Theory And Application Research On Dynamic Portfolio Choice With Time-varying Financial Markets

Financial market is full of all kinds of uncertainty, investors must face risk in the investment. How to effectively control and manage risk and how to form the optimal portfolio through diversified financial investment to effectively reduce the risk have become a big challenge and problem faced by investors. In reality, different investment may produce different risk. In particular, the optimal investment demand between short-term investors and long-term investors is totally different. Short-term investors usually only care about the expected return and variance of asset(portfolio) return within a single period and ignore possible variation of the investment opportunity set in the next period. The investment behavior of short-term investors in this case is considered to be short-sighted. The traditional portfolio choice theory ignores the impact of changes in the investment opportunity set on investment decisions.Plenty of empirical stylized facts, such as the cointegration effect, momentum effect, stochastic interest rate, stochastic volatility, and random transformation in macroeconomic condition indicate investment opportunity set is not fixed, but stochastic and time-varying. In practice, the investment behavior of investors is usually dynamic and multi-period, they are not only concerned about the shock of current investment opportunity set to wealth, but also the randomness of future investment opportunity set for the intertemporal impact of wealth. As the investment opportunity set changes over time, in addition to short-term demand for financial assets, investors have intertemporal hedging demand, namely the use of financial assets to hedge the intertemporal impact of stochastic investment opportunity set on wealth.Therefore, from a long-term perspective, the study of dynamic portfolio selection problem is not only of great theoretical value, but also has important practical significance.In this context, this dissertation considers several issues originating from dynamic portfolio choice under different financial market characteristics. Based on utility function theory, stochastic control theory and stochastic differential game theory, this paper establishes strict mathematical financial models and systermatically discusses model features, applicable conditions, investment and consumption behavior, intertemporal hedging demand, paired trading and momentum investment strategy and so on. The main research results are summarized as follows.Firstly, according to the cointegration effect between equity assets, this paper studies the optimal investment and consumption problem by assuming there exists cointegration relation between stocks in financial market. The investment objective is to maximize the expected discounted finite horizon, time-additive utility of intermediate consumption and terminal wealth. By using stochastic control theory, this paper derives high-dimensional, nonlinear, the non-homogeneous Hamilton-Jacobi-Bellman(HJB) equation for the optimal value function under power utility and logarithm utility respectively. By variable substitution, the optimal investment and consumption are explicitly given. For comparison, it further discusses four special cases of co-integration model: 1. investors only care about the maximization of terminal wealth; 2. investors only care about the maximization of total consumption utility during finite horizon;3. There is no co-integration relationship between stock prices; 4. Risk aversion coefficient is zero. Based on the above four cases, this paper has carried on the comparative static analysis about the influence of the speed rate of cointegration vector on the welfare of investors, optimal consumption and investment under three market scenarios respectively. The research results show that, under the market scenario 1, when the prices of two risky assets are undervalued, investors tend to buy two undervalued risky assets through financial leverage. Under the market scenario 2, when the prices of two risky assets are overvalued, investors tend to be short selling all the cointegration risk assets. Under the market scenario 3, of which the price of an asset is overvalued, another asset price is undervalued, investors with mild degree of risk aversion tend to adopt a "long-short" investment patterns, This corresponds to pair trading strategy in the financial practice.Secondly, in view of the momentum effect in stock returns in a short period, this paper studies the optimal consumption and portfolio choice problem when there exists momentum in the stock returns. The investment objective is to maximize consumption utility in the whole life. Under these assumptions, this paper derives exact explicit expressions for the optimal consumption and portfolio choice when investors have unit elasticity of intertemporal substitution of consumption and approximate solution to value function optimal consumption and optimal investment through log-linear approximation method for elasticities of intertemporal substitution different from one. Under these assumptions, the model is calibrated to Chinese stock market data and furthermore numerically analyze the impact of momentum effect on optimal investment and consumption pattern.The research results show that for the influence of personal preference parameters on the optimal investment strategy, relative risk aversion levels is far more important than the elasticity of intertemporal substitution. When the initial value of momentum state variable takes moderate negative value, the optimal investment demand is greater than zero and intertemporal hedging demand is less than zero. In addition, when the risk aversion coefficient is greater than 1, the intertemporal hedging demand plays a very important role in the total investment demand. When the latest levels of stock returns are positive or moderate negative, intertemporal hedging demand motives greatly decrease the portfolio demand for stocks by investors whose risk aversion exceed one. Further intertemporal hedging demand greatly increases the portfolio demand for stocks when the levels of stock returns are sufficient negative. Moreover, for the influence of personal preference parameters on the optimal consumption-wealth ratio, the elasticity of intertemporal substitution is far more important than relative risk aversion levels. Furthermore, given the risk aversion coefficient, the optimal consumption-wealth ratio is monotone decreasing function of the intertemporal substitution elasticity. Meanwhile, given the intertemporal substitution elasticity coefficient, the ratio of the optimal consumption to wealth is monotone decreasing function of risk aversion.Thirdly, this paper considers the optimal interactive decision problem when two investors face the same investment opportunity set. Stock price is assumed to follow constant elasticity of variance stochastic volatility model in order to describe the volatility smile phenomenon existed in financial market. The investment objective is to maximize the utility of the weighted average of personal terminal wealth and the difference of wealth between investors. By using stochastic control theory, this paper derives Hamilton-Jacobi-Bellman(HJB) equations for the two optimal value function. Furthermore, the dissertation derives explicit expressions for optimal equlibrium strategy under power utility and exponential utility. Under the above theory results, the dissertation proceeds the comparative static analysis about the equilibrium strategies obtained under the exponential utility function according to different model parameters. The results show that the equilibrium strategy is a monotone decreasing function of related volatility parameters, elastic coefficient, the investor’s own risk aversion coefficient and risk-free interest rate. However, equilibrium strategy is a monotone increasing function of the investment period and the initial stock price. Furthermore, equilibrium strategy first increases with stock expected return and then decreases.Finally, in order to study the impact of inflation and random transformation of macroeconomic condition on investment decisions It is necessary to assume that stock prices and inflation are described by generalized geometric Brownian motion driven by the markov process. By using stochastic differential game theory, this paper considers the optimal interactive decision problem when two investors face correlated but different investment opportunity set. The investment objective is to maximize the utility of the sum of two terminal wealth. By using stochastic control theory, this paper derives Hamilton-Jacobi-Bellman(HJB) equations for the two value function. Thus, when the investor has power utility and financial model has inflation, this paper deduces Feynman-Kac expression for value function and explicit expression for the optimal equilibrium strategies. Meanwhile, when the investor has exponential utility and financial model does not have inflation, this paper also derives Feynman-Kac expression for value function and explicit expression for the optimal equilibrium strategies. Based on the above results, in order to depict dynamic effect of regime switching on equilibrium strategies, the paper discusses one special case with two regimes and then proceeds the comparative static analysis about the equilibrium strategies obtained under the exponential utility function according to different model parameters. the research results show that the regime switching in the model parameters has a significant impact for the optimal portfolio strategies.