The Research Of The Dynamic Portfolio Models In A Stochastic Market Environments

With the rapid development of technology, the popularity of theinternet, the acceleration of economic globalization, the world economichas been fully developed recent years. The global financial system hasalso undergone a process of change in such market environments. Thefinancial crisis, originated from the U.S. Wall Street, is the result ofignoring such changes. Therefore, it is crucial to consider the investmentportfolio model in a stochastic market environment. Under continuous-time framework and discrete-time framework, this thesis discusses fivesorts of portfolio models according to the di?erent risk aversion (or riskpreference) of di?erent investors. It gives the analytical forms for someoptimal portfolios, and it expresses the others in the form of expectationsof random variables. The feasibilities of the some models are examinedthrough numerical calculation and Monte Carlo simulation. The primarycontents and harvests of research will be generalized as follows:(1) Using continuous-time model to describe the dynamic characteristicof stochastic market environment and the returns of assets, we consideran investment strategy problem for maximizing expected terminal wealthutility with random horizon. We assume that the state of market envi-ronment follows a Brown motion with drift; and the risk assets returns follow stochastic diffusion processes which are affected by the stochasticmarket environment. The horizon of investment depends on the stateof the market environment, i.e., investors predefine some terminal con-straints according to the market state during investment process. Thecurrent investment strategy is stopped if the market state hits the pre-defined constraints. Here three constrains are discussed that are upsideconstrain, downside constrain and up-downside constrain, which repre-sent respectively the investment strategy focusing on a catastrophic riskcaused by excessive market status, the investment strategy paying atten-tion to the opportunities for profit at market bottom and both. Usingtechniques of dynamic programming and Feynman-Kac representationtheorem, we obtain analytical form of the optimal portfolio for the threecases. Furthermore, we compute numerically the proposed models andthe classical Merton model. The results show that the higher the cor-relation between the asset return and market environment, the greaterthe difference between the portfolio derived here and that derived fromMerton model. And the numerical results reveal also the impacts ofconstrains on the portfolios.(2) Assume that the state of stochastic market environment and theprices of risk assets follow stochastic diffusion process. Under the sameterminal constrains as in(1), we consider mean-variance portfolio prob-lem, and give a suffcient condition for the optimal portfolio. In order toderive the optimal portfolio, we reduce the problem to a nonlinear partialdifferential equation problem. Two kinds of special cases are discussed. First, if all coeffcients of market state and assets price dynamic processare constants, by introducing a transformation of function, we reducethe nonlinear partial differential equation into a linear one. Second, ifthe Brown motions, which trigger the market state and price of risk as-sets, are independent, the nonlinear partial differential equation problemdegenerates into linear one. Studying the solution to the linear partialdifferential equation problems, we derive the conditions of optimal port-folio and express the optimal portfolio in the form of expectations ofrandom variables.(3) The states of stochastic market environment follow stochastic diffu-sion processes. The underlying assets consist of risk assets and a default-able bond, where the risk assets follow stochastic diffusion process, andthe defaultable bond follows a Poisson process. We consider a portfolioproblem for maximization of the risk sensitized growth rate with infinitehorizon of investment, and give a suffcient condition for the optimalportfolio. Under a special assumption, we conclude that the optimalportfolio satisfies an algebra equation.(4) We consider a multi-period discrete-time mean-variance portfoliomodel imposed by a bankruptcy constraint. The market process is as-sumed to follow a Markov chain with finite state space. The mean vectorand covariance matrix of the random returns of risky assets all dependon the state of the market during any period. And the selection of port-folio satisfies the probability constraints that the wealth falls below thebankruptcy level during the investment. Primal-dual method and dy- namic programming is used to derive an optimal portfolio policy. Wecompare the model with bankruptcy constraint proposed here with themodel without bankruptcy constraint through numerical calculation anda Monte Carlo simulation. The results show the function of the proposedmodel on controlling over bankruptcy.(5) Assume characteristic of stochastic market environment is sameas that in (4). The asset returns are affected by the market state. Weconsider a minimax mean-variance portfolio model, where the expectedreturn of each underlying asset varies in an interval while the covari-ance between any two asset returns is given and fixed under per marketstate. Numerical calculation and Monte Carlo simulations demonstratethe distinct advantage for the proposed model under the worst marketenvironment.In summary, this thesis discusses the assets allocation strategy ac-cording to market environment state. The objective is to help investorsnot only achieve an optimal return but also have a risk control, andto provide some approaches of investment for different risk preferenceinvestors.