稳定分布及投资组合研究
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摘要
正确的投资决策是建立在对收益率与风险的可靠预测之上的，而可靠的预测只能通过基于现实假定上的统计模型而得到。稳定分布能够描述金融数据中的两个重要经验特征：厚尾以及倾斜。本文对深圳成分指数（SZSI）以及上海综合指数（SHCI）的稳定分布、概率准则投资组合问题以及机会约束投资组合问题进行了研究，主要内容及研究结果如下：
1. 介绍了一元稳定分布以及多元稳定分布与稳定随机过程的基本理论。
2. 对SZSI以及SHCI建立了非对称稳定幂α-GARCH模型，对其标准稳定新息进行了稳定分布拟合、稳定性检验以及预测精确性的实证检验。利用预测的稳定随机变量，得到了SZSI和SHCI的联合稳定分布下密度函数的图示。
3. 对于投资组合的均值方差问题，给出了求解不允许卖空时有效边界的一种精确算法，并与通常的二次规划方法进行了比较。
4. 研究了正态分布下的概率准则投资组合问题，给出了允许卖空时最优解的解析表达式以及不允许卖空时求解最优解的方法；研究了完备标准动态金融市场中允许卖空时的概率准则投资组合问题，给出了准则函数的上确界及任一非零测度集合的贴现资产过程和投资组合过程,得到了无论投资者的预期贴现资产有多大，投资者一定能以任意接近于1的概率在任一正区间实现其目标的结论，预期贴现资产与起始资产间的差异越大，风险就越大；研究了完备标准动态金融市场中容许投资组合时的概率准则投资组合问题，给出了准则函数，贴现资产过程及最优允许投资组合过程的解析表达式，期望贴现资产越大，准则函数越小、贴现资产的均值及风险越大；研究了稳定分布下两个风险资产、两个风险资产和一个无风险资产的概率准则投资组合问题，给出了求解最优解数值算法的具体步骤。
5. 研究了正态分布下机会约束的最小风险投资组合问题，给出了允许卖空时最优解的解析表达式，不允许卖空时求解最优解的方法；研究了稳定分布下两个风险资产、两个风险资产和一个无风险资产的机会约束投资组合问题，给出了可行集及求解最优解数值算法的具体步骤。
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 admissibleportfolio 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.
1. 介绍了一元稳定分布以及多元稳定分布与稳定随机过程的基本理论。
2. 对SZSI以及SHCI建立了非对称稳定幂α-GARCH模型，对其标准稳定新息进行了稳定分布拟合、稳定性检验以及预测精确性的实证检验。利用预测的稳定随机变量，得到了SZSI和SHCI的联合稳定分布下密度函数的图示。
3. 对于投资组合的均值方差问题，给出了求解不允许卖空时有效边界的一种精确算法，并与通常的二次规划方法进行了比较。
4. 研究了正态分布下的概率准则投资组合问题，给出了允许卖空时最优解的解析表达式以及不允许卖空时求解最优解的方法；研究了完备标准动态金融市场中允许卖空时的概率准则投资组合问题，给出了准则函数的上确界及任一非零测度集合的贴现资产过程和投资组合过程,得到了无论投资者的预期贴现资产有多大，投资者一定能以任意接近于1的概率在任一正区间实现其目标的结论，预期贴现资产与起始资产间的差异越大，风险就越大；研究了完备标准动态金融市场中容许投资组合时的概率准则投资组合问题，给出了准则函数，贴现资产过程及最优允许投资组合过程的解析表达式，期望贴现资产越大，准则函数越小、贴现资产的均值及风险越大；研究了稳定分布下两个风险资产、两个风险资产和一个无风险资产的概率准则投资组合问题，给出了求解最优解数值算法的具体步骤。
5. 研究了正态分布下机会约束的最小风险投资组合问题，给出了允许卖空时最优解的解析表达式，不允许卖空时求解最优解的方法；研究了稳定分布下两个风险资产、两个风险资产和一个无风险资产的机会约束投资组合问题，给出了可行集及求解最优解数值算法的具体步骤。
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
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.
引文
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