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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Paweł Kliber 
Poznan University of Economics 

Jumps Activity and Singularity Spectra for Instruments  
in the Polish Financial Market† 

A b s t r a c t. In the paper we try to measure the activity of jumps in returns of some instruments  
from the Polish financial market. We use Blumenthal-Getoor index β for Lévy processes as 
a measure of jumps’ activity. This allows us to distinguish between processes with rare and sharp 
jumps and the processes with infinitely-active jump component. We use three different methods. 
First we use activity signature plots to estimate the activity patterns of jumps. Then we estimate 
the Blumenthal-Getoor index with Aït-Sahalia and Jacod threshold estimator.Then we use meth-
ods based on singularity spectra of Lévy processes. Finally, we compare the results.  

K e y w o r d s: Blumenthal-Getoor index, singularity spectrum, Lévy exponential models.  

Introduction 

The classical models of assets’ returns are based on the assumption of the nor-
mality of returns. This is for example the case of classical portfolio theory, de-
veloped by Markowitz (1952) and Sharpe (1963), and the option pricing formu-
la, derived by Black and Scholes (1973) and by Merton (1973). However it is 
well-known that the normality assumption does not hold. Many well-established 
stylized facts about assets’ returns contradict this. The observed returns reveal 
characteristics such as heavy tails, high kurtosis or volatility clustering, which is 
not consistent with Gaussian distribution1. 

 There are several methods to deal with non-normality of returns and to 
make models better fitted to observed data. The most popular approach is the 
modeling the conditional volatility with some kind of ARCH/GARCH model. 

                                                 
† The research was undertaken in the project sponsored by the Polish Ministry of Science and 

Higher Education, N N111 436 534. The author would like to thank two anonymous referees for 
valuable comments. 

1 The survey of stylized facts concerning assets’ returns can be found in (Cont, 2001). 



Paweł Kliber 172 

When using non-Gaussian distributions of error terms, such models seem to be 
well-fitted to the data. 

 In this work we take another approach, which lately becomes more and 
more popular, namely we use Lévy processes in the modeling. Although this 
line of modeling is as old as the seminal paper of Maldenbrot (1963), in which 
models with stable distributions were proposed, it became more popular at the 
beginning of XXI century. 

 Lévy processes can be represented as a sum of continuous diffusion (Wiener 
process) and discontinuous jumps. Thus, one of the main questions is if there 
are jumps in the returns of financial instruments and if there are, how much 
intensive are the jumps. Formally the intensity of jumps of Lévy process is de-
scribed by Blumenthal-Getoor index. In the paper we try to estimate the value 
of this index for the suitably-chosen sample of instrument from Polish financial 
market. We use three different methods of estimation and then we compare the 
results. 

 The paper is organized as follows. In the section 1 we present basic facts 
about Lévy processes and Lévy exponential models of prices. In this section we 
also introduce the concept of Blumenthal-Getoor index of jumps. In section 2 
we try to analyze the type of processes using activity signature plots. In section 
3 we estimate Blumenthal-Getoor indexes using threshold estimator. In section 
4 we analyze the type of processes using its singularity spectrum. Section 5 
concludes. 

1. Lévy Processes and Lévy Exponential Models 

The Lévy process L  is the stochastic process with continuous time that starts at 
zero (i.e. 0 0L  ) and fulfils the following conditions: 

1. for any 1 2 3 40 t t t t     the random variables 
12t tL L  and 

4 3t tL L  are 

independent and the distribution of t h tL L   depends only on h  and not on 

t , 

2. the process is stochastically continuous, i.e. for all 0t   and all 0   

 
0

lim 0
h

t h tP L L 
    

3. the trajectories of the process are cadlag (i.e. they are right-continuous with 
left limits). 

We should stress that the second condition does not imply that the process is 
continuous. In fact the Lévy processes typically have discontinuities  
(of “jumps”) and some of them are discontinuous at each point t . The condition 
2 means only that the process L  does not have the jump of arbitrary size   at 
any pre-specified moment t . 



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 173

 The Lévy processes can be seen as the extension of Wiener process. In fact 
if we change the condition 2 and require the process L  to be continuous, then 
the only processes that fulfill the definition are Wiener processes with drift. 
There is also another connection between Wiener process and Lévy processes. 
The increments of a Wiener process are normally-distributed, while the incre-
ments of Lévy process belong to the family of infinitely-divisible distributions. 
It is the biggest family of distributions that can serve as limits for the sums of 
independent variables (see Feller, 1971). Thus, if we believe that the returns of 
the assets in financial markets result from many independent shock, then the 
Lévy processes are natural choice as an engine in the model. 

 There are two fundamental theorems that reveal the structure of the Lévy 
processes. The first one is the Lévy-Itô decomposition, which states that any 
Lévy process L  can be uniquely represented as the sum of Wiener process with 
drift, Poisson process and purely discontinuous martingale: 

l s
t t t tL XW Xt    , (1) 

where   and 0   are constants, W is a standardized Wiener process, lX  is 

a compound Poisson process of large jumps and sX  is purely discontinuous 

martingale (i.e. it is discontinuous at each point t ). The process sX  represents 
small jumps – in every finite interval there are infinitely many of them, but they 

are very small, so that the process sX  does not explode. 

 The second theorem characterizes the characteristic function of the Lévy 
processes. According to Lévy-Khintchine representation the characteristic func-
tion of the Lévy process is: 

[ ] exp[ ( )]tiuLE e t u , (2) 

where the characteristic exponent   equals: 

 
2

2
11 1( ) ( )

2
iux

x
R

e iuxu i u u xd
       . (3) 

The first two terms in the sum (3) are the same as in the characteristic function 
for the Gaussian distribution. These terms describes continuous part of the pro-
cess (diffusion). The measure  (called Lévy measure) describes the jumps of 
the process L . The value of ( )R  is the intensity of jumps. If finite, it is the 
average number of jumps in the unit of time – the process is then said to be 
“finitely active”. If ( )v R   , then the process is infinitely active – in any in-

terval the number of jumps is infinite. The values (( , ))v a b  give the relative 

intensity of jumps with sizes between a  and b  (i.e. jumps such that 
( , )t t tLL aL b   ). Thus the Lévy measure contains information of both the 



Paweł Kliber 174 

intensity of jumps and distribution of jumps’ sizes. The triple ( , , )   , called 
“characteristic triple”, gives unique characterization of the process. 

 The measure  , which contains all information of jumps and its structure, 
can vary, depending on the type of the process or on the probability distribu-
tions of increments of the process. There exists however one synthetic index 
that divides measures   into certain categories and gives characteristics of 
jump behavior. Blumenthal-Getoor index is defined as 

1

1

inf 0 : ( )
b

db x x 


      
  

 . (4) 

The index   takes values in the interval [0,2) . Higher values of   mean that 
jumps are more intensive and smaller and the process L  resemble continuous 
process. If 0   the process is finitely active. All other values mean that the 

activity of the process is infinite. In the case 0   the discontinuous part of the 

process L  is the compound Poisson process, i.e. 0sX   in the decompo-  
sition (1). 

 The models of assets prices driven by Lévy process usually take the form of 
exponential Lévy models: 

0
tL

tS S e , (5) 

where tS  denotes the asset price at time t . The models can be also formulated 
as stochastic differential equation (as in classis Black-Scholes model): 

t t tdS S dV , (6) 

where V  is a Lévy process, whose characteristics can be derived from L 2. The 
logarithms of the prices are described by Lévy process L  and thus the logarith-
mic returns are increments of Lévy process. According to (1) the logarithmic 
price is given by: 

ln l s
t t t t tts XWS X     . (7) 

In the literature one considers also the generalization of (7), assuming that the 
volatility of continuous part is not constant. Such a model can be specified as 
follows: 

d
t t t ts Wt L   , (8) 

where t  is a process or a function representing volatility and dL  is discontinu-
ous part of the process L  and represents jumps. 

                                                 
2 Kallsen (2000) has shown that specifications (5) and (6) are equivalent and gave the formu-

lae how to express V  in terms of L  and vice versa. 



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 175

 Let us select some time-scale, i.e. choose some frequency in which we sam-
ple the process. Suppose that we sample every h  units of time (in practice it can 
be a period from several seconds to one day). The increments of the process at 
the specified frequency are logarithmic returns of the assets. We denote them 
by ( )ix h : 

( 1) ( 1)( )i i h ih i h ihh s sx L L    . (9) 

If there is no ambiguity about frequency we omit brackets and denote returns 
simply by ix . In the next three sections we will use data about returns to calcu-

late index  , using three different methods. 

2. Estimating Jumps Activity with Activity Signatures 

The first method is based on power variation of stochastic process, defined as: 

1

( , ) ( )
p

i
i

N

V p h x h


 . (10) 

In the case 2p  , the (2, )V h  is well-known realized volatility, which tends to 
the quadratic variation of the process, as the sampling frequency tends to infini-
ty. The behavior of ( , )V p h  in other cases depends on the type of the process. 
As Barndorff-Nielsen and Shephard (2002) have shown, if the process L  con-

tains no jumps (i.e. 0dL   in (8)), then: 

1 /2

0 0

plim ( , )
T

pp
s

h
pph shV d 


  , (11) 

where p  is appropriate constant. If the process L  contains no diffusion part  

(  0t  ), then as 0h   the sum (10) diverge for p  . On the other hand for 
p   the sum is convergent: 

0 0

(plim , )
T

p
s

h s

p h LV
 

  . (12) 

If the process contains both diffusion and jump parts, then the following equa-
tions holds: 



Paweł Kliber 176 

1

2

/2

0 0

2

0 00

0 0

plim fo( , ) , (0,2)r 

pli

,

(2, ) , ,m for 2

pli (m , for 2, .)

T
pp

p s

T T

s s
s

T
p

s

h

h s

h
p h

h

h V ds p

L

p h L

V ds p

V p

 







 

 


 


   


  













 (13) 

 Based on these limit behavior of the power variation, Todorov and Tauchen 
(2009) proposed a qualitative test of the type of the process. They have defined 

activity signature function ˆ( ; , )p k h  as: 

lnˆ( ; , )
ln ln ( , ) ln ( , )

p k
p k h

k V p kh V p h
 

 
.  (14) 

The graph of ˆ( ; , )p k h  with respect to p  are called “activity signature plot”.  

If the process is continuous, then for all 0p  : 

0

ˆplim ( ; , ) 2
h

p k h


 .  (15) 

For processes of pure jumps we have: 

0

, for (0, ),ˆplim ( ; , )
, for .h

p
p k h

p p

 





  

 (16) 

In the case, when the process contains both jumps and diffusion: 

0

2, for (0,2),ˆplim ( ; , )
, for .h

p
p k h

p p





  

 (17) 

 We apply method of activity signatures to several instruments from Polish 
financial markets. The sample contains of four stocks: two liquid ones (PKN 
Orlen and KGHM) and two less liquid (Agora and BRE Bank), three stock 
market indexes (WIG, WIG20 and MWIG40), one future contract (FWIG) and 
two currencies (euro and US dollar). The sample was chosen as to contain pos-
sibly wide range of different instruments. We have used intraday data for the 
period from the beginning of 2009 to the March of 2011. In the computation we 
used 10-minutes returns. We have tried some other frequencies and decided that 
this frequency is high enough to justify the usage of the limits (15)-(17). On the 
other hand it is not so high, that the market microstructure noise affects the re-

sults3. In the computations of ˆ( ; , )p k h  we took 2k  , as in the original work 

                                                 
3 We have used methods of Zhang, Mykland, Aït-Sahalia (2005) to control the microstructure 

noise. For sampling period of 10 minutes the difference between “two time scales” estimator of 



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 177

of Todorov and Tauchen (2009). Thus we have worked with two time scales:  
10 minutes and 20 minutes returns. 

 We do not present results for each instrument, but only show three typical 
cases. Figure 1 shows activity signature plot for Agora. Such a graph is typical 
for less-liquid stocks. It represents the case (16) of pure jump process with 

0  , which means that the prices are driven by compound Poisson processes.  

 
Figure 1.  The activity signature plot for Agora (vertical axis – exponents p , horizon-

tal axis – the values of ˆ( ;2, )p h ) 

 

 
Figure 2.  The activity signature plot for PKN Orlen (vertical axis – exponents p , 

horizontal axis – the values of ˆ( ;2, )p h ) 

                                                                                                                        
quadratic variance and realized variance was small, so we decide that microstructure effect for 
this frequency is negligible.  



Paweł Kliber 178 

The Figure 2 represents the activity signature plot for PKN Orlen, but the graph 
is similar to the plots for liquid stock, future contract and currency prices. The 
plot resemble the case (17), when the process contains both jump and diffusion 
parts. The plot for index WIG, shown on the Figure 3, is typical for all indexes. 
In this case the process is continuous. 

 
Figure 3.  The activity signature plot for WIG  (vertical axis – exponents p , horizontal 

axis – the values of ˆ( ;2, )p h ) 

3. Estimating Blumenthal-Getoor Index Using Threshold Estimator 

The method of activity signatures allows us to identify the type of process that 
underlines prices, but usually does not allow estimating the value of Blumen-
thal-Getoor index. To estimate the values of this index we use threshold estima-
tor proposed by Aït-Sahalia and Jacod (2009). The estimator is given by the 
formula: 

ln ( , ) ln ( , )ˆ( , , )
ln

U h U k h
k h

k

   
 , (18) 

where ,( )U h  is counting function which counts the exceedances of the 
threshold: 

 ( )
1

( ), 1
i

N

x h h
i

hU 





 , (19) 

with (0,1 / 2) . Estimator (18) uses two time scales, as activity signature 
method, but the latter method is based on limit properties of power variation, 
while the threshold estimator is based on different exceedance rates in different 
time scales. It allows us to estimate Blumenthal-Getoor index in the case when 
the process contains diffusion (continuous) part. It also allows for testing the 
accuracy of the estimation. The asymptotic standard error of the estimator 
equals: 



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 179

1 1 1

ln ( ,) (, )k hk U U h 
 . (20) 

 We have calculated the estimator ˆ( , , )k h   for all instruments in the sam-
ple. As in the original work of Aït-Sahalia and Jacod (2009) we have taken 

1 / 5   and 2k  . As for the threshold level   it was taken as seven times 
the estimated standard error of the continuous part of the process. The results 
are shown in Table 1. 

Table 1. Estimators of Blumenthal-Getoor index 

Instrument ˆ( , , )k h   Std. dev. 

AGO 0.7382 0.0087 
BRE 1.0888 0.0074 
KGH 1.5486 0.0052 
PKN 1.5405 0.0085 
WIG 2.6208 0.0285 

WIG20 1.9684 0.0097 
MWIG40 2.2870 0.0185 
FWIG20 1.9280 0.0024 

EUR 1.9835 0.0097 
USD 2.0491 0.0069 

 The results are only partially consistent with the ones obtained with activity 
signature method in the previous section. For the less-liquid stocks (AGO and 
BRE) the estimators of   are significantly lower than for the other instruments. 
They are however greater than 0. In case of stock indexes (WIG, WIG20 and 
MWIG40) the estimators are high (close to 2 or even greater than 2), what is 
consistent with previous results, that trajectories of these instruments are con-
tinuous. Liquid stocks have   between 1 and 2. As for the currencies (EUR, 
USD) and futures contract (FWIG20), the obtained values are close to 2, what 
stands in contradiction with the results from the previous section. The last col-
umn in the Table 1 contains estimators’ errors. However they were calculated 
with asymptotic formula (20) and it is dubious if they give the true errors of 
estimators, especially if some values are greater than 2.  

4. Singularity Spectra and Jump Activity 

The third method of estimating Blumenthal-Getoor index is based on singularity 
spectra of observed trajectories of prices. It is non-statistical methods. It is 
based on the fact that the trajectories of Lévy processes with different   almost 
surely (i.e. with probability 1) reveals different types of continuity. Let us first 
introduce the concept of singularity spectrum. 



Paweł Kliber 180 

 Take any function :f R R . We say that the function is a -Hölder contin-

uous in the point 0t  if there exists a polynomial ( )P t  of order lower then  

a  such that in some neighborhood of 0t : 

0( ) ( )
a

f t P t K t t    (21) 

for some 0K  . Let   be a number such that for a   function f  is  

a -Hölder continuous at t  and for all a   f  is not a -Hölder continuous  
at t . The   is called local Hölder exponent of the function f  at the point t  

and is denoted by ( )fh t . It is the measure of “regularity” of f  in the neighbor-

hood of t . The higher ( )fh t , the more regular the function is. For example the 

trajectories of Wiener motion almost surely have local Hölder exponent equal to 
0.5 at each point. Let ( )f   be the set of all points in which the function f  

has local Hölder exponent equal to  : 

 ) :( ( )f ft h t    . (22) 

Singularity spectrum it is the mapping which for all   returns the Hausdorff 
dimension (see Mallat, 2003; Falconer, 2003) of the set ( )f  : 

H( ) dim ( )f fD    . (23) 

 As was shown by Jaffard (1999) the Lévy processes with different Blumen-
thal-Getoor index almost surely have different singularity spectra. If the Lévy 
process L  does not contain diffusion part, then: 

1
(dim ) for H L   


     (24) 

and ( )L    for 1/  . The shape of singularity spectrum for such 
a process is show in the Figure 4. If the process L  contains diffusion part, then:  

1
( ) for dim

2H L      ,  (25) 

(1di / 2)m 1H L   and ( )L    for 1 / 2  . For the Wiener process (with-

out jumps) (1di / 2)m 1H L   and ( )L    for 1 / 2  . 



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 181

 
Figure 4.  Singularity spectrum for Lévy process with Blumenthal-Getoor index   

 To calculate singularity spectra from discretely sampled data one uses so 
called “multifractal formalism”, introduced by Frisch and Parsi (1985) and de-

veloped by Jaffard (1997a, 1997b). It can be shown that the function *
fD , de-

fined as: 

0

* 1
( ) liminf ln ( ) ( )

ln
q

f
h

D q f x h f x dx
h

    (26) 

is the Legendre transform4 of the singularity spectrum D , i.e.: 

 * ( ) inf ( )f f
R

D q q D





  . (27) 

If the function )(fD   is convex (as it is generally assumed in the literature), 

then one can obtain fD  by performing Legendre transform on *
fD . The func-

tion *
fD  can be estimated from sample moments. One has to calculate power 

variation (10) for multiple of sampling times h  and then to use the regression: 
*ln ( , ) ( ) lnfV q h C qD h  . (28) 

This method however is not stable numerically and some better methods were 
proposed. Most of them use wavelet transform. The review of them can be 
found in Turiel, Pérez-Vincente, Grazzini (2006) or Oświęcimek (2005).  
In our research we have used method based on “modulus maxima” of wavelet 
coefficients, implemented in Matlab package Fraclab. The method was pro-

                                                 
4 Lagendre transform is well-established method of convex analysis. See for example Rocke-

faller (1970), ch. 26. 

α 

Df(α) 

1/β 

1 



Paweł Kliber 182 

posed by Jaffard (1997a) and very good description can be found in Mallat 
(2003, ch. 6). 

 
Figure 5.  Estimated singularity spectrum for PKN Orlen 

 The Figure 5 shows the results for one stock (PKN Orlen). The estimated 
singularity spectra for all other instrument revealed very similar pattern, so we 
omit them. One should also stress that the results bears a great resemblance to 
the results of Cont and Tankov (2004) for the stock from US market. The graph 
does not star at the origin, which is inconsistent with typical shape depicted on 
the Figure 4. This however can be due to numerical inaccuracy. The main fea-
ture of the graph is the value at which the function reaches its peak, as it is the 
reciprocal of Blumenthal-Getoor index.  For all instrument in the sample the 
peak values lie in the interval [0.6, 0.8], which means that the indexes   are 
between 1.2 and 1.8. 

Conclusions 

The estimation of the Blumenthal-Getoor index is a complicated problem. We 
have used three different methods and as one can see the results are in many 
cases inconsistent.  

 As for the less-liquid stocks, according to the activity signature method the 
price process is driven by compound Poisson process, while the two other 
methods reveal positive value of Blumenthal-Getoor index. This inconsistency 
may be due to the fact of low liquidity of these stocks. There were days when 
the prices did not change for several hours. The estimators we have used take 
advantage of limit properties of price changes as time scale tends to 0. It is thus 
dubious if they gave correct estimators for such illiquid stocks.  

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.2

0.4

0.6

0.8

1



Jumps Activity and Singularity Spectra for Instruments in the Polish Financial Market 183

 As for the liquid stocks all three estimators gave similar results. Moreover 
the results are consistent with similar results for other markets (  at the level 
about 1.5). According to the first two methods (activity signature and threshold 
estimator) the indexes are continuous processes, while the third method (singu-
larity spectrum) revealed 2  . Probably in fact the former result holds, as the 
method of singularity spectrum is the most prone to numerical errors. 

 The results for currencies and futures contract are ambiguous. This can be 
a result of active process of jumps with Blumenthal-Getoor index close to 2.  
As it was pointed out by Zhang (2007) the jump process is then hardly distin-
guishable from continuous diffusion. 

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Aktywność skoków i spectrum osobliwości dla instrumentów  
z polskiego rynku finansowego 

Z a r y s  t r e ś c i. W artykule podejmujemy próbę oszacowania aktywności skoków w procesach 
cen kilku instrumentów z polskiego rynku finansowego. Jako miarę aktywności skoków przyjmu-
jemy indeks β Blumenthala-Getoora dla procesów Lévy’ego. Pozwala nam to na rozróżnienie 
procesów charakteryzujących się rzadkimi i dużymi skokami i procesów o nieskończonej aktyw-
ności procesu skoków. Aktywność skoków szacujemy trzema różnymi metodami. Wykorzystu-
jemy wykresy podpisu aktywności (activity signature plots) do zbadania typu procesu. Następnie 
korzystamy z estymatora Aït-Sahalii i Jacod, opartego na liczbie przekroczeń, do oszacowania 
wartości indeksu  . Wreszcie korzystamy ze spektrum ciągłości oraz z odpowiednich twierdzeń 

na temat przebiegu tej funkcji dla procesów Lévy’ego z różnymi wartościami indeksu  . 

S ł o w a  k l u c z o w e: wykładnicze modele Lévy’ego, indeks Blumenthala-Getoora, spektrum 
osobliwości.  


