统计物理学在我国股票市场特性研究中的应用
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摘要
自二十世纪八十年代以来,人们逐渐发现一些金融理论赖以成立的基本假设与实证结果不符,而要修正现有理论,或要构建新的理论模型,就必须对金融市场表现出来的各种特征作全面的分析研究。股票市场是个复杂的、多体系统,价格的变化是多方作用的结果。统计物理学是研究多体系统相互作用的宏观结果及其内部原因的。由鉴于此,本文以我国沪深股票市场在不同时间标度的收盘指数序列作为研究对象,借助统计物理学的理论和方法,对我国股票市场的变化特征进行全面、系统的分析研究,对其变化的微观机理进行初步探索,从而为研究金融市场价格波动的动力学机理奠定实证基础。主要内容包括以下五个方面:
第一,标度特性的研究。主要通过重标度极差分析法和消除趋势波动分析法,分析了我国股市收益波动的标度特性。结果表明:我国股票市场的收益波动具有长期记忆性,沪深两市的赫斯特指数分别为0.6282和0.6358,并存在284天左右的长记忆周期和115天左右的短记忆周期;波动持续性随着标度的减小而减弱,随时间跨度的缩短而增强;波动性随着时间标度的减小而越趋复杂,说明高频数据更能反映波动的精细结构;收益的绝对值序列及其平方序列的波动具有更明显的长程幂律相关特性,而且随着标度的减小,收益绝对值序列的幂律相关特性逐渐增强,但收益平方序列的幂律相关特性不随标度的变化而变化。
第二,收益特性的研究。主要分析了我国股市收益概率分布函数的标度特性以及其遵循的微观动力学规律。通过对高频数据的初步实证分析,我们认为利维稳定分布能较好地描述收益概率分布的中间区域,沪深两市的特征指数分别为1.26和1.74,而且分布的渐近行为遵循截尾利维分布。
第三,波动率特性研究。分析了上海证券市场波动率的标度特性、相关特性及其遵循的分布规律,发现无论是波动率的概率分布,还是累积概率分布,都具有标度不变性;波动率的概率分布服从对数正态分布;波动率具有明显的长程幂律相关特性,并有4天左右的分岔特征;上海证券市场具有明显的日效应现象,呈“W”型波动,即日开盘和日收盘时波动较大,午时休盘和开盘时有较小波动。
第四,多重分形特性的研究。主要提出了多重分形的三种研究方法,分析了上海证券市场的多重分形特性及其随时间标度的变化,发现多重分形的形状不随时间标度的改变而改变,但其强度随标度的减小而减弱;广义赫斯特指数是配分阶数的减函数,而且随着标度的减小,衰减趋势增强;奇异谱随配分阶数的变化不连续,其规律类似于统计物理学中序参量在临界点处的变化规律。
第五,股票市场中的序参量及其特性研究。我们将统计物理学中序参量的概念引入证券市场,通过类比分析的方法,认为证券市场多重分形中的广义维数Dq或广义赫斯特指数h (q)相当于相变过程中的序参量,并分析了它们都具有负双曲正切函数的变化规律。
Since 1980s, people have found that some basic hypotheses on which financial theories rely do not accord with the empirical results. If we want to revise them or establish new theory, we should first analyse comprehensively the peoperties of financial market. Stock market is a complicated multi-body system and so the price fluctuations are the results of interactions of multi-bodies. Statistical physics is the subject that studies the results and fundamental principles of interactions of multi-bodies. According to the closing stock index of different time scales in Shanghai and Shenzhen and with the help of the theories and methods of statistical physics, this dissertation makes a thorough and systematic study on the properties of Chinese stock market and provides empirical results for dynamic mechanism of price fluctuations. It is made of five parts.
The first is the scaling properties. By using the Rescaled Range Analysis and the Detrended Fluctuation Analysis, we study the scaling properties of return series and find there is long-range memory. The Hurst exponents are 0.6282 and 0.6358 respectively in Shanghai stock market and Shenzhen stock market. We also find there are 284-day long memory period and 115-day short memory period. The persistence becomes weak when scale shortening, but it becomes strong when time range shortening. The fluctuation becomes complicated when the scale becomes short. As to the absolute value series and square value series of returns, there are stronger long-range persistence.
The second is the properties of returns. According to the high-frequency closing stock index, we study the probability distributions of returns and find the middle part of the distribution is well described by the Lévy stable distribution with Lévy Exponents 1.26 in Shanghai and 1.74 in Shenzhen. The asymptotic behaviour of the distribution is truncated Lévy distribution.
The third is the volatility. We analyse the statistical properties of volatility in Shanghai stock market and find there are scale invariance both in probability distribution and cumulative probability distribution. The former can be well described by a log-normal function and there are power law and crossover time, which is about four days. The Shanghai stock market has obvious day-effect whick shape is like the word“W”.
The forth is the multifractal. We put forward three methods of multifractal and have conclusions that Shanghai stock market has weak multifractal and its shape does not change with scales, but its strength weakens with the scale shortening. The general Hurst exponent decreases when the order of partition function increases. The shorter the scale, the stronger it decreases. The singular spectrum function does not change continuously with partition order. It is like the behaviour of the order parameter at its critical point.
The last one is the order parameter of stock market. With analogy, we believe the general dimension D q or the general Hurst exponent h (q) as the order parameter, the important parameter in phase transition and analyse its properties.
第一,标度特性的研究。主要通过重标度极差分析法和消除趋势波动分析法,分析了我国股市收益波动的标度特性。结果表明:我国股票市场的收益波动具有长期记忆性,沪深两市的赫斯特指数分别为0.6282和0.6358,并存在284天左右的长记忆周期和115天左右的短记忆周期;波动持续性随着标度的减小而减弱,随时间跨度的缩短而增强;波动性随着时间标度的减小而越趋复杂,说明高频数据更能反映波动的精细结构;收益的绝对值序列及其平方序列的波动具有更明显的长程幂律相关特性,而且随着标度的减小,收益绝对值序列的幂律相关特性逐渐增强,但收益平方序列的幂律相关特性不随标度的变化而变化。
第二,收益特性的研究。主要分析了我国股市收益概率分布函数的标度特性以及其遵循的微观动力学规律。通过对高频数据的初步实证分析,我们认为利维稳定分布能较好地描述收益概率分布的中间区域,沪深两市的特征指数分别为1.26和1.74,而且分布的渐近行为遵循截尾利维分布。
第三,波动率特性研究。分析了上海证券市场波动率的标度特性、相关特性及其遵循的分布规律,发现无论是波动率的概率分布,还是累积概率分布,都具有标度不变性;波动率的概率分布服从对数正态分布;波动率具有明显的长程幂律相关特性,并有4天左右的分岔特征;上海证券市场具有明显的日效应现象,呈“W”型波动,即日开盘和日收盘时波动较大,午时休盘和开盘时有较小波动。
第四,多重分形特性的研究。主要提出了多重分形的三种研究方法,分析了上海证券市场的多重分形特性及其随时间标度的变化,发现多重分形的形状不随时间标度的改变而改变,但其强度随标度的减小而减弱;广义赫斯特指数是配分阶数的减函数,而且随着标度的减小,衰减趋势增强;奇异谱随配分阶数的变化不连续,其规律类似于统计物理学中序参量在临界点处的变化规律。
第五,股票市场中的序参量及其特性研究。我们将统计物理学中序参量的概念引入证券市场,通过类比分析的方法,认为证券市场多重分形中的广义维数Dq或广义赫斯特指数h (q)相当于相变过程中的序参量,并分析了它们都具有负双曲正切函数的变化规律。
Since 1980s, people have found that some basic hypotheses on which financial theories rely do not accord with the empirical results. If we want to revise them or establish new theory, we should first analyse comprehensively the peoperties of financial market. Stock market is a complicated multi-body system and so the price fluctuations are the results of interactions of multi-bodies. Statistical physics is the subject that studies the results and fundamental principles of interactions of multi-bodies. According to the closing stock index of different time scales in Shanghai and Shenzhen and with the help of the theories and methods of statistical physics, this dissertation makes a thorough and systematic study on the properties of Chinese stock market and provides empirical results for dynamic mechanism of price fluctuations. It is made of five parts.
The first is the scaling properties. By using the Rescaled Range Analysis and the Detrended Fluctuation Analysis, we study the scaling properties of return series and find there is long-range memory. The Hurst exponents are 0.6282 and 0.6358 respectively in Shanghai stock market and Shenzhen stock market. We also find there are 284-day long memory period and 115-day short memory period. The persistence becomes weak when scale shortening, but it becomes strong when time range shortening. The fluctuation becomes complicated when the scale becomes short. As to the absolute value series and square value series of returns, there are stronger long-range persistence.
The second is the properties of returns. According to the high-frequency closing stock index, we study the probability distributions of returns and find the middle part of the distribution is well described by the Lévy stable distribution with Lévy Exponents 1.26 in Shanghai and 1.74 in Shenzhen. The asymptotic behaviour of the distribution is truncated Lévy distribution.
The third is the volatility. We analyse the statistical properties of volatility in Shanghai stock market and find there are scale invariance both in probability distribution and cumulative probability distribution. The former can be well described by a log-normal function and there are power law and crossover time, which is about four days. The Shanghai stock market has obvious day-effect whick shape is like the word“W”.
The forth is the multifractal. We put forward three methods of multifractal and have conclusions that Shanghai stock market has weak multifractal and its shape does not change with scales, but its strength weakens with the scale shortening. The general Hurst exponent decreases when the order of partition function increases. The shorter the scale, the stronger it decreases. The singular spectrum function does not change continuously with partition order. It is like the behaviour of the order parameter at its critical point.
The last one is the order parameter of stock market. With analogy, we believe the general dimension D q or the general Hurst exponent h (q) as the order parameter, the important parameter in phase transition and analyse its properties.
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