极值统计的理论及其在风险管理中的应用
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摘要
在高科技飞速发展的今天,人们一方面享受着信息社会带来的便利条件,另一方面又不得不承受着极端事件发生所带来的各种各样的风险.这些极端事件不一定是完全相关的,但在某种程度上又几乎都有一定的相关性.本文主要研究极值统计的理论及其在风险管理中的应用.
和谐性度量是刻画两个变量之间相关性的理想指标.对于连续型随机变量,和谐性度量仅仅与Copula有关,而与边缘分布无关;但对于离散型随机变量,该结论不成立.本文推导了计算任意两个离散型随机变量的和谐性度量的一般公式,讨论了最小和最大次序统计量的和谐性度量的计算方法,还分析了离散型随机变量的和谐性度量与边缘分布有关的原因.
在金融风险管理中,经常会碰到损失与收益不对称的情况,因此在建模时,要特别注意非对称性的度量.本文提出了非径向对称的度量方法,给出了最大非径向对称的概念;通过构造4个奇异Copula,将最大非径向对称的Copula分成4类;并研究了最大非径向对称Copula的性质.
在深入研究一元极值和Copula理论的基础上,本文建立了平稳序列阈值模型、多元meta-t分布模型和多元阈值模型,分别探讨了它们在网络流量控制、保险准备金的确定和外汇收益率风险管理中的应用.首先,根据阈值模型建立了网络流量的分布函数,由此可以了解网络流量的情况,进行实时监测和控制,避免网络崩溃.然后,将广义Pareto分布和对数正态分布相结合作为边缘分布,将t-Copula作为各项保险业务之间的相关结构,构造了各项保险业务之间的联合分布.根据该分布,计算了保险公式的准备金.与传统方法相比较,该方法至少可为保险公司减少10%的准备金,达到了既规避风险又节约资金的目的.最后,将一元超阈值分布与非对称Logistic Copula相结合,构造了多元阈值模型,得到在超阈值情况下,两只汇率的收益率下降的联合分布,从而对未来两只汇率的走向进行分析和预测.
In the technic time, people benefit from the information society. On theother hand, they have to take on various risks induced by extreme events.These extreme events are not completely dependent, but in a way, almost allextreme events have some dependence. This paper mainly talks about thetheory of Statistics of Extremes and its applications in risk management.
Measure of concordance is an ideal index that describes the dependencebetween two variables. For continuous random variable, measure of concor-dance is only related to Copula, while it is independent of marginal distri-butions; for discrete random variable, however, this conclusion is not valid.This paper induces the general formula which computes concordance measurebetween any two discrete random variables, it also talks about the way to com-pute the concordance measure between minimal and maximal order statistics.In addition, it analysis reasons why concordance measure between discreterandom variables is related to marginal distributions.
During the analysis of finance risk, people usually come up against thecase that loss and income are asymmetric. Therefore, they should pay atten-tion to measurement of asymmetry. This paper puts forward a method whichmeasures radially asymmetry, provides the concept of maximal radially asym-metric Copula. Through constructing four singular Copulas, it also classifiesmaximal radially asymmetrical Copula to four sorts among which each classcrosses two given points. In addition, this paper explores the properties ofmaximal radially asymmetric Copula.
Bases on the further study of univariate Extreme and Copula theory,this paper founds the threshold model of stationary sequence、multiplemeta-t distribution and multiple threshold model, studies their applicationsin network ?ow control、deciding insurance reserve and analyzing the risk ofexchange rate yield, respectively. First of all, it establishes the distribution of network ?ow according to threshold model. Subsequently, situation of network?ow can be researched, inspected and controlled real time, further to avoidnetwork blowing up. This model provides credible evidences for assigningbandwidth of network ?ow. Then, it uses combination of Pareto distributionand Log-normal distribution as marginal distribution, uses t-Copula asthe dependent structure among insurance operations, constructs the jointdistribution among insurance operations. According to this distribution,reserve of insurance formula can be computed. Compared with traditionalmethod, this method can reduce at least ten percents of reserve for insurancecompany and attains the purposes that avoids risk and saves fund. Lastly,through combining univariate excess threshold distribution and nonsymmetriclogistic Copula, this paper constructs multiple threshold model. Under thecase of excess threshold, this paper gains the joint distribution of yields oftwo exchange rates. According to this, the trend of two exchange rates canbe analyzed and predicted.
和谐性度量是刻画两个变量之间相关性的理想指标.对于连续型随机变量,和谐性度量仅仅与Copula有关,而与边缘分布无关;但对于离散型随机变量,该结论不成立.本文推导了计算任意两个离散型随机变量的和谐性度量的一般公式,讨论了最小和最大次序统计量的和谐性度量的计算方法,还分析了离散型随机变量的和谐性度量与边缘分布有关的原因.
在金融风险管理中,经常会碰到损失与收益不对称的情况,因此在建模时,要特别注意非对称性的度量.本文提出了非径向对称的度量方法,给出了最大非径向对称的概念;通过构造4个奇异Copula,将最大非径向对称的Copula分成4类;并研究了最大非径向对称Copula的性质.
在深入研究一元极值和Copula理论的基础上,本文建立了平稳序列阈值模型、多元meta-t分布模型和多元阈值模型,分别探讨了它们在网络流量控制、保险准备金的确定和外汇收益率风险管理中的应用.首先,根据阈值模型建立了网络流量的分布函数,由此可以了解网络流量的情况,进行实时监测和控制,避免网络崩溃.然后,将广义Pareto分布和对数正态分布相结合作为边缘分布,将t-Copula作为各项保险业务之间的相关结构,构造了各项保险业务之间的联合分布.根据该分布,计算了保险公式的准备金.与传统方法相比较,该方法至少可为保险公司减少10%的准备金,达到了既规避风险又节约资金的目的.最后,将一元超阈值分布与非对称Logistic Copula相结合,构造了多元阈值模型,得到在超阈值情况下,两只汇率的收益率下降的联合分布,从而对未来两只汇率的走向进行分析和预测.
In the technic time, people benefit from the information society. On theother hand, they have to take on various risks induced by extreme events.These extreme events are not completely dependent, but in a way, almost allextreme events have some dependence. This paper mainly talks about thetheory of Statistics of Extremes and its applications in risk management.
Measure of concordance is an ideal index that describes the dependencebetween two variables. For continuous random variable, measure of concor-dance is only related to Copula, while it is independent of marginal distri-butions; for discrete random variable, however, this conclusion is not valid.This paper induces the general formula which computes concordance measurebetween any two discrete random variables, it also talks about the way to com-pute the concordance measure between minimal and maximal order statistics.In addition, it analysis reasons why concordance measure between discreterandom variables is related to marginal distributions.
During the analysis of finance risk, people usually come up against thecase that loss and income are asymmetric. Therefore, they should pay atten-tion to measurement of asymmetry. This paper puts forward a method whichmeasures radially asymmetry, provides the concept of maximal radially asym-metric Copula. Through constructing four singular Copulas, it also classifiesmaximal radially asymmetrical Copula to four sorts among which each classcrosses two given points. In addition, this paper explores the properties ofmaximal radially asymmetric Copula.
Bases on the further study of univariate Extreme and Copula theory,this paper founds the threshold model of stationary sequence、multiplemeta-t distribution and multiple threshold model, studies their applicationsin network ?ow control、deciding insurance reserve and analyzing the risk ofexchange rate yield, respectively. First of all, it establishes the distribution of network ?ow according to threshold model. Subsequently, situation of network?ow can be researched, inspected and controlled real time, further to avoidnetwork blowing up. This model provides credible evidences for assigningbandwidth of network ?ow. Then, it uses combination of Pareto distributionand Log-normal distribution as marginal distribution, uses t-Copula asthe dependent structure among insurance operations, constructs the jointdistribution among insurance operations. According to this distribution,reserve of insurance formula can be computed. Compared with traditionalmethod, this method can reduce at least ten percents of reserve for insurancecompany and attains the purposes that avoids risk and saves fund. Lastly,through combining univariate excess threshold distribution and nonsymmetriclogistic Copula, this paper constructs multiple threshold model. Under thecase of excess threshold, this paper gains the joint distribution of yields oftwo exchange rates. According to this, the trend of two exchange rates canbe analyzed and predicted.
引文
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