Stable Distribution AndPortfolio Research

Right investment decision requires reliable predictions of return and risk, and reliable predictions can only be obtained if the underlying statistical model rests on realistic assumptions. Stable laws are able to capture the two main characteristics of empirical evidence that returns follow a heavy-tailed and sometimes even skewed distribution. The stable distributions of Shenzhen Stock Sub-index (SZSI) and Shanghai Stock Composite Index (SHCI) are discussed, and the portfolio problems of probability criterion and chance-constrained programming are also analyzed. The main contents and results are as follows:1. The basic theories of univariate stable distribution and multivariate stable distributions and stable stochastic processes are introduced.2. The stable Asymmetric Powerα-GARCH model is applied to SZSI and SHCI and stable law is fitted into the empirical distributions. The stability of standardized stable innovation is checked and the evaluation of prediction accuracy is performed. Furthermore, by the predicted stable random variables, the plot of joint stable density function of SZSI and SHCI is provided.3. For mean-variance efficient frontier, an exact algorithm and its comparison with quadratic programming method are given.4. The optimal portfolio selection of probability criterion with normal distribution is considered, the analytic representation of the optimal portfolio with short selling allowed is obtained and the method for solving the optimal portfolio with no short selling allowed is given; The optimal portfolio selection of probability criterion in dynamic financial market with short selling allowed is analyzed. The upper limit of criterion function and the corresponding discounted wealth process and hedging portfolio process are obtained. An investor can realize his claim in any positive time interval with probability arbitrarily close to 1 no matter how great his expected discounted wealth is. The greater the differences between the expected discounted wealth and initial wealth, the greater the risk; The optimal portfolio selection of probability criterion in dynamic financial market with admissible portfolio is discussed. The analytic representations of the maximum of probability criterion function and the discounted wealth process and the optimal admissible portfolio process are obtained. The greater the expected discounted wealth at terminal time, the smaller the probability criterion function and the greater the mean and the variance of the discounted wealth at terminal time; The optimal portfolio selection of probability criterion with stable distribution for two risky assets or two risky assets and one riskless asset is considered. The steps of numerical method are given.5. The optimal chance-constrained portfolio selection with normal distribution is discussed, the analytic representation of the optimal portfolio with short selling allowed is obtained and the method for solving the optimal portfolio with no short selling allowed is given; The optimal chance-constrained portfolio selection with stable distribution for two risky assets or two risky assets and one riskless asset is considered, the permission set is given and the steps of numerical method are provided.