Pareto-Zipf’s Law in Variability of Financial Time Series
ROBERT KITT, JAAN KALDA
Department of Mechanics an Applied Mathematics
Institute of Cybernetics at Tallinn Technical University
Akadeemia tee 21, 12618, Tallinn
ESTONIA
Abstract: The goal of the paper is to reveal some new facts about financial time series. It is generally accepted that financial time series are not following the Brownian random walk process, but rather (multi-)fractional Brownian motion i.e. the fluctuations of financial time series tend to have a memory. While the (multi)fractal description is adequate for the analysis of long-term dynamics, certain aspects of the variability of prices has been left out of focus. In this paper the long-term correlations in short-term variability are studied. The trading data are divided into two categories: “large variability” and “small variability”. These definitions are based on the relative difference between the current price and the periods’ sliding average, using a (adjustable) threshold value T. The sequential “low” volatility trading days make up a “silent” period; the “silent” periods are “ranked” according to their length (measured in the number of trading days N). The rank R of a “silent” period equals to the number of “silent” periods longer than N. The relationship between the rank R and N was studied for various financial data. The analysis was conducted using different threshold values T ranging from 0.25% to 3.00%. The time series studied included daily closings of five equity and five currency series for a relatively long time (each series containing at least 8000 data points). It appears that all the time series studied showed relatively good Pareto-Zipf’s power-law: the amount of periods is decreasing according to power law. Besides, the currency time series exhibited “super-universality”: the scaling exponent was similar in all currencies and all definitions of “large changes” (i.e. T-values), with a»1.75. For equity time series, the scaling exponent was sensitive with respect to the value of T; however, for a fixed T, different equities were described by similar scaling exponents a. Finally note that there is a simple implication of the very existence of such a power law: the probability of terminating the “silence” tomorrow is inversely proportional to the length of the current “silent” period.
Key Words: Pareto-Zipf’s law, scaling, econophysics, multifractality, power law
1. Introduction
Recent decade has opened a new era in financial analysis. It is known that methods and assumptions done half-century ago have been opposed and new methods are introduced. Even a new term: Econophysics is invented to describe the new interdisciplinary field of physics. Finance has gained from econophysics a lot: the econometrics and time series analysis have got the new methods. A lot of research is done, but even more is most likely still ahead.
Econophysics is not only pioneering new field of science. New methods of non-linear time-series analysis, developed for econophysics, have made significant contributions to many fields of physics. Vice versa, the methods of intermittent time-series analysis, developed in a different context, can be successfully used to improve the understanding of financial time-series.
Some stylized facts about financial time series can be listed as follows. The time-series exhibit multifractal structure [1-4]; the increments have non-Gaussian distribution [5-6]; the autocorrelation of returns drops quickly to zero [7-8]. This paper is aimed to search for yet-unknown properties of these time-series and is motivated by the recent results about heart-rate variability indicating that the low-variability periods of heart rate follow a Zipf’s law [9].
2. Statement of the Problem
Pareto-Zipf’s law is a well-known phenomenon in various fields of science. Italian economist Pareto suggested [10] that in different countries and times, the distribution of income and wealth amount follows a logarithmic pattern: log N = log A + m log x, where N is the number of income earners who receive income higher than x and A and m are constants.
Zipf found that similar relation is valid for word distribution in language [11]: suppose every word has assigned a rank, according to its “size” f, defined as a relative number of occurrences in some long text. Then, there is a power law (actually, inverse proportionality law) between the rank and size of a word. Such power laws have been found in a vast variety of systems. A recent example of Pareto-Zipf’s law in econophysics is provided by Fujiwara, who found power law distribution in companies’ bankruptcy events [12]. Also some other developments are recently done by using Pareto-Zipf’s law [13-15].
2.1 The Method
For our analysis we are using the method developed by Kalda et al [9].
I. We define a local average, which in financial community is often referred as moving average:
. (1)
Here: Pt denotes security’s price at time t, d denotes the length of averaging period. In our analysis we took d = 5.
II. We evaluate each trading day and label them as follows:
(2)
where θ(x) denotes the Heaviside function and T is a threshold to describe a “large” price fluctuation
III. Following steps I and II results in a new time-series lt: a sequence of "0"-s and "1"-s. In this sequence, "0" means that we have a "silent" day and "1" means that we have a day with a large fluctuation. As a step III we measure the lengths of the periods of subsequent days with lt=0. These are the "silent" periods where price fluctuations stayed below the threshold T.
IV Further we define a function R(n), which gives the number of such “silent” periods, the length of which is at least n days. For example, R(1) is the overall number of “silent” periods; R(5) is the number of “silent” periods with fluctuations staying below the threshold T for at least 5 days.
V Finally, we plot R(n) against n in log-log coordinates, see Fig.1.-3. The linear part of the curve at the right-hand-side of the plot indicates the presence of a power law
. (3)
The scaling exponent is calculated by using the least square fit for various values of the threshold parameter T.
2.2 Data and Analysis
We use daily closing data of different stock exchange indices and also currencies. Table 1 describes the data used for our analysis.
Table 1 Description of the input data
The examples of rank-length curves R(n) are provided in Figures 1.-3. In Figure 1, USDJPY time series and T = 0.75% is used. The least-square fit line corresponds to the scaling exponent .
Fig.1 USDJPY R(n) – n plot in log-log scale for T=0,75%
In figure 2, S&P500 equity index is used; the plots are given for T=0.5%, 1.0%. The scaling exponent value increases with decreasing T. In figure 3, USD/DEM exchange rate data are studied using the same threshold parameter values. The scaling exponent value is almost independent of T.
Fig.2 S&P 500 R(n) – n plot in log-log scale for T=0.5% and T=1.0%
Fig 3 USDDEM R(n) – n plot in log-log scale using T = 0.5%, T = 1.0%
Now, the following natural questions arise:
1. How universal is such a power-law for financial time-series?
2. How dispersed are the scaling exponent values for different time-series, but for a fixed threshold parameter value?
3. For a given time-series, how does the scaling exponent depend on the threshold parameter T?
It is clear that choice of the threshold is one of the important issues. If T is too large, then we would not see any movements outside the threshold, and R(n)-curve would be equivalently zero. On the other hand, if T is too small, the whole time-series would be a single "large fluctuation" period. A non-trivial scaling behavior of R(N)-curve appeared to be provided by the values TÎ{0.25%, 0.5%, 0.75%, 1.0%, 1.5%, 3.0%}. For all these values, the scaling behavior was reasonably well described by a power-law. Power law held extremely well in a region 0.5%<T<1.5%. Therefore we conclude that fluctuations of financial time series are described by Pareto-Zipf’s power-law.
The respective scaling exponent values are given in Tables 2 and 3 (in order to get idea about the “intrinsic” degree of fluctuations, this table provides also the standard deviation of the input data).
Table 2 Measured values of a for Equity time series
Table 3 Measured values of a for Currency time series
Regarding the question 2, the scaling exponent values appeared to be universal within both classes of data (equities and currencies), exhibiting very limited fluctuations for a fixed threshold parameter T. However, the scaling exponents’ values between the two classes were clearly different. Therefore, one can conclude that this scaling law has captured certain universal feature of the underlying data.
As for questions 3, one could expect that a is a decreasing function of T, because larger values of T means that there are fewer long periods of "silence", and therefore a less steep fall-off of the R(n)-curve. While such a dependence was observed, indeed, for equity indices, in the case of currencies (except for CAD, which seemed to be a special case, due to a strong coupling to USD), an unexpected super-universality was observed: the scaling exponent was almost constant, a ~ 1.74 for the range 0.25% £T£1%.
3. Probability of “Silence breaking”
As we have shown, the length-distribution of the “silent” periods in stock- and currency markets follows a power law. The very presence of such a power law has an interesting consequence for the “silence-breaking” probability. Suppose today is the n-th day of a “silent” period. What is the probability p(n) that tomorrow will be a “non-silent” day with lt=1? This can be calculated as the number of “silence-breaking” days at the end of those “silent” periods, which are not shorter than n, divided by the overall number M(n) of such “silent” days, which follow at least n-days-long “silent” period. Since each “silent” period is terminated by exactly one “non-silent” day, the first number is equal to R(n). The second number is calculated as
Assuming , the sum can be substituted by integral; according to the power law,
This integral converges for , yielding . Thus, the “silence-breaking” probability
.
Therefore, we arrived at a super-universal law: assuming the presence of a power law (3) with , the “silence-breaking” probability is inversely proportional to the observed “silence” length.
4. Conclusion
We have shown that the length-distribution of the "silent" periods in currency and equity markets follows the Pareto-Zipf’s power law. The “silent” periods are defined as sequences of such subsequent days, for which the local index variability stayed below a threshold level T. It was established that within the two groups, equities and currencies, the scaling exponent values were very similar for a fixed threshold parameter T. For currencies, a super-universality was observed: the scaling exponent was also (almost) independent of T, the values being scattered around a~1.75. Finally we have shown that the very existence of a power law for the length-distribution of "silent" periods implies that the “silence-breaking” probability (the probability of terminating the “silence” tomorrow) is inversely proportional to the length of the current “silent” period.
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