估计比局部多项式估计更稳健。另一方面,考虑使用非参数对VaR进行建模,重点给出了使用局部多项式非参分位数计算VaR值的详细过程；通过实证分析,发现使用该方法计算出的VaR值比一些现存的方法更准确。
第五章给出了一种在考虑交易量影响情况下直接计算VaR值的方法。以中国股票市场为背景,基于变系数建模思想,研究了一种以交易量作为附加指标的VaR预测模型。本章首先给出了变系数模型的具体表达式并介绍使用局部多项式和多项式样条来近似它的方法。接着,详细阐述了B样条分位数估计方法并对B样条的基本概念和性质、模型估计的渐近性质以及涉及到的变量选择问题都分别进行了叙述。同时,还设计了一个蒙特卡洛模拟实验来检验B样条分位数估计的有效性。这个实验结果与预期的一样,就是B样条分位数估计是稳健有效的。由于金融资产价格和成交量是金融市场上两个最根本指标,所以成交量是否对风险有影响需要检验。因此,在VaR建模时,把资产成交量作为指标加入到模型中推导出了一个变系数VaR模型。之后,根据B样条分位数估计方法,给出了该VaR模型的估计程序。最后,把该方法应用于实证研究,所得的结果表明成交量的大小对VaR估计值产生直接影响。
第六章引入加权分位数Copula方法去计算VaR值。由于资产间的相关结构在很大程度上影响着VaR值的准确性,所以相关性研究在金融风险分析上相当重要。近些年发展起来的Copula函数理论,作为一种衡量变量之间相依关系的新工具,与传统VaR计算方法相比,具有更加准确和灵活的优势。该章首先介绍了Copula函数的定义、基本性质,并给出了Copula函数的分类和不同参数下的表达式。其次,对由Copula函数导出的相关性测度指标做了较深入的探讨,说明了这些相关性指标可以捕捉到变量间非线性的相关关系,特别是变量间尾部的相关关系,所以比常见的相关测度应用范围更广接着,对Copula函数的参数估计和模型选择方法作了介绍并应用蒙特卡洛模拟法对各种估计方法进行了检验。再次,本章提出使用加权分位数回归方法估计Copula函数中的未知参数并推导出几种常见Copula分位数曲线,然后应用模拟研究证明了该估计方法的精确性。最后,应用加权分位数Copula方法研究沪深股市的相关结构并得到了资产组合的VaR值。
The financial innovations and financial globalizations make the business activities of modern financial institutions face more market risks. In2007, the subprime mortgage crisis was triggered by the rapid development of the U.S. real estate market, which pushed five major Wall Street investment banks to be collapsed. The crisis was rapidly evolving into a global financial crisis. It has led to that the global economy has experienced the most severe recession since the1930s. In fact, except for the subprime mortgage crisis, the financial market is never calm. From the history of several financial crises, it has been seen that financial risks and financial development are accompanied. At the same time, the financial regulation experienced from freedom to preliminary control, from the preliminary control to strict comprehensive control and from the strict comprehensive control again towards the free development. Without a doubt, financial liberalization has greatly promoted the development of financial sectors and financial markets. However, when financial liberalization does not synchronize with strengthening financial supervision or institutional innovation lacks of corresponding system innovation, especially, regulatory innovation, there will undoubtedly increase the financial risk. With the emergence and innovation of financial derivatives, the variety of financial risks are rapidly growthing, which has brought many difficulties to the risk management. Thus, it is an urgent requirement to strengthen financial supervision. China's financial market is reforming continuously and accelerating the opening. It not only brings us great opportunities, but also makes our financial institutions face more risks. Therefore, raising the level of market risk management and strengthening risk control are essential for the stability and development of Chinese financial markets.
The measure of risk is the key of risk management and control. Along with the development of financial markets, it has attracted a high degree of attention about how to accurately measure the market risk. In many of the risk metrics model, VaR approach has been acquired the wide range of application and promotion, which is regarded as an important tool of risk measurement in all financial institutions and is considered to be the standard measure of international financial risk. Therefore, the vigorous development and application of VaR method have become significance for preventing financial risks and ensuring the effectiveness and rationality of risk management. Although the concept of VaR is simple, one has a variety of different viewpoints about how to compute its value. The core of the method is to estimate the statistical distribution or probability density function of the future earnings of financial position. This paper attempts to use the quantile regression method to calculate the VaR. This method is different from the traditional methods, which directly estimates models and needs not to consider the specific distribution pattern of models. In addition, the quantile method is a robust regression method and is particularly effective for studying the thick-tailed distribution of financial data. According to simulational and empirical analysis, the advantages and limitations of the quantile regression with other methods are compared in the calculation of VaR, so as to provide an alternative method for the measure of actual risk in the future.
Generally speaking, this thesis consists of six chapters. The structure and contents are described as follows.
The first chapter is the introduction of this article. It describes the background and significance of this study. After that, the background of the VaR method is briefly described and some views of this method that needs further to be improved are also pointed. Meanwhile, research summaries and literature reviews are given on the research progress and practical application of VaR in the domestic and foreign financial markets. Finally, the ideas, methods and innovation of this research are introduced, which details the quantile regression method for highlighting its strengths and advantages in the VaR estimation.
The second chapter begins to expound the general framework of the financial risk management, which tries to find and specify the foundational and central position of the VaR in risk management. Firstly, a detailed explanation of financial market risks is presented, which mainly contains the affect, management and common measurement methods of financial market risk. Secondly, for describing the characteristics of financial markets, some basic statistics are given which involve the calculation of financial return and the properties of related statistics. In the end of this chapter, a brief introduction is presented on the development and current situation of China's stock market. It takes an example of Shanghai and Shenzhen composite index to analyse their characteristics and properties of return distributions.
A more comprehensive study and discussion on VaR theories and methods are given in Chapter three. This chapter explains the subject that VaR satisfies the requirement of the modern market risk measurement. Then, it details the VaR starting from its definition. For measuring risk, the difficulty for using the VaR is how to select the appropriate estimation method to calculate its value. Some estimation techniques of VaR are introduced by the order of theries and empirical research. Firstly, the basic principles and processes are explained for estimateing the value of VaR, which contains the parametric and non-parametric methods. Form two aspects of volatility and accuracy, it begins to explain how to make a backtest for VaR, in which an empirical analysis is performed. Later, the current popular VaR method based on volatility of modeling is mainly disscussed. Also, some common methods dealing with the financial data with heavy tail and asymmetric distribution are examined. At the end, several volatility methods are compared by employing the data of composite index in Shanghai stock exchange. The empirical results show that using these methods to calculate the VaR is relatively effective.
The fourth chapter applies the technology of local polynomial with nonparametric quantile to compute the value of VaR. This chapter contains mainly two aspects. On one hand, the nonparametric model is introduced. The main advantage that nonparametric modeling can avoid the errors caused by specifying models is pointed out. Then, the local polynomial steps are given for the nonparametric model in which one problem related to variable selection is discussed. Subsequently, the method of local polynomial with nonparametric quantile is suggested to estimate nonparametric model. In the same time, the statistical properties and the variable selection method of this technology are also discussed. Furthermore, the finite sample properties of local polynomial with nonparametric quantile estimator are examined by Monte Carlo simulation. Simulation results prove that the proposed estimate is more robust than the local polynomial estimate. On the other hand, the nonparametric VaR modeling is considered, where the detailed procedure about how to use the local polynomial with nonparametric quantile method to calculate the VaR is emphatically described. Through empirical analysis, it is testified that this method is more accurate than some existing methods in calculating the VaR value.
Considering the influence of trading volume, a direct calculation method of the VaR value is discussed in the fifth chapter. In the setting of Chinese stock market, a VaR forecasting model based on the varying coefficients modeling is recommended, whose variables include the additional indicator of trading volume. This chapter first gives the specific expression of varying coefficients model and describes some approximate methods of this model by local polynomials and polynomial splines. Then, a detailed explanation of the B-spline quantile method is presented and the specific steps that how to use this method to estimate the varying coefficient models are elaborated. During this process, the basic concepts, asymptotic properties and variable selection problems of B-spline are described respectively. At the same time, a Monte Carlo experiment is designed to test the effectiveness of B-spline quantile estimate. As expected, the experimental results show that the B-spline quantile estimates are robust and effective. Because the prices and trading volume of financial asset are the two most fundamental indicators of financial markets, whether the volume has an impact for risk needs to be inspected. Therefore, in the modeling of VaR, the index of asset turnover is added to the model, which deduces a varying coefficients VaR model. After that, an estimation procedures of this VaR model is given by the B-spline quantile method. Finally, the method is applied to the empirical research and the results demonstrate that the size of trading volume has a direct impact on VaR estimates.
The sixth Chapter introduces the approach of weighted quantile Copula to compute the VaR value. Because the correlation structure between the assets does greatly influence the accuracy of the VaR value, the studies of correlation are very important in financial risk analysis. Copula function theory recently developed is viewed to be a new tool for studying the dependencies between variables. It proves that the Copula methos is more accurate and flexible than some traditional VaR methods. Firstly, the definition, basic properties, classification and expression with different parameters of Copula function are introduced. Secondly, some correlational indicators derived from Copula function are discussed deeply. Since those indicators can capture the non-linear relationship between variables, especially, the tailed relationship, they have more extensive adaptability than some common measure indicators. Then, the methods of Copula's parameter estimation and model selection are recommended and investigated by Monte Carlo simulation. Whereafter, the technology of weighted quantile regression is proposed to estimate the unknown parameters in the Copula Function. Meanwhile, the quantile curves of several common Copula are deduced. Subsequently, the accuracy of the suggested methods is proved by applying a simulation study. At last, the method of weighted quantile Copula is applied for examining the correlation structure between Shangzheng and Shenzhen stock market and the VaR value of portfolio is obtainted.
风险管理和控制的关键是风险度量。伴随着金融市场的发展,如何准确对市场风险进行测量受到了高度重视。在众多风险度量模型中,VaR作为一个重要的风险测量工具在各金融机构中获得了广泛应用和推广并且被认为是国际金融风险度量的标准。因此,大力开发和应用VaR方法,对金融风险的防范以及保证金融风险管理的有效性与合理性具有相当重要的意义。尽管VaR的概念比较简单,但是如何计算却存在着各种不同观点和方法。大多数方法的核心都在于估计金融头寸未来收益的统计分布或概率密度函数。本文尝试采用分位数回归方法来计算VaR。与传统方法不同的是,该方法直接对模型进行估计并且无需考虑模型的具体分布形式。除此之外,分位数方法是一种稳健的回归方法,这对于研究具有厚尾分布的金融数据来说尤为有效。通过模拟和实证分析,试图比较分位数回归与其它方法在计算VaR上的优越性和局限性,从而为以后的实际风险度量提供备选方法。
全文共分六个部分,具体结构和内容如下：
第一章为绪论。该章阐述了本文研究的背景和意义,介绍了VaR方法的产生和发展背景,并说明了该方法需要进一步完善的地方。同时,还对VaR技术在国内外的研究进展和实际应用做出了总结和文献综述。最后,给出了本文的研究思路和研究方法以及研究的创新点,其中详细介绍了分位数回归方法,说明了该方法在VaR估计中的优势和长处。
第二章从金融风险的定义、基本特征和分类等相关概念出发,讨论了金融市场风险在众多金融风险中占据着特殊的重要地位。其次,对金融市场风险进行了阐述,其中主要包含了金融市场风险影响、金融市场风险管理以及金融市场风险常用的度量方法。然后,对于金融市场风险中用于描述金融市场特征的一些基本统计量给出了介绍,这里涉及到金融资产收益率的计算和相关统计量的性质。在本章结尾,对中国股票市场的发展和现状做一个简要概述,并以上证和深证综合指数为例,分析了它们收益率分布的特征和性质。
第三章介绍了VaR理论与方法以及其在研究金融市场风险中的应用。大家知道VaR方法是研究和讨论现代市场风险测量的主流方法,从VaR的定义出发对其进行具体介绍。使用VaR度量风险的难点就是如何选择合适的估计方法来计算VaR值。本章依照先理论后实证的顺序,对当前VaR估计技术进行叙述。首先介绍了VaR计算的基本原理与过程,这里包含了估计VaR所使用的参数方法和非参数方法。接着,从易变性和精确性两方面说明如何对VaR进行事后检验,并做出了实证分析。之后,重点分析了当前流行的基于波动性的VaR建模方法并探讨了处理金融数据厚尾和不对称现象的常用方法。在结尾处,使用深证综合指数数据对几种波动方法进行比较说明,实证结果表明使用这些方法计算出的VaR值比较有效。
第四章应用局部多项式非参分位数方法来计算VaR值。这一章主要包含两个方面：一方面,先从非参数模型开始,说明了非参数建模的主要优势是可以避免模型设定误差问题；接着,给出了非参数模型的局部多项式估计步骤并对其中涉及到的变量选择问题进行探讨。随后,使用局部多项式非参分位数估计方法对非参数模型进行估计。期间,也介绍了该估计方法的统计性质和所涉及到的变量选择方法。进一步,使用蒙特卡洛模拟检验了局部多项式非参分位数估计的有限样本性质,证明了局部多项式分位数估计比局部多项式估计更稳健。另一方面,考虑使用非参数对VaR进行建模,重点给出了使用局部多项式非参分位数计算VaR值的详细过程；通过实证分析,发现使用该方法计算出的VaR值比一些现存的方法更准确。
第五章给出了一种在考虑交易量影响情况下直接计算VaR值的方法。以中国股票市场为背景,基于变系数建模思想,研究了一种以交易量作为附加指标的VaR预测模型。本章首先给出了变系数模型的具体表达式并介绍使用局部多项式和多项式样条来近似它的方法。接着,详细阐述了B样条分位数估计方法并对B样条的基本概念和性质、模型估计的渐近性质以及涉及到的变量选择问题都分别进行了叙述。同时,还设计了一个蒙特卡洛模拟实验来检验B样条分位数估计的有效性。这个实验结果与预期的一样,就是B样条分位数估计是稳健有效的。由于金融资产价格和成交量是金融市场上两个最根本指标,所以成交量是否对风险有影响需要检验。因此,在VaR建模时,把资产成交量作为指标加入到模型中推导出了一个变系数VaR模型。之后,根据B样条分位数估计方法,给出了该VaR模型的估计程序。最后,把该方法应用于实证研究,所得的结果表明成交量的大小对VaR估计值产生直接影响。
第六章引入加权分位数Copula方法去计算VaR值。由于资产间的相关结构在很大程度上影响着VaR值的准确性,所以相关性研究在金融风险分析上相当重要。近些年发展起来的Copula函数理论,作为一种衡量变量之间相依关系的新工具,与传统VaR计算方法相比,具有更加准确和灵活的优势。该章首先介绍了Copula函数的定义、基本性质,并给出了Copula函数的分类和不同参数下的表达式。其次,对由Copula函数导出的相关性测度指标做了较深入的探讨,说明了这些相关性指标可以捕捉到变量间非线性的相关关系,特别是变量间尾部的相关关系,所以比常见的相关测度应用范围更广接着,对Copula函数的参数估计和模型选择方法作了介绍并应用蒙特卡洛模拟法对各种估计方法进行了检验。再次,本章提出使用加权分位数回归方法估计Copula函数中的未知参数并推导出几种常见Copula分位数曲线,然后应用模拟研究证明了该估计方法的精确性。最后,应用加权分位数Copula方法研究沪深股市的相关结构并得到了资产组合的VaR值。
The financial innovations and financial globalizations make the business activities of modern financial institutions face more market risks. In2007, the subprime mortgage crisis was triggered by the rapid development of the U.S. real estate market, which pushed five major Wall Street investment banks to be collapsed. The crisis was rapidly evolving into a global financial crisis. It has led to that the global economy has experienced the most severe recession since the1930s. In fact, except for the subprime mortgage crisis, the financial market is never calm. From the history of several financial crises, it has been seen that financial risks and financial development are accompanied. At the same time, the financial regulation experienced from freedom to preliminary control, from the preliminary control to strict comprehensive control and from the strict comprehensive control again towards the free development. Without a doubt, financial liberalization has greatly promoted the development of financial sectors and financial markets. However, when financial liberalization does not synchronize with strengthening financial supervision or institutional innovation lacks of corresponding system innovation, especially, regulatory innovation, there will undoubtedly increase the financial risk. With the emergence and innovation of financial derivatives, the variety of financial risks are rapidly growthing, which has brought many difficulties to the risk management. Thus, it is an urgent requirement to strengthen financial supervision. China's financial market is reforming continuously and accelerating the opening. It not only brings us great opportunities, but also makes our financial institutions face more risks. Therefore, raising the level of market risk management and strengthening risk control are essential for the stability and development of Chinese financial markets.
The measure of risk is the key of risk management and control. Along with the development of financial markets, it has attracted a high degree of attention about how to accurately measure the market risk. In many of the risk metrics model, VaR approach has been acquired the wide range of application and promotion, which is regarded as an important tool of risk measurement in all financial institutions and is considered to be the standard measure of international financial risk. Therefore, the vigorous development and application of VaR method have become significance for preventing financial risks and ensuring the effectiveness and rationality of risk management. Although the concept of VaR is simple, one has a variety of different viewpoints about how to compute its value. The core of the method is to estimate the statistical distribution or probability density function of the future earnings of financial position. This paper attempts to use the quantile regression method to calculate the VaR. This method is different from the traditional methods, which directly estimates models and needs not to consider the specific distribution pattern of models. In addition, the quantile method is a robust regression method and is particularly effective for studying the thick-tailed distribution of financial data. According to simulational and empirical analysis, the advantages and limitations of the quantile regression with other methods are compared in the calculation of VaR, so as to provide an alternative method for the measure of actual risk in the future.
Generally speaking, this thesis consists of six chapters. The structure and contents are described as follows.
The first chapter is the introduction of this article. It describes the background and significance of this study. After that, the background of the VaR method is briefly described and some views of this method that needs further to be improved are also pointed. Meanwhile, research summaries and literature reviews are given on the research progress and practical application of VaR in the domestic and foreign financial markets. Finally, the ideas, methods and innovation of this research are introduced, which details the quantile regression method for highlighting its strengths and advantages in the VaR estimation.
The second chapter begins to expound the general framework of the financial risk management, which tries to find and specify the foundational and central position of the VaR in risk management. Firstly, a detailed explanation of financial market risks is presented, which mainly contains the affect, management and common measurement methods of financial market risk. Secondly, for describing the characteristics of financial markets, some basic statistics are given which involve the calculation of financial return and the properties of related statistics. In the end of this chapter, a brief introduction is presented on the development and current situation of China's stock market. It takes an example of Shanghai and Shenzhen composite index to analyse their characteristics and properties of return distributions.
A more comprehensive study and discussion on VaR theories and methods are given in Chapter three. This chapter explains the subject that VaR satisfies the requirement of the modern market risk measurement. Then, it details the VaR starting from its definition. For measuring risk, the difficulty for using the VaR is how to select the appropriate estimation method to calculate its value. Some estimation techniques of VaR are introduced by the order of theries and empirical research. Firstly, the basic principles and processes are explained for estimateing the value of VaR, which contains the parametric and non-parametric methods. Form two aspects of volatility and accuracy, it begins to explain how to make a backtest for VaR, in which an empirical analysis is performed. Later, the current popular VaR method based on volatility of modeling is mainly disscussed. Also, some common methods dealing with the financial data with heavy tail and asymmetric distribution are examined. At the end, several volatility methods are compared by employing the data of composite index in Shanghai stock exchange. The empirical results show that using these methods to calculate the VaR is relatively effective.
The fourth chapter applies the technology of local polynomial with nonparametric quantile to compute the value of VaR. This chapter contains mainly two aspects. On one hand, the nonparametric model is introduced. The main advantage that nonparametric modeling can avoid the errors caused by specifying models is pointed out. Then, the local polynomial steps are given for the nonparametric model in which one problem related to variable selection is discussed. Subsequently, the method of local polynomial with nonparametric quantile is suggested to estimate nonparametric model. In the same time, the statistical properties and the variable selection method of this technology are also discussed. Furthermore, the finite sample properties of local polynomial with nonparametric quantile estimator are examined by Monte Carlo simulation. Simulation results prove that the proposed estimate is more robust than the local polynomial estimate. On the other hand, the nonparametric VaR modeling is considered, where the detailed procedure about how to use the local polynomial with nonparametric quantile method to calculate the VaR is emphatically described. Through empirical analysis, it is testified that this method is more accurate than some existing methods in calculating the VaR value.
Considering the influence of trading volume, a direct calculation method of the VaR value is discussed in the fifth chapter. In the setting of Chinese stock market, a VaR forecasting model based on the varying coefficients modeling is recommended, whose variables include the additional indicator of trading volume. This chapter first gives the specific expression of varying coefficients model and describes some approximate methods of this model by local polynomials and polynomial splines. Then, a detailed explanation of the B-spline quantile method is presented and the specific steps that how to use this method to estimate the varying coefficient models are elaborated. During this process, the basic concepts, asymptotic properties and variable selection problems of B-spline are described respectively. At the same time, a Monte Carlo experiment is designed to test the effectiveness of B-spline quantile estimate. As expected, the experimental results show that the B-spline quantile estimates are robust and effective. Because the prices and trading volume of financial asset are the two most fundamental indicators of financial markets, whether the volume has an impact for risk needs to be inspected. Therefore, in the modeling of VaR, the index of asset turnover is added to the model, which deduces a varying coefficients VaR model. After that, an estimation procedures of this VaR model is given by the B-spline quantile method. Finally, the method is applied to the empirical research and the results demonstrate that the size of trading volume has a direct impact on VaR estimates.
The sixth Chapter introduces the approach of weighted quantile Copula to compute the VaR value. Because the correlation structure between the assets does greatly influence the accuracy of the VaR value, the studies of correlation are very important in financial risk analysis. Copula function theory recently developed is viewed to be a new tool for studying the dependencies between variables. It proves that the Copula methos is more accurate and flexible than some traditional VaR methods. Firstly, the definition, basic properties, classification and expression with different parameters of Copula function are introduced. Secondly, some correlational indicators derived from Copula function are discussed deeply. Since those indicators can capture the non-linear relationship between variables, especially, the tailed relationship, they have more extensive adaptability than some common measure indicators. Then, the methods of Copula's parameter estimation and model selection are recommended and investigated by Monte Carlo simulation. Whereafter, the technology of weighted quantile regression is proposed to estimate the unknown parameters in the Copula Function. Meanwhile, the quantile curves of several common Copula are deduced. Subsequently, the accuracy of the suggested methods is proved by applying a simulation study. At last, the method of weighted quantile Copula is applied for examining the correlation structure between Shangzheng and Shenzhen stock market and the VaR value of portfolio is obtainted.
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