Research On A Direct Method For Optimal Portfolio Problem In The Security Market

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.