Memory effect and multifractality of cross-correlations in financial markets
aa r X i v : . [ q -f i n . S T ] A p r Memory effect and multifractality ofcross-correlations in financial markets
Tian Qiu a , ∗ , Guang Chen a , Li-Xin Zhong b , Xiao-Wei Lei c a School of Information Engineering, Nanchang Hangkong University, Nanchang, 330063,China b School of Journalism, Hangzhou Dianzi University, Hangzhou, 310018, China c Department of Physics, Chongqing University of Arts and Sciences, Chongqing 402160,China
Abstract
Abstract: An average instantaneous cross-correlation function is introduced to quantifythe interaction of the financial market of a specific time. Based on the daily data of theAmerican and Chinese stock markets, memory effect of the average instantaneous cross-correlations is investigated over different price return time intervals. Long-range time-correlations are revealed, and are found to persist up to a month-order magnitude of theprice return time interval. Multifractal nature is investigated by a multifractal detrendedfluctuation analysis.
Key words:
Econophysics; Stock market; Detrended fluctuation analysis
PACS:
In recent years, dynamics of financial markets has drawn much attention of physi-cists [1–19]. Financial market is a complex system with many interacting compo-nents. From the view of many-body systems, interactions among components maylead the system to collective behavior, and therefore result in the so-called dynamicscaling behavior. Based on large amounts of historical data, some stylized factshave been revealed in the past years, such as the ’fat tail’ distribution of the pricereturn, and the long-range time-correlation of the magnitude of returns [1, 10–12]. ∗ Corresponding author. Address: 696 South Fenghe Avenue, School of Information En-gineering, Nanchang Hangkong University, Nanchang, 330063, China.
Email address: [email protected] (Tian Qiu ).Preprint submitted toPhysica A 17 November 2018 ifferent models and theoretical approaches have been developed to describe fi-nancial markets[4, 11, 12, 20–24].The cross-correlation function is an important indicator to quantify the interac-tion between stocks, and therefore has attracted much attention of physicists inrecent years. Random and nonrandom properties of the cross-correlation and therelevant economic sectors are revealed [5, 17, 25–33]. Correlation-based hierarchi-cal or network structures are studied with the graph or complexity theory [34–40].The so-called pull effect is found with a time-dependent cross-correlation func-tion [41]. These lines of work are mainly based on a static definition of the cross-correlation function. The equal-time or the time-dependent cross-correlation is usu-ally defined as C ij = < r i ( t ′ ) r j ( t ′ ) > or C ij = < r i ( t ′ ) r j ( t ′ + τ ) > , with r i ( t ′ ) = lny i ( t ′ + ∆ t ′ ) − lny i ( t ′ ) being the return of stock i ′ s price y i over a time interval ∆ t ′ , τ being the time lag, and < ... > taking time average over t ′ . The static cross-correlation function can not reveal the dynamic behavior of the cross-correlationsbetween stocks. More recently, a Detrended Cross-Correlation Analysis(DCCA) isproposed to investigate the memory effect of the cross-correlations between twotime series [42–45]. Long-range time-correlation of the cross-correlations is char-acterized by a power-law scaling of the DCCA function. The DCCA method con-centrates on dynamics of two series’ cross-correlations.In this paper, we introduce an Instantaneous Cross-correlation( IC ) and an AverageCross-correlation( AIC ) function by considering the cross-correlations of a singletime step. The IC and AIC function describes the current interaction betweenstocks with local information. Our purpose is to investigate the dynamics of the IC and AIC series, based on the daily data of the American and Chinese stock markets.More importantly, we examine the memory effect of the
AIC over different pricereturn time intervals. The multifractal nature of the
AIC is also revealed.The organization of this paper is as follows. In the next section, the datasets and thedefinition of the IC and AIC functions are presented. In Sec. 3, we investigate thememory effect of the IC and AIC for a shorter price return time interval. In Sec. 4,the memory effect of the
AIC is detected over different scales of price return timeintervals. In section 5, we examine the multifractal nature of the
AIC . Finally, Sec.6 contains the conclusion.
To obtain a comprehensive study, we analyze two different databases, the New YorkStock Exchange (NYSE) and the Chinese Stock Market(CSM). The two marketscover the mature and the emerging markets. The NYSE is one of the oldest stockexchanges, whereas the CMS is a newly set up market in . We investigate thedaily data of individual stocks, with data points from the year to2 for the NYSE, and the daily data of individual stocks, with datapoints from the year to for the CSM. To compare different stocks, wedefine the normalized the price return as R i ( t ′ , ∆ t ′ ) = r i ( t ′ ) − < r i ( t ′ ) >σ i (1)where r i ( t ′ ) is the price return of stock i at time t ′ , and ∆ t ′ is the price return timeinterval. The σ i = q < r i > − < r i > is the standard deviation of r i , and < . . . > takes time average over t ′ . In order to quantify the current cross-correlation betweenstocks, we introduce an IC function between two stocks by IC ij ( t ′ ) = R i ( t ′ ) R j ( t ′ ) , (2)The IC function indicates the instantaneous cross-correlation between two indi-vidual stocks. However, it does not depict the average interaction of the financialmarket. Therefore, we define an AIC function as
AIC ( t ′ ) = 2 N ( N − N − X i =1 N X j = i +1 C ij ( t ′ ) (3)where N is the number of stocks. The AIC function indicates the average instan-taneous cross-correlation of a number of stocks with the stock size to be N . Asthe stock number N is large enough, the AIC function can be then considered asan indicator to quantify the average interaction of the financial market at a specifictime step. For N = 2 , the AIC function reduces to the IC function. IC and AIC for a shorter ∆ t ′ It is important to measure the memory effect of the time series during the dynamicevolution. We investigate the memory effect of the IC and AIC by computingthe time-correlations. The autocorrelation function is widely adopted to measurethe time-correlation. However, it shows large fluctuations for nonstationary timeseries. Therefore, we apply the DFA method [46, 47].Considering a fluctuating dynamic series A ( t ′ ) , one can construct B ( t ′ ) = t ′ X t ”=1 A ( t ”) , (4)3ividing the total time interval into windows N t with a size of t , and linearly fit B ( t ′ ) to a linear function B t ( t ′ ) in each window. The DFA function of the k th window box is then defined as: f k ( t ) = 1 t kt X t ′ =( k − t +1 [ B ( t ′ ) − B t ( t ′ )] , (5)The overall detrended fluctuation is estimated as F ( t ) = 1 N t N t X k =1 [ f k ( t )] , (6)In general, F ( t ) will increase with the window size t and obey a power-law be-havior F ( t ) ∼ t H . If . < H < . , A ( t ′ ) is long-range correlated in time; if < H < . , A ( t ′ ) is temporally anti-correlated; H = 0 . corresponds to theGaussian white noise, while H = 1 . indicates the /f noise. If H is bigger than . , the time series is considered to be unstable.The DFA functions of the IC and the AIC are computed with the price return timeinterval ∆ t ′ = 1 day. To illustrate the results, we take ICs and
AICs as exam-ples, with the stocks randomly chosen from the NYSE and the CSM, respectively.As shown in Fig. 1(a), the DFA exponents of the
ICs are estimated to be from . to . for the NYSE. The exponent . is close to the Gaussian behavior,while . is the long-range correlation. Similarly, as shown in Fig. 1(c) for the ICs of the CSM, the DFA exponents range from . to . , also correspondingto the Gaussian behavior and the long-range correlation, respectively. It suggeststhat the long-range time-correlation does not hold for all ICs . However, when wecompute the DFA of the
AIC , robust long-range time-correlations are observed forboth the NYSE and the CSM. As shown in Fig. 1(b) and (d), the DFA functions of6
AICs are shown as examples with N = 50 . The DFA exponents are estimatedto be around 0.73 for the NYSE, and 0.67 for the CSM, both in the long-rangetime-correlation range (0 . , . . The DFA exponents of the AIC take the similarvalue for a larger stock number N from our databases. It implies that, for both theNYSE and the CSM, even though the absence of the long-range time-correlationof the instantaneous cross-correlation between two individual stocks, the averageinstantaneous cross-correlation of a number of stocks is long-range correlated, i.e.,the average interaction of the financial market shows long-term memory. The resultis reasonable. For example, it is possible for two correlated companies to break uptheir relationship during the time evolution for some reason. With the end of thecorrelation, the memory of the cross-correlation then also ends up. However, thefluctuation from the endogenous events would not influence the average interac-tion of the whole market, i.e., as a collective, the cross-correlation of the financialmarket is always characterized by a long-range memory.4 Memory effect of
AIC for different ∆ t ′ Scalings observed in the financial market has been found to always evolve with thedifferent value of the price return intervals. For example, the ’fat tail’ of the prob-ability density function of the price returns can not be found for a big return timeinterval [10]. To further understand the memory effect of the cross-correlations, wethen investigate the DFA functions of the
AIC with different price return time in-terval ∆ t ′ . The return time interval ∆ t ′ covers three magnitude orders, the day, theweek and the month time scales.The AIC is computed with N = 249 for the NYSE, and N = 259 for the CSM. InFig. 2(a), the DFA function of the AIC is shown for the NYSE, with ∆ t ′ = 1 , , , , and days, approximate to a working day, a week, half a month, a month,and two months. For ∆ t ′ = 1 day, clean power-law behavior is observed, with theexponent estimated to be . , consistent with the exponents obtained in Fig. 1.For the return time interval ∆ t ′ bigger than day, two-stage power-law scalingsare observed, with a crossover in between. Such a two-stage behavior has beenwidely found in the DFA function of the financial series, such as the volatilities,intertrade durations, etc [48–50]. The crossover time is about t c ∼ days. For thesmaller window size, the DFA exponents take the value around . for ∆ t ′ = 5 , andbigger than the 1.0 for ∆ t ′ = 10 , days, which correspond to the /f noise andunstable time series, respectively. Due to the narrow range of the smaller windowsize, we care more about the DFA exponents of the larger window size. For thelarger window size from t = 35 to 100 days, the exponents are measured to be . , . , . for ∆ t ′ = 5 , and days, with all the exponent value ranged inthe long-range time-correlation. The estimated exponents also remain unchangedfor a relatively larger window size than 100 days. However, due to the finite size ofthe time series, it will show large fluctuation for a large window size. For ∆ t ′ = 44 days, both the smaller and the larger window size do not show long-range time-correlations. Therefore, the long-range time-correlation of the AIC persists up toa working month magnitude of the price return time interval for the NYSE for thelarge window size. Similar behavior is also observed for the CSM. For ∆ t ′ = 1 day, clean power-law behavior is observed, with the exponent estimated to be 0.68,around 0.67 found in Fig. 1. Also, two-stage power-law scalings are observed for ∆ t ′ = 5 , days, with the smaller window size showing /f noise and unstabletime series, and the larger window size showing long-range time-correlations. Thecross-over time t c ∼ days. The long-range memory persists up to half a monthmagnitude of the price return time interval for the CSM for the larger window size.5 multifractal nature of AIC
Financial time series such as the price returns and the intertrade durations has beenrevealed to present multifractal feature [50–52]. The multifractal detrended fluctu-ation analysis(MF-DFA) has been successfully applied to detect multifractal char-acteristic of nonstationary time series [53]. We then apply the MF-DFA into the
AIC , with N = 249 for the NYSE, and N = 259 for the CSM. The MF-DFAis a generalization of the DFA method by considering different order of detrendedfluctuation. For the q th order of the detrended fluctuation, we have F q ( t ) = { N t N t X k =1 [ f k ( t )] q } /q , (7)where q can take any real number except q = 0 . For q = 0 , we have F ( t ) = exp { N t N t X k =1 ln[ f k ( t )] } , (8)The MF-DFA function F q ( t ) scales with the window size t : F q ( t ) ∼ t h ( q ) , (9)where h ( q ) is the MF-DFA exponent, with q = 2 recovering the DFA exponent.Due to the finite size of the time series, the F q ( t ) shows large fluctuations for thelarge values of | q | . Here we take q ∈ [ − , , and ∆ t ′ = 1 day as examples toshow the multifractal properties. The F q ( t ) of the AIC is shown for the NYSE andCSM in Fig. 3. Clean power law scalings are observed for q = − , , and , withthe exponents estimated to be . , . , . , . for the NYSE, and . , . , . , . for the CSM. The dependence of the F q ( t ) on q suggests that the AIC shows a multifractal characteristic. The MF-DFA exponent h ( q ) versus different q is shown in Fig. 4(a) and (d), respectively for the NYSE and the CSM.The scaling exponent function τ ( q ) based on partition function is widely adoptedto reveal the multifractality, τ ( q ) = qh ( q ) − D f , (10)where D f is the fractal dimension, with D f = 1 in our case. As shown in Fig. 4(b)and (e), the τ ( q ) of the NYSE and the CSM presents a strong nonlinearity, whichis consistent with multifractal characteristic. By the Legendre transformation, thelocal singularity exponent α and its spectrum f ( α ) can be calculated as [54],6 = dτ ( q ) /dq, (11) f ( α ) = qα − τ ( q ) , (12)The difference between the maximum and the minimum of the local singularityexponent ∆ α , α max − α min is widely used to quantify the width of the extractedmultifractal spectrum. The larger the ∆ α , the stronger the multifractality. Fig. 4(c)and Fig. 4(f) illustrate the multifractal singularity spectra f ( α ) , with the width ofthe extracted multifractal spectrum ∆ α measured to be . and . respectivelyfor the NYSE and the CSM. It indicates the CSM shows stronger multifractalitythan the NYSE. We have investigated the memory effect of the instantaneous cross-correlations andthe average instantaneous cross-correlations based on the daily data of the NYSEand the CSM. It is interesting to find that, in spite of the absence of the long-range time-correlation of the instantaneous cross-correlations between two individ-ual stocks, the average instantaneous cross-correlation of a set of stocks is long-range correlated for the price return time interval ∆ t ′ = 1 day. The long-rangetime-correlation persists up to a month price return time interval for the NYSE, andhalf a month time interval for the CSM for the large time window.Multifractal nature is revealed for the average instantaneous cross-correlations bythe MF-DFA. By examining the MF-DFA function F q ( t ) , the scaling exponentfunction τ ( q ) , and the extracted multifractal spectrum f ( α ) , multifractal featuresare revealed for both the NYSE and the CSM. Acknowledgments:
This work was supported by the National Natural Science Foundation of China(Grant Nos. 10805025, 10774080) and the Foundation of Jiangxi Educational Com-mittee of China.
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F (t)F (t) t t (a) NYSE (b) NYSE(c) CSM (d) CSM t t Fig. 1. The DFA functions of the IC and the AIC are displayed on a log-log scale, with thecircles, triangles, crosses, diamonds, pluses and squares being six samples. The dashed linesare the power law fits. For clarity, some curves have been shifted downwards or upwards.(a)for the IC of the NYSE. The exponents are measured to be . , . , . , . , . and . . (b) for the AIC of the NYSE with N = 50 . The exponents are measured to be . , . , . , . , . and . . (c) for the IC of the CSM. The exponents are measured tobe . , . , . , . , . and . . (d) for the AIC of the CSM with N = 50 . Theexponents are measured to be . , . , . , . , . and . . (a) NYSE(b) CSMF (t) tF (t) t t c t c Fig. 2. The DFA functions of the
AIC are displayed on a log-log scale, with the dashedlines being the power law fits. Some curves have also been shifted downwards or upwardsfor clarity. (a) For the NYSE and N = 249 , the circles, triangles, crosses, diamonds, andpluses are for ∆ t ′ = 1 , , , , and days, respectively. The power-law exponent for ∆ t ′ = 1 day is measured to be . for the whole window size. Two stage power-lawexponents for ∆ t ′ = 5 , , and days are respectively measured to be . , . , . , . for the smaller window size, and . , . , . , . for the larger windowsize. (b)For the CSM and N = 259 , the circles, triangles, crosses, and diamonds are for ∆ t ′ = 1 , , , and days, respectively. The power-law exponent for ∆ t ′ = 1 dayis measured to be . for the whole window size. Two stage power-law exponents for ∆ t ′ = 5 , and days are respectively measured to be . , . , . for the smallerwindow size, and . , . , . for the larger window size. (a) NYSE(b) CSMF (t) q tF (t) t q Fig. 3. The MF-DFA functions of the
AIC are displayed on a log-log scale, with the dashedlines being the power law fits. The circles, triangles, crosses and diamonds are for q = − , , and , with curves being shifted downwards or upwards for clarity. (a) For the NYSEand N = 249 , with the exponents estimated to be . , . , . , and . . (b) For theCSM and N = 259 , with the exponents estimated to be . , . , . , and . . -2 -1 0 1 2 3 40.40.60.81-2 -1 0 1 2 3 40.50.60.70.80.9 -2 -1 0 1 2 3 4-4-3-2-1012 0 0.5 1 1.50.20.40.60.81-2 -1 0 1 2 3 4-3-2-1012 0 0.5 1 1.500.20.40.60.81 q α f( )qqqh(q) (a) NYSE h(q) q qq (d) CSM αα α αατ( )τ( ) f( ) (f) CSM(e) CSM(b) NYSE (c) NYSE Fig. 4. The multifractal analysis of the
AIC is displayed with N = 249 for the NYSEand N = 259 for the CSM. (a) and (d) are the MF-DFA exponents h ( q ) versus q for theNYSE and the CSM. (b) and (e) are the scaling exponent function τ ( q ) for the NYSE andthe CSM. (c) and (f) are the multifractal spectrum f ( α ) for the NYSE and the CSM.for the NYSE and the CSM.