极值统计理论及其在金融风险管理中的应用
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摘要
极值事件很少出现在人们的生产和生活中,但是它一旦发生所带来的影响是非同寻常的,所以近年来人们开始关注对极值事件出现规律的研究。极值统计就是研究这种小概率事件风险的模型技术,它的意义在于对极端风险事件的预测和评估。本文主要对极值统计模型的特性、复合极值分布参数的估计方法以及极值统计模型在金融风险管理领域的应用进行研究。论文的主要工作如下:
1.作为被广泛应用于海况研究的Poisson-Gumbel复合极值分布,论文给模型变量赋予具体的金融含义并引入金融风险管理领域,提出采用概率权矩法进行参数估计,且将其结果与极大似然法和复合矩法做比较研究,结果表明:概率权矩法估计效果很好且表现稳定,与极大似然法结果差别不大,但远比复合矩法好。在此基础之上,对美元/英镑的汇率数据进行了实证分析,结果显示模型的适用性较好。
2.论文结合广义Pareto(GP)分布拟合底分布尾部的原理与复合极值分布理论,构建Poisson-GP复合超阈值分布,并给出了极大似然法、复合矩法和概率权矩法的估计结果。结合实例,将Poisson-Gumbel和Poisson-GP两模型进行比较分析,结果表明:当重现期比较短时,适宜选择Poisson-Gumbel模型,当重现期比较长时,适宜选择Poisson-GP模型。
3.论文提出风险价值VaR误差模型,分别讨论了Poisson-Gumbel复合极值模型和Poisson-GP复合超阈值模型中参数误差传递系数和弹性系数对VaR的误差的影响,并比较分析了两个模型拟合效果的优劣性,结果表明:从参数的误差传递系数角度来讲,用Poisson-Gumbel复合极值分布模型度量VaR要优于Poisson-GP复合超阈值分布模型,但是从弹性系数角度来讲,两个模型的优劣性没有明显差别。
4.投资组合日益复杂,原有单参数Copula族不能充分刻画金融数据之间的相关结构。论文讨论了对称Bernstein Copula,这类多项式形式的多参数Copula族,根据实例将其用于拟合相关结构较为对称的两组数据,并与常用单参数Copula族和一般Bernstein Copula进行了比较分析,结果指出:常用单参数Copula族不能很好地拟合这种相关性,对称Bernstein Copula和一般的Bernstein Copula拟合效果很好,但是一般的Bernstein Copula待估参数多,效率低。
Extreme events rarely appear in our daily life, but it will bring tremendous impact once it happens. In recent years people began to focus on extreme events study. Extreme value theory is the model technology to study such events risk with small probability. The theory can predict and assess the risk of extreme events. In this dissertation, the properties of extreme value models, parametric estimations of compound extreme value distribution and their applications in financial risk management fields are studied intensively. The main achievements are listed as follows:
1. Poisson-Gumbel compound extreme value distribution is widely used on sea conditions. In this paper, the model variables have been given the financial means and the model is introduced to the area of financial risk management. Maximum likelihood method(MLE), compound moment method (CME)and probability- weighted moment method (PWM)are used to estimate the parameters of the distribution function respectively. Through Monte Carlo simulation, it compares the statistical characters of these methods and draws a conclusion that there is little difference between the results of PWM and MLE. PWM is a good estimation method and it behaves steadily. Finally, it gives an example of foreign exchange rate and shows that the model has a good applicability.
2. Combining the principle of generalized Pareto(GP)distribution fitting the tail of a distribution with compound extreme value distribution theory, this paper puts forward Poisson-GP compound threshold distribution model, and gives the results estimated by MLE, CMM and PWM respectively. With the example, Poisson-Gumbel model and Poisson-GP model are analyzed comparatively. The empirical results show that we will choose Poisson-Gumbel model for short time prediction and choose Poisson-GP model for long time prediction.
3. The thesis comes up with value at risk (VaR) error model. The parametric error transfer coefficients and elasticity coefficients in Poisson-Gumbel model and Poisson-GP model are studied to compare their fitting efficiency. The conclusions drawn are as follows: from the parametric error transfer coefficient point of view, VaR estimated by Poisson-Gumbel model is more efficient than that estimated by Poisson-GP model; from the parametric elasticity coefficient point of view, there is no significant difference between the two models.
4. Portfolio increasingly complex, the original copulas with single parameter have already could not describe the dependent structure among portfolios sufficiently. Symmetrical Bernstein Copula in form of multinomial is one of the copulas with multiple parameters. On the basis of foreign exchange rates example, the paper fits the dependent structure by Symmetrical Bernstein Copula, common Bernstein Copula and copulas with single parameter respectively. When the dependent structure between two exchange rates is fairly symmetric, copulas with single parameter can not fit the dependent well, but Symmetrical Bernstein Copula and common Bernstein Copula do well. At the same time Symmetrical Bernstein Copula is more efficient than common Bernstein Copula.
1.作为被广泛应用于海况研究的Poisson-Gumbel复合极值分布,论文给模型变量赋予具体的金融含义并引入金融风险管理领域,提出采用概率权矩法进行参数估计,且将其结果与极大似然法和复合矩法做比较研究,结果表明:概率权矩法估计效果很好且表现稳定,与极大似然法结果差别不大,但远比复合矩法好。在此基础之上,对美元/英镑的汇率数据进行了实证分析,结果显示模型的适用性较好。
2.论文结合广义Pareto(GP)分布拟合底分布尾部的原理与复合极值分布理论,构建Poisson-GP复合超阈值分布,并给出了极大似然法、复合矩法和概率权矩法的估计结果。结合实例,将Poisson-Gumbel和Poisson-GP两模型进行比较分析,结果表明:当重现期比较短时,适宜选择Poisson-Gumbel模型,当重现期比较长时,适宜选择Poisson-GP模型。
3.论文提出风险价值VaR误差模型,分别讨论了Poisson-Gumbel复合极值模型和Poisson-GP复合超阈值模型中参数误差传递系数和弹性系数对VaR的误差的影响,并比较分析了两个模型拟合效果的优劣性,结果表明:从参数的误差传递系数角度来讲,用Poisson-Gumbel复合极值分布模型度量VaR要优于Poisson-GP复合超阈值分布模型,但是从弹性系数角度来讲,两个模型的优劣性没有明显差别。
4.投资组合日益复杂,原有单参数Copula族不能充分刻画金融数据之间的相关结构。论文讨论了对称Bernstein Copula,这类多项式形式的多参数Copula族,根据实例将其用于拟合相关结构较为对称的两组数据,并与常用单参数Copula族和一般Bernstein Copula进行了比较分析,结果指出:常用单参数Copula族不能很好地拟合这种相关性,对称Bernstein Copula和一般的Bernstein Copula拟合效果很好,但是一般的Bernstein Copula待估参数多,效率低。
Extreme events rarely appear in our daily life, but it will bring tremendous impact once it happens. In recent years people began to focus on extreme events study. Extreme value theory is the model technology to study such events risk with small probability. The theory can predict and assess the risk of extreme events. In this dissertation, the properties of extreme value models, parametric estimations of compound extreme value distribution and their applications in financial risk management fields are studied intensively. The main achievements are listed as follows:
1. Poisson-Gumbel compound extreme value distribution is widely used on sea conditions. In this paper, the model variables have been given the financial means and the model is introduced to the area of financial risk management. Maximum likelihood method(MLE), compound moment method (CME)and probability- weighted moment method (PWM)are used to estimate the parameters of the distribution function respectively. Through Monte Carlo simulation, it compares the statistical characters of these methods and draws a conclusion that there is little difference between the results of PWM and MLE. PWM is a good estimation method and it behaves steadily. Finally, it gives an example of foreign exchange rate and shows that the model has a good applicability.
2. Combining the principle of generalized Pareto(GP)distribution fitting the tail of a distribution with compound extreme value distribution theory, this paper puts forward Poisson-GP compound threshold distribution model, and gives the results estimated by MLE, CMM and PWM respectively. With the example, Poisson-Gumbel model and Poisson-GP model are analyzed comparatively. The empirical results show that we will choose Poisson-Gumbel model for short time prediction and choose Poisson-GP model for long time prediction.
3. The thesis comes up with value at risk (VaR) error model. The parametric error transfer coefficients and elasticity coefficients in Poisson-Gumbel model and Poisson-GP model are studied to compare their fitting efficiency. The conclusions drawn are as follows: from the parametric error transfer coefficient point of view, VaR estimated by Poisson-Gumbel model is more efficient than that estimated by Poisson-GP model; from the parametric elasticity coefficient point of view, there is no significant difference between the two models.
4. Portfolio increasingly complex, the original copulas with single parameter have already could not describe the dependent structure among portfolios sufficiently. Symmetrical Bernstein Copula in form of multinomial is one of the copulas with multiple parameters. On the basis of foreign exchange rates example, the paper fits the dependent structure by Symmetrical Bernstein Copula, common Bernstein Copula and copulas with single parameter respectively. When the dependent structure between two exchange rates is fairly symmetric, copulas with single parameter can not fit the dependent well, but Symmetrical Bernstein Copula and common Bernstein Copula do well. At the same time Symmetrical Bernstein Copula is more efficient than common Bernstein Copula.
引文
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