基于Copula理论的金融市场相依结构研究
|
摘要
经济全球化和金融市场国际化导致了金融市场之间的联系越来越紧密,彼此的关系更加复杂。由于准确刻画金融市场之间的相依结构是研究投资组合以及风险管理的基础,在金融决策中,为了提高决策的准确性,降低决策风险,研究分析金融市场之间的相依结构是非常必要的。
Copula函数是将随机向量的联合分布函数与其对应各分量的边缘分布函数连接在一起的函数,是描述多个金融市场之间相依结构的重要工具。运用它构造联合分布函数时不受边缘分布函数的限制,可以将随机向量的边缘分布函数及其相依结构分开研究。本文运用Copula函数研究金融市场相依结构的建模问题,探讨关于Copula函数的一些理论问题及其在风险管理、投资组合中的应用。
运用Copula函数来构建金融资产相依结构的基础是在众多的Copula函数族中选择一个合适的Copula函数,常用的方法是AIC或BIC准则,该准则在Copula函数密度函数存在的条件下有效。为了解决Copula函数的密度函数不存在或密度函数没有显式表达式时Copula函数的选择问题,本文提出了基于非参数核密度估计的Copula函数选择原理,并用蒙特卡罗模拟方法,在金融资产边缘分布函数属于不同类型的情况下,将AIC准则与基于非参数核密度估计的Copula函数选择原理进行了系统的比较。
系统地研究了目前存在的一些基于Copula函数的相依性测度,并在此基础上,利用门限Copula函数,分别给出了下门限相依性测度与上门限相依性测度的概念。
由于现实金融市场间的相依结构通常随市场的调节不断变化,而且市场“利好消息”与“利坏消息”对金融市场相依结构的影响通常具有不对称性,因此,本文基于门限GARCH模型思想,提出了具有门限结构的时变Copula模型,并利用该模型研究了世界及地区股票市场对我国股票市场间相依结构的影响问题。
在金融风险管理及投资组合方面,系统研究了基于VaR和ES的金融风险管理方法,构建了一元APD-GARCH模型,并将该模型与多元Copula相结合,研究了不同相依结构下的均值—ES有效前沿问题。实证结果表明,在研究投资组合以及风险管理问题时,考虑金融市场间的非对称尾部相依结构是十分必要。
本论文是国家自然科学基金资助项目《多变量矩序列长期均衡关系及动态金融风险规避策略研究》(No:70471050)的组成部分。
Dependencies among financial markets have significantly increased and relationships among them become more complex. This phenomenon is a direct consequence of economic globalization and the globalization of the financial market. As the foundation of risk management and portfolio investment, description on dependence structure correctly in financial markets is very important. So it is necessary to research on the dependence structure among financial markets in financial decision-making in order to increase the correction of decision and to decrease the risk of decision.
Copula functions, which are important tool to describe the dependence structure of financial markets, are functions which connect the joint distribution function with its marginal distributions. It is possible to study the dependence structure of random vectors irrespective of their margins by use of the Copula function to construct the multivariate joint distribution. This paper study dependence structure among financial markets based on the Copula theory and discuss some academic question of Copula function and its application in risk management and portfolio investment.
The foundation of modeling the dependence among the financial assets by Copula function is the choice of an appropriate parametric copula in a given set of copulas. The commonly used approach to selecting copula is AIC or BIC criterion which relies on the density function of each Copula in the given set of Copulas. However, the explicit expressions of density function of some Copulas may be very complicated or very difficult to obtain, which cause great inconvenience in the application of AIC or BIC criterion. In this sense, a new Copula selection method based on nonparametric kernel density estimation is proposed. Under different marginal distributions, the Copula selection method based on nonparametric kernel density estimation is compared with AIC by means of Monte Carlo simulation Methods.
Dependence measures based on Copula functions, which are used commonly, are systemically studied in this paper. From the analysis, the definitions of upper threshold dependence measures and lower threshold of dependence measures are proposed by threshold Copula functions.
The dependence structures of financial markets are incessantly changing with the accommodation of market. And asymmetry lies in the influence of positive and negative innovations on the dependence structures of financial market. So, a time-varying Copula model with threshold structure is proposed based on the idea of the threshold GARCH model. By this model, the influence of international and regional stock markets on the dependence structures of domestic financial market is studied.
In the aspect of risk management and portfolio investment, financial risk management methods based on VaR and ES are studied. Furthermore, the APD-GARCH model is established. With this model and multivariate Copula functions, the efficient frontier of portfolio optimization based on mean-ES is studied. The experiential result indicates that it is very necessary to consider the asymmetry tail dependence structure in research on risk management and portfolio investment.
The research is sponsored by National Natural Science Foundation of China: Research on Long Run Equilibrium in Multivariate Moments Series and Avoiding Tactics of Dynamic Financial Risk (No.70471050).
Copula函数是将随机向量的联合分布函数与其对应各分量的边缘分布函数连接在一起的函数,是描述多个金融市场之间相依结构的重要工具。运用它构造联合分布函数时不受边缘分布函数的限制,可以将随机向量的边缘分布函数及其相依结构分开研究。本文运用Copula函数研究金融市场相依结构的建模问题,探讨关于Copula函数的一些理论问题及其在风险管理、投资组合中的应用。
运用Copula函数来构建金融资产相依结构的基础是在众多的Copula函数族中选择一个合适的Copula函数,常用的方法是AIC或BIC准则,该准则在Copula函数密度函数存在的条件下有效。为了解决Copula函数的密度函数不存在或密度函数没有显式表达式时Copula函数的选择问题,本文提出了基于非参数核密度估计的Copula函数选择原理,并用蒙特卡罗模拟方法,在金融资产边缘分布函数属于不同类型的情况下,将AIC准则与基于非参数核密度估计的Copula函数选择原理进行了系统的比较。
系统地研究了目前存在的一些基于Copula函数的相依性测度,并在此基础上,利用门限Copula函数,分别给出了下门限相依性测度与上门限相依性测度的概念。
由于现实金融市场间的相依结构通常随市场的调节不断变化,而且市场“利好消息”与“利坏消息”对金融市场相依结构的影响通常具有不对称性,因此,本文基于门限GARCH模型思想,提出了具有门限结构的时变Copula模型,并利用该模型研究了世界及地区股票市场对我国股票市场间相依结构的影响问题。
在金融风险管理及投资组合方面,系统研究了基于VaR和ES的金融风险管理方法,构建了一元APD-GARCH模型,并将该模型与多元Copula相结合,研究了不同相依结构下的均值—ES有效前沿问题。实证结果表明,在研究投资组合以及风险管理问题时,考虑金融市场间的非对称尾部相依结构是十分必要。
本论文是国家自然科学基金资助项目《多变量矩序列长期均衡关系及动态金融风险规避策略研究》(No:70471050)的组成部分。
Dependencies among financial markets have significantly increased and relationships among them become more complex. This phenomenon is a direct consequence of economic globalization and the globalization of the financial market. As the foundation of risk management and portfolio investment, description on dependence structure correctly in financial markets is very important. So it is necessary to research on the dependence structure among financial markets in financial decision-making in order to increase the correction of decision and to decrease the risk of decision.
Copula functions, which are important tool to describe the dependence structure of financial markets, are functions which connect the joint distribution function with its marginal distributions. It is possible to study the dependence structure of random vectors irrespective of their margins by use of the Copula function to construct the multivariate joint distribution. This paper study dependence structure among financial markets based on the Copula theory and discuss some academic question of Copula function and its application in risk management and portfolio investment.
The foundation of modeling the dependence among the financial assets by Copula function is the choice of an appropriate parametric copula in a given set of copulas. The commonly used approach to selecting copula is AIC or BIC criterion which relies on the density function of each Copula in the given set of Copulas. However, the explicit expressions of density function of some Copulas may be very complicated or very difficult to obtain, which cause great inconvenience in the application of AIC or BIC criterion. In this sense, a new Copula selection method based on nonparametric kernel density estimation is proposed. Under different marginal distributions, the Copula selection method based on nonparametric kernel density estimation is compared with AIC by means of Monte Carlo simulation Methods.
Dependence measures based on Copula functions, which are used commonly, are systemically studied in this paper. From the analysis, the definitions of upper threshold dependence measures and lower threshold of dependence measures are proposed by threshold Copula functions.
The dependence structures of financial markets are incessantly changing with the accommodation of market. And asymmetry lies in the influence of positive and negative innovations on the dependence structures of financial market. So, a time-varying Copula model with threshold structure is proposed based on the idea of the threshold GARCH model. By this model, the influence of international and regional stock markets on the dependence structures of domestic financial market is studied.
In the aspect of risk management and portfolio investment, financial risk management methods based on VaR and ES are studied. Furthermore, the APD-GARCH model is established. With this model and multivariate Copula functions, the efficient frontier of portfolio optimization based on mean-ES is studied. The experiential result indicates that it is very necessary to consider the asymmetry tail dependence structure in research on risk management and portfolio investment.
The research is sponsored by National Natural Science Foundation of China: Research on Long Run Equilibrium in Multivariate Moments Series and Avoiding Tactics of Dynamic Financial Risk (No.70471050).
引文
[1] Markowitz H M, Portfolio selection [J], Joumal of Finance, 1952, 7:77-91
[2] Rudiger Frey, Alexander J McNeil, VaR and expected shortfall in portfolios of dependent credit risks: conceptual and practical insights [J], Journal of Banking & Finance, 2002, 26(7):1317-1334
[3] Stefano Benati, The optimal portfolio problem with coherent risk measure constraints [J], European Journal of Operational Research, 2003, 150(3):572-584
[4] Giorgio Szego, Measures of risk [J], European Journal of Operational Research, 2005, 163(1):5-19
[5] Yasuhiro Yamai, Toshinao Yoshiba, Value-at-risk versus expected shortfall: A practical perspective [J], Journal of Banking & Finance, 2005, 29(4): 997-1015
[6] Sklar A,Fonctions de repartitionàn dimensions et leurs marges [J], Publication de l' Institut de Statistique de l'Universitéde Paris, 1959, 8:229-231
[7] Genest C, Rivest L P, On the multivariate probability integral transformation [J], Statistics and Probability Letters, 2001, 53(4):391-399
[8] Rodríguez-Lallena JoséA,úbeda-Flores Manuel, Distribution functions of multivariate copulas [J], Statistics and Probability Letters, 2003, 64(1):41-50
[9] Frahm G, Junker M, Szimayer A, Elliptical copulas: applicability and limitations [J], Statistics and Probability Letters, 2003, 63(3):275-286
[10] Vaz de Melo Mendes Beatriz, Martins de Souza Rafael, Measuring financial risks with copulas [J], International Review of Financial Analysis, 2004 13(1):27-45
[11] Akaike H, A new look at the statistical model identification, IEEE Transactions on Automatic Control [J], 1974, 19 (6): 716-723
[12] Chen X, Fan Y, Pseudo-likelihood ratio tests for semiparametric multivariatecopula model selection [J], The Canadian Journal of Statistics, 2005, 33(2):389-414
[13] Genest C, Rivest L P, Statistical inference procedures for bivariate Archimedean Copulas [J], Journal of The American Statistical Association,1993, 8:1034-1043
[14] Durrleman V, Nikeghbali A, Roncalli T, Which copula is the right one? Working document [J], Groupe de Recherche Opérationnelle, Crédit Lyonnais, 2000
[15] Kole Erik, Koedijk Kees, Verbeek Marno, Selecting copulas for risk management [J], Journal of Banking and Finance, 2007, 31(8):2405-2423
[16] Chen X, Fan Y, Patton A, Simple tests for models of dependence between multiple financial time series, with applications to U,S, equity returns and exchange rates, Discussion Paper 483, Financial Markets Group, International Asset Management, http://fmg.lse.ac.uk/upload_file/204_dp483.pdf
[17] Jean-David Fermanian, Goodness-of-fit tests for copulas [J], Journal of Multivariate Analysis, 2005, 95(1): 119-152
[18] Diebold F X, T Gunther, A S Tay, Evaluating density forecasts with applications to financial risk management [J], International Economic Review, 1998, 39, 863-883
[19] Diebold F X, J Hahn, A S Tay, Multivariate density forecast and calibration in financial risk management: High-frequency returns on foreign exchange [J], The Review of Economics and Statistics, 1999, 81(4):661-673
[20] Hong Y, Evaluation of out-of-sample density forecasts with applications to S&p 500stock prices, Working Paper, Department of Economics and Department of StatisticalScience, Cornell University, 2000
[21] Berkowitz J, Testing density forecasts, with applications to risk management [J], Journal of Business and Economic Statistics, 2001, 19: 465-474
[22] Genest C, J F Quessy, B R′emillard, Goodness-of-fit procedures for copula models based on the probability integral transform [J], Scandinavian Journal of Statistics, 2006, 33(2): 337-366
[23] Dobri? Friedrich Schmid, A goodness of fit test for copulas based onRosenblatt's transformation [J], Computational Statistics and Data Analysis, 2007, 51(9): 4633-4642
[24] Dobri? J, F Schmid, Testing goodness of fit for parametric families of copulas-application to financial data [J], Communications in Statistics-Simulation and Computation, 2005, 34(4): 1053-1068
[25] David Huard, Guillaumeévin, Anne-Catherine Favre, Bayesian copula selection [J], Computational Statistics & Data Analysis, 2006, 51(2):809-822
[26]王沁,王璐,何平,基于截断tau的copula模型选择及应用[J],数理统计与管理, 2008, 27(1):118-123
[27]张军舰,刘冬香,相依结构数据的ACE拟合[J],数理统计与管理, 2008, 27(2):285-291
[28] Patton A J, Estimation of Copula models for time series of possibly different,University of California at San Diego, Working Paper, 2001
[29] Patton A J, Modelling time-varying exchange rate dependence using the conditional copula, University of California at San Diego, Working Paper,2001
[30] Patton A J, Applications of copula theory in financial econometrics [D], University of California, San Diego, 2002
[31] Rob W J, van den Goorbergh, Christian Genest etc,, Multivariate option pricing using dynamic copula models, Tilburg University, Working Paper, 2003,
[32] Rob W J, van den Goorbergh, Christian Genest etc,, Bivariate option pricing using dynamic copula models [J], Insurance: Mathematics and Economics 2005,37:101-114
[33] Thierry Ane, Chiraz Labidi, Spillover effects and conditional dependence [J], International Review of Economics and Finance, 2006, 15:417-442
[34] J Zhang, D Guegan, Pricing bivariate option under GARCH processes with time-varying copula [J], Insurance: Mathematics and Economics, 2008, 42:1095-1103
[35]韦艳华,张世英,金融市场非对称尾部相关结构的研究[J],管理学报, 2005, 2(5):601-605
[36]罗付岩,邓光明,基于时变Copula的VaR估计[J],系统工程, 2007, 25(8):28-33
[37]任仙玲,张世英,基于Copula函数的金融市场尾部相关性分析[J],统计与信息论坛, 2008, 23(6):66-71
[38]张瑞锋,金融市场波动溢出研究[博士学位论文],天津;天津大学, 2006
[39] Y Hamao, RW Masulis, V Ng, Correlations in price changes and volatility across international stock markets [J], Review of Financial Studies, 1990, 3(2):281-307,
[40] W L Lin, R F Engle, T Ito, Do Bull and Bears Move across Borders? International Transmission of Stock Return and Volatility [J], The Review of Financial Studies, 1994, 7(3):507-538
[41] R F Engle, Stock Volatility and the Crash of 87: Discussion [J] ,The Review of Financial Studies, 1990, 3(1):103-106
[42] Cheung Y, Ng L K, A causality-in-variance test and its application to financial market prices [J], Journal of Econometrics, 1996, 72(1):33-48
[43] Geert Bekaert, Campbell R, Harvey, Emerging equity market volatility [J], Journal of Financial Econometrics, 1997, 43(1):29-77
[44] Andrew J Patton, Modeling asymmetric exchange rate dependence [J], International Economic Review, 2006, 47(2):527-556
[45] Zhang R F, Zou Q W, Zhang SY, Volatility spillover analysis and empirical study on the financial market based on Copula theory [C], 2007 International Conference on Management Science and Engineering, 2007, 1874-1881
[46]蒋翠侠,动态金融风险测度及管理研究[博士学位论文],天津;天津大学, 2007
[47] Ang A, Chen J, Asymmetric correlation of equity portfolio [J], Journal of Financial Economics, 2002, 63(3): 443-494
[48] Erb Claude B, Harvey Campbell R,, Viskanta Tadas E, Forecasting international equity correlation [J], Financial Analysis Journal, 1994, 50:32-45
[49] Login F, Solnik B,Extreme correlation of international equity markets [J],Journal of Finance, 2001, 56(2): 649-676
[50] Anderw J Patton, On the out-of-sample importance of skewness and asymmetric dependence for asset allocation [J], Journal of Financial Econometrics, 2004, 2(1):130-168
[51] Breymann W, Dias, Embrechts P, Dependence structure for multivariate high-frequency data in finance [R], Working paper, Department of Mathematics, ETH Zurich, 2003
[52] Mashal R, Zeevi A, Comparing the dependence structure of equity and asset returns [R], Working paper, Columbia University, New York, 2002
[53] Mashal R, Zeevi A, Beyond correlation: extreme co-movements between financial assets [R], Working paper, Columbia University, New York, 2002
[54]曾健,陈俊芳, Copula函数在风险管理中的应用研究-以上证A股与B股的相关结构分析为例[J],当代财经, 2005(2) : 34-38
[55] Embrechts P, Mcneil A J, Straumann D, Correlation and dependence in risk management: Properties and pitfalls [M], Dempster M,Risk Management: Value at Risk and Beyond1, Cambridge University Press, 1999:176-223
[56] Claudio Romano, Calibrating and Simulating Copula Functions: an Application to the Italian Stock Market [R], Working paper, CIDEM, 2002
[57] Joshua V Rosenberg, Til Schuermann, A general approach to integrated risk management with skewed, fat-tailed risks [J], Journal of Financial Economics, 2006, 79(3):569-614
[58] Helder Parra Palaro, Luiz Koodi Hotta, Using conditional copula to estimate value at risk [J], Journal of Data Science, 2006, 4(1):93-115
[59] Nikolai Kolev, Ulisses dos Anjos, Beatriz Vaz de M Mendes, Copulas: a review and recent developments [J], Stochastic Models, 2006, 22(4): 617-660
[60]张尧庭,连接函数(Copula)技术与金融风险分析[J],统计研究, 2002,(4):48-51
[61]张明恒,多金融资产风险价值的Copula计量方法研究[J],数量经济技术经济研究, 2004, 21(4):67-70
[62]韦艳华,张世英,金融市场的相关性分析-Copula-GARCH模型及其应用[J],系统工程, 2004, 22(4):7-12
[63]吴振翔,陈敏,叶五一等,基于Copula的外汇投资组合风险分析[J],中国管理科学, 2004, 12(4):1-5
[64]吴振翔,陈敏,叶五一等,基于Copula-GARCH的投资组合风险分析[J],系统工程理论与实践, 2006, 26(3):45-52
[65]梁冯珍,史道济,基于copula函数的保险准备金的确定方法[J],统计与决策, 2006, (12):142-144
[66]韦艳华,张世英,金融市场动态相关结构的研究[J],系统工程学报, 2006, 21(3):313-317
[67]任仙玲,张世英,基于核估计及多元阿基米德Copula的投资组合风险分析[J],管理科学, 2007, 20(5):92-97
[68]应益荣,詹炜,资产组合ES风险测度的Copula-EV算法[J],系统管理学报, 2007, 16(6):602-606
[69]刘志东,度量收益率的实际分布的实际分布和相关性对资产组合选择的绩效影响[J],系统管理学报, 2007, 16(6):628-635
[70] Nelson R B, An introduction to Copulas [M], Springer, New York, 2006
[71] McNeil A, R Frey, P Embrechts, Quantitative risk management: concepts, techniques and tools [M], Princeton University Press, 2005
[72] Joe H, Multivariate models and dependence concepts [M], Chapman and Hall, London, 1997
[73] Durrlenman V, Nikeghbali A, Roncalli T, Copulas approximation and new families [R], Groupe de Recherche Opèrationelle, Crédit Lyonnais, Working paper, 2000
[74] Durrlenman V, Nikeghbali A, Roncalli T, Which Copula is the right one? [R], Groupe de Recherche Opèrationelle, Crédit Lyonnais, Working paper, 2000
[75] Umberto Cherubini, Elisa Luciano, Walter Vecchiato, Copula methods in finance [M], John Wiley & Sons Ltd, 2004
[76] Joe H, Xu J J, The estimation method of inference functions for margins formultivariate models[R], Dept of Statistics University of British Columbia, Tech Report, 1996
[77] C Genest, K Ghoudi, L P Rivest, A semiparametric estimation procedure of dependence parameters in multivariate families of distributions [J], Biometrika, 1995, 82:543-552
[78]李松臣,多元非线性平稳时间序列的建模方法研究:[博士学位论文],天津;天津大学,2007
[79] Bowman A W, A Azzalini, Applied smoothing techniques for data analysis [M], Oxford University Press, 1997
[80]叶阿忠,非参数计量经济学[M],天津:南开大学出版社, 2003
[81] Fan J, Yao Q, Nonlinear time series: nonparametric and parametric methods [M], Springer-Verlag, New York, 2003
[82] Charpentier A, Dependence structures and limiting results, with applications in finance and insurance [D], Leuven: Katholieke University, 2005
[83] Scarsini M, On measures of concordance [J], Stochastica, 1984 8:201-208
[84] Dominique Drouet Mari, Samuel Kotz, Correlation and dependence [M], London: Imperial College Press, 2001
[85] Juri A, M V Wuthrich, Copula convergence theorems for tail events [J], Insurance: Mathematics and Ecnomics, 2002, 30:411-427
[86] Juri A, M V Wuthrich, Tail denpendence from adistributional point of view [J], Extremes, 2003, 6:213-246
[87] Stefano Demarta, The copula approach to modelling multivariate extreme values [D], Zurich: ETH, 2007
[88] Fabrizio Durante, Rachele Foschi, Fabio Spizzichino, Threshold copulas and positive dependence [J], Statistics and Probability Letters, 2008
[89] Thierry Ane, Loredana Ureche-Rangau, Chiraz Labidi-Makni, Time-varying conditional dependence in Chinese stock markets [J], Applied Financial Economics, 2008, 18:895-916
[90] Wing Lon Ng, Modeling duration clusters with dynamic copulas [J], FinanceResearch Letters, 2008, 5:96-103
[91] John Y Campbell, Andrew W Lo, A Craig MacKinlay, The econometrics of financial markets [M], Princeton: Princeton University Press, 1997
[92] Rue S TSAY, Analysis of financial time series [M], John Wiley & Sons Ltd, 2005
[93] Engle R, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation [J], Econometrica, 1982, 50(4):987-1008
[94] Bollerslev T, Generalized autoregressive conditional heteroscedasticity [J], Journal of Econometrics, 1986, 31(3):307-327
[95] Ivana Komunjer, Asymmetric power distribution: theory and applications to risk measurement [J], Journal of Applied Econometrics, 2007, 22(5): 891-921
[96] Hansen Bruce E, Autoregressive conditional density estimation [J], International Economic Review, 1994, 35(3):705-730
[97]弗兰克?H奈特著,安佳译,风险、不确定性与利润[M],北京:商务印书馆, 2006
[98] Jorion Philippe, Value as risk: the new benchmark for managing financial risk-2nd edition [M], McGraw-Hill, 2001
[99] P Artzner, U L Pasteur, Strasbourg, Coherent measures of risk [J], Mathematical Finance, 1999, 9:203-228
[100] Rochafeller R T, Uryasev S, Optimization of conditional value-at-risk [J], Journal of Risk, 2000, 2(3):21-41
[2] Rudiger Frey, Alexander J McNeil, VaR and expected shortfall in portfolios of dependent credit risks: conceptual and practical insights [J], Journal of Banking & Finance, 2002, 26(7):1317-1334
[3] Stefano Benati, The optimal portfolio problem with coherent risk measure constraints [J], European Journal of Operational Research, 2003, 150(3):572-584
[4] Giorgio Szego, Measures of risk [J], European Journal of Operational Research, 2005, 163(1):5-19
[5] Yasuhiro Yamai, Toshinao Yoshiba, Value-at-risk versus expected shortfall: A practical perspective [J], Journal of Banking & Finance, 2005, 29(4): 997-1015
[6] Sklar A,Fonctions de repartitionàn dimensions et leurs marges [J], Publication de l' Institut de Statistique de l'Universitéde Paris, 1959, 8:229-231
[7] Genest C, Rivest L P, On the multivariate probability integral transformation [J], Statistics and Probability Letters, 2001, 53(4):391-399
[8] Rodríguez-Lallena JoséA,úbeda-Flores Manuel, Distribution functions of multivariate copulas [J], Statistics and Probability Letters, 2003, 64(1):41-50
[9] Frahm G, Junker M, Szimayer A, Elliptical copulas: applicability and limitations [J], Statistics and Probability Letters, 2003, 63(3):275-286
[10] Vaz de Melo Mendes Beatriz, Martins de Souza Rafael, Measuring financial risks with copulas [J], International Review of Financial Analysis, 2004 13(1):27-45
[11] Akaike H, A new look at the statistical model identification, IEEE Transactions on Automatic Control [J], 1974, 19 (6): 716-723
[12] Chen X, Fan Y, Pseudo-likelihood ratio tests for semiparametric multivariatecopula model selection [J], The Canadian Journal of Statistics, 2005, 33(2):389-414
[13] Genest C, Rivest L P, Statistical inference procedures for bivariate Archimedean Copulas [J], Journal of The American Statistical Association,1993, 8:1034-1043
[14] Durrleman V, Nikeghbali A, Roncalli T, Which copula is the right one? Working document [J], Groupe de Recherche Opérationnelle, Crédit Lyonnais, 2000
[15] Kole Erik, Koedijk Kees, Verbeek Marno, Selecting copulas for risk management [J], Journal of Banking and Finance, 2007, 31(8):2405-2423
[16] Chen X, Fan Y, Patton A, Simple tests for models of dependence between multiple financial time series, with applications to U,S, equity returns and exchange rates, Discussion Paper 483, Financial Markets Group, International Asset Management, http://fmg.lse.ac.uk/upload_file/204_dp483.pdf
[17] Jean-David Fermanian, Goodness-of-fit tests for copulas [J], Journal of Multivariate Analysis, 2005, 95(1): 119-152
[18] Diebold F X, T Gunther, A S Tay, Evaluating density forecasts with applications to financial risk management [J], International Economic Review, 1998, 39, 863-883
[19] Diebold F X, J Hahn, A S Tay, Multivariate density forecast and calibration in financial risk management: High-frequency returns on foreign exchange [J], The Review of Economics and Statistics, 1999, 81(4):661-673
[20] Hong Y, Evaluation of out-of-sample density forecasts with applications to S&p 500stock prices, Working Paper, Department of Economics and Department of StatisticalScience, Cornell University, 2000
[21] Berkowitz J, Testing density forecasts, with applications to risk management [J], Journal of Business and Economic Statistics, 2001, 19: 465-474
[22] Genest C, J F Quessy, B R′emillard, Goodness-of-fit procedures for copula models based on the probability integral transform [J], Scandinavian Journal of Statistics, 2006, 33(2): 337-366
[23] Dobri? Friedrich Schmid, A goodness of fit test for copulas based onRosenblatt's transformation [J], Computational Statistics and Data Analysis, 2007, 51(9): 4633-4642
[24] Dobri? J, F Schmid, Testing goodness of fit for parametric families of copulas-application to financial data [J], Communications in Statistics-Simulation and Computation, 2005, 34(4): 1053-1068
[25] David Huard, Guillaumeévin, Anne-Catherine Favre, Bayesian copula selection [J], Computational Statistics & Data Analysis, 2006, 51(2):809-822
[26]王沁,王璐,何平,基于截断tau的copula模型选择及应用[J],数理统计与管理, 2008, 27(1):118-123
[27]张军舰,刘冬香,相依结构数据的ACE拟合[J],数理统计与管理, 2008, 27(2):285-291
[28] Patton A J, Estimation of Copula models for time series of possibly different,University of California at San Diego, Working Paper, 2001
[29] Patton A J, Modelling time-varying exchange rate dependence using the conditional copula, University of California at San Diego, Working Paper,2001
[30] Patton A J, Applications of copula theory in financial econometrics [D], University of California, San Diego, 2002
[31] Rob W J, van den Goorbergh, Christian Genest etc,, Multivariate option pricing using dynamic copula models, Tilburg University, Working Paper, 2003,
[32] Rob W J, van den Goorbergh, Christian Genest etc,, Bivariate option pricing using dynamic copula models [J], Insurance: Mathematics and Economics 2005,37:101-114
[33] Thierry Ane, Chiraz Labidi, Spillover effects and conditional dependence [J], International Review of Economics and Finance, 2006, 15:417-442
[34] J Zhang, D Guegan, Pricing bivariate option under GARCH processes with time-varying copula [J], Insurance: Mathematics and Economics, 2008, 42:1095-1103
[35]韦艳华,张世英,金融市场非对称尾部相关结构的研究[J],管理学报, 2005, 2(5):601-605
[36]罗付岩,邓光明,基于时变Copula的VaR估计[J],系统工程, 2007, 25(8):28-33
[37]任仙玲,张世英,基于Copula函数的金融市场尾部相关性分析[J],统计与信息论坛, 2008, 23(6):66-71
[38]张瑞锋,金融市场波动溢出研究[博士学位论文],天津;天津大学, 2006
[39] Y Hamao, RW Masulis, V Ng, Correlations in price changes and volatility across international stock markets [J], Review of Financial Studies, 1990, 3(2):281-307,
[40] W L Lin, R F Engle, T Ito, Do Bull and Bears Move across Borders? International Transmission of Stock Return and Volatility [J], The Review of Financial Studies, 1994, 7(3):507-538
[41] R F Engle, Stock Volatility and the Crash of 87: Discussion [J] ,The Review of Financial Studies, 1990, 3(1):103-106
[42] Cheung Y, Ng L K, A causality-in-variance test and its application to financial market prices [J], Journal of Econometrics, 1996, 72(1):33-48
[43] Geert Bekaert, Campbell R, Harvey, Emerging equity market volatility [J], Journal of Financial Econometrics, 1997, 43(1):29-77
[44] Andrew J Patton, Modeling asymmetric exchange rate dependence [J], International Economic Review, 2006, 47(2):527-556
[45] Zhang R F, Zou Q W, Zhang SY, Volatility spillover analysis and empirical study on the financial market based on Copula theory [C], 2007 International Conference on Management Science and Engineering, 2007, 1874-1881
[46]蒋翠侠,动态金融风险测度及管理研究[博士学位论文],天津;天津大学, 2007
[47] Ang A, Chen J, Asymmetric correlation of equity portfolio [J], Journal of Financial Economics, 2002, 63(3): 443-494
[48] Erb Claude B, Harvey Campbell R,, Viskanta Tadas E, Forecasting international equity correlation [J], Financial Analysis Journal, 1994, 50:32-45
[49] Login F, Solnik B,Extreme correlation of international equity markets [J],Journal of Finance, 2001, 56(2): 649-676
[50] Anderw J Patton, On the out-of-sample importance of skewness and asymmetric dependence for asset allocation [J], Journal of Financial Econometrics, 2004, 2(1):130-168
[51] Breymann W, Dias, Embrechts P, Dependence structure for multivariate high-frequency data in finance [R], Working paper, Department of Mathematics, ETH Zurich, 2003
[52] Mashal R, Zeevi A, Comparing the dependence structure of equity and asset returns [R], Working paper, Columbia University, New York, 2002
[53] Mashal R, Zeevi A, Beyond correlation: extreme co-movements between financial assets [R], Working paper, Columbia University, New York, 2002
[54]曾健,陈俊芳, Copula函数在风险管理中的应用研究-以上证A股与B股的相关结构分析为例[J],当代财经, 2005(2) : 34-38
[55] Embrechts P, Mcneil A J, Straumann D, Correlation and dependence in risk management: Properties and pitfalls [M], Dempster M,Risk Management: Value at Risk and Beyond1, Cambridge University Press, 1999:176-223
[56] Claudio Romano, Calibrating and Simulating Copula Functions: an Application to the Italian Stock Market [R], Working paper, CIDEM, 2002
[57] Joshua V Rosenberg, Til Schuermann, A general approach to integrated risk management with skewed, fat-tailed risks [J], Journal of Financial Economics, 2006, 79(3):569-614
[58] Helder Parra Palaro, Luiz Koodi Hotta, Using conditional copula to estimate value at risk [J], Journal of Data Science, 2006, 4(1):93-115
[59] Nikolai Kolev, Ulisses dos Anjos, Beatriz Vaz de M Mendes, Copulas: a review and recent developments [J], Stochastic Models, 2006, 22(4): 617-660
[60]张尧庭,连接函数(Copula)技术与金融风险分析[J],统计研究, 2002,(4):48-51
[61]张明恒,多金融资产风险价值的Copula计量方法研究[J],数量经济技术经济研究, 2004, 21(4):67-70
[62]韦艳华,张世英,金融市场的相关性分析-Copula-GARCH模型及其应用[J],系统工程, 2004, 22(4):7-12
[63]吴振翔,陈敏,叶五一等,基于Copula的外汇投资组合风险分析[J],中国管理科学, 2004, 12(4):1-5
[64]吴振翔,陈敏,叶五一等,基于Copula-GARCH的投资组合风险分析[J],系统工程理论与实践, 2006, 26(3):45-52
[65]梁冯珍,史道济,基于copula函数的保险准备金的确定方法[J],统计与决策, 2006, (12):142-144
[66]韦艳华,张世英,金融市场动态相关结构的研究[J],系统工程学报, 2006, 21(3):313-317
[67]任仙玲,张世英,基于核估计及多元阿基米德Copula的投资组合风险分析[J],管理科学, 2007, 20(5):92-97
[68]应益荣,詹炜,资产组合ES风险测度的Copula-EV算法[J],系统管理学报, 2007, 16(6):602-606
[69]刘志东,度量收益率的实际分布的实际分布和相关性对资产组合选择的绩效影响[J],系统管理学报, 2007, 16(6):628-635
[70] Nelson R B, An introduction to Copulas [M], Springer, New York, 2006
[71] McNeil A, R Frey, P Embrechts, Quantitative risk management: concepts, techniques and tools [M], Princeton University Press, 2005
[72] Joe H, Multivariate models and dependence concepts [M], Chapman and Hall, London, 1997
[73] Durrlenman V, Nikeghbali A, Roncalli T, Copulas approximation and new families [R], Groupe de Recherche Opèrationelle, Crédit Lyonnais, Working paper, 2000
[74] Durrlenman V, Nikeghbali A, Roncalli T, Which Copula is the right one? [R], Groupe de Recherche Opèrationelle, Crédit Lyonnais, Working paper, 2000
[75] Umberto Cherubini, Elisa Luciano, Walter Vecchiato, Copula methods in finance [M], John Wiley & Sons Ltd, 2004
[76] Joe H, Xu J J, The estimation method of inference functions for margins formultivariate models[R], Dept of Statistics University of British Columbia, Tech Report, 1996
[77] C Genest, K Ghoudi, L P Rivest, A semiparametric estimation procedure of dependence parameters in multivariate families of distributions [J], Biometrika, 1995, 82:543-552
[78]李松臣,多元非线性平稳时间序列的建模方法研究:[博士学位论文],天津;天津大学,2007
[79] Bowman A W, A Azzalini, Applied smoothing techniques for data analysis [M], Oxford University Press, 1997
[80]叶阿忠,非参数计量经济学[M],天津:南开大学出版社, 2003
[81] Fan J, Yao Q, Nonlinear time series: nonparametric and parametric methods [M], Springer-Verlag, New York, 2003
[82] Charpentier A, Dependence structures and limiting results, with applications in finance and insurance [D], Leuven: Katholieke University, 2005
[83] Scarsini M, On measures of concordance [J], Stochastica, 1984 8:201-208
[84] Dominique Drouet Mari, Samuel Kotz, Correlation and dependence [M], London: Imperial College Press, 2001
[85] Juri A, M V Wuthrich, Copula convergence theorems for tail events [J], Insurance: Mathematics and Ecnomics, 2002, 30:411-427
[86] Juri A, M V Wuthrich, Tail denpendence from adistributional point of view [J], Extremes, 2003, 6:213-246
[87] Stefano Demarta, The copula approach to modelling multivariate extreme values [D], Zurich: ETH, 2007
[88] Fabrizio Durante, Rachele Foschi, Fabio Spizzichino, Threshold copulas and positive dependence [J], Statistics and Probability Letters, 2008
[89] Thierry Ane, Loredana Ureche-Rangau, Chiraz Labidi-Makni, Time-varying conditional dependence in Chinese stock markets [J], Applied Financial Economics, 2008, 18:895-916
[90] Wing Lon Ng, Modeling duration clusters with dynamic copulas [J], FinanceResearch Letters, 2008, 5:96-103
[91] John Y Campbell, Andrew W Lo, A Craig MacKinlay, The econometrics of financial markets [M], Princeton: Princeton University Press, 1997
[92] Rue S TSAY, Analysis of financial time series [M], John Wiley & Sons Ltd, 2005
[93] Engle R, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation [J], Econometrica, 1982, 50(4):987-1008
[94] Bollerslev T, Generalized autoregressive conditional heteroscedasticity [J], Journal of Econometrics, 1986, 31(3):307-327
[95] Ivana Komunjer, Asymmetric power distribution: theory and applications to risk measurement [J], Journal of Applied Econometrics, 2007, 22(5): 891-921
[96] Hansen Bruce E, Autoregressive conditional density estimation [J], International Economic Review, 1994, 35(3):705-730
[97]弗兰克?H奈特著,安佳译,风险、不确定性与利润[M],北京:商务印书馆, 2006
[98] Jorion Philippe, Value as risk: the new benchmark for managing financial risk-2nd edition [M], McGraw-Hill, 2001
[99] P Artzner, U L Pasteur, Strasbourg, Coherent measures of risk [J], Mathematical Finance, 1999, 9:203-228
[100] Rochafeller R T, Uryasev S, Optimization of conditional value-at-risk [J], Journal of Risk, 2000, 2(3):21-41