连续时间投资组合优化理论方法研究
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摘要
连续时间投资组合问题是数理金融的重点研究内容,是投资人或者投资机构进行资产套期保值和风险对冲的重要理论方法.通过研究不同投资环境下的投资组合优化问题,一方面可以应用数理方法创造性的解决人们在实际投资过程中所遇到的问题,为投资人进行科学投资提供理论依据;另一方面可以为投资学的理论方法有更加广泛的应用提供科学依据.本文主要对连续时间投资组合优化问题进行了一些扩展性研究,取得了一些研究成果.针对实际投资环境的的多样性和金融市场的不确定性,本文主要侧重于四个方面的研究: (1)不完全市场下动态资产分配的扩展性研究; (2)随机环境下资产-负债管理问题研究; (3)限制性投资组合优化问题的扩展性研究;(4)随机环境下投资-消费问题研究.具体研究成果详述如下:
第二章主要对不完全市场下的动态投资组合优化问题进行了扩展性研究.首先,对不完全市场下基于效用最大化的动态资产分配问题进行了研究.通过降低布朗运动的维数将不完全市场转化为完全市场,并在转化后的完全市场下应用鞅方法得到了指数效用和对数效用函数下最优投资策略的解析表达式.应用完全市场与原不完全市场间参数关系得到不完全市场下的最优投资策略.算例解释了模型的结论,分析了从完全市场到不完全市场下最优投资策略的变化情况,并把指数效用和对数效用函数下的最优投资策略与幂效用函数下的最优投资策略进行了比较.其次,对不完全市场下基于二次效用函数的投资组合优化问题进行了研究,应用鞅方法得到了最优投资组合的解析表达式.分析了均值-方差模型下不完全市场的最优投资策略问题,为进一步全面探讨均值-方差模型提供了理论基础.第三,对不完全市场下的投资-消费问题进行了研究,应用动态规划原理和HJB方程方法得到了幂效用、指数效用和对数效用函数下最优投资-消费策略的解析表达式.第四,对不完全市场下的资产-负债管理问题进行了研究,通过构造指数鞅方法和引入二次优化问题解决了指数效用函数下的最优投资组合问题.所有这些研究扩展了不完全市场下动态投资组合优化方面的研究,丰富和发展了Zhang的研究内容.
第三章主要研究随机环境下的资产-负债管理问题,如随机利率模型和随机波动率模型等.首先,在常数利率环境下对效用最大化下的资产-负债管理问题进行了研究,应用动态规划原理和Legendre变换-对偶解法得到了幂效用、指数效用和对数效用函数下最优投资策略的解析表达式,并给出算例分析了市场参数对最优投资组合的影响.其次,假设无风险利率、股票收益率和波动率均为一致有界随机过程,应用向后随机微分方程理论和随机线性二次规划方法得到了最优投资策略的解析表达式.第三,假设利率是服从Ho-Lee利率模型的随机过程,应用动态规划原理对资产-负债管理进行了研究,得到了幂效用和指数效用函数下最优投资策略的解析表达式.第四,对Vasicek利率模型下的资产-负债管理问题进行了研究,结合动态规划原理和Legendre变换-对偶解法得到了幂效用和指数效用函数下最优投资策略的解析表达式.将负债过程引入到投资组合优化问题中,并研究了此类问题的最优投资组合与风险管理的问题,是现阶段资产-负债管理的新的研究内容.本章的研究内容丰富和发展了资产-负债管理方面的理论方法,尤其是解决了随机利率模型下的最优投资组合问题,为随机利率模型下带有负债的投资机构进行资产套期保值和对冲风险提供了理论依据.
第四章主要对不同借贷利率限制下的动态投资组合优化问题进行了扩展性研究.首先,对效用最大化下的投资组合选择问题进行了研究,应用动态规划原理和HJB方程方法得到了幂效用、指数效用和对数效用函数下最优投资策略的解析表达式,并给出算例对不同借贷利率限制下投资人的投资行为进行了分析;其次,对负债情形下的均值-方差问题进行了研究,应用拉格朗日对偶定理和动态规划原理得到了最优投资策略和有效前沿的解析表达式;最后,将几何布朗运动扩展至CEV模型,对CEV模型下的投资组合优化问题进行了研究,得到了最优投资策略和有效前沿的解析表达式.本章的研究工作进一步丰富和发展了不同借贷利率限制下投资组合优化问题的理论方法,扩展了Fu和Lari-Lavassani等人的研究工作,为进一步研究负债和CEV模型下的投资组合优化模型提供了理论基础.
第五章主要研究了随机环境下的投资-消费问题.首先,我们假设金融市场中存在两种资产,一种资产是无风险资产,其中无风险利率是服从Ho-Lee利率模型的随机过程,且与风险资产价格存在线性相关性,以投资人有限投资周期内终端财富和累积消费的期望贴现效用作为目标函数,应用动态规划原理和HJB方程对幂效用和对数效用函数下的最优投资-消费策略进行了研究,得到了两种效用函数下最优投资-消费策略的解析表达式.其次,我们将Ho-Lee利率模型扩展至Vasicek利率模型,应用动态规划原理和Legendre变换-对偶方法得到了幂效用和对数效用函数下最优投资-消费策略的解析表达式.我们的这些研究将Merton的投资-消费模型扩展到随机环境下,并着重研究了随机利率模型下的投资-消费模型,解决了Ho-Lee利率模型、Vasicek利率模型下的最优投资-消费策略问题.
Continuous-time portfolio optimization problem is an important research content in the mathematical finance, which is also important theory and methodology for investors and investment institutions in the asset and risk hedge strategy. There are two main venues for portfolio optimization in the different investment environments. On the one hand, the problems in the real-world situations the investors are faced with are deal with by applying mathematical method. On the other hand, scientific evidence is supported for more widely application of theory and method of investment science. Some extensive researches on continuous-time portfolio optimization problem are obtained and some instructive and fruitful conclusions are achieved in this thesis. According to the different investment environment this thesis mainly investigate four problems: (1) extensive researches on dynamic asset allocation in an incomplete market; (2) asset and liability management problem in the stochastic environment; (3) portfolio selection with constraints; (4) the investment and consumption problems in the stochastic environment. Research conclusions are summarized as follows.
Chapter two investigates some extensive model on dynamic portfolio selection in an incomplete market. Firstly, we study a dynamic asset allocation problem in an incomplete market for utility maximizing criteria. A complete market is created by reducing the dimension of Brownian motion and martingale approach is used to obtain the analytic solutions for exponential utility and logarithm utility. According to the parameters relationships between the completed market and the original incomplete market, we achieve the optimal investment strategies in an incomplete market. A numerical example is given to illustrate the results obtained and analyze changing situation of the optimal trading strategies from a complete market to an incomplete market and compare the optimal investment strategies under exponential utility and logarithm utility with those under power utility. Secondly, we investigate the optimal investment strategy for quadratic utility in an incomplete market and apply martingale method to get the analytical solution to the optimal portfolio. As the same time, we also get the optimal investment strategy under mean-variance model. More importantly, our work support the theory ground for investigating the mean-variance model comprehensively. Thirdly, the optimal investment and consumption strategies in an incomplete market are studied and we obtain the explicit solution to the investment and consumption strategies for power utility and exponential utility and logarithm utility by applying dynamic programming principle and HJB equation. Finally, we introduce liability process into portfolio selection and investigate an asset and liability management problem in an incomplete market. We obtain the optimal portfolio for exponential utility by constructing exponential martingale approach and introducing quadratic optimization problem. All our works extend the models in an incomplete market and enrich the model of Zhang(2007).
Chapter three investigates asset and liability management problem in the stochastic environment, for example stochastic interest rate model and stochastic volatility model. Firstly, we study an asset and liability management problem for utility maximizing in a constant interest rate framework. The closed-form solutions to the optimal investment strategies under power utility and exponential utility and logarithm utility are derived by applying dynamic programming principle and Legendre transform. A numerical example is given to analyze the effect of the market parameters on the optimal portfolio. Secondly, we assume that risk-free interest rate and appreciation rate and volatility rate are allowed to be uniformly bounded stochastic processes. We apply stochastic linear quadratic control technique and results from backward stochastic differential equations(BSDEs) theory to the explicit solution to the optimal investment strategy. Thirdly, we introduce liability process into portfolio selection and suppose interest rate to be driven by the Ho-Lee model. The closed-form solutions to the optimal investment strategies for power utility and exponential utility are obtained by applying dynamic programming principle and HJB equation. Fourthly, we assume that interest rate dynamics is the Vasicek model and apply dynamic programming principle and Legendre transform to investigate an asset and liability management problem. The closed-form solutions under power utility and exponential utility are obtained. Introducing liability into portfolio selection and investigating the optimal investment strategy and risk management problem are the more important content in the assent and liability management. Our works enrich theory and method of asset and liability management. Our solving the optimal investment strategy under stochastic interest rate offers the theoretical supports for some investment institutions in the asset hedge and risk hedge problem.
Chapter four investigates the extensive models for dynamic portfolio selection with borrowing constraints. Firstly, we focus on a continuous-time dynamic portfolio selection problem with different interest rates for borrowing and lending. The closed-form solutions to the optimal portfolios under power utility and exponential utility and logarithm utility are derived by solving HJB equations and introducing the borrowing curve. The situations of borrowing and lending for the investors are analyzed under three different utility functions and a numerical example is given to illustrate the results obtained. Secondly, we introduce liability process into mean-variance model and obtain the closed-form solutions to the optimal investment strategy and the effective frontier are derived by applying Lagrange duality theorem and dynamic programming principle. Finally, we extend geometric Brownian motion to the constant elasticity of variance (CEV) model and investigate portfolio selection with borrowing constraints. The closed-form solutions to the optimal investment strategy and effective frontier are obtained. Our models enrich the theory and methodology for portfolio selection with borrowing constraints and extend the model in the Fu and Lari-Lavassani (2010) paper. Our research results present the theoretical techniques for portfolio selection problems with liability and CEV model.
Chapter five mainly investigates the investment and consumption problem in stochastic environments. Firstly, we suppose that there exists two assets, namely a risk-free asset and a risky asset and risk-free interest rate dynamic is Ho-Lee model. There is the correlation between interest rate and stock price. We take the expected discount utility of consumption and terminal wealth in the finite horizon as our objective function. By applying dynamic programming principle and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategies when the risky preference of the investor is given by power utility and logarithm utility function. Secondly, we extend the Ho-Lee model to the Vasicek model and investigate an investment and consumption problem. The closed-form expressions of the optimal investment and consumption strategies are derived by applying dynamic programming principle and Legendre transform and variable change techniques. Our work enriches and extends the theory and method on the investment and consumption problem and investigates investment and consumption problems with stochastic interest rates. The optimal investment and consumption strategies are obtained when risk-free interest rate dynamics is driven by the Ho-Lee model and the Vasicek model respectively.
第二章主要对不完全市场下的动态投资组合优化问题进行了扩展性研究.首先,对不完全市场下基于效用最大化的动态资产分配问题进行了研究.通过降低布朗运动的维数将不完全市场转化为完全市场,并在转化后的完全市场下应用鞅方法得到了指数效用和对数效用函数下最优投资策略的解析表达式.应用完全市场与原不完全市场间参数关系得到不完全市场下的最优投资策略.算例解释了模型的结论,分析了从完全市场到不完全市场下最优投资策略的变化情况,并把指数效用和对数效用函数下的最优投资策略与幂效用函数下的最优投资策略进行了比较.其次,对不完全市场下基于二次效用函数的投资组合优化问题进行了研究,应用鞅方法得到了最优投资组合的解析表达式.分析了均值-方差模型下不完全市场的最优投资策略问题,为进一步全面探讨均值-方差模型提供了理论基础.第三,对不完全市场下的投资-消费问题进行了研究,应用动态规划原理和HJB方程方法得到了幂效用、指数效用和对数效用函数下最优投资-消费策略的解析表达式.第四,对不完全市场下的资产-负债管理问题进行了研究,通过构造指数鞅方法和引入二次优化问题解决了指数效用函数下的最优投资组合问题.所有这些研究扩展了不完全市场下动态投资组合优化方面的研究,丰富和发展了Zhang的研究内容.
第三章主要研究随机环境下的资产-负债管理问题,如随机利率模型和随机波动率模型等.首先,在常数利率环境下对效用最大化下的资产-负债管理问题进行了研究,应用动态规划原理和Legendre变换-对偶解法得到了幂效用、指数效用和对数效用函数下最优投资策略的解析表达式,并给出算例分析了市场参数对最优投资组合的影响.其次,假设无风险利率、股票收益率和波动率均为一致有界随机过程,应用向后随机微分方程理论和随机线性二次规划方法得到了最优投资策略的解析表达式.第三,假设利率是服从Ho-Lee利率模型的随机过程,应用动态规划原理对资产-负债管理进行了研究,得到了幂效用和指数效用函数下最优投资策略的解析表达式.第四,对Vasicek利率模型下的资产-负债管理问题进行了研究,结合动态规划原理和Legendre变换-对偶解法得到了幂效用和指数效用函数下最优投资策略的解析表达式.将负债过程引入到投资组合优化问题中,并研究了此类问题的最优投资组合与风险管理的问题,是现阶段资产-负债管理的新的研究内容.本章的研究内容丰富和发展了资产-负债管理方面的理论方法,尤其是解决了随机利率模型下的最优投资组合问题,为随机利率模型下带有负债的投资机构进行资产套期保值和对冲风险提供了理论依据.
第四章主要对不同借贷利率限制下的动态投资组合优化问题进行了扩展性研究.首先,对效用最大化下的投资组合选择问题进行了研究,应用动态规划原理和HJB方程方法得到了幂效用、指数效用和对数效用函数下最优投资策略的解析表达式,并给出算例对不同借贷利率限制下投资人的投资行为进行了分析;其次,对负债情形下的均值-方差问题进行了研究,应用拉格朗日对偶定理和动态规划原理得到了最优投资策略和有效前沿的解析表达式;最后,将几何布朗运动扩展至CEV模型,对CEV模型下的投资组合优化问题进行了研究,得到了最优投资策略和有效前沿的解析表达式.本章的研究工作进一步丰富和发展了不同借贷利率限制下投资组合优化问题的理论方法,扩展了Fu和Lari-Lavassani等人的研究工作,为进一步研究负债和CEV模型下的投资组合优化模型提供了理论基础.
第五章主要研究了随机环境下的投资-消费问题.首先,我们假设金融市场中存在两种资产,一种资产是无风险资产,其中无风险利率是服从Ho-Lee利率模型的随机过程,且与风险资产价格存在线性相关性,以投资人有限投资周期内终端财富和累积消费的期望贴现效用作为目标函数,应用动态规划原理和HJB方程对幂效用和对数效用函数下的最优投资-消费策略进行了研究,得到了两种效用函数下最优投资-消费策略的解析表达式.其次,我们将Ho-Lee利率模型扩展至Vasicek利率模型,应用动态规划原理和Legendre变换-对偶方法得到了幂效用和对数效用函数下最优投资-消费策略的解析表达式.我们的这些研究将Merton的投资-消费模型扩展到随机环境下,并着重研究了随机利率模型下的投资-消费模型,解决了Ho-Lee利率模型、Vasicek利率模型下的最优投资-消费策略问题.
Continuous-time portfolio optimization problem is an important research content in the mathematical finance, which is also important theory and methodology for investors and investment institutions in the asset and risk hedge strategy. There are two main venues for portfolio optimization in the different investment environments. On the one hand, the problems in the real-world situations the investors are faced with are deal with by applying mathematical method. On the other hand, scientific evidence is supported for more widely application of theory and method of investment science. Some extensive researches on continuous-time portfolio optimization problem are obtained and some instructive and fruitful conclusions are achieved in this thesis. According to the different investment environment this thesis mainly investigate four problems: (1) extensive researches on dynamic asset allocation in an incomplete market; (2) asset and liability management problem in the stochastic environment; (3) portfolio selection with constraints; (4) the investment and consumption problems in the stochastic environment. Research conclusions are summarized as follows.
Chapter two investigates some extensive model on dynamic portfolio selection in an incomplete market. Firstly, we study a dynamic asset allocation problem in an incomplete market for utility maximizing criteria. A complete market is created by reducing the dimension of Brownian motion and martingale approach is used to obtain the analytic solutions for exponential utility and logarithm utility. According to the parameters relationships between the completed market and the original incomplete market, we achieve the optimal investment strategies in an incomplete market. A numerical example is given to illustrate the results obtained and analyze changing situation of the optimal trading strategies from a complete market to an incomplete market and compare the optimal investment strategies under exponential utility and logarithm utility with those under power utility. Secondly, we investigate the optimal investment strategy for quadratic utility in an incomplete market and apply martingale method to get the analytical solution to the optimal portfolio. As the same time, we also get the optimal investment strategy under mean-variance model. More importantly, our work support the theory ground for investigating the mean-variance model comprehensively. Thirdly, the optimal investment and consumption strategies in an incomplete market are studied and we obtain the explicit solution to the investment and consumption strategies for power utility and exponential utility and logarithm utility by applying dynamic programming principle and HJB equation. Finally, we introduce liability process into portfolio selection and investigate an asset and liability management problem in an incomplete market. We obtain the optimal portfolio for exponential utility by constructing exponential martingale approach and introducing quadratic optimization problem. All our works extend the models in an incomplete market and enrich the model of Zhang(2007).
Chapter three investigates asset and liability management problem in the stochastic environment, for example stochastic interest rate model and stochastic volatility model. Firstly, we study an asset and liability management problem for utility maximizing in a constant interest rate framework. The closed-form solutions to the optimal investment strategies under power utility and exponential utility and logarithm utility are derived by applying dynamic programming principle and Legendre transform. A numerical example is given to analyze the effect of the market parameters on the optimal portfolio. Secondly, we assume that risk-free interest rate and appreciation rate and volatility rate are allowed to be uniformly bounded stochastic processes. We apply stochastic linear quadratic control technique and results from backward stochastic differential equations(BSDEs) theory to the explicit solution to the optimal investment strategy. Thirdly, we introduce liability process into portfolio selection and suppose interest rate to be driven by the Ho-Lee model. The closed-form solutions to the optimal investment strategies for power utility and exponential utility are obtained by applying dynamic programming principle and HJB equation. Fourthly, we assume that interest rate dynamics is the Vasicek model and apply dynamic programming principle and Legendre transform to investigate an asset and liability management problem. The closed-form solutions under power utility and exponential utility are obtained. Introducing liability into portfolio selection and investigating the optimal investment strategy and risk management problem are the more important content in the assent and liability management. Our works enrich theory and method of asset and liability management. Our solving the optimal investment strategy under stochastic interest rate offers the theoretical supports for some investment institutions in the asset hedge and risk hedge problem.
Chapter four investigates the extensive models for dynamic portfolio selection with borrowing constraints. Firstly, we focus on a continuous-time dynamic portfolio selection problem with different interest rates for borrowing and lending. The closed-form solutions to the optimal portfolios under power utility and exponential utility and logarithm utility are derived by solving HJB equations and introducing the borrowing curve. The situations of borrowing and lending for the investors are analyzed under three different utility functions and a numerical example is given to illustrate the results obtained. Secondly, we introduce liability process into mean-variance model and obtain the closed-form solutions to the optimal investment strategy and the effective frontier are derived by applying Lagrange duality theorem and dynamic programming principle. Finally, we extend geometric Brownian motion to the constant elasticity of variance (CEV) model and investigate portfolio selection with borrowing constraints. The closed-form solutions to the optimal investment strategy and effective frontier are obtained. Our models enrich the theory and methodology for portfolio selection with borrowing constraints and extend the model in the Fu and Lari-Lavassani (2010) paper. Our research results present the theoretical techniques for portfolio selection problems with liability and CEV model.
Chapter five mainly investigates the investment and consumption problem in stochastic environments. Firstly, we suppose that there exists two assets, namely a risk-free asset and a risky asset and risk-free interest rate dynamic is Ho-Lee model. There is the correlation between interest rate and stock price. We take the expected discount utility of consumption and terminal wealth in the finite horizon as our objective function. By applying dynamic programming principle and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategies when the risky preference of the investor is given by power utility and logarithm utility function. Secondly, we extend the Ho-Lee model to the Vasicek model and investigate an investment and consumption problem. The closed-form expressions of the optimal investment and consumption strategies are derived by applying dynamic programming principle and Legendre transform and variable change techniques. Our work enriches and extends the theory and method on the investment and consumption problem and investigates investment and consumption problems with stochastic interest rates. The optimal investment and consumption strategies are obtained when risk-free interest rate dynamics is driven by the Ho-Lee model and the Vasicek model respectively.
引文
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