投资选择及资产定价数学模型研究
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摘要
现代证券组合投资理论是以金融资产的配置与选择为研究对象，以不确定情况下，资产配置的风险与收益的计量为内容的金融理论，该理论自20世纪50年代形成以来，就一直处在当代金融投资理论的前沿，特别是随着以现代数学方法研究金融经济的金融数学的问世，标志着金融投资决策开始摆脱纯粹描述性的研究和单纯经验操作的状态，进入到了定量分析的高级阶段。随着我国经济发展和经济体制改革的深化，特别是金融、证券体制的转轨和参与国际市场竞争，金融投资决策面临着越来越多错综复杂的理论和实践问题，投资选择和资产定价的数学模型研究越来越具有重要的理论价值和现实意义。
金融数学研究随着金融市场的发展而发展，20世纪50年代已有许多杰出的成果比如，著名的Mdigliani-Miller定理，70年代以来日趋活跃，逐步形成比较完整的数理经济理论。1988年D．Duffie《证券市场：随机模型》和1989年CF．Huang《金融经济学基础》的出版，总结了前提的工作并提出了许多新的学术思想。与此同时，证券投资研究日益深化，并逐步与数理金融研究融合。证券分析方法和证券投资数学模型研究，成为当前国际上金融研究和证券研究的结合点和前沿。在这方面，卓有建树的几位学者如H．Markowitz、W．Sharpe、K．Merton、M．Scholes先后于1990年和1997年荣膺诺贝尔经济学奖。
我国的金融数学研究起步较晚，最近几年才开始比较系统的研究，虽然在某些方面取得了一些进展，但整体而言，与国际先进水平的差距还很大，实用的数学模型数量较少。本文在深入研究现代金融经济学、投资学等有关理论的基础上，运用现代数学工具和数学方法，对投资组合优化和资本资产定价等重要理论进行了较为系统的研究，试图得到一些对我国资本市场建设及金融投资实践具有借鉴意义的数学模型。
全文共分四章，第一章着重分析研究单阶段静态投资组合理论，并结合我国资本市场的投资实际，建立了一些富有意义的投资组合优化模型：第二章分析了资本市场均衡定价的的经典模型CAPM及其扩展模型，在自由竞争的条件下，市场价格波动起伏，随机变化，但始终在均衡价格这一准绳附近变化，CAPM说明了均衡条件下，资本资产定价的方式，是投资组合理论的发展。但由于经典CAPM与现实资本
投资选择及资产定价数学模型研究
市场存在有较大差距，本文寻求在更一般情况下的均衡定价模型，特别是针对我国
证券市场是新兴市场的特点，研究了CAPM的衍生模型;第三章着重分析APT的基
本思想，研究揭示了APT定价模型的特征及其与CAPM的异同，并分析了APT在
我国实际应用中的几个问题。APT由于其较小的假设及宽阔的思路，使其对实际的
解释能力强于CAPM，但由于没有明确资产收益受哪些因素的影响，致使其无法动
摇C妙M的地位;第四章研究了在动态环境下的资产定价机理，分析了连续时间资
产投资组合优化模型及在变系数下的扩展，研究了随着证券品种的增加，投资绩效
的变化及其M一V有限边界的漂移。
Modem portfolio management and investment theory is a financial theory, which studies the measurement of risk and return of financial assets under uncertainties. Since the theory was established in 1950s, it has been on the frontier of modern financial investment theories. Specially, the emergence of financial mathematics, as a method of studying financial economics by means of modern mathematics, is a symbol which indicates that financial investment decisions are no more only describing studies or pure empirical researches but reaching to a highly stage of numerical analysis. In our country, with the economic growth and the innovations of economic systems, especially with the shunt of financial and security system and with the competition with international markets, financial investment decisions face more and more complicated theoretical and practical problems. So the researches on mathematic models of investment-selecting and assets-pricing become more and more important.
Mathematical finance blossoms with the development of financial market. In 1950s there emerged many outstanding achievements, such as the famous Mdigliani-Miller theorem. Mathematical finance flourished from 1970s and gradually developed to a integrated mathematically economic theory. "Security Market: the Random Model" written by D. Duffie published in 1988. "Financial Economics Basis" written by CF. Huang published in 1989. In their books presented lots of new science thoughts. At the same time, the researches on security investment deepened and associated with mathematical finance researches. The studies on security analysis and mathematical model of security investment become the frontier and combination of financial research and security research. Famous scholars in this domain such as Markowitz, Sharpe and Miller (in 1990 ; Merton and Scholes (in 1997 have been awarded Nobel Memorial Prize in Economic Science for their contributions to modern financial theory.
The researches on mathematical finance started relatively late in our country. There are not systematic studies until recent years. Though we achieved some developments in some aspects, there is a long distance from foreign advanced researches and few practical mathematical models. On the basis of deeply and thoroughly studies in modern financial economics, investment theories and other relevant theories, we make systematic researches on portfolio optimization and assets-pricing theory with the application of mathematical techniques. Furthermore, we try to get some mathematical models with
referential meaning to the construction of capital markets and the practice of financial investment in our country.
This book is organized in four chapters. Chapter one puts emphasis on one-stage stationary portfolio management theory. Considering the characteristics of the present capital markets in our country, we put forward several useful portfolio optimization models. Chapter two mainly analyzes the classical model of capital market equilibrium-pricing theory (CAPM and some other extensions. Under the assumption of free competition, the market price normally fluctuates stochastically around the equilibrium price. CAPM shows that under equilibrium conditions, the method of pricing capital and assets is a development of portfolio optimization theory. But the classical CAPM is very different from the practical capital market; an equilibrium-pricing model is presented in this book under more usual conditions. The secure market in our country is very young, so a derivative model of CAPM is also put forward to describe its characteristics. Chapter three includes the analysis of arbitrary pricing principle, the difference b
etween APT and CAPM, and some problems existing in the application of APT model in our country. There are fewer hypothesis and broader thoughts in APT, so it can explain the reality better than CAPM. On the other hand, APT is ambiguous in answering which factors affect the capital profit, so it will not take the place of CAPM. Chapter four is mainly about the assets-pricing
金融数学研究随着金融市场的发展而发展，20世纪50年代已有许多杰出的成果比如，著名的Mdigliani-Miller定理，70年代以来日趋活跃，逐步形成比较完整的数理经济理论。1988年D．Duffie《证券市场：随机模型》和1989年CF．Huang《金融经济学基础》的出版，总结了前提的工作并提出了许多新的学术思想。与此同时，证券投资研究日益深化，并逐步与数理金融研究融合。证券分析方法和证券投资数学模型研究，成为当前国际上金融研究和证券研究的结合点和前沿。在这方面，卓有建树的几位学者如H．Markowitz、W．Sharpe、K．Merton、M．Scholes先后于1990年和1997年荣膺诺贝尔经济学奖。
我国的金融数学研究起步较晚，最近几年才开始比较系统的研究，虽然在某些方面取得了一些进展，但整体而言，与国际先进水平的差距还很大，实用的数学模型数量较少。本文在深入研究现代金融经济学、投资学等有关理论的基础上，运用现代数学工具和数学方法，对投资组合优化和资本资产定价等重要理论进行了较为系统的研究，试图得到一些对我国资本市场建设及金融投资实践具有借鉴意义的数学模型。
全文共分四章，第一章着重分析研究单阶段静态投资组合理论，并结合我国资本市场的投资实际，建立了一些富有意义的投资组合优化模型：第二章分析了资本市场均衡定价的的经典模型CAPM及其扩展模型，在自由竞争的条件下，市场价格波动起伏，随机变化，但始终在均衡价格这一准绳附近变化，CAPM说明了均衡条件下，资本资产定价的方式，是投资组合理论的发展。但由于经典CAPM与现实资本
投资选择及资产定价数学模型研究
市场存在有较大差距，本文寻求在更一般情况下的均衡定价模型，特别是针对我国
证券市场是新兴市场的特点，研究了CAPM的衍生模型;第三章着重分析APT的基
本思想，研究揭示了APT定价模型的特征及其与CAPM的异同，并分析了APT在
我国实际应用中的几个问题。APT由于其较小的假设及宽阔的思路，使其对实际的
解释能力强于CAPM，但由于没有明确资产收益受哪些因素的影响，致使其无法动
摇C妙M的地位;第四章研究了在动态环境下的资产定价机理，分析了连续时间资
产投资组合优化模型及在变系数下的扩展，研究了随着证券品种的增加，投资绩效
的变化及其M一V有限边界的漂移。
Modem portfolio management and investment theory is a financial theory, which studies the measurement of risk and return of financial assets under uncertainties. Since the theory was established in 1950s, it has been on the frontier of modern financial investment theories. Specially, the emergence of financial mathematics, as a method of studying financial economics by means of modern mathematics, is a symbol which indicates that financial investment decisions are no more only describing studies or pure empirical researches but reaching to a highly stage of numerical analysis. In our country, with the economic growth and the innovations of economic systems, especially with the shunt of financial and security system and with the competition with international markets, financial investment decisions face more and more complicated theoretical and practical problems. So the researches on mathematic models of investment-selecting and assets-pricing become more and more important.
Mathematical finance blossoms with the development of financial market. In 1950s there emerged many outstanding achievements, such as the famous Mdigliani-Miller theorem. Mathematical finance flourished from 1970s and gradually developed to a integrated mathematically economic theory. "Security Market: the Random Model" written by D. Duffie published in 1988. "Financial Economics Basis" written by CF. Huang published in 1989. In their books presented lots of new science thoughts. At the same time, the researches on security investment deepened and associated with mathematical finance researches. The studies on security analysis and mathematical model of security investment become the frontier and combination of financial research and security research. Famous scholars in this domain such as Markowitz, Sharpe and Miller (in 1990 ; Merton and Scholes (in 1997 have been awarded Nobel Memorial Prize in Economic Science for their contributions to modern financial theory.
The researches on mathematical finance started relatively late in our country. There are not systematic studies until recent years. Though we achieved some developments in some aspects, there is a long distance from foreign advanced researches and few practical mathematical models. On the basis of deeply and thoroughly studies in modern financial economics, investment theories and other relevant theories, we make systematic researches on portfolio optimization and assets-pricing theory with the application of mathematical techniques. Furthermore, we try to get some mathematical models with
referential meaning to the construction of capital markets and the practice of financial investment in our country.
This book is organized in four chapters. Chapter one puts emphasis on one-stage stationary portfolio management theory. Considering the characteristics of the present capital markets in our country, we put forward several useful portfolio optimization models. Chapter two mainly analyzes the classical model of capital market equilibrium-pricing theory (CAPM and some other extensions. Under the assumption of free competition, the market price normally fluctuates stochastically around the equilibrium price. CAPM shows that under equilibrium conditions, the method of pricing capital and assets is a development of portfolio optimization theory. But the classical CAPM is very different from the practical capital market; an equilibrium-pricing model is presented in this book under more usual conditions. The secure market in our country is very young, so a derivative model of CAPM is also put forward to describe its characteristics. Chapter three includes the analysis of arbitrary pricing principle, the difference b
etween APT and CAPM, and some problems existing in the application of APT model in our country. There are fewer hypothesis and broader thoughts in APT, so it can explain the reality better than CAPM. On the other hand, APT is ambiguous in answering which factors affect the capital profit, so it will not take the place of CAPM. Chapter four is mainly about the assets-pricing
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