基于Copula熵的资本资产定价研究
摘要
投资组合问题在现代金融研究中占有重要地位,资本资产定价理论一直被认为是现代金融的核心内容之一,该理论从诞生之初至今已经经历了多个阶段的改进和变化,论文将以资本资产定价理论中的三方面问题为中心展开。
首先,文章针对投资前期的市场选择问题,提出了以区域证券指数的联合波动情况代表市场因素变化的思路,建立了更加完备可信的市场因素。以此为基础所提出的Copula熵的概念,用以作为市场内部相关程度的衡量指标,以评估市场内部风险。通过对比分析Copula熵、相关系数及互信息,证明了Copula熵在度量相关度问题上的有效性及优势。数据试验中,以Copula熵为衡量指标度量了三大经济区域市场内部风险的时变性及结构差异性,以经济理论对分析的有效性进行了验证。
第二,针对投资组合问题中的基金投资问题,提出了两阶段分析法：基金成分股联合波动分析以及基金组合分析。根据所建立的联合熵优化模型推导出针对不同情况的求解方法：联合熵对偶法和Copula熵数值法。论文将Jayness极大熵优化理论进行扩展,借鉴熵优化问题的对偶方法推导出最大联合熵及最小联合叉熵的优化问题求解方法。通过Copula熵与熵理论之间的联系,建立了联合熵与Copula熵之间的转化关系,提出了基于Copula熵的求解方法。利用Copula函数的构建思想将熵优化扩展到多元的情况,拓宽了熵理论在投资组合问题中的适用范围。由于基金等封装类的金融产品在我国属于新兴金融产品,缺少其本身的历史数据作为投资参考,该方法可以帮助投资者通过成分股信息分析基金组合,并构建适当的投资策略。
最后一部分,文章针对资本资产定价模型中的投资期限问题提出了贝叶斯Copula估计方法,用以获得参数系统风险值与投资期限比的后验分布。在样本数据相关性明显存在的有力证据下,使用半相依回归(Seemingly Unrelated Regressions, SUR)替代普通估计方法进行参数估计；使用贝叶斯Copula估计替代原有贝叶斯估计方法进行了贝叶斯估计。半相依回归的使用主要用以考虑残差之间相关性对参数的影响,贝叶斯Copula方法将基于Copula函数的联合函数作为似然函数用以获得后验分布。数据试验分析了6个工业产业证券收益所受系统风险的影响与其投资期限比之间的关系,并对其进行了灵敏度分析。
Investment portfolio is taken at the foremost position in the financial research and the capital asset pricing model is considered as one of the modern financial core contents. The CAPM keeps developing from the moment it born through many years. My dissertation includes three issues.
First of all, the dissertation proposes the method which builds the market factor with joint distribution of the local stock index before we choose the investment field. The method solves the problem and we can build a market factor completely. We define the copula entropy which is considered as the measure index of correlation and the risk within the market. Comparing copula entropy with correlation coefficient and the mutual information, we can find that copula entropy is reasonable and has many advantages about measuring dependence. The copula entropy is used as the correlation index to measure the time-varying characteristics of the correlation and the risk within the market. In economic theory analyses, we also find out that the copula entropy is very significant.
Second, we build the joint entropy optimization model to solve the fund investment portfolio, in which we propose two steps approach:constituent stocks analysis and fund portfolio analysis. There are two methods to solve the different joint entropy optimization models named dual joint entropy approach and copula entropy approach. We extend Jaynes's maximization entropy principle. The dual theory is taken to get the solution of both the maximum joint entropy model and the minimum relative joint entropy model. The ideas of the copula entropy let us to transfer the joint entropy model to copula entropy model in entropy optimization problem. The copula entropy approach is proposed. The application to the fund portfolio is broadened. The packaged financial product such as the fund investment is still the new financial product and the history data are probably limited in China. The method we propose can build the possible strategy base on the constituent stocks of the fund.
The last part of the dissertation, we propose the Bayesian copula estimation to get the posterior distribution of the parameters of the system risk beta and the investment horizon lambada. We find out the data are correlated each other strongly and choose Seemingly Unrelated Regressions method to get the result of the parameter estimation. The Bayesian copula estimation is chosen to consider the correlation of the data instead of regular Bayesian estimation. The reason we chose SUR is concerning the affect of the correlation of residuals and the Bayesian copula estimation using copula function as the likelihood function is proposed with the similar argument. The experiment analyzes the interaction of the system risk and the investment horizon in six different industries.
首先,文章针对投资前期的市场选择问题,提出了以区域证券指数的联合波动情况代表市场因素变化的思路,建立了更加完备可信的市场因素。以此为基础所提出的Copula熵的概念,用以作为市场内部相关程度的衡量指标,以评估市场内部风险。通过对比分析Copula熵、相关系数及互信息,证明了Copula熵在度量相关度问题上的有效性及优势。数据试验中,以Copula熵为衡量指标度量了三大经济区域市场内部风险的时变性及结构差异性,以经济理论对分析的有效性进行了验证。
第二,针对投资组合问题中的基金投资问题,提出了两阶段分析法：基金成分股联合波动分析以及基金组合分析。根据所建立的联合熵优化模型推导出针对不同情况的求解方法：联合熵对偶法和Copula熵数值法。论文将Jayness极大熵优化理论进行扩展,借鉴熵优化问题的对偶方法推导出最大联合熵及最小联合叉熵的优化问题求解方法。通过Copula熵与熵理论之间的联系,建立了联合熵与Copula熵之间的转化关系,提出了基于Copula熵的求解方法。利用Copula函数的构建思想将熵优化扩展到多元的情况,拓宽了熵理论在投资组合问题中的适用范围。由于基金等封装类的金融产品在我国属于新兴金融产品,缺少其本身的历史数据作为投资参考,该方法可以帮助投资者通过成分股信息分析基金组合,并构建适当的投资策略。
最后一部分,文章针对资本资产定价模型中的投资期限问题提出了贝叶斯Copula估计方法,用以获得参数系统风险值与投资期限比的后验分布。在样本数据相关性明显存在的有力证据下,使用半相依回归(Seemingly Unrelated Regressions, SUR)替代普通估计方法进行参数估计；使用贝叶斯Copula估计替代原有贝叶斯估计方法进行了贝叶斯估计。半相依回归的使用主要用以考虑残差之间相关性对参数的影响,贝叶斯Copula方法将基于Copula函数的联合函数作为似然函数用以获得后验分布。数据试验分析了6个工业产业证券收益所受系统风险的影响与其投资期限比之间的关系,并对其进行了灵敏度分析。
Investment portfolio is taken at the foremost position in the financial research and the capital asset pricing model is considered as one of the modern financial core contents. The CAPM keeps developing from the moment it born through many years. My dissertation includes three issues.
First of all, the dissertation proposes the method which builds the market factor with joint distribution of the local stock index before we choose the investment field. The method solves the problem and we can build a market factor completely. We define the copula entropy which is considered as the measure index of correlation and the risk within the market. Comparing copula entropy with correlation coefficient and the mutual information, we can find that copula entropy is reasonable and has many advantages about measuring dependence. The copula entropy is used as the correlation index to measure the time-varying characteristics of the correlation and the risk within the market. In economic theory analyses, we also find out that the copula entropy is very significant.
Second, we build the joint entropy optimization model to solve the fund investment portfolio, in which we propose two steps approach:constituent stocks analysis and fund portfolio analysis. There are two methods to solve the different joint entropy optimization models named dual joint entropy approach and copula entropy approach. We extend Jaynes's maximization entropy principle. The dual theory is taken to get the solution of both the maximum joint entropy model and the minimum relative joint entropy model. The ideas of the copula entropy let us to transfer the joint entropy model to copula entropy model in entropy optimization problem. The copula entropy approach is proposed. The application to the fund portfolio is broadened. The packaged financial product such as the fund investment is still the new financial product and the history data are probably limited in China. The method we propose can build the possible strategy base on the constituent stocks of the fund.
The last part of the dissertation, we propose the Bayesian copula estimation to get the posterior distribution of the parameters of the system risk beta and the investment horizon lambada. We find out the data are correlated each other strongly and choose Seemingly Unrelated Regressions method to get the result of the parameter estimation. The Bayesian copula estimation is chosen to consider the correlation of the data instead of regular Bayesian estimation. The reason we chose SUR is concerning the affect of the correlation of residuals and the Bayesian copula estimation using copula function as the likelihood function is proposed with the similar argument. The experiment analyzes the interaction of the system risk and the investment horizon in six different industries.
引文
[1]Bachelier L. Theorie de la speculation [J]. Annales Scientif iques del Ecole Normale Superieure,1900,3, (17):21-86.
[2]Markowitz H. Portfolio selection [J].The Journal of Finance,1952,7:77-91.
[3]Sharpe W F. Capital asset prices:a theory of market equilibrium under conditions risk [J]. Journal of Finance,1964,19:425-442.
[4]Lintner J. The valuation of risk asset and the selection of risky investments in stock portfolio and capital budgets [J]. Review of Economics and Statistics,1965, 47:13-37.
[5]Mossin J. Equilibrium in a capital asset market [J]. Econometric,1966,34:768-783.
[6]Fama E. F. The Behavior of Stock Market Prices [J]. Journal of Business,1965, 38:34-105
[7]Samuelson P A. Proof that Properly Anticipated Prices Fluctuate Randomly [J]. Industrial Management Review,1965,6:41-49
[8]Ross S A. The arbitrage theory of capital asset pricing [J]. Journal of Economic Theory,1976,13:341-360.
[9]Black F S., Scholes M S. The Pricing of Options and Corporate Liabilities [J]. Journal of Political Economy,1973,3:637-659
[10]Merton R C. Theory of Rational Option Pricing [J]. Bell Journal of Economics and Management Science,1973,4:141-183
[11]Black F. Capital market equilibrium with restricted borrowing [J]. Journal of Business,1972,457:444-455.
[12]Mayers D, Michael C. Nonmarketable asset and capital market equilibrium under uncertainty,1972 [C]. Studies in the Theory of Capital Market, Praeger Publishers, New York:223-248.
[13]Merton R C. An intertemporal capital asset pricing model [J]. Econometrica,1973,41: 867-887.
[14]Brennen B M. Market valuation and corporate financial policed [J]. National Tax Journal,1970,23:417-427.
[15]Jensen M C. Risk, the pricing of capital asset, and the evaluation of investment portfolio [J]. Journal of Business,1969,42:167-247.
[16]Solnik B. An equilibrium model of intemational capital markets [J]. Journal of Economies Theory,1974,8:500-524.
[17]Breeden D T. An inter-temporal asset pricing model with stochastic consumption and investment opportunities [J]. Journal of Financial Economics,1979,73:265-296.
[16]Fama E F. Multi-period consumption-investment decision [J]. American Economic Review,1970,60:163-174.
[17]Blume M. On the assessment of risk [J]. Journal of Finance,1971,26:1-10.
[19]Blume M. Betas and their regression tendencies:some further evidence [J]. Journal of Finance,1979,34:265-275.
[20]Levy R A. Beta coefficients as predictors of returns [J]. Financial Analysts Journal,1974,30:61-69.
[21]Fisher L, Kamin J. On the estimation of a systematic risk [R]. Univ. of Chicago, 1972.
[22]Franzoni F. Learning about beta:time-varying factor loadings, expected returns, and the conditional CAPM [J]. Journal of Empirical Finance,2009,164:537-556.
[23]Barberis N, Shleifer A. Style investing [J]. Journal of Financial Economics,2003, 68:161-199.
[24]Lewellen J, Nagel S, Shanken J. A skeptical appraisal of asset pricing tests [J]. Journal of Financial Economics,2010,96:175-194.
[25]Roll R. A critique of the asset pricing theory's tests; part I:on past and potential testability of the theory [J]. Journal of Financial Economics,1977,4: 129-176.
[26]Fama E F, French K R. The cross-section of expected stock returns [J]. Journal of Finance,1992,47:427-465.
[27]Kothari S, Shanken J, Sloan R. Another look at the cross-section of expected stock. Returns [J]. Journal of Finance,1995,50:185-224.
[28]Clare A, Priestley R, Thomas S H. Reports of beta's death are premature:evidence from the UK [J].Journal of Banking and Finance,1998,22(9):1207-1229.
[29]Banz R W. The relationship between return and market value of common stocks [J]. Journal of Financial Economics,1981,9:3-18.
[30]Rosenberg B, Reid K, Lanstein R. Inefficiency, persuasive evidence of market [J]. Journal of Portfolio Management,1985,11:9-17.
[31]Chan L K C, Hamao Y, Lakonishok J. Fundamentals and stock returns in Japan [J]. Journal of Finance,1991,46:1739-1764.
[32]Bollerslev T, Engle F R, Wooldridge J M. A capital asset pricing model with time-varying covariances [J]. Journal of Political Economy,1988,96:116-131.
[33]Galagedera D U A, Silvapulle P. Conditional relation between higher-order co-moments and stock return:evidence from Australian data,2002 [C]. Proceedings of the Econometric Society Australasian Meeting, Australia:Queensland University of Technology press.
[34]Ane T, Geman H. Order flow, transaction clock, and normality of asset returns [J]. Journal of Finance,2000,55:2259-2284.
[35]Chung Y P, Johnson H, Schill M J. Asset prcing when returns are non-normal: fama-french factors vs higher-order systematic co-moments [C]. Univ. of California, Riverside.2001.
[36]Kraus A, Litzenberger R H. Skewness preference and the valuation of risk assets [J]. Journal of Finance,1976,31:1085-1100.
[37]Friend I, Westerfield R. Co-skewness and capital asset pricing [J]. Journal of Finance,1980,35:897-913
[38]Sears R S, Wei K C J. Asset pricing, higher moments, and the market risk premium: a note [J]. Journal of Finance,1985,40:1251-1253.
[39]Faff R, Ho Y K, Zhang L. A generalised method of moment test of the three-moment capital asset pricing model in the Australian equity market [J]. Asia Pacific Journal of Finance,1998,1:45-60.
[40]Harvey C, Siddique A. Conditional skewness in asset pricing tests [J]. Journal of Finance,2000,55:1263-1295.
[41]Fang H, Lai T Y. Co-kurtosis and capital asset pricing [J]. The Financial Review, 1997,32:293-307.
[42]Christie D R, Chaudhry M. Currency futures, news releases, and uncertainty resolution [J]. Global Finance Journal,2000,11:109-127.
[43]Samuelson P A. Lifetime portfolio selection by dynamic stochastic programming [J]. The Review of Economics and Statistics,1969,50:239-246.
[44]Embrechts P. Copulas:A personal view [J]. Journal of Risk and Insurance,2009, 76:639-650.
[45]Levhari D, Levy H. The capital asset pricing model and the investment horizon [J]. Review of Economics and statistics,1976,59:356-363.
[46]Levy H. Portfolio performance and the investment horizon [J]. Management Science 1972,18:645-653.
[47]Cheng P L, Deet M K. Systematic risk and the horizon problem [J]. Journal of Financial and Quantitative Analysis,1973,3:299-316.
[48]Gressis N, Philippators G C, Hayya J C. Multiperiod portfolio analysis and the inefficiency of the market portfolio [J]. Journal of Finance,1976,31:1115-1126.
[49]Gilster J E J. Autocorrelation investment horizon and efficient frontier composition [J]. The Financial Review,1979,14:23-29.
[50]Gilster J E J. Capital market equilibrium with divergent investment horizon length assumption [J]. Journal of Financial and Quanti tative Analysis,1983,18:257-268.
[51]Lee C F. Investment horizon and the functional form of the capital asset pricing model[J]. Review of Economics and Statistics,1976,58:356-363.
[52]Lin T W, Chen H Y. Investment horizon and beta coefficient [J]. Journal of Business Research,1990,21:19-37.
[53]Sklar A. Fonctions de repartition an dimensions et leurs marges [J]. Publications de 1'Institut de Statistique de 1'University de Paris,1959,8:229-231.
[54]Schweizer B, Sklar A. Probabilistic metric spaces[M]. New York:North Holland, 1983.
[55]Genest C, Mackay J. The joy of copulas:bivariate distributions with uniform marginals [J]. The American Statistician,1986,40,280-283.
[56]Joe H. Multivariate models and dependence concepts [M]. London:Chapman & Hall, 1997.
[57]Nelsen B R. An introduction to copulas [M]. New York:Springer, First edition,1999; Second edition,2006.
[58]Molinaa Q J J, Sempi C. Discrete quasi-copulas [J]. Insurance:Mathematics and Economics,2005,37:27-41.
[59]Denuita M, Lamberta P. Constraints on concordance measures in bivariate discrete data [J]. Journal of Multivariate Analysis,2005,93:40-57.
[60]Kolesarova A, Mesiar R, Mordelova J et al. Discrete copulas [J]. IEEE Transactions on Fuzzy Systems,2006,14(5):698-705.
[61]Patton J A. Applications of copula theory in financial econometrics [D] Univ. of California, San Diego,2002.
[62]Hu L. Essays in econometrics with applications in macroeconomic and financial modeling [D] New Haven:Yale Univ.,2002.
[63]Savu C, Trede M. Hierarchical archimedean copulas [R] Institute of Econometrics, University of Munster,2006.
[64]Mattei s R D. Fitting copulas to data [R] Diploma thesis, Institute of Mathematics, Zurich University,2001.
[65]Clemen R T. Correlations and copulas for decision and risk analysis [J]. Management Science,1999,45(2):208-224.
[66]Embrechts P, McNeil A, Straumann D.Correlation:pitfalls and alternatives [J]. Risk,1999,12(1):69-71.
[67]Embrechts P, McNeil A, Straumann D. Correlation and dependence in risk management [J]. Properties and Pitfalls,2002 (29):114-117.
[68]Embrechts P, Hoeing A, Juri A. Using copulae to bound the Value-at-Risk for functions of dependent risks [J]. Finance & Stochastics,2003,7(2):145-167.
[69]Coles S, Heffernan J. Dependence measures for extreme value analyses [J]. Extremes,1999,2(4):339-365.
[70]Cizeau P, Potters M, Bouchaud J. Correlation structure of extreme stock returns [J]. Quantitative Finance,2001,1:217-222.
[71]Engle F R. Dynamic conditional correlation-a simple class of multivariate GARCH models [J]. Journal of Business and Economic Statistics,2002,20(3):339-350.
[72]Clive W J G, Terasvirta T, Patton A. Common factors in conditional distributions for bivariate time series [J]. Journal of Econometrics,2006,132(1):43-57.
[73]Patton A. Estimation of multivariate models for time series of possibly different lengths [J]. Journal of Applied Econometrics,2006,21(2):147-173.
[74]Xu Y. Applications of copula-based models in portfolio optimization [D] Miami: Univ. of Miami,2005.
[75]Hull J, White A. Valuing credit derivatives using an implied copula approach [J]. Journal of Derivatives,2006,14(2):8-28.
[76]Hull J, White A. Forwards and European options on CDO tranches [J]. Journal of Credit Risk,2007,3(2):63-73.
[77]Dempster M A H, Medova E A, Yang S W. Empirical copulas for CDO tranche pricing using relative entropy [R] Univ. of Cambridge,2007.
[78]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.
[79]韦艳华Copula理论及其在多变量金融时间序列分析上的应用研究[D].天津：天津大学,2004.
[80]韦艳华,张世英,孟利锋Copula理论在金融上的应用[J].西北农林科技大学学报(社会科学版),2003,3(5)：97-101.
[81]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)：4-52.
[82]吴振翔,叶五一,缪柏其.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.
[83]朱世武.基于Copula函数度量违约相关性[J].统计研究,2005,4：61-64.
[84]叶五一,缪柏其,吴振翔.基于Copula方法的条件VaR估计[J].中国科学技术大学学报2006,36(9)：917-922.
[85]Shannon C E. The mathematical theory communication [J]. Bell Systems Techcical Journal,1948,27(4):279-423 and 623-656.
[86]Jaynes E T. Information theory and statistical mechanics [J]. The Physics Review Ⅰ, 1957a,106:620-630; The Physics Review Ⅱ,1957b,108:171-190.
[87]Kullback S. Information theory and statistics [M]. New York:John Wiley,1959.
[88]Philippatos G C, Wilsont C J. Entropy, market risk, and the selection of efficient portfolios [J]. Applied Economics,1972,4:209-220.
[89]Nawrocki D N, Harding W H. State-value weighted entropy as a measure of investment risk [J]. Applied Economics,1986,18:411-419.
[90]Maasoumi E, Theil H. The elect of the shape of the income distribution on two inequality measures [J]. Economics Letters,1979,4:209-220.
[91]McCauley J. Thermodynamic analogies in economics and finance:instability of markets [J]. Physica A,2003,329:199-212.
[92]李华,李兴斯.证券投资组合理论的一种新模型及其应用[J].运筹与管理,2003,12(6)：83-86.
[93]李华,李兴斯.证券投资组合中的熵优化模型研究[J].大连理工大学学报,2005,45(1)：153-156.
[94]丰雪,李兴斯,张阚.不完全信息下损失风险的研究[J]数学的实践与认识,2008,38(15)：86-90.
[95]姜昱汐.保险精算的信息熵方法[D].大连：大连理工大学,2005.
[96]Ebrahimi N, Maasoumi E, Soofi E S. Ordering univariate distributions by entropy and variance [J]. Journal of Econometrics,1999,90:317-336.
[97]Chandra M T, Singpurwalla N D. Relationships between some notions which are common to reliability theory and economics [J]. Mathematics of Operations Research,1981, 6:113-121.
[98]Gulko B L. The entropy pricing theory [D]. New Haven:Yale Univ.,1998.
[99]Gulko B L. The entropic market hypotheses [J]. International Journal of Theoretical and Applied Finance,1999,2:293-329.
[100]Gulko B L. The entropy theory of stock option pricing [J]. International Journal of Theoretical and Applied Finance,1999,2:331-355.
[101]Gulko B L. The entropy theory of bond option pricing [J]. International Journal of Theoretical and Applied Finance,2002,2:355-384.
[102]Buchen P W, Kelly M. The maximum entropy distribution of an asset inferred from option pries [J]. Journal of Financial and Quantitative Analysis,1996,31(1): 143-159.
[103]Samperi D. Inverse problems, model selection and entropy in derivative security pricing [D]. New York:New York Univ.,1998.
[104]Reesor R M. Relative entropy, distortion, the bootstrap and risk [D]. Waterloo: University of Waterloo,2001.
[105]Stutzer M. A simple nonparametric approach to deriative security valuation [J]. Journal of Finance,1996,5:1633-1652.
[106]Branger N. Pricing derivative securities using cross-entropy:an economic analysis [J]. International Journal of Theoretical and Applied Finance,2004,7(1): 63-81.
[107]Kitamura Y, Stutzer M. An information theoretic alternative to generalized method of moments estimation [J]. Econometrica,1997,65(4):861-874.
[108]Huang G, Wan J. A nonparametric approach for European option valuation [J]. Physica A:Statistical Mechanics and its Applications,2007,10:2306-2316.
[109]Michael M. The minimal entropy measure and an esscher transform in an incomplete market model [J]. Statistics & Probability Letters,2007,11:1070-1076.
[110]Cover T M, Thomas J A,阮吉寿等.信息论基础[M].北京：机械工业出版社,2005.
[111]朱雪龙.应用信息论基础[M].北京：清华大学出版社,2001.
[112]沈世镒,吴忠华.信息论基础与应用[M].北京：高等教育出版社,2004.
[113]Ebrahimi N, Maasoumi E, Soofi E S. Ordering univariate distributions by entropy and variance [J]. Journal of Econometrics,1999,90:317-336.
[114]Ane T, Kharoubi C. Dependence structure and risk measure [J]. Journal of Business, 2003,76:411-438.
[115]Jenison R L, Reale R A. The shape of neural dependence [J]. Neural Computation. 2004,16:665-672.
[116]Kolesarova A, Mesiar R, Mordelova J, et al. Discrete copulas [J]. IEEE Transfer Fuzzy System.14 (2006) 698-705.
[117]Parzen E. On estimation of a probability density function and mode [J]. The Annals of Mathematical Statistics,1962,33(3),1065-1076.
[118]Silverman B W. Density estimation for statistics and data analysis[M]. London: Chapman & Hall.1986.
[119]Fan J, Yao Q. Nonlinear time series:nonparametric and parmetric methods [M]. Springer, New York,2003.
[120]X. Li. Entropy and optimization, [D]. Merseyside:Univ. of Liverpool,1987.
[121]陈德照,安和芬,王鼎咏.世界三大经济圈：漫谈世界区域经济集团化趋势[M].北京,世界知识出版社,1995.
[122]Piantadosi J, Howlett P G, Boland J W. Matching the grade correlation coefficient using a copula with maximum disorder [J]. Journal of Industrial and Management Optimization,2007,3(2):305-312.
[123]Piantadosi J, Howlett P, Jonathan B. Copulas with maximum entropy [JOL]. Optimization Letters,2011, October 27.
[124]Bayes T R. An essay towards solving a problem in the doctrine of chances [J]. Philosophical Transactions,1763,53:370-418.
[125]张尧庭,陈汉峰.贝叶斯统计推断[M].北京：科学出版社,1991.
[126]Jeffreys H. Theory of probability [M]. Oxford:Clarendon Press, third edition,1961.
[127]Heidbreder R G. Maximum entropy and bayesian methods:proceedings of the thirteenth international workshop on maximum entropy and bayesian methods [M]. California:Santa Barbara,1993.
[128]Schlaifer R. Applied statistical decision theory [M]. Cambridge, Mass:MIT Press, 1961.
[129]Pitt M, Chan D, Kohn R. Efficient bayesian inference for gaussian copula regression models[J]. Biometrika,2006,93:537-554.
[130]Lee F C. Investment horizon and the functional form of the capital asset pricing[J]. The Review of Economics and Statistics,1976,58(3):356-363.
[131]Zellner A. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias [J]. Journal of the American Statistical Association,1962,57:348-368.
[2]Markowitz H. Portfolio selection [J].The Journal of Finance,1952,7:77-91.
[3]Sharpe W F. Capital asset prices:a theory of market equilibrium under conditions risk [J]. Journal of Finance,1964,19:425-442.
[4]Lintner J. The valuation of risk asset and the selection of risky investments in stock portfolio and capital budgets [J]. Review of Economics and Statistics,1965, 47:13-37.
[5]Mossin J. Equilibrium in a capital asset market [J]. Econometric,1966,34:768-783.
[6]Fama E. F. The Behavior of Stock Market Prices [J]. Journal of Business,1965, 38:34-105
[7]Samuelson P A. Proof that Properly Anticipated Prices Fluctuate Randomly [J]. Industrial Management Review,1965,6:41-49
[8]Ross S A. The arbitrage theory of capital asset pricing [J]. Journal of Economic Theory,1976,13:341-360.
[9]Black F S., Scholes M S. The Pricing of Options and Corporate Liabilities [J]. Journal of Political Economy,1973,3:637-659
[10]Merton R C. Theory of Rational Option Pricing [J]. Bell Journal of Economics and Management Science,1973,4:141-183
[11]Black F. Capital market equilibrium with restricted borrowing [J]. Journal of Business,1972,457:444-455.
[12]Mayers D, Michael C. Nonmarketable asset and capital market equilibrium under uncertainty,1972 [C]. Studies in the Theory of Capital Market, Praeger Publishers, New York:223-248.
[13]Merton R C. An intertemporal capital asset pricing model [J]. Econometrica,1973,41: 867-887.
[14]Brennen B M. Market valuation and corporate financial policed [J]. National Tax Journal,1970,23:417-427.
[15]Jensen M C. Risk, the pricing of capital asset, and the evaluation of investment portfolio [J]. Journal of Business,1969,42:167-247.
[16]Solnik B. An equilibrium model of intemational capital markets [J]. Journal of Economies Theory,1974,8:500-524.
[17]Breeden D T. An inter-temporal asset pricing model with stochastic consumption and investment opportunities [J]. Journal of Financial Economics,1979,73:265-296.
[16]Fama E F. Multi-period consumption-investment decision [J]. American Economic Review,1970,60:163-174.
[17]Blume M. On the assessment of risk [J]. Journal of Finance,1971,26:1-10.
[19]Blume M. Betas and their regression tendencies:some further evidence [J]. Journal of Finance,1979,34:265-275.
[20]Levy R A. Beta coefficients as predictors of returns [J]. Financial Analysts Journal,1974,30:61-69.
[21]Fisher L, Kamin J. On the estimation of a systematic risk [R]. Univ. of Chicago, 1972.
[22]Franzoni F. Learning about beta:time-varying factor loadings, expected returns, and the conditional CAPM [J]. Journal of Empirical Finance,2009,164:537-556.
[23]Barberis N, Shleifer A. Style investing [J]. Journal of Financial Economics,2003, 68:161-199.
[24]Lewellen J, Nagel S, Shanken J. A skeptical appraisal of asset pricing tests [J]. Journal of Financial Economics,2010,96:175-194.
[25]Roll R. A critique of the asset pricing theory's tests; part I:on past and potential testability of the theory [J]. Journal of Financial Economics,1977,4: 129-176.
[26]Fama E F, French K R. The cross-section of expected stock returns [J]. Journal of Finance,1992,47:427-465.
[27]Kothari S, Shanken J, Sloan R. Another look at the cross-section of expected stock. Returns [J]. Journal of Finance,1995,50:185-224.
[28]Clare A, Priestley R, Thomas S H. Reports of beta's death are premature:evidence from the UK [J].Journal of Banking and Finance,1998,22(9):1207-1229.
[29]Banz R W. The relationship between return and market value of common stocks [J]. Journal of Financial Economics,1981,9:3-18.
[30]Rosenberg B, Reid K, Lanstein R. Inefficiency, persuasive evidence of market [J]. Journal of Portfolio Management,1985,11:9-17.
[31]Chan L K C, Hamao Y, Lakonishok J. Fundamentals and stock returns in Japan [J]. Journal of Finance,1991,46:1739-1764.
[32]Bollerslev T, Engle F R, Wooldridge J M. A capital asset pricing model with time-varying covariances [J]. Journal of Political Economy,1988,96:116-131.
[33]Galagedera D U A, Silvapulle P. Conditional relation between higher-order co-moments and stock return:evidence from Australian data,2002 [C]. Proceedings of the Econometric Society Australasian Meeting, Australia:Queensland University of Technology press.
[34]Ane T, Geman H. Order flow, transaction clock, and normality of asset returns [J]. Journal of Finance,2000,55:2259-2284.
[35]Chung Y P, Johnson H, Schill M J. Asset prcing when returns are non-normal: fama-french factors vs higher-order systematic co-moments [C]. Univ. of California, Riverside.2001.
[36]Kraus A, Litzenberger R H. Skewness preference and the valuation of risk assets [J]. Journal of Finance,1976,31:1085-1100.
[37]Friend I, Westerfield R. Co-skewness and capital asset pricing [J]. Journal of Finance,1980,35:897-913
[38]Sears R S, Wei K C J. Asset pricing, higher moments, and the market risk premium: a note [J]. Journal of Finance,1985,40:1251-1253.
[39]Faff R, Ho Y K, Zhang L. A generalised method of moment test of the three-moment capital asset pricing model in the Australian equity market [J]. Asia Pacific Journal of Finance,1998,1:45-60.
[40]Harvey C, Siddique A. Conditional skewness in asset pricing tests [J]. Journal of Finance,2000,55:1263-1295.
[41]Fang H, Lai T Y. Co-kurtosis and capital asset pricing [J]. The Financial Review, 1997,32:293-307.
[42]Christie D R, Chaudhry M. Currency futures, news releases, and uncertainty resolution [J]. Global Finance Journal,2000,11:109-127.
[43]Samuelson P A. Lifetime portfolio selection by dynamic stochastic programming [J]. The Review of Economics and Statistics,1969,50:239-246.
[44]Embrechts P. Copulas:A personal view [J]. Journal of Risk and Insurance,2009, 76:639-650.
[45]Levhari D, Levy H. The capital asset pricing model and the investment horizon [J]. Review of Economics and statistics,1976,59:356-363.
[46]Levy H. Portfolio performance and the investment horizon [J]. Management Science 1972,18:645-653.
[47]Cheng P L, Deet M K. Systematic risk and the horizon problem [J]. Journal of Financial and Quantitative Analysis,1973,3:299-316.
[48]Gressis N, Philippators G C, Hayya J C. Multiperiod portfolio analysis and the inefficiency of the market portfolio [J]. Journal of Finance,1976,31:1115-1126.
[49]Gilster J E J. Autocorrelation investment horizon and efficient frontier composition [J]. The Financial Review,1979,14:23-29.
[50]Gilster J E J. Capital market equilibrium with divergent investment horizon length assumption [J]. Journal of Financial and Quanti tative Analysis,1983,18:257-268.
[51]Lee C F. Investment horizon and the functional form of the capital asset pricing model[J]. Review of Economics and Statistics,1976,58:356-363.
[52]Lin T W, Chen H Y. Investment horizon and beta coefficient [J]. Journal of Business Research,1990,21:19-37.
[53]Sklar A. Fonctions de repartition an dimensions et leurs marges [J]. Publications de 1'Institut de Statistique de 1'University de Paris,1959,8:229-231.
[54]Schweizer B, Sklar A. Probabilistic metric spaces[M]. New York:North Holland, 1983.
[55]Genest C, Mackay J. The joy of copulas:bivariate distributions with uniform marginals [J]. The American Statistician,1986,40,280-283.
[56]Joe H. Multivariate models and dependence concepts [M]. London:Chapman & Hall, 1997.
[57]Nelsen B R. An introduction to copulas [M]. New York:Springer, First edition,1999; Second edition,2006.
[58]Molinaa Q J J, Sempi C. Discrete quasi-copulas [J]. Insurance:Mathematics and Economics,2005,37:27-41.
[59]Denuita M, Lamberta P. Constraints on concordance measures in bivariate discrete data [J]. Journal of Multivariate Analysis,2005,93:40-57.
[60]Kolesarova A, Mesiar R, Mordelova J et al. Discrete copulas [J]. IEEE Transactions on Fuzzy Systems,2006,14(5):698-705.
[61]Patton J A. Applications of copula theory in financial econometrics [D] Univ. of California, San Diego,2002.
[62]Hu L. Essays in econometrics with applications in macroeconomic and financial modeling [D] New Haven:Yale Univ.,2002.
[63]Savu C, Trede M. Hierarchical archimedean copulas [R] Institute of Econometrics, University of Munster,2006.
[64]Mattei s R D. Fitting copulas to data [R] Diploma thesis, Institute of Mathematics, Zurich University,2001.
[65]Clemen R T. Correlations and copulas for decision and risk analysis [J]. Management Science,1999,45(2):208-224.
[66]Embrechts P, McNeil A, Straumann D.Correlation:pitfalls and alternatives [J]. Risk,1999,12(1):69-71.
[67]Embrechts P, McNeil A, Straumann D. Correlation and dependence in risk management [J]. Properties and Pitfalls,2002 (29):114-117.
[68]Embrechts P, Hoeing A, Juri A. Using copulae to bound the Value-at-Risk for functions of dependent risks [J]. Finance & Stochastics,2003,7(2):145-167.
[69]Coles S, Heffernan J. Dependence measures for extreme value analyses [J]. Extremes,1999,2(4):339-365.
[70]Cizeau P, Potters M, Bouchaud J. Correlation structure of extreme stock returns [J]. Quantitative Finance,2001,1:217-222.
[71]Engle F R. Dynamic conditional correlation-a simple class of multivariate GARCH models [J]. Journal of Business and Economic Statistics,2002,20(3):339-350.
[72]Clive W J G, Terasvirta T, Patton A. Common factors in conditional distributions for bivariate time series [J]. Journal of Econometrics,2006,132(1):43-57.
[73]Patton A. Estimation of multivariate models for time series of possibly different lengths [J]. Journal of Applied Econometrics,2006,21(2):147-173.
[74]Xu Y. Applications of copula-based models in portfolio optimization [D] Miami: Univ. of Miami,2005.
[75]Hull J, White A. Valuing credit derivatives using an implied copula approach [J]. Journal of Derivatives,2006,14(2):8-28.
[76]Hull J, White A. Forwards and European options on CDO tranches [J]. Journal of Credit Risk,2007,3(2):63-73.
[77]Dempster M A H, Medova E A, Yang S W. Empirical copulas for CDO tranche pricing using relative entropy [R] Univ. of Cambridge,2007.
[78]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.
[79]韦艳华Copula理论及其在多变量金融时间序列分析上的应用研究[D].天津：天津大学,2004.
[80]韦艳华,张世英,孟利锋Copula理论在金融上的应用[J].西北农林科技大学学报(社会科学版),2003,3(5)：97-101.
[81]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)：4-52.
[82]吴振翔,叶五一,缪柏其.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.
[83]朱世武.基于Copula函数度量违约相关性[J].统计研究,2005,4：61-64.
[84]叶五一,缪柏其,吴振翔.基于Copula方法的条件VaR估计[J].中国科学技术大学学报2006,36(9)：917-922.
[85]Shannon C E. The mathematical theory communication [J]. Bell Systems Techcical Journal,1948,27(4):279-423 and 623-656.
[86]Jaynes E T. Information theory and statistical mechanics [J]. The Physics Review Ⅰ, 1957a,106:620-630; The Physics Review Ⅱ,1957b,108:171-190.
[87]Kullback S. Information theory and statistics [M]. New York:John Wiley,1959.
[88]Philippatos G C, Wilsont C J. Entropy, market risk, and the selection of efficient portfolios [J]. Applied Economics,1972,4:209-220.
[89]Nawrocki D N, Harding W H. State-value weighted entropy as a measure of investment risk [J]. Applied Economics,1986,18:411-419.
[90]Maasoumi E, Theil H. The elect of the shape of the income distribution on two inequality measures [J]. Economics Letters,1979,4:209-220.
[91]McCauley J. Thermodynamic analogies in economics and finance:instability of markets [J]. Physica A,2003,329:199-212.
[92]李华,李兴斯.证券投资组合理论的一种新模型及其应用[J].运筹与管理,2003,12(6)：83-86.
[93]李华,李兴斯.证券投资组合中的熵优化模型研究[J].大连理工大学学报,2005,45(1)：153-156.
[94]丰雪,李兴斯,张阚.不完全信息下损失风险的研究[J]数学的实践与认识,2008,38(15)：86-90.
[95]姜昱汐.保险精算的信息熵方法[D].大连：大连理工大学,2005.
[96]Ebrahimi N, Maasoumi E, Soofi E S. Ordering univariate distributions by entropy and variance [J]. Journal of Econometrics,1999,90:317-336.
[97]Chandra M T, Singpurwalla N D. Relationships between some notions which are common to reliability theory and economics [J]. Mathematics of Operations Research,1981, 6:113-121.
[98]Gulko B L. The entropy pricing theory [D]. New Haven:Yale Univ.,1998.
[99]Gulko B L. The entropic market hypotheses [J]. International Journal of Theoretical and Applied Finance,1999,2:293-329.
[100]Gulko B L. The entropy theory of stock option pricing [J]. International Journal of Theoretical and Applied Finance,1999,2:331-355.
[101]Gulko B L. The entropy theory of bond option pricing [J]. International Journal of Theoretical and Applied Finance,2002,2:355-384.
[102]Buchen P W, Kelly M. The maximum entropy distribution of an asset inferred from option pries [J]. Journal of Financial and Quantitative Analysis,1996,31(1): 143-159.
[103]Samperi D. Inverse problems, model selection and entropy in derivative security pricing [D]. New York:New York Univ.,1998.
[104]Reesor R M. Relative entropy, distortion, the bootstrap and risk [D]. Waterloo: University of Waterloo,2001.
[105]Stutzer M. A simple nonparametric approach to deriative security valuation [J]. Journal of Finance,1996,5:1633-1652.
[106]Branger N. Pricing derivative securities using cross-entropy:an economic analysis [J]. International Journal of Theoretical and Applied Finance,2004,7(1): 63-81.
[107]Kitamura Y, Stutzer M. An information theoretic alternative to generalized method of moments estimation [J]. Econometrica,1997,65(4):861-874.
[108]Huang G, Wan J. A nonparametric approach for European option valuation [J]. Physica A:Statistical Mechanics and its Applications,2007,10:2306-2316.
[109]Michael M. The minimal entropy measure and an esscher transform in an incomplete market model [J]. Statistics & Probability Letters,2007,11:1070-1076.
[110]Cover T M, Thomas J A,阮吉寿等.信息论基础[M].北京：机械工业出版社,2005.
[111]朱雪龙.应用信息论基础[M].北京：清华大学出版社,2001.
[112]沈世镒,吴忠华.信息论基础与应用[M].北京：高等教育出版社,2004.
[113]Ebrahimi N, Maasoumi E, Soofi E S. Ordering univariate distributions by entropy and variance [J]. Journal of Econometrics,1999,90:317-336.
[114]Ane T, Kharoubi C. Dependence structure and risk measure [J]. Journal of Business, 2003,76:411-438.
[115]Jenison R L, Reale R A. The shape of neural dependence [J]. Neural Computation. 2004,16:665-672.
[116]Kolesarova A, Mesiar R, Mordelova J, et al. Discrete copulas [J]. IEEE Transfer Fuzzy System.14 (2006) 698-705.
[117]Parzen E. On estimation of a probability density function and mode [J]. The Annals of Mathematical Statistics,1962,33(3),1065-1076.
[118]Silverman B W. Density estimation for statistics and data analysis[M]. London: Chapman & Hall.1986.
[119]Fan J, Yao Q. Nonlinear time series:nonparametric and parmetric methods [M]. Springer, New York,2003.
[120]X. Li. Entropy and optimization, [D]. Merseyside:Univ. of Liverpool,1987.
[121]陈德照,安和芬,王鼎咏.世界三大经济圈：漫谈世界区域经济集团化趋势[M].北京,世界知识出版社,1995.
[122]Piantadosi J, Howlett P G, Boland J W. Matching the grade correlation coefficient using a copula with maximum disorder [J]. Journal of Industrial and Management Optimization,2007,3(2):305-312.
[123]Piantadosi J, Howlett P, Jonathan B. Copulas with maximum entropy [JOL]. Optimization Letters,2011, October 27.
[124]Bayes T R. An essay towards solving a problem in the doctrine of chances [J]. Philosophical Transactions,1763,53:370-418.
[125]张尧庭,陈汉峰.贝叶斯统计推断[M].北京：科学出版社,1991.
[126]Jeffreys H. Theory of probability [M]. Oxford:Clarendon Press, third edition,1961.
[127]Heidbreder R G. Maximum entropy and bayesian methods:proceedings of the thirteenth international workshop on maximum entropy and bayesian methods [M]. California:Santa Barbara,1993.
[128]Schlaifer R. Applied statistical decision theory [M]. Cambridge, Mass:MIT Press, 1961.
[129]Pitt M, Chan D, Kohn R. Efficient bayesian inference for gaussian copula regression models[J]. Biometrika,2006,93:537-554.
[130]Lee F C. Investment horizon and the functional form of the capital asset pricing[J]. The Review of Economics and Statistics,1976,58(3):356-363.
[131]Zellner A. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias [J]. Journal of the American Statistical Association,1962,57:348-368.