Multiscaling behavior in the volatility return intervals of Chinese indices

a r X i v :0809.0250v 1 [p h y s i c s .s o c -p h ] 1 S e p 2008EPJ manuscript No.(will be inserted by the editor)Multiscaling behavior in the volatility return intervals of Chinese indicesFei Ren 1,2,a and Wei-Xing Zhou 1,2,3,4,b1School of Business,East China University of Science and Technology,Shanghai 200237,China2Research Center for Econophysics,East China University of Science and Technology,Shanghai 200237,China 3School of Science,East China University of Science and Technology,Shanghai 200237,China4Research Center of Systems Engineering,East China University of Science and Technology,Shanghai 200237,ChinaReceived September 1,2008/Revised version:Abstract.We investigate the probability distribution of the return intervals τbetween successive 1-min volatilities of two Chinese indices exceeding a certain threshold q .The Kolmogorov-Smirnov (KS)tests show that the two in-dices exhibit multiscaling behavior in the distribution of τ,which follows a stretched exponential form f q (τ/ τ )∼e −a (τ/ τ )γwith different correlation exponent γfor different threshold q ,where τ is the mean return interval cor-responding to a certain value of q .An extended self-similarity analysis of the moments provides further evidence of multiscaling in the return intervals.PACS.89.65.Gh Economics;econophysics,financial markets,business and management –89.75.Da Systems obeying scaling laws –05.45.Tp Time series analysis1IntroductionThe analysis of the waiting time between two successive events is helpful to understand the dynamics of stock markets,which has drawn much attention.A variety of waiting time variables have been raised by different definitions of event to characterize the stock markets from different view angles,such as the per-sistence probability [1,2,3],the exit time [4,5,6,7,8,9],and the intertrade duration [10,11,12,13].Recently the return intervals between successive extreme events exceeding a certain thresh-old q have been investigated for numerous complex systems,including rainfalls,floods,temperatures and earthquakes [14,15,16,17,18].Similar analysis was subsequently carried out concerning the volatility return intervals,which are defined as the waiting times between successive volatilities exceeding a certain threshold in stock markets.Yamasaki et al.and Wang et ed the daily data and in-traday data of US stocks to study the properties of the volatility return intervals [19,20,21,22].They found that the distribution of return intervals τbetween successive volatilities greater than a certain threshold q showed scaling behavior.This scaling be-havior is expected to be of great importance for the risk assess-ment of large price fluctuations.Similar scaling behavior was observed in the return intervals of daily and 1-min volatilities of thousands Japanese stocks [23].Qiu,Guo and Chen analyzed the high-frequency intraday data of four liquid stocks traded in the emerging Chinese market,and found that the returninterval2Fei Ren,Wei-Xing Zhou:Multiscaling behavior in volatility return intervals of Chinese indicesThe paper is organized as follows.In Section2,we explain the database analyzed and how the volatility return intervals are calculated.Section 3examines the scaling behavior and the curve fitting of the return interval distributions using the KS tests.In Section 4,we further study the multiscaling behavior by analyzing the moments of the scaled return intervals.Sec-tion 5concludes.2Preprocessing the data setsOur analysis is based on the high-frequency intraday data of two Chinese indices,the Shanghai Stock Exchange Composite Index (SSEC)and the Shenzhen Stock Exchange Composite Index (SZCI).Each composite index is constructed based on all the stocks listed on the corresponding exchange.The indices are recorded every six to eight seconds from January 2004to June 2006.We define the volatility as the magnitude of loga-rithmic index return between two consecutive minutes,that is R (t )=|ln Y (t )−ln Y (t −1)|,where the index Y is the closest tick to a minute mark.Thus the sampling time is one minute,and the volatility data size is about 140,000.The intraday volatilities of both indices exhibit a L-shaped intraday pattern [30],similar to the individual stocks [30,29].When dealing with intraday data,this pattern should be re-moved [20,21,22,26,24].Otherwise,the return intervals dis-tribution will exhibit daily periodicity for large thresholds.The intraday pattern A (s )is defined asA (s )=1A (s ).(2)Then we normalize the volatility by dividing its standard devi-ationv (t )=R ′(t )τf q (τ/ τ ),(4)where τ is the mean return interval that depends on the thresh-old q .If the function f q (x )is independent of q ,there exists a universal function f (x )such that f q (x )=f (x )for differ-ent values of q .In other words,the probability distributions f q (τ/ τ )of the scaled return intervals collapse onto a single curve f (τ/ τ )and the return intervals exhibit scaling behav-ior.To investigate whether the return interval distributions of the two Chinese indices exhibit scaling behavior,we plot in Figure 1the empirical probability distributions f q (τ/ τ )=P q (τ) τ as a function of the scaled return intervals τ/ τ for a wide range of threshold q =2,3,4,5.It is evident that the curves for different thresholds q show systematic deviations from each other and do not collapse onto a single curve,es-pecially for the Shanghai Composite Index.This indicates that the distributions of return intervals for both indices could not be approximated by a scaling relation.With the increase of the threshold q ,there are more large scaled return intervals and the distribution becomes broader.Fig.1.(Color online)Empirical probability distributions of scaled return intervals for different threshold q =2,3,4,5for the Shanghai Composite Index and the Shenzhen Composite Index.The solid curvesare the fitted functions ce −ax γwith the parameters listed in Table 2.Fei Ren,Wei-Xing Zhou:Multiscaling behavior in volatility return intervals of Chinese indices 3The observation that there is no scaling behavior in the volatility return interval distributions is consistent with there-sults of a previous study ofindividual Chinesestocks [29].TheKolmogorov-Smirnov test shows that only 12stocks out of 30most liquid Chinese stocks exhibit scaling behaviors in the re-turn interval distributions for different thresholds q ,while the other 18stocks do not show scaling behavior [29].3.2Kolmogorov-Smirnov test of scaling in return interval distributionsThe eyeballing of the probability distributions offers a qualita-tive way of distinguishing scaling and nonscaling behaviors.Here we further adopt a quantitative approach based on the Kolmogorov-Smirnov test.The standard KS test is designed to test the hypothesis that the distribution of the empirical data is equal to a particular distribution by comparing their cumulative distribution functions (CDFs).Our hypothesis is that the two return interval distributions for any two different q values do not differ at least in the common region of the scaled return in-tervals [23].Suppose that F q i is the CDF of return intervals for q i and F q j is the CDF of return intervals for q j ,where q i =q j .We calculate the KS statistic by comparing the two CDFs in the overlapping region:KS =max|F q i −F q j | ,q i =q j .(5)When the KS statistic is less than a critical value CV ,the hy-pothesis is accepted and we can assume that the distribution forq i is coincident with the distribution for q j .The critical value at the significance level of 5%is CV =1.36/q i q j KS CV 230.03210.0211240.07200.0296250.05350.0408340.04360.0329350.04440.0432450.04450.050620.802.050.590.800.673.510.470.0232.1314.200.380.631.5514.850.370.5440.925.790.430.581.7122.010.340.7251.0514.190.350.741.5625.060.330.85The p value could be regarded as the probability that the empirical distribution consists with its best fit.Consider the4Fei Ren,Wei-Xing Zhou:Multiscaling behavior in volatility return intervals of Chinese indicessignificance level of 1%.If the p -value of an index for a cer-tain threshold q is less than 1%,then the null hypothesis that the empirical PDF of this index can be wellfitted by a stretched ex-ponential is rejected.According to Table 2,the null hypotheses for all the q values are accepted for both two Chinese indices.It is noteworthy to point out that the p -values for all the q values (except for q =2for the Shenzhen Composite Index)are very large,implying high goodness-of-fit of the stretched exponen-tial to the empirical PDFs.At the significance level of 5%,the stretched exponential is rejected when q =2for the Shenzhen Composite Index.To show how good the stretched exponen-tial fits the data,we illustrate in Figure 1the fitted stretched exponential with the parameters listed in Table 2.It is obvi-ous that the empirical PDFs could be well fitted by a stretched exponential.In principle,the stretched exponential fits the em-pirical PDF better when the p -value is larger.For instance,the stretched exponential fits the empirical PDF for the Shanghai Composite Index better than the Shenzhen Composite Index when q =2.According to Table 2,the parameters differ from one an-other,providing further evidence supporting our conclusion that the return interval distributions do not have a scaling form.On average,the exponent γdecreases with increasing threshold q ,which is in line with the US stocks [27].4Moments of scaled return intervalsThe distributions of return intervals exhibit multiscaling behav-ior and show a systematic tendency with the threshold q .To fur-ther study this tendency of the interval distribution with q ,we compute the moments of the scaled return intervals x =τ/ τ defined asµm = (τ/ τ )m 1/m = ∞0x mf q (x )dx 1/m.(9)where the mean interval τ is dependent of the threshold q .When m =1,we have µ1=1by definition,independent of q .If there is a scaling behavior that f q (x )=f (x ),the m -th moment µm is a univariate function of the order m and is in-dependent of any other variables including the threshold q and the mean return interval τ .On the contrary,the m -th moment µm is not constant with respect to τ for m =1,when there is no scaling in the return interval distributions.4.1Dependence of moment on mean return interval We first investigate the relation between µm and τ .To better quantify the dependence of µm on τ ,we calculate the mo-ments in a certain medium range of τ to avoid the finite size effect and discreteness effect [26].Figure 2illustrates the mo-ments µm for m =0.25,0.5,1.5,2.0versus τ for the two Chinese indices.We investigate µm for a range of τ corre-sponding to 1≤q ≤5.Each curve of the moments µm sig-nificantly deviate from a horizontal line,confirming the multi-scaling behavior in the return interval distributions.The mo-ment function µm decreases with the increase of τ when m <1,and shows an increasing tendency with the increaseof τ when m >1.These two types of moment functions are delimited by the horizontal line µ1=1.Fig.2.(Color online)Moment µm vs τ for the two Chinese indices:Shanghai Composite Index and Shenzhen Composite Index.For each data set,four moments m =0.25,0.5,1.5,2.0are presented.Take a careful look at µm in Figure 2,for m <1(m >1)µm first decreases (increases)rapidly with the increase of τ ,and then starts to decrease (increase)relatively slowly at τ =10.The discreteness of the records of τresponds to the rapid increase (decrease)of µm for small τ ( τ <10).The moment for extremely large τ ,i.e., τ >100( τ is in units of standard deviations),will increase (decrease)for m <1(m >1)due to the finite size effect.We choose to study µm in a medium region 10< τ <100where in the effects of finite size and discreteness are small and µm shows a clear power-law-like trend with τ .We use a power law to fit the moment in this medium region,µm ∼ τ α.(10)If the PDF of return intervals follow a scaling form,µm is inde-pendent of τ according to Eq.(9)and the exponent αshould be some value very close to 0.If the exponent αis significantly different from 0,it implies that the PDF of return intervals may show multiscaling behavior.Fei Ren,Wei-Xing Zhou:Multiscaling behavior in volatility return intervals of Chinese indices 5Figure 3plots the exponent αas a function of order m for the two Chinese indices.This figure shows that the exponent αdiffers from 0in a systematic fashion.The exponents for the two indices are very close to each other when m is small.For m <1,αis negative.The exponent αincreases with m when m <3and decreases afterwards owning to the finite size effect.For large order m ,the αvalue for the Shenzhen Com-posite Index is greater than that for the Shanghai Composite Index.This implies that large τ tends to occur with greater probability for the Shenzhen Composite Index than the Shang-hai Composite Index,since large τ contributes more for high order µm .Fig.3.(Color online)Exponent αof µm in the region 10< τ <100for the two Chinese indices:Shanghai Composite Index and Shen-zhen Composite Index presented by circles and squares respectively.The above analysis according to Eq.(10)can be related to the extended self-similarity (ESS)analysis [33],which readsτm ∼ τnξ(m,n ).(11)If the generalized variable of µmµm,n =ττn 1/n(12)scales asµm,n ∼τn 1/nα(13)together with Eq.(11),we have(α+1)/n =ξ(m,n )/m.(14)If the return interval distribution can be scaled as followsP q (τ)=1τn 1/n,(15)we obtain thatξ(m,n )=m/n.(16)In this case,we haveα=0.(17)In other words,µm,n is independent of τn 1/n .Our empirical test focuses on the case that n =1.This ESS framework was also used to investigate scaling in the exit times in turbulence [34]and intertrade durations [35].According to Eq.(14)α(m )=ξ(m,1)/m −1.(18)Since ξ(1,1)=1,we have α(1)=0when m =1.This is wellverified by Figure 3.4.2Dependence of moment on order mThe moment µm not only display a significant dependence on τ ,but also shows a systematic tendency with m as shown in Figure 2.It is interesting to investigate the relation between µm and m directly.For a fixed τ ,one can study the moment of τof various orders m .If the return interval distribution strictly obeys a scaling form,µm should not depend on τ ,and the µm curves for different τ should all collapse onto a single curve.We plot in Figure 4(a)the moment µm as a function of m for fixed τ =10,30,100corresponding to q =2.0,3.2,4.8for the Shenzhen Composite Index.One sees the curves for dif-ferent τ exhibit substantial deviations from a single curve,Fig.4.(Color online)(a)Moment µm vs m for Shenzhen Composite Index for τ =10,30,100corresponding to q =2.0,3.2,4.8,pre-sented by circles,squares,triangles respectively.(b)Analytical mo-ments obtained from the stretched exponential distributions with pa-rameters fitted from empirical data of Shenzhen Composite Index for q =2,3,5.6Fei Ren,Wei-Xing Zhou:Multiscaling behavior in volatility return intervals of Chinese indices which demonstrates the multiscaling behavior of return inter-vals.For small m(m<1),µm decreases with the increase ofτ ,while for big m(m>1)µm tends to increase with theincrease of τ .This is not difficult to understand since smallτdominatesµm for small order m and largeτdominatesµm forlarge order m.The situation for the Shanghai Composite Indexis very similar.We have demonstrated that the return interval distributionfollows a stretched exponential form with different parametersfor various thresholds q in Section3.3.For the data perfectlyfollow a stretched exponential distribution,we can calculatethe analytical result of the momentµm by substituting Eq.(6)to Eq.(9)and considering the normalization condition of prob-ability density.It follows immediately that[26]µ=1Γ(1/γ)1/m.(19)In Figure4(b),the analytical curves ofµm versus m for three stretched exponential distributionsfitted from empirical data of Shenzhen Composite Index for q=2,3,5are plotted.As one can see that the analytical results are similar to that of the em-pirical data,which supports the multiscaling of the empirical return intervals.5Summary and conclusionsWe have studied the multiscaling properties of the distributions of volatility return intervals for two Chinese indices,together with their moments.The Kolmogorov-Smirnov test is adopted to examine the scaling behavior of the return interval distribu-tions as well as the particular form of the distribution.Wefind that the return intervals of the two indices exhibit multiscal-ing behaviors,and their distributions for different thresholds q can be well approximated by stretched exponential functions f q(x)∼e−axγbut with different values of the correlation ex-ponentγ.An ESS-like moment analysis confirms the existence of multiscaling rather than monoscaling.This result is consis-tent with previous analysis on individual Chinese stocks[29] and help us better understand the properties of volatility return intervals in the Chinese stock markets.This work was partially supported by the Shanghai Educational De-velopment Foundation(No.2008CG37),the National Natural Science Foundation of China(No.70501011),the Fok Ying Tong Education Foundation(No.101086),and the Program for New Century Excel-lent Talents in University(No.NCET-07-0288).References1. 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