medium and smallscale analysis of financial data中小规模的金融数据分析外文翻译学位论文.docx

1、medium and smallscale analysis of financial data中小规模的金融数据分析外文翻译学位论文Medium and small-scale analysis of financial dataAbstractA stochastic analysis of financial data is presented. In particular we investigate how the statistics of log returns change with different time delays t. The scale-dependent be

2、haviour of financial data can be divided into two regions. The first time range, the small-timescale region (in the range of seconds) seems to be characterised by universal features. The second time range, the medium-timescale range from several minutes upwards can be characterised by a cascade proc

3、ess, which is given by a stochastic Markov process in the scale . A corresponding FokkerPlanck equation can be extracted from given data and provides a non-equilibrium thermodynamical description of the complexity of financial data.Keywords: Econophysics; Financial markets; Stochastic processes; Fok

4、kerPlanck equation1.IntroductionOne of the outstanding features of the complexity of financial markets is that very often financial quantities display non-Gaussian statistics often denoted as heavy tailed or intermittent statistics . To characterize the fluctuations of a financial time series x(t),

5、most commonly quantities like returns, log returns or price increments are used. Here, we consider the statistics of the log return y() over a certain timescale t, which is defined as y()=log x(t+) - log x(t), (1)where x(t) denotes the price of the asset at time t. A common problem in the analysis o

6、f financial data is the question of stationarity for the discussed stochastic quantities. In particular we find in our analysis that the methods seem to be robust against nonstationarity effects. This may be due to the data selection. Note that the use of (conditional) returns of scale corresponds t

7、o a specific filtering of the data. Nevertheless the particular results change slightly for different data windows, indicating a possible influence of nonstationarity effects. In this paper we focus on the analysis and reconstruction of the processes for a given data window (time period). The analys

8、is presented is mainly based on Bayer data for the time span of 19932003. The financial data sets were provided by the Karlsruher Kapitalmarkt Datenbank (KKMDB) .2. Small-scale analysisOne remarkable feature of financial data is the fact that the probability density functions (pdfs) are not Gaussian

9、, but exhibit heavy tailed shapes. Another remarkable feature is the change of the shape with the size of the scale variable . To analyse the changing statistics of the pdfs with the scale t a non-parametric approach is chosen. The distance between the pdf p(y() on a timescale and a pdf pT(y(T) on a

10、 reference timescale T is computed. As a reference timescale, T=1 s is chosen, which is close to the smallest available timescale in our data sets and on which there are still sufficient events. In order to be able to compare the shape of the pdfs and to exclude effects due to variations of the mean

11、 and variance, all pdfs p(y() have been normalised to a zero mean and a standard deviation of 1.As a measure to quantify the distance between the two distributions p(y() and pT(y(T), the KullbackLeibler entropy is used.dK()= (2)The evolution of dK with increasing t is illustrated. This quantifies th

12、e change of the shape of the pdfs. For different stocks we found that for timescales smaller than about 1 min a linear growth of the distance measure seems to be universally present. If a normalised Gaussian distribution is taken as a reference distribution, the fast deviation from the Gaussian shap

13、e in the small-timescale regime becomes evident. For larger timescales dK remains approximately constant, indicating a very slow change of the shape of the pdfs.3. Medium scale analysisNext the behaviour for larger timescales (1 min) is discussed. We proceed with the idea of a cascade. it is possibl

14、e to grasp the complexity of financial data by cascade processes running in the variable . In particular it has been shown that it is possible to estimate directly from given data a stochastic cascade process in the form of a FokkerPlanck equation. The underlying idea of this approach is to access s

15、tatistics of all orders of the financial data by the general joint n-scale probability densities p(y1, 1;y2, 2;yN, N). Here we use the shorthand notation y1=y(1) and take without loss of generality ii+1. The smaller log returns y(i) are nested inside the larger log returns y(i+1) with common end poi

16、nt t.The joint pdfs can be expressed as well by the multiple conditional probability densities p(yi, tiyi+1, ti+1; . . . ; yN, tN). This very general n-scale characterisation of a data set, which contains the general n-point statistics, can be simplified essentially if there is a stochastic process

17、in t, which is a Markov process. This is the case if the conditional probability densities fulfil the following relations:p(y1, 1y2, 2;y3, 3; . . . ; yN, N)p(y1, 1y2) (3)Consequently,p(y1, 1;yN, N)= p(y1, 1y2)p(yN-1, N-1yN, N)p(yN, N) (4)holds. Eq. (4) indicates the importance of the conditional pdf

18、 for Markov processes. Knowledge of p(y, y0, 0) (for arbitrary scales and 0 with 0) is sufficient to generate the entire statistics of the increment, encoded in the N-point probability density p(y1, 1;y2, 2;yN, N).For Markov processes the conditional probability density satisfies a master equation,

19、which can be put into the form of a KramersMoyal expansion for which the KramersMoyal coefficients D(K)(y, ) are defined as the limit 0 of the conditional moments M(K)(y, , ): (5) (6)For a general stochastic process, all KramersMoyal coefficients are different from zero. According to Pawulas theorem

20、, however, the KramersMoyal expansion stops after the second term, provided that the fourth order coefficient D(4)(y, ) vanishes. In that case, the KramersMoyal expansion reduces to a FokkerPlanck equation (also known as the backwards or second Kolmogorov equation): (7)D(1) is denoted as drift term,

21、 D(2) as diffusion term. The probability density p(y, ) has to satisfy the same equation, as can be shown by a simple integration of Eq. (7).4. DiscussionThe results indicate that for financial data there are two scale regimes. In the small-scale regime the shape of the pdfs changes very fast and a

22、measure like the KullbackLeibler entropy increases linearly. At timescales of a few seconds not all available information may be included in the price and processes necessary for price formation take place. Nevertheless this regime seems to exhibit a well-defined structure, expressed by the very sim

23、ple functional form of the KullbackLeibler entropy with respect to the timescale . The upper boundary in timescale for this regime seems to be very similar for different stocks. Based on a stochastic analysis we have shown that a second time range, the medium scale range exists, where multi-scale jo

24、int probability densities can be expressed by a stochastic cascade process. Here, the information on the comprehensive multi-scale statistics can be expressed by simple conditioned probability densities. This simplification may be seen in analogy to the thermodynamical description of a gas by means

25、of statistical mechanics. The comprehensive statistical quantity for the gas is the joint n-particle probability density, which describes the location and the momentum of all the individual particles. One essential simplification for the kinetic gas theory is the single particle approximation. The B

26、oltzmann equation is an equation for the time evolution of the probability density p(x; p; t) in one-particle phase space, where x and p are position and momentum, respectively. In analogy to this we have obtained for the financial data a FokkerPlanck equation for the scale t evolution of conditiona

27、l probabilities, p(yi, iyi+1, i+1). In our cascade picture the conditional probabilities cannot be reduced further to single probability densities, p(yi, i), without loss of information, as it is done for the kinetic gas theory.As a last point, we would like to draw attention to the fact that based

28、on the information obtained by the FokkerPlanck equation it is possible to generate artificial data sets. The knowledge of conditional probabilities can be used to generate time series. One important point is that increments y() with common right end points should be used. By the knowledge of the n-

29、scale conditional probability density of all y(i) the stochastically correct next point can be selected. We could show that time series for turbulent data generated by this procedure reproduce the conditional probability densities, as the central quantity for a comprehensive multi-scale characterisa

30、tion.Andreas P-Nawroth, Joachim Peinke. Carl-von-Ossietzky 奥尔登堡大学, D-26111奥尔登伯格，德国J. 2008年3月30日.中小规模的金融数据分析摘 要财务数据随机分析已经被提出，特别是我们探讨如何统计在不同时间里记录返回的变化。财务数据的时间规模依赖行为可分为两个区域：第一个时间范围是被描述为普遍特征的小时就区域（范围秒）。第二个时间范围是增加了几分钟的可以被描述为随机的级联过程的中期时间范围。相应的Fokker-Planck方程可以从特定的数据提取，并提供了一个非平衡热力学描述的复杂的财务数据。关键词：经济物理学；金融市场

31、；随机过程；Fokker-Planck方程第一章 前言复杂的金融市场的其中一个突出特点是资金数量显示非高斯统计往往被命名为重尾或间歇统计。描述金融时间序列x(t) 的波动 ，最常见的就是log函数或价格增量的使用。在这里我们认为，log函数y()超过一定时间t的统计，被定义为：y(r)=logx(t+r)-logx(t) （1）其中x(t)是指在时间t时资产的价格。在财务分析数据中一个常见的问题是讨论随机数量的平稳性，尤其是我们发现在我们的分析中采用什么样的方法似乎是强大的非平稳性的影响，这可能是由于数据的选择。请注意，有条件的应用相当于一个特定的数据过滤。尽管如此，特殊的结果略微改变了不同的

32、数据窗口，显示出非平稳性影响的可能性。在本文中，对于一个特定的数据窗口（时间段）我们侧重于分析和重建进程。目前已有的分析主要是基于1993至2003年的拜耳数据，财务数据集是由Kapitabmarkt Datenbank （KKMDB）提供。第二章 小规模分析财务数据的一个突出特点是事实上概率密度函数（pdfs）不是Gaussian，而是展览重尾形状。另一个显著的特点是形状伴随着可变规模的大小而变化。分析pdfs伴随着规模的变化的统计，非参数方法是一种选择。Pdf p(y()的时间T和PT（y(T)）的参考时间T之间的差距是可以计算的。作为一个参考的时间，在我们的数据集上接近最小的可用时间但仍然有足够的活动，T=1 s是选择。为了能够比较pdfs，并排除由于不同的均值和方差的影响 ，所有的pdfs p(y()正常化为零平均，标准偏差为1 。作为衡量量化两个分布p (y() 和PT (y(T) 之间的距离，需使用Kullback Leibler：dK()= （2）dK 随着t的增加而变化，量化的改变pdfs的形状。对于不同的股票，目前我们发现时间小于1