西安交通大学：多期风险度量与多阶段投资组合选择问题（博士学位论文）Multi-period Risk Measures and Multi-stage Portfolio Selection Problems.pdf_大学文库

摘要论文题目:多期风险度量与多阶段投资组合选择问题学科名称:数学学位申请人:刘嘉指导教师:陈志平教授摘要金融活动的本质是为了财富的快速增值,而金融活动的基本手段为投资运作.因此,关于最优投资策略的研究一直都是金融学和运筹学的重点研究领域.随着静态投资组合选择理论与方法的日趋完善,关于多期最优投资组合选择问题的研究近年来越来越受到金融学者和运筹学工作者的关注.尤其在我国,受近几年证券市场不景气的影响,以往带有投机色彩的短期交易行为很难再发挥作用,因此寻求稳健的多期投资策略已是一个急迫的研究课题.然而,随着金融市场全球化的发展,现实的市场环境越来越复杂,且时常会发生剧烈的变化,而投资者在选择投资策略时可能得不到完全的市场信息.这些不确定因素促使我们考虑如何在复杂的市场环境下,构建能刻画市场环境动态性的多期风险度量,由此构建能稳定获取收益且规避极端风险的多阶段投资组合选择模型.综合运用统计方法与多种现代优化技术,本文将就这些问题展开如下的研究 1)作为构建复杂环境下新型多期风险度量的基础,我们首先总结多期风险度量应该满足的基本性质,特别是多期风险度量特有的一个重要性质,时间相容性.同时,我们综述并归纳已有文献中出现的多期风险度量的形式,并按其构造方式将多期风险度量分为三类:终期财富风险度量,可加型风险度量,和递归型风险度量,进而给出这三类多期风险度量的一般形式.对于每类风险度量,我们分别总结分析其数学性质,然后讨论这三类多期风险度量之间的关系 (②)针对市场环境的时变特性,我们将机制转换模型,因子模型和时间序列模型相结合,构建了一种新的联合信息框架.基于这个信息框架,我们提出了一个基于机制转换的时间相容递归风险函数.当联合信息框架和基于机制转换的递归风险度量应用于多期投资问题时,我们论述了如何有效转换和求解该多期投资问题.实证结果表明了联合信息框架和相应多期投资模型的优异表现. (3)针对投资者不能精确获取市场完整信息的情况,我们使用分布式鲁棒优化技术来处理一类随机损失的具体分布未知而只知道分布的矩信息的问题.我们提出了一种基于可分期望条件函数的多期最坏情况风险度量.通过运用动态规划方法,我们导出了相应多阶段鲁棒投资组合选择问题的解析最优解.实证结果表明这种多期最坏情况风险度量和相应的多阶段鲁棒投资组合选择模型能规避市场最坏环境中的极端损失,做出稳健的投资策略. 本研究得到国家自然科学基金(编号:70971109,71371152,11571270)资助

ABSTRACT Title: Multi-period Risk Measures and Multi-stage Portfolio Selection Prob lems Descipline: Mathematics Applicant: Jia Liu Supervisor: Prof. Zhiping Chen ABSTRACT The essence of financial decision making problems is to pursuit higher profits, which is accomplished through choosing the optimal investment policy. Therefore, the optimal portfolio selection problem is always an important topic in financial optimization and operations research. Single-period portfolio selection problems have been extensively investigated, while the multi-period portfolio selection problem is attracting more and more attention in fields such as financial engineering and operations research. Espe. cially in China, the short-term investment can not earn stable profits any more due to the market depression. Hence, finding robust and optimal multi-stage portfolio becomes an urgent issue. Meanwhile, with the development of economic globalization, the market environment becomes more complicated, and the market state changes rapidly, it is thus very difficult for investors to obtain the complete market information. All these facts re- quire us to construct new multi-period risk measures which can capture the dynamics of market environment, and to construct resulting multi-stage portfolio selection models to gain stable returns while avoiding extreme risks. By simultaneously utilizing statistical methods and some modern optimization techniques, we will systematically investigate these issues from the following perspectives (1)As the basis for constructing new multi-period risk measures under complex market environment, we first review fundamental properties of multi-period risk measures, es pecially its time consistency. According to their structural properties, we divide current multi-period risk measures into three classes: terminal wealth risk measures, additive risk measures and recursive risk measures. Then we derive their generic forms. For each class, we survey typical formulations of multi-period risk measures in the literature, dis- cuss their mathematical properties and relationships (2)In order to describe the time-varying property of stochastic market, we propose a joint information framework by combining the regime switching model, the factor model The work is supported by the National Natural Science Foundation of China(Grant numbers: 70971109 71371152,11571270)

ABSTRACT Title: Multi-period Risk Measures and Multi-stage Portfolio Selection Problems Descipline: Mathematics Applicant: Jia Liu Supervisor: Prof. Zhiping Chen ABSTRACT The essence of financial decision making problems is to pursuit higher profits, which is accomplished through choosing the optimal investment policy. Therefore, the optimal portfolio selection problem is always an important topic in financial optimization and operations research. Single-period portfolio selection problems have been extensively investigated, while the multi-period portfolio selection problem is attracting more and more attention in fields such as financial engineering and operations research. Especially in China, the short-term investment can not earn stable profits any more due to the market depression. Hence, finding robust and optimal multi-stage portfolio becomes an urgent issue. Meanwhile, with the development of economic globalization, the market environment becomes more complicated, and the market state changes rapidly, it is thus very difficult for investors to obtain the complete market information. All these facts require us to construct new multi-period risk measures which can capture the dynamics of market environment, and to construct resulting multi-stage portfolio selection models to gain stable returns while avoiding extreme risks. By simultaneously utilizing statistical methods and some modern optimization techniques, we will systematically investigate these issues from the following perspectives* . (1) As the basis for constructing new multi-period risk measures under complex market environment, we first review fundamental properties of multi-period risk measures, especially its time consistency. According to their structural properties, we divide current multi-period risk measures into three classes: terminal wealth risk measures, additive risk measures and recursive risk measures. Then we derive their generic forms. For each class, we survey typical formulations of multi-period risk measures in the literature, discuss their mathematical properties and relationships. (2) In order to describe the time-varying property of stochastic market, we propose a joint information framework by combining the regime switching model, the factor model *The work is supported by the National Natural Science Foundation of China (Grant numbers: 70971109, 71371152, 11571270). III

西安交通大学博士学位论文 and the time series model. Based on the joint information framework, we define the regime-based time consistent nested risk measure. moreover we show how to establish and efficiently solve the corresponding multi-stage portfolio selection models by using the time consistent multi-period risk measure. We carry out a series of empirical tests to illustrate the superior performance of the proposed joint information framework and the corresponding multi-stage portfolio selection models (3)When an investor does not know the complete information about the distribution of the random loss, but only its moments information, we consider the distribution ally robust counterpart of the separate expectation conditional function, namely, the multi-period worst-case risk measure. By using the dynamic programming technique we derive the explicitly optimal investment strategy for the multi-stage robust portfo- lio selection problem under the multi-period worst-case risk measure. Numerical results demonstrate that the multi-period worst-case risk measure and the corresponding multi- stage portfolio selection model can help investors make robust decision to avoid extreme risks in worst-case scenarios (4)In more complex markets, not only the distribution of the random loss is unknown but its moments information is also unknown. To deal with the complex uncertainties in terms of both the distribution and the moments information, we propose two new uncertainty sets, and apply them to the multi-stage portfolio selection problems with the additive mean-CVar risk measure and the nested mean-CVar risk measure, respec tively. We find the closed form of the optimal portfolio or give an efficient solution method for these multi-stage robust portfolio selection problems. Numerical results il- lustrate that the robust mean-CVaR models with unknown moment information can steadily gain high returns and control extreme losses (5) To better describe the time-varying property of the multi-period investment risk with only partial information, we further propose two multi-period robust risk measures under the regime switching framework. We show that the corresponding multi-period robust portfolio selection problems under the regime dependent multi-period robust risk measures can be efficiently solved by using the scenario tree technique. Numerical results show the necessariness and efficiency of considering regime switching in multi-period ro- bust risk measures and multi-stage robust portfolio selection problems (6)For a class of investors who have pre-specified investment targets, we consider the effects of stochastic volatility on the time when the targets can be achieved. We define the earliest probabilistic target reaching time and propose a new risk measure which

‹SœåÆÆ¨Æ†ÿ© and the time series model. Based on the joint information framework, we define the regime-based time consistent nested risk measure. Moreover, we show how to establish and efficiently solve the corresponding multi-stage portfolio selection models by using the time consistent multi-period risk measure. We carry out a series of empirical tests to illustrate the superior performance of the proposed joint information framework and the corresponding multi-stage portfolio selection models. (3) When an investor does not know the complete information about the distribution of the random loss, but only its moments information, we consider the distributionally robust counterpart of the separate expectation conditional function, namely, the multi-period worst-case risk measure. By using the dynamic programming technique, we derive the explicitly optimal investment strategy for the multi-stage robust portfolio selection problem under the multi-period worst-case risk measure. Numerical results demonstrate that the multi-period worst-case risk measure and the corresponding multistage portfolio selection model can help investors make robust decision to avoid extreme risks in worst-case scenarios. (4) In more complex markets, not only the distribution of the random loss is unknown, but its moments information is also unknown. To deal with the complex uncertainties in terms of both the distribution and the moments information, we propose two new uncertainty sets, and apply them to the multi-stage portfolio selection problems with the additive mean-CVaR risk measure and the nested mean-CVaR risk measure, respectively. We find the closed form of the optimal portfolio or give an efficient solution method for these multi-stage robust portfolio selection problems. Numerical results illustrate that, the robust mean-CVaR models with unknown moment information can steadily gain high returns and control extreme losses. (5) To better describe the time-varying property of the multi-period investment risk with only partial information, we further propose two multi-period robust risk measures under the regime switching framework. We show that the corresponding multi-period robust portfolio selection problems under the regime dependent multi-period robust risk measures can be efficiently solved by using the scenario tree technique. Numerical results show the necessariness and efficiency of considering regime switching in multi-period robust risk measures and multi-stage robust portfolio selection problems. (6) For a class of investors who have pre-specified investment targets, we consider the effects of stochastic volatility on the time when the targets can be achieved. We define the earliest probabilistic target reaching time and propose a new risk measure which IV

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