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

Joanna Bruzda† 
Nicolaus Copernicus University in Toruń 

European Equity Market Integration and Optimal  
Investment Horizons – Evidence from Wavelet Analysis 

A b s t r a c t. In the paper the process of equity market integration in Europe is examined from 
the wavelet perspective. The method applied is the Continuous Discrete Wavelet Transform that 
enables to perform global and local wavelet variance and correlation decompositions. In particu-
lar, questions about changes of the investment risk and the possibility of international portfolio 
diversification under different investment horizons are addressed. The study documents both 
convergence of the Central and Eastern European equity markets as well as their segmentation on 
the European market. The latter enables reduction of portfolio returns variability by an interna-
tional portfolio diversification, especially for long investment horizons. 

K e y w o r d s: equity market integration, time-scale analysis, wavelet variance, wavelet correla-
tions  

1. Introduction 
 One consequence of financial globalization are comovements of prices on 
different stock markets. Investigation of these processes is important due to 
both investors allocation decisions and policy-makers actions. From the point of 
view of investors the ongoing integration of capital markets increases the im-
portance of a sectoral portfolio diversification at the expense of an international 
diversification. From a global perspective integration of financial markets is 
fundamentally related to economic growth via improvement of allocative effi-
ciency, risk sharing and reduction of macroeconomic variability (see Kim et al., 
2005). Convergence of stock markets, through the income channel, influences 
also the effectiveness of monetary policy and – as such – should be of consider-
able interest to policy-makers. 

                                                 
† The author acknowledges the financial support from the Polish Ministry of Science and 

Higher Education under the grant no. N N111 285135. 



Joanna Bruzda 16

 Studies by Longin and Solnik (1995) point out increasing integration of 
major world stock exchanges over the period 1960–1990 and find also an addi-
tional rise of correlation in the time of high volatility. Also more recent studies 
document convergence of stock markets, although the process is non-uniform 
both in time and market segments (Kim et al., 2005), asymmetric, i.e. negative 
shocks are more strongly transmitted via borders (Fratzscher, 2002) and leaves 
place for partial segmentation of certain stock markets (Bessler, Yang, 2003). 
Behind capital market integration stands real and nominal macroeconomic con-
vergence, and therein reduction of currency risk and convergence of monetary 
policy with respect to interest rates and inflation (Fratzscher, 2002; Phengpis et 
al., 2004). Integration of major European equity markets and the rise in their 
world significance first of all results from formation of the common currency 
area (Fratzscher, 2002; Kim et al., 2005; Hardouvelis et al., 2006). 
 Only a couple of papers undertakes the task of examining the convergence 
of  Central and Eastern European (CEE) capital markets and their integration 
with the world capital market (Scheicher, 2001; Voronkova, 2004; Chelley-
Steeley, 2005; Gilmore et al., 2008; Harrison, Moore, 2009). Chelley-Steeley 
(2005) documents that for the period 1994–1999 correlations of stock index 
returns between CEE and developed European markets were usually below 0.3, 
while at the same time correlations between indices on developed European 
markets are often above 0.5. The majority of empirical investigations points out 
a certain kind of segmentation of Central-Eastern European markets and the 
lack of uniformly increasing integration with the rest of Europe, although CEE 
markets remain under a significant influence of the world capital market. 
 Recent theoretical studies underline the importance of agents heterogeneity 
in asset pricing models (see, e.g., the fractal market hypothesis of Peters, 1994). 
Also survey studies confirm that investors acting on financial markets have 
different investment strategies and different investment horizons – from one day 
to several years (see the discussion in Demary, 2009). Agents with long invest-
ment horizons concentrate on economic fundamentals driving trends, while 
speculators want to beat the market in the short run and often resort to methods 
of technical analysis. Heterogeneity of agents in a natural way gives rise to ana-
lyze stock prices according to different time scales (investment horizons). 
A method which enables investigation of stochastic processes decomposed ac-
cording to scales is time-scale (wavelet) analysis. The aim of this study is to 
apply wavelet analysis to investigate the convergence process between Central 
European capital markets and developed European markets. As by design the 
subject of this investigation are not the directly observed results of the globali-
zation process, and therein the so-called contagion effects on financial markets, 
but rather their time-scale consequences for stock investors, stock indices have 
been denominated in one currency (Polish Zloty) in order to compensate for 
foreign currency exposure.  



European Equity Market Integration and Optimal Investment Horizons… 17

 The empirical investigation spans indices from three developed European 
stock markets – Frankfurt (DAX), London (FTSE1000) and Paris (CAC40) – as 
well as six emerging markets from the Central and Eastern Europe – Prague 
(PX50), Bratislava (SAX), Budapest (BUX), Sofia (SOFIX), Bucharest (BET) 
and Warsaw (WIG). The study concentrates on the following questions: 
− Do European stock markets converge (emerging markets alone as well as 

both developing and developed European markets), and if so, is the process  
uniform in time and across investment horizons (scales)? 

− For which investment horizons is international portfolio diversification most 
efficient?  

− Is the process of the fast development of CEE stock markets accompanied 
by the rise of investment risk?  

− Is there an EU’s effect – an increase in stock exchange comovements caused 
by policy coordination, the rise in trade and investment and the opening of 
labor markets that foster the process of real  macroeconomic convergence?  

 The rest of the paper is organized as follows. Section 2 describes briefly the 
tools of wavelet analysis that are used in the study, Section 3 presents main 
empirical results, while the last section shortly concludes.  

2. Methodology 
 Wavelet analysis consists in decomposing a signal into shifted and scaled 
versions of a basis function, )(⋅ψ , called the mother wavelet. The decomposi-
tion can be continuous or discrete depending on the kind of the wavelet trans-
form applied. The Discrete Wavelet Transform (DWT) provides a parsimonious  
representation of the data and is particularly useful in noise reduction and in-
formation compression, while the Continuous Wavelet Transform (CWT) is 
more helpful in recognizing local features of signals, especially those that are 
defined over the entire real axis, although this results in excessive redundancy 
of information.  
 The Continuous Wavelet Transform of a function )(⋅f  is defined as fol-
lows: 

∫
∞

∞−
= dxxfxtW t )()(),( ,λψλ , (1)  

where: 

 ⎟
⎠
⎞

⎜
⎝
⎛ −

=
λ

ψ
λ

ψ λ
txxt

1)(, , 0>λ . (2) 

By applying the CWT we obtain a set of wavelet coefficients, )(, xtλψ , depend-

ing on scale λ  and time t. Let us consider a vector of length JN 2=  in the 



Joanna Bruzda 18

form  ),,,( 110 ′= −Nxxx …x . For Jj ,,2,1 …=  and 12,,1,0 −= − jJt …  we de-
fine the Discrete Wavelet Transform of vector x: 

( )∑ −

=
= 1

0 ,,
N
n tjntj nxW ψ , (3) 

where )(, ⋅tjψ  are shifted and scaled versions of the mother wavelet with dyadic 
shifts and scales, i.e.:  

( )txx jj
tj −= −− 22)( 2/

, ψψ . (4) 

For a given j the coefficients tjW ,  correspond to scale 12 −= j
jλ . The DWT 

results from a critical sampling of the CWT, which means that it contains the 
minimal amount of wavelet coefficients for complete reconstruction of the sig-
nal.  
 Here in the paper, following, e.g., Percival and Walden (2002), we concen-
trate exclusively on the DWT considering it as a more natural way of analyzing 
discrete time series. Among wavelet tools based on the discrete transform are 
the wavelet variances and the wavelet correlations (also known as the wavelet 
coherences – see Sanderson et al., 2009).1 For a stochastic process tY  the time-
dependent wavelet variance is defined as: 

)Var(
2

1)( ,
2

tj
j

jt W
λ

λσ = . (5) 

Assuming that (5) does not depend on time2, the following variance decomposi-
tion according to time scales is obtained (see Percival, Walden, pp. 296–298): 

∑ ∑
∞

=

∞

=

==
1 1

2
, )()Var(1

2
1)Var(

j j
jtj

j
t WY λσ

λ
. (6) 

The wavelet variance at level j corresponding to scale 12 −= j
jλ , )(2

jλσ , in-
forms about variation of oscillations with period lengths approximately in the 

                                                 
1 Other tools based on the DWT are: multiresolution analysis taking advantage of both the 

DWT and its inversion, wavelet cross-correlations computed via the Continuous Discrete (Max-
imal Overlap) Wavelet Transform operating on discrete scale and continuous time as well as  
methods from complex-valued wavelet analysis, especially the discrete wavelet phase angle tak-
ing advantage of the Maximal Overlap Discrete Hilbert Wavelet Transform (MODHWT) – see 
Gençay et al. (2002); Whitcher, Craigmile (2004). The measures mentioned above can also be 
applied to only a portion of wavelet coefficients, what results in local (short-time) versions of the 
tools – see, e.g., Sanderson et al. (2009).  

2 Such an assumption is fulfilled also for nonstationary processes provided that they are inte-
grated of order d and the width of the Daubechies wavelet filter, L, is sufficient to eliminate non-
stationarity (i.e. L ≥ 2d) – see Percival, Walden (2000), p. 304. In what follows we concentrate 
exclusively on Daubechies filters, although they are not the only ones that are interpretable in 
terms of generalized differences of weighted averages.  



European Equity Market Integration and Optimal Investment Horizons… 19

interval j2 – 12 +j . Similarly, the wavelet covariance and wavelet correlation are 
introduced. For stochastic processes tY1  and tY2  the wavelet covariance for scale 

jλ  is defined as: 

),Cov(
2

1)( ,,2,,1 tjtj
j

j WW
λ

λγ = . (7) 

As in the case of variance decomposition (6), wavelet covariances are obtained 
by decomposing the covariance between tY1  and tY2  according to different scales 

jλ . Next, let us define the wavelet correlation coefficient for scale jλ  via: 

)()(
)(

)(
21 jj

j
j λσλσ

λγ
λρ = . (8) 

The quantity (8) is normalized in the interval [-1, 1] and indicates the strength 
and direction of a relationship between two processes for a given resolution 
level (i.e. for a given time scale jλ ).  

 In practice, when estimating the wavelet variance, covariance and correla-
tion, instead of the DWT their modification in the form of the so-called Maxim-
al Overlap (Continuous Discrete) DWT is used. The MODWT can be thought 
of as a subsampling of the CWT at dyadic scales, but leaving all times t instead 
of only those that are multiples of powers of 2. For this reason it is called also 
the Non-decimated Wavelet Transform. This enables to eliminate certain arti-
facts produced by the DWT resulting from the lack of time-invariance, does not 
require data sets of length J2  and – what is important from the point of view of 
this study – provides more statistically efficient estimators of the wavelet va-
riance (Gençay et al., 2002, p. 135). An unbiased estimator of the wavelet va-
riance is then defined as: 

∑
−

−=

=
1

1

2
,

2 ~
~
1)(~ N

Lt
tj

j
j

j

W
N

λσ , (9) 

where tjW ,
~  are the MODWT coefficients, 1)1)(12( +−−= LL j

j  is the width 
of the wavelet filter for scale jλ  (L is the width of the basic wavelet filter at the 
first stage of the pyramid algorithm that computes the DWT) and 

1~
+−= jj LNN  is the number of wavelet coefficients not affected by the 

boundary. By using (9) we assume that the wavelet filter applied eliminates all 
deterministic components of the process under scrutiny. If the process has con-
stant, nonzero mean value, the formula (9) is modified appropriately. An ap-
proximate (1–α)% confidence interval for )(2

jλσ  is computed as follows:  



Joanna Bruzda 20

5,0

,~
2

~
)0(ˆ

)(~
2 ⎟⎟

⎟

⎠

⎞

⎜⎜
⎜

⎝

⎛
±

j

jW
j N

f
αςλσ , (10) 

where 
2
ας  is the (1–α/2) quantile of the standard normal distribution and 

)0(ˆ
,~ jW

f  is an estimator of the spectral density of scale jλ  squared wavelet 

coefficients at frequency 0. Estimates of wavelet covariances and wavelet corre-
lations are computed via the following formulas:  

tj

N

Lt
tj

j
j WW

N j

,,2

1

1
,,1

~~
~
1)(~ ∑

−

−=

=λγ , (11) 

)(~)(~
)(~

)(~
21 jj

j
j λσλσ

λγ
λρ = . (12) 

An approximate (1–α)% CI for )( jλγ  is obtained as previously with )0(ˆ
,~ jWf  

being an estimate of the cross-spectrum at frequency 0. In the case of wavelet 
correlations Gençay et al. (2002), p. 259–260, suggest an approach taking ad-
vantage of the fact that the DWT approximately decorrelates even long memory 
processes. In such a case, in order to obtain confidence limits that are placed in 
the interval ±1 the Fisher z-transformation can be applied, which under the 
Gaussian assumption leads to the following (1–α)% CI for scale jλ : 

[ ]
⎪⎭

⎪
⎬
⎫

⎪⎩

⎪
⎨
⎧

⎟
⎟

⎠

⎞

⎜
⎜

⎝

⎛

−
±−

5,0

1

3ˆ
1)(~tanhtanh

2
j

j N
αςλρ , (13) 

where jN̂  is the number of (conventional) DWT coefficients associated with 
scale jλ  that is treated here as a measure of the scale-dependent sample size. 

3. Empirical Results 
 In the empirical examination daily closing prices of nine indices denomi-
nated in Polish Zloty at the National Bank of Poland (NBP) exchange rates have 
been used. We decided on examining prices instead of returns due to the fact 
that, firstly, using wavelets we do not need to transform data to stationarity prior 
to the analysis, secondly, examining indices provides better interpretation in 
terms of synchronization of stock market cycles and, thirdly, wavelet coeffi-
cients obtained with the Haar wavelet filter can be thought of being local esti-
mates of multi-period returns and – as such – our results of wavelet variance 
and correlation analysis will have practical implications for portfolio construc-



European Equity Market Integration and Optimal Investment Horizons… 21

tion. The main advantage of using wavelets over more traditional approaches is 
their efficiency in data exploration resulting from optimal time-frequency reso-
lution and the simplicity in analyzing scale-dependent phenomena. 
 The indices examined here are from both developed European capital mar-
kets (DAX, FTSE, CAC) and CEE markets (PX, SAX, BUX, SOFIX, BET and 
WIG) and span the period 2.01.2002–30.04.2009 (1912 observations), except 
for SAX, for which the sample starts on 2.01.2003, as well as SOFIX and BET, 
where the samples begin on 3.04.2007. The reason for  the differences in data 
length is the lack of daily quotations of Slovak Koruna, Bulgarian Lev and Ro-
manian Leu to the beginnings of our samples. After linear interpolation of miss-
ing observations the quotations have been transformed into logarithms. The 
computations were performed on the entire sample and in three subsamples: 
01.2002–04.2004 (608 observations), 05.2004–03.2007 (761 observations), 
04.2007–04.2009 (543 observations). The first subsample spans the period prior 
to the enlargement of the European Union in 2004, the second comprises first 
three years after the enlargement, while the last covers the most recent period 
that includes also the data on the latest financial crisis and is the only one span-
ning all nine indices.  
 The results of group unit root tests on an unbalanced panel comprising all 
indices and all observations performed with lag length chosen on the base of the 
Schwartz criterion, spectrum estimation at frequency 0 with the Bartlett kernel, 
Newey-West bandwidth selection, individual intercepts for the levels and no 
intercept for first differences are given in Table 1. The results (except for the t* 
statistic) allow to treat our multivariate process as integrated of order 1. 

Table 1. Results of group unit root tests 
Test Statistic p-value Statistic p-value 

                                                                              Level                                               First difference 
                                      H0: common unit root 

 Levin, Lin, Chu t* -3.029 0.001 -92.054 0.000 
Breitung t 3.199 0.999 87.593 0.000 

                                    H0: Individual unit root 
Im, Pesaran, Shin W 0.596 0.725   

ADF-Fisher χ2 15.915 0.599 1153.20 0.000 
PP-Fisher χ2 16.155 0.582 529.606 0.000 

                                      H0: no common unit root 
Hadri Z 66.89 0.000   

 Depending on data length a 6- or 8-level Maximal Overlap Discrete Wave-
let Transforms were executed. One of the Daubechies least asymmetric wavelet 
filters – LA(8) – was applied, which is nearly linear in phase and has the 
width 8. Besides, in the case of wavelet correlations the examination was also 
performed with the Haar wavelet filter3. The decomposition levels 1–6 corres-

                                                 
3 As they confirm results obtained with the LA(8) filter, we do not present them here. 



Joanna Bruzda 22

pond to oscillations with period lengths approximately in the following inter-
vals: 2–4, 4–8 (up to one and a half weeks), 8–16 (up to 3 weeks), 16–32 (up to 
6 weeks), 32–64 (up to one quarter) and 64–128 (up to 2 quarters). When the 
whole samples were analyzed, two further decomposition levels were added that 
correspond to fluctuations with periods 128–256 (up to one year) and 256–512 
(up to 2 years), respectively. The computations ware executed in Matlab after 
modifying codes developed by B. Whitcher (the WaveCov package at 
www2.imperial.ac.uk/~bwhitche/software) as  well as D. B. Percival and A. T. 
Walden (the WMTSA toolkit at www.atmos.washington.edu/~wmtsa) and sup-
plementing them with own programs. 
  Figures 1–3 present decompositions of the wavelet variances with the help 
of LA(8) wavelet. The results point out that for the three mature markets the 
investment risk in the second period was significantly below its level in the first 
and third part of the sample. In the third period it rose for all investment hori-
zons (for the London Stock Exchange even above its level in the first subsam-
ple). In the case of the three examined emerging markets – Prague, Budapest 
and Warsaw – the situation looks differently: in the pre- and post-accession 
periods the wavelet variances do not differ at all scales, while in the last period 
they rose significantly above their previous levels, except for the longest in-
vestment horizons (above 6 weeks in the case of BUX and WIG and one quarter 
in the case of PX). This lack of a significant increase in the investment risk for 
the longest horizons can be thought of as a sign that Central and Eastern Euro-
pean capital markets were more robust to the financial crisis at the beginning of 
it than the developed markets. A more detailed examination, of which we 
present here only the comparison between DAX and WIG (see Figure 3), per-
formed with the help of the local wavelet variance4, shows, however, a quite 
similar level of volatility for medium and long horizon investments on the 
Polish market as compared to Frankfurt, with periods of a rapid increase of risk, 
except for the highest level of examination. A similar situation was observed for 
the other developing and developed markets. 
 Figures 4–10 show decompositions of the wavelet correlations. There are  
very strong dependences present for the developed capital markets at all in-
vestment horizons. The three markets can be thought of as a one investment 
possibility. Analyzing correlations between the developed and developing mar-
kets it is seen that the mature markets significantly influence the CEE stock 
indices, except for Bratislava and for longer horizons also Sofia. One important 
finding is the lack of systematic convergence of the mature and the CEE mar-

                                                 
4 The local wavelet variances were computed for windows of width 100 after aligning them 

with the original data and exclusion of all wavelet coefficients affected by the boundary (the 
circular filtering), what cut results at the beginning and the end of the period and is especially 
pronounced for higher resolution levels. The number of affected coefficients at the beginning 
(end) of the sample for the eight decomposition levels is, respectively, 3 (4), 10 (11), 24 (25), 52 
(53), 108 (109), 220 (221), 444 (445), 892 (893).   



European Equity Market Integration and Optimal Investment Horizons… 23

kets: in the majority of cases the highest correlations occur in the last period, 
what can be attributed to the contagion effects on the financial markets, while 
the lowest are usually in the middle subsample, although they do not differ sig-
nificantly from those computed for the first period. Generally, our conclusion is 
that we do not observe a uniformly increasing dependence between these two 
types of markets. This finding has been also confirmed with a more detailed 
examination with the help of the local wavelet correlations analysis – see Figure 
9 – that was executed for data windows of length 200. 

 
Figure 1.  Results of 6-level wavelet variance decomposition using the LA(8) wavelet 

filter in subsamples: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– – ), 
04.2007–04.2009 (–∗– ) 

 On the other hand, a systematic rise in the degree of dependence is found 
for three of the emerging markets: Prague, Budapest and Warsaw, what is espe-
cially pronounced for the lowest decomposition levels (see Figure 10). Besides 
there are also significant relationships between these indices and BET, while 
SAX seem to be uncorrelated with all the other indices and SOFIX is best corre-
lated with DAX and BET. 



 

 

Figure 2.  Comparison of wavelet variance in subsamples together with the 95% confi-
dence intervals using the LA(8) wavelet filter: 01.2002–04.2004 (–ο–), 
05.2004–03.2007 (– –), 04.2007–04.2009(–∗–); thick lines correspond to 
the later periods 

  



European Equity Market Integration and Optimal Investment Horizons… 25

 
Figure 3.  Local wavelet variance at 8 decomposition levels for DAX (thick solid line) 

and WIG  (dashed line) using the LA(8) wavelet filter 

 
Figure 4.  Results of wavelet correlation decompositions together with the 95% confi-

dence intervals using the LA(8) wavelet filter 

 As for international portfolio diversification very promising seem to be the 
relationships between FTSE and WIG as well as CAC and WIG, especially for 
long investment horizons. Besides, we notice insignificant wavelet correlations 
of PX, BUX and WIG with the indices from Sofia and Bratislava. For the ma-
jority of pairs of indices the wavelet correlations seem to be homogenous across 
scales, except for the longest investment horizons.  

500 1000 1500
0

1

2

3

4
x 10-4 Scale 1

500 1000 1500
0

1

2

3

4
x 10-4 Scale 2

500 1000 1500
0

2

4

6
x 10-4 Scale 4

500 1000 1500
0

0.5

1
x 10-3 Scale 8

500 1000 1500
0

0.5

1

1.5

2
x 10-3 Scale 16

500 1000 1500
0

1

2

3
x 10-3 Scale 32

500 1000 1500
0

1

2

3
x 10-3 Scale 64

500 1000 1500
0

0.005

0.01

0.015
Scale 128



 

 
Figure 5.  Results of wavelet correlation decompositions (continued) 

 
Figure 6.  Results of wavelet correlation decompositions in subsamples using the 

LA(8) wavelet filter: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– – ), 
04.2007–04.2009 (–∗– ) 



 

 
Figure 7.  Comparison of wavelet correlations in subsamples together with the 95% 

confidence intervals using the LA(8) wavelet filter: 01.2002–04.2004 (–ο–), 
05.2004–03.2007 (– –), 04.2007–04.2009(–∗–); thick lines correspond to 
the later periods 

 
Figure 8.  Comparison of wavelet correlations in subsamples (continued) 



Joanna Bruzda 28

 
Figure 9.  Local wavelet correlations at 6 decomposition levels for the DAX-WIG 

(thick solid line) and FTSE-WIG (dashed line) relationships using the LA(8) 
wavelet filter 

 
Figure 10. Local wavelet correlations at 6 decomposition levels for the BUX-WIG 

(thick solid line) and PX-WIG (dashed line) relationships using the LA(8) 
wavelet filter 

4. Conclusions 
 Of the countries investigated an approximately uniform rise in integration 
takes place for the ‘big three’ emerging CEE equity markets: the Czech Repub-
lic, Hungary and Poland. We do not observe systematic convergence among 
emerging capital markets and the three mature stock exchanges (Frankfurt, 
London and Paris), as the rise in the wavelet correlations documented for the 
most recent data investigated can be explained by contagion effects on financial 

500 1000 1500
-1

-0.5

0

0.5

1
Scale 1

500 1000 1500
-1

-0.5

0

0.5

1
Scale 2

500 1000 1500
-1

-0.5

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0.5

1
Scale 4

500 1000 1500
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-0.5

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0.5

1
Scale 8

500 1000 1500
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-0.5

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Scale 16

500 1000 1500
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-0.5

0

0.5

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Scale 32

600 800 1000 1200
-1

-0.5

0

0.5

1
Scale 64

500 1000 1500
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-0.5

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0.5

1
Scale 1

500 1000 1500
-1

-0.5

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-0.5

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-0.5

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-0.5

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600 800 1000 1200
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Scale 64



European Equity Market Integration and Optimal Investment Horizons… 29

markets. Our empirical results point out a certain kind of segmentation of CEE 
capital markets, although these markets remain under a significant influence of 
the developed stock exchanges. So, it seems that in the rather short post-
accession period the developing capital markets do not face the EU’s effect yet. 
The distribution of wavelet correlations across scales is relatively homogenous. 
Departures from homogeneity take usually place for longer horizons and have 
the form of bidirectional deviations. For certain pairs of indices we found sig-
nificantly negative wavelet correlations for the longest investment horizons 
investigated. This makes it possible substantially to reduce portfolio returns 
variability by international portfolio diversification. For shorter investments 
zero or even negative wavelet correlations have been found with the SAX and 
SOFIX indices. Generally, the fast development of Central and Eastern Euro-
pean equity markets is accompanied by a relatively moderate risk for medium-
term investments, with rapid changes in volatility. 

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Integracja giełd europejskich i optymalne horyzonty inwestycyjne           
w świetle analizy falkowej 

Z a r y s  t r e ś c i. W artykule prezentuje się wyniki badania procesu integracji giełd europejskich 
przeprowadzonego z użyciem narzędzi Ciągło-Dyskretnej Transformaty Falkowej, a dokładniej 
globalnych i lokalnych (krótkookresowych) wariancji i korelacji falkowych. W szczególności 
odpowiada się na pytania o zmiany ryzyka inwestycyjnego oraz możliwość międzynarodowej 
dywersyfikacji portfeli przy uwzględnieniu różnych horyzontów inwestycyjnych. Badanie poka-
zuje, że ma miejsce proces konwergencji giełd środkowoeuropejskich, ale rynki te jako całość 
wykazują pewną segmentację. Daje to możliwość międzynarodowej dywersyfikacji portfeli, 
przede wszystkim dla dłuższych horyzontów inwestycyjnych. 

S ł o w a  k l u c z o w e: integracja giełd, analiza czasowo-skalowa, wariancje falkowe, korelacje 
falkowe. 




