| This thesis is devoted to the mean-risk portfolio optimization problem in a continuous-time financial market, where we want to minimize the risk of the investment and at the same time ensure that a given expected return level is obtained. Three topics are studied in this thesis. (1) The first topic is the mean-variance portfolio selection problem with bankruptcy prohibition in a complete continuous-time market. The problem is completely solved using a decomposition approach. Specifically, when bankruptcy is prohibited, we find that the efficient policy for a mean-variance investor is simply to purchase a European put option that is chosen, according to his or her risk preferences, from a particular class of options. Moreover, we obtain the efficient frontier by a system of parameterized equations. (2) The second topic is the mean-variance portfolio selection problem with or without constraints in an incomplete continuous-time market. Four models are discussed: portfolios unconstrained, shorting prohibited, bankruptcy prohibited, and both shorting and bankruptcy prohibited. A duality method is used to solve all the models, and explicit solution are obtained when parameters of the market are all deterministic. (3) The third topic is the general mean-risk portfolio selection problem in a complete continuous-time market. In this mean-risk problem, we measure the risk by the expectation of a certain function of the deviation of the terminal payoff from its mean. First of all, the weighted mean-variance problem is solved explicitly. The limit of this weighted mean-variance problem, as the weight on the upside variance goes to zero, is the mean-semivariance problem which is shown to admit no optimal solution. This negative result is further generalized to a mean-downside-risk portfolio selection problem where the risk has non-zero value only when the terminal payoff is lower than its mean. Finally, a general model is investigated where the risk function is convex. Sufficient and necessary conditions for the existence of optimal portfolios are given. Moreover, optimal portfolios are obtained when they do exist, and asymptotic optimal portfolios are obtained when optimal portfolios do not exist. |
