Physica A 391 (2012) 3484–3495

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Physica A

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Price–volume multifractal analysis and its application in Chinesestock marketsYing Yuan ∗, Xin-tian Zhuang, Zhi-ying LiuSchool of Business Administration, Northeastern University, Shenyang, 110819, China

a r t i c l e i n f o

Article history:Received 2 October 2011Received in revised form 18 November2011Available online 6 February 2012

Keywords:Trading volumeGeneralized Hurst exponentsPrice LimitsReform of Non-tradable SharesFinancial crisis

a b s t r a c t

An empirical research on Chinese stock markets is conducted using statistical tools.First, the multifractality of stock price return series, ri(ri = ln(Pt+1) − ln(Pt)) andtrading volume variation series, vi(vi = ln(Vt+1) − ln(Vt)) is confirmed using multifractaldetrended fluctuation analysis. Furthermore, a multifractal detrended cross-correlationanalysis between stock price return and trading volume variation in Chinese stockmarketsis also conducted. It is shown that the cross relationship between them is also found tobe multifractal. Second, the cross-correlation between stock price Pi and trading volumeVi is empirically studied using cross-correlation function and detrended cross-correlationanalysis. It is found that both Shanghai stock market and Shenzhen stock market showpronounced long-range cross-correlations between stock price and trading volume. Third,a composite index R based on price and trading volume is introduced. Comparedwith stockprice return series ri and trading volume variation series vi, R variation series not onlyremain the characteristics of original series but also demonstrate the relative correlationbetween stock price and trading volume. Finally,we analyze themultifractal characteristicsof R variation series before and after three financial events in China (namely, Price Limits,Reform of Non-tradable Shares and financial crisis in 2008) in the whole period of sampleto study the changes of stock market fluctuation and financial risk. It is found that theempirical results verified the validity of R.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The financial market can be thought of as a complex system and thus the operating law of the financial market is difficultto understand and describe. Traditional financial market theory relies on the Efficient Market Hypothesis (EMH) [1]. Underthe frame of EMH, stock returns follow a normal distribution and price behaviors obey ‘random-walk’ hypothesis. However,in some recent papers, a deviation fromEMHwas recently reported for both transition and somedeveloped economies [2–4].For example, the deviation from EMH was reported for the European transition economies by Podobnik et al. [4]. Theresults demonstrated that many time series of major indices exhibit power-law correlations in their returns. Therefore,it is necessary to develop an alternative method to extract the characteristics of the price fluctuation to enable an accurateestimate for risk prevention and control purpose.

In 1963, Mandelbrot observed the process of returns showing time scaling property in his analysis of cotton prices [5].After that,Mandelbrot& Stantely introduced themethodof scaling invariance from the complexity science into the economicsystems for the first time [6]. The fractal theory was applied to redescribe the scaling invariance property theoretically. Theexistence of scaling invariance characteristics was further confirmed by some prominent approaches, including rescaledrange analysis [7], the Levy stable distribution [8] and detrended fluctuation analysis [9–11]. However, these models couldnot be used to address the scaling behavior of the probability distributions in financial time series due to the limitation ofretrieving recapitulatory information.

∗ Corresponding author. Tel.: +86 24 83686784; fax: +86 24 23891569.E-mail address: yuanying1980@126.com (Y. Yuan).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.01.034

Y. Yuan et al. / Physica A 391 (2012) 3484–3495 3485

In order to make up the limitation of single fractal methods, Mandelbrot advanced a new method, namely, multifractalmethod [12]. Multifractal analysis can be used to divide a complex system into various regions according to their complexityand therefore is able to describe market volatilities. Therefore, many scholars analyzed multifractality of different marketsusingmultifractal tools. In fact, the presence ofmultifractality has been a ‘‘stylized fact’’ in financial markets [13]. It has beenverified that multifractality widely existed in financial markets such as stock markets, future markets, spot markets, foreignexchangemarkets, derivativemarkets, interest ratemarkets and so on [14–22]. Prior researches concentrated on confirmingthe multifractality of different financial markets, and nowadays some scholars try to explore the complicated statisticalcharacteristics. Ho et al. conducted a multifractal analysis on the Taiwan stock price index. He found that one scalingexponent is insufficient to capture the time dynamics of Taiwan stock market and the nature of multifractal phenomenain Taiwan stock market might be interpreted using themultiplicative cascade process of stock market information [23]. Sunet al. investigated statistically the correlations between the parameters of the multifractal spectrum and the variation ofclose returns [24]. Chen et al. predicted the price movements using two kinds of sign sequences as given conditions. One isthe parameter of themultifractal spectrum1f based on theminute indices, and the other is the variation of the closing index.Results showed that large fluctuations of the closing price and the conditions are strongly connected in these two methodsand some sign sequences of the parameter 1f can be used to predict the probability of near future price movements [25].Wei et al. proposed amultifractal volatility (MFV)model based on themultifractal spectrumof one trading day. His empiricalresults showed that the MFV model has the forecasting accuracy [26]. We also measured the multifractality of stock pricefluctuation and advanced two risk measures based on generalized Hurst exponents [27]. In addition, another robust andpowerful technique isMultifractal Detrended Fluctuation Analysis (MF-DFA), which is proposed by Kantelhardt in 2002 [28].The advantages of MF-DFA over many techniques are that it permits the detection of long-range correlations embedded inseemingly non-stationary time series, and also avoids the spurious detection of apparent long-range correlations that arean artifact of nonstationarity.

In addition, it should be noted that although some statistical characteristics of stock market data such as stock returns,price series and volatility series have become stylized fact, behavior of other related variables such as trading volumeis less studied. In fact, as a most important index, the trading volume contains much of useful information about thedynamics of price formation, which can help us understand the behavior of financial markets. Mu & Zhou studied the longmemory and multifractality in trading volumes for Chinese stocks and got some meaningful results [29]. Cross-correlationfunction is a well-known statistical method used to establish the degree of correlation between two time series. This isdone considering that stationarity characterizes both time series under investigation. However, most time series are hardlystationary. Therefore, when the time series need to be seen and analyzed as a whole cross-correlation function is not alwaysa valid choice. Podobnik and Stanley advanced a new method that deals with nonstationary time series, named DetrendedCross-Correlation Analysis (DCCA) [9]. Then long-range power-law volatility cross-correlations in finance were reportedbetween couple of time series by using DCCA and between large number of simultaneously recorded time series [30–34].For example, Podobnik et al. analyzed the cross-correlations between volume change and price change and power-lawcross-correlations were reported between the absolute values in price changes and volume changes [30]. On the basis ofDCCA, Zhou proposed a Multifractal DCCA (MF-DCCA) method to combine multifractal detrended fluctuation analysis andDCCA method [10]. Furthermore, Arianos and Carbone advanced a method for estimating the cross-correlation Cxy(τ ) oflong-range correlated series x(t) and y(t), at varying lags τ and scales n [35].

On the basis of previous literature, in this paper, we conduct an empirical research on Chinese stock market usingstatistical tools. This study extends previous work in several respects. Firstly, the multifractality of stock price return seriesand trading volume variation series are confirmed using MF-DFA. Furthermore, a multifractal detrended cross-correlationanalysis between stock price return series and trading volume variation series in Chinese stock markets is also conducted.It is shown that the cross relationship between them are also found to be multifractal. Secondly, the cross-correlationbetween stock price Pi and trading volume Vi is empirically studied using cross-correlation function and detrended cross-correlation analysis. It is found that both Shanghai stock market and Shenzhen stock market show pronounced long-rangecross-correlations between stock price and trading volume. Thirdly, a composite index R based on price and trading volumeis advanced. Compared with stock price return series ri and trading volume variation series vi, R variation series not onlyremain the characteristics of original series but also demonstrate the relative correlation between stock price and tradingvolume. Finally, we analyze the multifractal characteristics of R variation series before and after three financial events inChina (namely, Price Limits, Reform of Non-tradable Shares and financial crisis in 2008) in the whole period of sample tostudy the changes of stockmarket fluctuation and financial risk. It is found that the empirical results verified the validity of R.

The structure of the paper is as follows. Section 2 describes the researchmethod. Section 3 introduces the empirical data.Section 4 presents the empirical results. Section 5 is a brief conclusion of the paper.

2. Research methods

2.1. Multifractal Detrended Fluctuation Analysis (MF-DFA)

Kantelhardt introduced a multifractal method called Multifractal Detrended Fluctuation Analysis (MF-DFA) [28]. Theoperation of MF-DFA on the series x(i), where i = 1, 2, . . . ,N and N is the length of the series, is as follows. With x weindicate the mean value of series x(i).

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We assume that x(i) are increments of a random walk process around the average x, thus the ‘‘trajectory’’ or ‘‘profile’’ isgiven by the integration of the signal

y(i) =

ik=1

[x(k) − x] , i = 1, 2, . . . ,N. (1)

Next, the integrated series is divided into Ns = int(N/s) nonoverlapping segments of equal length s. Since the lengthN of the series is often not a multiple of the considered time scale s, a short part at the end of the profile y(i) may remain.In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Thereby,2Ns segments are obtained altogether. We calculate the local trend for each of the 2Ns segments by a least-square fit of theseries. Then we determine the variance

F 2(s, v) =1s

si=1

{y[(v − 1)s + i] − yv(i)}2 (2)

for each segment v, v = 1, . . . ,Ns and

F 2(s, v) =1s

si=1

{y [N − (v − Ns)s + i] − yv(i)}2 (3)

for v = Ns + 1, . . . , 2Ns. Here, yv(i) is the fitting line in segment v. Then, we average over all segments to obtain the q-thorder fluctuation function

Fq(s) =

1

2Ns

2Nsv=1

[F 2(s, v)]q2

1q

(4)

where, in general, the index variable q can take any real value except zero. Repeating the procedure described above, forseveral time scales s, Fq(s) will increase as s increases. By analyzing log–log plots Fq(s) versus s for each value of q, wedetermine the scaling behavior of the fluctuation functions. If the series x(i) is long-range power-law correlated, Fq(s) willincrease for large values of s as a power-law

Fq(s) ≈ shq . (5)

The value h0 corresponds to the limit hq as q → 0, and cannot be determined directly by using the averaging procedureof Eq. (4) because of the diverging exponent. Instead, a logarithmic averaging procedure has to be employed,

F0(s) = exp

1

4Ns

2Nsv=1

ln[F 2(s, v)]

≈ sh0 . (6)

In general the exponent hq will depend on q. For stationary series, h2 is the well-defined Hurst exponent H . Thus, wecall hq the generalized Hurst exponent. hq is independent from q, which characterizes monofractal series. The differentscaling of small and large fluctuations will yield a significant dependence of hq on q. For positive q, the segments v withlarge variance (i.e. large deviation from the corresponding fit) will dominate the average Fq(s). Therefore, if q is positive, hqdescribes the scaling behavior of the segments with large fluctuations; and generally, large fluctuations are characterized bya smaller scaling exponent hq for multifractal time series. For negative q, the segments v with small variance will dominatethe average Fq(s). Thus, for negative q values, the scaling exponent hq describes the scaling behavior of segments with smallfluctuations, usually characterized by large scaling exponents.

Furthermore, in order to measure the degree of multifractality, we define 1h:

1h = hmax(q) − hmin(q) (7)

where 1h is the range of generalized Hurst exponents h(q). The greater is 1h, the stronger is the degree of multifractality,therefore the greater is the stock risk, and vice versa.

2.2. Detrended cross-correlation analysis and multifractal detrended cross-correlation analysis

Detrended cross-correlation analysis was proposed by Podobnik to quantify long-range cross-correlations between timeseries [9]. This method represents a generalization of the DFA and it is designed to analyze power-law cross-correlationsbetween two time series.

Consider two time series xi, i = 1, 2, 3, . . . ,N and yi, i = 1, 2, 3, . . . ,N . They are integrated to produceX(k) =k

i=1 x(i)and Y (k) =

ki=1 y(i), where k is an integer between 1 and N . Next, the integrated series are divided into Nn segments of

equal length n and in each segment a linear (or higher order polynomial) regression is performed to capture the local trend.

Y. Yuan et al. / Physica A 391 (2012) 3484–3495 3487

The integrated series X(k) and Y (k) are then detrended by subtracting the local trends Xn,s(k) and Yn,s(k) (ordinates of thestraight line or polynomials within each segment) from the data in each box and the detrended covariance is calculated as

F 2DCCA(n) =

1nNn

Nn−1s=0

n(s+1)k=ns+1

X(k) − Xn,s+1(k)

Y (k) − Yn,s+1(k)

. (8)

Repeating this calculation for all segment sizes provides the relationship between FDCCA(n) and the segment size n. Whenonly one series is analyzed (X(k) = Y (k)) the detrended covariance F 2

DCCA. If the original series xi and yi are power-lawcross-correlated, then FDCCA(n) ∼ nλ, and the scaling exponent λ are determined from the linear regression of log[FDCCA(n)]versus log(n). The interpretation of λ is similar to that of the DFA exponent α. Long-range cross-correlations between twoseries imply that each series has longmemory of its own previous values and additionally has a longmemory of the previousvalues of the other series [9].

Multifractal detrended cross-correlation analysis is proposed by Zhou et al. and it is a multifractal modification ofDCCA [10].

3. Data analyzed

The original data are taken from Chinese stock markets, which are composed Shanghai Stock Exchange (SHSE) and theShenzhen Stock Exchange (SZSE). The Chinese stock markets were newly set up markets and have only a history of morethan 20 years. The Shanghai Stock Exchange was established on November 26, 1990, and put into operation on December19. Shortly after, the SZSE was established on December 1, 1990 and put into operation on April 3, 1991. On a trading dayboth Stock Exchanges open from 9:30 to 11:30 am and then from 1:00 to 3:00 pm, so there are 4 trading hours on one day.

The data analyzed are daily closing price series, trading volume series and their logarithmic variation of Shanghai StockIndex (SHSI) and Shenzhen Component Index (SZCI). The sample period for SHSI is fromDecember 20, 1990 to December 30,2010 and the sample period for SZCI is fromApril 3, 1991 to December 30, 2010. In this paper, whenwe conductmultifractalanalysis we focus on the stock price return series (that is ln(Pt+1)− ln(Pt)), logarithmic trading volume variation series (thatis ln(Vt+1) − ln(Vt)) and logarithmic R variation series (that is ln(Rt+1) − ln(Rt)) of Chinese stock markets.

4. Empirical results

4.1. Multifractal analysis of Chinese stock markets

4.1.1. Multifractal characteristics of stock price return series in Chinese stock marketsWe performed the MF-DFA1 over the stock price return series of SHSI and SZCI, respectively. We calculated the

fluctuations Fq(s) for scales s ranging from 10 events to N/4, where N is the total length of the series. Fig. 1 shows thefluctuation functions Fq(s) of stock price return series for SHSI for q = −10, q = −8 · · · and q = +10, where the upperand the lower curves are the curves of q = 10 and q = −10. As can be seen from Fig. 1, there is a crossover time scales∗ (about s∗ ≈ 28) in the double logarithmic plots of Fq(s) versus s. This shows that there are different scaling laws andscaling exponents for time scales s > s∗ and s < s∗. Fig. 2(a) shows the q-dependence of the generalized Hurst exponent hqdetermined by fits in the regime 10 < s < N/4. As shown in Fig. 2(a), when q varies from −10 to 10, h(q) decreases from1.057 to 0.2817. h(q) is not a constant, indicating multifractality in time series. Similarly, Fig. 2(b) shows the generalizedHurst exponent plot of stock price return series of SZCI. As can be seen from Fig. 2(b), when q varies from −10 to 10, h(q)decreases from 1.1477 to 0.4538. h(q) is also not a constant, indicatingmultifractality in time series. The values of hq of stockprice return series in the regime 10 < s < N/4 for SHSI and SZCI are illustrated in Table 1, respectively.

4.1.2. Multifractal characteristics of trading volume variation series in Chinese stock marketsIn order to better understand the dynamics of price formation, it is very useful to understand the structure of trading

volumes. Similarly, we performed the MF-DFA1 over the trading volume variation series of SHSI and SZCI, respectively.Fig. 3(a) and (b) show the q-dependence of the generalized Hurst exponents hq determined by fits in the regime 10 < s <N/4 for SHSI and SHCI, respectively. As shown in Fig. 3, when q varies from −10 to 10, h(q) decreases monotonously forboth of them, indicating multifractality in both time series. The values of h(q) in the regime 10 < s < N/4 for SHSI and SZCIare illustrated in Table 2, respectively.

4.1.3. Multifractal detrended cross-correlation analysis between stock price return and trading volume variation in Chinese stockmarkets

Furthermore, we also conducted an empirical analysis on price return series and trading volume variation series usingmultifractal DCCA proposed by Zhou [10]. For detailed procedures, please see the Refs. [10,37]. Fig. 4(a) and (b) display therelationship between cross-correlation exponent h(q) and q. As can be seen from Fig. 4(a) and (b), both of cross-correlationrelationships for Shanghai stock markets and Shenzhen stock markets are found to be multifractal because for different

3488 Y. Yuan et al. / Physica A 391 (2012) 3484–3495

Fig. 1. The multifractal fluctuation function Fq(s) of stock price return series for SHSI.

(a) SHSI. (b) SZCI.

Fig. 2. The generalized Hurst exponents h(q) of stock price return series.

Table 1Values of the generalized qth-order Hurst exponentsh(q) of the stock price return series.

Order SHSI SZCIq k = 1 k = 1

−10 1.057 1.1477−8 1.0348 1.1246−6 1.0004 1.0876−4 0.942 1.0212−2 0.8187 0.8820 0.6096 0.64742 0.5439 0.6264 0.413 0.55856 0.3422 0.5088 0.3047 0.475310 0.2817 0.45381h 0.7753 0.6939

q, there are different exponents h(q). The values of h(q) for SHSI and SZCI are illustrated in Table 3, respectively. Moreclearly, the generalized Hurst exponents of stock price return series, trading volume variation series and cross-correlationrelationships between them are shown in Fig. 5(a) and (b).

Y. Yuan et al. / Physica A 391 (2012) 3484–3495 3489

(a) SHSI. (b) SZCI.

Fig. 3. The generalized Hurst exponents h(q) of trading volume variation series.

Table 2Values of the generalized qth-order Hurst exponentsh(q) of the trading volume variation series.

q SHSI SZCI

−10 0.3982 0.4229−8 0.3823 0.4048−6 0.3612 0.3807−4 0.3331 0.3488−2 0.2911 0.30540 0.2145 0.2352 0.1643 0.2034 0.0992 0.13136 0.0591 0.07118 0.0352 0.03410 0.0196 0.01091h 0.3786 0.412

(a) SHSI. (b) SZCI.

Fig. 4. The generalized Hurst exponents h(q) by MF-DCCA.

3490 Y. Yuan et al. / Physica A 391 (2012) 3484–3495

Table 3Values of the generalized qth-order Hurst exponentsh(q) by MF-DCCA.

q SHSI SZCI

−10 0.7026 0.8281−8 0.6713 0.7976−6 0.6235 0.7481−4 0.5604 0.6571−2 0.5079 0.52560 0.477 0.47362 0.444 0.43124 0.3801 0.36796 0.3152 0.30278 0.2724 0.260410 0.2451 0.23481h 0.4575 0.5933

(a) SHSI. (b) SZCI.

Fig. 5. The generalized Hurst exponents h(q) for stock price return, trading volume variation and cross-correlation relationship between them.

4.2. Cross-correlation between stock price and trading volume

As two of the most important variables of stock market data, price and trading volume interact each other and formtogether the dynamics of stock market. As we all known, trading volume can have a large impact on the stock prices. It isan established fact that trading volume is a major propelling force for stock prices. Therefore, it is necessary to research thecorrelation between price and trading volume.

4.2.1. Cross-correlation analysis between stock price and trading volumeThe conditional dependence of trading volume on stock price explains the correlation between trading volume and stock

price to a certain degree. In order to further understand the relation between trading volume and price, we investigate thecross-correlation between trading volume and price. The volume–price cross-correlation function is defined as ⟨V (t ′)P(t ′ +t)⟩, where t is the lagged time window [38].

Fig. 6 shows the volume–price cross-correlation function of the SHSI and SZCI. Each of Fig. 6(a) and (b) show a positivecross-correlation between trading volume and price for both the positive and the negative time direction. For the positivetime direction, it indicates that high volume triggers high price, and small volume triggers low price. Large volume indicateshigh liquidity ofmarkets, whichmay result in sharp rise ofmarkets, whereas small volume leadsmarkets to become inactiveand display a low price. For the negative time direction, it suggests also that high price triggers a big trading volume, andsmall price triggers a small volume [38]. The volume–price cross-correlation function shows that there exists a positivelong-range correlation between trading volume and price.

4.2.2. Detrended cross-correlation analysis between stock price and trading volumeIt should be noted that cross-correlation function is done considering that stationarity characterizes both time series

under investigation. However, most time series are hardly stationary. Therefore, when the time series need to be seen and

Y. Yuan et al. / Physica A 391 (2012) 3484–3495 3491

(a) SHSI. (b) SZCI.

Fig. 6. The volume–price cross-correlation functions.

analyzed as a whole cross-correlation is not always a valid choice. Podobnik and Stanley advanced a newmethod that dealswith nonstationary time series, named Detrended Cross-Correlation Analysis (DCCA) [9].

Podobnik et al. proposed a cross-correlation statistic in analogy to the Ljung–Box test [30]. For two series, {xt , t = 1,2, 3, . . . ,N} and {yt , t = 1, 2, 3, . . . ,N}, the test statistic

Qcc(m) = N2mi=1

Xi2

N − i. (9)

Here, the cross-correlation function

Xi =

Nk=i+1

xkyk−iN

k=1xk2

Nk=1

yk2. (10)

The cross-correlation statistic QCC (m) is approximately χ2(m) distributed with m degrees of freedom. The statistic canbe used the null hypothesis that none of the first m cross-correlation coefficient is different from zero. If for a broad rangeof m the test statistic of Eq. (9) exceeds the critical values of χ2(m)(QCC(m) > χ0.95

2(m)), we claim that there are not onlycross-correlations, but there are long-range cross-correlations [30,39].

Fig. 7(a) and (b) demonstrated the cross-correlation statistics between stock price and trading volume in Shanghai stockmarket and Shenzhen stock market. The degrees of freedom vary from 100 to 103. As a comparison, we also plot the criticalvalues withm increasing. As can be seen from Fig. 7(a) and (b), for the whole range ofm, all of the test statistics exceed thecritical values of χ2(m)(QCC(m) > χ0.95

2(m)), indicating that there are not only cross-correlations, but there are long-rangecross-correlations between stock price and trading volume for both Shanghai stock market and Shenzhen stock market.

Furthermore, the DCCA should be used to test the presence of cross-correlations quantitatively [30]. We conducted anempirical analysis on stock price and trading volume using DCCA. The empirical results are shown in Fig. 8(a) and (b).As can be seen from Fig. 8, both Shanghai stock market and Shenzhen stock market show pronounced long-range cross-correlations between stock price and trading volume because both of DCCA exponents are greater than 0.5(DCCA _SHSI ≈

1.3 and DCCA _SZCI ≈ 1.3).

4.3. Composite index based on price and trading volume and its multifractality

As seen from the empirical results in Section 4.2, there is a stable correlation between the trading volume and price.Indeed, an important quantity that characterizes the dynamics of price movements is the number of shares traded (tradingvolume) in a period of time. Based on the above-mentioned discussion, it is suggested that price and trading volume shouldbe used as a whole to analyze the Chinese stock market. Therefore, a composite index based on price and trading volume isadvanced according to the way of Ref. [36]:

R =VP

(11)

where R is the ratio trading volume to stock price, V is the trading volume and P is the closing price of stock price index.

3492 Y. Yuan et al. / Physica A 391 (2012) 3484–3495

(a) SHSI. (b) SZCI.

Fig. 7. The volume–price cross-correlation statistics.

(a) SHSI. (b) SZCI.

Fig. 8. The volume–price detrended cross-correlation analysis.

Therefore, a new composite index series for each of SHSI and SZCI can be calculated through Eq. (11). Furthermore, itis necessary to test the multifractality of the R series and compare the difference between R series and each of the originaldata.

We performed the MF-DFA1 over the logarithmic variation of R series of SHSI and SZCI, respectively. Fig. 9(a) and (b)show the q-dependence of the generalized Hurst exponents h(q) determined by fits in the regime 10 < s < N/4 for SHSIand SHCI, respectively. As shown in Fig. 9 when q varies from −10 to 10, h(q) decreases monotonously for both of them,indicating multifractality in both time series. The values of hq in the regime 10 < s < N/4 for SHSI and SZCI are illustratedin Table 4, respectively. More clearly, the generalized Hurst exponents of price return series, trading volume variation seriesand R variation series are shown in Fig. 10(a) and (b). As can be seen in Fig. 10(a) and (b), for both SHSI and SZCI, comparedwith original series, R variation series basically remain the same trends and display similar characteristics. In another words,R variation series do not change the characteristics of original series and meanwhile demonstrate the relative correlationbetween stock price and trading volume. Therefore, more valuable information can be revealed about the dynamics of stockmarket if the new index is valid.

4.4. Applications of the composite index in financial practices

Taking Chinese stockmarket as an example, we compared themultifractal characteristics before and after three financialevents in China (namely, Price Limits, Reform of Non-tradable Shares and financial crisis in 2008) in the whole period ofsample to study the changes of stock market fluctuation and financial risk. Price Limits started on December 27, 1996, Non-tradable Shares Reformwas on April 30, 2005 and Financial crisis in 2008 started on September, 15, 2008. (Because Lehman

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(a) SHSI. (b) SZCI.

Fig. 9. The generalized Hurst exponents h(q) of R variation series.

Table 4Values of the generalized qth-order Hurst exponentsh(q) of R variation series.

q SHSI SZCIk = 1 k = 1

−10 0.4364 0.503−8 0.4185 0.4836−6 0.3941 0.4574−4 0.3603 0.422−2 0.3082 0.37080 0.2126 0.306152 0.1543 0.24154 0.0836 0.16726 0.0421 0.11128 0.0183 0.079210 0.003 0.05971h 0.4334 0.4433

(a) SHSI. (b) SZCI.

Fig. 10. Comparison of the generalized Hurst exponents h(q) for different series.

Brothers filed for bankruptcy protection on September, 15, 2008, and this means the financial crisis has begun to break outformally.) Therefore, the three financial events divide the whole series into four periods. The first one is before Price Limits;the second one is after Price Limits and before Reform; and third one is after Reform and before financial crisis; the last oneis after financial crisis.

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Table 5Comparison of 1h before and after three events in Chinese stock markets.

Market states Before Price Limits After Price Limits before Reform After Reform before crisis After crisis

Time periods 12/19/1990–12/26/1996 12/27/1996–04/29/2005 05/09/2005–09/12/2008 09/16/2008–12/30/20101h (SHSI) 0.4614 0.4521 0.4911 0.55821h (SZCI) 0.597 0.4317 0.4647 0.6553

The1h based on generalizedHurst exponents of differentmarket states are shown in Table 5. As can be seen fromTable 5,the strength of multifractality after Price Limits is weaker than that of multifractality before Price Limits, and the 1h basedon generalized Hurst exponents after Price Limits is smaller than that of before Price Limits. These results indicate that afterthe Price Limits, the stock price fluctuation decreases and the Price Limits have played a positive role in stabilizing sharpfluctuations of market prices and stabilizing the stock market. In addition, the strength of multifractality after Reform ofNon-tradable Shares is stronger than that of multifractality before Reform of Non-tradable Shares (but after Price Limits),and the1h based on generalized Hurst exponents after Reform of Non-tradable Shares (but before financial crisis) is greaterthan that of before Reform of Non-tradable Shares (but after Price Limits). These results indicate that after Reform of Non-tradable Shares, the stock price fluctuation increases and the Reform of Non-tradable Shares increases the market risk.Finally, the strength of multifractality after financial crisis is stronger than that of multifractality before financial crisis (butafter Reform), and the 1h based on generalized Hurst exponents after financial crisis is greater than that of before financialcrisis. All of these findings are consistent with the real situations. These empirical results verify the validity of R.

5. Conclusions

For multifractal model describes the local characteristics of the asset price process through a range of scalings, itis acknowledged as the most appropriate model for price variations. The multifractal detrended fluctuation analysis,performed in this study, leads to a better understanding of such complex stock market.

Our study extends previous work in several respects. Firstly, the multifractality of stock price return series andtrading volume variation series are confirmed usingmultifractal detrended fluctuation analysis. Furthermore, a multifractaldetrended cross-correlation analysis between stock price return and trading volume variation in Chinese stock marketsis also conducted. It is shown that the cross relationship between them are also found to be multifractal. Secondly, thecross-correlation between stock price Pi and trading volume Vi is empirically studied using cross-correlation functionand detrended cross-correlation analysis. It is found that both Shanghai stock market and Shenzhen stock market showpronounced long-range cross-correlations between stock price and trading volume. Thirdly, a composite index R based onprice and trading volume is advanced. Compared with stock price return series ri and trading volume variation series vi, Rvariation series not only remain the characteristics of original series but also demonstrate the relative correlation betweenstock price and trading volume. Finally, we analyze the multifractal characteristics of R variation series before and afterthree financial events in China (namely, Price Limits, Reform of Non-tradable Shares and financial crisis in 2008) in thewhole period of sample to study the changes of stock market fluctuation and financial risk. It is found that the empiricalresults verified the validity of R.

However, the potential ofmultifractal analysis is far frombeing fully exploited, since only very recently has attention beendrawn to the need for a thorough testing of the multifractal tools. How to deeply understand the essence of the multifractalcharacteristics in capital markets and how to reveal more valuable information about market changes are two key issues inthe future. Study on these issues will help to make more accurate estimates for risk prevention and control.

Acknowledgments

We sincerely thank our anonymous referee. We also thank Zhou W.X. and Wei Y. for comments and discussion on thisarticle. We are appreciative for financial support by National Science Foundation of China (Nos 70901017, 70101022),Science Foundation of Postdoctors in China (No. 20080441095), Special Science Foundation of Postdoctors in China(200902546) and Fundamental Research Funds for the Central Universities (N100406003).

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