金融市场的复杂性与投资组合选择研究

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摘要

在经典资本市场理论中，有效市场假说一直是资本市场理论的基石．它在资

产定价、风险管理、投资决策等方面都发挥着重要作用．然而，经典资本市场理

论的线性化分析方法有其内在的局限性，对现实情况的解释乏力，它不能解释现

实金融市场资产价格的复杂多变的行为．造成这种情况的根本原因在于金融市场

大多是以非线性方式对外界作用起反应的．同时金融市场在运行中显示了局部的

随机性与全局的决定性特征，这是非线性系统混沌特征的具体表现，混沌始终伴

随着系统的运行．这无疑对经典资本市场理论产生巨大的冲击．在这样的背景下，

金融市场的研究出现了从线性转向非线性分析，从均衡走向演化的新趋势．

本文将混沌分形理论、复杂网络理论和经典资本市场理论相结合，从金融市

场的非线性复杂性入手，实证研究了中国股票市场的非线性特征和混沌特性；并

在此基础上，用复杂网络技术对证券市场网络稳定性进行实证分析；并用崭新的

视角研究了投资组合选择问题，为投资者的投资决策提供一种新的分析工具，为

建立适用于我国金融市场发展的一种新市场理论做一些有益的探索．这样可以使

投资者的投资行为和投资管理更加符合金融市场的实际情况．文章的主要研究内

容和创新之处可归纳如下：

（1）结合金融市场的复杂性及混沌、分形等非线性特征，通过非线性动力

学分析从微观结构层面研究金融市场中的基于混沌理论的相空间预测方法及混

沌时间序列的局域预测方法．

（2）对重标极差分析法（R/S）作出了改进，求出了沪深两市的 Hurst 指数

和我国股市的平均循环长度，论证了我国金融市场表现出显著的非线性动力学特

征，沪深两市具有显著的分形特征与长期记忆效应，股票价格行为服从分形布朗

运动．

（3）利用支持向量机这种新的学习算法对上证指数和深证成指收益率的周

波动序列进行了预测研究，研究结果表明，支持向量机具有很强的函数拟合能力，

且预测精度较高．

（4）利用复杂网络理论研究证券市场网，根据复杂网络的拓扑结构及其动

力学特征，发现证券市场网络的成长具有无标度特性，证券市场网络具有对随机

攻击的鲁棒性，又具有对恶意破坏的脆弱性，并得出了网络的稳定性是由一些关

键的节点（股票）的稳定性决定的结论，我们应该关注这类股票的价格走势．

（5）用分形分布蒙特卡洛模拟法来计算中国证券市场的 VaR 值，考虑了股

指收益率的尖峰和厚尾特性，事后检验中所采用的 LR 统计量非常低，这反映了

在我国证券市场目前所处的无效状态下，用分形分布蒙特卡洛模拟法来度量我国

股市风险是非常适合的．

（6）运用内点算法计算风险投资者效用最大化的二次规划问题，用算例验

证了该方法的有效性；此外,构建了带有交易成本等摩擦因素的投资组合选择的

极大极小模型，并用数值算法求解，算例分析表明了该方法具有良好的实用价值．

（7）把

VaR

和最佳投资组合的概念结合起来，建立并求解了最优均值-

VaR

投资组合选择模型．

VaR

约束下的最佳投资组合问题确切地说是寻求在满足

VaR

约束条件下获得最大收益的投资组合，加入约束后，这种模型的风险防御能力大

大加强．从实证分析的结果可以看出，应用均值-

VaR

模型进行资产配置具有比

Markowitz均值方差模型更高的效率．

关键词：金融市场非线性动力学混沌分形理论支持向量机复杂网

络投资组合选择模型

ABSTRACT

In classical Capital Market Theory (CMT), Efficient Market Hypothesis (EMH)

is the headstone of CMT at all times. It plays an important role in capital pricing, risk

management and investment decision-making. However, the linear methodology of

CMT has limitations inherently as they are invalid to capture complicated patterns in

stock prices, which can’t interpret a lot of practices in financial market. It is causation

that the realistic financial markets always react exoteric action with a nonlinear mode.

And financial markets show local random and whole deterministic, which are the

chaotic characters of nonlinear system. Financial system is chaotic. Modern capital

theory was impacted immensely. On this background, a new research trend, from the

point of nonlinearity and evolution instead of in a linear and equilibrium view,

emerges.

In this paper we combine chaos and fractal theory, complex network theory and

classical capital markets theory, and start our research from the non-linearity of

financial markets and empirical study both nonlinear character and chaos character of

china stock market. On the basis of it, we carry out demonstration analysis of network

stability in securities market with the technology of complex network. We also

research portfolio selection model with new method which help investors make

decision in order to establish new capital theory in our financial markets, which may

makes both investment behavior and portfolio management of investor to accord with

reality of financial market. The results and innovation of the dissertation are in several

perspectives:

(1) We combine complexity of financial market and chaos, fractal and other

non-linear characteristics, through the analysis of nonlinear dynamics which study

prediction methods of the phase-space and local prediction method of chaotic time

series based on chaos theory from micro-structure in financial markets.

(2) We improve R/S analysis method, and obtain Hurst index and average cycle

length in Shanghai and Shenzhen stock markets, and demonstrate significant

non-linear dynamics characteristics which is fractal features and long-term memory

effect in Shanghai and Shenzhen stock markets. It also describe stock price obey

fractal Brownian motion.

(3) We utilize support vector machine (SVM) to study prediction of weekly

fluctuations sequence of yield on Shanghai and Shenzhen Stock Index. It shows that

support vector machine has a strong ability to function fitting and high prediction

accuracy.

(4) We utilize complex network theory to research security market network.

According to topological structure and dynamics characteristics of complex networks,

it is found that the security market networks have robustness of random attack and

vulnerability of malicious destruction, i.e., scale-free property．Stability of networks

are determined by critical nodes(stocks)．We should pay attention to price trend of

these stocks.

(5) With fractal distribution of Monte Carlo simulation method to calculate the

VaR value of security market in China index has given full consideration to the peak

yield and fat-tail characteristics. It is very low that later test has been adopted in LR

statistic, which reflect to use fractal distribution of Monte Carlo simulation method to

measure the risk of China's security market is very appropriate in invalid state of

security market at present.

(6) Using interior-point algorithms to calculates the quadratic programming

problem of utility maximization for investors, and verifies validity of the algorithm

with practical example. In addition, minimax model of portfolio selection is built with

the transaction costs of friction factors and is solved by numerical algorithms.

Example shows that the method has good practical value.

(7) Combining optimum portfolio and VaR we set up and solve optimal

mean-portfolio selection model. Under VaR restriction optimal portfolio problem

seeks to obtain maximum benefit from portfolio. Adding VaR restriction, the

capabilities of defensive risk in this model greatly enhance. From demonstration

analysis we can see that asset allocation of mean-VaR model has greater efficiency

than Markowitz mean-variance model’s.

Key Words ：Financial Market, Nonlinear Dynamics, Chaos and

Fractal Theory, Support Vector Machine, Complex

中文摘要

ABSTRACT

第一章绪论...................................................................................................................1

§1.1 研究背景及意义..................................................................................................1

§1.2 国内外相关研究综述..........................................................................................2

§1.2.1 金融市场的混沌与分形理论研究现状.......................................................2

§1.2.2 复杂网络理论在社会经济管理领域中的研究现状...................................4

§1.2.3 现代投资组合理论研究现状.......................................................................5

§1.3 论文研究的思路和技术路线..............................................................................8

§1.4 论文研究的主要内容与创新点........................................................................10

§1.4.1 研究内容.....................................................................................................10

§1.4.2 论文的创新点.............................................................................................11

§1.5 本章小结............................................................................................................12

第二章金融市场的分形特征研究...............................................................................13

§2.1 引言....................................................................................................................13

§2.2 分形布朗运动....................................................................................................13

§2.2.1 布朗运动.....................................................................................................14

§2.2.2 分形布朗运动.............................................................................................15

§2.3 分形时间序列的特征量....................................................................................16

§2.3.1 分形维.........................................................................................................16

§2.3.2 赫斯特指数与相关性度量.........................................................................17

§2.3.3 分形分布.....................................................................................................17

§2.4 重标极差分析方法.............................................................................................19

§2.4.1 重标极差分析的基本原理..........................................................................19

§2.4.2

－统计量.................................................................................................20

§2.4.3 显著性检验.................................................................................................20

§2.4.4 打乱检验.....................................................................................................21

§2.4.5 修正重标极差分析.....................................................................................21

§2.5 金融市场时间序列重标极差分析的实证研究................................................22

§2.5.1 样本数据选取.............................................................................................22

§2.5.2 重标极差分析的具体步骤..........................................................................23

§2.5.3 样本数据处理结果与分析.........................................................................23

§2.6 本章小结............................................................................................................27

第三章金融市场的混沌现象及波动性研究...............................................................28

§3.1 引言....................................................................................................................28

§3.2 混沌的定义与基本特征....................................................................................29

§3.3 相空间重构........................................................................................................31

§3.3.1 嵌入定理.....................................................................................................32

§3.3.2 相空间重构理论.........................................................................................32

§3.3.3 G-P 算法....................................................................................................33

§3.3.4 时间序列中延迟时间



的确定................................................................. 34

§3.3.4.1 自相关函数法......................................................................................35

§3.3.4.2 C-C 算法.............................................................................................35

§3.4 李雅谱诺夫指数..............................................................................................36

§3.5 实证分析............................................................................................................37

§3.5.1 数据样本选取.............................................................................................37

§3.5.1 数据处理及分析.........................................................................................37

§3.6 金融市场的波动性.............................................................................................38

§3.6.1 金融市场波动性概述..................................................................................38

§3.6.2 股市波动性的度量......................................................................................39

§3.7 金融市场波动的预测.........................................................................................40

§3.7.1 预测方法......................................................................................................40

§3.7.2 基于支持向量机的股市波动预测.............................................................40

§3.8 支持向量机对股市波动序列预测的实证分析.................................................43

§3.8.1 数据及处理..................................................................................................43

§3.8.2 实证结果分析..............................................................................................44

§3.9 本章小结............................................................................................................45

第四章基于复杂网络理论的证券市场网稳定性研究...............................................46

§4.1 引言....................................................................................................................46

§4.2 复杂网络概述.....................................................................................................47

§4.3 基于复杂网络理论的证券市场网络拓扑建模及拓扑特征参数....................50

§4.3.1 证券市场网络拓扑建模.............................................................................50

§4.3.2 证券市场网络的拓扑特征参数.................................................................51

§4.4 复杂网络稳定性度量........................................................................................52

§4.4.1 复杂网络连通性测度.................................................................................53

§4.4.2 复杂网络抗毁性测度.................................................................................53

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摘要：

摘要在经典资本市场理论中，有效市场假说一直是资本市场理论的基石．它在资产定价、风险管理、投资决策等方面都发挥着重要作用．然而，经典资本市场理论的线性化分析方法有其内在的局限性，对现实情况的解释乏力，它不能解释现实金融市场资产价格的复杂多变的行为．造成这种情况的根本原因在于金融市场大多是以非线性方式对外界作用起反应的．同时金融市场在运行中显示了局部的随机性与全局的决定性特征，这是非线性系统混沌特征的具体表现，混沌始终伴随着系统的运行．这无疑对经典资本市场理论产生巨大的冲击．在这样的背景下，金融市场的研究出现了从线性转向非线性分析，从均衡走向演化的新趋势．本文将混沌分形理论、复杂网络理论和经典资本...

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