证券市场中最优投资组合问题的直接方法研究
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摘要
随着概率论和随机过程等近代数学理论的发展和应用,利用随机分析方法研究最优投资与消费问题已成为金融数学中定量研究的热门领域之一.Merton提出的连续时间下投资组合及消费模型标志着连续时间投资组合理论的真正开始.他通过把随机控制理论应用到最优投资组合问题中,得到了一些特殊情形下的显式解.
本文研究证券市场中不同模型的随机最优投资组合问题.分别选取了常数相对风险厌恶函数、绝对风险厌恶函数和幂效用函数,通过运用直接构造的方法和动态规划的方法得到了最优投资策略,并给出了相应的值函数.
本文的具体安排如下.
第一章介绍最优投资组合问题的背景、发展状况以及研究意义.比较系统地给出了关于最优投资组合问题的一些基础知识.第二章研究投资者投资两种期望收益和风险都不相同的股票的最优投资组合问题,在影响股票价格的随机干扰源又相互关联的情况下,运用伊藤公式,采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.
第三章研究投资股票和外汇存款的最优投资组合问题.这里允许股票卖空和从银行贷款,因此,分为了三种情况进行讨论.通过采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.
第四章研究投资者在投资股票和外汇这样有风险的资产的同时,又将一部分资产投入到无风险的债券.针对幂效用函数,运用动态规划方法,得到了最优投资组合及消费的显式解,并给出了值函数.
With the development and application of probability theory and stochastic process, studying optimal portfolio and consumption problems by using stochastic analysis methods has become one of the most common research methods in financial mathematics. It is re-garded as the real starting point of continuous-time portfolio theory that Merton formulated the continuous-time portfolio consumption model. By applying results of stochastic control theory to the portfolio problem he has able to obtain explicit solutions for some special ex-amples.
In this paper, stochastic optimal portfolio problems for different models are studied. We choose constant relative risk aversion utility function, hyperbolic absolute risk aversion utility and power utility function respectively, using a direct method and dynamic program-ming approach, the optimal strategies are obtained, and the corresponding value functions are presented.
This paper is organized as follows.
In chapter 1, we introduce some background information, present development situation and research significance about the optimal portfolio problem. We systematically present some elementary knowledge about optimal portfolio problem.
In chapter 2, we study the case in which the investor may choose two kinds of stocks with different expected returns and risks. Under the condition of correlated random interferences that affect the price of stocks, using Ito formula and a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.
In chapter 3, we investigate the problem that the investor has two different investments. One is a stock, the other is a foreign exchange deposit. The investor is allowed to short-sell and loan from bank. Therefore, we divide into three kinds of case to discuss the problem. Using a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.
In chapter 4, we research the case that the investor invest part of his wealth in the risky assets and the rest in the risk-free assets. Using dynamic programming approach, the explicit optimal portfolio and consumption choice are obtained for the power utility function case, and the corresponding value functions are presented.
本文研究证券市场中不同模型的随机最优投资组合问题.分别选取了常数相对风险厌恶函数、绝对风险厌恶函数和幂效用函数,通过运用直接构造的方法和动态规划的方法得到了最优投资策略,并给出了相应的值函数.
本文的具体安排如下.
第一章介绍最优投资组合问题的背景、发展状况以及研究意义.比较系统地给出了关于最优投资组合问题的一些基础知识.第二章研究投资者投资两种期望收益和风险都不相同的股票的最优投资组合问题,在影响股票价格的随机干扰源又相互关联的情况下,运用伊藤公式,采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.
第三章研究投资股票和外汇存款的最优投资组合问题.这里允许股票卖空和从银行贷款,因此,分为了三种情况进行讨论.通过采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.
第四章研究投资者在投资股票和外汇这样有风险的资产的同时,又将一部分资产投入到无风险的债券.针对幂效用函数,运用动态规划方法,得到了最优投资组合及消费的显式解,并给出了值函数.
With the development and application of probability theory and stochastic process, studying optimal portfolio and consumption problems by using stochastic analysis methods has become one of the most common research methods in financial mathematics. It is re-garded as the real starting point of continuous-time portfolio theory that Merton formulated the continuous-time portfolio consumption model. By applying results of stochastic control theory to the portfolio problem he has able to obtain explicit solutions for some special ex-amples.
In this paper, stochastic optimal portfolio problems for different models are studied. We choose constant relative risk aversion utility function, hyperbolic absolute risk aversion utility and power utility function respectively, using a direct method and dynamic program-ming approach, the optimal strategies are obtained, and the corresponding value functions are presented.
This paper is organized as follows.
In chapter 1, we introduce some background information, present development situation and research significance about the optimal portfolio problem. We systematically present some elementary knowledge about optimal portfolio problem.
In chapter 2, we study the case in which the investor may choose two kinds of stocks with different expected returns and risks. Under the condition of correlated random interferences that affect the price of stocks, using Ito formula and a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.
In chapter 3, we investigate the problem that the investor has two different investments. One is a stock, the other is a foreign exchange deposit. The investor is allowed to short-sell and loan from bank. Therefore, we divide into three kinds of case to discuss the problem. Using a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.
In chapter 4, we research the case that the investor invest part of his wealth in the risky assets and the rest in the risk-free assets. Using dynamic programming approach, the explicit optimal portfolio and consumption choice are obtained for the power utility function case, and the corresponding value functions are presented.
引文
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[2]叶中行,林建忠.数理金融-资产定价与金融决策理论[M].科学出版社,2004.
[3]Tobin J.. Liquidity preference as behaviour towards risk [J]. Review of Economic Studies,1958. Vol.25(2):65-86
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[8]Jiongmin Yong, Xunyu Zhou. Stochastic Controls:Hamiltonian Systems and HJB Equations [M]. New York:Springer-verlag.1999.
[9]Zhen Wu, Wensheng Xu. A direct method in optimal portfolio and consumption choice [J]. Appl Math JCU(B),1996. Vol.11:349-354
[10]Weiyin F.. A study on dynamical economic model for optimal consumption and investment [J]. Journal of Applied Probability and Statistics,1999. Vol.15(1):35-42
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[12]Harrison J.M.. Pliska S.R.. Martingales and stochastic integrals in the theory of continuous trading [J]. Stochastic Processes and Aplicaton,1981. Vol.11:215-260
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[14]Karatzas I., Lehoczky J.P., Shreve S.E.. Optimal portfolio and consumption decisions for a "small investor" on a finite horizon [J]. SIAM Journal on Control and Optimization,1987. Vol.27:1157-1186
[15]Karatzas I.. Optimization problems in the theory of continuous trading [J]. SIAM Journal on Control and Optimization,1989, Vol.27:1221-1259
[16]Cox J., Chifu H.. Optimal consumption and portfolio policies when asset prices follow a diffusion process [J]. Journal of Economic Theory,1989, Vol.49:33-83
[17]Korn R.. Optimal Portfolios:Stochastic Models for Optimal investment and Risk Management in Continuous Time [M]. World Scientific, Singapore,1997.
[18]Musiela, Rutkowski M.. Martingale Mathods in Financial Modelling [M]. Berlin: Springer,1997.
[19]Preman E.. Sethi S.. Distribution of bankruptcy time in a consumption portfolio problem [J]. Journal of Economics Dynamics and Control,1998. Vol.22:471-477
[20]Wonham W.M.. On a matrix Riccati equation of stochastic control [J]. SIAM J. Control,1968, Vol.6:681-697.
[21]Bensoussan A.. Stochastic Control of Partially Observable Systems [M]. Cambridge University Press, Cambridge.1992.
[22]Davis M.H.A.. Linear Estimation and Stochastic Control [M]. Chapman and Hall, London,1977.
[23]Shuping Chen. Xunjing Li. Xunyu Zhou. Stochastic linear quadratic regulators with indefinite control weight costs [J]. SIAM J. Control and Optim.,1998, Vol.36:1685-1702.
[24]Soner H.M.. Optimal control with state space constraints [J]. SIAM Journal on Control and Optimization.1986. Vol.24:552-561
[25]Yunhong Y.. Existence of optimal consumption and portfolio rules with portfolio constaints and stochastic income. durability and habit formation [J]. Journal of Mathematical Economics. 2000. Vol.33:135-153
[26]Pham H.. Smooth solutions to optimal investment models with stochasties and portfolio con-straina [J]. Appl Math Optim,2002. Vol.46:55-78
[27]Zariphopoulou T.. Optimal investment and consumption models with non-linear stock dynamics [J]. Math Meth Operations Res,1999, Vol.50:271-296
[28]Zariphopoulou T.. A solution approach to valuation with unhodgeable risks [J]. Finance Stochast, 2001, Vol.5:61-82
[29]Magill M.J.P., Constantinides G.M.. Portfolio selection with transaction costs [J]. Journal of Economic Theory.1976. Vol.13:245-263
[30]Bcrnt Φkscndal. Stochastic Differential Equations 6th ed [M]. Springer.2006.
[31]史敬涛,吴臻.一类证券市场中投资组合及消费选择的最优控制问题[J].高校应用数学学报,2005,20(1)：1-8.
[32]王光臣,史敬涛.一类关于投资组合和消费选择的最优控制问题[J].山东大学学报(理学版).2004,39(4)：1-6.
[2]叶中行,林建忠.数理金融-资产定价与金融决策理论[M].科学出版社,2004.
[3]Tobin J.. Liquidity preference as behaviour towards risk [J]. Review of Economic Studies,1958. Vol.25(2):65-86
[4]Tobin J.. The theory of portfolio selection [M]. The Theory of Interest Rates,1965.
[5]Duffie D., Richardson H.R.. Mean variance hedging in continuous time [J]. Annals of Applied Probability.1991, Vol.1:1-15
[6]Merton R.. Life portfolio selection under uncertainty:the continuous case [J]. Rev Econom Statis-tics.1969. Vol.51:247-257
[7]Merton R.. Optimal consumption and portfolio rules in continuous-time mode [J]. Econom The-ory,1971, Vol.3:373-413
[8]Jiongmin Yong, Xunyu Zhou. Stochastic Controls:Hamiltonian Systems and HJB Equations [M]. New York:Springer-verlag.1999.
[9]Zhen Wu, Wensheng Xu. A direct method in optimal portfolio and consumption choice [J]. Appl Math JCU(B),1996. Vol.11:349-354
[10]Weiyin F.. A study on dynamical economic model for optimal consumption and investment [J]. Journal of Applied Probability and Statistics,1999. Vol.15(1):35-42
[11]Harrison J.M., Kreps D.. Martingales and arbitrage in multiperiod securities markets [J]. Journal of Economic Theory,1979. Vol.20:381-408
[12]Harrison J.M.. Pliska S.R.. Martingales and stochastic integrals in the theory of continuous trading [J]. Stochastic Processes and Aplicaton,1981. Vol.11:215-260
[13]Harrison J.M., Pliska S.R.. A stochastic calculus model of continuous trading [J]. Stochastic Processes and Aplicaton,1983, Vol.15:313-316
[14]Karatzas I., Lehoczky J.P., Shreve S.E.. Optimal portfolio and consumption decisions for a "small investor" on a finite horizon [J]. SIAM Journal on Control and Optimization,1987. Vol.27:1157-1186
[15]Karatzas I.. Optimization problems in the theory of continuous trading [J]. SIAM Journal on Control and Optimization,1989, Vol.27:1221-1259
[16]Cox J., Chifu H.. Optimal consumption and portfolio policies when asset prices follow a diffusion process [J]. Journal of Economic Theory,1989, Vol.49:33-83
[17]Korn R.. Optimal Portfolios:Stochastic Models for Optimal investment and Risk Management in Continuous Time [M]. World Scientific, Singapore,1997.
[18]Musiela, Rutkowski M.. Martingale Mathods in Financial Modelling [M]. Berlin: Springer,1997.
[19]Preman E.. Sethi S.. Distribution of bankruptcy time in a consumption portfolio problem [J]. Journal of Economics Dynamics and Control,1998. Vol.22:471-477
[20]Wonham W.M.. On a matrix Riccati equation of stochastic control [J]. SIAM J. Control,1968, Vol.6:681-697.
[21]Bensoussan A.. Stochastic Control of Partially Observable Systems [M]. Cambridge University Press, Cambridge.1992.
[22]Davis M.H.A.. Linear Estimation and Stochastic Control [M]. Chapman and Hall, London,1977.
[23]Shuping Chen. Xunjing Li. Xunyu Zhou. Stochastic linear quadratic regulators with indefinite control weight costs [J]. SIAM J. Control and Optim.,1998, Vol.36:1685-1702.
[24]Soner H.M.. Optimal control with state space constraints [J]. SIAM Journal on Control and Optimization.1986. Vol.24:552-561
[25]Yunhong Y.. Existence of optimal consumption and portfolio rules with portfolio constaints and stochastic income. durability and habit formation [J]. Journal of Mathematical Economics. 2000. Vol.33:135-153
[26]Pham H.. Smooth solutions to optimal investment models with stochasties and portfolio con-straina [J]. Appl Math Optim,2002. Vol.46:55-78
[27]Zariphopoulou T.. Optimal investment and consumption models with non-linear stock dynamics [J]. Math Meth Operations Res,1999, Vol.50:271-296
[28]Zariphopoulou T.. A solution approach to valuation with unhodgeable risks [J]. Finance Stochast, 2001, Vol.5:61-82
[29]Magill M.J.P., Constantinides G.M.. Portfolio selection with transaction costs [J]. Journal of Economic Theory.1976. Vol.13:245-263
[30]Bcrnt Φkscndal. Stochastic Differential Equations 6th ed [M]. Springer.2006.
[31]史敬涛,吴臻.一类证券市场中投资组合及消费选择的最优控制问题[J].高校应用数学学报,2005,20(1)：1-8.
[32]王光臣,史敬涛.一类关于投资组合和消费选择的最优控制问题[J].山东大学学报(理学版).2004,39(4)：1-6.