Models And Algorithms For Some Kinds Of Portfolio Optimization Problems

Portfolio optimization, which addresses the ideal assignment of resources to thefinancial assets to balance the assets returns and the assets risks, is one of the coreresearch fields in modern financial management. In1952, Harry M. Markowitz, theAmerican economist, introduced the variance of the assets returns as the measure ofrisks in the paper “Portfolio Selection” which published in “The Journal of Finance”.This paper is considered as the beginning of the modern investment theory and thefoundation of numerical analysis of financial investment. Along with the improvementof the modern mathematical methods and the emergence of mathematical finance, thestudy of modern financial investment theory are no more only describing studies or pureempirical researches, but the numerical analysis guiding the behavior of investors.Today, the world’s economics grow rapidly, but the financial crisis and volatility arealso increased. And in our country, although the financial market has been developedgreatly, it is still not so perfect that the investors face more and more complicatedtheoretical and practical problems. So the researches on portfolio optimization becomemore and more important.The present dissertation makes a deep and systematic investigation on candidateconstraint portfolio optimization problem, dynamic portfolio optimization problem andscenario generation methods in portfolio optimization problem. The main researchesand creative results of this dissertation are shown as follows:1. The Artificial Bee Colony algorithm, proposed for the high dimension andmulti-modal problem, is a recently introduced optimization algorithm. Considering theadvantages of Artificial Bee Colony algorithm, the cardinality constrainedmean-variance model is solved by using this algorithm. The experimental results showthat the proposed algorithm performs well for the portfolio optimization problem.2. To tackle the cardinality constrained portfolio optimization problem, anmodified Artificial Bee Colony algorithm is designed. The Deb’s selection rule isintroduced to guarantee the feasibility of optimal solution. To improve the convergentspeed, a new search strategy is proposed. Furthermore, the Bolzmann selectionprobability is employed to maintain the population diversity. The experiment resultsindicate that the proposed algorithm is efficient and effective for the portfoliooptimization problem, which can obtain a better portfolio strategy and diversify theportfolio risk efficiently.3. The general multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs are investigated. According to the dynamicprogramming approach, the optimal strategies, the boundaries of the no-transactionregion and the efficient frontier are given in the explicit form. Therefore, the long terminvestment strategies for the investors are given. Numerical result shows that themethod provided in this paper works well.4. A mean-variance portfolio selection problem in continuous time with fixed andproportional transaction costs is investigated. Utilizing the dynamic programming, theHamilton-Jacobi-Bellman equation is derived, and the explicit closed form solution isobtained. Furthermore, the optimal strategies and efficient frontiers are also proposedfor the original mean-variance problem. Numerical experiments present the variation ofthe transaction region and efficient frontier with the transaction costs change, whichdemonstrate the proposed method performs effectively.5. The effectiveness of forecasting and decision-making by using four scenariogeneration methods are compared. The results of in-sample and out-of-sample propertyof the portfolios, obtained by using these methods to generate the rates of return, showthat for the Chinese stock market, the scenario generation methods and optimizationmodel are useful for forecasting and decision-making. Moment matching method canbetter reflect the market’s downward trend and multivariate GARCH method can betterreflect the market’s upward trend.At last, the questions of the portfolio optimization and its future developingtendency are summarized.