Study On Portfolio Optimization Models With Characteristics Of Incomplete Markets And Its Algorithm

The present complex situations of the real financial market have prompted theinternational communities to pay more attention to the theory of portfolio selection, which isone of the most important financial theories. So far, the classic models including Markowitz’smean–variance model, are discussed and proposed under the assumption of complete market.Due to the affects of market frictions, background risk and asymmetric information, the realfinancial market is usually incomplete. The friction factors in the real markets mainly includetransaction costs and minimum trading unit constraints. Besides, asymmetric information andimperfect rationality make the investors’ decision-making behaviors subjective andambiguous. Based on these considerations, we regard CVaR and safety first criteria as themeasures of risk, then establish portfolio optimization models with characteristics ofincomplete markets and their algorithms. In this paper, we do our work from the followingaspects, so that we can try to make the new portfolio selection theory more closer to the realfinancial markets, and gain a more profound understanding of the actual decision-makingbehaviors of the investors.Firstly, under the framework of mean-CVaR model and variable safety-first criteria, weimprove the risk measurement definition by adding the minimum trading unit andtransaction cost constraints. By taking forbidding short selling and restricting investment ratiointo consideration, we propose mean-CVaR model and variable safety first model both withtrading constraints. Then we solve the two models by using the algorithms which are designedwith quasi-Monte Carlo, Matlab optimization tool, genetic algorithm and stochasticsimulation.Secondly, we introduce the logistic membership function, and regards the maximizationof portfolio expected return and the minimization of CVaR under transaction constraints as itsgoal, and establish the Mean-CVaR fuzzy portfolio adjustment model with transactionconstraints. After transforming the complex model with nonlinear constrains into a simpleplanning problem which is much easier to calculate than before, we also solve the model byusing quasi-Monte Carlo method.Thirdly, based on the idea of safety-first criteria, Huang have consider several modelsunder different fuzzy environment. Therefore, considering the impact of trading constraints,we establish the variable safety-first portfolio model under the probability theory, then use“Chebyshev inequality under the credibility theory” to tighten the constraint of the credibilityrisk measure. Thus, basing on the variable safety-first criteria, we improve Huang’s fuzzy portfolio theory system.Finally, regarding CVaR as the risk measure indicator, we introduce the concept ofbackground risk, and discuss the formula and the shape of the mean-CVaR efficient frontierwith background risk. Then considering the real situation that background assets are related tothe financial assets, we propose the mean-CVaR portfolio optimal model with both tradingconstrains and background risk, and choose the appropriate programming software andoptimal toolbox to solve a numerical example to illustrate the validity and feasibility of themodel.