使用GARCH-EVT和藤式Copula进行极端值依赖性建模和在险价值估计
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摘要
依赖性建模和VaR估计是金融风险管理中的重要概念。但是,必须注意VaR高度依赖我们要研究的金融收益的分布假设的合理性和精度。因此,这篇论文使用GARCH-EVT和藤式Copula对极端值依赖性建模,并且进行了VaR和期望损失估计。在这两项任务中,首先要做的就是在极端值分析之前对收益率使用GARCH类模型进行过滤。
关于VaR(在险价值,Value at Risk)的估计,分析中使用了边缘分布函数推断(IMF)方法。在此,估计被分为两个阶段。第一步是边缘分布函数的建模。这一步是使用了半参数方法,其中超阈值峰值法(peak-over threshold,POT)用于对每一个残差序列尾部的分布进行建模(参数建模部分),而每一个序列的主体部分则通过核函数光滑进行实证建模(非参数建模部分)。IFM的第二步是依赖性建模。这一步是通过建立藤状copula实现,其中任意两个时间序列之间的配对copula作为构建藤状copula的基础。为了进行对照,我们也给出了使用其它方法对VaR的估计。我们建议的方法和其它方法对比的有效性评价基于VaR和尾部期望的后向检验结果性能。
关于使用藤状copula对极端值的依赖性建模,超阈值峰值法用于挑选出资产组合中每一类资产的极端收益数据集合和极端损失数据集合。在本论文的依赖性建模中我们考虑了3种资产。为此我们的依赖性建模中总共使用了6个数据集合。在C类和D类藤状copula模型之间进行选择的时候,选择最佳依赖性模型的基础是看它们在统计检验中的表现。
关于VaR的估计,基于后向检验结果的实证证据表明：使用半参数边缘分布,GARCH-EVT结合混合D类藤状copula模型的方法比其它方法的效果都要好一些,因为在1%和5%的显著性水平情况下,以最少的VaR越界数量通过了条件和非条件覆盖检验。
关于依赖性建模,基于数据的实证证据表明,C类藤状copula模型更适用于构建极端值之间的依赖性关系。高端尾部和低端尾部的依赖性模型的有关参数显著不等于0。这些依赖性参数的大多数为负数,这表明资产组合中资产尾巴对的依赖性关系显著存在,这有助于有关人员进行资产管理规划。
Dependence modeling and estimation of Value-at-Risk are vital concepts in financial risk management. However, it is important to note that the validity and accuracy of VaR highly depends on the distribution assumption of the financial returns under study. This thesis therefore evaluates the effectiveness of using GARCH-EVT and Vine-Copula in modeling dependence in extreme financial returns andthe estimation of value at risk (VaR). In both cases, the returns are first filtered using GARCH-type models before the Extreme Value analysis.
For the Value at Risk (VaR) estimation, the inference function for margin (IFM) approach is used in the analysis, where the estimation is done in two stages. The first stage is the modeling of the marginal distributions. This is done using the semi-parametric method, where the peak-over threshold (POT) approach is used to model the tails of each residual series (parametric) and the center of each series modeled empirically using kernel smoothing (non-parametric). The second stage of the IFM is the dependence modeling. This is done using vine copula with pair-copula as building blocks. For comparison, other methods of VaR estimation are also considered. The performance of the proposed method relative to the other methods is assessed based on the out of sample performance of each method as indicated by the VaR and ES backtest results.
For the dependence modeling of extremes (Extreme gains and Losses) using Vine-Copula, The peak over threshold approach is use to identify the sets of extreme gains and extreme losses in each asset contain in the portfolio. Three assets are considered for the dependence modeling. For the three assets considered, a total of six data sets (3sets of extreme gains and3sets of extreme losses) are use in the dependence modeling. Between the two special classes (C-and D-vine) of vine copula models, the best model for the dependence modeling is chosen base on statistical tests.
For the estimation of VaR, empirical evidence base on the backtest results shows that the GARCH-EVT approach using Mix D-vine copula model with semiparametric margins outperforms all the other models, as it passed both the conditional and unconditional coverage tests with the least number of VaR violation for both upper and lower tails at the1%and5%significance levels.
For the dependence modeling, Empirical evidence (base on data) shows that, the C-vine copula is more appropriate for modeling the dependence in the extremes.
The dependence parameters for the upper and lower tails are negative for most pairs.This shows that, some form of dependence relationships exist between pairs of tails in the portfolio that worth knowing for good management planning.
关于VaR(在险价值,Value at Risk)的估计,分析中使用了边缘分布函数推断(IMF)方法。在此,估计被分为两个阶段。第一步是边缘分布函数的建模。这一步是使用了半参数方法,其中超阈值峰值法(peak-over threshold,POT)用于对每一个残差序列尾部的分布进行建模(参数建模部分),而每一个序列的主体部分则通过核函数光滑进行实证建模(非参数建模部分)。IFM的第二步是依赖性建模。这一步是通过建立藤状copula实现,其中任意两个时间序列之间的配对copula作为构建藤状copula的基础。为了进行对照,我们也给出了使用其它方法对VaR的估计。我们建议的方法和其它方法对比的有效性评价基于VaR和尾部期望的后向检验结果性能。
关于使用藤状copula对极端值的依赖性建模,超阈值峰值法用于挑选出资产组合中每一类资产的极端收益数据集合和极端损失数据集合。在本论文的依赖性建模中我们考虑了3种资产。为此我们的依赖性建模中总共使用了6个数据集合。在C类和D类藤状copula模型之间进行选择的时候,选择最佳依赖性模型的基础是看它们在统计检验中的表现。
关于VaR的估计,基于后向检验结果的实证证据表明：使用半参数边缘分布,GARCH-EVT结合混合D类藤状copula模型的方法比其它方法的效果都要好一些,因为在1%和5%的显著性水平情况下,以最少的VaR越界数量通过了条件和非条件覆盖检验。
关于依赖性建模,基于数据的实证证据表明,C类藤状copula模型更适用于构建极端值之间的依赖性关系。高端尾部和低端尾部的依赖性模型的有关参数显著不等于0。这些依赖性参数的大多数为负数,这表明资产组合中资产尾巴对的依赖性关系显著存在,这有助于有关人员进行资产管理规划。
Dependence modeling and estimation of Value-at-Risk are vital concepts in financial risk management. However, it is important to note that the validity and accuracy of VaR highly depends on the distribution assumption of the financial returns under study. This thesis therefore evaluates the effectiveness of using GARCH-EVT and Vine-Copula in modeling dependence in extreme financial returns andthe estimation of value at risk (VaR). In both cases, the returns are first filtered using GARCH-type models before the Extreme Value analysis.
For the Value at Risk (VaR) estimation, the inference function for margin (IFM) approach is used in the analysis, where the estimation is done in two stages. The first stage is the modeling of the marginal distributions. This is done using the semi-parametric method, where the peak-over threshold (POT) approach is used to model the tails of each residual series (parametric) and the center of each series modeled empirically using kernel smoothing (non-parametric). The second stage of the IFM is the dependence modeling. This is done using vine copula with pair-copula as building blocks. For comparison, other methods of VaR estimation are also considered. The performance of the proposed method relative to the other methods is assessed based on the out of sample performance of each method as indicated by the VaR and ES backtest results.
For the dependence modeling of extremes (Extreme gains and Losses) using Vine-Copula, The peak over threshold approach is use to identify the sets of extreme gains and extreme losses in each asset contain in the portfolio. Three assets are considered for the dependence modeling. For the three assets considered, a total of six data sets (3sets of extreme gains and3sets of extreme losses) are use in the dependence modeling. Between the two special classes (C-and D-vine) of vine copula models, the best model for the dependence modeling is chosen base on statistical tests.
For the estimation of VaR, empirical evidence base on the backtest results shows that the GARCH-EVT approach using Mix D-vine copula model with semiparametric margins outperforms all the other models, as it passed both the conditional and unconditional coverage tests with the least number of VaR violation for both upper and lower tails at the1%and5%significance levels.
For the dependence modeling, Empirical evidence (base on data) shows that, the C-vine copula is more appropriate for modeling the dependence in the extremes.
The dependence parameters for the upper and lower tails are negative for most pairs.This shows that, some form of dependence relationships exist between pairs of tails in the portfolio that worth knowing for good management planning.
引文
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[2]Aas, K., Czado, C., Frigessi, A. and BaKKen, H. (2009). Pair-copula constructions of multivariate dependence. Insurance:Mathematics and Economics,44,182-198.
[3]Aas. K., Czado. C., Frigessi, A. and BaKKen, H. (2009). Pair-copula constructions of multivariate dependence. Insurance:Mathematics and Economics,44,182-198.
[4]Acerbi, C. and Tashe, D., On the coherence of Expected shortfall. Journal of Banking & Finance,26(7):1487-1503,2002. ISSN 0378-4266.
[5]AKaike, H. (1974). A new look at statistical model identification. IEEE Transactions on Automatic Control,19,716-723.
[6]An introduction to Copulas.2nd edition. New York:Springer. Artzner, P.; Delbaen, F.:Eber, J.M.; Heath, D. (1999).
[7]Balkema, A.A. and de Haan, L. (1974). Residual life time at great age. Annals of Probability,2,792-804
[8]Balkima, A. A. and de Haan, L. (1974). Residual life time at great age. Annals of Probability,2,792-804
[9]Balkima, A. A. and de Haan, L. (1974). Residual life time at great age. Annals of Probability,2,792-804
[10]Basel Committee on Banking and supervision (2005a), International Convergence of Capital Measurement and Capital standards. A Revised Framework.
[11]Basel Committee on Banking Supervision (2005b), Studies on the validation of internal rating systems (revised), working paper, Bank for international settlements.
[12]Bedford, T and R. M Cooke (2001) "probability density decomposition for conditionally dependent random variables modeled by vines", Annals of Mathematics and Artificial Intelligence 32,245-268
[13]Bedford, T and R. M. Cooke (2001) "probability density decomposition for conditionally dependent random variables modeled by vines", Annals of Mathematics and Artificial Intelligence 32,245-268
[14]Bedford, T. and R. M. Cooke (2002) "Vines-a new graphical model for dependent random variables". Annals of Statistics 30,1031-1068.
[15]Bedford, T. and R. M. Cooke (2002) "Vines-a new graphical model for dependent random variables". Annals of Statistics 30,1031-1068.
[16]Berkowitz, J. and O'Brien, J.,2002," How accurate are the Value-at-Risk Models at Commercial Banks?" Journal of Finance, Vol.57, Issue 3, pp. 1011-1111.
[17]Black, F.,1976, Studies of stock Prices Votality Changes. Proceedings of the American statistical Association, Business and Economics Section,177-181
[18]Box, G.E.P. and Pierce,D.A.1970, Distribution Residual Autocorrelations in Autoregressive-Integral Moving Average Time series. Journal of the America Statistical Association,65,1509-1526.
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