基于遗传算法的CVaR模型研究
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摘要
风险值(Value-at-Risk, VaR)是一种以统计技术全面度量市场风险的方法,是指在正常的市场环境下,给定一定的时间周期和置信水平,预期最大损失的测度。条件风险值(Conditional Value-at-Risk, CVaR)是指损失额超过VaR部分的期望损失值或平均损失值。它具有VaR模型的优点,同时在理论上又具有良好的性质,如具有次可加性、凸性等。
目前,在建立CVaR的数学模型方面有连续型CVaR和离散型CVaR模型,本文主要讨论连续型CVaR,在研究的过程中,运用了系统理论、归纳演绎、比较与实证分析等研究方法。
全文共分五章进行。第一章介绍了本文的研究背景,VaR、CVaR和遗传算法的研究概况。第二章概述了VaR模型和CVaR模型,第三章研究连续型单损失CVaR模型及其基于改进遗传算法的解,第四章研究连续型多损失CVaR模型及其基于改进多目标遗传算法的解,第五章对全文进行了总结。
本文所取得的研究成果主要有以下几点:
1.研究了连续型单损失CVaR模型。通过对线性损失函数的改进,建立了CVaR的一个非线性规划模型,推广了CVaR已有的线性规划模型。通过一种改进的遗传算法求出新的CVaR模型的近似最优解,得到更优的VaR和CVaR值,有效降低了风险。
2.研究了连续型多损失CVaR模型,并建立了CVaR的一个多目标优化模型。根据一种新的关系算子,利用求解多目标规划问题的Pareto多目标遗传算法对连续型多损失CVaR模型进行了求解,得到更优的VaR和CVaR值,有效降低了风险。
3.利用深证成份股6只股票(深发展、深科技、深万科、世纪星源、深华新和深天地)对新的模型进行了实证分析,通过MATLAB编程进行了数值实验,结果表明对模型和算法的改进是有效的。
The value at risk (VaR) is a statistic method to measure the risk of stock markets or portfolio, and the conditional value at risk (CVaR) is another kind of statistic method to measure the risk of portfolio, and it measures the risk by calculating the mean of the values which exceed the VaR. CVaR overcomes several limitations of VaR and has good properties, especially its good computability.
The research on CVaR mainly focuses on discrete type of CVaR and continuous type of CVaR. This thesis studies continuous type of CVaR based on the methods of systems theory, induction and deduction, comparison and empirical analysis etc.
The thesis is organized as follows. In Chapter 1, we introduce the background and the concepts of VaR, CVaR and genetic algorithms. In Chapter 2, we introduce the research status on VaR and CVaR. In Chapter 3, the continuous type of CVaR model with single loss is studied. In Chapter 4, the continuous type of CVaR model with multiple losses is studied. Chapter 5 concludes the work.
The main results obtained in this thesis are as follows.
1. The continuous type of CVaR models with single loss is studied. The nonlinear loss functions are designed first. Based on this, a nonlinear programming model for CVaR problem is proposed, which generalized the existing linear CVaR models. Then an improved genetic algorithm is designed to solve the proposed nonlinear programming model. The simulation results indicate the proposed method can decrease the values of both CVaR and VaR.
2. The continuous type of CVaR model with multiple losses is studied and a multi-objective optimization model is proposed. To overcome the limitations of traditional multi-objective optimization methods, A Pareto multi-objective genetic algorithm for multi-objective programming problems is used. The proposed model and algorithm can decrease the values of both CVaR and VaR.
3. The simulations are made for the proposed algorithms by using some data on Shenzhen
目前,在建立CVaR的数学模型方面有连续型CVaR和离散型CVaR模型,本文主要讨论连续型CVaR,在研究的过程中,运用了系统理论、归纳演绎、比较与实证分析等研究方法。
全文共分五章进行。第一章介绍了本文的研究背景,VaR、CVaR和遗传算法的研究概况。第二章概述了VaR模型和CVaR模型,第三章研究连续型单损失CVaR模型及其基于改进遗传算法的解,第四章研究连续型多损失CVaR模型及其基于改进多目标遗传算法的解,第五章对全文进行了总结。
本文所取得的研究成果主要有以下几点:
1.研究了连续型单损失CVaR模型。通过对线性损失函数的改进,建立了CVaR的一个非线性规划模型,推广了CVaR已有的线性规划模型。通过一种改进的遗传算法求出新的CVaR模型的近似最优解,得到更优的VaR和CVaR值,有效降低了风险。
2.研究了连续型多损失CVaR模型,并建立了CVaR的一个多目标优化模型。根据一种新的关系算子,利用求解多目标规划问题的Pareto多目标遗传算法对连续型多损失CVaR模型进行了求解,得到更优的VaR和CVaR值,有效降低了风险。
3.利用深证成份股6只股票(深发展、深科技、深万科、世纪星源、深华新和深天地)对新的模型进行了实证分析,通过MATLAB编程进行了数值实验,结果表明对模型和算法的改进是有效的。
The value at risk (VaR) is a statistic method to measure the risk of stock markets or portfolio, and the conditional value at risk (CVaR) is another kind of statistic method to measure the risk of portfolio, and it measures the risk by calculating the mean of the values which exceed the VaR. CVaR overcomes several limitations of VaR and has good properties, especially its good computability.
The research on CVaR mainly focuses on discrete type of CVaR and continuous type of CVaR. This thesis studies continuous type of CVaR based on the methods of systems theory, induction and deduction, comparison and empirical analysis etc.
The thesis is organized as follows. In Chapter 1, we introduce the background and the concepts of VaR, CVaR and genetic algorithms. In Chapter 2, we introduce the research status on VaR and CVaR. In Chapter 3, the continuous type of CVaR model with single loss is studied. In Chapter 4, the continuous type of CVaR model with multiple losses is studied. Chapter 5 concludes the work.
The main results obtained in this thesis are as follows.
1. The continuous type of CVaR models with single loss is studied. The nonlinear loss functions are designed first. Based on this, a nonlinear programming model for CVaR problem is proposed, which generalized the existing linear CVaR models. Then an improved genetic algorithm is designed to solve the proposed nonlinear programming model. The simulation results indicate the proposed method can decrease the values of both CVaR and VaR.
2. The continuous type of CVaR model with multiple losses is studied and a multi-objective optimization model is proposed. To overcome the limitations of traditional multi-objective optimization methods, A Pareto multi-objective genetic algorithm for multi-objective programming problems is used. The proposed model and algorithm can decrease the values of both CVaR and VaR.
3. The simulations are made for the proposed algorithms by using some data on Shenzhen
引文
[1]王春峰.金融市场风险管理[M].天津:天津大学出版社,2001. 70-73.
[2]王建华,李楚霖. 度量与控制金融风险的新方法[J].武汉:武汉理工大学学报信息与管理工程版,2002, 24(2). 60-83.
[3]李海涛. 基于 CVaR 的衍生证券投资组合优化模型研究[D]. 武汉理工大学硕士学位论文.2005.1. 58-66.
[4]Rockafellar R T, Uryasev S. Optimization of Conditional Value-at-Risk[J].Journal of Risk, 2002.2, 21~24.
[5]Sack R. Model-based simulation, Whiter paper, National Science foundation, Arlington, 1999. 14~17.
[6]戴国强,徐龙炳. CVaR 方法对我国金融风险管理的借鉴和运用[J]. 金融研究. 2000(7), 8-10.
[7]王春峰,万海晕等. 金融市场风险测量模型——VaR[J]. 系统工程学报. 2000, 15(1).
[8]Boumal W. An expected gain-confidence limit criterion for portfolio selection[J]. Management Science 1963(10):174~182.
[9]J P Morgan. Riskmetrics-Technical document. 3thed. New York: Morgan Guaranty Trust Company Global Research, 1994.
[10]Jorion P. Value-at-Risk: The new benchmark for controlling market risk[M]. The McGraw-hill Companies,Inc,1997.
[11]Dowd K. Beyond Value-at-Risk[M]. New York: John Wiley&Sons, Inc, 1998.
[12]Best P. Implementing Value-at-Risk[M]. New York: John Wiley&Sons, Inc, 1998.
[13]Basle Committee on Banking Supervision. Amendment to the capital accord to corporate market risks. Bank for International Settlement, Basle,1996.
[14]Rflung G Ch. Some Remarks on the Value-at-Risk and the Conditional Value-at- Risk. In: Ed. Uryasev S. Probabilistic Constrained Optimization: Methodology and Application. Boston: Kluwer. Academic Publishers[M].2000.
[15]Anderson F, Mausser H, Rosen D, Urasev S. Credit risk optimization with Conditional Value-at-Risk criterion[M]. Math Progam. Ser. B .2000:273~291.
[16]Nikolas T, Hercules V, Stavros A. CVaR models with selective hedging international assets allocation[J]. Journal of Banking&Finance. 2002(26):1535~1561.
[17]Pamquist J, Uryasev S, Krokhmal P. Portfolio optimization with Conditional Value-at-Risk objective and constrains[DB/OL].1999.
[18]陈金龙, 张维 CVaR 与投资组合优化统一模型. 《系 统 工 程 理 论 方 法 应 用》 第 11 卷第 1 期.
[19]王建华,李楚霖. 度量与控制金融风险的新方法. 武汉理工大学学报. 2002,11(1) . 68-71.
[20]岳瑞锋等 风险管理的 CVaR 方法及其简化模型. 《河北省科学院学报》. 第 20 卷第 3 期.2003.8.
[21]黄向阳,陈学华,杨辉耀 基于条件风险价值的投资组合优化模型. 《西南交通大学学报》第 39 卷 第 4 期.
[22]吴军,汪寿阳. CVaR 与供应链风险管理. 中国运筹学第七届学术交流会论文集. 2004. 633-640.
[23]Holland J H. Adaptation in Nature and Artificial System[M].Ann Arbor, Michigan: University of Michigan Press,1975.
[24]王小平,曹立祖. 遗传算法理论、应用与软件实现[M]. 西安:西安交通大学出版社,2002.
[25]Schaffer J D. Multiple objective optimization with vector evaluated genetic algorithms[A]. Proc of International Conference on Genetic Algorithms and their Application[C]. 1985.
[26]Goldberg D E. Genetic algorithms in search, optimization, and machine learning [M]. Reading, MA:Addison Weelev. 1989.
[27]Fonseca C M, Fleming P J. Multiobjective optimization and multiple constraint handling with evolutionary algorithms-Part I:a unified formulation[J]. IEEE Transaction on Systems, Man and Cybernetics, 1998.
[28]蒋敏. 条件风险值(CVaR)模型的理论研究.西安电子科技大学博士学位论文[D]. 2005.4.
[29] 魏一鸣,曾嵘等. 北京市人口、资源、环境与经济协调发展的多目标规划模型[J]. 系统工程理论与实践. 2002, (2): 74-83.
[30]盛昭翰,曹忻. 最优化算法基本教程[M].南京:东南大学出版社,1992.
[31]Kirkpatrick S. Optimization by simulated annealing[J]. Science.1983.
[32]Evans M, Hasting N, Peacock B. Statistical distributions[M].New York. John Wiley& Sons, 1993.
[33]Pritsker M. Evaluating value at risk methodologies: Accuracy versus computational time[P]. Working Paper496-48, Wharton Financial Institution Center, University of Pennsylvania, 1996.
[34] Kall P, Wallace.S.W. Stochastic Programming. John Wiley & Sons, 1995.
[35]姚新领. 基于CVaR风险度量的证券组合投资决策模型研究. 安徽理工大学学报(自然 科学版). 2004 (6) 67-69.
[36]田新民,黄海平 基于条件VaR(CVaR)的投资组合优化模型及比较研究. 《数学的实践与认识》. 第 34 卷第 7 期.
[37]岳瑞锋等. 风险管理的 CVaR 方法及其简化模型[J]. 河北省科学院学报. 第 20 卷第 3 期, 2003, 8.
[38]李旭东等. 遗传算法中引入非自然规则的研究[J]. 计算机工程与应用. 2003, 34.
[39]赖红松, 董品杰等. 求解多目标规划问题的 Pareto 多目标遗传算法. 《系统工程》. 第 21 卷第 5 期(总第 119 期) 2003 年 9 月.
[40]RockfellerT, UryasevS. Conditional Value-at-Risk for general loss distribution[J]. Journal of Banking&Finance-2002(26)1445-1471.
[41]Pan Z J, Kang L S. An Adaptive Evolutionary Algorithms for Numerical Optimization[J].In Lecture Notes in Artificial Intelligence,1997.
[42]Y.W. Leung & Yuping Wang, An orthogonal genetic algorithm with quantization for global numerical optimization, IEEE Trans. Evolutionary Computation, vol.5, no. 1, 2001.
[43]C.Y. Lee and X. Yao, Evolutionary Programming Using Mutations Based on the Levy Probability Distribution, IEEE Trans. Evolutionary Computation, vol.8, no.1, pp.1--13, 2004.
[44]Kellerer H, Mansini R, Speranza M G. On Selecting a Portfolio with Fixed Cost and Minimum Lots[J].Annals of Operations Reseach,2000.99(3):287～233.
[45]Campbell R, Huisman R, Koedijk K. Optional Portfolio Selection in a Value at Risk Framework[J].Journal of Banking and Finance,2001.25.
[46]Fredrik Andersson, Helmut Mausser. Dan Rosen. Credit Risk Optimization with Conditional Value-at-Risk Criterion[J].Math Program,2001.Ser.B 89:273～291.
[47]A.I.Kibzun, E.A.Kuznetsov. Comparison of VaR and CVaR Criteria[J].Automation and Remote Control.2003,64(7).1154～1164.
[48]曲圣宁,田新时. 投资组合风险管理中VaR模型的缺陷以及CVaR模型研究. 《统计与决策》.2005年5月下.
[49]李磊,秦超英等. CVaR的灵敏度分析. 《西南民族大学学报·自然科学版》.31卷第5期.
[50]范龙振,余世典. 中国股票市场的三因子模型. 《系 统 工 程 学 报》.2004年6月.
[51]张 莉, 唐万生等. 概率准则下组合投资的整数规划模型. 《天 津 大 学 学 报 (社 会 科 学 版)》. 2003年4月.
[52]徐金红, 徐维军等. 概率原则下基于投资者偏好的组合投资选择. 《系统工程学报》. 第20卷第5期.
[53]王燕青,唐万生,韩其恒 基于遗传算法的概率准则组合证券模拟求解. 《管理科学学报》.第5卷第6期.
[54]李旭东,涂奉生. 基于遗传算法的多类人力资源计划最优决策. 《系统工程学报》.2005年8月, 第20卷第4期.
[55]吴新余,马敏肖. 遗传算法在多目标规划中的应用. 《南京邮电学院学报》. 1996年6月,第16卷第2期.
[56]刘漫丹,钱 锋. 基于递进制编码的遗传算法研究. 《信息与控制》.2004年10月,第33卷第5期.
[57]万定生,余长海等. 基于位变异防止遗传算法过早收敛的算法. 《微电子学与计算机》. 2005年, 第22卷第8期.
[2]王建华,李楚霖. 度量与控制金融风险的新方法[J].武汉:武汉理工大学学报信息与管理工程版,2002, 24(2). 60-83.
[3]李海涛. 基于 CVaR 的衍生证券投资组合优化模型研究[D]. 武汉理工大学硕士学位论文.2005.1. 58-66.
[4]Rockafellar R T, Uryasev S. Optimization of Conditional Value-at-Risk[J].Journal of Risk, 2002.2, 21~24.
[5]Sack R. Model-based simulation, Whiter paper, National Science foundation, Arlington, 1999. 14~17.
[6]戴国强,徐龙炳. CVaR 方法对我国金融风险管理的借鉴和运用[J]. 金融研究. 2000(7), 8-10.
[7]王春峰,万海晕等. 金融市场风险测量模型——VaR[J]. 系统工程学报. 2000, 15(1).
[8]Boumal W. An expected gain-confidence limit criterion for portfolio selection[J]. Management Science 1963(10):174~182.
[9]J P Morgan. Riskmetrics-Technical document. 3thed. New York: Morgan Guaranty Trust Company Global Research, 1994.
[10]Jorion P. Value-at-Risk: The new benchmark for controlling market risk[M]. The McGraw-hill Companies,Inc,1997.
[11]Dowd K. Beyond Value-at-Risk[M]. New York: John Wiley&Sons, Inc, 1998.
[12]Best P. Implementing Value-at-Risk[M]. New York: John Wiley&Sons, Inc, 1998.
[13]Basle Committee on Banking Supervision. Amendment to the capital accord to corporate market risks. Bank for International Settlement, Basle,1996.
[14]Rflung G Ch. Some Remarks on the Value-at-Risk and the Conditional Value-at- Risk. In: Ed. Uryasev S. Probabilistic Constrained Optimization: Methodology and Application. Boston: Kluwer. Academic Publishers[M].2000.
[15]Anderson F, Mausser H, Rosen D, Urasev S. Credit risk optimization with Conditional Value-at-Risk criterion[M]. Math Progam. Ser. B .2000:273~291.
[16]Nikolas T, Hercules V, Stavros A. CVaR models with selective hedging international assets allocation[J]. Journal of Banking&Finance. 2002(26):1535~1561.
[17]Pamquist J, Uryasev S, Krokhmal P. Portfolio optimization with Conditional Value-at-Risk objective and constrains[DB/OL].1999.
[18]陈金龙, 张维 CVaR 与投资组合优化统一模型. 《系 统 工 程 理 论 方 法 应 用》 第 11 卷第 1 期.
[19]王建华,李楚霖. 度量与控制金融风险的新方法. 武汉理工大学学报. 2002,11(1) . 68-71.
[20]岳瑞锋等 风险管理的 CVaR 方法及其简化模型. 《河北省科学院学报》. 第 20 卷第 3 期.2003.8.
[21]黄向阳,陈学华,杨辉耀 基于条件风险价值的投资组合优化模型. 《西南交通大学学报》第 39 卷 第 4 期.
[22]吴军,汪寿阳. CVaR 与供应链风险管理. 中国运筹学第七届学术交流会论文集. 2004. 633-640.
[23]Holland J H. Adaptation in Nature and Artificial System[M].Ann Arbor, Michigan: University of Michigan Press,1975.
[24]王小平,曹立祖. 遗传算法理论、应用与软件实现[M]. 西安:西安交通大学出版社,2002.
[25]Schaffer J D. Multiple objective optimization with vector evaluated genetic algorithms[A]. Proc of International Conference on Genetic Algorithms and their Application[C]. 1985.
[26]Goldberg D E. Genetic algorithms in search, optimization, and machine learning [M]. Reading, MA:Addison Weelev. 1989.
[27]Fonseca C M, Fleming P J. Multiobjective optimization and multiple constraint handling with evolutionary algorithms-Part I:a unified formulation[J]. IEEE Transaction on Systems, Man and Cybernetics, 1998.
[28]蒋敏. 条件风险值(CVaR)模型的理论研究.西安电子科技大学博士学位论文[D]. 2005.4.
[29] 魏一鸣,曾嵘等. 北京市人口、资源、环境与经济协调发展的多目标规划模型[J]. 系统工程理论与实践. 2002, (2): 74-83.
[30]盛昭翰,曹忻. 最优化算法基本教程[M].南京:东南大学出版社,1992.
[31]Kirkpatrick S. Optimization by simulated annealing[J]. Science.1983.
[32]Evans M, Hasting N, Peacock B. Statistical distributions[M].New York. John Wiley& Sons, 1993.
[33]Pritsker M. Evaluating value at risk methodologies: Accuracy versus computational time[P]. Working Paper496-48, Wharton Financial Institution Center, University of Pennsylvania, 1996.
[34] Kall P, Wallace.S.W. Stochastic Programming. John Wiley & Sons, 1995.
[35]姚新领. 基于CVaR风险度量的证券组合投资决策模型研究. 安徽理工大学学报(自然 科学版). 2004 (6) 67-69.
[36]田新民,黄海平 基于条件VaR(CVaR)的投资组合优化模型及比较研究. 《数学的实践与认识》. 第 34 卷第 7 期.
[37]岳瑞锋等. 风险管理的 CVaR 方法及其简化模型[J]. 河北省科学院学报. 第 20 卷第 3 期, 2003, 8.
[38]李旭东等. 遗传算法中引入非自然规则的研究[J]. 计算机工程与应用. 2003, 34.
[39]赖红松, 董品杰等. 求解多目标规划问题的 Pareto 多目标遗传算法. 《系统工程》. 第 21 卷第 5 期(总第 119 期) 2003 年 9 月.
[40]RockfellerT, UryasevS. Conditional Value-at-Risk for general loss distribution[J]. Journal of Banking&Finance-2002(26)1445-1471.
[41]Pan Z J, Kang L S. An Adaptive Evolutionary Algorithms for Numerical Optimization[J].In Lecture Notes in Artificial Intelligence,1997.
[42]Y.W. Leung & Yuping Wang, An orthogonal genetic algorithm with quantization for global numerical optimization, IEEE Trans. Evolutionary Computation, vol.5, no. 1, 2001.
[43]C.Y. Lee and X. Yao, Evolutionary Programming Using Mutations Based on the Levy Probability Distribution, IEEE Trans. Evolutionary Computation, vol.8, no.1, pp.1--13, 2004.
[44]Kellerer H, Mansini R, Speranza M G. On Selecting a Portfolio with Fixed Cost and Minimum Lots[J].Annals of Operations Reseach,2000.99(3):287～233.
[45]Campbell R, Huisman R, Koedijk K. Optional Portfolio Selection in a Value at Risk Framework[J].Journal of Banking and Finance,2001.25.
[46]Fredrik Andersson, Helmut Mausser. Dan Rosen. Credit Risk Optimization with Conditional Value-at-Risk Criterion[J].Math Program,2001.Ser.B 89:273～291.
[47]A.I.Kibzun, E.A.Kuznetsov. Comparison of VaR and CVaR Criteria[J].Automation and Remote Control.2003,64(7).1154～1164.
[48]曲圣宁,田新时. 投资组合风险管理中VaR模型的缺陷以及CVaR模型研究. 《统计与决策》.2005年5月下.
[49]李磊,秦超英等. CVaR的灵敏度分析. 《西南民族大学学报·自然科学版》.31卷第5期.
[50]范龙振,余世典. 中国股票市场的三因子模型. 《系 统 工 程 学 报》.2004年6月.
[51]张 莉, 唐万生等. 概率准则下组合投资的整数规划模型. 《天 津 大 学 学 报 (社 会 科 学 版)》. 2003年4月.
[52]徐金红, 徐维军等. 概率原则下基于投资者偏好的组合投资选择. 《系统工程学报》. 第20卷第5期.
[53]王燕青,唐万生,韩其恒 基于遗传算法的概率准则组合证券模拟求解. 《管理科学学报》.第5卷第6期.
[54]李旭东,涂奉生. 基于遗传算法的多类人力资源计划最优决策. 《系统工程学报》.2005年8月, 第20卷第4期.
[55]吴新余,马敏肖. 遗传算法在多目标规划中的应用. 《南京邮电学院学报》. 1996年6月,第16卷第2期.
[56]刘漫丹,钱 锋. 基于递进制编码的遗传算法研究. 《信息与控制》.2004年10月,第33卷第5期.
[57]万定生,余长海等. 基于位变异防止遗传算法过早收敛的算法. 《微电子学与计算机》. 2005年, 第22卷第8期.