分形与小波的集成研究及其在股票市场波动分析中的应用

发布时间：2018-05-18 23:25

本文选题：多重分形 + 小波　；参考：《华南理工大学》2012年博士论文

【摘要】：作为现代金融理论的基石，有效市场假说对金融理论的发展起着至关重要的作用。有效市场假说把市场当作一个线性孤立的系统，投资者对市场信息的反应是线性的，然而大量实证研究显示：市场并非一直都处于均衡状态，有时市场也会发生动荡甚至崩溃。不同于有效市场假说，分形市场假说则认为市场是一个非线性、开放、耗散的系统，投资者对市场信息的反应是非线性的。因此，可以说有效市场假说只是分形市场假说的一个特例。在分形市场假说中，市场被认为是同时具备整体的确定性与局部的随机性，市场的分形结构可以揭示出价格波动的动力学特征。分形理论与小波理论在尺度性能上具有很多相似性，所以小波理论非常适合刻画系统的分形特性。本文从分形理论、多重分形理论以及小波理论出发，详细阐述了基于小波理论的分形分析方法，并首次提出二分递归小波变换模极大值法（WTMM）来计算多重分形。随后文章应用这些理论，依次分析了股票市场的单重分形特性、多重分形特性，且以分析多重分形性的演化特征为主。在多重分形分析中，不仅采用了基于统计物理的配分函数法（PF）与基于数值分析的多重分形消除趋势波动分析法（MF-DFA），还采用了目前国际上广泛使用的小波分析法，包括小波变换模极大值法（WTMM）与小波领袖法（WL）。首先，研究对中国股市的正态性进行了检验，并应用不同的方法对沪深股市的长期记忆性进行了考察，研究结果显示：中国股市的收益率序列具有较明显的“尖峰肥尾”特征，而所有方法计算得到的沪深股指的Hurst指数都大于0.5，说明沪深股指存在着正持续性；随着收益尺度增大，Hurst指数也逐渐增大，说明两市股指的长期收益具有更强的正持续性。接着研究更多地考察了股票市场的多重分形特性，分为以下四个部分：一、同时应用PF、MF-DFA考察了二十一世纪以来，中国股市与美、英、法、德、日五个主要股市的多重分形性，两种方法均显示：中国股市均显示出具有更强的多重分形性，与其它各股市相比，中国股指在低价位徘徊的时间更频繁，且大波动也较小波动更频繁。二、基于多重分形消除趋势波动分析法MF-DFA，对日本七个经济时期以及中国股市自建立以来三个经济阶段的股票市场指数进行实证研究。研究结果显示：不同经济发展时期日、中两国的股票市场均具有明显的多重分形特性；但各自不同的经济时期多重分形特性差异显著，且与当时经济发展的状况存在着一定联系。最后，通过对照日中两国不同时期股票市场的多重分形性，得出一些对中国经济发展有益的启示。三、与其他方法不同，小波变换模极大值法（WTMM）不但可以从数据自身结构侦测出系统的突变点，，还可以基于突变点计算系统的多重分形特性。研究应用本文提出的二分递归小波变换模极大值法（WTMM）先通过建立道琼斯工业指数（DJI）与东京证交所股价指数（TPX）的模极大值线来定位金融危机发生的时点，然后选取道琼斯工业指数模极大值线上的奇异点系数对其进行多重分形分析。研究结果显示：小波变换模极大值法不仅可以准确定位金融危机发生的时点，还能刻画危机前后股市多重分形特性的变化。四、研究通过应用小波领袖（WL）多重分析法刻画市场波动的多重分形特性来衡量市场的有效性，提出一种应用市场最大波动点集分形维数的演化来侦测金融风险发生时点的新方法，并与最大波动点集的奇异性指数结合起来对金融风险进行测量。研究结果表明：中、美、日三国在不同时期市场的有效性具有明显的差异，近年来中国市场有效性得到了显著提高，而美、日两国市场有效性则与金融风险的发生密切相关；此外，借助多重分形参数的演变能准确定位出金融风险发生的时点并对其大小进行测量。综上所述，与基于均衡模型的有效市场假说理论相比，分形市场假说认为市场看作是一个复杂的非线性系统，所以它不仅可以刻画平稳运行时的市场，也可以考察市场在稳定与动荡之间的变换。对于市场的监督者与投资者来说，分形市场假说不仅有益于市场监管与投资决策，同时也有助于更有效地维护市场稳定与管理金融风险。
[Abstract]:As the cornerstone of modern financial theory, the effective market hypothesis plays a vital role in the development of financial theory. The effective market hypothesis regards the market as a linear isolated system, and the response of investors to market information is linear. However, a large number of empirical studies show that the market is not always in equilibrium, and sometimes the market is also Different from the effective market hypothesis, the fractal market hypothesis holds that the market is a nonlinear, open, dissipative system, and the investor's response to market information is nonlinear. Therefore, the effective market hypothesis is only a special case of the fractal market hypothesis. In the fractal market hypothesis, the market is considered to be At the same time, it has the overall certainty and local randomness. The fractal structure of the market can reveal the dynamic characteristics of price fluctuation.
Fractal theory and wavelet theory have a lot of similarity in scale performance, so the wavelet theory is very suitable to describe the fractal characteristics of the system. From the fractal theory, the multifractal theory and the wavelet theory, the fractal analysis method based on the wavelet theory is elaborated in detail, and the two recursion wavelet transform modulus maxima method is proposed for the first time. (WTMM) to calculate the multifractal. Then the paper applies these theories to analyze the single fractal and multifractal characteristics of the stock market in order to analyze the evolution characteristics of multifractal. In the multifractal analysis, not only the partition function method based on Statistical Physics (PF) and the multifractal elimination based on numerical analysis are used in the multifractal analysis. The trend fluctuation analysis (MF-DFA) method also adopts the widely used wavelet analysis methods, including the wavelet transform modulus maxima (WTMM) and the wavelet leader method (WL).
First, the research on the normality of the Chinese stock market is tested, and the long-term memory of the Shanghai and Shenzhen stock market is examined by different methods. The results show that the return sequence of the Chinese stock market has a distinct "peak fat tail" feature, and the Hurst index of the Shanghai and Shenzhen Stock index is more than 0.5, indicating that the stock index of the stock market is more than 0.5. The Shanghai and Shenzhen stock index has a positive continuity. With the increase of income scale, the Hurst index also gradually increases, indicating that the long-term returns of the two cities have stronger positive persistence.
Next, we study the multifractal characteristics of the stock market and divide them into four parts.
First, using PF, MF-DFA examines the multifractal nature of the five main stock markets in China's stock market and the United States, Britain, France, Germany and Japan since twenty-first Century. The two methods show that the Chinese stock market has a stronger multifractal nature. Compared with the other stock markets, the Chinese stock market is more frequent in the low price and more fluctuating than the other stock markets. Small fluctuations are more frequent.
Two, based on the multi fractal elimination trend analysis method MF-DFA, the empirical study on the stock market index of the seven economic periods and the three economic stages since the establishment of the Chinese stock market has been carried out. The results show that the stock markets in the two countries have obvious multifractal characteristics in different economic development periods, but they are not different. There are significant differences in multifractal characteristics in the same economic period, and there is a certain connection with the situation of economic development at that time. Finally, some useful revelations to China's economic development are obtained by comparing the multifractal nature of the stock market in the different periods of the two countries.
Three, different from other methods, the wavelet transform modulus maxima method (WTMM) can not only detect the mutation points of the system from the structure of the data, but also calculate the multifractal characteristics of the system based on the mutation point. In this paper, the Dow Jones industrial index (DJI) and the East are first established by the two recursive wavelet transform modulus maxima method (WTMM) proposed in this paper. The peak value line of the stock index of the Beijing stock exchange (TPX) is used to locate the time point of the financial crisis, and then the multi fractal analysis of the Dow Jones industrial index modulus maximum line is selected. The results show that the wavelet transform modulus maxima method can not only determine the time points of the financial crisis, but also can depict the danger. Changes in the multifractal characteristics of the stock market before and after the machine.
Four, the study uses the multifractal analysis of the wavelet leader (WL) multiple analysis to describe the multifractal characteristics of the market volatility to measure the effectiveness of the market. A new method is proposed to detect the time points of the occurrence of financial risk by using the evolution of the fractal dimension of the largest fluctuation point of the market to detect the time points of the financial risk, and the financial risk is combined with the singularity index of the maximum wave set set. The results show that the effectiveness of the three countries in different periods has obvious differences. In recent years, the effectiveness of China's market has been significantly improved, while the effectiveness of the United States and Japan is closely related to the occurrence of financial risks. In addition, the evolution of the multi fractal parameters can accurately locate the financial wind. The time points of the risk are measured and the size of the risk is measured.
To sum up, compared with the efficient market hypothesis theory based on equilibrium model, the fractal market hypothesis thinks that the market is a complex nonlinear system, so it can not only describe the market in the stable operation, but also the transformation between the market stability and the turbulence. The field hypothesis not only benefits market supervision and investment decisions, but also helps to maintain market stability and manage financial risks more effectively.
【学位授予单位】：华南理工大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F832.51;F224

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