Research Of Theory And Methodology On Continuous-time Portfolio Optimization

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.