Physica A 390 (2011) 3815–3825

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A multifractal detrended fluctuation analysis of trading behavior ofindividual and institutional traders in Tehran stock marketMeysam Bolgorian ∗, Reza RaeiFaculty of management, Department of financial management, University of Tehran, Tehran, 14155, Iran

a r t i c l e i n f o

Article history:Received 22 May 2010Received in revised form 1 March 2011Available online 30 June 2011

Keywords:MultifractalityIndividual and institutional investorsTrading volume

a b s t r a c t

Employing the multifractal detrended fluctuation analysis (MF-DFA), the multifractalproperties of trading behavior of individual and institutional traders in the Tehran StockExchange (TSE) are numerically investigated. Using daily trading volume time seriesof these two categories of traders, the scaling exponents, generalized Hurst exponents,generalized fractal dimensions and singularity spectrum are derived. Furthermore, twomain sources of multifractality, i.e. temporal correlations and fat-tailed probabilitydistributions are also examined. We also compare our results with data of S&P 500. Resultsof this paper suggest that for both classes of investors in TSE, multifractality is mainly dueto long-range correlation while for S&P 500, the fat-tailed probability distribution is themain source of multifractality.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Econophysics is a newscience evolved from the combination of other sciences includingphysics,mathematics, economicsand finance. Despite the significant advances, because of the vital role of human factor in social sciences like economics,econophysics is growing with the faster step than before. Many signals generated by economic systems provide interestingstructures which can be characterized in terms of the theory of multifractals. The concept of ‘‘fractal world’’ was proposedby Mandelbrot in 1980s and was based on scale-invariant statistics with power-law correlations [1]. This new theory wasprogressively developed in recent years, and finally it brought a more general concept of multiscaling. It allows one to studythe global and local behavior of a singular measure or in other words, the mono- and multifractal properties of a system. Ineconomy, multifractality is one of the well-known stylized facts which characterizes non-trivial properties of financial timeseries [2].

Recently, the detrended fluctuation analysis (DFA) method [3] has become a widely used technique for thecharacterization of fractal scaling properties and the detection of long-range correlations in noisy, nonstationary timeseries [4–7]. It has successfully been applied to a variety of fields such as DNA sequences, heart rate dynamics, neuronspiking, human gait, long-time weather data, studying the cloud structure, geology, ethnology, economics time series, andsolid state physics [8–18]. One reason to employ the DFA method is to avoid spurious detection of correlations that areartifacts of nonstationarities in the time series. Many records do not exhibit a simple monofractal scaling behavior, whichcan be accounted for by a single scaling exponent [19].

Themultifractal analysis in its simplest form is based on the standard partition functionmultifractal formalism,which hasbeen devised for themultifractal characterization of normalized, stationarymeasurements. But this simple framework doesnot give reliable results for nonstationary time series that are affected by trends or that cannot be normalized. This is whileexperimental data often are affected by nonstationarities like trends, which have to be well distinguished from the intrinsic

∗ Corresponding author. Tel.: +98 21 44425592; fax: +98 21 88776122.E-mail addresses:meysam_ng2009@yahoo.com, nyroglob@yahoo.ie (M. Bolgorian).

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.06.017

3816 M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825

fluctuations of the system in order to find the correct scaling behavior of the fluctuations.Multifractal Detrended FluctuationAnalysis (MF-DFA) is a well-establishedmethod for determining the scaling behavior of noisy data in the presence of trendswithout knowing their origin and shape [20].

Many researchers applied MF-DFA technique for economically special financial data but generally most of theapplications of this technique in financial markets have focussed on the stock price time series [21–24]. This is because it iswidely believed that in financial markets the performance of a company is compactly characterized by a single number, thestock price,which results froma large number of interactions between differentmarket participants. Despite this rationality,it should be noted that using stock price returns prevents extracting information from separate forces acting in financialmarkets, say individual and institutional investors. Individual traders that usually are not professional investors often haveshort investment horizon. This group of traders more rely on the trends, i.e. if they believe that there is no indication ofa trend reversal they will buy the asset which leads to increase in price of that asset. This is while institutional investorswhich usually are called fundamental investors watch deviations from an equilibrium level of the price which implied bya fundamental model. If the prices are above (below) the equilibrium price the asset is sold (bought) and leads to a decline(rise) in the prices. The importance of the interplay of these two classes of investors has been stressed by several recentworks to be essential in order to retrieve the important stylized facts of stock market price statistics [25–33]. What happensto the price of asset depends on the net effect of the forces supply and demand by institutional and individual investors.

The main purpose of this paper is to characterize the complex behavior of time series of trading volume of two group ofinvestors, i.e. individual and institutional traders in Tehran stock market through the computation of the signal parameters– scaling exponents – which quantifies the correlation exponents and multifractality of the signal.

The rest of paper is organized as follows. In Section 2, MF-DFAmethod is described and other related theoretical conceptsare reviewed. In Section 3, data are provided and numerical results are presented. Finally, conclusion is given in Section 4.

2. Method description

The generalized Multifractal Detrended Fluctuation Analysis (MF-DFA) procedure introduced by Kantelhardt et al.consists of five steps. Here we briefly discuss these steps. For more detailed discussion, please see Ref. [19]. The first threesteps are essentially identical to the conventional DFA Procedure. Let us assume that xk is a series of length N , and this seriesis of compact support, i.e. xk = 0 for an insignificant fraction of values only:

Step 1: Determine the ‘‘profile’’

Y (i) ≡

i−k=1

[xk − ⟨x⟩]. (1)

Subtraction of ⟨x⟩ is not compulsory because it would be eliminated by the latter detrending in the third step.Step 2: Divide the profile Y (i) into Ns ≡

Ns non-overlapping segments of equal length s. Since the length N of the series is

not often amultiple of considered time scale s, short part at the end of the profile may remain. In order to disregard this partof the series, the same procedure is repeated starting from the end of series. Thus, 2Ns segments are obtained eventually.

Step 3: Calculate the local trend for each of the 2Ns segments by a least-square fit of the series. Then determine thevariance

F 2(s, ν) ≡1s

s−i=1

{Y [(ν − 1)s + i] − yν(i)}2 (2)

for each segment ν, ν = 1, 2, . . . ,Ns and

F 2(s, ν) ≡1s

s−i=1

{Y [N − (ν − Ns)s + i] − yν(i)}2 (3)

for ν = Ns, . . . , 2Ns. Here, yν(i) is the fitting polynomial in segment ν. Linear, quadratic, cubic or higher order polynomialscan be used for fitting.

Step 4: Average over all segments to obtain the qth order fluctuation function

Fq(s) ≡

1

2Ns

2Ns−ν=1

[F 2(s, ν)]q/2

1/q

. (4)

We are interested in how the generalized q dependent fluctuation function Fq(s) depend on the time scale s for differentvalues of q. Hence wemust repeat Steps 2–4 for several time scales s. It is apparent that Fq(s) will increase with increasing s.

Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log–log plots of Fq(s) versus s for eachvalue of q. If the series xi are long-range power-law correlated, Fq(s) increases, for large values of s, as a power-law

Fq(s) ∼ sh(q). (5)In general, the exponent h(q) may depend on q. For stationary time series, h(2) is identical to the well-known Hurst

exponentH . Thus we call the function h(q) generalized Hurst exponent. For monofractal times series with compact support,

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10–2

10–1

10–4

10–3

10–2

10–1

100

Pr(

X≥

x)

Normalized returns

Fig. 1. Cumulative Distribution Function (CDF) of TSE returns. The tail power-law exponent is µ ≃ 3.38. Red dashed line represents the best power-lawfitting to the empirical data.

h(q) is independent of q, since the scaling behavior of the variances F 2(s, ν) is identical for all segments ν, and the averagingprocedure in Eq. (4) will give just this identical scaling behavior for all values of q. The family of the exponents h(q) describethe scaling of the qth order fluctuation function.

The h(q) obtained from MF-DFA is related to the Renyi exponent τ(q) by

qh(q) = τ(q) + 1. (6)

Therefore, another way to characterize a multifractal series is the singularity spectrum f (α) defined by Feder [34]

α = h(q) + qh′(q) (7)

and

f (α) = q[α − h(q)] + 1 (8)

where h′(q) stands for the derivative of h(q) with respect to q. α is the Holder exponent or singularity strength whichcharacterizes the singularities in a time series. The singularity spectrum f (α) describes the singularity content of the timeseries. Finally, it must be noted that h(q) is different from the generalized multifractal dimensions

D(q) ≡τ(q)q − 1

=qh(q) − 1q − 1

(9)

that are used instead of τ(q) in some papers. While h(q) is independent of q for a monofractal time series with compactsupport, D(q) depends on q in that case [22].

3. Data analysis and results

The data which is analyzed in this study include the time series of individual and institutional daily trading volumelogarithmic variations (i.e. ln(V (t + 1)/V (t)) for the time period 1998–2009 which are obtained from www.irbourse.com).We distinguished between these traders by assigning the financial transactions done by individual investors and thosecarried out by large funds and investment companies in Tehran Stock Exchange (TSE) to individual and institutional tradersrespectively. Data related to the S&P 500 trading volume is downloaded from the www.finance.yahoo.com for the period1980–2009.

In Fig. 1, Cumulative Distribution Function (CDF) of Tehran Stock Exchange price index (TEPIX) returns is presented. InFig. 2 CDF of TSE individual and institutional trading and S&P 500 trading volume in log–log plots are presented. In first stepwe are to determine whether return and volume in TSE scale the same or not. Previous studies analyzing the distributionfunction of returns for the 1000 largest US stocks and several major international indices have shown that

P(|r| > x) ∼ x−µ. (10)

The tail distribution of returns and trading volumes has been analyzed in many studies which use an ever increasingnumber of data points. Based on these studies it becomes a stylized fact that tail exponent for market returns is µret ≃ 3[35–39]. Plerou et al. analyzingmajor financialmarkets reported that the number of trades in thesemarkets display a power-law asymptotic behavior with tail exponent of µN ≃ 3 [40,41]. Gopikrishnan et al. find that the size of individual trades forthe 1000 largest US stocks are also power-law distributed with the tail scaling exponent of µind = 1.53 ± 0.07 [42].

3818 M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825

100

101

10 –4

10 –3

10 –2

10 –1

100

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

10 –4

10 –3

10 –2

10 –1

100

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

10 –4

10 –3

10 –2

10 –1

100

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

Normalized log variations

100

101

Normalized log variations

100

101

Normalized log variations

a

b

c

Fig. 2. Cumulative Distribution Function (CDF) of TSE (a) individual (µ ≃ 3.68) (b) institutional and (c) S&P 500 trading volume log variation (µ ≃ 3.65).Red dashed lines represent the best power-law fitting to the empirical data.

Power-law regression fit yields estimate of the tail exponents for µret ≃ 3.38, µind ≃ 3.68 and µS&P ≃ 3.65 respectivelyfor returns and individual trading volume of TSE and trading volume of S&P 500. One can easily see that the CDF ofinstitutional trading volume (Fig. 2(b)) does not show the power-law tail. Although tail scaling exponent of individual tradesin Tehran Stock Exchange (TSE) is somewhat different from what obtained in US market, we can see that in TSE individualtrading volume and stock market return approximately scale the same.

In Figs. 3(a)–5(a) the MF-DFA2 fluctuation functions Fq(s) for various q’s for the time series of the total daily tradingvolume of individual and institutional traders in TSE and also trading volume of S&P 500 are shown. As can be seen, thereare crossovers for negative q values in the range 4 < s < 10 for both class of investors. It is also seen that for s = 400 thedata of individual traders shows strange oscillations and deviates from the initial scaling behavior (Fig. (a)). This indicates

M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825 3819

100

101

102

103

10–2

10–1

100

101

102

s

100

101

102

103

s

100

101

102

103

s

Fq(s

)

10–2

10–1

100

101

102

Fq(s

)

10–2

10–1

100

101

102

Fq(s

)q=6q=4q=2q=–2q=–4q=–6

q = 6q = 4q = 2q = –2q = –4q = –6

q = 6q = 4q = 2q = –2q = –4q = –6

a

b

c

Fig. 3. TheMF-DFA2 functions Fq(s) of individual trading volume log variation time series versus the time scale s in log–log plot for (a) original, (b) shuffledand (c) surrogate data. Vertical arrows roughly specify the range from which the scaling exponents were approximated.

that individual trading tends to lose its memory after a period of about 400 days. This behavior with lower intensificationand longer period exists in institutional trading. The results for S&P 500 trading volume shows a slight crossover in Fq(s)curve around s = 400 for positive q values suggesting a multiscaling behavior.

For analyzing the source of multifractality in time series we have analyzed the modified times series including shuffledand surrogated times series. The main rationale behind this approach is that generally two different types of sources formultifractality in time series are identified: (i) multifractality due to different long-range temporal correlations for smalland large fluctuations, and (ii) multifractality related to the fat-tailed probability distributions of variations. shuffling, and

3820 M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825

100

101

102

103

10–2

10–1

100

101

102

s

100

101

102

103

s

100

101

102

103

s

Fq(s

)

10–2

10–1

100

101

102

Fq(s

)

10–2

10–1

100

101

102

Fq(s

)q=6q=4q=2q=–2q=–4q=–6

q = 6q = 4q = 2q = –2q = –4q = –6

q = 6q = 4q = 2q = –2q = –4q = –6

a

b

c

Fig. 4. The MF-DFA2 functions Fq(s) of institutional trading volume log variation time series versus the time scale s in log–log plot for (a) original,(b) shuffled and (c) surrogate data. Vertical arrows roughly specify the range from which the scaling exponents were approximated.

phase randomization (surrogated data) are the main procedures to find the contributions of two sources of multifractalityand to indicate the multifractality strength. Shuffling preserves the distribution of the variations but destroys any temporalcorrelations. In fact, one can destroy the temporal correlations by randomly shuffling the corresponding time series ofvariations. What then remains are data with exactly the same fluctuation distributions but without any correlation. Onthe other hand, surrogate data is a method for testing the Gaussianity and one can eliminate any sort of nonlinearities inoriginal time series. In fact, the non-Gaussianity of the distributions can be weakened by creating the phase-randomizedsurrogates. The Phase randomization steps are:

M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825 3821

100 101 102 10310–2

10–1

100

101

102

s

100 101 102 103s

100 101 102 103s

F q(s)

10–2

10–1

100

101

102

F q(s)

10–2

10–1

100

101

102

F q(s)

q=6q=4q=2q=–2q=–4q=–6

q=6q=4q=2q=–2q=–4q=–6

q=6q=4q=2q=–2q=–4q=–6

a

c

b

Fig. 5. The MF-DFA2 functions Fq(s) of S&P trading volume log variation time series versus the time scale s in log–log plot for (a) original, (b) shuffled and(c) surrogate data. Vertical arrows roughly specify the range from which the scaling exponents were approximated.

(1) Take discrete Fourier transform of time series.(2) Multiply the discrete Fourier transform of the data by random phases.(3) Perform an inverse Fourier transform to create a phase-randomized surrogates.

Phase randomization preserves the amplitudes of the Fourier transform but randomizing the Fourier phases [43,44].In Fig. 3(b)–(c), Fig. 4(b)–(c) and Fig. 5(b)–(c) theMF-DFA2 fluctuations Fq(s) for various q’s for the shuffled and surrogated

time series for the individual and institutional traders in TSE and trading volume of S&P 500 are shown respectively.

3822 M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825

q

h(q)

OriginalShuffledSurrogate

q

h(q)

OriginalShuffledSurrogate

q

h(q)

OriginalShuffledSurrogate

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–6 –4 –2 0 2 4 6

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–6 –4 –2 0 2 4 6

–6 –4 –2 0 2 4 6

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

a

b

c

Fig. 6. Generalized Hurst exponent, h(q) as a function of q for (a) individual (b) institutional and (c) S&P 500 trading volume log variation time series.Vertical lines are error bars of each time series.

The h(q) spectra have been shown for original, reshuffled and surrogate series in Fig. 6(a) and (b). One can see thatfor both the individual and institutional trading volume time series the q dependence of h(q) for the original time seriesis higher than the two other randomized time series. The main feature of these two plots is that for both time series qdependence of h(q) is lowest for shuffled time series which indicates that the multifractality nature of this time series isdue to long-range correlation. This is while examining the correspondent time series for S&P 500 trading volume 6(c), itis found that multifractality of this time series is more related to the broadness of its Probability Distribution Function(PDF).

M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825 3823

Originalshulffledsurrogate

α

OriginalShuffledSurrogate

α

OriginalShuffledSurrogate

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

f(α

)f(

α)

f(α

)

1

0.5

0.6

0.7

0.8

0.9

1.1

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6α

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

a

b

c

Fig. 7. Singularity spectrum f (α) for (a) individual (b) institutional and (c) S&P 500 trading volume log variation time series.

In order to better analyzing the strength of multifractality for original, reshuffled and surrogate data, the singularityspectra of two time series are shown in Fig. 7. Singularity spectrum for the individual and institutional trading volumetime series is presented in Fig. 7(a) and (b) respectively. There is a clear difference between spectra for the original andthe modified time series for both class of investors. The position of maximum for the original fluctuations is localizedapproximately at αind

max ≃ 1.115 for the individual trading whereas for the institutional trading the maximum is placed atαinsmax ≃ 1.044. The width of the singularity spectra for the individual trading for the original data is ∆αind

org ≃ 0.599 while forshuffled and surrogate time series this value is∆αind

shuf ≃ 0.087 and∆αindsurr ≃ 0.234 respectively. For the institutional trading

time series, on the other hand, the width of the singularity spectrum for original time series is ∆αinsorg ≃ 0.361 whereas for

shuffled and surrogated time series this value is ∆αinsshuf ≃ 0.120 and ∆αins

surr ≃ 0.187 respectively. This indicates that the

3824 M. Bolgorian, R. Raei / Physica A 390 (2011) 3815–3825

q

OriginalShuffledSurrogate

–6 –4 –2 0 2 4 6q

OriginalShuffledSurrogate

q

OriginalShuffledSurrogate

–6 –4 –2 0 2 4 6

τ(q)

τ(q)

τ(q)

–12

–10

–8

–6

–4

–2

0

2

4

6

8

–12

–10

–8

–6

–4

–2

0

2

4

6

–6 –4 –2 0 2 4 6–12

–10

–8

–6

–4

–2

0

2

4

6

8

a

c

b

Fig. 8. Renyi exponent τ(q) for (a) individual (b) institutional and (c) S&P 500 trading volume log variation time series.

main multifractality source in individual and institutional trading is long-range correlation. Doing similar analysis for S&P500, we found that ∆αS&P

org ≃ 0.281, ∆αS&Pshuf ≃ 0.227 and ∆αS&P

surr ≃ 0.161. This demonstrates that multifractality in S&P timeseries is due to a broad probability density function.

In order to study the scaling character of the data, in Fig. 8, the multifractal scaling spectra τ(q) for both time series ofTSE and also trading volume of S&P 500 is shown. Fig. 8(a) and (b) show three curves of τ(q) for the original, shuffled andsurrogate data for the individual and institutional trading volume time series in TSE respectively. It is well-established factthat monofractal time series are associated with a linear plot τ(q)while multifractal ones possess the spectra nonlinear in q.The more the nonlinearity of the spectrum, the stronger the multifractality nature in time series. It can be seen that for bothclass of investors, nonlinearity of τ(q) is much stronger for the original time series in comparison with two other modifiedtime series. Again interestingly for both category of traders in TSE, shuffled time series have lowest nonlinearity of τ(q).Fig. 8(c) shows the same spectra τ(q) for S&P 500 trading volumes.

4. Conclusion

Weapplied theMF-DFA technique to show a difference in the fractal properties of the individual and institutional tradingbehavior fluctuations. Our results suggest that a more persistent behavior and often richer multifractality is associated withthe individual trading volume time series. It is found that main sources of multifractality in both categories of investors inTSE is temporal correlations while for S&P 500, broad probability distribution function plays a more important role.

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