Chaos, Solitons & Fractals 45 (2012) 838–845

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

Time-clustering behavior of sharp fluctuation sequences in Chinesestock markets

Ying Yuan ⇑, Xin-tian Zhuang, Zhi-ying Liu, Wei-qiang HuangSchool of Business Administration, Northeastern University, Shenyang 110819, China

a r t i c l e i n f o

Article history:Received 4 January 2012Accepted 29 February 2012Available online 24 March 2012

0960-0779/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.chaos.2012.02.020

⇑ Corresponding author.E-mail address: yyuan@mail.neu.edu.cn (Y. Yuan

a b s t r a c t

Sharp fluctuations (in particular, extreme fluctuations) of asset prices have a great impacton financial markets and risk management. Therefore, investigating the time dynamics ofsharp fluctuation is a challenge in the financial fields. Using two different representationsof the sharp fluctuations (inter-event times and series of counts), the time clusteringbehavior in the sharp fluctuation sequences of stock markets in China is studied with sev-eral statistical tools, including coefficient of variation, Allan Factor, Fano Factor as well as R/S (rescaled range) analysis. All of the empirical results indicate that the time dynamics ofthe sharp fluctuation sequences can be considered as a fractal process with a high degree oftime-clusterization of the events. It can help us to get a better understanding of the natureand dynamics of sharp fluctuation of stock price in stock markets.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Sharp fluctuations (in particular, extreme fluctuations)of asset prices have a great impact on financial marketsand risk management. Therefore, investigating the timedynamics of sharp fluctuation is a challenge in the financialfields.

Many scholars showed their interests on this issue andfound some meaningful conclusions. Chen et al. [1] ana-lyzed the daily Hang Seng index in the Hong Kong stockmarket. They predicted the future price movements usingtwo kinds of sign sequences as given conditions. One isthe parameter of multifractal spectrum Df based on the in-dexes recorded in every minute, and the other is the vari-ation of the close index Di. Results show that correlationbetween large fluctuations of the close price and the condi-tion in these two methods is strong and some sign se-quences of the parameter Df can be used to predict theprobability of the near future price movements. Muchniket al. [2] studied the long term memory in extreme returnsof financial time series and revealed that the returns exhi-bit pronounced long-term memory. This ‘‘stylized fact’’ can

. All rights reserved.

).

shed further insight on price dynamics that might be usedfor risk estimation. We also studied the sharp fluctuation ofstock price index in Chinese stock market using multifrac-tal spectrum and multifractal detrended fluctuation analy-sis (MF-DFA), respectively, and found that when the stockprice index fluctuates sharply, a strong variability is clearlycharacterized by multifractal parameters and the general-ized Hurst exponents [3,4]. Zhang et al. [5] tried to capturethe fluctuations caused by the extreme event on crude oilprices variation during the analyzed period using anempirical mode decomposition (EMD-based) event analy-sis approach. It was found that this method provided a fea-sible solution to estimate the impact of extreme events oncrude oil prices variation. In addition, many relative re-searches have been performed in order to capture the mainfeatures of extreme fluctuations [6–12], all of these resultsare meaningful and important and can lead to a betterunderstanding of complex stock markets.

However, it is noted that most of the previous researchesconcentrated on the time series analysis of capital price ser-ies and capturing the statistical characteristics of financialtime series, and few focused on the temporal dynamics offinancial sharp fluctuation (or extreme fluctuation) se-quences. In fact, the temporal dynamics of sharp fluctuationsequences are very useful information for understanding

Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845 839

possible causes of sharp fluctuation, improving sharpfluctuation prediction and financial risk management aswell as for improving predictive models. Nevertheless, thetemporal dynamics of sharp fluctuation are still seldomknown because the dynamics of sharp fluctuation resultfrom many complex interaction factors and also factorsinvolving human activities. It is very important and mean-ingful to study the temporal behavior and dynamics charac-teristics of sharp fluctuation of financial markets. Therefore,a further research work will help us to get a better under-standing of the nature of stock price dynamics.

On the other hand, Telesca et al. [13–22] studied thecharacterization of temporal fluctuations in other complexsystems, such as seismic sequences, car accident sequenceand forest fire sequence, etc. The fundamental principle ofthese researches is that extreme event sequence can be as-sumed to be a realization of a point process. The discrete-time process can be derived from the stochastic point pro-cess in two equivalent ways (1) using the inter-event timeseries or (2) forming its relative counting process. In thefirst representation a discrete-time series is formed bythe rule si = ti+1 � ti, where ti indicates the time of eventnumbered by the index i. In the second representation,the time axis is divided into equally spaced contiguouscounting windows of duration T. The duration T of the win-dow is called counting time or timescale. The latter ap-proach considers the extreme events as the events ofinterest and assumes that there is an objective clock forthe timing of the events. The former approach emphasizesthe interspike intervals and uses the event number as anindex of the time [16,17]. This research idea can also be ap-plied in the temporal fluctuation analysis on sharp fluctu-ation in stock markets, for sharp fluctuation (or extremefluctuation) sequences can also be viewed as a realizationof a stochastic point process.

In this context, our study aims to analyze the time-clus-tering characteristics of sharp fluctuation sequences ofstock markets in China. We performed a detailed statisticalanalysis to investigate the time-clustering properties of thesharp fluctuation sequences of stock markets in China. Thestatistical tools include coefficient of variation, Allan Factor,Fano Factor as well as R/S (rescaled range) analysis. All ofthese empirical results suggest that the sequences are char-acterized by a high degree of time-clusterization. It can helpus to get a better understanding of the nature and dynamicsof sharp fluctuation of stock price in stock markets.

2. Methods

Sharp fluctuation sequences can be viewed as a realiza-tion of a stochastic point process. A stochastic point processdescribes events that occur at some random locations intime and is completely defined by the set of the event times.The series can be represented by a finite sum of Dirac’s deltafunctions centered on the occurrence time ti:

yðtÞ ¼XN

i¼1

dðt � tiÞ ð1Þ

where N represents the number of events recorded. Thendividing the time axis into equally spaced contiguous

counting windows of duration s, which is called timescale,we produce a sequence of counts {Nk(s)}, with Nk(s) denot-ing the number of events in the kth window:

NkðsÞ ¼Z tk

tk�1

XN

j¼1

dðt � tjÞdt ð2Þ

This sequence is a discrete-random process of non-nega-tive integers. The importance of this representation is thatit preserves the correspondence between the discrete timeaxis of the counting process {Nk} and the ‘‘real’’ time axis ofthe underlying point process, and the correlation in theprocess {Nk} refers to correlation in the underlying pointprocess. Such a process may be called fractal when a num-ber of relevant statistics exhibit scaling with related scal-ing exponents, which indicate that the representedphenomenon contain clusters of points over a relativelylarge set of timescales.

In order to analyze time behavior and time-clusteringproperties of sharp fluctuation (or extreme fluctuation) se-quences, we used several statistical measures to featurethe time properties. These methods have been extensivelyused to analyze the time-clustering properties in complexsystems such as seismic sequences, car accident sequenceand forest fire sequence, etc. [13–24]. These methods in-clude coefficient of variation (CV), Allan Factor (AF), FanoFactor (FF) and rescaled range (R/S) analysis. Two of them(CV, R/S analysis) are related to the inter-event intervalrepresentation, while the remaining two (AF, FF) are re-lated to the counting process representation. We will givea brief introduction of these measures.

2.1. Coefficient of variation

The Cv is a commonly used measure to evaluate theclustering behavior of a point process, it is defined as

Cv ¼rs

hsi ð3Þ

where hsi is the mean inter-event time and rs is its standarddeviation, a Poissonian process (completely random) has aCv = 1, but a clusterized process is characterized by aCv > 1. This coefficient does not give information about thetimescale ranges where the process can be reliably charac-terized as a clustered process. Nevertheless, a complex phe-nomenon can be deeply known only if the differenttimescales governing its dynamics are well understood [16].

2.2. Fano Factor

The Fano Factor (FF) is defined as the variance of thenumber of events in a specified counting time or timescales divided by the mean number of events in that countingtime; that is

FFðsÞ ¼N2

kðsÞD E

� hNkðsÞi2

hNkðsÞið4Þ

where hi indicates the average value. In order to evaluatethe presence of scaling, the timescale T is varied and a rela-tionship FF(s) � s is obtained [17].

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

1000

2000

3000

4000

5000

6000

7000

N(days)

P

(a) SHSI

1.2

1.4

1.6

1.8

2x 10

4

840 Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845

The FF(s) of a fractal point process with 0 < a < 1 variesas a function of counting time s as:

FFðsÞ ¼ 1þ ss0

� �a

ð5Þ

The scaling exponent a is the so-called fractal exponent. Ifa > 0 then the represented phenomenon contains clustersof points over a relatively large set of timescales. If a � 0,the process is Poisonian and the occurrence times areuncorrelated. The crossover timescale s0 is the fractal onsettime for the FF and is estimated as the crossover timescalebetween Poissonian and scaling behaviors, so that fors� s0 the clustering property becomes negligible at thistimescales. Thus, s0 is estimated as the time scale overwhich the FF increases as a power-law function of the time-scale s. FF assumes values near unity for Poisson processes.

2.3. Allan Factor

AF is defined as the variation of successive counts for aspecified counting time s divided by twice the mean num-ber of events in that counting time

AFðsÞ ¼ hðNkþ1ðsÞ � NkðsÞÞ2i2hNkðsÞi

ð6Þ

where hi indicates the average value. In order to evaluatethe presence of scaling, the timescale s is varied and a rela-tionship AF(s) � s is obtained.

This measure reduces the effect of possible nonstationa-rity of the point process, because it is defined in terms of thedifference of successive counts. The variation of the time-scale s allows us to produce a relationship between AF(s)and s, useful to detect scaling behavior in the sequence.

The AF of a fractal point process varies with the count-ing time s with a power-law form:

AFðsÞ ¼ 1þ ss1

� �a

ð7Þ

with 0 < a < 3 over a large range of counting times s[15,21].If a > 0 then the represented phenomenon contains

clusters of points over a relatively large set of timescales.If a � 0, the sharp fluctuation occurrence process is Poisso-nian and the occurrence times are uncorrelated. s1 is thefractal onset time and marks the lower limit for significantscaling behavior in the AF, so that for s� s1 the clusteringproperty becomes negligible within these timescales. FromEq. (7), the calculation of a can be performed by estimatedthe slope of the straight line that fits in a least square sensethe AF curve, plotted in log–log scales.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.2

0.4

0.6

0.8

1

N(days)

P

(b) SZCI

Fig. 1. Evolution of the stock price index for the whole sample periods.

2.4. Rescaled range analysis

The rescaled range analysis (R/S analysis) is a techniqueproposed by Hurst in 1951 to test presence of correlationsin empirical time series. The procedure is as follows:

r The time series of length L has to be divided into d subseries (Zi,m) of length n, and for each sub seriesm = 1, . . . ,d. Then, it is necessary to find the mean (Em)and the standard deviation (Sm) of the sub series (Zi,m).

s The data of the sub series (Zi,m) has to be normalized bysubtracting the sample mean Xi,m = Zi,m � Em fori = 1, . . . ,n.

t Creating the cumulative time series Yi;m ¼Pi

j¼1Xj;m, fori = 1, . . . ,n.

u Find the range Rm = max {Y1,m, . . . , Yn,m}-min{Y1,m, . . . ,Yn,m}.

v Rescaled the range (Rm/Sm).w Calculate the mean value (R/S)n of the rescaled range for

all sub series of length n.

Considering that the R/S statistic asymptotically followsthe relation (R/S)n � nH, the value of H can be estimated bymeans of the slope of the line fitting the (R/S)n plotted onlog–log scales. When the process is a white noise, H hasto be 0.5, when it is persistent (namely, long memory) Hwill be greater than 0.5, and finally when it is anti-persis-tent H will be less than 0.5. Note that H must lie between 0and 1 [25].

3. Data description

3.1. Original data

The original data are taken from Chinese stock market,which is composed Shanghai Stock Exchange (SHSE) and

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

100

200

300

400

500

600

700

N(days)

V

m=3m=1

m=2

(a) SHSI

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

200

400

600

800

1000

1200

1400

N(days)

V

m=3m=2m=1

(b) SZCI

Fig. 2. Daily price variation series along with horizontal lines corre-sponding to the threshold.

Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845 841

the Shenzhen Stock Exchange (SZSE). The Chinese stockmarket was a newly set up market and it has only a historyof 21 years. The Shanghai Stock Exchange was establishedon November 26, 1990, and put into operation on Decem-ber 19. Shortly after, the SZSE was established on Decem-ber 1, 1990 and put into operation on April 3, 1991. Oneach trading day both Stock Exchanges open from 9:30 to11:30 am and then from 1:00 pm to 3:00 pm, so thereare 4 trading hours on one day.

The original data are daily closing price series of Shang-hai Stock Index (SHSI) and Shenzhen Component Index(SZCI). The sample period for SHSI is from December 20,1990 to December 30, 2010 and the sample period for SZCI

Table 1The number of sharp fluctuation events for each threshold.

Threshold m = 1 m = 1.5 m = 2 m = 2.5 m = 3 T

Number of events (SHSI) 1481 942 670 475 364 N

Table 2Results of Cv under different thresholds.

Threshold m = 1 m = 1.5 m = 2 m = 2.5 m = 3 T

Cv(SHSI) 1.5925 2.4376 2.9297 3.5119 3.5861 C

is from April 3, 1991 to Dec 30, 2010. The daily closingprice series of SHSI and SZCI over time for the whole sam-ple periods are shown in Fig. 1(a) and (b), respectively.

3.2. Data analysis

Actually, it is difficult to define a sharp fluctuation be-cause it is only a relative concept. Therefore, in this articlewe describe the sharp fluctuation by defining differentthresholds. Firstly, the daily price variation series is formedby Vi = jPi+1 � Pij, where Pi is the stock price index on ithday. Then calculate the average value of Vi,hVii.Thirdly,we define threshold as m times of hVii, where m = 1, 1.5,2, 2.5, 3. Finally, we define the sharp fluctuation like this,namely, if the daily variation is greater than threshold thenit can be regarded as a sharp fluctuation. It is noted that thenumber of sharp fluctuation events will decrease withincreasing m. In order to ensure that the number of eventsare ample we choose m = 1, 1.5, 2, 2.5, 3, respectively.

The daily price variation series, along with horizontallines corresponding to the thresholds for SHSI and SZCIare shown in Fig. 2(a) and (b),respectively. And the numberof sharp fluctuation events for each threshold is shown inTable 1.

4. Empirical results

In this section, the time-scaling properties of the distri-bution of the sharp fluctuation sequences in China from1990 to 2010 were studied.

Firstly, we calculated the coefficient of variation underdifferent thresholds according to Eq. (3). Table 2 showsthe results of Cv.

From Table 2, all values of Cv > 1 under different thresh-olds for SHSI and SZCI, indicating that all the sequences arecharacterized by a time-clustering behavior. It is importantto highlight that Cv is a dimensionless number which al-lows us to compare, the degree of variation from one se-quence to another characterized by means that aredrastically different from each other [16]. It is also foundthat the greater threshold is, the greater is the Cv, thereforethe stronger time-clustering behavior. In other words, ahigh value of Cv indicates that the sharp fluctuation se-quences, associated to higher threshold, seem to be moreclustered. It is also noted that the values of Cv under differ-ent thresholds for SZCI are higher than those of corre-sponding Cv for SHSI, which indicates the clusterization

hreshold m = 1 m = 1.5 m = 2 m = 2.5 m = 3

umber of events (SZCI) 1289 853 637 502 391

hreshold m = 1 m = 1.5 m = 2 m = 2.5 m = 3

v (SZCI) 2.6098 3.1497 4.1022 4.7471 4.7654

842 Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845

degree for SZCI is higher than that of SHSI. Namely, com-pared with Shanghai stock market, Shenzhen stock marketshows a stronger time-clusterization of the sharp fluctua-tion events. The coefficient of variation represents a firstindicator of the presence of clustering in a point process.It only discriminates between clusterized and Poisson se-quences, but it does not convey any information aboutwhat timescales are involved in the clustering behavior[16,17]. In order to obtain such information the AF was ap-plied to the sharp fluctuation sequences.

1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

τ(days)

AF(

τ) slope2=1.3168

slope1=0.709

m=1

τc=80

(a) Doubly logarithmic AF plot of sharp fluctuation sequences for SHSI under threshold m=1

1 2 3 4 5 6 7-1

0

1

2

3

4

5

τ(days)

AF(

τ)

slope2=1.4961

slope1=0.6979

m=2

τc=70

(b) Doubly logarithmic AF plot of sharp fluctuation sequences for SHSI under threshold m=2

1 2 3 4 5 6 7-1

0

1

2

3

4

5

τ(days)

AF(

τ)

m=3

slope2=1.5932

slope1=0.5144

τc=68

(c) Doubly logarithmic AF plot of sharp fluctuation sequences for SHSI under threshold m=3

Fig. 3. Results of Allan Factor for Shanghai stock market in China, sc iscrossover.

Fig. 3(a)–(c) show the AF results for SHSI under differentthresholds, for timescales N from 5 to N/7, where N is the to-tal period of each sequence. As can be seen from Fig. 3(a)–(c),the AF plots under different thresholds display similartrends and show similar clustering behavior. All of the AFplots clearly show fractal behavior in the sharp fluctuationsequences. It is noted that there is a crossover time scale sc

in the doubly logarithmic plots of AF versus s in each of AFplots. This shows that there are different scaling laws andscaling exponents for time scales s < sc and s > sc. And atapproximately the time scales �70 days ðMaxsc ¼80; Minsc ¼ 59Þ, the AF is characterized by a crossover, indi-cating the presence of a quarter periodicity in the countingdynamics of the sequence (For there are 22–23 trading daysfor one month). In addition, all of the scaling exponents aAF,estimated as the slope of the line that fits by the least squaremethod the linear region, are greater than 0, which indicatesthe presence of time-clustering behavior in the sharp fluctu-ation dynamics. These results imply the presence of time-correlation structures in the time distribution of the sharpfluctuation sequences. Meanwhile different values of thescaling exponent indicate different underlying dynamicsof different markets.

1 2 3 4 5 6 7-1

0

1

2

3

4

5

τ(days)

AF(

τ)

m=3

slope2=1.7249

slope1=0.439

τc=59

(a) Doubly logarithmic AF plot of original series for SZCI under threshold m=3

1 2 3 4 5 6 7-1.5

-1

-0.5

0

0.5

1

1.5

τ(days)

AF(

τ)

(b) Doubly logarithmic AF plot of shuffled series for SZCI under threshold m=3

Fig. 4. AF plots for the original series and its randomly shuffled series forShenzhen stock market.

Table 3Results of AF for original data and shuffled data under different thresholds.

Threshold m = 1 m = 2 m = 3

Series Original (�) Shuffled Original (�) Shuffled Original (�) Shuffleds < sc s > sc (mean ± std) s < sc s > sc (mean ± std) s < sc s > sc (mean ± std)

aAF 0.709 1.3168 �0.0596 0.6979 1.4961 �0.0609 0.5144 1.5932 �0.0536(SHSI) (s < 80) (s > 80) ±(0.1738) (s < 70) (s > 67) ±(0.1729) (s < 68) (s > 68) ± (0.1918)aAF 0.6759 1.679 �0.0566 0.6031 1.8284 �0.0360 0.439 1.7249 �0.0570(SZCI) (s < 67) (s > 67) ±(0.1820) (s < 63) (s > 63) ±(0.1832) (s < 59) (s > 59) ±(0.1863)

1 2 3 4 5 6 70.5

1

1.5

2

2.5

3

3.5

4

4.5

5

τ(days)

FF

( τ)

slope=0.6114

m=3

(a) Doubly logarithmic FF plot of sharp fluctuation sequences for SHSI under threshold m=3

1 2 3 4 5 6 70.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

τ(days)

FF

( τ)

slope=0.7183

m=3

(b) Doubly logarithmic FF plot of sharp fluctuation sequences for SHSZ under threshold m=3

Fig. 5. Results of Fano Factor of sharp fluctuation events for stock marketsin China.

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 41

1.5

2

2.5

Ln(n)

Ln(R/S)

m=3

slope=0.6262

(a) Doubly logarithmic R/S plot of sharp fluctuation events for SHSI under threshold m=3

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.21

1.5

2

2.5

Ln(n)

Ln(R/S)

slope=0.645

m=3

(b) Doubly logarithmic R/S plot of sharp fluctuation events for SZCI under threshold m=3

Fig. 6. R/S analysis results of sharp fluctuation events for stock markets inChina.

Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845 843

In order to verify that the results are not random butsignificant, we performed the same analysis on randomlyshuffled versions of the original series. Taking Shenzhenstock market as an example for comparison, Fig. 4(a) and

Table 4Results of FF for original data and shuffled data under different thresholds.

Threshold m = 1 m = 2

Series Original (�) Shuffled (mean ± std) Original (�)aFF(SHSI) 0.5992 �0.0685(±0.1287) 0.6446aFF ( SZCI) 0.7187 �0.1079 (±0.1595) 0.7527

(b) present the AF results for an original series under thresh-old m = 3 and its corresponding randomly shuffled series.We can clearly observe that the values of AF for the originalseries (see Fig. 4(a)) differ significantly from those for theshuffled series (see Fig. 4(b)). In addition, it seems that allof the AF values for the shuffled series fluctuate up anddown around zero within a certain narrower range.

m = 3

Shuffled (mean ± std) Original (�) Shuffled (mean ± std)�0.0978(±0.1596) 0.6114 �0.1164(±0.1593)0.0975( 0.1533) 0.7183 �0.0745(±0.1044)

Table 5Results of R/S for original data and shuffled data under different thresholds.

Threshold m = 1 m = 2 m = 3

Series Original (�) Shuffled (mean ± std) Original (�) Shuffled (mean ± std) Original (�) Shuffled (mean ± std)H(SHSI) 0.7433 0.5444(±0.0302) 0.7485 0.5446(±0.0339) 0.6262 0.5429(±0.0386)H ( SZCI) 0.7228 0.5357(±0.0303) 0.652 0.5401(±0.0258) 0.645 0.5389 (±0.0303)

844 Y. Yuan et al. / Chaos, Solitons & Fractals 45 (2012) 838–845

A summary of AF results for original data and their corre-sponding shuffled data under different thresholds as well asdifferent stock markets is shown in Table 3. As can be seenfrom Table 3, all of values of aAF for original series within dif-ferent time scales are greater than 0, showing pronouncedtime-clustering properties, while all of values of aAF for shuf-fled series are around 0. These results verified the signifi-cance of the time-clustering properties of stock markets.

Fig. 5(a) and (b) show the FF results for SHSI and SZCIunder threshold m = 3, for timescales N from 5 to N/7,where N is the total period of each sequence. FromFig. 5(a) and (b), both of the FF plots clearly show fractalbehavior in the sharp fluctuation sequences. In addition,both of the scaling exponents aFF, estimated as the slopeof the line that fits by the least square method the linearregion, are greater than 0 (aFF � 0.6114 for SHSI, seeFig. 5(a) and aFF � 0.7183 for SZCI, see Fig. 5(b)),whichindicates the presence of time-clustering behavior in thesharp fluctuation dynamics. These results imply the pres-ence of time-correlation structures in the time distributionof the sharp fluctuation events. Meanwhile different valuesof the scaling exponent for SHSI and SZCI indicate differentunderlying dynamics of different markets.

A summary of FF results for original data and their corre-sponding shuffled data under different thresholds as well asdifferent stock markets is shown in Table 4. As can be seenfrom Table 4, all of values of aFF for original series are greaterthan 0, showing pronounced time-clustering properties,while all of values ofaFF for shuffled series are around 0. Theseresults verified the significance of the time-clustering prop-erties of stock markets. In addition, it is found that Shenzhenstock market shows a stronger time-clusterization of thesharp fluctuation events than Shanghai stock markets underthe same threshold because the aFF of Shenzhen stock marketis greater than that of Shanghai stock market.

Comparing results of AF (see Fig. 4) and those of FF (seeFig. 5), we observed that the AF plot is more irregular androugher than the FF plot. However, both the AF and FF plotspresent a consistent conclusion that sharp fluctuation se-quences show pronounced time-clustering behavior.

Finally, we conduct R/S analysis on the inter-event timeseries set for scales n from 10 to N/7, where N is the totalperiod of each sequence. The R/S results under thresholdm = 3 for SHSI and SZCI are shown in Fig. 6(a) and (b),respectively. As can be seen from Fig. 6(a) and (b), boththe inter-event time series for different stock marketsshow pronounced long memory for each coefficient of ser-ies is greater than 0.5 (H � 0.6262 for SHSI, see Fig. 6(a)and H � 0.645 for SZCI, see Fig. 6(b)). It indicates that theinter-event time series of sharp fluctuation events showpronounced long memory properties.

A summary of R/S analysis for original data and theircorresponding shuffled data under different thresholds as

well as different stock markets is shown in Table 5. Ascan be seen from Table 5, all of values of H for original ser-ies are greater than 0.5, showing pronounced time-cluster-ing properties, while all of values of H for shuffled seriesare around 0.5. These results also verified the significanceof the time-clustering properties of stock markets.

All of these results are consistent with the conclusionthat sharp fluctuation sequences show time-clusteringbehavior and long-range correlations in Chinese stock mar-kets. Meanwhile, these empirical results can also give use-ful information on the dynamical features of sharpfluctuation events.

5. Conclusions

Using two different representations of the sharp fluctu-ations (inter-event times and series of counts), the timeclustering behavior in the sharp fluctuation sequences ofstock markets in China is studied. All of empirical resultsdemonstrate that the sharp fluctuation sequences in Chi-nese stock markets are characterized by a time-clusteringbehavior and show long-range correlation. The character-ization of time clustering in sharp fluctuation sequencesprovides valuable information to better understand thesharp fluctuation behavior.

However, the potential of time dynamical features ofsharp fluctuation (especially extreme fluctuation) se-quences is far from being fully exploited. How to deeplyunderstand the essence of the time-clustering behaviorand long memory effects in capital markets and how to re-veal more valuable information about market changes, fur-thermore, how to use information on these features in riskmanagement, are key issues in the future. Research onthese issues will help to make more accurate risk estima-tion and suggest effective measures for risk preventionand control in stock market.

Acknowledgement

We are appreciative for financial support by NationalScience Foundation of China (No. 70901017, 70101022)and Science Foundation of Postdoctors in China (No.20080441095), Special Science Foundation of Postdoctorsin China (200902546) and Fundamental Research Fundsfor the Central Universities (N100406003).

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