本文選題：分形理論 + 多分形測度　； 參考：《西南財(cái)經(jīng)大學(xué)》2013年碩士論文

【摘要】：金融市場與自然界中的很多現(xiàn)象一樣,瞬息萬變,充滿了不確定性,然而人們對(duì)事物運(yùn)行規(guī)律的好奇心和探索從來沒有停止,總是試圖從繁亂復(fù)雜的現(xiàn)象中總結(jié)出簡單而富有解釋力的成因。由于金融市場往往關(guān)系到一個(gè)國家的經(jīng)濟(jì)發(fā)展程度和健康水平,保持它的平穩(wěn)運(yùn)行關(guān)系到國計(jì)民生甚至生活水平的基本保障,因此對(duì)金融市場的把握和預(yù)測顯得尤為重要。特別是近幾十年來,金融危機(jī)頻頻爆發(fā),市場的大幅異常波動(dòng)時(shí)有發(fā)生,波及范圍越來越廣,影響程度也隨之加深,給投資者和監(jiān)管機(jī)構(gòu)的金融風(fēng)險(xiǎn)管理工作帶來了新的挑戰(zhàn)。 自從法國人Bachelier將隨機(jī)游走模型引入金融市場中資產(chǎn)價(jià)格的變化行為后,一個(gè)多世紀(jì)以來,對(duì)金融市場理論的探索不斷發(fā)展,主要經(jīng)歷了三個(gè)階段：“舊金融學(xué)”(Old Finance)階段,“現(xiàn)代金融學(xué)”(Modern Finance)階段以及“新金融學(xué)”(New Finance)階段(魏宇,2004)。20世紀(jì)60年代以前的金融學(xué)研究稱為“舊金融學(xué)”階段,這一時(shí)期主要研究對(duì)象著重針對(duì)單個(gè)的市場參與者,利用其公司或機(jī)構(gòu)的財(cái)務(wù)報(bào)表一類的會(huì)計(jì)信息進(jìn)行分析,而沒有對(duì)市場整體做出系統(tǒng)的研究,也缺乏嚴(yán)謹(jǐn)?shù)睦碚擉w系。20世紀(jì)60年代后,“現(xiàn)代金融學(xué)”迅速崛起,很快就全面占領(lǐng)了金融市場研究的前線。這一類研究方法主要是基于一系列的假定,其中包括投資參與者理性、價(jià)格的隨機(jī)游走、收益率服從獨(dú)立同分布等等。通過這些看似合理簡單的假設(shè),將市場信息和市場資產(chǎn)的運(yùn)動(dòng)規(guī)律結(jié)合起來,產(chǎn)生了許多經(jīng)典實(shí)用的模型和理論,至今仍為理論界和實(shí)務(wù)界所廣泛運(yùn)用。Fama將經(jīng)典的現(xiàn)代金融學(xué)研究進(jìn)行總結(jié)歸納,提出了“有效市場假說”(Efficient Market Hypothesis, EMH),眾多以此為基礎(chǔ)、研究范圍涉及公司金融、資產(chǎn)定價(jià)以及投資決策的理論模型蓬勃發(fā)展,形成了一整套現(xiàn)代金融市場研究的體系框架。 但隨著市場創(chuàng)新發(fā)展的加速和實(shí)證研究的深入,人們發(fā)現(xiàn)了許多在經(jīng)典金融市場理論條件下無法得到解釋的金融市場“異象”(Anomalies)。這些違反有效市場假說的異常現(xiàn)象并不僅僅存在于某個(gè)特殊市場,而是普遍表現(xiàn)在各個(gè)不同地區(qū)、不同發(fā)展程度的市場中,因此動(dòng)搖了EMH理論的成立基礎(chǔ)。由此引發(fā)的對(duì)金融市場運(yùn)行規(guī)律的重新認(rèn)識(shí)和探討,便開創(chuàng)了“新金融學(xué)”研究的時(shí)代。 “新金融學(xué)”的主要代表是行為金融學(xué)(Behavioral Finance)和經(jīng)濟(jì)物理學(xué)(Econophysics)。二者都擯棄了現(xiàn)代金融學(xué)中的投資者理性假說,試圖從新的理論基礎(chǔ)開始解釋市場運(yùn)行的規(guī)律。行為金融學(xué)主要考察投資者在市場中的心理變化與市場中資產(chǎn)價(jià)格走勢的聯(lián)系,包含了心理學(xué)、社會(huì)學(xué)等學(xué)科的知識(shí)。而經(jīng)濟(jì)物理學(xué)則把金融市場看作是一個(gè)非線性的復(fù)雜系統(tǒng),結(jié)合物理學(xué)、幾何學(xué)和經(jīng)濟(jì)金融學(xué)的知識(shí),特別是物理學(xué)中的統(tǒng)計(jì)物理學(xué)(Statistic Physics)和復(fù)雜系統(tǒng)(Complex System)等理論方法來解釋市場運(yùn)動(dòng)的特征及規(guī)律。 經(jīng)濟(jì)物理學(xué)中比較突出的研究是利用分形(Fractal)理論來研究市場。分形定義的是自然界和社會(huì)中廣泛存在、復(fù)雜無序而又具有某些規(guī)律的圖形和現(xiàn)象。它能夠探討事物在多個(gè)測度空間上的自相似性,從局部的隨機(jī)到整體的確定,是幫助認(rèn)識(shí)許多復(fù)雜運(yùn)動(dòng)現(xiàn)象的有力工具。分形中的多分形(Multifractal)更是可以讓我們?cè)诓煌潭鹊臏y度范圍內(nèi)觀察事物的多種變化特征,而這種方式正好符合金融風(fēng)險(xiǎn)管理中對(duì)資產(chǎn)價(jià)格(收益率)的波動(dòng)特征進(jìn)行細(xì)致全面剖析的要求。通過多分形的測度工具和分形市場理論對(duì)市場的重新認(rèn)識(shí),對(duì)市場運(yùn)行規(guī)律的把握和解釋便擁有了全新的視角和方法。 本文基于分形市場理論的研究方法,對(duì)我國金融市場中的價(jià)格(收益率)分形運(yùn)動(dòng)特征進(jìn)行了探討和分析,形成了具有代表性的分形和多分形測度工具,并將之用于金融風(fēng)險(xiǎn)管理的實(shí)際指標(biāo)構(gòu)建中。文章的主體結(jié)構(gòu)安排如下： 第一章為緒論。主要概括介紹了選題背景,選題目的和意義,以及文章的內(nèi)容與結(jié)構(gòu)。 第二章介紹理論基礎(chǔ)及研究現(xiàn)狀。梳理了本文涉及的有效市場假說理論,多分形的概念和分形市場理論以及風(fēng)險(xiǎn)管理相關(guān)理論。同時(shí)將國內(nèi)外在這些方面的研究成果及本文的研究基礎(chǔ)做了簡要匯報(bào)。 第三章介紹金融市場收益率的分布特征。主要針對(duì)以上證綜指(SSCI)為代表的我國金融市場資產(chǎn)價(jià)格收益率做了描述性統(tǒng)計(jì)和正態(tài)檢驗(yàn)。 第四章介紹金融市場的多分形特征。主要用使用重標(biāo)極差分析法(R/S方法)計(jì)算了H指數(shù)來驗(yàn)證我國市場收益率的持久性特征,然后使用結(jié)構(gòu)化分割函數(shù)(Structure Partition Function)檢驗(yàn)了多分形形態(tài)的存在性。 第五章介紹金融市場的多分形測度研究和風(fēng)險(xiǎn)管理研究。用數(shù)盒子(Box-counting)的方法計(jì)算了多分形測度中的兩個(gè)重要指標(biāo)：奇異指數(shù)和多分形譜,并運(yùn)用它們建立了新的風(fēng)險(xiǎn)測度指標(biāo),對(duì)其在金融風(fēng)險(xiǎn)管理中的作用和前景做了介紹。 第六章為結(jié)論、啟示和展望�？偨Y(jié)論文研究結(jié)果以及得到的啟示和不足,對(duì)以后的研究方向和方法提出建議和展望。 本文的創(chuàng)新之處主要有：一是同時(shí)考察了市場收益率的分形和多分形特征,不同于以前多數(shù)文章只注意到了市場的單分形形態(tài),本文通過實(shí)證驗(yàn)證檢驗(yàn)了我國市場的確具有多分形的特征。其次是對(duì)多分形風(fēng)險(xiǎn)管理指標(biāo)的綜合創(chuàng)新,將奇異指數(shù)的寬度、多分形譜的寬度和奇異指數(shù)的標(biāo)準(zhǔn)差放在一起進(jìn)行比較研究,考察了基于不同測度指標(biāo)的風(fēng)險(xiǎn)管理辦法。 本文的不足之處主要是：文中所使用數(shù)據(jù)的及時(shí)可得性有一定的局限。這是由于股市高頻數(shù)據(jù)庫的數(shù)據(jù)比較難以獲得,指數(shù)價(jià)格的高頻數(shù)據(jù)更有一定的滯后。此外,文中對(duì)分形理論的基礎(chǔ)證明并未展開詳細(xì)闡述,只是做了基本介紹。由于分形理論設(shè)計(jì)的范圍包括物理學(xué)、幾何學(xué)、金融學(xué)等多個(gè)學(xué)科的融合,其方法的說明受到了作者知識(shí)的局限,因而略去了基礎(chǔ)理論而直接將其運(yùn)用到了金融市場的實(shí)證計(jì)算當(dāng)中。最后,文章對(duì)于多分形測度指標(biāo)在風(fēng)險(xiǎn)管理中的運(yùn)用雖然進(jìn)行了闡述,卻未能對(duì)其在實(shí)際市場中的指導(dǎo)能力和有效性做實(shí)證方面的檢驗(yàn),這是文章較為遺憾的地方,也是作者下一步打算研究的主要方向。
[Abstract]:The financial market, like many phenomena in nature, is rapidly changing and full of uncertainty. However, the curiosity and exploration of the rules of the operation of things have never stopped, and they always try to sum up a simple and explanatory cause from the complicated and complicated phenomena. Because the financial market is often related to the economic development of a country. The level and health level, keeping its smooth operation related to the basic guarantee of the national economy, the livelihood of the people and even the standard of living, so it is particularly important to grasp and predict the financial market. In depth, it brings new challenges to financial risk management for investors and regulators.
Since French Bachelier introduced the random walk model to the change of asset price in the financial market, the exploration of financial market theory has been developing for more than a century. It has undergone three stages: the "Old Finance" stage, the "Modern Finance" stage and "the new finance" (N). The EW Finance) phase (Wei Yu, 2004) the financial studies before the 60s of the.20 century is called the "old finance" stage. This period mainly focuses on individual market participants, analyzing accounting information such as the financial statements of their companies or institutions, without systematic research on the market as a whole, and lack of a systematic study. A rigorous theoretical system, after the 60s.20 century, the rapid rise of "modern finance" has quickly occupied the front line of financial market research. This kind of research method is mainly based on a series of assumptions, including the rationality of the investment participants, the random walk of the price, the rate of return obeying the independent distribution, and so on. By combining the market information with the movement of market assets, many classical and practical models and theories are produced. So far, the classical modern finance research is summarized by.Fama, which is widely used by the theory and practice circles, and the "Efficient Market Hypothesis (EMH)" is put forward. On the basis of this, the theoretical models of the scope of research involving corporate finance, asset pricing and investment decision-making have flourished and formed a whole system framework for the research of modern financial markets.
However, with the rapid development of market innovation and the deepening of empirical research, many financial markets "Anomalies" which can not be explained under the classical financial market theory are found. These abnormal phenomena which violate the effective market hypothesis are not only in a particular market, but generally in different regions. In the market of different levels of development, the foundation of the EMH theory has been shaken up, and the re recognition and discussion of the rules of the operation of the financial market have initiated the era of "new finance".
The main representatives of "new finance" are behavioral finance (Behavioral Finance) and economic Physics (Econophysics). All two discarded the rational hypothesis of investor in modern finance and try to explain the law of market operation from the new theoretical basis. The behavioral finance owners should examine the psychological changes and market of investors in the market. The relationship between the price trend of the asset price includes the knowledge of psychology, sociology and other disciplines. Economic physics regards the financial market as a nonlinear complex system, combining the knowledge of physics, geometry and economics and finance, especially the Statistic Physics and the complex system (Complex System) in physics. Theoretical methods are used to explain the characteristics and laws of market movements.
The more prominent research in economic physics is to use the Fractal theory to study the market. The fractal definition is the graphics and phenomena that exist in nature and society, which are complex and unordered and have some laws. It can discuss the self similarity of things in multiple measure spaces, and it is helpful to determine from local random to whole. The multifractal (Multifractal) in fractal can also allow us to observe a variety of changes in the range of measurement in a different degree of measurement, and this way is exactly in line with the requirements of the detailed and comprehensive analysis of the volatility characteristics of the asset price (rate of return) in the financial risk management. The multifractal measure tool and the fractal market theory have a new perspective and method to understand the market and grasp and explain the operation rules of the market.
Based on the research method of fractal market theory, the fractal movement characteristics of price (rate of return) in China's financial market are discussed and analyzed, and a representative fractal and multi fractal measure tool is formed and used in the construction of the actual index of financial risk management. The main body structure of the article is as follows:
The first chapter is the introduction, which mainly introduces the background of the topic selection, the purpose and significance of the topic, and the content and structure of the article.
The second chapter introduces the theoretical basis and the research status. It combs the theory of effective market hypothesis, the concept of multifractal, the fractal market theory and the related theory of risk management, and briefly reports the research results at home and abroad and the foundation of this study.
The third chapter introduces the distribution characteristics of the rate of return in the financial market. This paper makes a descriptive statistics and a normal test on the rate of return on the asset price of our financial market, which is represented by the SSCI.
The fourth chapter introduces the multi fractal characteristics of the financial market. The H index is used to verify the persistence characteristics of the market returns in our country by using the R/S method (R/S method), and then the existence of the multifractal morphology is tested by using the structured partition function (Structure Partition Function).
The fifth chapter introduces the research of multifractal measurement and risk management in financial markets. By using Box-counting, the two important indexes in multifractal measure are calculated: singular index and multifractal spectrum, and a new index of risk measurement is established by using them, and the role and prospect in the financial risk management are introduced.
The sixth chapter is the conclusion, inspiration and prospect. It summarizes the research results and the enlightenment and shortage of the paper, and puts forward suggestions and prospects for future research directions and methods.
The main innovations of this paper are as follows: first, the fractal and multi fractal characteristics of market returns are examined at the same time. Unlike most of the previous articles, only the single fractal morphology of the market is paid attention to. In this paper, the characteristics of multi fractal in China's market are verified by empirical verification. Secondly, the comprehensive innovation of the multi fractal risk management index will be made, and the second is the comprehensive innovation of the multi fractal risk management index. The width of the singular exponent, the width of the multifractal spectrum and the standard deviation of the singular exponent are compared, and the risk management method based on the different measure indexes is investigated.
The shortcoming of this paper is that the timely availability of data used in this paper is limited. This is because the data comparison of the high frequency database in the stock market is difficult to obtain, and the high frequency data of the index price is lagging behind. In addition, the basic proof of the fractal theory is not expounded in detail, but the basic introduction is made. The scope of fractal theory design includes the integration of many disciplines such as physics, geometry, finance and other disciplines. The explanation of the method is limited by the author's knowledge, so the basic theory is omitted and applied directly to the empirical calculation of the financial market. Finally, the article is on the use of the multi fractal measure index in risk management. However, it has been expounded, but not the empirical test of its guidance ability and effectiveness in the actual market, which is a more regrettable place in the article, and the main direction of the author's next study.

【學(xué)位授予單位】：西南財(cái)經(jīng)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2013
【分類號(hào)】：F830.9;F224

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