Eur. Phys. J. B (2014) 87: 129
DOI: 10.1140/ epjb/ e2014-50064-x
Regular Article
THE EUROPEAN
PHYSICAL JOURNAL B
The origins of multifractality in financial time series
and t he eff ect of ext reme event s
Elena Green
1,a
, William Hanan
1
, and Daniel Heff ernan
1,2
1
Department of Mat hematical P hysics, National University of Ireland Maynoot h, Maynoot h, Co. Kildare, Ireland
2
School of Theoretical Physics, Dublin Institute for Advanced Studies, 4 Dublin, Ireland
Received 27 J anuary 2014 / Received in final form 31 March 2014
P ublished online 11 J une 2014 –
c
EDP Sciences, Societ `a Italiana di Fisica, Springer-Verlag 2014
Abstract. This paper presents the results of multifractal testing of two sets of financial data: daily data
of the Dow Jones Industrial Average (DJIA) index and minutely data of the Euro Stoxx 50 index. Where
multifractal scaling is found, the spectrum of scaling exponents is calculated via Multifractal Detrended
Fluctuation Analysis. In both cases, further investigations reveal that the temporal correlations in the data
are a more significant source of the multifractal scaling than are the distributions of the returns. It is also
shown that the extreme events which make up the heavy tails of the distribution of the Euro Stoxx 50
log returns distort the scaling in the data set . T he most ext reme event s are inimical to t he scaling regime.
T his result is in contrast t o previous findings that extreme event s cont ribut e t o multifractality.
1 Introduction
Mult ifract al analysis has proved to be a valuable method
of capturing the underlying scaling structure present in
many types of systems via generalised dimensions [1]and
f (α) spectra [2]. These systems include diff usion limited
aggregation [3–5], fluid flow through random porous me-
dia [6], atomic spectra of rare-earth elements [7], clu st er-
cluster aggregation [8] and turbulent flow [9]. In phys-
iology, multifractal structures have been found in heart
rate variability [10] and brain dynamics [11], an d mult i-
fract al analysis has been helpful in distinguishing between
healthy and pathological patients [12]. Multifractal mea-
sures have also been found in man-made phenomena such
as the Internet [13], a r t [14] and the stock market [15–17].
The concept of multifractality was first int roduced
in the context of turbulence. It was soon applied to fi-
nance because of its heavy tails and long-term depen-
dence. These two feat ures are also argued t o be present in
financial data [18,19].
Performing multifractal analysis helps to increase our
knowledge about the financial system and further charac-
terise it. Many studies have found multifractal scaling in
financial data [20–23]. An understanding of this multifrac-
tal structure can enable deeper understanding of the dy-
namics of financial markets. If it is found to be a universal
feat ure of financial data, it provides an additional bench-
mark by which to measure the fitness of financial models.
Thisinturncanhelpinthedesignofwellperforming
portfolios and in risk management [17].
a
e-mail: elena.s.green@nuim.ie
The Multifractal Model of Asset Returns (MMAR)
was introduced by Mandelbrot et al. [24]asanexpla-
nation of the volatility clusters in financial data and to
include “outliers”, large deviations which make up the
fat tails of the return distribution. The MMAR was pre-
sented as an alternative to Autoregressive Conditional
Heteroscedasticity (ARCH) models which were introduced
by Engle [25] to account for volatility clust ering. The
MMAR incorporates fat tails, fractional Brownian motion
B
H
1
and t he concept of “trading t ime” being distinct from
physical time [24].
The main assumpt ion of the MMAR is that t he dis-
tinct trading time warps the financial time series into a
mult ifractal struct ure. It t akes t he multifractality of the
financial time series as a given. It also rejects the concept
of outliers, insisting that even the most extreme events
should be account ed for by a decent model. The results
presented in this paper add credence to the assumption of
mult ifract ality as a stylised fact s of financial data. How-
ever they also cast doubt on the inclusion of the most
extreme events which was advocated by Mandelbrot and
others [19].
Two distinct empirical data sets are examined in this
paper. They are distinct in location and time scale. One is
an American index wit h prices recorded daily (Dow Jones
Industrial Average (DJIA)) and the other is a European
index with prices recorded each minute (Euro St oxx 50).
The test for mult ifractality is carried out on the log returns
1
Where Brownian mot ion has Hurst exponent H =
1
/ 2, B
H
has Hurst exponent H ,0<H<1. H<
1
/ 2 for an ant ipersis-
tent process, H>
1
/ 2 for a persistent process. Brownian motion
with H =
1
/ 2 has no memory.
Page 2 of 9 Eur. Phys. J. B (2014) 87: 129
which were construct ed from the price t ime series by:
Z (t) = log(S(t + ∆ t)) − log(S(t))
where S(t) is the price at time t. ∆ t is one day for DJ IA
and one minute for Euro Stoxx 50.
Themethodusedtofind t he scaling in t he data is Mul-
tifractal Detrended Fluctuation Analysis (MF-DFA) [26].
The data and method are further described below.
The rest of this paper is laid out in the following way:
Section 2 outlines the method used to uncover the multi-
fractal structure of the data. Section 3 describes the em-
pirical data and how the parameters of MF-DFA were set.
Section 4 presents the results of t he analysis. In Section 5
some further analysis is conducted to find the source of the
multifractal structures found in the data. Finally Section 6
contains a summary and conclusions.
2 M ult ifractal detrended fl uctuat ion analysis
There are a number of numerical methods by which to
find the multifractal spectrum of time series. Two of
the most well-known are the Wavelet Transform Modulus
Maxima (WTMM) method [27,28]andMultifractalDe-
trended Fluctuation Analysis (MF-DFA) [26]. It has been
shown that for data where the true fractal structure is
unknown, MF-DFA is the recommended method of these
two, showing less bias and being less likely to give a false
positive result [29–31]. This is t he method used in this
paper.
MF-DFA is well suited t o time series analysis because
it is designed for data of a finite length N , without re-
quiring an N →∞ approximation for validity [26]. Also
this method treats the data simply as a one-dimensional
line and assigns new values to each portion of the time se-
ries. T his deals wit h the dat a having direct ion-dependent
scaling properties and t he nonequivalence of the t ime and
value axes [26]. The assigned values are then assessed for
mult ifractality.
The method involves the following steps, beginning
with a disaggregated t ime series X such as a set of
financial log returns.
1. Transform X into its mean-reduced cumulative sums
Y , Y
j
=
j
i= 1
X
i
−
¯
X
. This new data set is aggre-
gated, resembling a random walk rather than a noise
series, and has mean 0.
2. Starting from the beginning, divide Y into non-
overlapping segments of length s.Sinces may not
divide evenly into N , make another set of segments
starting at the end of the data and coming back so
that no piece of the data is left out. This results in
2[
N
/ s]= 2N
s
boxes covering the entire data set. Find
the least-squares polynomial fit y
v
of order m to the
data in each segment v =1,...,2N
s
.
3. Find the root-mean-square error or fluctuation be-
tween the fit and the data in each segment. This is
the value F
2
(v, s)ofsegmentv of size s;
F
2
(v, s)=
1
s
s
i= 1
(Y [( v − 1) s + i] − y
v
[i ])
2
for v =1,...,N
s
and
F
2
(v, s)=
1
s
s
i= 1
(Y [N − (v − N
s
)s + i ] ...
...− y
v
[i ])
2
for v = N
s
+1,...,2N
s
.
4. Introduce a parameter q.Findtheqth order variance
F
q
for a range of both positive and negative q for each
segment size s.
F
q
(s)=
1
2N
s
2N
s
v= 1
F
2
(v, s)
q
/ 2
1
/ q
.
For q = 0, use the quenched average F
0
(s)=
exp[
1
4N
s
2N
s
v= 1
ln(F
2
(v, s))].
5. Repeat steps 2, 3 and 4 for diff erent segment lengt hs s,
finding a new set of values F
q
(s) in each case.
6. For each value of q,plotF
q
(s)versuss on a doubly
logarithmic scaled graph and find the least-squares lin-
ear fit t o each curve. If an appropriate linear region
(more than one order of magnitude of s) is found for
all values of q, it can be concluded t hat there is scaling
in the dat a and the slopes h(q) can be calculated. If
h(q) varies with q, one can conclude t hat t he scaling
is multifractal.
7. Find the multifractal exponent τ (q),
τ (q)= qh(q) − 1 − qH
where H
= h(1) − 1 is called the nonconservat ion
parameter
2
and proceed t o the f (α)spectrumviathe
Legendre transforms:
α(q)=
dτ (q)
dq
f (α(q)) = α(q)q −τ(q).
Aplotoff (α)versusα is the multifractal spectrum
for the time series data X .
Mult ifractality has been reported in cases where there
is only t he spurious scaling which can arise in non- or
monofractal t ime series [31,34–36], and so caution is re-
quired. It is crit ically important t o check t he linearity
of the logarithmic plots as described in Step 6. Plotting
the slope of the line over a moving window should reveal
roughly constant slope over the length of the line before
linearity is accepted. Oscillation about a straight line is
2
This is an adjustment to the original definition of τ given
by Kantelhardt et al. [26], τ (q)= qh(q) − 1. It account s for
the fact that F
2
(v, s) is not strictly speaking a measure on the
time series Y . For further det ails, see references [32,33].
Eur. Phys. J. B (2014) 87: 129 Page 3 of 9
Table 1. Summary statistics for the log return data examined in this paper for multifractal properties. N is the sample size of
the data,
μ is the sample mean and σ the sample standard deviation. H is the est imat ed Hurst exponent of t he sample.
Data ∆ tN Min Max μ σ Skewness Kurtosis H
DJ IA 1 day 20922 − 0.2563 0.1427 1.89 × 10
− 4
0.0117 − 0.5931 27.2784 0.5146
Euro Stoxx 50 1 min 109545 − 0.0935 0.0610 − 4.5257 × 10
− 6
0.0011 − 2.1397 1.0335 × 10
3
0.448
60
40
20
0
-20
-40
-60
-80
-100
0
12
3
4
56
7
8910
x10
4
normalised log returns
minutes
DJIA Euro Stoxx 50
15
10
-5
-10
-15
-20
-25
5
0
00.20.40.60.8
11.21.4
1.6 1.8
2
days
x10
4
Fig. 1. Graph of the daily log return data of the DJIA and of the minutely log return data of Euro Stoxx 50 whose multifractal
properties are examined in this paper. The log returns are given in units of standard deviation for ease of comparison in this
figure.
to be expect ed as these are statistical fractals. However,
if t here is no significant linear region revealed by t he lo-
cal slopes, we cannot conclude t hat t here is multifractal
scaling in t he dat a.
Finite-size eff ects are also an important considera-
tion [35]. Short monofractal time series can appear mult i-
fractal due t o linear correlat ions. Since t he log return data
considered here have negligible linear correlations (Hurst
exponent H ≈
1
/ 2,seeTab.1), t his is not a concern for
our analysis.
Mult iscaling Mult ifractal Analysis [37], a n ext ension
to t he MF-DFA method, has recently been recommended
to pick up information from any cross-overs that might be
in the dat a. A crossover is a point where the slopes change
on the graph of log(F
q
)vs.log(s).Sinceweseenosuch
crossover points in our data, there is no need for this extra
analysis.
3 Data and implementation
The first data examined is the daily log returns of the
DJIA from 1928 to 2012 which cont ains 20 922 point s.
This is a weighted average of the prices of 30 companies
based in t he Unit ed States. Its normalised form is shown in
Figure 1. The dramatic downturn of late 2007 and 2008 is
included in this data set and the major “Black Monday”
crash of Oct ober 19th 1987 is obvious at approximately
1.5 × 10
4
days.
The Dow Jones Euro Stoxx 50 was also examined
and the normalised log returns for the time period of
interest are also shown in Figure 1.Thisisanindexof
50 Blue-chip sector leaders from 12 Eurozone countries
which was launched in 1998. The data is minutely and
runs for a year, from the start of May 2008 until the end
of April 2009. T here are 109 545 points in this time series.
The high volatility that can be seen in the middle of t he
time series corresponds to t he time around the Lehman
Brother’s collapse in Sept ember 2008.
TheMF-DFAmethodwasappliedtobothlogreturn
time series. Summary st atistics for the log return data of
DJIA and Euro Stoxx 50 are presented in Table 1.The
exclusion of overnight returns in the minutely time series
made no diff erence to the results of the analysis and so
they have been retained. All time outside of trading hours
has been omitt ed.
For the implementation of MF-DFA, certain parame-
ters have t o be chosen. Both data set s were detrended by
order m = 1 polynomials as this led to the best scaling
results. The length scale s takes small steps from a min-
imum of 10 to a maximum of
N
/ 4 = N
4
,whereN is the
length of the time series. T his means that at the largest
scale there are 8 boxes since there are 2N
s
boxes for each s.
This range of scales is proposed by Kantelhardt et al. [26].
A wide range of 1001 equally spaced values of the pa-
rameter q was chosen, with q ∈ [− 50, 50]. This is a very
wide range in comparison with other studies [12,20,26,38]
where it is standard to use q ∈ [− 5, 5]. However, for
smaller ranges of q, less of the multifractal spectrum is
revealed. It is found that f (α) ≈ 0fortheexamineddata
as q → ± 50, and this captures the full spectrum.
4Results
The plots of F
q
(s)versuss on a doubly logarithmic scale
for t he DJ IA data for selected values of q are shown in Fig-
ure 2a. Although 1001 values of q were used in the analysis,
it is not practical t o show all of t hem on t he graph. The
segment size s t akes 59 values from 10 to 5230. By check-
ing the local slopes of these lines (Fig. 2b) it is possible to
identify a scaling region over more than two orders of mag-
nitude from s =10tos = 2000. This region of scaling was
then used to construct the multifractal spectrum which is
displayed in Figure 3.
Page 4 of 9 Eur. Phys. J. B (2014) 87: 129
s
log(F
q
)
0
-1
-2
-3
10 100 1000 2000
q = 50
q = 5
q = 0
q = -5
q = -50
(a)
s
local slopes of log(F
q
)
1
0.8
0.6
0.4
0.2
0
10 100 1000 2000 10000
q = 50
q = 5
q = 0
q = -5
q = -50
(b)
Fig. 2. DJIA: (a) Graph of log(F
q
)versuslog(s) for selected
values of q as shown on the graph. (b) Graph of the local slopes
of the lines in (a) calculat ed over 15 p oint s for t he same values
of q. The slopes remain reasonably constant for s ∈ [10, 2000].
1
0.8
0.6
0.4
0.2
0
0.8 0.9
11.11.2
f
(
α
)
α
F ig. 3. Graph of t he multifractal spect rum, f (α)versusα,for
DJIA calculated for the length scales s ∈ [10, 2000] and with
q ∈ [− 50, 50].
The results of the initial check for scaling for the Euro
St oxx 50 dat a are shown in F igure 4. It is not obvious
whether or not t here is scaling in this data. The slopes are
not of the quality of those for DJIA observed in Figure 2b.
The mult ifractality is less certain in this case. It could be
argued that the local slopes in Figure 4aarenotconstant
over a suffi cient range of s and so indicate a lack of scaling
in the Euro Stoxx 50 data. In this case, this data could be
presented as a counterexample to the stylised fact of the
presence of multifractality in financial return data [39].
It could also be argued that scaling is present over
more than two orders of magnitude; for 65 s≤ 10 000.
It breaks down for small segment sizes (s65) when q is
negative. T he abrupt change in F
q
(s) can be explained by
the presence of a sect ion of consecutive zeroes in the log
returns. Since F is a measure of the distance of the data
10 100 1000 10000
s
-10
-8
-6
-4
-2
0
-12
-14
-16
q = 50
q = 5
q = 0
q = -5
q = -50
log(F
q
)
(a)
q = 50
q = 5
q = 0
q = -5
q = -50
1
0.8
0.6
0.4
0.2
0
10 100 1000 10000
q
local slopes of log(F )
s
(b)
F ig. 4. Euro Stoxx 50: (a) Graph of log(F
q
)versuslog(s)for
selected values of q as shown on the graph. (b) Graph of the
local slopes of t he lines in (a) calculated over 15 points for t he
same values of q.
f
(
α
)
1
0.8
0.6
0.4
0.2
0
0.7 0.8 0.9
11.11.2
1.3
α
F ig. 5. Graph of t he multifractal spect rum, f (α)versusα,for
Euro Stoxx 50 calculat ed for the length scales s ∈ [65, 10 000]
and with q ∈ [− 50, 50].
in any segment from a linear fit, when a segment ν lies
within t his interval of zeroes, F (ν) is close t o zero. T he
smallest F dominates in F
q
when q< 0 which explains
the drop in log(F
q
)ass decreases for q< 0.
The multifractal spectrum for the range 65 s≤
10 000 is shown in Figure 5. The left side of the spec-
trum is stretched out and f (α) < 0forα 0.63. The
left side represents the areas of high F
q
and so this is ev-
idence of poor scaling, and possibly even a breakdown in
scaling, of the most volatile segments. As we shall show
in Sect ion 5.2, it is t he extreme return event s which are
responsible for these phenomena.
The fact that Figure 4 seems to indicat e a lack of scal-
ing and yet the spectrum in Figure 5 can still be produced
shows that real caution is required when conducting mul-
tifractal analysis. A wide smoot h spectrum does not imply
that the data actually has multifractal scaling.
Eur. Phys. J. B (2014) 87: 129 Page 5 of 9
Table 2. Summary of the main results of MF-DFA on the daily DJIA and minutely Euro Stoxx 50 data for a range of
values of q ∈ [− 50, 50]. The truncated data has extreme events replaced with smaller ones. It is discussed in Section 5.2.Here
∆α = α
max
−α
min
.
Data f (− 50) α(− 50) f (0) α(0) f (50) α(50) ∆α
DJ IA 0.049882 1.2124 1 1.0126 0.058373 0.78155 0.43087
Euro Stoxx 50 − 0.023382 1.2437 1 1.0184 − 0.0981 0.6226 0.62162
Truncated Euro Stoxx 50, c =15 − 0.021838 1.265 1 1.0169 0.017184 0.78068 0.48431
The spectra in t his paper seem shifted t o t he right
in comparison to those in the lit erature [20,23,38]. T h is
can be accounted for by the updated definition of τ (q)in
Step 7 in Section 2. A summary of the results of MF-DFA
for both data sets is contained in Table 2.
5 The origins of multifractality
It is generally accepted that there are two possible sources
of multifractal scaling in t ime series data [26]. It cou ld b e
predominantly due to (1) the long-term correlations of
small and large fluctuations or (2) the data being drawn
from a heavy-tailed probability dist ribution. Both of these
influences can individually be removed from the data to
reveal what impact they have on the multifractality of the
time series.
Other recent work has shown that multifractality can
be viewed as the result of the T weedie Convergence T heo-
rem, similarly to how Gaussian noise can be seen as the re-
sult of the Cent ral Limit T heorem [40,41]. However, since
financial time series are not sequences ofindependent iden-
tically distributed random variables, the convergence the-
orem does not apply. Here we will examine the traditional
sources: correlations and the shape of the distribution.
5.1 Source of scaling – correlations
A simple way to check if correlations in the data pro-
duce any scaling is to shuffl e the data as suggested by
Kantelhardt et al. [26]. Shuffl ing removes t ime correlations
and any scaling that remains must be due to the proba-
bility dist ribut ion from which the dat a is drawn. T he dis-
tribution of the values is not aff ect ed by reordering the
series.
Any individual shuffl e may st ill contain some corre-
lations, so to be sure to completely rid the data of all
correlations, both the DJIA and the Euro Stoxx 50 data
were shuffl ed 100 t imes, each random permut ation be-
ginning with a new random number generator seed in
MatLab. The function F
q
was found for each of the shuf-
fled data set s. These were then averaged to find
F
q
(s)=
1
100
100
i= 1
F
q,i
(s), where the index i ident ifies the shuffl ed
data sequence. The doubly logarithmic plots of
F
q
(s)ver-
sus s for diff erent q were then checked for linearity. T he
results are shown in Figures 6 and 7. The same analy-
sis was conducted wit h the quenched average,
log(F
q
(s)),
with very similar results.
For both time series, t here is no significant linear re-
gion in the plots of log(
F
q
(s)) versus log(s). Thus we do
(a)
(b)
Fig. 6. Shuffl ed DJIA data: (a) Graph of the log of the av-
eraged scaling function, log(
F
q
), versus the log of the scale,
log(s), for selected values of q as shown on the graph. (b) Graph
of the local slopes of the lines in (a) calculated over 15 points
for t he sam e values of q.
not have the rationale to proceed t o calculate h(q)and
must instead conclude that multifract al scaling is absent
in t hese shuffl ed data set s.
Other studies [16,20,22,42,43] have found multifractal
scaling in shuffl ed financial dat a. However, as no explicit
invest igation of the logarithmic plots and t heir local slopes
was conducted, the conclusion that multifractal scaling is
present is not justified.
Diff erent degrees of shuffl ing were also employed so
that correlations of diff erent length scales can be re-
moved [31]. Rather than reordering every point in the
data, the data was divided int o intervals of length l.
Then each set of l adjoining point s were kept t oget her
while t he order of the intervals was shuffl ed. T his helps to
reveal how robust the scaling is to the presence of tempo-
ral correlations.
The result of this analysis for the DJIA data is
shown in Figure 8. Intervals of lengths l =10, 50,
100, 500, 1000, and 5000 were kept int act and only the