本文選題：多重分形 + 小波　； 參考：《華南理工大學(xué)》2012年博士論文

【摘要】：作為現(xiàn)代金融理論的基石，有效市場(chǎng)假說(shuō)對(duì)金融理論的發(fā)展起著至關(guān)重要的作用。有效市場(chǎng)假說(shuō)把市場(chǎng)當(dāng)作一個(gè)線性孤立的系統(tǒng)，投資者對(duì)市場(chǎng)信息的反應(yīng)是線性的，然而大量實(shí)證研究顯示：市場(chǎng)并非一直都處于均衡狀態(tài)，有時(shí)市場(chǎng)也會(huì)發(fā)生動(dòng)蕩甚至崩潰。不同于有效市場(chǎng)假說(shuō)，分形市場(chǎng)假說(shuō)則認(rèn)為市場(chǎng)是一個(gè)非線性、開(kāi)放、耗散的系統(tǒng)，投資者對(duì)市場(chǎng)信息的反應(yīng)是非線性的。因此，可以說(shuō)有效市場(chǎng)假說(shuō)只是分形市場(chǎng)假說(shuō)的一個(gè)特例。在分形市場(chǎng)假說(shuō)中，市場(chǎng)被認(rèn)為是同時(shí)具備整體的確定性與局部的隨機(jī)性，市場(chǎng)的分形結(jié)構(gòu)可以揭示出價(jià)格波動(dòng)的動(dòng)力學(xué)特征。 分形理論與小波理論在尺度性能上具有很多相似性，所以小波理論非常適合刻畫(huà)系統(tǒng)的分形特性。本文從分形理論、多重分形理論以及小波理論出發(fā)，詳細(xì)闡述了基于小波理論的分形分析方法，并首次提出二分遞歸小波變換模極大值法（WTMM）來(lái)計(jì)算多重分形。隨后文章應(yīng)用這些理論，依次分析了股票市場(chǎng)的單重分形特性、多重分形特性，且以分析多重分形性的演化特征為主。在多重分形分析中，不僅采用了基于統(tǒng)計(jì)物理的配分函數(shù)法（PF）與基于數(shù)值分析的多重分形消除趨勢(shì)波動(dòng)分析法（MF-DFA），還采用了目前國(guó)際上廣泛使用的小波分析法，包括小波變換模極大值法（WTMM）與小波領(lǐng)袖法（WL）。 首先，研究對(duì)中國(guó)股市的正態(tài)性進(jìn)行了檢驗(yàn)，并應(yīng)用不同的方法對(duì)滬深股市的長(zhǎng)期記憶性進(jìn)行了考察，研究結(jié)果顯示：中國(guó)股市的收益率序列具有較明顯的“尖峰肥尾”特征，而所有方法計(jì)算得到的滬深股指的Hurst指數(shù)都大于0.5，說(shuō)明滬深股指存在著正持續(xù)性；隨著收益尺度增大，Hurst指數(shù)也逐漸增大，說(shuō)明兩市股指的長(zhǎng)期收益具有更強(qiáng)的正持續(xù)性。 接著研究更多地考察了股票市場(chǎng)的多重分形特性，分為以下四個(gè)部分： 一、同時(shí)應(yīng)用PF、MF-DFA考察了二十一世紀(jì)以來(lái)，中國(guó)股市與美、英、法、德、日五個(gè)主要股市的多重分形性，兩種方法均顯示：中國(guó)股市均顯示出具有更強(qiáng)的多重分形性，與其它各股市相比，中國(guó)股指在低價(jià)位徘徊的時(shí)間更頻繁，且大波動(dòng)也較小波動(dòng)更頻繁。 二、基于多重分形消除趨勢(shì)波動(dòng)分析法MF-DFA，對(duì)日本七個(gè)經(jīng)濟(jì)時(shí)期以及中國(guó)股市自建立以來(lái)三個(gè)經(jīng)濟(jì)階段的股票市場(chǎng)指數(shù)進(jìn)行實(shí)證研究。研究結(jié)果顯示：不同經(jīng)濟(jì)發(fā)展時(shí)期日、中兩國(guó)的股票市場(chǎng)均具有明顯的多重分形特性；但各自不同的經(jīng)濟(jì)時(shí)期多重分形特性差異顯著，且與當(dāng)時(shí)經(jīng)濟(jì)發(fā)展的狀況存在著一定聯(lián)系。最后，通過(guò)對(duì)照日中兩國(guó)不同時(shí)期股票市場(chǎng)的多重分形性，得出一些對(duì)中國(guó)經(jīng)濟(jì)發(fā)展有益的啟示。 三、與其他方法不同，小波變換模極大值法（WTMM）不但可以從數(shù)據(jù)自身結(jié)構(gòu)偵測(cè)出系統(tǒng)的突變點(diǎn)，，還可以基于突變點(diǎn)計(jì)算系統(tǒng)的多重分形特性。研究應(yīng)用本文提出的二分遞歸小波變換模極大值法（WTMM）先通過(guò)建立道瓊斯工業(yè)指數(shù)（DJI）與東京證交所股價(jià)指數(shù)（TPX）的模極大值線來(lái)定位金融危機(jī)發(fā)生的時(shí)點(diǎn)，然后選取道瓊斯工業(yè)指數(shù)模極大值線上的奇異點(diǎn)系數(shù)對(duì)其進(jìn)行多重分形分析。研究結(jié)果顯示：小波變換模極大值法不僅可以準(zhǔn)確定位金融危機(jī)發(fā)生的時(shí)點(diǎn)，還能刻畫(huà)危機(jī)前后股市多重分形特性的變化。 四、研究通過(guò)應(yīng)用小波領(lǐng)袖（WL）多重分析法刻畫(huà)市場(chǎng)波動(dòng)的多重分形特性來(lái)衡量市場(chǎng)的有效性，提出一種應(yīng)用市場(chǎng)最大波動(dòng)點(diǎn)集分形維數(shù)的演化來(lái)偵測(cè)金融風(fēng)險(xiǎn)發(fā)生時(shí)點(diǎn)的新方法，并與最大波動(dòng)點(diǎn)集的奇異性指數(shù)結(jié)合起來(lái)對(duì)金融風(fēng)險(xiǎn)進(jìn)行測(cè)量。研究結(jié)果表明：中、美、日三國(guó)在不同時(shí)期市場(chǎng)的有效性具有明顯的差異，近年來(lái)中國(guó)市場(chǎng)有效性得到了顯著提高，而美、日兩國(guó)市場(chǎng)有效性則與金融風(fēng)險(xiǎn)的發(fā)生密切相關(guān)；此外，借助多重分形參數(shù)的演變能準(zhǔn)確定位出金融風(fēng)險(xiǎn)發(fā)生的時(shí)點(diǎn)并對(duì)其大小進(jìn)行測(cè)量。 綜上所述，與基于均衡模型的有效市場(chǎng)假說(shuō)理論相比，分形市場(chǎng)假說(shuō)認(rèn)為市場(chǎng)看作是一個(gè)復(fù)雜的非線性系統(tǒng)，所以它不僅可以刻畫(huà)平穩(wěn)運(yùn)行時(shí)的市場(chǎng)，也可以考察市場(chǎng)在穩(wěn)定與動(dòng)蕩之間的變換。對(duì)于市場(chǎng)的監(jiān)督者與投資者來(lái)說(shuō)，分形市場(chǎng)假說(shuō)不僅有益于市場(chǎng)監(jiān)管與投資決策，同時(shí)也有助于更有效地維護(hù)市場(chǎng)穩(wěn)定與管理金融風(fēng)險(xiǎn)。
[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.
【學(xué)位授予單位】：華南理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2012
【分類號(hào)】：F832.51;F224

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