西安交通大学 博士学位论文 多期风险度量与多阶段投资组合选择问题 学位申请人：刘嘉 指导教师：陈志平 学科名称：数学 2017年2月
ƨƆÿ© ıœºx›˛Üı „›]|‹¿JØK Ɔû<: 4W çì: ùì² Æⶰ: ÍÆ 2017 c 2
Multi-period Risk Measures and Multi-stage Portfolio Selection problems a dissertation submitted to Xi'an Jiaotong University in partial fulfillment of the requirement for the degree of Doctor of Natural so B Jia liu Supervisor: Prof. Zhiping Chen (Mathematics) February 2017
Multi-period Risk Measures and Multi-stage Portfolio Selection Problems A dissertation submitted to Xi’an Jiaotong University in partial fulfillment of the requirement for the degree of Doctor of Natural Science By Jia Liu Supervisor: Prof. Zhiping Chen (Mathematics) February 2017
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
£ á ÿ©K8: ıœºx›˛Üı „›]|‹¿JØK Æⶰ: ÍÆ Æ†û<:4W çì: ùì² « £ á 7K¹ƒ ü¥è „LØÑOä, 7K¹ƒƒ Äè›]$ä. œ d, 'uÅ`›]¸—ÔƒòÜ—¥7KÆ⁄$ Æ:Ôƒ+ç. ëX·›] |‹¿JnÿÜê{F™ı, 'uıœÅ`›]|‹¿JØKÔƒCc55 …7Kƈ⁄$ ÆÛäˆ'5. cŸ3·I, …CAcy ½|ÿµÌK è, ± ëk›Å⁄Á·œ¥1èÈJ2uûä^, œdœ¶Ëıœ›]¸— Æ¥òá:½ÔƒëK. , , ëX7K½| •zu–, y¢½|Ǹ5 E,, Öû~¨u)ÏCz, ›]ˆ3¿J›]¸—ûåUÿ ½|& E. ˘ ÿ(½œÉr¶·ÇƒX¤3E,½|Ǹe, ÔUèx½|Ǹƒ 5ıœºx›˛, ddÔU½º¬ÃÖ5;4‡ºxı „›]|‹¿J .. n‹$^⁄Oê{Üı´yì`zE‚, ©Ú“˘ ØK–mXeÔƒ* : (1) äèÔE,Ǹe#.ıœºx›˛ƒ:, ·Çƒko(ıœºx›˛A T˜vƒ 5ü, AO¥ıœºx›˛Akòáá5ü, ûmÉN5. ”û, · Çn„ø8BÆk©z•—yıœºx›˛/™, øUŸEê™Úıœºx› ˛©èna: ™œ„Lºx›˛, å\.ºx›˛, ⁄48.ºx›˛, ? â—˘n aıœºx›˛òÑ/™. Èuzaºx›˛, ·Ç©Oo(©¤ŸÍÆ5ü, , ?ÿ˘naıœºx›˛Ém'X. (2) Ƚ|ǸûCA5, ·ÇÚÅõ=Ü ., œf .⁄ûmS .É (‹, Ô ò´#È‹&Eµe. ƒu˘á&Eµe, ·ÇJ— òáƒuÅõ= ÜûmÉN48ºxºÍ. È‹&Eµe⁄ƒuÅõ=Ü48ºx›˛A^u ıœ›]ØKû, ·Çÿ„ X¤k=Ü⁄¶)Tıœ›]ØK. ¢y(JL² È ‹&Eµe⁄ÉAıœ›] .`…Ly. (3) È›]ˆÿU°(º½|&Eú¹, ·Ç¶^©Ÿ™°ï`zE‚ 5?nòaëÅõî‰N©Ÿô ê©Ÿ›&EØK. ·ÇJ— ò´ ƒuå©œ"^áºÍıœÅÄú¹ºx›˛. œL$^ƒ5yê{, ·Ç— ÉAı „°ï›]|‹¿JØK)¤Å`). ¢y(JL²˘´ıœÅÄú¹º x›˛⁄ÉAı „°ï›]|‹¿J .U5;½|ÅÄǸ•4‡õî, â —Ë›]¸—. * ÔƒI[g,âƃ7 (?“: 70971109, 71371152, 11571270) ]œ. I
西安交通大学博士学位论文 (4)在复杂的市场环境中,不仅随机损失的具体分布不确定,其分布矩信息的精确 值亦未知.为了刻画分布矩信息不确定的更复杂情况,我们构造了两种矩信息不确定的 不确定集,并将其分别应用于基于可加型均值-CVaR度量和基于递归型均值-CVaR 度量的多期鲁棒投资策略选择模型.我们分别珨给岀了这些多期鲁棒投资策略选择模型 的解析最优解或者有效的求解方法.实证结果表明矩信息未知的鲁棒均值-CVaR模 型在获取稳健收益的同时更能够控制极端损失的岀出现 (5)针对市场信息不完备且市场环境动态变化的情况,我们将分布式鲁棒技术与机 制转换相结合,提出了两种基于机制转换的多期鲁棒风险度量.我们进而构建了相应的 多期鲁棒投资组合选择模型,并运用情景树技术设计了有效求解方法.实证研究表明了 在多期鲁棒风险度量和多期鲁棒投资组合选择中考虑机制转换的必要性和有效性. (6)最后,针对一类投资目标已预先给定的投资者,我们考虑市场随机环境对其实 现目标期限的影响,提出了概率目标首达时的概念,并将预定完成时间与概率目标首达 时之间的时间成本定义为目标达成型风险度量.这种新型的多期风险度量可以看作是 VaR在动态市场环境下的推广.进而,我们构造了基于目标达成型风险度量的多期投 资策略选择模型,并给出了求解最优概率目标首达时的有效方法.实证研究表明基于目 标达成型风险度量的多期投资策略选择模型能有效地帮助投资者尽早完成其投资目标 关键词:多期风险度量;多阶段投资组合;机制转换;随机规划;鲁棒优化 论文类型:应用基础
‹SœåÆƨƆÿ© (4) 3E,½|Ǹ•, ÿ=ëÅõî‰N©Ÿÿ(½, Ÿ©Ÿ›&E°( ä½ô. è èx©Ÿ›&Eÿ(½çE,ú¹, ·ÇE ¸´›&Eÿ(½ ÿ(½8, øÚŸ©OA^uƒuå\.˛ä - CVaR ›˛⁄ƒu48.˛ä - CVaR ›˛ıœ°ï›]¸—¿J .. ·Ç©Oâ— ˘ ıœ°ï›]¸—¿J . )¤Å`)½ˆk¶)ê{. ¢y(JL²›&Eô°ï˛ä - CVaR .3ºË¬Ã”ûçU õõ4‡õî—y. (5) Ƚ|&EÿÖ½|ǸƒCzú¹, ·ÇÚ©Ÿ™°ïE‚ÜÅ õ=ÜÉ(‹, J— ¸´ƒuÅõ=Üıœ°ïºx›˛. ·Ç? Ô ÉA ıœ°ï›]|‹¿J ., ø$^úµ‰E‚O k¶)ê{. ¢yÔƒL² 3ıœ°ïºx›˛⁄ıœ°ï›]|‹¿J•ƒÅõ=Ü7á5⁄k5. (6) Å, Èòa›]8IÆ˝kâ½›]ˆ, ·Çƒ½|ëÅǸȟ¢ y8IœÅKè, J— V«8IƒàûVg, øÚ˝½§ûmÜV«8Iƒà ûÉmûm§ ½¬è8Ià§.ºx›˛. ˘´#.ıœºx›˛å±wä¥ VaR 3ƒ½|ǸeÌ2. ? , ·ÇE ƒu8Ià§.ºx›˛ıœ› ]¸—¿J ., øâ— ¶)Å`V«8Iƒàûkê{. ¢yÔƒL²ƒu8 Ià§.ºx›˛ıœ›]¸—¿J .Uk/ꜛ]ˆ¶@§Ÿ›]8I. ' Ö c: ıœºx›˛; ı „›]|‹; Åõ=Ü; ëÅ5y; °ï`z ÿ©a.: A^ƒ: II
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
‹SœåÆƨƆÿ© 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
ABSTRACT takes into account the extra time value between the pre-set time and the earliest proba bilistic target reaching time. The introduced target reaching risk measure can be viewed as a dynamic generalization of Value-at-Risk. Moreover, we consider its application to the multi-period portfolio selection problem and find an efficient method to find the op- timal earliest probabilistic target reaching time. Numerical results show the multi-stage portfolio selection with the target reaching risk measure can help investors reach their investment targets as early as possible KEY WORDS: Multi-period risk measure: Multi-stage portfolio selection; Regime switching; Stochastic programming; Robust optimization TYPE OF DISSERTATION: Applied Fundamentals
ABSTRACT takes into account the extra time value between the pre-set time and the earliest probabilistic target reaching time. The introduced target reaching risk measure can be viewed as a dynamic generalization of Value-at-Risk. Moreover, we consider its application to the multi-period portfolio selection problem and find an efficient method to find the optimal earliest probabilistic target reaching time. Numerical results show the multi-stage portfolio selection with the target reaching risk measure can help investors reach their investment targets as early as possible. KEY WORDS: Multi-period risk measure; Multi-stage portfolio selection; Regime switching; Stochastic programming; Robust optimization TYPE OF DISSERTATION: Applied Fundamentals V
目录 目录 1绪论 1.1研究背景 12多期风险度量 1.3多阶段投资组合选择 1.3.1情景树方法 1.32统计方法 1.3.3分布式鲁棒方法 14本文的主要工作和组织 2多期风险度量 2.1风险度量与多阶段投资组合选择… 22多期风险度量的性质 23时间相容性 231动态时间相容性 2.3.2弱时间相容性 2.3.3最优投资策略的时间相容性 24多期风险度量的分类 2.4.1终期财富风险度量 24.2可加型风险度量 43递归型风险度量 3基于混合信息框架的递归CVaR风险度量及其在多期投资组合选择中的应用…24 31联合信息框架… 311联合信息过程 312收益率与因子的动态性 32基于机制转换的递归风险度量 33基于机制转换的多期投资组合选择 34实证研究 341数据集和参数估计 36 3.4.2最优投资组合选择 343样本外表现 34.4联合信息框架的优越性 3.5小结
8 ¹ 8 ¹ 1 Xÿ ··························································································································· 1 1.1 Ôƒµ ············································································································· 1 1.2 ıœºx›˛ ······································································································ 1 1.3 ı „›]|‹¿J ·························································································· 2 1.3.1 úµ‰ê{ ··································································································· 4 1.3.2 ⁄Oê{ ······································································································ 4 1.3.3 ©Ÿ™°ïê{ ··························································································· 6 1.4 ©ÃáÛä⁄|Ñ ······················································································ 7 2 ıœºx›˛ ··········································································································· 10 2.1 ºx›˛Üı „›]|‹¿J ······································································· 10 2.2 ıœºx›˛5ü ·························································································· 12 2.3 ûmÉN5 ········································································································· 14 2.3.1 ƒûmÉN5 ··························································································· 14 2.3.2 fûmÉN5 ······························································································· 15 2.3.3 Å`›]¸—ûmÉN5 ········································································ 15 2.4 ıœºx›˛©a ·························································································· 16 2.4.1 ™œ„Lºx›˛ ······················································································· 16 2.4.2 å\.ºx›˛ ··························································································· 18 2.4.3 48.ºx›˛ ··························································································· 20 3 ƒu·‹&Eµe48 CVaR ºx›˛9Ÿ3ıœ›]|‹¿J•A^ ··· 24 3.1 È‹&Eµe ······································································································ 24 3.1.1 È‹&ELß ······························································································· 24 3.1.2 ¬Ã«Üœfƒ5 ··············································································· 25 3.2 ƒuÅõ=Ü48ºx›˛ ··········································································· 27 3.3 ƒuÅõ=Üıœ›]|‹¿J ··································································· 28 3.4 ¢yÔƒ ············································································································· 36 3.4.1 Í‚8⁄ÎÍO ······················································································· 36 3.4.2 Å`›]|‹¿J ······················································································· 38 3.4.3 Ly ··································································································· 43 3.4.4 È‹&Eµe`5 ··············································································· 45 3.5 ( ····················································································································· 47 VII
西安交通大学博士学位论文 4多期最坏情况风险度量及其在多阶段投资组合选择中的应用 4.1多期最坏情况风险度量的定义 42基于 w CVaR的多阶段鲁棒投资组合选择 4.3数值实验 44小结 5矩信息未知的可加型鲁棒风险度量及其在多期投资策略选择中的应用 5.1矩信息未知的单期鲁棒投资策略选择 52基于可加型风险度量的多期矩信息未知鲁棒投资策略选择…… 53实证研究 531数据集 532不确定集中参数的估计 533样本外表现 54小结 6矩信息未知的递归型鲁棒风险度量及其在多期投资组合选择中的应用 6.1新不确定集下的单期鲁棒投资组合选择模型 62基于递归型风险度量的多期矩信息未知鲁棒投资组合选择模型 63实证研究 64小结 7基于机制转换的多期鲁棒风险度量及其在多期投资问题中的应用 7.1多期最坏机制风险度量和多期混合最坏情况风险度量 72基于wr(VaR和 mw CVaR的多期鲁棒投资模型 7.3实证研究 74小结 8目标达成型风险度量及其在多期投资策略选择中的应用 599% 8.1概率目标首达时和新型动态风险度量 82目标达成型风险度量的性质 83目标达成型风险度量与VaR和方差的关系 8.4基于目标达成型风险度量的多期投资策略选择 106 84.1给定概率目标首达时的子问题 8.4.2辅助问题的解析解 110 843最优乘子… 844最优概率目标首达时 114 85实证研究 851在资本市场中的应用 8.52与动态MV模型的比较 116 8.6小结 …117 VIII
‹SœåÆƨƆÿ© 4 ıœÅÄú¹ºx›˛9Ÿ3ı „›]|‹¿J•A^ ······························ 48 4.1 ıœÅÄú¹ºx›˛½¬ ··········································································· 48 4.2 ƒu wCVaR ı „°ï›]|‹¿J ························································ 50 4.3 Íä¢ ············································································································· 55 4.4 ( ····················································································································· 56 5 ›&Eôå\.°ïºx›˛9Ÿ3ıœ›]¸—¿J•A^ ··············· 57 5.1 ›&Eô¸œ°ï›]¸—¿J ······························································· 57 5.2 ƒuå\.ºx›˛ıœ›&Eô°ï›]¸—¿J ···························· 60 5.3 ¢yÔƒ ············································································································· 64 5.3.1 Í‚8 ·········································································································· 65 5.3.2 ÿ(½8•ÎÍO ··············································································· 65 5.3.3 Ly ··································································································· 66 5.4 ( ····················································································································· 67 6 ›&Eô48.°ïºx›˛9Ÿ3ıœ›]|‹¿J•A^ ··············· 69 6.1 #ÿ(½8e¸œ°ï›]|‹¿J . ··················································· 69 6.2 ƒu48.ºx›˛ıœ›&Eô°ï›]|‹¿J . ····················· 74 6.3 ¢yÔƒ ············································································································· 77 6.4 ( ····················································································································· 81 7 ƒuÅõ=Üıœ°ïºx›˛9Ÿ3ıœ›]ØK•A^ ······················ 82 7.1 ıœÅÄÅõºx›˛⁄ıœ·‹ÅÄú¹ºx›˛ ···································· 82 7.2 ƒu wrCVaR ⁄ mwCVaR ıœ°ï›] . ············································ 85 7.3 ¢yÔƒ ············································································································· 89 7.4 ( ····················································································································· 94 8 8Ià§.ºx›˛9Ÿ3ıœ›]¸—¿J•A^ ······································ 95 8.1 V«8Iƒàû⁄#.ƒºx›˛ ······························································· 95 8.2 8Ià§.ºx›˛5ü ·············································································· 97 8.3 8Ià§.ºx›˛Ü VaR ⁄ê 'X ······················································ 102 8.4 ƒu8Ià§.ºx›˛ıœ›]¸—¿J ················································ 106 8.4.1 â½V«8IƒàûfØK ···································································· 107 8.4.2 9œØK)¤) ······················································································· 110 8.4.3 Å`¶f ······································································································ 113 8.4.4 Å`V«8Iƒàû ··················································································· 114 8.5 ¢yÔƒ ············································································································· 114 8.5.1 3] ½|•A^ ··················································································· 114 8.5.2 ܃ MV .' ··············································································· 116 8.6 ( ····················································································································· 117 VIII
目录 9总结与展望 118 9.1总结 92展望… ……………120 致谢 …121 参考文献 攻读博士学位期间的研究成果…… 129
8 ¹ 9 o(Ü–" ··············································································································· 118 9.1 o( ····················································································································· 118 9.2 –" ····················································································································· 120 ó ··························································································································· 121 Ωz ······················································································································· 122 Ù÷ƨƆœmÔƒ§J ···················································································· 129 IX
