连续时间投资组合优化理论方法研究

发布时间：2018-05-20 07:13

本文选题：不完全市场 + 资产-负债管理问题　；参考：《天津大学》2012年博士论文

【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.
[Abstract]:The problem of continuous time investment portfolio is the key research content of mathematical finance. It is an important theoretical method of asset hedging and risk hedging by investors or investment institutions. By studying the problem of portfolio optimization under different investment environment, on the one hand, we can creatively solve people's investment process by mathematical method. The problems encountered in this study provide a theoretical basis for investors to invest in scientific investment; on the other hand, we can provide a scientific basis for more extensive application of the theory and methods of investment science. This paper mainly studies the extension of the problem of continuous time portfolio optimization, and has obtained some research results. The uncertainty of diversity and financial market is mainly focused on four aspects: (1) the expansibility of dynamic asset allocation under incomplete market; (2) research on asset liability management in random environment; (3) extensibility research on the problem of restrictive portfolio optimization; (4) research on investment consumption in random environment. The results of the study are described in detail as follows:
The second chapter mainly studies the expansibility of the dynamic portfolio optimization problem under incomplete market. First, the dynamic asset allocation problem based on the utility maximization under incomplete market is studied. By reducing the dimension of the Brown movement, the incomplete market is transformed into a complete market, and the application is applied to the complete market after the transformation. The analytic expression of the optimal investment strategy under the exponential utility and the logarithmic utility function is obtained by the martingale method. The optimal investment strategy under incomplete market is obtained by using the relation between the complete market and the original incomplete market. The calculation example explains the conclusion of the model and analyzes the change of the optimal investment strategy from the complete market to the incomplete market. The optimal investment strategy under the exponential utility and logarithmic utility function is compared with the optimal investment strategy under the power utility function. Secondly, the portfolio optimization problem based on the two utility function under incomplete market is studied. The analytic expression of the optimal portfolio is obtained by the martingale method. The optimal investment strategy of incomplete market under the variance model provides a theoretical basis for further study of the mean variance model. Third, the investment consumption problem under incomplete markets is studied. The optimal investment under the power utility, exponential utility and logarithmic utility function is obtained by using the dynamic programming principle and the HJB equation method. The analytical expression of the consumption strategy. Fourth, the problem of asset liability management under incomplete market is studied. The optimal portfolio problem under the exponential utility function is solved by constructing the exponential martingale method and introducing the two optimization problem. All these studies extend the study of dynamic portfolio optimization under incomplete markets. It enriches and develops the research content of Zhang.
The third chapter mainly studies the management of assets and liabilities under random environment, such as the stochastic interest rate model and the stochastic volatility model. First, the asset liability management problem under the utility maximization is studied under the constant interest rate environment. The power utility, the exponential utility and the Legendre change dual solution are applied to the problem of asset liability management under the constant interest rate environment. The analytic expression of the optimal investment strategy under the logarithmic utility function is given, and an example is given to analyze the effect of the market parameters on the optimal portfolio. Secondly, it is assumed that the risk free interest rate, the stock return rate and the volatility are uniformly bounded random processes, and the optimal investment is obtained by using the backward stochastic differential equation theory and the stochastic linear two programming method. The analytic expression of the capital strategy. Third, assuming that the interest rate is a stochastic process that obeys the Ho-Lee interest rate model, the dynamic programming principle is used to study the asset liability management, and the analytic expression of the optimal investment strategy under the power utility and the exponential utility function is obtained. Fourth, the asset liability management problem under the Vasicek interest rate model is carried out. The analytical expression of the optimal investment strategy under power utility and exponential utility function is obtained by combining the dynamic programming principle and the Legendre transformation dual method. The debt process is introduced into the portfolio optimization problem, and the optimal portfolio and risk management of this kind of problem are studied. The research content of this chapter enriches and develops the theoretical method of asset liability management, especially solving the optimal portfolio problem under the stochastic interest rate model, which provides a theoretical basis for the asset hedging and hedging risk of the investment institutions with liabilities under the stochastic interest rate model.
The fourth chapter mainly studies the expansibility of the dynamic portfolio optimization problem under the different lending rates. First, the portfolio selection problem under the utility maximization is studied. The dynamic programming principle and the HJB equation method are used to obtain the power utility, the exponential utility and the logarithmic utility function to analyze the optimal investment strategy. An example is given and an example is given to analyze the investor's investment behavior under the restriction of different lending rates. Secondly, the mean variance problem under the debt situation is studied. The Lagrange dual theorem and the dynamic programming principle are used to obtain the optimal investment strategy and the analytic expression of the effective frontier. Finally, the geometric Brown movement is carried out. This chapter extends to the CEV model, studies the portfolio optimization problem under the CEV model, and obtains the optimal investment strategy and the analytical expression of the effective frontier. This chapter further enriches and develops the theory square method of the portfolio optimization with different lending rate restrictions, and extends the research work of Fu and Lari-Lavassani. It provides a theoretical basis for further research on portfolio optimization models under liabilities and CEV models.
The fifth chapter mainly studies the investment and consumption problem in random environment. Firstly, we assume that there are two kinds of assets in the financial market, one kind of assets are risk-free assets, and the risk free interest rate is a random process that obeys the Ho-Lee interest rate model, and there is a linear correlation with the risk asset price, and the terminal money in the investor's limited investment cycle is in the end. The expected discounted utility of rich and cumulative consumption is the objective function, the optimal investment consumption strategy under power utility and logarithmic utility function is studied with dynamic programming principle and HJB equation, and the analytic expression of the optimal investment consumption strategy under two kinds of utility functions is obtained. The second, we extend the Ho-Lee interest rate model to Vasicek The interest rate model, using the dynamic programming principle and the Legendre transformation dual method, obtains the analytic expression of the optimal investment consumption strategy under the power utility and the logarithmic utility function. These studies extend the investment and consumption model of Merton to the random environment, and focus on the investment consumption model under the stochastic interest rate model and solve the problem. The Ho-Lee interest rate model and the optimal investment consumption strategy under the Vasicek interest rate model.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F224;F830.59

【引证文献】