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© 2013 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2013.006  Vol. 13 (2013) 107−125 

Submitted June 30, 2013  ISSN 
Accepted December 14, 2013 1234-3862 

Agnieszka Kapecka* 

Fractal Analysis of Financial Time Series Using Fractal 
Dimension and Pointwise Hölder Exponents∗∗ 

A b s t r a c t. This paper presents a fractal analysis application to the verification of assump-
tions of Fractal Market Hypothesis and the presence of fractal properties in financial time 
series. In this research, the box-counting dimension and pointwise Hölder exponents are used. 
Achieved results lead to interesting observations related to nonrandomness of price series and 
occurrence of relationships binding fractal properties and variability measures with the pres-
ence of trends and influence of the economic situation on financial instruments’ prices. 

K e y w o r d s: fractal analysis, fractal dimension, box-counting dimension, pointwise Hölder 
exponents, Hurst exponent. 

J E L Classification: G14, G15, G17. 

Introduction 
 Ever since Mandelbrot had published his works on the application of R/S 
analysis to long-memory dependencies in time series (Mandelbrot, Wallis, 
1969; Mandelbrot, 1972) and since Peters had presented his Fractal Market 
Hypothesis (Peters, 1991) as an alternative to commonly acknowledged Ef-
ficient Market Hypothesis, this approach is being explored with regard to 
financial time series. Mulligan examined the use of Lo’s modified rescaled 
range analysis on foreign exchange markets (Mulligan, 2000), proving the 

                                                 
* Correspondence to: Agnieszka Kapecka, Warsaw School of Economics, The Collegium 

of Economic Analysis, Al. Niepodległości 162, 02-554 Warsaw, Poland, e-mail: a.kapec-
ka@gmail.com. 

∗∗ This work was financed by the author. 



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DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 

108

presence of long memory dependencies and fractal structure of analysed 
price series. Another published research assumed Hurst exponent estimation 
using geometrical interpretation (Granero, Segovia, Pérez, 2008) applied to 
stock market indices or the search for periodic and nonperiodic components 
in S&P 500 time series (Bohdalová, Greguš, 2010). Another group of studies 
focused on the analysis of variation of Hurst exponent over time, showing 
the possible impact of capital flow and trading volume on the decrease of 
Hurst exponent values (Cajueiro, Tabak, 2004), the influence that the end of 
Bretton Woods System had on efficiency of US stock markets (Alvarez-
Ramirez, Alvarez, Rodriguez, Fernandez-Anaya, 2008), or the relationship 
between local Hurst exponent and stock market crashes with example of the 
Warsaw Stock Exchange Index (Grech, Pamuła, 2008). Due to certain limi-
tations of classical R/S analysis approach and Hurst exponent itself, some of 
the authors explored Hölderian pointwise regularity of some major stock 
market indices (Bianchi, Pantanella, 2010) and usage of multifractal spectra 
analysis in order to discover patterns of change in price series before the 
1987 market crash and other significant market drawdowns (Los, Yalamova, 
2004). 
 In the presented paper, the initial assumption is that the markets are not 
efficient, but are fractal in their nature. Despite the fact that Efficient Market 
Hypothesis (EMH) has been commonly accepted as a default theory explain-
ing the fundamentals of financial markets’ behavior, plenty of criticism and 
doubts have been adressed towards it. The critical remarks are mainly related 
to too strong assumptions underlying this hypothesis, vastly mangling the 
real world behavior of the markets. In terms of real market behavior, a chaos 
theory based Fractal Market Hypothesis (FMH) seems to be much more 
appriopriate. It assumes – on the contrary to EMH, which uses linear differ-
ential equation – that the market is a nonlinear dynamic system, which al-
lows to suppose that „real feedback systems involve long-term correlations 
and trends, because memories of long-past events can still affect the deci-
sions made in the present” (Peters, 1997, p. 6). Actions of market partici-
pants usually generate nonlinear behavior of financial processes. Functions 
describing investors’ attitute towards risk, their expectations towards stock 
market returns or financial instruments pricing formulae are nonlinear as 
well (Osińska, 2006, p. 118). The most characteristic property of FMH is 
acknowledgement that stock market returns time series have fractal (self-
similar) structure. FMH also allows chaotic behavior of the market during 
particular periods and under certain conditions (Peters, 1994, pp. 46–48). 
 FMH was described by Peters based on the results of long studies con-
ducted by Hurst in the first half of the 20th century (Peters, 1991). The main 



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109

conclusion was that most of natural systems do not follow the random walk 
model, but are subject to fractional Brownian motion. Such theory is in line 
with Peters’ assumption about the markets, which says that globally (in the 
long-term) the market is deterministic, while locally (in the short-term), due 
to randomly occuring information and emotional reactions of market partici-
pants, the market is random (Peters, 1997, pp. 45, 64). Empirical confirma-
tion of this hypothesis is presented in the section 4 of this paper. 

1. Fractal Dimension 
 One of the most substantial characteristics of geometric object is its di-
mension. However, an attempt to analyse this matter is nontrivial due to the 
fact that so far the scientists provided many different definitions of dimen-
sion, namely: the topological dimension, the Hausdorff dimension, the frac-
tional dimension, the box-counting dimension, the self-similarity dimension 
etc. The reasoning behind using particular types of dimension depends on 
certain conditions and while sometimes using different types of dimensions 
can lead to similar results, it might as well show varying results for the same 
object (Peitgen, Jürgens, Saupe, 2002, p. 274). 
 In brief, a dimension describes the way in which a geometric object (or 
time series)  fills the space. A common characteristic of all fractal objects is 
presence of self-similarity, which means that there is a relation between the 
reduction coefficient (a scale of similarity) and the amount of reduced frag-
ments similar to the original object. While analyzing zigzag-shaped financial 
instrument time series chart in terms of dimension, it is easy to conclude that 
its dimension falls into range ( )1, 2 .  As zigzag is not a straight line, it has 
dimension which is definitely distinct from 1, but it is not two-dimensional 
as it does not fill the entire plane.  
 There are several kinds of fractal dimensions. One of them is the box-
counting dimension, which – in view of its use as a research tool – is worth 
presenting. Below description of the box-counting dimension was written 
based on: Peitgen, Jürgens, Saupe (2002), Mastalerz-Kodzis (2003), 
Kudrewicz (2007) and Borys (2011). 
 Let F  be a certain geometric object embedded in n -dimensional Eu-
clidean space ,nR  covered with a set of small cubes (boxes called 
hypercubes) which sides are equal to ε  (e.g., for 1=n  it will be segments, 
for 2=n  squares, for 3=n  cubes). Theoretically these cubes can be discre-
tionary oriented with respect to the axes of a coordinate system, but they can 
be as well spheres or other convex solids with a diameter .ε  Let ( )ε,FN  be 



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110

a minimum amount of cubes that can completely cover the entire object .F  
Having satisfied this, below relations are true: 
− ( )ε,FN  ~ ε/1   – for F  being a segment of a smooth line (amount of 

cubes is approximately inversely proportional to the length of cube side), 
− ( )ε,FN  ~ ( )2/1 ε  – for F  being a piece of smooth plane, 

− ( )ε,FN  ~ ( )3/1 ε – for F  being an area contained in .3R  
 With respect to the above, a generalization can be made that there are 
such geometric objects, for which – assuming small values of ε  (which is 
a scale of similarity, so that the smaller the values of ,ε the better the approx-
imation) – below commensurateness is approximately met: 

( )ε,FN  ~ ( ) ,/1 Dε  (1) 

whereas – while in majority of cases there is only an approximation of com-
mensurateness between the two sides of the equation – in case of strictly 
self-similar objects an equality sign can be placed so that: 

( )( , ) 1/ ,DN F ε ε=  (2) 

where D  (not necessarily being an integer number) can be treated as a di-
mension of the object .F  Then the limit: 

( ) ( )
( ) ,

/1log
,loglim

0 ε
ε

ε

FNFDf →
=            (3) 

if it exists, is called a fractal dimension (in this case, a box-counting dimen-
sion)1.  
 It is easy to conclude from (3), which is applicable to exact fractals, that 
a dimension of such objects is defined by: 

( ) ( )
( ) .

/1log
,log
ε
εFNFDf =  (4) 

 A possibility to apply the box-counting dimension led to creation of al-
gorithms capable of estimating the dimension of time series. In this method, 
the analyzed structure is placed on a regular grid with a size of ,ε followed by 

                                                 
1 If a given unit gets reduced ε  times, the measured line will approximately contain ε  

times more units than in the previous iteration step (approximation is in this case a result of 
a possible presence of small curves disturbing the existing commensurateness). Given com-
mensurateness is exact only in the limit, because after reducing the length of the sunit it is 
possible to achieve a infinitely small unit only in the limit (Tempczyk, 1995, p. 135). 



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111

counting all the cubes (boxes), which contain the fragments of analyzed 
structure. The amount of fragments is obviously dependent on the value of 

.ε  In the next stage it is necessary to repeat the calculations for smaller val-
ues of ε  and plot the obtained values on the logarithmic chart (marking 

( )log ,N F ε  with respect to ( )log 1/ ε ). The slope of the straight line fitted 
to the points marked on the plot determines the box-counting dimension of 
the time series (Daros, 2010, p. 13).  

2. Measures of Variability of the Graph of the Function 
 One of the most popular and useful measures of variability (irregularity) 
are Hurst exponent and Hölder exponents. The Hurst exponent is a numeri-
cal characteristic of the entire price series, whereas Hölder exponents can be 
used to analyze the complexity of the function and the trajectory of some 
stochastic processes in the vicinity of any point of the graph of the function 
(Mastalerz-Kodzis, 2003, p. 37). 
 The most important turning point in the subject of long-term dependency 
analysis of time series was without any doubt the creation of rescaled range 
analysis n)S/R(  method (Hurst, 1951). A starting point of his analysis was 
Einstein’s work on Brownian motion (Einstein, 1908), which presented an 
equation for the distance R that a particle travels in time ,T  which is defined 
by ,TcR = where c  is a nonnegative constant. This equation was applica-
ble when the series of increments of the distance travelled by the particle in 
time was a random walk, characterised by the independency of normally 
distributed random variables (Weron, Weron, 1998, p. 323). However, dur-
ing almost fourty years of research, Hurst has reached a conclusion that the 
majority of natural phenomena is not subject to Gaussian random walk, but 
rather to processes with „long memory”, later called by the name of fraction-
al Brownian motion, which is a combination of a trend and noise (Peters, 
1997, p. 64; Mastalerz-Kodzis, 2003, pp. 37–38).  
 Derivation of a formula for rescaled range allowed the comparison of 
different types of time series. Creation of this dimensionless indicator, which 
should increase over time, allowed to formulate the following equation being 
an extension of Brownian motion model proposed by Einstein. 

,nc)S/R( H
n ⋅=      (5) 

where )S/R(  – rescaled range, n  – number of observations, c  – positive 
constant, H – Hurst exponent. 



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112

 In order to calculate Hurst exponent, one has to calculate the average 
value of n)S/R(  for different n  and then solve the following equation us-
ing linear regression: 

log ( / ) log( ) log( ),nE R S H n c= +      (6) 

where n)S/R(E  – expected value of rescaled range. 
 In above equation, the Hurst exponent can be treated as the regression 
coefficient and estimated using the least squares method (Jajuga, Papla, 
1997). 
 The Hurst exponent is strictly linked to the fractal dimension of time 
series, therefore the search for the Hurst exponent is in fact a search for the 
fractal properties of the series. This relation is described by the following 
equation (Grech, 2012, p. 10): 

.2 HDf −=      (7) 

 This equation has a huge practical importance as it can be used to classi-
fy the type of a time series depending on the fractal dimension of a given 
object.  
 Following cases can be distinguished based on Hurst exponent values 
(Peters, 1997, p. 76–77): 
− if ,5.00 << H  then 25.1 << D  (antipersistent time series), 
− if ,5.0=H  then 5.1=D  (random walk), 
− if ,15.0 << H  then 5.11 << D  (persistent time series). 

 First case ( 5.00 <≤ H ) applies to antipersistent (ergodic) time series. 
Such a series has a mean reversion tendency. If in a given period the value of 
the series increased, then in the following period it will most probably de-
crease and vice versa. The closer the value of H to ,0 the more ergodic the 
behavior of the system and the time series graph has more jagged line, which 
is a result of a frequent trend reversion. In such case the fractal dimension of 
the series 2→fD  as the series fills the plane more and more. The lower the 
H value, the more noise can be observed in the system. Speaking in the 
probability language, if e.g. ,2.0=H then there is 80% probability that in 
the future the market will change the direction, which will be equal to trend 
reversion (Stawicki, Janiak, Müller-Frączek, 1997, p. 37). Despite the fact 
that mean reversion plays a dominant role in economic and financial litera-
ture, so far only a few antipersistent time series have been observed. 
 The second case applies to a situation, when with ∞→n  ,5.0=H
which corresponds to a random walk (the consecutive elements of the series 



Fractal Analysis of Financial Time Series Using Fractal Dimension… 

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113

are independent). The fractal dimension of the series equals ,5.1=D the 
series itself is unpredictable, the present does not influence the future, and 
the past did not influence the present. The probability distribution function 
can be Gaussian, but not necessarily. Both in natural and economic phenom-
ena the H exponent’s value usually differs from ,5.0  and the natural pro-
cesses most often have a long-term data dependency. 
 When ,15.0 ≤< H  the time series are persistent, which means that they 
bolster the trend. It is caused by the presence of long-term data dependency. 
When ,1→H the trend gets stronger. As Hurst exponent defines the proba-
bility of consecutive rises or drops of the prices, with ,1→H  there are more 
consecutive rises or drops and the level of noise becomes smaller. For exam-
ple, if  ,8.0=H  then there is 80% probability that a given trend will be sus-
tained in the future. The fractal dimension ( )5.1,1∈D  as the more persistent 
the time series, the less it fills the plane and the smoother are the curves cre-
ated by a given system. Fractal time series is obviously not purely determi-
nistic, it is rather an intermediate form between a completely random time 
series and a deterministic system. Persistent time series are fractional 
Brownian motions, which means that their important feature is a biased ran-
dom walk, and the strength of bias increases when ,1→H  so when the 
Hurst exponent value recedes from .5.0  
 Despite n)S/R(  analysis is a very useful tool, it poses a major disad-
vantage – it does not take into consideration the changes in particular 
subperiods. For example, if a particular stock is subject to rapid price chang-
es, whereas the prices of other stocks show only minor price changes, it is 
not possible to detect this periodical changes using Hurst exponent. There-
fore, a good measure of variability of the graph of a function over time is 
a Hölder function2, which values in particular points are equal to pointwise 
Hölder exponents (Mastalerz-Kodzis, 2003, p. 115).  
 In order to further discuss pointwise Hölder exponents, let us first define 
the Hölder function3. Let )d,X( X  and )d,Y( Y  be metric spaces, then func-
tion YX:f →  is called a Hölder function with exponent ,α  where ,0>α  if 

                                                 
2 Although pointwise Hölder exponents are considered to be the best measure of function 

regularity in the vicinity of a certain point, other measures used include: local box-counting 
dimension, local Hausdorff dimension, the degree of fractional differentiability. 

3 The definition of Hölder function and pointwise Hölder exponents was written based on: 
Mastalerz-Kodzis 2003, pp. 49–51; Kuperin, Schastlivtsev, 2008, pp. 4–6. 



Agnieszka Kapecka 

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114

for each Xy,x ∈  such that 1)y,x(dX <  the function satisfies the following 
inequality: 

,),()](),([ αyxcdyfxfd XY ≤      (8) 

where c  – a positive constant. 
 Assuming that function RD:f →  and that parameter ),1,0(∈α  func-

tion f  is a αC  class ( α∈Cf ) Hölder function,  if such 0c >  and 0h0 >  
constants exist that for every x and every ( )0h,0h∈  the following inequality 
is satisfied: 

.ch)x(f)hx(f α≤−+      (9) 

 Assuming that 0x  is an arbitrary point from the domain of function ,f  

so that ,RDx0 ⊂∈  function RD:f →  is a α
0xC  class ( α∈

0xCf ) Hölder 

function in ,x0  if such 0c >  and 0>ε  constant exist that for every 
)x,x(x 00 ε+ε−∈  the following inequality is satisfied: 

.xxc)x(f)x(f 00
α−≤−      (10) 

  By definition Hölder function is continous in its entire domain and when 
this assumption is satisfied, the graph of the function has fractal nature 
(Gabryś, 2005, p. 24). If the Hölder function is not continuous, it is called 
generalized Hölder function. It is worth noting that thanks to its time-varying 
values, Hölder function can take different types of random walks in different 
ranges  (Kutner, 2009, p. 36).  
 After providing a Hölder function definition it is possible to define 
a pointwise Hölder exponent of function f  in .x0  By definition it is a num-
ber )x( 0fα  given by the following equation: 

{ }. Cf:sup)x(
0x0f
α∈α=α      (11) 

 Approximated pointwise Hölder exponent is not a flawless measure, 
however its major advantage is the ability to accomodate the stationarity of 
the series. Interpretation of pointwise Hölder exponent is the same as for 
Hurst exponent with a difference that pointwise Hölder exponent estimates 
local, not global value. It is also worth mentioning that Hölder functions are 
not constant as they are time-varying (Mastalerz-Kodzis, 2003, p. 121).  



Fractal Analysis of Financial Time Series Using Fractal Dimension… 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 

115

3. Tools and Analysis Methodology 

3.1. Data 

 All monthly and daily price series were downloaded from http://stooq.pl. 
Table 1 presents the list of stock market indices and forex currency pairs 
used in this research. 

Table 1. Financial instruments chosen for the analysis 
Symbol Available Data Period Market Type 

DJIA 1896.05–2012.12 Mature market 
S&P 500 1923.01–2012.12 Mature market 

DAX 1959.10–2012.12 Mature market 
Nikkei225 1949.05–2012.12 Mature market 

Hang Seng 1969.11–2012.12 Mature market 
WIG20 1991.04–2012.12 Emerging market 

Bovespa 1992.01–2012.12 Emerging market 
RTS 1995.09–2012.12 Emerging market 

SENSEX30 1979.04–2012.12 Emerging market 
SCI 1990.12–2012.12 Emerging market 

XU100 1990.01–2012.12 Emerging market 
EUR/USD 1980.01–2012.12 Major currency pair 
GBP/USD 1971.01–2012.12 Major currency pair 
USD/JPY 1971.01–2012.12 Major currency pair 
CHF/PLN 1990.01–2012.12 Exotic currency pair 
EUR/PLN 1990.01–2012.12 Exotic currency pair 
USD/BRL 1995.01–2012.12 Exotic currency pair 
USD/RUB 1995.10–2012.12 Exotic currency pair 
USD/INR 1973.01–2012.12 Exotic currency pair 
USD/CNY 1984.01–2012.12 Exotic currency pair 
USD/TRY 1984.01–2012.12 Exotic currency pair 

The above selection is reasoned by analysis of both mature and emerging 
markets and the associated currency pairs in order to check if any relation-
ships interesting from the fractal analysis point of view are present. 

3.2. Tools 

 During the analysis, Microsoft Excel was used to calculate the common 
logarithms of price values and plot the charts of logarithmic price series and 
pointwise Hölder exponents. For the purposes of box-counting dimension 
estimation and pointwise Hölder exponents calculation, FracLab 2.0 was 
used. As quoted from FracLab homepage (http://fraclab.saclay.inria.fr/): 
„FracLab is a general purpose signal and image processing toolbox based on 
fractal and multifractal methods. (…) A large number of procedures allow to 



Agnieszka Kapecka 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 

116

compute various fractal quantities associated with 1D or 2D signals, such as 
dimensions, Hölder exponents or multifractal spectra. (…) FracLab is a free 
software developed in the Regularity team at Inria Saclay/Ecole Centrale de 
Paris.” 

3.3. Analysis Methodology 

 Of the fractal analysis methods described in sections 1 and 2, in this re-
search the box-counting dimension, Hurst exponent and pointwise Hölder 
exponents were used for long-term dependency analysis of chosen financial 
time series. 

Table 2.  Values of fractal dimension D and Hurst exponent H calculated on the 
entire data range of monthly and daily price series of chosen financial in-
struments 

Symbol Period Monthly data Daily data 
D H D H 

DJIA 1896.05–2012.12 1.44 0.56 1.43 0.57 
S&P 500 1923.01–2012.12 1.40 0.60 1.40 0.60 

DAX 1959.10–2012.12 1.50 0.50 1.47 0.53 
Nikkei225 1949.05–2012.12 1.36 0.64 1.44 0.56 

Hang Seng 1969.11–2012.12 1.47 0.53 1.50 0.50 
WIG20 1991.04–2012.12 1.43 0.53 1.48 0.52 

Bovespa 1992.01–2012.12 1.22 0.78 1.48 0.52 
RTS 1995.09–2012.12 1.47 0.53 1.44 0.56 

SENSEX30 1979.04–2012.12 1.37 0.63 1.46 0.54 
SCI 1990.12–2012.12 1.50 0.50 1.46 0.54 

XU100 1990.01–2012.12 1.31 0.69 1.46 0.54 
EUR/USD 1980.01–2012.12 1.49 0.51 1.47 0.53 
GBP/USD 1971.01–2012.12 1.56 0.44 1.44 0.56 
USD/JPY 1971.01–2012.12 1.49 0.51 1.42 0.58 
CHF/PLN 1990.01–2012.12 1.32 0.68 1.44 0.56 
EUR/PLN 1990.01–2012.12 1.35 0.65 1.43 0.57 
USD/BRL 1995.01–2012.12 1.34 0.66 1.40 0.60 
USD/RUB 1995.10–2012.12 1.20 0.80 1.36 0.64 
USD/INR 1973.01–2012.12 1.23 0.77 1.32 0.68 
USD/CNY 1984.01–2012.12 1.16 0.84 1.36 0.64 
USD/TRY 1984.01–2012.12 1.08 0.92 1.39 0.61 

 
  



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4. Fractal Analysis Using Fractal Dimension and Pointwise 
Hölder Exponents 

4.1. Presentation of Results 

 The research was divided into two parts. In the first part, the dimensions 
of chosen price series were estimated and equation (7) was used to derive 
Hurst exponent from the estimated dimension. First, the dimensions of price 
series were calculated for the entire historical data range (see Table 2). 

Table 3.  Values of fractal dimension D and Hurst exponent H calculated on the Oct 
1995 to Dec 2012 data range of monthly and daily price series of chosen 
financial instruments 

Symbol Period Monthly data Daily data 
D H D H 

DJIA 1995.10–2012.12 1.41 0.59 1.48 0.52 
S&P 500 1995.10–2012.12 1.36 0.64 1.45 0.55 

DAX 1995.10–2012.12 1.39 0.61 1.44 0.56 
Nikkei225 1995.10–2012.12 1.50 0.50 1.48 0.52 

Hang Seng 1995.10–2012.12 1.60 0.40 1.51 0.49 
WIG20 1995.10–2012.12 1.52 0.48 1.48 0.52 

Bovespa 1995.10–2012.12 1.46 0.54 1.49 0.51 
RTS 1995.10–2012.12 1.46 0.54 1.44 0.56 

SENSEX30 1995.10–2012.12 1.51 0.49 1.47 0.53 
SCI 1995.10–2012.12 1.53 0.47 1.44 0.56 

XU100 1995.10–2012.12 1.37 0.63 1.45 0.55 
EUR/USD 1995.10–2012.12 1.53 0.47 1.49 0.51 
GBP/USD 1995.10–2012.12 1.52 0.48 1.52 0.48 
USD/JPY 1995.10–2012.12 1.50 0.50 1.49 0.51 
CHF/PLN 1995.10–2012.12 1.51 0.49 1.50 0.50 
EUR/PLN 1995.10–2012.12 1.57 0.43 1.49 0.51 
USD/BRL 1995.10–2012.12 1.36 0.64 1.40 0.60 
USD/RUB 1995.10–2012.12 1.20 0.80 1.36 0.64 
USD/INR 1995.10–2012.12 1..40 0.60 1.39 0.61 
USD/CNY 1995.10–2012.12 1.26 0.74 1.56 0.44 
USD/TRY 1995.10–2012.12 1.21 0.79 1.40 0.60 

In the second part, pointwise Hölder exponents were calculated for six cho-
sen markets. Figures 1–6 present pointwise Hölder exponents plotted togeth-
er with common logarithms of time series values. The range of values taken 
by pointwise Hölder exponents’ values is presented on the right of each 
graph. 
  



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118

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Figure 3

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DYNAMIC ECON

ntwise Hölder

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DYNAMIC

120

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gnieszka Kapec

2013) 107–12

markets in th
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25 

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Nikkei225 a
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Germany. 
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DYNAMIC ECON

or the rando
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values of 



Agnieszka Kapecka 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 

122

the pointwise Hölder exponents are significantly higher than 5.0  during 
most of the entire data range, which proves the presence of long-term de-
pendencies in the investigated time series. Moreover, the graphs expose the 
relationship between the trend strength and Hölder exponents’ values as they 
tend to considerably decrease and converge to 5.0  during recent years side-
ways price movements (which is especially visible for Nikkei225, DAX or 
EUR/USD). 
 One of the pros of using the box-counting dimension method in this arti-
cle is that due to its nature, it does not require assumption on the 
selfsimilarity of the analysed object, which allows to avoid making a possi-
bly false statement even before the beginning of the research. Another con-
siderable advantage of the methods used in this article was their algorithmic 
accessibility, thus making it affordable from the calculation complexity point 
of view and available even for the users which do not have a high-end pro-
cessor cluster at their disposal. Nevertheless, as the estimation method as-
sumes that the Hölder function is constant over each of the intervals, it might 
introduce some error of method to the results. 

Conclusions 
 The analysis conducted in this research provides solid empirical evi-
dence in favour of Fractal Market Hypothesis, with the presence of long-
term dependencies in the financial time series and the confirmation of the 
global determinism and local randomness of the markets being the most 
important ones.  
 An important observation supporting the nonrandomness of the markets 
is a relationship between fractal properties of the investigated time series and 
the underlying economic situation. One of the most interesting researches 
that could be made in the future based on this would be a repetition of the 
calculations at some fixed intervals of time – after ten, twenty or thirty years. 
Such approach could possibly verify if the opinions stated by economists 
a posteriori major changes in economic conditions are in line with the con-
clusions drawn from the fractal characteristics of analysed markets. 
 Another study that could be made as a continuation of research conduct-
ed in this article could involve inclusion of additional test data, this time not 
limited to market time series. Inclusion of some regularly published econom-
ic indicators like gross domestic product, consumer price index or money 
supply could potentially reveal some additional relationships and regularities 
and shed more light on the macroeconomic processes. 



Fractal Analysis of Financial Time Series Using Fractal Dimension… 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 

123

 Moreover, the analysis of results suggests that fractal analysis can be 
a valuable tool for the evaluation of market trends, which might be of practi-
cal use for institutional and individual investors. 

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125

Fraktalna analiza finansowych szeregów czasowych z wykorzysta-
niem wymiaru fraktalnego oraz punktowych wykładników Höldera 

Z a r y s  t r e ś c i. Artykuł przedstawia propozycję zastosowania analizy fraktalnej w celu 
weryfikacji niektórych założeń hipotezy rynku fraktalnego oraz występowania fraktalnych 
właściwości w finansowych szeregach czasowych. W celu przeprowadzenia badań wykorzy-
stany został wymiar pudełkowy oraz punktowe wykładniki Höldera. Rezultaty osiągnięte dla 
badanych rynków pozwoliły dokonać interesujących obserwacji dotyczących nielosowości 
szeregów cenowych oraz występowania relacji między fraktalnymi właściwościami i miarami 
zmienności a obecnością trendów i wpływem sytuacji ekonomicznej na ceny instrumentów 
finansowych. 

S ł o w a  k l u c z o w e: analiza fraktalna, wymiar fraktalny, wymiar pudełkowy, punktowe 
wykładniki Höldera, wykładnik Hursta. 






