Stock market return distributions: from past to present
Stock market return distributions: from past
to present
S. Drożdż1,2, M. Forczek1, J. Kwapień1, P. Oświȩcimka1, R. Rak2
1Institute of Nuclear Physics, Polish Academy of Sciences, PL–31-342 Kraków,
Poland
2 Institute of Physics, University of Rzeszów, PL–35-959 Rzeszów, Poland
Abstract
We show that recent stock market fluctuations are characterized by the cumu-
lative distributions whose tails on short, minute time scales exhibit power scaling
with the scaling index α > 3 and this index tends to increase quickly with de-
creasing sampling frequency. Our study is based on high-frequency recordings of
the S&P500, DAX and WIG20 indices over the interval May 2004 - May 2006. Our
findings suggest that dynamics of the contemporary market may differ from the one
observed in the past. This effect indicates a constantly increasing efficiency of world
markets.
Key words: Financial markets, Inverse cubic power law, q-Gaussian distributions,
Multifractality
PACS: 89.20.-a, 89.65.Gh, 89.75.-k
The so-called financial stylized facts are among the central issues of econo-
physics research. Much effort has been devoted on both the empirical and the
theoretical level to such phenomena like fat-tailed distributions of financial
fluctuations, persistent correlations in volatility, multifractal properties of re-
turns etc. Specifically, the interest in the return distributions can be traced
back to an early work of Mandelbrot [1] in which he proposed a Lévy process
as the one governing the logarithmic price fluctuations. Much later this issue
was revisited in [2] based on data with much better statistics and a new model
of exponentially-truncated Lévy flights was introduced. Then, in an extensive
systematic study of the largest American stock markets [3] the distribution
tails for both the prices and the indices were shown to be power-law with the
scaling exponent α ≃ 3. The most striking outcome of that study was that de-
spite the fact that the tails were well outside the Lévy-stable regime (α ≤ 2),
they were apparently stable under time aggregation up to several days for in-
dices and up to a month for stocks. The existence of return distributions with
Preprint submitted to Elsevier 11 October 2018
http://arxiv.org/abs/0704.0664v1
scaling tails was also reported in other markets like e.g. London [4], Frank-
furt [5], Paris [6], Oslo [7], Tokyo [3], and Hong Kong [3,8] but sometimes with
a slightly different value of the scaling index. This empirical property of price
and index returns led to the formulation of the so-called inverse cubic power
law [3], which was soon followed by an attempt of formulating its theoretical
foundation [6] (see also [4]).
Subsequent related study [9] revealed that, opposite to the earlier outcomes
of [3], the tail shape of the return distributions might no longer be so stable
along time axis. After comparison of the results obtained from the American
stock market data in years 1994-95 and in 1998-99, it turned out that in
more recent data the scaling tails with α ≃ 3 for individual companies are
preserved up to the time scales (sampling intervals) ∆t of less than one hour
instead of one month. This earlier crossover for 1998-99 data can easily be seen
in Fig. 1. This result was obtained by extending our previous analysis [9] over
a set of 1000 largest American companies [10] in order to enable more direct
comparison with outcomes of ref. [3] based on the same number of stocks.
The inverse cubic scaling is still evident in Fig. 1 for short time scales up to
∆t = 4 min, but the scaling index starts rising already for data with ∆t = 16
min and for longer time scales the tail behaviour is clearly governed by the
Central Limit Theorem. The difference of the results for the same market
but for different time intervals might suggest that the scaling behaviour is
not stable and depends on some crucial factors as, for example, the speed of
information processing which constantly increases from past to present.
This possibility can further be examined by considering even more recent
data from the American market. First, we look at the S&P500 index which
already was the subject of an analysis in [3]. Our data is a time series of
1 min returns covering the period May 2004 - May 2006 (in [3] the period
was 1984-1996). The c.d.f. for this data is presented in Fig. 2(a) for several
time scales up to 120 min. The most interesting feature is the lack of inverse
cubic scaling even for the shortest one-minute returns: in this case the actual
scaling index is slightly above 4 and it systematically increases with decreasing
sampling frequency (see Table 1). We leave open here the question what factor
underlies the evident absence of the α ≃ 3 type of scaling: the dynamics of
S&P500 returns could have changed sufficiently significantly since earlier half
of 1990s and the inverse cubic scaling no longer exists or it still exists but
is restricted to time scales shorter than 1 minute. That our observation is
more universal and can be made for other markets as well one may infer from
Fig. 2(b) and Fig. 2(c) presenting c.d.f. for the German index DAX and for
the Polish index WIG20, respectively, for the same period of time. While the
returns of DAX did not exactly comply with the inverse cubic scaling also in
the period 1998-99 [9], the ones of WIG20 indeed used to display this kind of
behaviour in the past as documented in [11]. However, nowadays WIG20 also
develops much thinner tails with α > 4 for ∆t = 1 min (Table 1).
abs. norm. returns
n 1 min
4 min
16 min
60 min
120 min
240 min
1 day
2 days
4 days
8 days
1000 stocks
α = 3.0
Gaussian
Fig. 1. Cumulative distributions of normalized stock returns averaged over 1000
highly-capitalized American companies in time interval Dec 1997 - Dec 1999 for
several different time scales from 1 min to 8 days. Gaussian distribution and inverse
cubic scaling are also shown for comparison. Best-fit power index α calculated by
means of log-log regression assumes the following values: 3.08± 0.05 (∆t = 1 min),
3.34 ± 0.05 (4 min), 4.00 ± 0.04 (16 min), 4.60 ± 0.05 (60 min), 4.95 ± 0.06 (120
min), 4.81± 0.15 (240 min), 5.90± 0.08 (1 day), 7.18± 0.28 (2 days), 9.17± 0.22 (4
days), and 8.32 ± 0.40 (8 days).
∆t 1 min 4 min 16 min 32 min 60 min
S&P500 4.12± 0.12 4.21± 0.08 5.18 ± 0.21 5.53 ± 0.20 6.10 ± 0.35
DAX 3.56± 0.12 3.76± 0.05 4.44 ± 0.16 5.14 ± 0.26 5.16 ± 0.66
WIG20 4.28± 0.16 5.24± 0.27 5.81 ± 0.78 5.61 ± 0.45 6.30 ± 0.72
Table 1
Values of the scaling index α obtained with a log-log regression fit for three different
market indices (S&P500, DAX, and WIG20) from the interval May 2004 - May 2006.
Recent studies [11,12] showed that in a wide range of returns the financial
return distributions can be approximated by a family of q-Gaussians [13] with
the parameter q depending on the sampling interval ∆t. The q-Gaussian dis-
tributions follow naturally from the nonextensive statistical mechanics [13,12]
and their c.d.f. can be written as [11]
P (X > x) = Nq
(3− q)β)
2Γ(β)
± (x− µ̄q)2F1(α, β; γ; δ)
, (1)
Fig. 2. Cumulative distributions of index returns for American S&P500 (top row),
German DAX (middle row) and Polish WIG20 (bottom row) recorded over the in-
terval May 2004 - May 2006. Data points for sampling intervals from 1 min to 120
min are denoted by different symbols. (Left column) Distributions for normalized
returns are shown together with Gaussian distribution and lines corresponding to
inverse cubic scaling and, approximately, the actual scaling. (Right column) Exper-
imental distributions are best-fitted by q-Gaussians with a free parameter q (fitted
values displayed) separately for negative and positive returns. Note the small asym-
metry between left and right tails.
0,2 0,3 0,4 0,5 0,6 0,7 0,8
WIG20
Fig. 3. Singularity spectra for 1 min index returns: S&P500 (top), DAX (middle),
and WIG20 (bottom). In each case, the actual data (solid line) is accompanied by
its randomized version averaged over 10 independent realizations (full symbols).
where α = 1
, β = 1/(q− 1), γ = 3
, δ = −Bq(q− 1)(µ̄q − x)2, 2F1(α, β; γ; δ) =
δk(α)k(β)k
k!(γ)k
is the Gauss hypergeometric function and Nq, µ̄q are, respec-
tively, the normalization factor and q-mean of the q-Gaussian p.d.f. [13].
Fig. 2(d)-2(f) exhibit cumulative distributions of returns for the same three
indices with the corresponding best fits in terms of Eq.(1). The theoretical
curves are in satisfactory agreement with the data for all the considered val-
ues of ∆t. It is noteworthy that, consistently with the left-hand side panels
of Fig. 2, the largest values of q are well below q = 3/2, which corresponds
to the inverse cubic scaling, and decrease with decreasing sampling frequency
towards the classic Gaussian distribution with q = 1.
Finally, we look at the singularity spectra f(α) (for numerical details of the
method see e.g. [15]) of our time series under study. Two cases are considered:
original data comprising the full variety of nonlinear temporal correlations
(solid lines in Fig. 3) and the randomized data in which all the correlations
are removed by shuffling the data points (full symbols in Fig. 3). Since both
the nonlinear dependencies and the fat-tailed distributions can be potential
sources of multifractality, the data shuffling can give some information of how
rich is the multiscaling behaviour due to each of these sources [14,15]. In the
present context the most interesting feature is that the Gaussian distribution
of uncorrelated data is associated with a monofractal f(α) spectrum. Fig. 3
shows the singularity spectra for S&P500, DAX and WIG20 (top to bot-
tom) together with their randomized-data counterparts. In all three cases, the
original data clearly represent multifractal processes with DAX and S&P500
showing richer multifractality than WIG20. This picture changes completely if
we look at the randomized data: we cannot detect sufficiently significant trace
of multifractality (f(α) is almost point-like). This is in agreement with the
observation that the distribution tails for the contemporary data tend to be
thinner than before. This result gives also an additional argument in favour of
the statement that the principal (here even the unique) source of multifractal
properties of the stock market data are the nonlinear correlations.
In this paper we have shown that c.d.f. for the most recent stock market data
represented by the index returns develops tails whose scaling index rises above
the value of 3 even for short, minute sampling intervals. This means that con-
temporary market dynamics significantly differs from the one observed 20 or
even 10 years ago and described by the inverse cubic power law [3]. That these
changes are a continuous process rather that a sudden transition we infer from
the existence of intermediate stages in which the inverse cubic scaling was ob-
served up to medium time scales of tens of minutes but was absent in daily
data [9,11]. This effect suggests a scenario of constantly increasing market
efficiency due to an acceleration of information processing in the world mar-
kets [9,16]. The related compression of the range of potential time correlations
between consecutive returns in the present analysis finds evidence in a faster
convergence towards a Gaussian distribution for aggregated returns. Such a
faster convergence indicates weeker time-correlations between returns. Further
argument in favour of an increasing stock market efficiency comes from the
autocorrelation analysis. The autocorrelation functions calculated exlicitely
from the 1 min returns for the three indices considered above drop down to
the noise level already for time-lags as small as 1-2 min. This is to be com-
pared to ∼ 5 min in the period 1998-99 [9], and to ∼ 20 min which according
to ref. [3] was characteristic for the period 1994-95.
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