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基于隐Markov模型的最优资产组合选择

张玲

摘 要 在具有可观测和不可观测状态的金融市场中,利用隐马尔可夫链描述不可观测状态的动态过程,研究了不完全信息市场中的多阶段最优投资组合选择问题.通过构造充分统计量,不完全信息下的投资组合优化问题转化为完全信息下的投资组合优化问题,利用动态规划方法求得了最优投资组合策略和最优值函数的解析解.作为特例,还给出了市场状态完全可观测时的最优投资组合策略和最优值函数.

关键词 不完全信息;隐马尔可夫链;充分统计量;基准准则;动态规划

中图分类号 F830.59, F224.3 文献标识码 A

1 引 言

多阶段最优投资组合选择问题的研究中,通常假定风险资产收益独立同分布且资产收益与市场状态无关[1].然而大量实证研究却发现,风险资产在各阶段的收益序列是相关的,且资产收益的这种相关性通过市场参数得以实现.金融市场中,股票价格不能独立于宏观经济之外,众多股票价格的时间序列表现出较强的相关性和较大的跳跃,且这些跳跃通常同一些事件联系相关,如牛市和熊市、金融危机、政府新的金融或经济政策等.Hardy[2]发现Markov链能显著地拟合这类金融市场状态的变化过程,且股票收益具有较强的Markov机制转移性质.此后,利用Markov链刻画金融市场状态变化过程成为经济和金融领域的一个研究热点.在资产组合选择问题的研究方面,Zhou和Yin[3]研究了Markov机制转移下的连续时间均值方差最优投资组合选择问题.Cakmak和zekici[4]用时齐的Markov过程描述市场状态的变化过程,得到了多阶段最优投资问题的最优投资策略的解析解.在文献[4]的基础上,Wei和Ye[5]引入了破产风险控制,Wu和Li[6]进一步考虑了投资终止时间不确定对最优投资决策的影响.Costa和Araujo[7] 研究了参数受Markov机制转移调制的多阶段均值方差资产组合选择问题,并将结果应用到了动态资产组合选择问题的破产风险控制中.Xie[8] 研究了风险资产价格和负债都是受到Markov机制转移调制的连续时间资产负债管理问题.在上述马尔科夫机制转移模型中,有一个基本的假设:Markov链的状态是完全可观测的,且状态转移矩阵是定常的.

然而,金融市场中不仅存在投资者可以观测到的状态信息(如利率、通货膨胀率、汇率等),还存在投资者无法观测到的市场状态信息,正是这些不可观测的金融市场状态信息导致了资产收益的变化,陈国华等[9] 研究了资产收益率为模糊数的投资组合选择问题.事实上,绝大多数投资者仅能依据从市场中观测得到的信息而非市场上全部的信息做出投资决策,这导致了不完全信息下的投资决策问题,隐马尔科夫链( hidden Markov chain)常用来刻画不可观测状态的变化过程.Sass和Haussmann[10]、Rieder和Buerle[11]、Putschgl 和Sass等[12]考虑了不可观测状态由隐马尔科夫链刻画的连续时间最优投资决策问题,分析了不可观测状态对最优投资策略的影响.基于优化问题的可解性,以往有关不完全信息下最优投资问题的研究集中于连续时间情形,离散时间最优投资组合选择问题的研究还很少.虽然Canakolu和zekici[13]考虑了隐Markov机制转移市场中的多阶段HARA效用最大化问题,但没有得到最优策略的解析解.连续时间模型在求解最优投资组合选择问题中具有非常好的便利性,然而离散时间多阶段模型更符合金融市场实际决策,且连续时间投资组合选择问题的实现依然需要借助于离散化的工具.所以,在具有不可观测市场状态的金融市场中,离散时间多阶段最优资产组合选择问题的研究具有重要的现实和理论意义.

基于以上研究和思考,本文利用离散时间有限状态隐Markov链刻画金融市场上不可观测状态的变化过程,建立了不完全信息下多阶段最优投资组合选择问题的基准准则模型.通过扩大状态空间和构造充分统计量,不完全信息下的优化问题转化为完全信息下的优化问题,利用动态规划方法求得了最优投资组合策略和最优值函数的解析表达式.

5 总 结

本文在具有可观测状态和不可观测状态的金融市场中,利用隐Markov链模拟不可观测市场状态的变化过程,研究了不完全信息市场中的多阶段最优资产组合选择问题.通过构造充分统计量,不完全信息下的最优资产组合选择问题转变为完全信息下的最优资产组合选择问题,采用动态规划方法求解优化问题,得到了各阶段最优资产组合策略和最优值函数的解析表达式.同时,给出了市场状态完全可观测时最优资产组合选择问题的最优组合策略和最优值函数.本文中所建立的模型还未考虑金融市场存在的各种约束,以后的工作中可以进一步考虑不完全信息市场中存在各种摩擦时的最优资产组合选择问题.

参考文献

[1] D LI, W L NG. Optimal dynamic portfolio selection: multiperiod mean-variance formulation [J]. Mathematical Finance, 2000, 10(3):387-406.

[2] M R HARDY. A regime-switching model of long-term stock return [J]. North American Actuarial Journal, 2001, 5(2):41-53.

[3] X Y ZHOU, G YIN. Markowitz's mean-variance portfolio selection with regime switching: a continuous-time model [J]. SIAM Journal on Control and Optimization, 2003, 42(4):1466-1482.

[4] U CAKMAK, S ZEKICI. Portfolio optimization in stochastic markets [J]. Mathematical Methods of Operations Research, 2006, 63(1):151-168.

[5] S Z WEI, Z X YE. Multi-period optimization portfolio with bankruptcy control in stochastic market [J]. Applied Mathematics and Computation, 2007, 186(1):414-425.

[6] H L WU, Z F LI. Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time horizon [J]. Journal of System Science and Complexity, 2011, 24(1):140-155.

[7] O L V COSTA, M V ARAUJO. A generalized multi-period mean-variance portfolio optimization with Markov switching parameters [J]. Automatica, 2008, 44(10):2487-2497.

[8] S X XIE. Continuous-time mean-variance portfolio selection with liability and regime switching [J]. Insurance: Mathematics and Economics, 2009, 45(1):148-155.

[9] 陈国华,陈收,房勇,汪寿阳. 基于模糊收益率的组合投资模型 [J]. 经济数学,2006,23(1):19-25.

[10]J SASS, U G HAUSSMANN. Optimizing the terminal wealth under partial information: the drift process as a continuous-time Markov chain [J]. Finance and Stochastics, 2004, 8(4):553-577.

[11]U RIEDER, N BUERLE. Portfolio optimization with unobservable Markov-modulated drift process [J]. Journal of Applied Probability, 2005, 42(2):362-378.

[12]W PUTSCHGL, J SASS. Optimal investment under dynamic risk constraints and partial information [J]. Quantitative Finance, 2011, 11(10): 1547-1564.

[13]E CANAKLU, S ZEKICI. Portfolio selection with imperfect information: a hidden Markov model [J]. Applied Stochastic Models in Business and Industry, 2011, 27(2):95-114.

[14]D P BERTSEKAS. Dynamic programming: deterministic and stochastic models [M]. Prentice-Hall, Englewood Cliffs, NJ, 1987.

[15]G E MONAHAN. A survey of incompletely observable Markov decision processes: theory, models and algorithms [J]. Management Science, 1982, 28(1): 1-16.

[4] U CAKMAK, S ZEKICI. Portfolio optimization in stochastic markets [J]. Mathematical Methods of Operations Research, 2006, 63(1):151-168.

[5] S Z WEI, Z X YE. Multi-period optimization portfolio with bankruptcy control in stochastic market [J]. Applied Mathematics and Computation, 2007, 186(1):414-425.

[6] H L WU, Z F LI. Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time horizon [J]. Journal of System Science and Complexity, 2011, 24(1):140-155.

[7] O L V COSTA, M V ARAUJO. A generalized multi-period mean-variance portfolio optimization with Markov switching parameters [J]. Automatica, 2008, 44(10):2487-2497.

[8] S X XIE. Continuous-time mean-variance portfolio selection with liability and regime switching [J]. Insurance: Mathematics and Economics, 2009, 45(1):148-155.

[9] 陈国华,陈收,房勇,汪寿阳. 基于模糊收益率的组合投资模型 [J]. 经济数学,2006,23(1):19-25.

[10]J SASS, U G HAUSSMANN. Optimizing the terminal wealth under partial information: the drift process as a continuous-time Markov chain [J]. Finance and Stochastics, 2004, 8(4):553-577.

[11]U RIEDER, N BUERLE. Portfolio optimization with unobservable Markov-modulated drift process [J]. Journal of Applied Probability, 2005, 42(2):362-378.

[12]W PUTSCHGL, J SASS. Optimal investment under dynamic risk constraints and partial information [J]. Quantitative Finance, 2011, 11(10): 1547-1564.

[13]E CANAKLU, S ZEKICI. Portfolio selection with imperfect information: a hidden Markov model [J]. Applied Stochastic Models in Business and Industry, 2011, 27(2):95-114.

[14]D P BERTSEKAS. Dynamic programming: deterministic and stochastic models [M]. Prentice-Hall, Englewood Cliffs, NJ, 1987.

[15]G E MONAHAN. A survey of incompletely observable Markov decision processes: theory, models and algorithms [J]. Management Science, 1982, 28(1): 1-16.

[4] U CAKMAK, S ZEKICI. Portfolio optimization in stochastic markets [J]. Mathematical Methods of Operations Research, 2006, 63(1):151-168.

[5] S Z WEI, Z X YE. Multi-period optimization portfolio with bankruptcy control in stochastic market [J]. Applied Mathematics and Computation, 2007, 186(1):414-425.

[6] H L WU, Z F LI. Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time horizon [J]. Journal of System Science and Complexity, 2011, 24(1):140-155.

[7] O L V COSTA, M V ARAUJO. A generalized multi-period mean-variance portfolio optimization with Markov switching parameters [J]. Automatica, 2008, 44(10):2487-2497.

[8] S X XIE. Continuous-time mean-variance portfolio selection with liability and regime switching [J]. Insurance: Mathematics and Economics, 2009, 45(1):148-155.

[9] 陈国华,陈收,房勇,汪寿阳. 基于模糊收益率的组合投资模型 [J]. 经济数学,2006,23(1):19-25.

[10]J SASS, U G HAUSSMANN. Optimizing the terminal wealth under partial information: the drift process as a continuous-time Markov chain [J]. Finance and Stochastics, 2004, 8(4):553-577.

[11]U RIEDER, N BUERLE. Portfolio optimization with unobservable Markov-modulated drift process [J]. Journal of Applied Probability, 2005, 42(2):362-378.

[12]W PUTSCHGL, J SASS. Optimal investment under dynamic risk constraints and partial information [J]. Quantitative Finance, 2011, 11(10): 1547-1564.

[13]E CANAKLU, S ZEKICI. Portfolio selection with imperfect information: a hidden Markov model [J]. Applied Stochastic Models in Business and Industry, 2011, 27(2):95-114.

[14]D P BERTSEKAS. Dynamic programming: deterministic and stochastic models [M]. Prentice-Hall, Englewood Cliffs, NJ, 1987.

[15]G E MONAHAN. A survey of incompletely observable Markov decision processes: theory, models and algorithms [J]. Management Science, 1982, 28(1): 1-16.

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