The Study On Mean-Variance Investment Problems In Incomplete Markets

The optimal portfolios and claim pricing and hedging are addressed under marker incompleteness and mean-variance preferences. On the one hand, the main contents and results on portfolios are as follows:The two-funds strong separation theorem holds and the mean-variance efficient frontier is a straight line in standard deviation-expected return space in the frictionless and constrained financial markets.The results obtained in the frictionless market are applied to study the effects of incomplete information to portfolios. The information affects not only the proportion of capital allocation in the riskless asset to that in risky ones, but also composition of risky assets. The less the information, the lower the expected payoff, risk and the risk price of market.The notion of the payoff-risk coefficient of information is proposed. The payoff-risk coefficient of information increases with the increase of information. The payoff and risk of the optimal risky portfolio is determined by both the payoff-risk coefficient of information and the risk aversion parameter. But the risk price of market is completely dominated by the risk aversion parameter. The payoff of the optimal risky portfolio is unique in the sense of constant times under equivalent information.3. The short-selling mechanism affects the optimal portfolio. The expected mean, the variance and the risk price of the payoff of the risky portfolio without short selling is less than with it. 4. Consider the effects of both information and short-selling mechanism. We conclude that the two-funds strong separation theorem holds and that the mean-variance efficient frontier is a straight line in standard deviation-expected return space under given partial information whether short selling is allowed or not.5. The numeral examples are given at each part.These results apply to static and dynamical portfolio problems although there are differences between them.On the other hand, the claim pricing and hedging are studied in an incomplete market in different criterions. The payoffs in different criterions are given and the performances of them are compared. And using the indifference argument, fair hedging price is derived in a new proof.