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DYNAMIC ECONOMETRIC MODELS 
Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 

Małgorzata Doman 
Poznań University of Economics 

Liquidity and Market Microstructure Noise:  
Evidence from the Pekao Data†  

A b s t r a c t. The availability of ultra-high frequency data justifies the use of a continuous-time 
approach in stock prices modeling. However, this data contain, apart from the information about 
the price process, a microstructure noise causing a bias in the realized volatility. This noise is 
connected with all the reality of trade. In the paper we separate the microstructure noise from the 
price process and determine the noise to signal ratio for the estimates of the realized volatility in 
the case of the shares of the Polish company Pekao S.A. The results are used to discover the 
optimal sampling frequency for the realized volatility calculation. Moreover, we check the lin-
kages between the noise and some liquidity measures.  

K e y w o r d s: market microstructure, volatility, realized variance, liquidity, stock market, trad-
ing volume, high frequency data.  

1. Introduction  
 Continuous-time econometric models are becoming now a standard tool for 
describing the financial market dynamics. They correspond well to the theoreti-
cal models of financial mathematics and can be quite easy estimated due to the 
availability of ultra-high frequency data. It seems natural that tick-by-tick data 
are the most useful in the context of continuous-time models. However, it is not 
the all truth. This kind of data contains apart from the useful information about 
the price process a noise which, for instance, causes a bias in the daily realized 
volatility estimates. The sources of the noise are connected with the reality of 
trade. Dealing with continuous-time models, we make many assumptions that 
are not satisfied in the real market. They concern time, price process, and mar-
ket mechanism. The departures of the observed process from these assumptions 

                                                 
† This work was financed from the Polish science budget resources in the years 2007-2010 as  

the research project NN 111 1256 33. 
The author would like to thank an anonymous referee for useful comments and suggestions. 



Małgorzata Doman 6 

are very often connected with the so-called market microstructure effects. The 
most known factors of market microstructure are liquidity, nonsynchronous 
trade, bid-ask spread, discrete-valued price, irregular time intervals between 
trades, and existence of diurnal pattern (Tsay, 2000). Usually these effects de-
pend on legal regulations, market electronic systems, and traders knowledge 
and behavior. 

Volatility is one of the most important parameter in risk management, de-
rivative pricing and portfolio allocation. Nowadays, one of the most popular and 
promising estimator of daily volatility is the daily realized variance (Andersen, 
Bollerslev, 1998; Barndorff-Nielsen, Shephard, 2002). It is calculated as a sum 
of squared intraday returns and so it depends on the chosen frequency of obser-
vations. The frequency of intraday data should be high enough to capture as 
much as possible of available information and small enough to avoid including 
a noise into the realized variance estimates. It seems rather obvious that the 
problem of separating the noise from the „true price” process is of great impor-
tance for quality of the daily volatility estimates. In this connection, the most 
significant microstructure phenomenon is liquidity. The microstructure noise is 
usually weaker for very liquid shares.  

The presented analysis applies the Aït-Sahalia and Yu (2009) approach to 
separate the microstructure noise from the price process in the case of  shares of 
the Polish company Pekao S.A.  Basing on the noise estimates, we determine 
the noise to signal ratio where the signal is the realized volatility. As a result of 
the analysis we obtain the optimal sampling frequency for the realized volatility 
calculation. Since liquidity is considered to be the crucial factor determining the 
noise level, we try to determine the dependencies between the noise to signal 
ratio and chosen liquidity measures. Moreover, we apply the signal to noise 
ratio to compare the strength of market microstructure effects observed in the 
analyzed Pekao data with that reported from more developed stock markets. 

2. Realized Volatility and Market Microstructure Noise  
 We consider a daily log-price process ))(ln()( tPtY =  where  t  is measured 
in days. Then the logarithmic returns are given by formula 

)()(),( htYtYhtr −−= , and the daily realized variance (volatility) (Andersen, 
Bollerslev, 1998; Barndorff-Nielsen, Shephard, 2002)  is defined as 

,)),1(()(
/1

1

2∑
=

+−=
h

j
t hjhtrhRV  (1) 

where h denotes time between two consecutive observations.  

The daily volatility 2
tσ  of a financial instrument is defined as the conditional 

variance of its daily return given the set of information 1−Ωt  available on day 
1−t , i.e.  



Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 7

).|))|((( 1
2

1
2

−− ΩΩ−= ttttt REREσ                       (2) 

Thus the volatility is an unobservable variable.  The realized variance (1) is 
a possible estimator of it. 

In the following discussion we assume that )(tY  is described by the follow-
ing stochastic differential equation 

).()()()( tdWtdtttdY σμ +=       (3) 
Here )(tW denotes a Brownian motion, )(tσ  is an instantaneous volatility and 

)(tμ is a drift function. In such a framework an ideal ex post measure of the 
daily volatility 2

tσ  is the integrated variance 

.)()(
1

2∫
−

=
t

t

duutIV σ       (4) 

From the quadratic variation theory it follows that  

.0  if  ,)()(
1

2 →→ ∫
−

hduuhRV
t

t
t σ           (5) 

It means that in absence of market microstructure noise the realized variance is 
a consistent estimator of the integrated variance. 

Following Aït-Sahalia and Yu (2009), we assume that the observed price 
tX  is a sum of  the “true price” tY  and the microstructure noise tε : 

,ttt YX ε+=                       (6) 

and we are interested in determining  the daily volatility 2
tσ  of  the tY  basing 

on discrete observations obtained in moments Tn =ΔΔ ,,,0 … . 
The model given by (6) is deep-rooted in the market microstructure theory. 

Many authors consider the noise tε  as a result of  bid-ask spread  (Roll, 1984; 
Huang Stoll, 1996), transaction costs (Huang, Stoll, 1996; Chan, Lakonishok, 
1997), discrete price changes (Gottlieb, Kalay, 1985).  Manganelli (2005) and 
Aït-Sahalia and Yu (2009) associate the noise with the low liquidity level. The 
framework of the presented investigation is based on  the Hasbrouck (1993) 
model according to which the standard deviation of tε  is a total measure of the 
market quality. 

In the following empirical analysis our main goal is to separate the micro-
structure noise from the fundamental price and evaluate the share of noise in 
observed values of the daily realized variance. We can use this result to deter-
mine the frequency for intraday returns allowing to minimize bias in the rea-
lized variance estimates. Moreover, we try to discover the dependencies be-



Małgorzata Doman 8 

tween liquidity and microstructure noise by modeling dependence of the later 
on a variety of liquidity measures. 

From now on we assume that the conditional mean of the return process is 
equal to 0. It means that (3) reduces to 

( ) ( ) ( ).σ=dY t t dW t         (7) 

Aït-Sahalia, Mykland and Zhang (2005) showed that in the parametric case 
this model is equivalent to that with constant σ . If 0=tε , i.e. if no microstruc-
ture noise is present, the observed log returns 

1−
−=

ii
XXri ττ  are i.i.d.

).,0( 2ΔσN  The daily realized volatility is then the maximum likelihood estima-

tor for 2σ and  

).2,0())(( 42 Δ⎯⎯ →⎯−Δ ∞→ σσ NRVT n    (8) 

In such a case the best estimates of volatility are obtained for the smallest poss-
ible Δ  (Aït-Sahalia, Yu, 2009; Aït-Sahalia, Mykland, Zhang, 2005). 

The situation changes in presence of the microstructure noise. Assume now 
that the noise tε  is i.i.d. with mean 0 and variance a. Thus the observed log-
returns process is MA(1) 

,)(
111 iii uuWWYYr

iiiiii
ηεεσ ττττττ +≡−+−=−=

−−−
   (9) 

with ),0.(i.i.d~ 2γiu , 22 2)var( ari +Δ=σ and 2)cov( ari −= . 

The above dependencies form a theoretical framework for the empirical analysis 
presented in Section 4. 

3. The Data  
 We consider the Polish bank Pekao S.A. stock returns. The period under 
scrutiny is from August 8, 2006 to February 13, 2009. The tick-by-tick data are 
provided by Stooq.pl.  

Table 1. Number of observations in the considered frequencies 
Type of observations Number of observations 

transactions 361 314 
tick by tick 325 177 
5 minute 53 520 
10 minute 27 160 

daily 629 

The analysis was performed for 10, 5, 2 and 1 minute observations and for 
the duration returns which are calculated from transaction data. The time be-



tween the
as equal t

For th
turns. Tab
considere
sented in 

Figure 1. D

4. Empi
 The s
volatility 
structure 
noise in 
noise to s

NS

Noise
what degr
structure 
Yu, 2009
friction. 

The n
crostructu
types of r

ta

Liquidity and M

e closing of t
to 0.  
he sake of pl
ble 1 contain
ed frequencie
Figure 1. 

Daily returns o

rical Resu
steps of the 
of the funda
noise tε  fo
the daily re

signal ratio (N

var(sig
var(noSR =

e to signal ra
ree the obse
theory it is

9) because in

next part of t
ure noise on 
regressions. T

0 1 1−= + +tc c x

Market Microstr

the stock ex

lace, we show
ns the inform
es. The plot 

of Pekao S.A.

lts  
presented a

amental price
or each cons
ealized volat
NSR) from th

.
)gnal

oise)            

atio is a mea
rved signal h
s often used
n some sense

the investiga
liquidity. T

The first one
,ν+ t        

ructure Noise: 

change and 

w here only 
mation about 

showing the

. Period: Augu

analysis are 
e process tX
idered day t
ility estimat
he following

asure commo
has been cor
d as a mark
e it allows to

ation deals w
o determine 

e is of the for

Evidence from 

its opening n

the results fo
the number 

e dynamics o

ust 8, 2006 to 

as follows. 
 and the var

t. To evalua
tes we calcu
g formula 

only used in 
rrupted by no
ket quality m
o evaluate th

with depende
the possible

rm  

the Pekao Data

next day is c

or 5 and 10 m
of observati

of daily retur

February 13, 

First we est
riance ta  of t
ate the partiti
ulate for eac

science to q
oise. In mark
measure (Aï
he level of t

ence of the m
e linkages w

a 9

considered 

minute re-
ions in the 
rns is pre-

 
2009 

timate the 
the micro-
ion of the 

ch day the 

(10) 

quantify to 
ket micro-
ït-Sahalia, 

the market 

market mi-
we run two 

(11) 



10

and captu
variance 

NS

allows us
quidity. T
(V), the d
during a d

The e
presented
contains m
noise and
5 minute 
result is s
2009). 

Table 2. M
 

ta  

tσ  

NSR 

 

Figure 2.  

ures the imp

ta . The seco

0 1SR = +t c c x

s to establish
The consider
daily mean tr
day (DTN). 
estimates of t
d in Figure 2
mean values
d noise to si
returns. The

similar to th

Mean and stand

The realized
(black line) r

M

act of liquid
ond one, 

1 ,ν− +t tx    

h the connec
red liquidity 
ransaction vo

the realized v
. Figure 3 sh
 and standar
ignal ratio.  
e mean level
hat observed 

dard deviation
Frequency

mean 
standard devia

mean 
standard devia

mean 
standard devia

d volatility esti
returns 

Małgorzata Dom

dity (measure

ctions betwee
measures ar

olume (DMT

volatility bas
hows the plo
rd deviations

The lowest
l of noise to 
in develope

n of noise and
y 

ation 

ation 

ation 

imates based 

man 

ed by a vari

en the noise 
re logarithm

TV), and the n

sed on 5 and
ot of  corresp
 of the daily

t values of n
signal ratio 

ed stock mar

d realized vola
5 min 

0.1123
0.0953
2.4584
1.4047
0.3381
0.2784

on 10 minute 

iable tx ) on

to signal rat
ms of the dail
number of tra

d 10 minute r
ponding nois
y volatility, v
noise are ob
is about 3/1

rkets (Aït-Sa

atility 
 1
3 0
3 0
4 2
7 1
1 0
4 0

(grey line) an

n the noise 

(12) 

tio and li-
ly volume 
ansactions 

returns are 
e. Table 2 

variance of 
btained for 
3  and this 

ahalia, Yu, 

10 min 
0.1468 
0.1408 
2.2690 
1.3096 
0.3314 
0.2977 

 
nd 5 minute 



Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 11

 
Figure 3. The market microstructure noise estimates for the realized volatility esti-

mates based on 10 minute (grey line) and 5 minute (black line) returns 

The results of analysis on the connections between the microstructure noise 
and liquidity are presented in Table 3.  

Table 3.  Parameter estimates for regressions (11) and (12) 

ta  
Explanatory 

variable 
5 min 10 min 

1c  2R  1c  2R  
Log(DNT) 0.012 

(0.008) 
0.01 0.042 

(0.011) 
0.03 

Log(MDTV) -0.005 
(0.009) 

0.004 0.004 
(0.0129) 

0.001 

Log(V) 0.006 
(0.006) 

0.002 0.0278 
(0.008) 

0.02 

NSR 
Explanatory 

variable 
5 min 10 min 

1c  2R  1c  2R  
Log(DNT) -0.095 

(0.019) 
0.1 -0.066 

(0.021) 
0.02 

Log(MVTD) -0.037 
(0.028) 

0.03 0.017 
(0.030) 

0.0005 

Log(V) -0.072 
(0.015) 

0.03 -0.037 
(0.016) 

0.01 

Surprisingly, the obtained estimates show rather weak connections between 
the both measures of the noise level and the considered liquidity measures. 
In the case of 10 minute returns there exists a positive and significant, though 
not very strong, dependence of the strength of noise and the number of transac-
tions during a day, and the transaction volume. The expectations were that these 
dependencies should be negative (the higher liquidity, the lower noise). As con-

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1 61 121 181 241 301 361 421 481 541 601



Małgorzata Doman 12

cerns the noise to signal ratio, a significant negative dependence on the daily 
number of transaction is in agreement with our early conjecture, but the results 
concerning the remaining liquidity measures are unexpected. It seems that in the 
case of  Pekao S.A. the measures based on trading volume are not good liquidi-
ty measures. Some explanation of this fact can be derived from the plots in Fig-
ures 4–5, which show a typical dynamics  of returns in days with high and low 
level of the noise. During the days with high noise to signal ratio the tick-by-
tick returns exhibit a very regular pattern caused probably by market makers 
activity. The high values of volume are presumably connected with this spu-
rious trade. On the other hand, during the days with the noise to signal ratio 
close to zero the dynamics of the returns is irregular and strong, which is cha-
racteristic for the days with high activity of uninformed traders. So, the conclu-
sion is that in the case of analyzed equities the microstructure noise is to a large 
extent connected with the market makers activity.  

 
Figure 4.  Tick-by-tick returns with the noise to signal ratio equal to 0.91 observed 

on May 13, 2008 

 
Figure 5.  Tick-by-tick returns with the noise to signal ratio equal to 0.03 observed 

on September 7, 2006 

-0,5

-0,3

-0,1

0,1

0,3

0,5

1 41 81 121 161 201 241 281 321 361 401 441 481 521 561

-0,5

-0,3

-0,1

0,1

0,3

0,5

1 41 81 121 161 201 241 281



Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 13

5. Conclusions  
 Due to the availability of ultra-high frequency data, a continuous-time ap-
proach to modeling the stock markets dynamics is still becoming more popular. 
In fact, many of fruitful research areas in financial econometrics are based on 
this methodology and use the realized variance as an estimator of true volatility. 
The daily realized variance is calculated as a sum of squared intraday returns. 
However, the estimates of volatility obtained in such a way are usually biased 
due to the presence of the market microstructure noise in the observed data. The 
market microstructure effects include all the phenomena connected with the 
reality of the trade that usually contradict the continuous-time model assump-
tions. 

In the paper we considered the quotations of the Polish stock company Pe-
kao S.A. and attempted to separate the market microstructure noise from the 
observed daily realized variance process. Our main findings are as follows. The 
best volatility estimates are obtained for 5-minute returns. The market micro-
structure noise is to a large extent connected with market makers activity. The 
analyzed liquidity measures (volume, mean volume of transaction, number 
of transactions during a day) poorly explain the market microstructure noise. 
The mean level of the noise to signal ratio in the case of the Pekao data is com-
parable to that observed in developed markets. This result seems to support the 
opinion about good quality of market regulations and procedures on the Warsaw 
Stock Exchange. 

References  
 Aït-Sahalia, Y., Yu, J. (2009), High Frequency Market Microstructure Noise Estimates and 

Liquidity Measures, Annals of Applied Statistics, 3, 422–457. 
Aït-Sahalia, Y., Mykland, P. A., Zhang, L. (2005), How Often to Sample a Continuous-Time 

Process in the Presence of Market Microstructure Noise, Review of Financial Studies 
18(2), 351–416.  

Andersen, T. G., Bollerslev, T. (1998),  Answering the Skeptics: Yes, Standard Volatility Models 
Do Provide Accurate Forecasts, International Economic Review, 39, 885–905. 

Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H. (2001), The Distribution of Realized 
Stock Return Volatility, Journal of Financial Economics, 61, 43–76. 

Barndorff-Nielsen, O. E., Shephard, N. (2002), Econometric Analysis of Realised Volatility and 
Its Use in Estimating Stochastic Volatility Models, Journal of the Royal Statistical Society, 
64, Series B,  253–280.  

Chan, L., Lakonishok, J. (1997), Institutional Equity Trading Costs: NYSE Versus Nasdaq, Jour-
nal of Finance, 52, 713–735. 

Gottlieb, G., Kalay, A. (1985), Implications of the Discreteness of Observed Stock Prices, Jour-
nal of Finance, 40, 135–153. 

Hasbrouck, J. (1993), Assessing the Quality of a Security Market: A New Approach to Transac-
tion Cost Measurement, Review of Financial Studies, 6, 191–212. 

Huang, R., Stoll, H. (1996), Dealer Versus Auction Markets: A Paired Comparison of Execution 
Costs on NASDAQ and the NYSE, Journal of Financial Economics,41 (3), 313–357. 

Manganelli, S. (2005), Duration, Volume and Volatility Impact of Trades, Journal of Financial 
Markets, 8, 377–399.  



Małgorzata Doman 14

Roll, R. (1984), A Simple Model of the Implicit Bid–Ask Spread in an Efficient Market, Journal 
of Finance, 39, 1127–1139. 

Tsay, R. S. (2002), Analysis of Financial Time Series, Wiley Series in Probability and Statistics, 
John Wiley& Sons, New York. 

Płynność a szum mikrostruktury rynku  
na przykładzie notowań spółki Pekao 

Z a r y s  t r e ś c i.  Dostępność danych giełdowych o bardzo wysokiej częstotliwości stanowi 
argument za stosowaniem do opisu dynamiki cen akcji modeli z czasem ciągłym. Jednak dane 
takie zawierają oprócz informacji na temat procesu ceny także szum mikrostruktury rynku, które-
go obecność powoduje obciążenie oszacowań zmienności. Szum ten jest związany z rzeczywi-
stymi warunkami, w jakich odbywa się handel. W pracy dokonano oszacowania szumu mikro-
struktury rynku w zmienności zrealizowanej cen akcji spółki Pekao SA oraz wyliczono stosunek 
sygnału do szumu. Wyniki badań wskazują, że optymalna częstotliwość wyliczania stóp zwrotu 
przy wyznaczaniu zmienności zrealizowanej to częstotliwość pięciominutowa, a obserwowany 
stosunek sygnału do szumu jest na poziomie zbliżonym do obserwowanego na rozwiniętych 
rynkach giełdowych. Ponadto, przeprowadzona została analiza powiązań pomiędzy wybranymi 
miarami płynności a poziomem szumu mikrostruktury rynku. 

S ł o w a  k l u c z o w e: mikrostruktura rynku, zmienność, wariancja zrealizowana, płynność, 
rynek giełdowy, wolumen obrotu, dane wysokiej częstotliwości.  

 




