The Epps effect revisited
Bence Tóth∗1,2 János Kertész2,3
December 2, 2008
1 ISI Foundation - Viale S. Severo, 65 - I-10133 Torino, Italy
2 Department of Theoretical Physics, Budapest University of Technology
and Economics - Budafoki út. 8. H-1111 Budapest, Hungary
3 Laboratory of Computational Engineering, Helsinki University of Tech-
nology - P.O.Box 9203, FI-02015, Finland
Abstract
We analyse the dependence of stock return cross-correlations on
the sampling frequency of the data known as the Epps effect: For
high resolution data the cross-correlations are significantly smaller
than their asymptotic value as observed on daily data. The former
description implies that changing trading frequency should alter the
characteristic time of the phenomenon. This is not true for the em-
pirical data: The Epps curves do not scale with market activity. The
latter result indicates that the time scale of the phenomenon is con-
nected to the reaction time of market participants (this we denote as
human time scale), independent of market activity. In this paper we
give a new description of the Epps effect through the decomposition of
cross-correlations. After testing our method on a model of generated
random walk price changes we justify our analytical results by fitting
the Epps curves of real world data.
1 Introduction
1979 Epps reported results showing that stock return correlations decrease
as the sampling frequency of data increases [1]. Since his discovery the
phenomenon has been detected in several studies of different stock markets
[2, 3, 4] and foreign exchange markets [5, 6].
∗E-mail: bence@maxwell.phy.bme.hu
http://arXiv.org/abs/0704.1099v2
Cross-correlations between the individual assets are the main factors in
classical portfolio management thus it is important to understand and give
their accurate description on different time scales. This is especially so, since
today the time scale in adjusting portfolios to events occuring may be in the
order of minutes.
Considerable effort has been devoted to uncover the phenomenon found
by Epps [7, 8, 9, 10, 11, 12]. However most of the works aim to construct a
better statistical measure for co-movements in prices in order to exclude bias
of the estimator by microstructure effects [13, 14, 15, 16, 17, 18, 19, 20], only
a few searching for the description of the microstructure dynamics.
Up to now two main factors causing the effect have been revealed: The
first one is a possible lead-lag effect between stock returns [21, 22, 23] which
can appear mainly between stocks of very different capitalisation and/or for
some functional dependencies between them. In this case the maximum of
the time-dependent cross-correlation function can be found at non zero time
lag, resulting in increasing cross-correlations as the sampling time scale gets
into the same order of magnitude as the characteristic lag. This factor can be
easily understood, morever, in a recent study [23] we showed that through
the years this effect becomes less important as the characteristic time lag
shrinks, signalising an increasing efficiency of stock markets. As the Epps
effect can also be found for the case when no lead-lag effect is present, in the
following we will focus only on other possible factors.
The second, more important factor is the asynchronicity of ticks in case
of different stocks [7, 8, 21, 24]. Empirical results [7] showed that taking into
account only the synchronous ticks reduces to a great degree the Epps effect,
i.e. measured correlations on short sampling time scale increase. Naturally
one would expect that for a given sampling frequency growing activity de-
creases the asynchronicity, leading to a weaker Epps effect. Indeed Monte
Carlo experiments showed an inverse relation between trading activity and
the correlation drop [7].
However, the analysis of empirical data showed [25] that the explanation
of the effect solely by asynchronicity is not satisfactory. After eliminating the
effect of changing asymptotic cross-correlations through the years (scaling
with the asymptotic value), the curves of cross-correlation as a function of
sampling time scale tend to collapse to one curve and surprisingly we do
not find a measurable reduction of the characteristic time of the Epps effect,
while the trading frequency grew by a factor of ∼ 5−10 in the period. These
results will be discussed further in details in Section 2.2.
The characteristic time of market phenomena can usually be split up into
three kinds of market time scales: the frequency of trading on the market
(which we will denote as activity), market periodicities and the reaction time
of traders to news, events. In Ref. [25] we showed that the characteristic time
of the Epps effect does not scale with changing market activity (this we will
discuss in Section 2.2), which points out that the characteristic time of the
Epps effect can not be determined solely by the market activity causing asyn-
chronicity. Market periodicities in high frequency data are the different types
of patterns, which can be found in intraday data, as well as on broader time
scales (see e.g. Refs. [26] and [27]). Market periodicities and intraday struc-
ture do not have a role in our results since we are averaging them out. Hence
we believe that the characteristic time of the Epps effect is the outcome of a
human time scale present on the market: The time that market participants
need to react to certain pieces of news. There are several studies in the litera-
ture about reaction time. The issue is connected both to behavioural finance
questions and to market efficiency. 1970, Fama defined an efficient market as
one in which prices fully reflect all available information [28]. This response
to information in practice can not happen instantaneously. There are several
results reporting that prices incorporate news within five to fifteen minutes
after news announcements [29, 30, 31, 32, 33]. More recent studies showed
similar results on the time that traders needed to react to news [34, 35, 36].
Supposing that the Epps effect is possibly the outcome of a human time
scale present on the market motivated us to separate the terms in the cross-
correlation function, in order to study their behaviour one by one. In this
paper we suggest an analytic decomposition of the cross-correlation function
of asynchronous events using time lagged correlations. As a second step we
demonstrate the efficiency of the formalism on the example of generated data.
Finally we describe and fit the empirically observed dependence of the cross-
correlations. We find that the origin of the independence of the characteristic
time of the Epps effect on the trading frequency is the presence of a human
time scale in the time lagged autocorrelation functions. Ref. [12] already
called the attention to the importance of lagged cross-influences of stock
returns in explaining the Epps-effect. Using a somewhat different formalism,
here we investigate thoroughly this relationship.
The paper is built up as follows: in Section 2 we present the data used
and discuss the problems of the former descriptions. Section 3 the decompo-
sition of the cross-correlation coefficient, Section 4 shows a simulation model
demonstrating the idea. In Section 5 we present the assumptions concerning
real data and show fits and statistics for the Epps curves. We finish the paper
with a discussion.
2 Empirical analysis
2.1 Data and methodology
In our analysis we used the Trade and Quote (TAQ) Database of the New
York Stock Exchange (NYSE) for the period of 4.1.1993 to 31.12.2003, con-
taining tick-by-tick data. The data used was adjusted for dividends and
splits.
We computed the logarithmic returns of stock prices:
rA∆t(t) = ln
pA(t)
pA(t −∆t)
, (1)
where pA(t) stands for the price of stock A at time t. The prices were deter-
mined using previous tick estimator on the high frequency data, i.e. prices
are defined constant between two consecutive trades. The time dependent
cross-correlation function C
∆t (τ) of stocks A and B is defined by
∆t (τ) =
rA∆t(t)r
∆t(t + τ)
rA∆t(t)
rB∆t(t + τ)
. (2)
The notion 〈· · · 〉 stands for the time average over the considered period:
〈r∆t(t)〉=
T −∆t
r∆t(i), (3)
where time is measured in seconds and T is the time span of the data.
The standard deviation σ of the returns reads as:
〈r∆t(t)2〉−〈r∆t(t)〉
, (4)
both for A and B in (2). We computed correlations for each day separately
and averaged over the set of days, this way avoiding large overnight returns
and trades out of the market opening hours. For pairs of stocks with a
lead–lag effect the function C
∆t has a peak at non-zero τ. The equal-time
cross-correlation coefficient is naturally: ρA/B∆t ≡C
∆t (τ = 0). In our notations
the Epps effect means the decrease of ρ∆t as ∆t decreases (see Figure 1). Since
the prices are defined as being constant between two consecutive trades, the
∆t time scale of the sampling can be chosen arbitrarily.
As stated above, we do not want to discuss the Epps effect originated
from the lead-lag effect in the correlations. Thus we consider only pairs of
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
∆t [sec]
Figure 1: The cross-correlation coefficient as a function of sampling time
scale for the period 1993–2003 for the Coca-Cola Pepsi pair. Several hours
are needed for the correlation to reach its asymptotic value.
stocks where the latter effect is negligible, i.e., for which the price changes
are highly correlated with the peak position of C
∆t of Equation (2) being at
τ ≈ 0. The results shown in this paper can be generalised for all stock pairs
(though in case of an empirical study one can never fit all data). To illus-
trate our results we will present results for some stock pairs and in Section
5 we will show statisitcs for a broader set of data. The stocks mentioned
in the paper are the following: Avon Products, Inc. (AVP), Caterpillar Inc.
(CAT), Colgate-Palmolive Company (CL), E.I. du Pont de Nemours & Com-
pany (DD), Deere & Company (DE), The Walt Disney Company (DIS), The
Dow Chemical Company (DOW), General Electric Co. (GE), International
Business Machines Corp. (IBM), Johnson & Johnson (JNJ), The Coca-Cola
Company (KO), 3M Company (MMM), Motorola Inc. (MOT), Merck &
Co., Inc. (MRK), PepsiCo, Inc. (PEP), Pfizer Inc. (PFE), The Procter &
Gamble Company (PG), Sprint Nextel Corp. (S), Vodafone Group (VOD),
Wal-Mart Stores Inc. (WMT).
2.2 Time evolution of the characteristic time
Previous studies claimed the asynchronicity of ticks for different stocks as the
main cause of the Epps effect [7, 8]. It is natural to assume that, for a given
sampling frequency, increasing trading activity should enhance synchronicity,
leading to a weaker Epps effect.
To study the trading frequency dependence of the cross-correlation drop,
we computed the Epps curve separately for different years. In Figure 2 the
cross-correlation coefficients can be seen as a function of the sampling time
scale for the years 1993, 1997, 2000 and 2003 for three example stock pairs.
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
Figure 2: The Epps curves for the CAT/DE (top), KO/PEP (middle) and
MRK/JNJ (bottom) pairs for the years 1993, 1997, 2000 and 2003. The
asymptotic value of the cross-correlations varies in time.
It is known that cross-correlation coefficients are not constant through
the years. The asymptotic values of cross-correlations (long sampling time
scale) depend on the economical situation, the state of the economic sectors
that the pairs of stocks belong to, and several other factors. We need to
take this into account and try to extract the effect of changing asymptotic
cross-correlations from the Epps phenomenon. In order to get comparable
curves, we scaled the cross-correlation curves with their asymptotic value:
The latter was defined as the mean of the cross-correlation coefficients for
the sampling time scales ∆t = 6000 seconds through ∆t = 9000 seconds, and
the cross-correlations were divided by this value. Figure 3 shows the scaled
curves for the same years and pairs as Figure 2.
The frequency of trades changed considerably in the last two decades:
Trading activity has grown almost monotonically, as it can be seen in Figure
4. This would infer the diminution of the Epps effect and a much weaker
decrease of the correlations as sampling frequency is increased. However,
after scaling with the asymptotic cross-correlation value, the curves give a
reasonable data collapse and no systematic trend can be seen. Surprisingly,
as it can be seen in Table 1, a rise of the trading frequency by a factor of
∼ 5−10 does not lead to a measurable reduction of the characteristic time
of the Epps effect (where we define the characteristic time the time scale for
which the cross-correlation reaches the 1− e−1 rate of its asymptotic value).
These results show that explaining the Epps effect merely as a result of
the asynchronicity of ticks is not satisfactory. It is also important to mention
that not even the changing tick sizes for the stocks (likely to change the
arrival rate of price changes) alter the characteristic time of the effect.
Table 1: The characteristic time of the Epps effect for the years 1993, 1997,
2000 and 2003 measured in seconds for the stocks pairs: CAT/DE, KO/PEP
and MRK/JNJ (characteristic time was defined as the time scale for which
the cross-correlation value reaches the 1− e−1 rate of its asymptotic value).
No clear trends can be seen in the charactersitic time while the activity is
growing rapidly.
CAT/DE KO/PEP MRK/JNJ
1993 940 920 800
1997 620 760 420
2000 1320 1040 880
2003 700 800 1060
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
Figure 3: The Epps curves scaled with the asymptotic cross-correlation values
for the CAT/DE (top), KO/PEP (middle) and MRK/JNJ (bottom) pairs
for the years 1993, 1997, 2000 and 2003. The scaled curves give a reasonable
data collapse in spite of the considerably changing trading frequency, showing
that the characteristic time of the Epps effect does not change with growing
activity.
1994 1996 1998 2000 2002
yearsa
Figure 4: The average intertrade time for the years 1993 to 2003 for some
example stocks (CAT, DE, KO, PEP, MRK, JNJ). The activity was growing
almost monotonically.
3 Decomposition of the cross-correlations
In this section we show calculations for the relation between the value of
cross-correlations on different time scales. We connect the cross-correlation
on a certain time scale (∆t) to lagged autocorrelations and cross-correlations
on smaller time scales (∆t0).
Returns in a certain time window ∆t are mere sums of returns in smaller,
non-overlapping windows ∆t0, where ∆t is a multiple of ∆t0:
r∆t(t) =
∆t/∆t0
r∆t0(t −∆t + s∆t0). (5)
Using this relationship the time average of the product of returns on the
large time scale (∆t) can be written in terms of the averages on the short
time scale (∆t0) in the following way:
rA∆t(t)r
∆t(t)
T −∆t
rA∆t(i)r
∆t(i) =
T −∆t
∆t/∆t0
rA∆t0(i−∆t + s∆t0)
∆t/∆t0
rB∆t0(i−∆t +q∆t0)
∆t/∆t0
∆t/∆t0
rA∆t0(i−∆t + s∆t0)r
∆t0(i−∆t +q∆t0)
. (6)
We can see that on the right side of Equation (6) the lagged time average
of return products appear on the short time scale, ∆t0, i.e., the non-trivial
part of the lagged cross-correlations. Naturally in the case of A = B, we get
the relation for
r∆t(t)
In order to apply Equation (6), we need to have information about the
lagged autocorrelation and cross-correlation functions. Writing out the sum
in Eq. (6) we get:
rA∆t(t)r
∆t(t)
x=− ∆t∆t0
rA∆t0(t)r
∆t0(t + x∆t0)
, (7)
and similarly
rA∆t(t)
x=− ∆t∆t0
rA∆t0(t)r
∆t0(t + x∆t0)
rB∆t(t)
x=− ∆t∆t0
rB∆t0(t)r
∆t0(t + x∆t0)
. (8)
Since the mean of returns is 1-2 orders of magnitude smaller than the
second moments in the correlation function, we can omit the expressions
rA∆t(t)
rB∆t(t + τ)
and 〈r∆t(t)〉
in Equation (2). As these terms are of second
order, this can even be done in case of slight price trends. Hence Equation
(2) becomes:
ρA/B∆t =
rA∆t(t)r
∆t(t)
rA∆t(t)
rB∆t(t)
. (9)
For simplicity, we introduce decay functions to describe lagged correla-
tions:
(x∆t0) =
rA∆t0(t)r
∆t0(t + x∆t0)
rA∆t0(t)r
〉 , (10)
defined for both positive and negative x values, and similarly f
(x∆t0) and
(x∆t0). Thus the correlation can be written in the following form:
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)r
∆t0(t)
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)
)−1/2
x=− ∆t∆t0
(x∆t0)
rB∆t0(t)
)−1/2
. (11)
Hence
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
x=− ∆t∆t0
(x∆t0)
)−1/2
x=− ∆t∆t0
(x∆t0)
)−1/2
ρA/B∆t0 . (12)
This way we obtained an expression of the cross-correlation coefficient
for any sampling time scale, ∆t, by knowing the coefficient on a shorter
sampling time scale, ∆t0, and the decay of lagged autocorrelations and cross-
correlations on the same shorter sampling time scale (given that ∆t is multiple
of ∆t0). Our method is to measure the correlations and fit their decay func-
tions on a certain short time scale and compute the Epps curve using the
above formula.
4 Model calculations
In this section we demonstrate the decomposition process on computer gen-
erated time series which should mimic two correlated return series. We will
demonstrate the Epps effect and see how the decomposition works for these
controlled cases. Our aim is to show that in case of generated “price” series,
the decomposition process leads to a very good description of the time scale
dependence of the cross-correlation coefficients. More discussion and details
on the analytic treatment of the model can be found in Ref. [38].
To mimic some properties of financial data, we simulate two correlated
but asynchronous price time series. As a first step we generate a core random
walk with unit steps up or down in each second with equal possibility (W (t)).
Second we sample the random walk, W (t), twice independently with waiting
times drawn from an exponential distribution. This way we obtain two time
series (pA(t) and pB(t)), which are correlated since they are sampled from the
same core random walk, but the steps in the two walks are asynchronous.
The core random walk is:
W (t) = W (t −1)+ ε(t),
where ε(t) is ±1 with equal probability (and W (0) is set high in order to
avoid negative values). The two price time series are determined by
pA(ti) =
W (ti) if ti = ∑ik=1 Xk
p(ti −1) otherwise
pB(ti) =
W (ti) if ti = ∑ik=1Yk
p(ti −1) otherwise
. (14)
from the core random walk, where, i = 1,2, · · · , and Xk and Yk are drawn from
an exponential distribution:
P(y) =
λe−λy if y ≥ 0
0 y < 0
with parameter λ = 1/60. A snapshot as an example of the generated time
series with exponentially distributed waiting times can be seen on Figure 5.
As a next step we create the logarithmic return time series (rA∆t(t) and
rB∆t(t)) of p
A(t) and pB(t) as defined by Equation (1). In case of the random
walk model of price changes we know that
rA∆t(t)
rB∆t(t)
= 0 without
having to make any assumptions. Of course having a random walk model,
the autocorrelation function of the steps is zero for all non-zero time lags:
(x∆t0) = f
(x∆t0) = δx,0, (16)
0 100 200 300 400
10000
10010
10020
pA(t)
pB(t)
Figure 5: A snapshot of the model with exponentially distributed waiting
times. The original random walk is shown with lines (black), the two sampled
series with dots and lines (red) and triangles and lines (blue).
rA∆t(t)
rA∆t0(t)
rB∆t(t)
rB∆t0(t)
. (17)
Hence the cross-correlation can be written in the following form:
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)r
∆t0(t)
rA∆t0(t)
rB∆t0(t)
)−1/2
x=− ∆t∆t0
(x∆t0)
ρA/B∆t0 . (18)
In the model case we set the smallest time scale ∆t0 = 1 time step. It
can be shown [38] that in the case of λ ≪ 1 (small density of ticks) the exact
analytical expression for the cross-correlations is identical to (18) with an
exponential decay function:
(x∆t0) = e−λ∆t0|x|, (19)
where λ is the parameter of the original exponential distribution used for
sampling. Further results and exact computations of the cross-correlations
for the model can be found in Ref. [38].
Figure 6 shows the computed cross-correlations of the generated time
series on several sampling time scales and the computed cross-correlations
using Equation (18) and the exponential decay function (19). The two curves
are in very good agreement showing that the decomposition procedure is able
to well capture the Epps effect for generated time series.
0 5000
measured correlations
analytically computed correlations
∆t [simulation steps]
Figure 6: The measured and the computed cross-correlation coefficients using
exponential decay function as a function of sampling time scale for the simu-
lated time series with exponentially distributed waiting times. The analytic
fit is in very good agreement with the Epps curve.
5 Application of the theory to the data
In this section we discuss the properties of the decay functions in case of real
world data, and inserting them into Equation (12) we derive analytical fits
for the measured Epps curves.
5.1 Decay functions
As discussed, we measure the equal-time cross-correlations and the decay of
cross and autocorrelations on a certain short sampling time scale and from
these we obtain the value of equal-time cross-correlations on larger sampling
time scales. To do this, in case of the toy model, we had the possibility
of using the smallest time scale available in the generated data as ∆t0, i.e.,
the resolution being one simulation step. When studying data from real
world markets, one has to make restrictions. As being the highest resolution
commonly used in financial analysis, it would be plausible to choose windows
of one second as ∆t0. However on this time scale one is only able to measure
noise, no valid correlations and decay functions can be found. Thus we had to
use less dense data for the smallest sampling time scale: in the results shown
below we set ∆t0 = 120 seconds. Using this resolution we get an acceptable
signal-to-noise ratio and we hope not to lose too much information compared
to higher frequencies.
To avoid new parameters in the model we use the raw decay functions in
the formula (12), without fitting them. Since it is an empirical approach to
determine the decay functions for real data, we have to distinguish the signal
from the noise in the decay functions. Concerning the sensitivity from the
input (decay function) we observed that the results are quite robust against
little changes in the input functions, however the noise in the tail can cause
significant deviations. According to this we take into account the decay
functions for correlations only for short time lags. For the decay of the cross-
correlations we take into account the function only up to the time lag where
the decaying signal reaches zero for the first time, for larger lags we assume
it to be zero. For the decay of autocorrelations we take into the account the
function only up to the time lag where after the negative overshoot of the
beginning it decays to zero from below for the first time, for larger lags we
assume it to be zero.
Figure 7 shows an example of the decay functions in case of the stock
pair KO/PEP (for other pairs the decay functions are very much similar).
The plot shows the decay functions up to the time lags of 1000 seconds, with
a vertical line showing how long we take the empirical decays into account.
We can see that the time lag for which the decay functions disappear is in
the order of a few minutes. In fact in case of all stock pairs studied we found
the decay disappearing after 5–15 minutes.
5.2 Fits
Inserting the empirical decay of lagged autocorrelations and cross-correlations
on the short time scale into the formula of Equation (12), we compare the
computed and the measured Epps curves. Figures 8, 9 and 10 show these
plots for a few example stock pairs.
One can see, that the fits are able to describe the change of cross-correlation
0 200 400 600 800 1000
x∆t0 [sec]
0 200 400 600 800 1000
x∆t0 [sec]
-1000 -750 -500 -250 0 250 500 750 1000
x∆t0 [sec]
Figure 7: Top: The decay of lagged autocorrelations for KO. Middle: The
decay of lagged autocorrelations for PEP. Bottom: The decay of lagged cross-
correlations for KO/PEP pair. A vertical lines show the threshold up to which
we take the decays into account, for larger lags we assume them to be zero.
Sampling time scale is ∆t0 = 120 seconds on all three plots.
with increasing sampling time scale. Note, that as it has been shown in
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 8: The measured and the analytically computed cross-correlation co-
efficients as a function of sampling time scale for the pairs CAT/DE and
KO/PEP. Note that using only the autocorrelations and cross-correlations
measured on the smallest time scale (∆t0 = 120 seconds) we are able to give
reasonable fits to the cross-correlations on all time scales. Details on the
goodness parameter of the measured and computed correlations can be found
in Table 2.
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 9: The measured and the analytically computed cross-correlation co-
efficients as a function of sampling time scale for the pairs WMT/S and
GE/MOT. Details on the goodness parameter of the measured and com-
puted correlations can be found in Table 2.
Section 3, in the analytical formula only the autocorrelations and cross-
correlations on the smallest time scale (∆t0) and the decay functions are
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 10: The measured and the analytically computed cross-correlation
coefficients as a function of sampling time scale for the pairs MRK/JNJ and
GE /IBM. Details on the goodness parameter of the measured and computed
correlations can be found in Table 2.
taken into account as input to compute the cross-correlations on all other
time scales, no additional parameters are used.
To show a broader set of results, we introduce a goodness parameter for
the agreement between the measured and the analytically determined Epps
curves. We define the goodness parameter as the absolute error between the
measured and the analytically computed points:
g(∆t) = 100
|ρmeasured∆t −ρ
computed
ρmeasured∆t
. (20)
Table 2 shows the maximum, the mean and the median of the goodness
parameters for a broader set of stocks. The results show that the absolute
mean error is very low, with a maximum around 7 percents and both a
mean and a median around 2 percents. It is important to mention, that
the maximal error is usually found for high frequency scales, for longer time
scales and especially for the asymptotic correlation value the aggrement is
very good.
These results show that the growing cross-correlations with decreasing
sampling frequency are due to finite time decay of the lagged autocorrelations
and cross-correlations in the high frequency sampled data.
The finite decay of the cross-correlations on the short time scale (∆t0)
is not caused by difference in the capitalisation of the two stocks or func-
tional dependencies between them. Instead, it is an artifact of the market
Table 2: The maximum, the mean and the median of the goodness parameters
a broader set of stocks. The results show that the absolute mean error is low.
Note that the maximal absolute error in general occurs for high frequency
scales.
stock pair max [%] mean [%] median [%]
CAT/DE 4.81 1.26 0.94
KO/PEP 10.67 2.46 1.23
WMT/S 7.66 2.32 1.74
GE/MOT 6.43 3.29 3.44
MRK/JNJ 5.26 1.51 0.95
GE/IBM 3.90 1.57 1.12
PG/CL 6.05 1.81 1.24
MRK/PFE 4.76 1.42 1.32
AVP/CL 10.37 7.75 9.53
DD/DOW 8.84 2.05 1.49
DD/MMM 5.76 2.17 1.93
MOT/VOD 9.73 2.57 1.82
DIS/GE 5.78 1.54 0.97
average 6.92 2.4 2.1
microstructure. Reaction to a certain piece of news is usually spread out on
an interval of a few minutes for the stocks [29, 30, 31, 32, 33, 34, 35, 36, 39, 40]
due to human trading nature, thus not scaling with activity, with ticks being
distributed more or less randomly. This means that correlated returns are
spread out for this interval (asynchronously), causing non zero lagged cross-
correlations on the short time scale and thus the Epps effect. This way, as
stated by Ref. [7], the asynchronicity is indeed important in describing the
Epps effect but only in promoting the lagged correlations. (Even in case of
completely synchronous, but randomly spread ticks we could have the finite
decay of lagged correlations on short time scale, and hence the Epps effect.)
6 Discussion
In our study we examined the causes of the Epps effect, the dependence
of stock return cross-correlations on sampling time scale. We showed that
explaining the effect solely through asynchronicity of price ticks is not sat-
isfactory. When scaling the Epps curves with their asymptotic value for
different years, we get a reasonable data collapse and a growing activity of
the order ∼ 5−10 does not affect the characteristic time of the Epps effect.
The main point of our calculations is that we connected the cross-correlations
on longer time scales to the lagged autocorrelations and cross-correlations on
any shorter time scale. We demonstrated the idea of these calculations on a
random walk asynchronous model of prices, getting a very good agreement
with the cross-correlation curves.
Assuming the time average of stock returns to be zero we were able to
decompose the expression for the cross-correlation coefficient deriving an an-
alytical formula of the cross-correlations on any time scale, given the decay
of the autocorrelations and cross-correlations on a certain short time scale.
With this analytical formula we were able to give fits to the Epps curves
of real stock pairs getting acceptable results. The fits show that the Epps
effect is caused by the finite time decay of the lagged correlations in the high
frequency sampled data. The reason for the characteristic time not chang-
ing with growing activity is a human time scale present in the phenomenon,
which does not scale with the changing inter-tick time. The finite decay of
lagged correlations on the short time scale is due to market microstructure
properties: different actors on the market have different time horizons of in-
terest resulting in the reactions to certain pieces of news being spread out
for a time interval of a few minutes. The correlated returns ranging over
this interval cause the finite time decay of lagged correlations on the short
time scale resulting in the Epps effect. Our results do not contradict to the
earlier observations on the importance of asynchronicity in the Epps-effect,
however, its role has been put into a new perspective.
Acknowledgments
Support by OTKA T049238 is acknowledged.
References
[1] T.W. Epps, Journal of the American Statistical Association 74, 291-298
(1979)
[2] G. Bonanno, F. Lillo, R.N. Mantegna, Quantitative Finance 1, 1-9
(2001)
[3] A. Zebedee, A closer look at co-movements among stock returns, San
Diego State University, working paper (2001)
[4] M. Tumminello, T. Di Matteo, T. Aste, R.N. Mantegna, Eur. Phys. J.
B. 55, 209-217 (2007)
[5] M. Lundin, M. Dacorogna, U. A. Müller, Correlation of high-frequency
financial time series. In P. Lequeux (Ed.), Financial Markets Tick by
Tick. Wiley & Sons.
[6] J. Muthuswamy, S. Sarkar, A. Low, E. Terry, Journal of Futures Markets
21(2), 127-144 (2001)
[7] R. Renò, International Journal of Theoretical and Applied Finance 6(1),
87-102 (2003)
[8] O. V. Precup, G. Iori, Physica A 344, 252-256 (2004)
[9] O. V. Precup, G. Iori, European Journal of Finance (2006)
[10] J. Kwapień, S. Drożdż, J. Speth, Physica A 337, 231-242 (2004)
[11] L. Zhang, Estimating Covariation: Epps Effect, Microstructure Noise
working paper (2006)
[12] M. Potters, J.-P. Bouchaud, L. Laloux, Acta Physica Polonica B, 36, 9,
(2005)
[13] N. Shephard, O.E. Barndorff-Nielsen, University of
Oxford, Department of Economics, Economics Se-
ries Working Papers No. 240. (2005) Available at:
http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf
[14] F. Corsi, F. Audrino, University of St. Gallen, Department of Eco-
nomics, Discussion Paper No. 2007-02 (2007) Available at SSRN:
http://ssrn.com/abstract=957997
[15] T. Hayashi, N. Yoshida, Bernoulli 11(2), 359-379 (2005)
[16] M. Martens, Working Paper, University of Rotterdam (2003)
[17] T.G. Andersen, T. Bollerslev, P.F. Christoffersen, F.X. Diebold, in G.
Elliot, C.W.J. Granger, and Allan Timmermann (eds.), Handbook of
Economic Forecasting, Amsterdam: North-Holland, 778-878 (2006)
[18] T. Andersen, T. Bollerslev, International Economic Review, 39, 885-905
(1998)
[19] F. de Jong, T. Nijman, Journal of Empirical Finance 4, 259-277 (1997)
[20] C. Ball, W. Torous, Journal of Empirical Finance 7, 373-388 (2000)
http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf
http://ssrn.com/abstract=957997
[21] A. Lo, A. C. MacKinlay, Rev. Finance Stud 3, 175-205 (1990)
[22] L. Kullmann, J. Kertész, K. Kaski, Phys. Rev. E 66, 026125 (2002)
[23] B. Tóth, J. Kertész, Physica A 360 505-515 (2006)
[24] A. Lo, A. C. MacKinlay, Journal of Econometrics 45, 181-211 (1990)
[25] B. Tóth, J. Kertész, Physica A 383, 54-58 (2007)
[26] R.A. Wood, T.H. McInish, J.K. Ord, Journal of Finance 40(3), 723-739
(1985)
[27] K. Chan, Y.P. Chung, H. Johnson, The Journal of Financial and Quan-
titative Analysis, 30, 3, 329-346 (1995)
[28] E. Fama, Journal of Finance 25, 383-417 (1970)
[29] L. Dann, D. Mayers, R. Raab, Journal of Financial Economics 4, 3-22
(1977)
[30] J. Patell, M. Wolfson, Journal of Financial Economics 13, 223-252 (1984)
[31] R. Jennings, L. Starks, Journal of Accounting Research 23, 336-350
(1985)
[32] M. Barclay, R. Litzenberger, Journal of Financial Economics 21, 71-99
(1988)
[33] S. Kim, J. Lin, M. Slovin, Journal of Financial and Quantitative Analysis
32, 507-524 (1997)
[34] J. Busse, C. Green, Journal of Financial Economics 65, 415-437 (2002)
[35] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics
76, 271-292 (2005)
[36] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics
87, 249-268 (2008)
[37] P.Ch. Ivanov, A. Yuen, B. Podobnik, Y. Lee, Phys. Rev. E 69, 056107
(2004)
[38] Bence Tóth, János Kertész, Bálint Tóth, Proceedings of SPIE Vol. 6601
(Fluctuations and Noise 2007)
[39] M.M. Dacorogna, R. Gençay, U.A. Müller, R.B. Olsen, O.V. Pictet, An
Introduction to High-Frequency Finance, Academic Press, 2001
[40] A. Almeida, C. Goodhart, R. Payne, The Journal of Financial and Quan-
titative Analysis, 33, 383-408 (1998)
	Introduction
	Empirical analysis
	Data and methodology
	Time evolution of the characteristic time
	Decomposition of the cross-correlations
	Model calculations
	Application of the theory to the data
	Decay functions
	Fits
	Discussion

  We analyse the dependence of stock return cross-correlations on the sampling
frequency of the data known as the Epps effect: For high resolution data the
cross-correlations are significantly smaller than their asymptotic value as
observed on daily data. The former description implies that changing trading
frequency should alter the characteristic time of the phenomenon. This is not
true for the empirical data: The Epps curves do not scale with market activity.
The latter result indicates that the time scale of the phenomenon is connected
to the reaction time of market participants (this we denote as human time
scale), independent of market activity. In this paper we give a new description
of the Epps effect through the decomposition of cross-correlations. After
testing our method on a model of generated random walk price changes we justify
our analytical results by fitting the Epps curves of real world data.

Introduction
1979 Epps reported results showing that stock return correlations decrease
as the sampling frequency of data increases [1]. Since his discovery the
phenomenon has been detected in several studies of different stock markets
[2, 3, 4] and foreign exchange markets [5, 6].
∗E-mail: bence@maxwell.phy.bme.hu
http://arXiv.org/abs/0704.1099v2
Cross-correlations between the individual assets are the main factors in
classical portfolio management thus it is important to understand and give
their accurate description on different time scales. This is especially so, since
today the time scale in adjusting portfolios to events occuring may be in the
order of minutes.
Considerable effort has been devoted to uncover the phenomenon found
by Epps [7, 8, 9, 10, 11, 12]. However most of the works aim to construct a
better statistical measure for co-movements in prices in order to exclude bias
of the estimator by microstructure effects [13, 14, 15, 16, 17, 18, 19, 20], only
a few searching for the description of the microstructure dynamics.
Up to now two main factors causing the effect have been revealed: The
first one is a possible lead-lag effect between stock returns [21, 22, 23] which
can appear mainly between stocks of very different capitalisation and/or for
some functional dependencies between them. In this case the maximum of
the time-dependent cross-correlation function can be found at non zero time
lag, resulting in increasing cross-correlations as the sampling time scale gets
into the same order of magnitude as the characteristic lag. This factor can be
easily understood, morever, in a recent study [23] we showed that through
the years this effect becomes less important as the characteristic time lag
shrinks, signalising an increasing efficiency of stock markets. As the Epps
effect can also be found for the case when no lead-lag effect is present, in the
following we will focus only on other possible factors.
The second, more important factor is the asynchronicity of ticks in case
of different stocks [7, 8, 21, 24]. Empirical results [7] showed that taking into
account only the synchronous ticks reduces to a great degree the Epps effect,
i.e. measured correlations on short sampling time scale increase. Naturally
one would expect that for a given sampling frequency growing activity de-
creases the asynchronicity, leading to a weaker Epps effect. Indeed Monte
Carlo experiments showed an inverse relation between trading activity and
the correlation drop [7].
However, the analysis of empirical data showed [25] that the explanation
of the effect solely by asynchronicity is not satisfactory. After eliminating the
effect of changing asymptotic cross-correlations through the years (scaling
with the asymptotic value), the curves of cross-correlation as a function of
sampling time scale tend to collapse to one curve and surprisingly we do
not find a measurable reduction of the characteristic time of the Epps effect,
while the trading frequency grew by a factor of ∼ 5−10 in the period. These
results will be discussed further in details in Section 2.2.
The characteristic time of market phenomena can usually be split up into
three kinds of market time scales: the frequency of trading on the market
(which we will denote as activity), market periodicities and the reaction time
of traders to news, events. In Ref. [25] we showed that the characteristic time
of the Epps effect does not scale with changing market activity (this we will
discuss in Section 2.2), which points out that the characteristic time of the
Epps effect can not be determined solely by the market activity causing asyn-
chronicity. Market periodicities in high frequency data are the different types
of patterns, which can be found in intraday data, as well as on broader time
scales (see e.g. Refs. [26] and [27]). Market periodicities and intraday struc-
ture do not have a role in our results since we are averaging them out. Hence
we believe that the characteristic time of the Epps effect is the outcome of a
human time scale present on the market: The time that market participants
need to react to certain pieces of news. There are several studies in the litera-
ture about reaction time. The issue is connected both to behavioural finance
questions and to market efficiency. 1970, Fama defined an efficient market as
one in which prices fully reflect all available information [28]. This response
to information in practice can not happen instantaneously. There are several
results reporting that prices incorporate news within five to fifteen minutes
after news announcements [29, 30, 31, 32, 33]. More recent studies showed
similar results on the time that traders needed to react to news [34, 35, 36].
Supposing that the Epps effect is possibly the outcome of a human time
scale present on the market motivated us to separate the terms in the cross-
correlation function, in order to study their behaviour one by one. In this
paper we suggest an analytic decomposition of the cross-correlation function
of asynchronous events using time lagged correlations. As a second step we
demonstrate the efficiency of the formalism on the example of generated data.
Finally we describe and fit the empirically observed dependence of the cross-
correlations. We find that the origin of the independence of the characteristic
time of the Epps effect on the trading frequency is the presence of a human
time scale in the time lagged autocorrelation functions. Ref. [12] already
called the attention to the importance of lagged cross-influences of stock
returns in explaining the Epps-effect. Using a somewhat different formalism,
here we investigate thoroughly this relationship.
The paper is built up as follows: in Section 2 we present the data used
and discuss the problems of the former descriptions. Section 3 the decompo-
sition of the cross-correlation coefficient, Section 4 shows a simulation model
demonstrating the idea. In Section 5 we present the assumptions concerning
real data and show fits and statistics for the Epps curves. We finish the paper
with a discussion.
2 Empirical analysis
2.1 Data and methodology
In our analysis we used the Trade and Quote (TAQ) Database of the New
York Stock Exchange (NYSE) for the period of 4.1.1993 to 31.12.2003, con-
taining tick-by-tick data. The data used was adjusted for dividends and
splits.
We computed the logarithmic returns of stock prices:
rA∆t(t) = ln
pA(t)
pA(t −∆t)
, (1)
where pA(t) stands for the price of stock A at time t. The prices were deter-
mined using previous tick estimator on the high frequency data, i.e. prices
are defined constant between two consecutive trades. The time dependent
cross-correlation function C
∆t (τ) of stocks A and B is defined by
∆t (τ) =
rA∆t(t)r
∆t(t + τ)
rA∆t(t)
rB∆t(t + τ)
. (2)
The notion 〈· · · 〉 stands for the time average over the considered period:
〈r∆t(t)〉=
T −∆t
r∆t(i), (3)
where time is measured in seconds and T is the time span of the data.
The standard deviation σ of the returns reads as:
〈r∆t(t)2〉−〈r∆t(t)〉
, (4)
both for A and B in (2). We computed correlations for each day separately
and averaged over the set of days, this way avoiding large overnight returns
and trades out of the market opening hours. For pairs of stocks with a
lead–lag effect the function C
∆t has a peak at non-zero τ. The equal-time
cross-correlation coefficient is naturally: ρA/B∆t ≡C
∆t (τ = 0). In our notations
the Epps effect means the decrease of ρ∆t as ∆t decreases (see Figure 1). Since
the prices are defined as being constant between two consecutive trades, the
∆t time scale of the sampling can be chosen arbitrarily.
As stated above, we do not want to discuss the Epps effect originated
from the lead-lag effect in the correlations. Thus we consider only pairs of
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
∆t [sec]
Figure 1: The cross-correlation coefficient as a function of sampling time
scale for the period 1993–2003 for the Coca-Cola Pepsi pair. Several hours
are needed for the correlation to reach its asymptotic value.
stocks where the latter effect is negligible, i.e., for which the price changes
are highly correlated with the peak position of C
∆t of Equation (2) being at
τ ≈ 0. The results shown in this paper can be generalised for all stock pairs
(though in case of an empirical study one can never fit all data). To illus-
trate our results we will present results for some stock pairs and in Section
5 we will show statisitcs for a broader set of data. The stocks mentioned
in the paper are the following: Avon Products, Inc. (AVP), Caterpillar Inc.
(CAT), Colgate-Palmolive Company (CL), E.I. du Pont de Nemours & Com-
pany (DD), Deere & Company (DE), The Walt Disney Company (DIS), The
Dow Chemical Company (DOW), General Electric Co. (GE), International
Business Machines Corp. (IBM), Johnson & Johnson (JNJ), The Coca-Cola
Company (KO), 3M Company (MMM), Motorola Inc. (MOT), Merck &
Co., Inc. (MRK), PepsiCo, Inc. (PEP), Pfizer Inc. (PFE), The Procter &
Gamble Company (PG), Sprint Nextel Corp. (S), Vodafone Group (VOD),
Wal-Mart Stores Inc. (WMT).
2.2 Time evolution of the characteristic time
Previous studies claimed the asynchronicity of ticks for different stocks as the
main cause of the Epps effect [7, 8]. It is natural to assume that, for a given
sampling frequency, increasing trading activity should enhance synchronicity,
leading to a weaker Epps effect.
To study the trading frequency dependence of the cross-correlation drop,
we computed the Epps curve separately for different years. In Figure 2 the
cross-correlation coefficients can be seen as a function of the sampling time
scale for the years 1993, 1997, 2000 and 2003 for three example stock pairs.
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
Figure 2: The Epps curves for the CAT/DE (top), KO/PEP (middle) and
MRK/JNJ (bottom) pairs for the years 1993, 1997, 2000 and 2003. The
asymptotic value of the cross-correlations varies in time.
It is known that cross-correlation coefficients are not constant through
the years. The asymptotic values of cross-correlations (long sampling time
scale) depend on the economical situation, the state of the economic sectors
that the pairs of stocks belong to, and several other factors. We need to
take this into account and try to extract the effect of changing asymptotic
cross-correlations from the Epps phenomenon. In order to get comparable
curves, we scaled the cross-correlation curves with their asymptotic value:
The latter was defined as the mean of the cross-correlation coefficients for
the sampling time scales ∆t = 6000 seconds through ∆t = 9000 seconds, and
the cross-correlations were divided by this value. Figure 3 shows the scaled
curves for the same years and pairs as Figure 2.
The frequency of trades changed considerably in the last two decades:
Trading activity has grown almost monotonically, as it can be seen in Figure
4. This would infer the diminution of the Epps effect and a much weaker
decrease of the correlations as sampling frequency is increased. However,
after scaling with the asymptotic cross-correlation value, the curves give a
reasonable data collapse and no systematic trend can be seen. Surprisingly,
as it can be seen in Table 1, a rise of the trading frequency by a factor of
∼ 5−10 does not lead to a measurable reduction of the characteristic time
of the Epps effect (where we define the characteristic time the time scale for
which the cross-correlation reaches the 1− e−1 rate of its asymptotic value).
These results show that explaining the Epps effect merely as a result of
the asynchronicity of ticks is not satisfactory. It is also important to mention
that not even the changing tick sizes for the stocks (likely to change the
arrival rate of price changes) alter the characteristic time of the effect.
Table 1: The characteristic time of the Epps effect for the years 1993, 1997,
2000 and 2003 measured in seconds for the stocks pairs: CAT/DE, KO/PEP
and MRK/JNJ (characteristic time was defined as the time scale for which
the cross-correlation value reaches the 1− e−1 rate of its asymptotic value).
No clear trends can be seen in the charactersitic time while the activity is
growing rapidly.
CAT/DE KO/PEP MRK/JNJ
1993 940 920 800
1997 620 760 420
2000 1320 1040 880
2003 700 800 1060
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
0 2000 4000 6000 8000
∆t [sec]
Figure 3: The Epps curves scaled with the asymptotic cross-correlation values
for the CAT/DE (top), KO/PEP (middle) and MRK/JNJ (bottom) pairs
for the years 1993, 1997, 2000 and 2003. The scaled curves give a reasonable
data collapse in spite of the considerably changing trading frequency, showing
that the characteristic time of the Epps effect does not change with growing
activity.
1994 1996 1998 2000 2002
yearsa
Figure 4: The average intertrade time for the years 1993 to 2003 for some
example stocks (CAT, DE, KO, PEP, MRK, JNJ). The activity was growing
almost monotonically.
3 Decomposition of the cross-correlations
In this section we show calculations for the relation between the value of
cross-correlations on different time scales. We connect the cross-correlation
on a certain time scale (∆t) to lagged autocorrelations and cross-correlations
on smaller time scales (∆t0).
Returns in a certain time window ∆t are mere sums of returns in smaller,
non-overlapping windows ∆t0, where ∆t is a multiple of ∆t0:
r∆t(t) =
∆t/∆t0
r∆t0(t −∆t + s∆t0). (5)
Using this relationship the time average of the product of returns on the
large time scale (∆t) can be written in terms of the averages on the short
time scale (∆t0) in the following way:
rA∆t(t)r
∆t(t)
T −∆t
rA∆t(i)r
∆t(i) =
T −∆t
∆t/∆t0
rA∆t0(i−∆t + s∆t0)
∆t/∆t0
rB∆t0(i−∆t +q∆t0)
∆t/∆t0
∆t/∆t0
rA∆t0(i−∆t + s∆t0)r
∆t0(i−∆t +q∆t0)
. (6)
We can see that on the right side of Equation (6) the lagged time average
of return products appear on the short time scale, ∆t0, i.e., the non-trivial
part of the lagged cross-correlations. Naturally in the case of A = B, we get
the relation for
r∆t(t)
In order to apply Equation (6), we need to have information about the
lagged autocorrelation and cross-correlation functions. Writing out the sum
in Eq. (6) we get:
rA∆t(t)r
∆t(t)
x=− ∆t∆t0
rA∆t0(t)r
∆t0(t + x∆t0)
, (7)
and similarly
rA∆t(t)
x=− ∆t∆t0
rA∆t0(t)r
∆t0(t + x∆t0)
rB∆t(t)
x=− ∆t∆t0
rB∆t0(t)r
∆t0(t + x∆t0)
. (8)
Since the mean of returns is 1-2 orders of magnitude smaller than the
second moments in the correlation function, we can omit the expressions
rA∆t(t)
rB∆t(t + τ)
and 〈r∆t(t)〉
in Equation (2). As these terms are of second
order, this can even be done in case of slight price trends. Hence Equation
(2) becomes:
ρA/B∆t =
rA∆t(t)r
∆t(t)
rA∆t(t)
rB∆t(t)
. (9)
For simplicity, we introduce decay functions to describe lagged correla-
tions:
(x∆t0) =
rA∆t0(t)r
∆t0(t + x∆t0)
rA∆t0(t)r
〉 , (10)
defined for both positive and negative x values, and similarly f
(x∆t0) and
(x∆t0). Thus the correlation can be written in the following form:
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)r
∆t0(t)
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)
)−1/2
x=− ∆t∆t0
(x∆t0)
rB∆t0(t)
)−1/2
. (11)
Hence
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
x=− ∆t∆t0
(x∆t0)
)−1/2
x=− ∆t∆t0
(x∆t0)
)−1/2
ρA/B∆t0 . (12)
This way we obtained an expression of the cross-correlation coefficient
for any sampling time scale, ∆t, by knowing the coefficient on a shorter
sampling time scale, ∆t0, and the decay of lagged autocorrelations and cross-
correlations on the same shorter sampling time scale (given that ∆t is multiple
of ∆t0). Our method is to measure the correlations and fit their decay func-
tions on a certain short time scale and compute the Epps curve using the
above formula.
4 Model calculations
In this section we demonstrate the decomposition process on computer gen-
erated time series which should mimic two correlated return series. We will
demonstrate the Epps effect and see how the decomposition works for these
controlled cases. Our aim is to show that in case of generated “price” series,
the decomposition process leads to a very good description of the time scale
dependence of the cross-correlation coefficients. More discussion and details
on the analytic treatment of the model can be found in Ref. [38].
To mimic some properties of financial data, we simulate two correlated
but asynchronous price time series. As a first step we generate a core random
walk with unit steps up or down in each second with equal possibility (W (t)).
Second we sample the random walk, W (t), twice independently with waiting
times drawn from an exponential distribution. This way we obtain two time
series (pA(t) and pB(t)), which are correlated since they are sampled from the
same core random walk, but the steps in the two walks are asynchronous.
The core random walk is:
W (t) = W (t −1)+ ε(t),
where ε(t) is ±1 with equal probability (and W (0) is set high in order to
avoid negative values). The two price time series are determined by
pA(ti) =
W (ti) if ti = ∑ik=1 Xk
p(ti −1) otherwise
pB(ti) =
W (ti) if ti = ∑ik=1Yk
p(ti −1) otherwise
. (14)
from the core random walk, where, i = 1,2, · · · , and Xk and Yk are drawn from
an exponential distribution:
P(y) =
λe−λy if y ≥ 0
0 y < 0
with parameter λ = 1/60. A snapshot as an example of the generated time
series with exponentially distributed waiting times can be seen on Figure 5.
As a next step we create the logarithmic return time series (rA∆t(t) and
rB∆t(t)) of p
A(t) and pB(t) as defined by Equation (1). In case of the random
walk model of price changes we know that
rA∆t(t)
rB∆t(t)
= 0 without
having to make any assumptions. Of course having a random walk model,
the autocorrelation function of the steps is zero for all non-zero time lags:
(x∆t0) = f
(x∆t0) = δx,0, (16)
0 100 200 300 400
10000
10010
10020
pA(t)
pB(t)
Figure 5: A snapshot of the model with exponentially distributed waiting
times. The original random walk is shown with lines (black), the two sampled
series with dots and lines (red) and triangles and lines (blue).
rA∆t(t)
rA∆t0(t)
rB∆t(t)
rB∆t0(t)
. (17)
Hence the cross-correlation can be written in the following form:
ρA/B∆t =
x=− ∆t∆t0
(x∆t0)
rA∆t0(t)r
∆t0(t)
rA∆t0(t)
rB∆t0(t)
)−1/2
x=− ∆t∆t0
(x∆t0)
ρA/B∆t0 . (18)
In the model case we set the smallest time scale ∆t0 = 1 time step. It
can be shown [38] that in the case of λ ≪ 1 (small density of ticks) the exact
analytical expression for the cross-correlations is identical to (18) with an
exponential decay function:
(x∆t0) = e−λ∆t0|x|, (19)
where λ is the parameter of the original exponential distribution used for
sampling. Further results and exact computations of the cross-correlations
for the model can be found in Ref. [38].
Figure 6 shows the computed cross-correlations of the generated time
series on several sampling time scales and the computed cross-correlations
using Equation (18) and the exponential decay function (19). The two curves
are in very good agreement showing that the decomposition procedure is able
to well capture the Epps effect for generated time series.
0 5000
measured correlations
analytically computed correlations
∆t [simulation steps]
Figure 6: The measured and the computed cross-correlation coefficients using
exponential decay function as a function of sampling time scale for the simu-
lated time series with exponentially distributed waiting times. The analytic
fit is in very good agreement with the Epps curve.
5 Application of the theory to the data
In this section we discuss the properties of the decay functions in case of real
world data, and inserting them into Equation (12) we derive analytical fits
for the measured Epps curves.
5.1 Decay functions
As discussed, we measure the equal-time cross-correlations and the decay of
cross and autocorrelations on a certain short sampling time scale and from
these we obtain the value of equal-time cross-correlations on larger sampling
time scales. To do this, in case of the toy model, we had the possibility
of using the smallest time scale available in the generated data as ∆t0, i.e.,
the resolution being one simulation step. When studying data from real
world markets, one has to make restrictions. As being the highest resolution
commonly used in financial analysis, it would be plausible to choose windows
of one second as ∆t0. However on this time scale one is only able to measure
noise, no valid correlations and decay functions can be found. Thus we had to
use less dense data for the smallest sampling time scale: in the results shown
below we set ∆t0 = 120 seconds. Using this resolution we get an acceptable
signal-to-noise ratio and we hope not to lose too much information compared
to higher frequencies.
To avoid new parameters in the model we use the raw decay functions in
the formula (12), without fitting them. Since it is an empirical approach to
determine the decay functions for real data, we have to distinguish the signal
from the noise in the decay functions. Concerning the sensitivity from the
input (decay function) we observed that the results are quite robust against
little changes in the input functions, however the noise in the tail can cause
significant deviations. According to this we take into account the decay
functions for correlations only for short time lags. For the decay of the cross-
correlations we take into account the function only up to the time lag where
the decaying signal reaches zero for the first time, for larger lags we assume
it to be zero. For the decay of autocorrelations we take into the account the
function only up to the time lag where after the negative overshoot of the
beginning it decays to zero from below for the first time, for larger lags we
assume it to be zero.
Figure 7 shows an example of the decay functions in case of the stock
pair KO/PEP (for other pairs the decay functions are very much similar).
The plot shows the decay functions up to the time lags of 1000 seconds, with
a vertical line showing how long we take the empirical decays into account.
We can see that the time lag for which the decay functions disappear is in
the order of a few minutes. In fact in case of all stock pairs studied we found
the decay disappearing after 5–15 minutes.
5.2 Fits
Inserting the empirical decay of lagged autocorrelations and cross-correlations
on the short time scale into the formula of Equation (12), we compare the
computed and the measured Epps curves. Figures 8, 9 and 10 show these
plots for a few example stock pairs.
One can see, that the fits are able to describe the change of cross-correlation
0 200 400 600 800 1000
x∆t0 [sec]
0 200 400 600 800 1000
x∆t0 [sec]
-1000 -750 -500 -250 0 250 500 750 1000
x∆t0 [sec]
Figure 7: Top: The decay of lagged autocorrelations for KO. Middle: The
decay of lagged autocorrelations for PEP. Bottom: The decay of lagged cross-
correlations for KO/PEP pair. A vertical lines show the threshold up to which
we take the decays into account, for larger lags we assume them to be zero.
Sampling time scale is ∆t0 = 120 seconds on all three plots.
with increasing sampling time scale. Note, that as it has been shown in
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 8: The measured and the analytically computed cross-correlation co-
efficients as a function of sampling time scale for the pairs CAT/DE and
KO/PEP. Note that using only the autocorrelations and cross-correlations
measured on the smallest time scale (∆t0 = 120 seconds) we are able to give
reasonable fits to the cross-correlations on all time scales. Details on the
goodness parameter of the measured and computed correlations can be found
in Table 2.
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 9: The measured and the analytically computed cross-correlation co-
efficients as a function of sampling time scale for the pairs WMT/S and
GE/MOT. Details on the goodness parameter of the measured and com-
puted correlations can be found in Table 2.
Section 3, in the analytical formula only the autocorrelations and cross-
correlations on the smallest time scale (∆t0) and the decay functions are
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
0 2000 4000 6000 8000
measured correlations
analytically computed correlations
∆t [sec]
Figure 10: The measured and the analytically computed cross-correlation
coefficients as a function of sampling time scale for the pairs MRK/JNJ and
GE /IBM. Details on the goodness parameter of the measured and computed
correlations can be found in Table 2.
taken into account as input to compute the cross-correlations on all other
time scales, no additional parameters are used.
To show a broader set of results, we introduce a goodness parameter for
the agreement between the measured and the analytically determined Epps
curves. We define the goodness parameter as the absolute error between the
measured and the analytically computed points:
g(∆t) = 100
|ρmeasured∆t −ρ
computed
ρmeasured∆t
. (20)
Table 2 shows the maximum, the mean and the median of the goodness
parameters for a broader set of stocks. The results show that the absolute
mean error is very low, with a maximum around 7 percents and both a
mean and a median around 2 percents. It is important to mention, that
the maximal error is usually found for high frequency scales, for longer time
scales and especially for the asymptotic correlation value the aggrement is
very good.
These results show that the growing cross-correlations with decreasing
sampling frequency are due to finite time decay of the lagged autocorrelations
and cross-correlations in the high frequency sampled data.
The finite decay of the cross-correlations on the short time scale (∆t0)
is not caused by difference in the capitalisation of the two stocks or func-
tional dependencies between them. Instead, it is an artifact of the market
Table 2: The maximum, the mean and the median of the goodness parameters
a broader set of stocks. The results show that the absolute mean error is low.
Note that the maximal absolute error in general occurs for high frequency
scales.
stock pair max [%] mean [%] median [%]
CAT/DE 4.81 1.26 0.94
KO/PEP 10.67 2.46 1.23
WMT/S 7.66 2.32 1.74
GE/MOT 6.43 3.29 3.44
MRK/JNJ 5.26 1.51 0.95
GE/IBM 3.90 1.57 1.12
PG/CL 6.05 1.81 1.24
MRK/PFE 4.76 1.42 1.32
AVP/CL 10.37 7.75 9.53
DD/DOW 8.84 2.05 1.49
DD/MMM 5.76 2.17 1.93
MOT/VOD 9.73 2.57 1.82
DIS/GE 5.78 1.54 0.97
average 6.92 2.4 2.1
microstructure. Reaction to a certain piece of news is usually spread out on
an interval of a few minutes for the stocks [29, 30, 31, 32, 33, 34, 35, 36, 39, 40]
due to human trading nature, thus not scaling with activity, with ticks being
distributed more or less randomly. This means that correlated returns are
spread out for this interval (asynchronously), causing non zero lagged cross-
correlations on the short time scale and thus the Epps effect. This way, as
stated by Ref. [7], the asynchronicity is indeed important in describing the
Epps effect but only in promoting the lagged correlations. (Even in case of
completely synchronous, but randomly spread ticks we could have the finite
decay of lagged correlations on short time scale, and hence the Epps effect.)
6 Discussion
In our study we examined the causes of the Epps effect, the dependence
of stock return cross-correlations on sampling time scale. We showed that
explaining the effect solely through asynchronicity of price ticks is not sat-
isfactory. When scaling the Epps curves with their asymptotic value for
different years, we get a reasonable data collapse and a growing activity of
the order ∼ 5−10 does not affect the characteristic time of the Epps effect.
The main point of our calculations is that we connected the cross-correlations
on longer time scales to the lagged autocorrelations and cross-correlations on
any shorter time scale. We demonstrated the idea of these calculations on a
random walk asynchronous model of prices, getting a very good agreement
with the cross-correlation curves.
Assuming the time average of stock returns to be zero we were able to
decompose the expression for the cross-correlation coefficient deriving an an-
alytical formula of the cross-correlations on any time scale, given the decay
of the autocorrelations and cross-correlations on a certain short time scale.
With this analytical formula we were able to give fits to the Epps curves
of real stock pairs getting acceptable results. The fits show that the Epps
effect is caused by the finite time decay of the lagged correlations in the high
frequency sampled data. The reason for the characteristic time not chang-
ing with growing activity is a human time scale present in the phenomenon,
which does not scale with the changing inter-tick time. The finite decay of
lagged correlations on the short time scale is due to market microstructure
properties: different actors on the market have different time horizons of in-
terest resulting in the reactions to certain pieces of news being spread out
for a time interval of a few minutes. The correlated returns ranging over
this interval cause the finite time decay of lagged correlations on the short
time scale resulting in the Epps effect. Our results do not contradict to the
earlier observations on the importance of asynchronicity in the Epps-effect,
however, its role has been put into a new perspective.
Acknowledgments
Support by OTKA T049238 is acknowledged.
References
[1] T.W. Epps, Journal of the American Statistical Association 74, 291-298
(1979)
[2] G. Bonanno, F. Lillo, R.N. Mantegna, Quantitative Finance 1, 1-9
(2001)
[3] A. Zebedee, A closer look at co-movements among stock returns, San
Diego State University, working paper (2001)
[4] M. Tumminello, T. Di Matteo, T. Aste, R.N. Mantegna, Eur. Phys. J.
B. 55, 209-217 (2007)
[5] M. Lundin, M. Dacorogna, U. A. Müller, Correlation of high-frequency
financial time series. In P. Lequeux (Ed.), Financial Markets Tick by
Tick. Wiley & Sons.
[6] J. Muthuswamy, S. Sarkar, A. Low, E. Terry, Journal of Futures Markets
21(2), 127-144 (2001)
[7] R. Renò, International Journal of Theoretical and Applied Finance 6(1),
87-102 (2003)
[8] O. V. Precup, G. Iori, Physica A 344, 252-256 (2004)
[9] O. V. Precup, G. Iori, European Journal of Finance (2006)
[10] J. Kwapień, S. Drożdż, J. Speth, Physica A 337, 231-242 (2004)
[11] L. Zhang, Estimating Covariation: Epps Effect, Microstructure Noise
working paper (2006)
[12] M. Potters, J.-P. Bouchaud, L. Laloux, Acta Physica Polonica B, 36, 9,
(2005)
[13] N. Shephard, O.E. Barndorff-Nielsen, University of
Oxford, Department of Economics, Economics Se-
ries Working Papers No. 240. (2005) Available at:
http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf
[14] F. Corsi, F. Audrino, University of St. Gallen, Department of Eco-
nomics, Discussion Paper No. 2007-02 (2007) Available at SSRN:
http://ssrn.com/abstract=957997
[15] T. Hayashi, N. Yoshida, Bernoulli 11(2), 359-379 (2005)
[16] M. Martens, Working Paper, University of Rotterdam (2003)
[17] T.G. Andersen, T. Bollerslev, P.F. Christoffersen, F.X. Diebold, in G.
Elliot, C.W.J. Granger, and Allan Timmermann (eds.), Handbook of
Economic Forecasting, Amsterdam: North-Holland, 778-878 (2006)
[18] T. Andersen, T. Bollerslev, International Economic Review, 39, 885-905
(1998)
[19] F. de Jong, T. Nijman, Journal of Empirical Finance 4, 259-277 (1997)
[20] C. Ball, W. Torous, Journal of Empirical Finance 7, 373-388 (2000)
http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf
http://ssrn.com/abstract=957997
[21] A. Lo, A. C. MacKinlay, Rev. Finance Stud 3, 175-205 (1990)
[22] L. Kullmann, J. Kertész, K. Kaski, Phys. Rev. E 66, 026125 (2002)
[23] B. Tóth, J. Kertész, Physica A 360 505-515 (2006)
[24] A. Lo, A. C. MacKinlay, Journal of Econometrics 45, 181-211 (1990)
[25] B. Tóth, J. Kertész, Physica A 383, 54-58 (2007)
[26] R.A. Wood, T.H. McInish, J.K. Ord, Journal of Finance 40(3), 723-739
(1985)
[27] K. Chan, Y.P. Chung, H. Johnson, The Journal of Financial and Quan-
titative Analysis, 30, 3, 329-346 (1995)
[28] E. Fama, Journal of Finance 25, 383-417 (1970)
[29] L. Dann, D. Mayers, R. Raab, Journal of Financial Economics 4, 3-22
(1977)
[30] J. Patell, M. Wolfson, Journal of Financial Economics 13, 223-252 (1984)
[31] R. Jennings, L. Starks, Journal of Accounting Research 23, 336-350
(1985)
[32] M. Barclay, R. Litzenberger, Journal of Financial Economics 21, 71-99
(1988)
[33] S. Kim, J. Lin, M. Slovin, Journal of Financial and Quantitative Analysis
32, 507-524 (1997)
[34] J. Busse, C. Green, Journal of Financial Economics 65, 415-437 (2002)
[35] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics
76, 271-292 (2005)
[36] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics
87, 249-268 (2008)
[37] P.Ch. Ivanov, A. Yuen, B. Podobnik, Y. Lee, Phys. Rev. E 69, 056107
(2004)
[38] Bence Tóth, János Kertész, Bálint Tóth, Proceedings of SPIE Vol. 6601
(Fluctuations and Noise 2007)
[39] M.M. Dacorogna, R. Gençay, U.A. Müller, R.B. Olsen, O.V. Pictet, An
Introduction to High-Frequency Finance, Academic Press, 2001
[40] A. Almeida, C. Goodhart, R. Payne, The Journal of Financial and Quan-
titative Analysis, 33, 383-408 (1998)
	Introduction
	Empirical analysis
	Data and methodology
	Time evolution of the characteristic time
	Decomposition of the cross-correlations
	Model calculations
	Application of the theory to the data
	Decay functions
	Fits
	Discussion
