文档介绍：该【medium-and-smallscale-analysis-of-financial-data(中小规模的金融数据分析外文翻译学士学位论文 】是由【知识徜徉土豆】上传分享，文档一共【11】页，该文档可以免费在线阅读，需要了解更多关于【medium-and-smallscale-analysis-of-financial-data(中小规模的金融数据分析外文翻译学士学位论文 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。Medium and small-scale analysis of financial data
Abstract
A 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 behaviour 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 process, which is given by a stochastic Markov process in the scale τ. A corresponding Fokker–Planck 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; Fokker–Planck equation
One 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), most commonly quantities like returns, log returns or price increments are used. Here, we consider the statistics of the log return y(τ) over a certain timescale t, which is defined as
y(τ)=log x(t+τ) - log x(t), (1)
where x(t) denotes the price of the asset at time t. A common problem in the analysis of 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 τ corresponds to 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 analysis presented is mainly based on Bayer data for the time span of 1993–2003. The financial data sets were provided by the Karlsruher Kapitalmarkt Datenbank (KKMDB) .
2. Small-scale analysis
One remarkable feature of financial data is the fact that the probability density functions (pdfs) are not Gaussian, but exhibit heavy tailed shapes. Another remarkable feature is the change of the shape with the size of the scale variable τ. 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(τ)) on a timescaleτ and a pdf pT(y(T)) on a 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 and variance, all pdfs p(y(τ)) 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(τ)) and pT(y(T)), the Kullback–Leibler entropy is used.
dK(τ)= (2)
The evolution of dK with increasing t is illustrated. This quantifies the 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 shape 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 analysis
Next the behaviour for larger timescales (τ>1 min) is discussed. We proceed with the idea of a cascade. it is possible to grasp the complexity of financial data by cascade processes running in the variable τ. 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 Fokker–Planck equation. The underlying idea of this approach is to access statistics of all orders of the financial data by the general joint n-scale probability densities p(y1, τ1;y2, τ2;…;yN, τN). Here we use the shorthand notation y1=y(τ1) and take without loss of generality τi<τi+1. The smaller log returns y(τi) are nested inside the larger log returns y(τi+1) with common end point t.
The joint pdfs can be expressed as well by the multiple conditional probability densities p(yi, ti│yi+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 in t, which is a Markov process. This is the case if the conditional probability densities fulfil the following relations:
p(y1, τ1│y2, τ2;y3, τ3; . . . ; yN, τN)＝p(y1, τ1│y2) (3)
Consequently,
p(y1, τ1;…;yN, τN)= p(y1, τ1│y2)……p(yN-1, τN-1│yN, τN)·p(yN, τN) (4)
holds. Eq. (4) indicates the importance of the conditional pdf for Markov processes. Knowledge of p(y, τ│y0, τ0) (for arbitrary scales τ and τ0 with τ<τ0) is sufficient to generate the entire statistics of the increment, encoded in the N-point probability density p(y1, τ1;y2, τ2;…;yN, τN).
For Markov processes the conditional probability density satisfies a master equation, which can be put into the form of a Kramers–Moyal expansion for which the Kramers–Moyal coefficients D(K)(y, τ) are defined as the limit △τ→0 of the conditional moments M(K)(y, τ, △τ):
(5)
(6)
For a general stochastic process, all Kramers–Moyal coefficients are different from zero. According to Pawula’s theorem, however, the Kramers–Moyal expansion stops after the second term, provided that the fourth order coefficient D(4)(y, τ) vanishes. In that case, the Kramers–Moyal expansion reduces to a Fokker–Planck equation (also known as the backwards or second Kolmogorov equation):
(7)
D(1) is denoted as drift term, D(2) as diffusion term. The probability density p(y, τ) has to satisfy the same equation, as can be shown by a simple integration of Eq. (7).
4. Discussion
The 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 measure like the Kullback–Leibler 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 simple functional form of the Kullback–Leibler entropy with respect to the timescale τ. 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 joint 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 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 Boltzmann 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 Fokker–Planck equation for the scale t evolution of conditional probabilities, p(yi, τi│yi+1, τi+1). In our cascade picture the conditional probabilities cannot be reduced further to single probability densities, p(yi, τi), 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 on the information obtained by the Fokker–Planck 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(τ) with common right end points should be used. By the knowledge of the n-scale conditional probability density of all y(τi) 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 characterisation.
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；随机过程；Fokker-Planck方程
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(τ)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τ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τ的大小而变化。分析pdfs伴随着规模τ的变化的统计，非参数方法是一种选择。Pdf p(y(τ))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(τ))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(τ)) 和PT (y(T)) 之间的距离，需使用Kullback – Leibler：
dK(τ)= （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分钟的线性增长的距离测度似乎是普遍的。如果正常化的Gaussian分布是作为参考分布的，在小型时间表制度中快速偏离Gaussian变得很明显。对于较大的时间规模dK仍然接近常数，这表明pdfs的形状改变的非常缓慢。
第三章 中等规模的分析
接下来，对于较大的时间尺度（τ﹥1分钟）进行讨论。我们从级联观点着手，有可能通过级联运行过程中的变量τ掌握复杂的财务数据，尤其是它已被证明，有可能从给出的随机级联过程Fokker - Planck方程的形式中直接估计数据。这一做法的基本意图是为了获取所有的财务数据的一般性联合正规模概率密度p(y1, τ1;y2, τ2;…;yN, τN)的订单统计。在这里，我们使用速记符号y1=y(τ1),采取完整的概括性的τi＜τi+1，包含在较大的y(τi+1)中的较小的y(τi)都取决于t 。
复合的pdfs可由多个条件概率密度p(yi, τi│yi+1, τi+1; . . . ; yN, τN)来表达，包含众多点n的数据集n大概的数值范围，基本上可以简化为马尔可夫过程中τ的一个随机变化过程。这种情况下，如果条件密度符合下列关系：
p(y1, τ1│y2, τ2;y3, τ3; . . . ; yN, τtN)＝p(y1, τ1│y2) （3）
因此，
p(y1, τ1;…;yN, τN)= p(y1, τ1│y2)……p(yN-1, τN-1│yN, τN)·p(yN, τN) （4）
公式4显示马尔可夫过程中有条件的pdf的重要性。p(y, τ│y0, τ0) (τ和τ0是任意数，τ<τ0) 足够产生整个统计的增量， 在点N的概率密度p(y1, τ1;y2, τ2;…;yN, τN)中编码。
马尔可夫过程的概率密度满足可放入被定义为有条件的时刻M(K)(y, τ, △τ),△τ→0 的Kramers-Moyal系数D(K)(y, τ) 主方程的条件：
（5）
（6）
对于一般的随机过程，所有的Kramers-Moyal系数都是以零作为分界点。根据Pawula定理，只要四阶系数D(4)(y, τ)消失，Kramers-Moyal在第二个周期内后停止扩大。在这种情况下，Kramers-Moyal的扩大降低到Fokker - Planck方程（也称为倒退或第二Kolmogorov方程）：
（7）
D(1)被命名为漂移时期，D(2)作为传播时期。概率密度p(y, τ)应满足相同的方程，简单的一体化显示在公式（7）中。