The Research On Portfolio Selection With Heterogeneous Beliefs In An Incomplete Market

The traditional portfolio theories are built on the complete market environment and assum the presence of homogeneous beliefs.Obviously,these assumptions don't match the actual situation in financial markets.The reality financial market is closer to an incomplete market.In Chinese reality financial market,as investors' economic statu,education level,knowledge background,degree of preference to returns and risks and ability to withstand risks are different,the heterogeneity of investors' belief is inevitable.Investors' having heterogeneous beliefs is an important feature of incomplete market.Heterogeneous beliefs will greatly influence the formation of asset prices and especially result in the differences of investors' investment strategies.Therefore,the research of portfolio selection problem with heterogeneous beliefs in an incomplete market has important practical significance.Incomplete market studied in this paper is that the number of underlying assets available for trading in the market is less than the dimensions of Brownian motion to meet prices of risky assets and investors have heterogeneous beliefs.Heterogeneous beliefs refer to that investors may well have different expected prices of risky assets because they have different prior beliefs based on the assumption that the investors get all the information for free at the same time,and then the heterogeneous beliefs can affect investors' investment decisions.Firstly,we apply information structure to describe heterogeneous beliefs studied in this paper,define the incomplete market,and make an analysis of the equity prices under heterogeneous beliefs.In incomplete market with heterogeneous beliefs,as HJB equation in stochastic optimal control method is often in the form of highly non-linear partial differential equations,we are more difficult to obtain explicit solutions.Especially when the market is incomplete or has additional restrictions,the use of stochastic optimal control method to solve the portfolio selection problem will be more difficult.And in incomplete market,there is no single equivalent martingale measure and martingale method will largely be limited.Therefore,this paper firstly transforms incomplete market into a complete market by dimensionality reduction method,and uses the martingale method to solve the portfolio selection problem under complete market after dimensionality reduction;then combines with investors' heterogeneous beliefs and parameters relationship between complete market after dimensionality reduction and original incomplete market to solve portfolio selection problem in incomplete market with heterogeneous beliefs.The results indicate that heterogeneous belief is an important factor of affecting portfolio selection in incomplete market and the ratio of risky assets invested by investors will increase with an increase in investor's optimism or confidence in the market.Finally,this paper presents numerical examples and draws a conclusion that in incomplete market,investors with different forms of the utility function select different optimal portfolio and have different sensitivity to changes in risk aversion coefficient.Through numerical analysis,we obtain that at the initial moment,if investors are more optimistic or more confidence in the market,their investment in risk assets will increase,and if investors are more pessimistic,or a lack of confidence in the market,their investment in risky assets will reduce.Therefore,heterogeneous prior beliefs of investors(investors' varying degrees of optimism or pessimism and different confidence in the market)will affect investors' portfolio selection.