基于RMT去噪法股票投资组合风险优化研究
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摘要
Markowitz证券投资组合理论是当代金融理论的重要支柱之一，它也是对实际证券投资具有最大指导价值的投资理论。作为Markowitz模型的关键输入参数，证券收益率历史相关矩阵和历史协方差矩阵的噪声会通过组合风险的增大和组合风险预测准确率的下降而导致组合风险的恶化。由于噪声对证券组合风险的影响随证券数的增加而增加，因此，在目前组合中证券数呈增大趋势的背景下，Markowitz模型在实践中应用的可行性正逐渐下降甚至接近于完全失效，噪声的影响已成为当今证券投资风险空前增大的重要原因之一。如何减小噪声对组合风险的影响是当前迫切需要关注的研究问题。越来越多的学者将对证券收益历史相关矩阵和历史协方差矩阵的去噪作为解决方法，并进行详细研究。
和针对金融相关矩阵和协方差矩阵的其他去噪方法相比，基于随机矩阵理论的去噪法（RMT去噪法）具有决定模型最佳维度、操作技术难度低和适用范围广泛等优点。然而，基于RMT相关或多元波动率的去噪及其在股票投资组合风险优化中的应用的研究目前尚处于较为初级的阶段，有很多研究工作有待于进一步开展。以随机矩阵理论和现代投资组合理论为理论基础，采用蒙特卡洛模拟法、toy模型法、模拟退火算法、主成分分析法、一般数理分析法、理论研究和实证研究相结合等多种研究方法深入研究应用RMT去噪法实现股票投资组合风险优化的问题不仅有助于完善RMT去噪理论，而且对扫除Markowitz模型的应用障碍，从而促进Markowitz证券投资组合理论的进一步发展有重要的理论意义。在证券投资已经成为全球社会经济生活的一个重要构成部分的今天，该研究明显也具有很强的现实意义。
本文从阐释Markowitz股票投资组合风险的含义入手，说明了以投资组合风险的大小和投资组合风险预测的准确率衡量投资组合风险的优劣，分析了股票收益历史相关矩阵和协方差矩阵的噪声对Markowitz投资组合风险优化水平的影响，并选取RMT去噪法作为解决噪声影响的方法，进而构建了应用RMT去噪法实现Markowitz股票投资组合风险优化的整体思路框架，其包括四个角度：第一，通过对面向股票收益相关矩阵的RMT去噪法原理和算法的改进实现股票投资组合风险优化；第二，通过建立对已有去噪研究未曾涉及的股票收益率多元波动率模型去噪的一般方法实现股票投资组合风险优化；第三，通过弥补面向收益样本协方差矩阵的现有RMT去噪法在小组合条件下的缺陷实现股票投资组合风险优化；第四，根据对总体协方差矩阵特征值进行估计的新去噪思路建立面向收益样本协方差矩阵的RMT去噪法来实现股票投资组合风险的优化。从这四个角度出发，论文的主要研究内容和成果包括如下方面。
首先，在介绍面向股票收益相关矩阵的现有RMT去噪法的基础上，在理论层面分析指出了KR方法是去噪原理最为合理和最有利于组合风险优化的RMT去噪法。对股票收益相关矩阵的特征向量最小扰动稳定性进行了数理推导，进而提出了KRMIN去噪法。KRMIN法吸纳了KR法的以相关矩阵特征向量稳定性的提高为核心的思想，弥补了KR法的原理和算法存在的缺陷，是一种更有利于改进股票投资组合风险的RMT去噪法。实证结果表明基于金融相关矩阵特征向量的Krzanowski稳定性的KR法和KRMIN法的组合风险优化效果好于其他RMT去噪法，且投资组合风险存在一种随收益相关矩阵特征向量的最小扰动稳定性提高而减小的趋势。
其次，基于RMT提出了对多元波动率模型去噪的一般方法，并通过定性分析和定量推导说明该方法对股票投资组合风险优化可能带来的好处。为了进一步验证该方法改进组合风险优化水平的有效性，建立了将基于RMT的相关矩阵估计和波动率结合在一起的两种多元波动率模型即SC-GARCH模型和IO-GARCH模型，并以这两种模型为去噪对象进行了股票投资组合风险优化的实证研究。结果表明RMT去噪法能对多元波动率模型的最佳维度进行正确的确定，从而实现股票组合风险的最优化。
再次，为了解决小组合风险优化条件下面向协方差矩阵的现有RMT去噪法因噪声特征值边界界定误差而产生的效力下降的问题，采用蒙特卡洛模拟法确定噪声特征值的边界，从而设计了蒙特卡洛RMT去噪法。通过实证分析方法，当股票收益序列长度和衰减因子不变时，在不同股票数量下对LCPB法、PG+法和KR法等已有RMT去噪法和蒙特卡洛RMT法的组合风险优化效果进行了对比研究。实证结果表明，在小组合条件下，蒙特卡洛RMT法能够弥补已有RMT方法的噪声特征值边界界定误差增大的缺陷，从而对已有RMT方法的组合风险优化作用的下降起到改进作用。
最后，以股票收益总体协方差矩阵和样本协方差矩阵特征谱矩的关系为理论基础，采用模拟退火算法估计总体协方差矩阵的特征值，进而提出了对样本协方差矩阵去噪的矩法。不同于已有的RMT去噪法，矩法并没有采取对噪声特征值替换的做法，而是通过估计总体协方差矩阵的特征值来实现协方差矩阵的降噪，从而引入了一种RMT去噪的新思路。通过构建toy模型，对矩法去噪效果进行了模拟研究。结果表明矩法对股票收益总体协方差矩阵特征值的估计误差一般都能被控制在小于10%的范围内，且矩法的去噪效果受到样本序列长度和样本协方差矩阵数量的影响。采用模拟方法，通过设定符合现实经济情况的总体协方差矩阵模型，在理论层面分析了矩法对股票投资组合风险优化水平的作用。结果表明矩法对股票投资组合风险的优化作用随噪声对组合风险影响的增大而提高，并且受到对总体协方差矩阵的部门数或特征值数的猜测的制约而存在饱和现象。使用bootstrap方法，在理想化和现实化条件下对矩法的组合风险优化效果进行了实证分析。结果表明，矩法的组合风险优化效果好于常用的RMT去噪法。
Markowitz Portfolio Theory is one of important pillars of modern financetheory， which is the investment theory giving maximum guidance for actualinvestments in securities. As the key input parameters of the Markowitz model,correlation and covariance matrices of historical security return rates contain a lotof noise, which will lead to the deterioration of portfolio risks by the increase ofportfolio risks and the decline of portfolio risk forecast accuracy. Because theimpact of noise on the securities portfolio risk increases with the increase in thenumber of securities, in the context of the present increasing trend of the number ofsecurities in the portfolio,the feasibility of the practical application of theMarkowitz model is gradually declining and even close to complete failure.Nowadays the effect of noise has become one of important reasons for anunprecedented increase of securities investment risks. How to reduce the impact ofnoise on portfolio risks is a research problem which urgently needs to be focusedon. More and more scholars regard denoising correlation matrices and covariancematrices of securities returns as a solution and carry out a detailed study.
Compared with other denoising methods of correlation matrices and covariancematrices, random matrix theory (RMT) filters have the advantages of deciding theoptimal model dimension, the low operation difficulty and a wide application range.However, the research on denoising correlation matrices and covariance matricesbased on RMT and applying RMT denoising in portfolio risk optimization is still ata relatively early stage, and there is a lot of work to be further carried out. Thispaper takes random matrix theory and modern portfolio theory as theoretical basis,uses many research techniques,such as Monte Carlo simulation method, toy model,simulated annealing algorithm, principal component analysis, general mathematicalannalysis, qualitative and quantitative analysis,etc., and deeply studies the keyquestions of adopting RMT denoising to achieve equity portfolio risk optimization,which not only helps to improve the denoising theory based on RMT, but also hasimportant theoretical significance for sweeping away the obstacles of applying theMarkowitz model to promote the further development of Markowitz portfoliotheory. Nowadays, as investment in the securities has become an important part ofglobal social and economic life, the study obviously has great practical significancetoo.
Start from interpreting the meaning of Markowitz portfolio risks, the paperexplains that the quality of portfolio risks should be measured by the size and forecast accuracy of portfolio risk, analyzes the impact of noise in historicalsecurity return correlation matrices and covariance matrices on the optimizationlevel of Markowitz stock portfolio risks, and selects RMT denoising as a solutionfor the noise effect. Then the paper constructs a theoretical framework ofoptimization of Markowitz stock portfolio risks based on RMT denoising whichconsists of four angles: First, equity portfolio risk optimization by improvingprinciples and algorithms of RMT denoising methods toward stock returncorrelation matrices. Second, equity portfolio risk optimization by establishing ageneral denoising method for multivariate volatility model which has not beeninvolved in recent research. Third, equity portfolio risk optimization by making upfor the disadvantage of the existing RMT denoising methods for sample covariancematrices of small portfolios. Fourth, equity portfolio risk optimization byestablishing a RMT denoising method toward sample covariance matrices based ona new denoising idea of estimating eigenvalues of population covariance matrices.From the above four angles, the paper includes the following research content andresults.
Firstly, on the basis of describing the existing RMT denoising methods towardsecurity return correlation matrices, the paper theoretically points out that amongthem the KR method is most reasonable and most beneficial to portfolio riskoptimization. After mathematically deducting the minimum perturbation of acertain eigenvector in a correlation matrix when the corresponding eigenvaluechanges, the paper proposes KRMIN denoising method. The KRMIN methodabsorbs the idea of the KR method focusing on improvement of eigenvectorstability of a correlation matrix and compensates the defects of the KR method inthe principles and algorithms. Therefore The KRMIN method is more conducive toequity portfolio risk optimization in contrast with other methods. The empiricalresults show that: the portfolio risk optimization effect of KR and KRMINconsidering Krzanowski stability of eigenvectors of earnings correlation matrices issuperior to other RMT methods. Portfolio risks have a decreasing tend with theimprovement of stability measured by minimum perturbation of correlation matrixeigenvectors.
Secondly, the paper presents a RMT denoising method for multivariate volatilitymodel and demonstrates its benefits to stock portfolio risk optimization throughqualitative analysis and quantitative derivation. In order to demonstrate theeffectiveness of the method for portfolio risk optimization, this paper constructstwo classes of multivariate volatility models combining the volatility process withthe correlation estimate, SC-GARCH model and IO-GARCH model. Taking the twokinds of models as denoising objects, an empirical study of stock portfolio risk optimization is carried out. It is shown in the experiment that the RMT denoisingmethod can precisely determine the optimal dimension of multivariate volatilitymodels and thus produce the best stock portfolio risk.
Thirdly, in order to solve the decline of effectiveness of existing RMT denoisingmethods toward covariance matrices of small portfolios due to the error indetermining the noise eigenvalue boundary, the Monte Carlo method is used todetermine the noise eigenvalue boundary and then the RMT denoising methodbased on Monte Carlo simulation is designed. Through empirical approach, whenstock return sequence length and attenuation factor are unchanged, the papercompares portfolio risk optimization effect of existing RMT denoising methodssuch as LCPB, PG+and KR etc and the RMT denoising method based on MonteCarlo simulation in a varying number of shares. Experimental results show thatunder the conditions of small portfolios the RMT method based on Monte Carlosimulation can cover the existing RMT denoising methods’inferiority indetermining the noise eigenvalue boundary and thus hinder the decline of their rolein risk improvement.
Finally, the paper takes the exact relation between the eigenvalue spectrummoments of a population covariance matrix and those of its estimator as theoreticalbasis, uses the simulated annealing algorithm to estimate eigenvalues of thepopulation covariance matrix and thereby presents the moment method used fordenoising sample covariance matrices.Unlike existing RMT denoising methods, themoment method does not replace noisy eigenvalues, but estimates eigenvalues ofthe population covariance matrix to denoise sample covariance matrices, therebyintroducing a new way of RMT denoising. Through a toy model, a simulation studyon denoising effect of the moment method is conducted. The results prove that byuse of the moment method, the error in the estimation of eigenvalues of stockreturns population covariance matrices can generally be controlled in the range ofless than10%and the denoising effect of the moment method is influenced by thesample sequence length and the number of sample covariance matrices. By settingthe population covariance matrix model in line with economic reality, the paperadopts the simulation method to theoretically analyze the effect of the momentmethod on the optimization level of stock portfolio risks. The results show that theoptimizing effect of the moment method on stock portfolio risks improves when theimpact of noise increases and the saturated phenomenon of the moment methodoccurs because of constraints of guess on the department number or the eigenvaluenumber of the population covariance matrix. Under idealized and realisticconditions, by using the bootstrap method the paper conducts empirical analysis ofoptimizing effect of the moment method on portfolio risks. It turns out that the optimizing effect of the moment method on portfolio risks is better than thefrequently used RMT denoising method.
和针对金融相关矩阵和协方差矩阵的其他去噪方法相比，基于随机矩阵理论的去噪法（RMT去噪法）具有决定模型最佳维度、操作技术难度低和适用范围广泛等优点。然而，基于RMT相关或多元波动率的去噪及其在股票投资组合风险优化中的应用的研究目前尚处于较为初级的阶段，有很多研究工作有待于进一步开展。以随机矩阵理论和现代投资组合理论为理论基础，采用蒙特卡洛模拟法、toy模型法、模拟退火算法、主成分分析法、一般数理分析法、理论研究和实证研究相结合等多种研究方法深入研究应用RMT去噪法实现股票投资组合风险优化的问题不仅有助于完善RMT去噪理论，而且对扫除Markowitz模型的应用障碍，从而促进Markowitz证券投资组合理论的进一步发展有重要的理论意义。在证券投资已经成为全球社会经济生活的一个重要构成部分的今天，该研究明显也具有很强的现实意义。
本文从阐释Markowitz股票投资组合风险的含义入手，说明了以投资组合风险的大小和投资组合风险预测的准确率衡量投资组合风险的优劣，分析了股票收益历史相关矩阵和协方差矩阵的噪声对Markowitz投资组合风险优化水平的影响，并选取RMT去噪法作为解决噪声影响的方法，进而构建了应用RMT去噪法实现Markowitz股票投资组合风险优化的整体思路框架，其包括四个角度：第一，通过对面向股票收益相关矩阵的RMT去噪法原理和算法的改进实现股票投资组合风险优化；第二，通过建立对已有去噪研究未曾涉及的股票收益率多元波动率模型去噪的一般方法实现股票投资组合风险优化；第三，通过弥补面向收益样本协方差矩阵的现有RMT去噪法在小组合条件下的缺陷实现股票投资组合风险优化；第四，根据对总体协方差矩阵特征值进行估计的新去噪思路建立面向收益样本协方差矩阵的RMT去噪法来实现股票投资组合风险的优化。从这四个角度出发，论文的主要研究内容和成果包括如下方面。
首先，在介绍面向股票收益相关矩阵的现有RMT去噪法的基础上，在理论层面分析指出了KR方法是去噪原理最为合理和最有利于组合风险优化的RMT去噪法。对股票收益相关矩阵的特征向量最小扰动稳定性进行了数理推导，进而提出了KRMIN去噪法。KRMIN法吸纳了KR法的以相关矩阵特征向量稳定性的提高为核心的思想，弥补了KR法的原理和算法存在的缺陷，是一种更有利于改进股票投资组合风险的RMT去噪法。实证结果表明基于金融相关矩阵特征向量的Krzanowski稳定性的KR法和KRMIN法的组合风险优化效果好于其他RMT去噪法，且投资组合风险存在一种随收益相关矩阵特征向量的最小扰动稳定性提高而减小的趋势。
其次，基于RMT提出了对多元波动率模型去噪的一般方法，并通过定性分析和定量推导说明该方法对股票投资组合风险优化可能带来的好处。为了进一步验证该方法改进组合风险优化水平的有效性，建立了将基于RMT的相关矩阵估计和波动率结合在一起的两种多元波动率模型即SC-GARCH模型和IO-GARCH模型，并以这两种模型为去噪对象进行了股票投资组合风险优化的实证研究。结果表明RMT去噪法能对多元波动率模型的最佳维度进行正确的确定，从而实现股票组合风险的最优化。
再次，为了解决小组合风险优化条件下面向协方差矩阵的现有RMT去噪法因噪声特征值边界界定误差而产生的效力下降的问题，采用蒙特卡洛模拟法确定噪声特征值的边界，从而设计了蒙特卡洛RMT去噪法。通过实证分析方法，当股票收益序列长度和衰减因子不变时，在不同股票数量下对LCPB法、PG+法和KR法等已有RMT去噪法和蒙特卡洛RMT法的组合风险优化效果进行了对比研究。实证结果表明，在小组合条件下，蒙特卡洛RMT法能够弥补已有RMT方法的噪声特征值边界界定误差增大的缺陷，从而对已有RMT方法的组合风险优化作用的下降起到改进作用。
最后，以股票收益总体协方差矩阵和样本协方差矩阵特征谱矩的关系为理论基础，采用模拟退火算法估计总体协方差矩阵的特征值，进而提出了对样本协方差矩阵去噪的矩法。不同于已有的RMT去噪法，矩法并没有采取对噪声特征值替换的做法，而是通过估计总体协方差矩阵的特征值来实现协方差矩阵的降噪，从而引入了一种RMT去噪的新思路。通过构建toy模型，对矩法去噪效果进行了模拟研究。结果表明矩法对股票收益总体协方差矩阵特征值的估计误差一般都能被控制在小于10%的范围内，且矩法的去噪效果受到样本序列长度和样本协方差矩阵数量的影响。采用模拟方法，通过设定符合现实经济情况的总体协方差矩阵模型，在理论层面分析了矩法对股票投资组合风险优化水平的作用。结果表明矩法对股票投资组合风险的优化作用随噪声对组合风险影响的增大而提高，并且受到对总体协方差矩阵的部门数或特征值数的猜测的制约而存在饱和现象。使用bootstrap方法，在理想化和现实化条件下对矩法的组合风险优化效果进行了实证分析。结果表明，矩法的组合风险优化效果好于常用的RMT去噪法。
Markowitz Portfolio Theory is one of important pillars of modern financetheory， which is the investment theory giving maximum guidance for actualinvestments in securities. As the key input parameters of the Markowitz model,correlation and covariance matrices of historical security return rates contain a lotof noise, which will lead to the deterioration of portfolio risks by the increase ofportfolio risks and the decline of portfolio risk forecast accuracy. Because theimpact of noise on the securities portfolio risk increases with the increase in thenumber of securities, in the context of the present increasing trend of the number ofsecurities in the portfolio,the feasibility of the practical application of theMarkowitz model is gradually declining and even close to complete failure.Nowadays the effect of noise has become one of important reasons for anunprecedented increase of securities investment risks. How to reduce the impact ofnoise on portfolio risks is a research problem which urgently needs to be focusedon. More and more scholars regard denoising correlation matrices and covariancematrices of securities returns as a solution and carry out a detailed study.
Compared with other denoising methods of correlation matrices and covariancematrices, random matrix theory (RMT) filters have the advantages of deciding theoptimal model dimension, the low operation difficulty and a wide application range.However, the research on denoising correlation matrices and covariance matricesbased on RMT and applying RMT denoising in portfolio risk optimization is still ata relatively early stage, and there is a lot of work to be further carried out. Thispaper takes random matrix theory and modern portfolio theory as theoretical basis,uses many research techniques,such as Monte Carlo simulation method, toy model,simulated annealing algorithm, principal component analysis, general mathematicalannalysis, qualitative and quantitative analysis,etc., and deeply studies the keyquestions of adopting RMT denoising to achieve equity portfolio risk optimization,which not only helps to improve the denoising theory based on RMT, but also hasimportant theoretical significance for sweeping away the obstacles of applying theMarkowitz model to promote the further development of Markowitz portfoliotheory. Nowadays, as investment in the securities has become an important part ofglobal social and economic life, the study obviously has great practical significancetoo.
Start from interpreting the meaning of Markowitz portfolio risks, the paperexplains that the quality of portfolio risks should be measured by the size and forecast accuracy of portfolio risk, analyzes the impact of noise in historicalsecurity return correlation matrices and covariance matrices on the optimizationlevel of Markowitz stock portfolio risks, and selects RMT denoising as a solutionfor the noise effect. Then the paper constructs a theoretical framework ofoptimization of Markowitz stock portfolio risks based on RMT denoising whichconsists of four angles: First, equity portfolio risk optimization by improvingprinciples and algorithms of RMT denoising methods toward stock returncorrelation matrices. Second, equity portfolio risk optimization by establishing ageneral denoising method for multivariate volatility model which has not beeninvolved in recent research. Third, equity portfolio risk optimization by making upfor the disadvantage of the existing RMT denoising methods for sample covariancematrices of small portfolios. Fourth, equity portfolio risk optimization byestablishing a RMT denoising method toward sample covariance matrices based ona new denoising idea of estimating eigenvalues of population covariance matrices.From the above four angles, the paper includes the following research content andresults.
Firstly, on the basis of describing the existing RMT denoising methods towardsecurity return correlation matrices, the paper theoretically points out that amongthem the KR method is most reasonable and most beneficial to portfolio riskoptimization. After mathematically deducting the minimum perturbation of acertain eigenvector in a correlation matrix when the corresponding eigenvaluechanges, the paper proposes KRMIN denoising method. The KRMIN methodabsorbs the idea of the KR method focusing on improvement of eigenvectorstability of a correlation matrix and compensates the defects of the KR method inthe principles and algorithms. Therefore The KRMIN method is more conducive toequity portfolio risk optimization in contrast with other methods. The empiricalresults show that: the portfolio risk optimization effect of KR and KRMINconsidering Krzanowski stability of eigenvectors of earnings correlation matrices issuperior to other RMT methods. Portfolio risks have a decreasing tend with theimprovement of stability measured by minimum perturbation of correlation matrixeigenvectors.
Secondly, the paper presents a RMT denoising method for multivariate volatilitymodel and demonstrates its benefits to stock portfolio risk optimization throughqualitative analysis and quantitative derivation. In order to demonstrate theeffectiveness of the method for portfolio risk optimization, this paper constructstwo classes of multivariate volatility models combining the volatility process withthe correlation estimate, SC-GARCH model and IO-GARCH model. Taking the twokinds of models as denoising objects, an empirical study of stock portfolio risk optimization is carried out. It is shown in the experiment that the RMT denoisingmethod can precisely determine the optimal dimension of multivariate volatilitymodels and thus produce the best stock portfolio risk.
Thirdly, in order to solve the decline of effectiveness of existing RMT denoisingmethods toward covariance matrices of small portfolios due to the error indetermining the noise eigenvalue boundary, the Monte Carlo method is used todetermine the noise eigenvalue boundary and then the RMT denoising methodbased on Monte Carlo simulation is designed. Through empirical approach, whenstock return sequence length and attenuation factor are unchanged, the papercompares portfolio risk optimization effect of existing RMT denoising methodssuch as LCPB, PG+and KR etc and the RMT denoising method based on MonteCarlo simulation in a varying number of shares. Experimental results show thatunder the conditions of small portfolios the RMT method based on Monte Carlosimulation can cover the existing RMT denoising methods’inferiority indetermining the noise eigenvalue boundary and thus hinder the decline of their rolein risk improvement.
Finally, the paper takes the exact relation between the eigenvalue spectrummoments of a population covariance matrix and those of its estimator as theoreticalbasis, uses the simulated annealing algorithm to estimate eigenvalues of thepopulation covariance matrix and thereby presents the moment method used fordenoising sample covariance matrices.Unlike existing RMT denoising methods, themoment method does not replace noisy eigenvalues, but estimates eigenvalues ofthe population covariance matrix to denoise sample covariance matrices, therebyintroducing a new way of RMT denoising. Through a toy model, a simulation studyon denoising effect of the moment method is conducted. The results prove that byuse of the moment method, the error in the estimation of eigenvalues of stockreturns population covariance matrices can generally be controlled in the range ofless than10%and the denoising effect of the moment method is influenced by thesample sequence length and the number of sample covariance matrices. By settingthe population covariance matrix model in line with economic reality, the paperadopts the simulation method to theoretically analyze the effect of the momentmethod on the optimization level of stock portfolio risks. The results show that theoptimizing effect of the moment method on stock portfolio risks improves when theimpact of noise increases and the saturated phenomenon of the moment methodoccurs because of constraints of guess on the department number or the eigenvaluenumber of the population covariance matrix. Under idealized and realisticconditions, by using the bootstrap method the paper conducts empirical analysis ofoptimizing effect of the moment method on portfolio risks. It turns out that the optimizing effect of the moment method on portfolio risks is better than thefrequently used RMT denoising method.
引文
[1]刘国芳.从证券化率看宏观经济管理[J].经济与管理研究,2009(7):25-30.
[2]戴志辉.证券公司股票投资风险的计量、预测和控制[D].西安:西北大学国民经济学学科博士学位论文,2007:3-5.
[3]肖新华,刘冬荣.个人股票投资风险的成因与防范分析[J].财经问题研究,2009(2):73-76.
[4] Muirhead R J.Aspects of Multivariate Statistical Theory[M]. New York:Wiley,1982:74-69.
[5] Efetov K B.Supersymmetry and the Theory of Disordered Metals[J].Adv.Phys.,1983(32):53-76.
[6] Efetov K B.Level Spacing Distribution in Small Metallic Particles[J]. Phys. G:Nucl. Phys.,1982(15): L909.
[7] Efetov K B.Supersymmetric Method in the Localization Theory[J].Zh. Eksp.Teor. Fiz.,1982,82:872-890.
[8] Hejhal D A. On the Triple Correlation of Zeros of the Zeta Function[J]. Int.Math. Res. Not.,1994(7):293-323.
[9] Bogomolny E B,Keating J P.Random Matrix Theory and the Riemann Zeros I:Three-and-Four-Point Correlations[J]. Nonlinearity,1995(8):1115-1143.
[10] Katz N M,Sarnak P.Random Matrices, Frobenius Eigenvalues andMonodromy[M].Providence, RI:American Mathematical Society,1999:156-191.
[11] Conrey J B,Farmer D W,Keating J P.Autocorrelation of Random MatrixPolynomials[J].Communications in Mathematical Physics,2003,237(3):365-395.
[12] Keating J P,Linden N,Rudnik Z.Random Matrix Theory, the Exceptional LieGroups and L-functions[J].Phys. A:Math. Gen.,2003(36):33-44.
[13] Keating J P,Snaith N C. Random Matrix Theory and ζ(1/2+it)[J]. Commun.Math. Phys.,2000(214):57-73.
[14] Keating J P,Snaith N C.Random Matrix Theory and L-functions at s=1/2[J].Commun. Math. Phys.,2000(214):91-114.
[15] Wegner F J.Algebraic Derivation of Symmetry-relations for DisorderedElectronic Systems[J]. Z. Phys. B,1983(49):297-302.
[16] Efetov K B.Supersymmetry in Disorder and Chaos[M].Cambridge: CambridgeUniversity Press,1997:56-79.
[17] Brody T A,Flores J,French J B,et al.Random-matrix Physics:Spectrum andStrength Fluctuations[J].Rev. Mod. Phys,1981(53):385-412.
[18] Webb R A,Washburn S,Umbach C P,et al.Observation of H/E Ahoronov–Bohm Oscillations in Normal-Metal Rings[J]. Phys. Rev. Lett.,1985(54):269-302.
[19] Eguchi T,Kawai H. Reduction of Dynamical Degrees of Freedom in the LargeN Gauge Theory[J]. Phys.Rev. Lett.,1982(48):106-145.
[20] Forrester P J. The Spectrum Edge of Random Matrix Ensembles[J].Nucl.Phys.B,1993(402):709-747.
[21] Zirnbauer M R.Riemannian Symmetric Superspaces and Their Origin inRandom-Matrix Theory[J].J. Math. Phys.,1996,37:498-523.
[22] Kazakov V A, Migdal A A. Induced QCD at Large N[J]. Nucl. Phys. B,1993(397):214-245.
[23] Douglas M R,Shenker S H. Strings in Less Than One-Dimension[J]. NuclearPhysics B,1990,335(3):635-654.
[24] Jimbo M,Miwa T,Mori Y,et al.Density Matrix of an Impenetrable Bose Gasand the Fifth Painleve Transcendent[J]. Physica D: Nonlinear Phenomena,1980,1(1):80-158.
[25] Tracy C A,Widom H.Level Spacing Distributions and the Airy Kernel[J].Communications in Mathematical Physics,1994,159(1):151-174.
[26] Berry M V.Semiclassical Theory of Spectral Rigidity[J].Proc. R. Soc. A,1985,400(1819):229-251.
[27] Laloux L, Cizeau P, Bouchaud J, et al. Noise Dressing of FinancialCorrelation Matrices[J]. Phys. Rev. Lett.,1999,83(7):1467-1470.
[28] Laloux L, Cizeau P, Potters M, et al. Random Matrix Theory and FinancialCorrelations[J]. Int. J. Theor. Appl. Finance,2000,3(3):391-397.
[29] Potters M, Bouchaud J, Laloux L. Financial Applications of Random MatrixTheory: Old Laces and New Pieces[C/OL].（2005-07-14）[2012-03-11].http://arxiv.org/abs/physics/0507111.
[30] Plerou V, Gopikrishnan P, Rosenow B, et al.Universal and NonuniversalProperties of Cross Correlations in Financial Time Series[J]. Phys. Rev.Lett.,1999,83(7):1471-1474.
[31] Plerou V, Gopikrishnan P, Rosenow B, et al. A Random Matrix TheoryApproach to Financial Cross-Correlations[J]. Physica A,2000,287(3-4):374-382.
[32] Stanley H E, Plerou V. Quantifying Economic Fluctuations[J]. Physica A,2001,302:126-137.
[33] Stanley H E, Plerou V. Quantifying Fluctuations in Economic Systems byAdapting Methods of Statistical Physics[J]. Physica A,2000,287:339-361.
[34] Plerou V, Gopikrishnan P, Rosenow B, et al. Collective Behaviour of StockPrice Movements-a Random Matrix Theory Approach[J].Physica A,2001,299:175-180.
[35] Rosenow B, Plerou V. Application of Random Matrix Theory to Study Cross-Correlations of Stock Prices[J]. International Journal of Theoretical andApplied Finance,2000,3(3):399-403.
[36] Plerou V, Gopikrishnan P, Rosenow B, et al. Econophysics:Financial TimeSeries from a Statistical Physics Point of View[J]. Physica A,2000,279:443-456.
[37] Lee Y H.Noise Detection from Financial Correlation Matrices[R/OL].(2001)[2012-04-11]. http://web.mit.edu/bazant/www/teach/18.325.
[38] Sadik C. Cross Correlations in an Emerging Market Financial Data[J].Physica A,2007,376:555-564.
[39] Utsugi A, Ino K, Oshikawa M. Random Matrix Theory Analysis of CrossCorrelations in Financial Markets[J]. Phys. Rev. E,2004,70(2):1-5.
[40] Nilantha K, Ranasinghe, Malmini P. Eigenvalue Density of Cross-correlationsin Sri Lankan Financial Market[J]. Physica A,2007,378(2):345-356.
[41] Wilcox D,Gebbie T. An Analysis of Cross-correlations in an EmergingMarket[J].Physica A,2007,375(2):584-598.
[42]李良新.中国股票市场有效性的随机矩阵实证分析[J].现代商业,2009(5):30-31.
[43] Conlon T, Ruskin H J,Crane M. Random Matrix Theory and Fund of FundsPortfolio Optimisation[J]. Physica A,2007,382(2):565-576.
[44] Malevergne Y,Sornette D. Collective Origin of the Coexistence of ApparentRandom Matrix Theory Noise and of Factors in Large Sample CorrelationMatrices[J]. Physica A,2004,331(3-4):660-668.
[45] Pafka S, Potters M,Kondor I. Exponential Weighting and Random-matrix-theory-based Denoising of Financial Covariance Matrices for PortfolioOptimization[J/OL].(2004-02-24)[2012-04-25]. http://eprintweb.org/S/article/cond-mat/0402573/cited.
[46] Wilcox D, Gebbie T. On the Analysis of Cross-correlations in South AfricanMarket Data[J]. Physica A,2004,344:294-298.
[47] Coronnello C. Sector Identification in a Set of Stock Return Time SeriesTraded at the London Stock Exchange[J].Acta Physica Polonica B,2005,36:2653-2676.
[48] Garas A, Argyrakis P. Correlation Study of the Athens Stock Exchange[J].Physica A,2007,380(1-2):399-410.
[49]郑波.金融动力学的时空关联与大波动特性——兼谈中西方金融市场的对比研究[J].物理,2010(2):95-100.
[50] Pafka S,Kondor I.Estimated Correlation Matrices and Portfolio Optimization[J]. Physica A,2004,343(1-4):623-634.
[51] Letchford A,Gao J,Zheng L.Filtering Financial Time Series by Least Squares[J].International Journal of Machine Learning and Cybernetics,2013,4(2):140-154.
[52] Hassani H,Dionisio A,Ghodsi M.The Effect of Noise Reduction in Measuringthe Linear and Nonlinear Dependency of Financial Markets[J]. NonlinearAnalysis:Real World Applications,2010,11(1):492-502.
[53] Lu C J, Lee T S, Chiu C C. Financial Time Series Forecasting UsingIndependent Component Analysis and Support Vector Regression[J]. DecisionSupport Systems,2009,47(2):115-125.
[54] Kang W, Kim K, Shin H. Denoising Monte Carlo Sensitivity Estimates[J].Operations Research Letters,2012,40(3):195-202.
[55] Neaime S.The Global Financial Crisis, Financial Linkages and Correlations inReturns and Volatilities in Emerging MENA Stock Markets[J].EMERGINGMARKETS REVIEW,2012,13(3):268-282.
[56] Elton E J, Gruber M J.Modern Portfolio Theory and Investment Analysis[M].New York: Wiley,1995:34-78.
[57] Chopra V K, Ziemba W T, Portf J. The Effect of Errors in Mean,Variances,andCovariance on Optimal Portfolio Choice[J]. The Journal of PortfolioManagement,1993,19(2):6-11.
[58] Livan G, Rebecchi L. Asymmetric Correlation Matrices:An Analysis ofFinancial Data[J]. European Physical Journal B,2012,85(6):1-11.
[59] Giacomo L,Simone A,Enrico S. Fine Structure of Spectral Properties forRandom Correlation Matrices: An Application to Financial Markets[J].Physical Review E,2011,84(1):1-13.
[60] Drozdz S, Kwapien J, Oswiecimka P. Empirics Versus RMT in FinancialCross-correlations[J]. Acta Physica Polonica B,2007,38(13):4027-4039.
[61] Akansu A N, Torun M U. Toeplitz Approximation to Empirical CorrelationMatrix of Asset Returns: A Signal Processing Perspective[J]. IEEE Journal onSelected Topics in Signal Processing,2012,6(4):319-326.
[62] Vincenzo T, Fabrizio L. Cluster Analysis for Portfolio Optimization[J].Journalof Economic Dynamics&Control,2008,32(1):235-258.
[63] Plerou V, Gopikrishnan P, Rosenow B, et al.Random Matrix Approach toCross Correlations in Financial Data[J]. Phys. Rev.E,2002,65(6):1-6.
[64] Sharifi S, Crane M, Shamaie A, et al. Random Matrix Theory for PortfolioOptimization: a Stability Approach[J]. Physica A,2004,335(3-4):629-643.
[65] Krzanowski.Sensitivity of Principal Components[J].J. Royal Stats. Soc. B,1984,46(3):558-563.
[66] Daly J, Crane M, Ruskin H J. Random Matrix Theory Filters in PortfolioOptimisation:A Stability and Risk Assessment[J].Physica A,2008,387(16):4248-4260.
[67]史宇峰.金融资产收益相关矩阵选取方法研究[J].证券市场导报,2008(5):65-68.
[68] Jiang J,Ma K,Cai X. Scaling and Correlations in Foreign Exchange Market[J].Physica A,2007,375(1):274-280.
[69] Jiang J,Ma K,Cai X. Non-linear Characteristics and Long-range Correlationsin Asian Stock Markets [J]. Physica A,2007,378(2):399-407.
[70]都国雄.经济物理学的研究内容与发展趋势[J].现代经济探讨,2006(4):47-49.
[71] Reese C, Rosenow B. Predicting Multivariate Volatility[J/OL].(2003-04-03)[2002-04-15]. http://arxiv.org/abs/cond-mat/0304082.
[72] Muirhead R B.Aspects of Multivariate Statistical Theory[M]. NewYork:Wiley,1982:45-65.
[73]王明进.高维波动率的预测[J].数量经济技术经济研究,2008(11):137-148.
[74]刘志东.多元GARCH模型结构特征、参数估计与假设检验研究综述[J].数量经济技术经济研究,2010(9):147-160.
[75]李文君,尹康.多元GARCH模型研究述评[J].数量经济技术经济研究,2009(10):138-147.
[76] Bauwens L, Laurent S, Rombouts J V K.Multivariate GARCH Models:aSurvey[J]. Journal of Applied Econometrics,2006,21(1):79-109.
[77] Pafka S,Kondor I.Noisy Covariance Matrices and Portfolio Optimization ii[J].Physica A,2002,319:487-494.
[78] Burda Z, Jurkiewicz J, Nowak M A. Is Econophysics a Solid Science[J].ActaPhys.Pol. B,2003,34(1):87-131.
[79] Kondor I, Pafka S,Nagy G. Noise Sensitivity of Portfolio Selection underVarious Risk Measures[J]. J. Banking Finance,2007,31(5):1545-1573.
[80] Varga-Haszonits I, Kondor I. Noise Sensitivity of Portfolio Selection inConstant Conditional Correlation GARCH Models[J].Physica A,2007,385:307-318.
[81] Jens S. The Asymptotic Spectrum of the EWMA Covariance Estimator[J].Physica A,2007,385(2):621-630.
[82] Eun C,Resnick B. Estimating the Correlation Structure of International SharePrices[J]. Journal of Finance,1984,39:1311-1324.
[83] Chan L, Karceski J, Lakonishok J. On Portfolio Optimization: ForecastingCovariances and Choosing the Risk Model[J]. Reviews of FinancialStudies,1999,12:937-974.
[84]王明进.多元波动率模型的一些新进展[J].数理统计与管理,2010,29(2):233-245.
[85] Ledoit O, Wolf M.A Well-conditioned Estimator for Large-dimensionalCovariance Matrices[J]. J. Multivar. Anal.,2004,88(2):365-411.
[86] Ledoit O, Wolf M. Improved Estimation of the Covariance Matrix of StockReturns with an Application to Portfolio Selection[J].J. Empirical Finance,2003,10(5):603-621.
[87] Ledoit O, Wolf M.Honey,I Shrunk the Sample Covariance Matrix[J]. TheJournal of Portfolio Management,2004,30(4):110-119.
[88] Burda Z, Goerlich A,Jarosz A.Signal and Noise in Correlation Matrix[J].Phys.A,2004,343:295-308.
[89] Bollerslev T.Modelling the Coherence in Short-run Nominal Exchange Rates:a Multivariate Generalized ARCH Model[J].Rev. Econ.Stat.,1990,72:498-505.
[90] Burda Z, Jarosz A, Jurkiewicz J.Applying Free Random Variables to RandomMatrix Analysis of Financial Data[J/OL].(2006-03-04)[2012-04-28].http://cdsweb.cern.ch/record/933639/.
[91] Daly J, Crane M,Ruskin H J.Random Matrix Theory Filters and CrrencyPortfolio Otimisation[J].Journal of Physics:Conference Series,2010,221(1):4-8.
[92]向骏文.基于马科维茨投资组合理论的股票投资策略[J].中国证券期货,2012,20(8):202.
[93]刘志东,薛莉.金融市场高维波动率的扩展广义正交GARCH模型与参数估计方法研究[J].中国管理科学,2010,18(6):33-41.
[94] Frei M, Kuensch H R. Sequential State and Observation Noise CovarianceEstimation Using Combined Ensemble Kalman and Particle Filters[J].Monthly Weather Review,2012,140(5):1476-1495.
[95] Menon R, Gerstoft P,Hodgkiss W S. Asymptotic Eigenvalue Density of NoiseCovariance Matrices[J]. IEEE Transactions on Signal Processing,2012,60(7):3415-3424.
[96] Bandiera F, Besson O, Ricci G.Knowledge-Aided Covariance MatrixEstimation and Adaptive Detection in Compound-Gaussian Noise [J].IEEETransactions on Signal Processing,2010,58(10):5390-5396.
[97] Mcaleer M, Medeiros M C.Realized Volatility:A Review[J].EconometricReviews,2008,27(1-3):10-45.
[98] Bandi F M,Russell J R.Separating Microstructure Noise from Volatility[J].Journal of Financial Economics,2006,79(3):655-692.
[99] Ghysels E,Sinko A.Volatility Forecasting and Microstructure Noise[J].Journal of Econometrics,2011,160(1):257-271.
[100] Boozer A D. The Role of Spatial Topology in a Toy Model of ClassicalElectrodynamics in (1+1) Dimensions[J]. Physics Letters A,2010,374(19-20):1901-1908.
[101] Eelbode D, Van de Voorde L.A Toy Model for Higher Spin DiracOperators[J]. Physics of Atomic Nuclei,2010,73(2):282-287.
[102] Kumar J.A Toy Model for Gauge-mediation in Intersecting Brane Models[J].Physical Review D,2009,79(6):1-6.
[103] Dimov I I, Kolm P N, Maclin L. Hidden Noise Structure and Random MatrixModels of Stock Correlations[J]. Quantitative Finance,2012,12(4):567-572.
[104] Martins A C R.Non-stationary Correlation Matrices and Noise[J].Physica A-Statistical Mechanics and its Applications,2007,379(2):552-558.
[105] Menon R, Gerstoft P, Hodgkiss W S.Asymptotic Eigenvalue Density of NoiseCovariance Matrices[J]. IEEE Transactions on Signal Processing,2012,60(7):3415-3424.
[106] Henker T, Husodo Z A. Noise and Efficient Variance in the Indonesia StockExchange[J].Pacific-basin Finance Journal,2010,18(2):199-216.
[107] Matteson D S, Ruppert D. Time-Series Models of Dynamic Volatility andCorrelation A Tutorial on Univariate and Multivariate Cases[J]. IEEE SignalProcessing Magazine,2011,28(5):72-82.
[108] Poon S H, Granger C W J. Forecasting Volatility in Financial Markets: AReview[J]. Journal of Economic Literature,2003,41(2):478-539.
[109] Olivier L,Michael W. Improved Estimation of the Covariance Matrix of StockReturns with an Application to Portfolio Selection[J]. Journal of EmpiricalFinance,2003(10):603-621.
[110] Sengupta A M, Mitra P P. Distributions of Singular Values for Some RandomMatrices[J]. Physical Review E,1999,60(3):3389-3392.
[111]王林,张蕾,刘连峰.用模拟退火算法寻找Heston期权定价模型参数[J].数量经济技术经济研究,2011(9):131-139.
[112]任世科,何正文,徐渝.基于联合视角的项目支付问题及其模拟退火启发式算法[J].系统工程,2012,30(6):82-89.
[113]罗大为.股票收益率影响因素的实证分析[J].人民论坛,2011(2):152-153.
[114]林立子,陈希敏.后危机时代我国股票收益率影响因素的实证研究[J].西安财经学院学报,2010,23(5):14-17.
[115]王大伟,王雪标.影响我国股票收益率的多因素实证研究—基于panel data模型分析[J].东北财经大学学报,2008(5):18-23.
[116]屠年松,徐光远.中国股票市场质量的国际比较[J].云南社会科学,2012(5):123-127.
[117]李进芳,王仁曾.Bootstrap方法在证券投资基金风险测量中的应用[J].统计研究,2010,27(3):66-69.
[118]张萍.基于Bootstrap方法的统计分析[J].宜宾学院学报,2011,11(12):31-33.
[119]徐正国,张世英.多维高频数据的“已实现”波动建模研究[J].系统工程学报,2006,21(1):6-11.
[120] Alexander C O. Orthogonal GARCH [J].Mastering Risk,2001(2):1-38.
[121]高岳,王家华,杨爱军.具有时变自由度的t-copula蒙特卡罗组合收益风险研究[J].中国管理科学,2011,19(2):10-15.
[122]刘金全,李楠,郑挺国.随机波动模型的马尔可夫链-蒙特卡洛模拟方法—在沪市收益率序列上的应用[J].数理统计与管理,2010,29(6):1026-1035.
[123]盛世明.基于连接函数的VaR的蒙特卡洛算法[J].统计与决策,2010(21):27-29.
[124] Davar K.Probability[M].America:American Mathematical Society,2007:60-67.
[125]张京.非负约束条件下组合证券投资决策的分枝定界法[J].数学的实践与认识,2004,34(5):18-23.
[2]戴志辉.证券公司股票投资风险的计量、预测和控制[D].西安:西北大学国民经济学学科博士学位论文,2007:3-5.
[3]肖新华,刘冬荣.个人股票投资风险的成因与防范分析[J].财经问题研究,2009(2):73-76.
[4] Muirhead R J.Aspects of Multivariate Statistical Theory[M]. New York:Wiley,1982:74-69.
[5] Efetov K B.Supersymmetry and the Theory of Disordered Metals[J].Adv.Phys.,1983(32):53-76.
[6] Efetov K B.Level Spacing Distribution in Small Metallic Particles[J]. Phys. G:Nucl. Phys.,1982(15): L909.
[7] Efetov K B.Supersymmetric Method in the Localization Theory[J].Zh. Eksp.Teor. Fiz.,1982,82:872-890.
[8] Hejhal D A. On the Triple Correlation of Zeros of the Zeta Function[J]. Int.Math. Res. Not.,1994(7):293-323.
[9] Bogomolny E B,Keating J P.Random Matrix Theory and the Riemann Zeros I:Three-and-Four-Point Correlations[J]. Nonlinearity,1995(8):1115-1143.
[10] Katz N M,Sarnak P.Random Matrices, Frobenius Eigenvalues andMonodromy[M].Providence, RI:American Mathematical Society,1999:156-191.
[11] Conrey J B,Farmer D W,Keating J P.Autocorrelation of Random MatrixPolynomials[J].Communications in Mathematical Physics,2003,237(3):365-395.
[12] Keating J P,Linden N,Rudnik Z.Random Matrix Theory, the Exceptional LieGroups and L-functions[J].Phys. A:Math. Gen.,2003(36):33-44.
[13] Keating J P,Snaith N C. Random Matrix Theory and ζ(1/2+it)[J]. Commun.Math. Phys.,2000(214):57-73.
[14] Keating J P,Snaith N C.Random Matrix Theory and L-functions at s=1/2[J].Commun. Math. Phys.,2000(214):91-114.
[15] Wegner F J.Algebraic Derivation of Symmetry-relations for DisorderedElectronic Systems[J]. Z. Phys. B,1983(49):297-302.
[16] Efetov K B.Supersymmetry in Disorder and Chaos[M].Cambridge: CambridgeUniversity Press,1997:56-79.
[17] Brody T A,Flores J,French J B,et al.Random-matrix Physics:Spectrum andStrength Fluctuations[J].Rev. Mod. Phys,1981(53):385-412.
[18] Webb R A,Washburn S,Umbach C P,et al.Observation of H/E Ahoronov–Bohm Oscillations in Normal-Metal Rings[J]. Phys. Rev. Lett.,1985(54):269-302.
[19] Eguchi T,Kawai H. Reduction of Dynamical Degrees of Freedom in the LargeN Gauge Theory[J]. Phys.Rev. Lett.,1982(48):106-145.
[20] Forrester P J. The Spectrum Edge of Random Matrix Ensembles[J].Nucl.Phys.B,1993(402):709-747.
[21] Zirnbauer M R.Riemannian Symmetric Superspaces and Their Origin inRandom-Matrix Theory[J].J. Math. Phys.,1996,37:498-523.
[22] Kazakov V A, Migdal A A. Induced QCD at Large N[J]. Nucl. Phys. B,1993(397):214-245.
[23] Douglas M R,Shenker S H. Strings in Less Than One-Dimension[J]. NuclearPhysics B,1990,335(3):635-654.
[24] Jimbo M,Miwa T,Mori Y,et al.Density Matrix of an Impenetrable Bose Gasand the Fifth Painleve Transcendent[J]. Physica D: Nonlinear Phenomena,1980,1(1):80-158.
[25] Tracy C A,Widom H.Level Spacing Distributions and the Airy Kernel[J].Communications in Mathematical Physics,1994,159(1):151-174.
[26] Berry M V.Semiclassical Theory of Spectral Rigidity[J].Proc. R. Soc. A,1985,400(1819):229-251.
[27] Laloux L, Cizeau P, Bouchaud J, et al. Noise Dressing of FinancialCorrelation Matrices[J]. Phys. Rev. Lett.,1999,83(7):1467-1470.
[28] Laloux L, Cizeau P, Potters M, et al. Random Matrix Theory and FinancialCorrelations[J]. Int. J. Theor. Appl. Finance,2000,3(3):391-397.
[29] Potters M, Bouchaud J, Laloux L. Financial Applications of Random MatrixTheory: Old Laces and New Pieces[C/OL].（2005-07-14）[2012-03-11].http://arxiv.org/abs/physics/0507111.
[30] Plerou V, Gopikrishnan P, Rosenow B, et al.Universal and NonuniversalProperties of Cross Correlations in Financial Time Series[J]. Phys. Rev.Lett.,1999,83(7):1471-1474.
[31] Plerou V, Gopikrishnan P, Rosenow B, et al. A Random Matrix TheoryApproach to Financial Cross-Correlations[J]. Physica A,2000,287(3-4):374-382.
[32] Stanley H E, Plerou V. Quantifying Economic Fluctuations[J]. Physica A,2001,302:126-137.
[33] Stanley H E, Plerou V. Quantifying Fluctuations in Economic Systems byAdapting Methods of Statistical Physics[J]. Physica A,2000,287:339-361.
[34] Plerou V, Gopikrishnan P, Rosenow B, et al. Collective Behaviour of StockPrice Movements-a Random Matrix Theory Approach[J].Physica A,2001,299:175-180.
[35] Rosenow B, Plerou V. Application of Random Matrix Theory to Study Cross-Correlations of Stock Prices[J]. International Journal of Theoretical andApplied Finance,2000,3(3):399-403.
[36] Plerou V, Gopikrishnan P, Rosenow B, et al. Econophysics:Financial TimeSeries from a Statistical Physics Point of View[J]. Physica A,2000,279:443-456.
[37] Lee Y H.Noise Detection from Financial Correlation Matrices[R/OL].(2001)[2012-04-11]. http://web.mit.edu/bazant/www/teach/18.325.
[38] Sadik C. Cross Correlations in an Emerging Market Financial Data[J].Physica A,2007,376:555-564.
[39] Utsugi A, Ino K, Oshikawa M. Random Matrix Theory Analysis of CrossCorrelations in Financial Markets[J]. Phys. Rev. E,2004,70(2):1-5.
[40] Nilantha K, Ranasinghe, Malmini P. Eigenvalue Density of Cross-correlationsin Sri Lankan Financial Market[J]. Physica A,2007,378(2):345-356.
[41] Wilcox D,Gebbie T. An Analysis of Cross-correlations in an EmergingMarket[J].Physica A,2007,375(2):584-598.
[42]李良新.中国股票市场有效性的随机矩阵实证分析[J].现代商业,2009(5):30-31.
[43] Conlon T, Ruskin H J,Crane M. Random Matrix Theory and Fund of FundsPortfolio Optimisation[J]. Physica A,2007,382(2):565-576.
[44] Malevergne Y,Sornette D. Collective Origin of the Coexistence of ApparentRandom Matrix Theory Noise and of Factors in Large Sample CorrelationMatrices[J]. Physica A,2004,331(3-4):660-668.
[45] Pafka S, Potters M,Kondor I. Exponential Weighting and Random-matrix-theory-based Denoising of Financial Covariance Matrices for PortfolioOptimization[J/OL].(2004-02-24)[2012-04-25]. http://eprintweb.org/S/article/cond-mat/0402573/cited.
[46] Wilcox D, Gebbie T. On the Analysis of Cross-correlations in South AfricanMarket Data[J]. Physica A,2004,344:294-298.
[47] Coronnello C. Sector Identification in a Set of Stock Return Time SeriesTraded at the London Stock Exchange[J].Acta Physica Polonica B,2005,36:2653-2676.
[48] Garas A, Argyrakis P. Correlation Study of the Athens Stock Exchange[J].Physica A,2007,380(1-2):399-410.
[49]郑波.金融动力学的时空关联与大波动特性——兼谈中西方金融市场的对比研究[J].物理,2010(2):95-100.
[50] Pafka S,Kondor I.Estimated Correlation Matrices and Portfolio Optimization[J]. Physica A,2004,343(1-4):623-634.
[51] Letchford A,Gao J,Zheng L.Filtering Financial Time Series by Least Squares[J].International Journal of Machine Learning and Cybernetics,2013,4(2):140-154.
[52] Hassani H,Dionisio A,Ghodsi M.The Effect of Noise Reduction in Measuringthe Linear and Nonlinear Dependency of Financial Markets[J]. NonlinearAnalysis:Real World Applications,2010,11(1):492-502.
[53] Lu C J, Lee T S, Chiu C C. Financial Time Series Forecasting UsingIndependent Component Analysis and Support Vector Regression[J]. DecisionSupport Systems,2009,47(2):115-125.
[54] Kang W, Kim K, Shin H. Denoising Monte Carlo Sensitivity Estimates[J].Operations Research Letters,2012,40(3):195-202.
[55] Neaime S.The Global Financial Crisis, Financial Linkages and Correlations inReturns and Volatilities in Emerging MENA Stock Markets[J].EMERGINGMARKETS REVIEW,2012,13(3):268-282.
[56] Elton E J, Gruber M J.Modern Portfolio Theory and Investment Analysis[M].New York: Wiley,1995:34-78.
[57] Chopra V K, Ziemba W T, Portf J. The Effect of Errors in Mean,Variances,andCovariance on Optimal Portfolio Choice[J]. The Journal of PortfolioManagement,1993,19(2):6-11.
[58] Livan G, Rebecchi L. Asymmetric Correlation Matrices:An Analysis ofFinancial Data[J]. European Physical Journal B,2012,85(6):1-11.
[59] Giacomo L,Simone A,Enrico S. Fine Structure of Spectral Properties forRandom Correlation Matrices: An Application to Financial Markets[J].Physical Review E,2011,84(1):1-13.
[60] Drozdz S, Kwapien J, Oswiecimka P. Empirics Versus RMT in FinancialCross-correlations[J]. Acta Physica Polonica B,2007,38(13):4027-4039.
[61] Akansu A N, Torun M U. Toeplitz Approximation to Empirical CorrelationMatrix of Asset Returns: A Signal Processing Perspective[J]. IEEE Journal onSelected Topics in Signal Processing,2012,6(4):319-326.
[62] Vincenzo T, Fabrizio L. Cluster Analysis for Portfolio Optimization[J].Journalof Economic Dynamics&Control,2008,32(1):235-258.
[63] Plerou V, Gopikrishnan P, Rosenow B, et al.Random Matrix Approach toCross Correlations in Financial Data[J]. Phys. Rev.E,2002,65(6):1-6.
[64] Sharifi S, Crane M, Shamaie A, et al. Random Matrix Theory for PortfolioOptimization: a Stability Approach[J]. Physica A,2004,335(3-4):629-643.
[65] Krzanowski.Sensitivity of Principal Components[J].J. Royal Stats. Soc. B,1984,46(3):558-563.
[66] Daly J, Crane M, Ruskin H J. Random Matrix Theory Filters in PortfolioOptimisation:A Stability and Risk Assessment[J].Physica A,2008,387(16):4248-4260.
[67]史宇峰.金融资产收益相关矩阵选取方法研究[J].证券市场导报,2008(5):65-68.
[68] Jiang J,Ma K,Cai X. Scaling and Correlations in Foreign Exchange Market[J].Physica A,2007,375(1):274-280.
[69] Jiang J,Ma K,Cai X. Non-linear Characteristics and Long-range Correlationsin Asian Stock Markets [J]. Physica A,2007,378(2):399-407.
[70]都国雄.经济物理学的研究内容与发展趋势[J].现代经济探讨,2006(4):47-49.
[71] Reese C, Rosenow B. Predicting Multivariate Volatility[J/OL].(2003-04-03)[2002-04-15]. http://arxiv.org/abs/cond-mat/0304082.
[72] Muirhead R B.Aspects of Multivariate Statistical Theory[M]. NewYork:Wiley,1982:45-65.
[73]王明进.高维波动率的预测[J].数量经济技术经济研究,2008(11):137-148.
[74]刘志东.多元GARCH模型结构特征、参数估计与假设检验研究综述[J].数量经济技术经济研究,2010(9):147-160.
[75]李文君,尹康.多元GARCH模型研究述评[J].数量经济技术经济研究,2009(10):138-147.
[76] Bauwens L, Laurent S, Rombouts J V K.Multivariate GARCH Models:aSurvey[J]. Journal of Applied Econometrics,2006,21(1):79-109.
[77] Pafka S,Kondor I.Noisy Covariance Matrices and Portfolio Optimization ii[J].Physica A,2002,319:487-494.
[78] Burda Z, Jurkiewicz J, Nowak M A. Is Econophysics a Solid Science[J].ActaPhys.Pol. B,2003,34(1):87-131.
[79] Kondor I, Pafka S,Nagy G. Noise Sensitivity of Portfolio Selection underVarious Risk Measures[J]. J. Banking Finance,2007,31(5):1545-1573.
[80] Varga-Haszonits I, Kondor I. Noise Sensitivity of Portfolio Selection inConstant Conditional Correlation GARCH Models[J].Physica A,2007,385:307-318.
[81] Jens S. The Asymptotic Spectrum of the EWMA Covariance Estimator[J].Physica A,2007,385(2):621-630.
[82] Eun C,Resnick B. Estimating the Correlation Structure of International SharePrices[J]. Journal of Finance,1984,39:1311-1324.
[83] Chan L, Karceski J, Lakonishok J. On Portfolio Optimization: ForecastingCovariances and Choosing the Risk Model[J]. Reviews of FinancialStudies,1999,12:937-974.
[84]王明进.多元波动率模型的一些新进展[J].数理统计与管理,2010,29(2):233-245.
[85] Ledoit O, Wolf M.A Well-conditioned Estimator for Large-dimensionalCovariance Matrices[J]. J. Multivar. Anal.,2004,88(2):365-411.
[86] Ledoit O, Wolf M. Improved Estimation of the Covariance Matrix of StockReturns with an Application to Portfolio Selection[J].J. Empirical Finance,2003,10(5):603-621.
[87] Ledoit O, Wolf M.Honey,I Shrunk the Sample Covariance Matrix[J]. TheJournal of Portfolio Management,2004,30(4):110-119.
[88] Burda Z, Goerlich A,Jarosz A.Signal and Noise in Correlation Matrix[J].Phys.A,2004,343:295-308.
[89] Bollerslev T.Modelling the Coherence in Short-run Nominal Exchange Rates:a Multivariate Generalized ARCH Model[J].Rev. Econ.Stat.,1990,72:498-505.
[90] Burda Z, Jarosz A, Jurkiewicz J.Applying Free Random Variables to RandomMatrix Analysis of Financial Data[J/OL].(2006-03-04)[2012-04-28].http://cdsweb.cern.ch/record/933639/.
[91] Daly J, Crane M,Ruskin H J.Random Matrix Theory Filters and CrrencyPortfolio Otimisation[J].Journal of Physics:Conference Series,2010,221(1):4-8.
[92]向骏文.基于马科维茨投资组合理论的股票投资策略[J].中国证券期货,2012,20(8):202.
[93]刘志东,薛莉.金融市场高维波动率的扩展广义正交GARCH模型与参数估计方法研究[J].中国管理科学,2010,18(6):33-41.
[94] Frei M, Kuensch H R. Sequential State and Observation Noise CovarianceEstimation Using Combined Ensemble Kalman and Particle Filters[J].Monthly Weather Review,2012,140(5):1476-1495.
[95] Menon R, Gerstoft P,Hodgkiss W S. Asymptotic Eigenvalue Density of NoiseCovariance Matrices[J]. IEEE Transactions on Signal Processing,2012,60(7):3415-3424.
[96] Bandiera F, Besson O, Ricci G.Knowledge-Aided Covariance MatrixEstimation and Adaptive Detection in Compound-Gaussian Noise [J].IEEETransactions on Signal Processing,2010,58(10):5390-5396.
[97] Mcaleer M, Medeiros M C.Realized Volatility:A Review[J].EconometricReviews,2008,27(1-3):10-45.
[98] Bandi F M,Russell J R.Separating Microstructure Noise from Volatility[J].Journal of Financial Economics,2006,79(3):655-692.
[99] Ghysels E,Sinko A.Volatility Forecasting and Microstructure Noise[J].Journal of Econometrics,2011,160(1):257-271.
[100] Boozer A D. The Role of Spatial Topology in a Toy Model of ClassicalElectrodynamics in (1+1) Dimensions[J]. Physics Letters A,2010,374(19-20):1901-1908.
[101] Eelbode D, Van de Voorde L.A Toy Model for Higher Spin DiracOperators[J]. Physics of Atomic Nuclei,2010,73(2):282-287.
[102] Kumar J.A Toy Model for Gauge-mediation in Intersecting Brane Models[J].Physical Review D,2009,79(6):1-6.
[103] Dimov I I, Kolm P N, Maclin L. Hidden Noise Structure and Random MatrixModels of Stock Correlations[J]. Quantitative Finance,2012,12(4):567-572.
[104] Martins A C R.Non-stationary Correlation Matrices and Noise[J].Physica A-Statistical Mechanics and its Applications,2007,379(2):552-558.
[105] Menon R, Gerstoft P, Hodgkiss W S.Asymptotic Eigenvalue Density of NoiseCovariance Matrices[J]. IEEE Transactions on Signal Processing,2012,60(7):3415-3424.
[106] Henker T, Husodo Z A. Noise and Efficient Variance in the Indonesia StockExchange[J].Pacific-basin Finance Journal,2010,18(2):199-216.
[107] Matteson D S, Ruppert D. Time-Series Models of Dynamic Volatility andCorrelation A Tutorial on Univariate and Multivariate Cases[J]. IEEE SignalProcessing Magazine,2011,28(5):72-82.
[108] Poon S H, Granger C W J. Forecasting Volatility in Financial Markets: AReview[J]. Journal of Economic Literature,2003,41(2):478-539.
[109] Olivier L,Michael W. Improved Estimation of the Covariance Matrix of StockReturns with an Application to Portfolio Selection[J]. Journal of EmpiricalFinance,2003(10):603-621.
[110] Sengupta A M, Mitra P P. Distributions of Singular Values for Some RandomMatrices[J]. Physical Review E,1999,60(3):3389-3392.
[111]王林,张蕾,刘连峰.用模拟退火算法寻找Heston期权定价模型参数[J].数量经济技术经济研究,2011(9):131-139.
[112]任世科,何正文,徐渝.基于联合视角的项目支付问题及其模拟退火启发式算法[J].系统工程,2012,30(6):82-89.
[113]罗大为.股票收益率影响因素的实证分析[J].人民论坛,2011(2):152-153.
[114]林立子,陈希敏.后危机时代我国股票收益率影响因素的实证研究[J].西安财经学院学报,2010,23(5):14-17.
[115]王大伟,王雪标.影响我国股票收益率的多因素实证研究—基于panel data模型分析[J].东北财经大学学报,2008(5):18-23.
[116]屠年松,徐光远.中国股票市场质量的国际比较[J].云南社会科学,2012(5):123-127.
[117]李进芳,王仁曾.Bootstrap方法在证券投资基金风险测量中的应用[J].统计研究,2010,27(3):66-69.
[118]张萍.基于Bootstrap方法的统计分析[J].宜宾学院学报,2011,11(12):31-33.
[119]徐正国,张世英.多维高频数据的“已实现”波动建模研究[J].系统工程学报,2006,21(1):6-11.
[120] Alexander C O. Orthogonal GARCH [J].Mastering Risk,2001(2):1-38.
[121]高岳,王家华,杨爱军.具有时变自由度的t-copula蒙特卡罗组合收益风险研究[J].中国管理科学,2011,19(2):10-15.
[122]刘金全,李楠,郑挺国.随机波动模型的马尔可夫链-蒙特卡洛模拟方法—在沪市收益率序列上的应用[J].数理统计与管理,2010,29(6):1026-1035.
[123]盛世明.基于连接函数的VaR的蒙特卡洛算法[J].统计与决策,2010(21):27-29.
[124] Davar K.Probability[M].America:American Mathematical Society,2007:60-67.
[125]张京.非负约束条件下组合证券投资决策的分枝定界法[J].数学的实践与认识,2004,34(5):18-23.
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