Multifractality in stock indexes: Fact or fiction?
aa r X i v : . [ q -f i n . S T ] J un Multifractality in stock indexes: Fact or fiction?
Zhi-Qiang Jiang a , b , Wei-Xing Zhou a , b , c , ∗ a School of Business, East China University of Science and Technology, Shanghai 200237,China b School of Science, East China University of Science and Technology, Shanghai 200237,China c Research Center of Systems Engineering, East China University of Science andTechnology, Shanghai 200237, China
Abstract
Multifractal analysis and extensive statistical tests are performed upon intraday minutelydata within individual trading days for four stock market indexes (including HSI, SZSC,S&P500, and NASDAQ) to check whether the indexes (instead of the returns) possessmultifractality. We find that the mass exponent τ ( q ) is linear and the singularity α ( q ) isclose to 1 for all trading days and all indexes. Furthermore, we find strong evidence showingthat the scaling behaviors of the original data sets cannot be distinguished from those ofthe shuffled time series. Hence, the so-called multifractality in the intraday stock marketindexes is merely an illusion. Key words:
Econophysics, Multifractal analysis, Bootstrapping, Stock markets
Econophysics is an emerging interdisciplinary field applying concepts, theories,and tools borrowed from statistical physics, nonlinear sciences, applied mathemat-ics, and complexity sciences to understand the complex self-organizing behaviorsof financial markets [1, 2, 3, 4]. This field has become to flourish since the pioneer-ing work of Mantegna and Stanley on the scaling behavior in the dynamics of theStandard & Poor’s 500 index [5], which is closely related to the Pareto-L´evy law ∗ Corresponding author. Address: 130 Meilong Road, School of Business, P.O. Box 114,East China University of Science and Technology, Shanghai 200237, China, Phone: +86 2164253634, Fax: +86 21 64253152.
Email address: [email protected] (Wei-Xing Zhou).Preprint submitted toPhysica A 8November 2018 roposed by Mandelbrot in the description of cotton price fluctuations [6]. Econo-physicists have uncovered remarkable similarities between financial markets andturbulent flows [1, 4]. Such analogues include (but not limited to) the evolution ofprobability densities of financial returns [7] based on the variational theory in turbu-lence [8, 9, 10, 11], inverse statistics in stock markets [12, 13, 14] motivated by theinverse structure function analysis of velocity [15, 16, 17, 18, 19], scale-invariantdistribution of multipliers defined from volatility of equities [20] and from dissi-pating energy [21, 22, 23, 24], and intermittency and multifractality of asset returns[7, 25].Indeed, the multifractal nature of equity returns is one of the most important styl-ized facts. A small part of this literature contains the studies on the foreign exchangerate [7, 25, 26, 27, 28, 29, 30, 31], gold price [28], commodity price [32], returnsof stock price or indexes [32, 33, 34, 35, 36, 37, 38, 39, 40, 41], and so on. Wenote that the quantity price (or its logarithm) in financial markets is the analogueof velocity in turbulence. Similarly, the counterpart of velocity difference in fluidmechanics is the asset return . In this framework, it is natural that numerous multi-fractal analyses have been carried out on the returns for financial equities similar tothe velocity differences for turbulent flows.However, there are exceptions, where analysis is performed on several indexes di-rectly rather than their variations (the returns) and the presence of multifractality inthe several indexes is claimed [42, 43, 44]. Specifically, they performed multifractalanalysis on the intraday high-frequency data of Hang Seng Index (HSI), ShanghaiStock Exchange Composite Index (SSEC), and Shenzhen Stock Exchange Com-posite Index (SZEC) within individual trading days. The extracted “multifractal”spectra f ( α ) were then utilized to predict abnormal price movements and serve asa risk measure in risk management. It seems to us that a careful scrutiny on theobtained multifractality should be undertaken based on the extremely narrow spec-tra of the singularity α . Two problems arise, casting doubts on the aforementionedanalysis [45].Firstly, based on the multifractal theory, there exists a constant α ( t ) for each mo-ment t such that the investigated measure µ on the neighbor B ( t, l ) of x scale with l when the scale l → , µ ( B ( t, l )) ∼ l α ( t ) . (1)The measure µ is singular at arbitrary moment t with the singularity strength being α ( t ) . When µ is defined as the sum of index prices within a given time interval, µ ( B ( t, l )) is approximately proportional to l , that is, α ( t ) ≈ for all t . This sug-gests that the measure µ does not possess multifractal nature. This inference is fur-ther supported by the fact that the span of singularity strength ∆ α = α max − α min ≈ in the real data [42, 43, 44].Secondly, in the analysis of multifractality in turbulence or high-frequency financialdata, the moment order q should not be greater than 8 in order to make the partition2unction converge. Specifically, it is shown that the size of a time series should beno less than one million to ensure the estimate of its eighth order partition functionstatistically significant [19, 46]. The situation is similar for high-frequency financialdata [20]. Hence, it is of little significance to compute partition function for higherorders. In the analysis of minutely (or five-minute) data within a time period of oneday [42, 43, 44], the size of the intraday high-frequency data is no more than 240while the moment order is taken to be − q . This usually broad intervalof q casts further doubts on the reported multifractality in the indexes.Despite of the specific considerations discussed above, it is worthwhile to put fur-ther comments in general on the investigation of multifractality in financial data.The multifractal features in financial series have attracted great interests, however,the origin and significance of the extracted “multifractality” is less concerned. Onone hand, it has been shown that an exact monofractal financial model can lead toan artificial multifractal behavior [47]. On the other hand, a time series of the pricefluctuations possessing multifractal nature usually has either fat tails in the distri-bution or long-range temporal correlation or both [48]. However, possessing longmemory is not sufficient for the presence of multifractality and one has to have anonlinear process with long-memory in order to have multifractality [49]. In manycases, the null hypothesis that the reported multifractal nature is stemmed from thelarge fluctuations of prices cannot be rejected [50].In this work, we focus on the presence of multifractal feature in stock market in-dexes and testing whether the obtained empirical multifractality stems from randomfluctuations. To address these issues, we adopt the bootstrap approach by shufflingthe intraday index series and perform multifractal analysis on them. The results arecompared with that from original data. This paper is organized as follows. In Sec. 2,we describe the data sets we investigate. The basic multifractal method is explainedin detail in Sec. 3. Multifractal analysis of the data sets is presented in Sec. 4. Sta-tistical bootstrapping tests are conducted in Sec. 5. Finally, Sec. 6 concludes. To gain a more profound insight into the multifractality in intraday stock marketindexes, we investigate four important indexes, i.e. , the Hang Seng Index (HSI),Shenzhen Stock Exchange Composite Index (SZSC), Standard & Poor’s 500 Index(S&P 500), and the National Association of Securities Dealers Automated Quota-tion (NASDAQ). HSI and SZSC are selected since they were used in the originalwork of this topic [42, 43, 44]. Both the Hongkong Stock Exchange and Shen-zhen Stock Exchange are emerging markets. The S&P 500 and NASDAQ that arerepresentative of mature stock markets are chosen for comparison.The data have been recorded at each minute in trading days. The HSI index covers3rom Jan. 2, 1997 to May 28, 1997, the SZSC index is from Nov. 12, 2001 to Aug.17, 2006, the S&P 500 index is recorded from Jan. 2, 1997 to Feb. 26, 1999, andthe NASDAQ index ranges from Aug. 18, 2000 to Oct. 30, 2000. Eliminating theweekend, holidays and, the days having recording errors, there are 101 days for theHSI data, 1149 days for the SZSC data, 448 days for the S&P 500 data, and 45 daysfor the NASDAQ data, respectively.
We use the box counting method following the work of [42, 43, 44] to investigatethe multifractal nature of the index series of each trading day. Denote the intradayindex series as { I ( t ) : t = 1 , , ··· , T } , where T = 240 for HSI and SZSC, T = 405 for S&P 500, and T = 390 for NASDAQ, respectively. For a given box size l , weobtain N = T /l boxes and construct a measure µ on each box as follows, µ ( n ; l ) = µ ([( n − l + 1 , nl ]) = l X i =1 I [( n − l + i ] , (2)where [( n − l +1 , nl ] is the n -th box and l ∈ [1 , , , , , , , , , , , , , for HSI and SZSC, l ∈ [1 , , , , , , , , , for S&P 500,and l ∈ [1 , , , , , , , , , , , , , for NASDAQ, respec-tively. The sizes l of the boxes are chosen such that the number of boxes of eachsize is an integer to cover the whole time series.We then construct the partition function χ q as χ q ( l ) = N X n =1 " µ ( n ; l ) P Nm =1 µ ( m ; l ) q , (3)and expect it to scale as χ q ( l ) ∼ l τ ( q ) , (4)which defines the exponent τ ( q ) . The local singularity exponent α of the measure µ and its spectrum f ( α ) are related to τ ( q ) through a Legendre transformation [51] ( α = d τ ( q ) / d qf ( α ) = qα − τ ( q ) . (5)In order to keep the comparability of our results with those in [44], we also pose − q .When µ ( n ; l ) / P µ ( m ; l ) ≪ and q ≫ , the estimate of the partition function χ will be very difficult since the value is so small that it is out of the memory.To overcome this problem, we can calculate the logarithm of the partition function4 n χ q ( l ) rather than the partition function itself. A simple manipulation results inthe following formula, ln χ q ( l ) = ln N X n =1 µ ( n ; l )max m { µ ( m ; l ) } q + q ln max m { µ ( m ; l ) } P µ ( m ; l ) , (6)where max m { µ ( m ; l ) } is the maximum of µ ( m ; l ) for m = 1 , , · · · , N . Four dates (Jan. 8, 1997 for HSI, Nov. 26, 2001 for SZSC, Feb. 10, 1997 for S&P500, and Aug. 22, 2000 for NASDAQ) are taken as examples to show the results ofmultifractal analysis. Figure 1 shows the dependence of the partition function χ q ( l ) on the box size l for different values of q in log-log coordinates. Excellent power-law scaling of χ q ( l ) with respect to l has been observed and the scaling range coversall the selected values of l . The solid lines are the best linear fits to the data. −1 −20 −10 q = −9 q = −6 q = −3 q = 0 q = 3 q = 6 q = 9 (a) l χ q ( l ) −1 −20 −10 q = −9 q = −6 q = −3 q = 0 q = 3 q = 6 q = 9 (b) l χ q ( l ) −1 −20 −10 q = −9 q = −6 q = −3 q = 0 q = 3 q = 6 q = 9 (c) l χ q ( l ) −1 −20 −10 q = −9 q = −6 q = −3 q = 0 q = 3 q = 6 q = 9 (d) l χ q ( l ) Fig. 1. Plots of χ q ( l ) as a function of the box size l for different values of q in log-logcoordinates. The solid lines are the least-squares fits to the data using linear regression (inlog-log coordinates) corresponding to power laws. (a) HSI, (b) SZSC, (c) S&P 500, and (d)NASDAQ. The scaling exponents τ ( q ) are given by the slopes of the linear fits to ln χ q ( l ) with respect to ln l for different values of q . Figure 2 plots the dependence of5he mass exponents τ ( q ) as a function of the moment order q . One observes thatthere is an evident linear relationship between τ ( q ) and q for all the four exam-ples. The solid lines are the least-squares fits to the data. The slopes of the linesare respectively ¯ α = 1 . ± . for HSI, ¯ α = 1 . ± . for SZSC, ¯ α = 1 . ± . for S&P 500, and ¯ α = 1 . ± . for NASDAQ, re-spectively. All the corresponding correlation coefficients of the linear fits are equalto . . Furthermore, the linear relationships are also hold for other trading days.Therefore, there is no evidence of nonlinearity in the functions τ and the intradaystock market index do not exhibit multifractal nature. Since α ( q ) = d τ ( q ) / d q , weexpect that α ( q ) ≈ for all q , as expected in our discussion in Sec. 1. −150−100 −50 0 50 100 150−150−100−50050100150 (a) q τ ( q ) −150−100 −50 0 50 100 150−150−100−50050100150 (b) q τ ( q ) −150−100 −50 0 50 100 150−150−100−50050100150 (c) q τ ( q ) −150−100 −50 0 50 100 150−150−100−50050100150 (d) q τ ( q ) Fig. 2. Dependence of the scaling exponent τ ( q ) on the order q . The solid lines are theleast-squares fits to the data. (a) HSI, (b) SZSC, (c) S&P 500, and (d) NASDAQ. Figure 3 presents the multifractal singularity spectra f ( α ) obtained through Leg-endre transformation of τ ( q ) defined by Eq. (5). The curves in Fig. 3 have thegeometrical features of the conservable multifractal spectra [4, 52], which makesthem look as if there is sound evidence for the presence of multifractality. However,when looking at the disperseness of the sigularity strength ∆ α , α max − α min , wefind that ∆ α is very close to zero. It is well-known that ∆ α is an important pa-rameter qualifying the width of the extracted multifractal spectrum. The larger isthe ∆ α , the stronger is the multifractality. According to Fig. 3, even in the caseof − q , ∆ α < . for NASDAQ. One can see that the values of ∆ α for other indexes are much smaller than that of NASDAQ. This observationindicates that there is no multifractality in stock market indexes.6 .999 0.9995 1 1.0005 1.0010.950.960.970.980.991 α f ( α ) HSISZSCS&P500NASDAQ
Fig. 3. (Color online) Multifractal spectra f ( α ) obtained by the Legendre transform of τ ( q ) for different indexes. (a) α f ( α ) OriginalShuffled (b) α f ( α ) (c) α f ( α ) (d) α f ( α ) Fig. 4. (color online) Comparison of multifractal spectra extracted from real and shuffledstock marker indexes. The solid lines are the real data, while the dotted lines are the shuffleddata. (a) HSI, (b) SZSC, (c) S&P 500, and (d) NASDAQ.
We access further the statistical significance of the empirical multifractality in thesprit of bootstrapping. For a given intraday time series, we reshuffle the series toremove any potential temporal correlation and carry out the same multifractal anal-ysis as for the original data. For the four examples discussed in Sec. 4, we computethe multifractal spectra of ten reshuffled time series for each index. The results areillustrated in Fig. 4, where the solid lines are associated with real stock market7ndexes, while the dotted lines are obtained from the shuffled data of the corre-sponding indexes. We find that the multifractal spectra of the real indexes f ( α ) andthat of the shuffled data f rnd ( α rnd ) are almost overlapping together in Fig.4 (b) and(c). Although the solid lines and the dotted line can be distinguished clearly in Fig.4(a) and (d), the differences between α and α rnd are ignorable. In other words, themultifractal nature in the real indexes is insignificant in these examples.For each intraday time series, we shuffle the data for 1000 times. The associatedmultifractal spectra are obtained. For each singularity spectrum, we calculate twocharacteristic quantity, ∆ α and F , [ f ( α min ) + f ( α max )] / . Figure 5 shows thescatter plots of F rnd for the shuffled data versus the corresponding ∆ α rnd for thefour example trading days. Clear linear relationship between F rnd and ∆ α rnd foreach case is observed and we have F rnd = k ∆ α rnd + b , (7)where k = − . and b = 1 . for HSI, k = − . and b = 1 . for SZSC, k = − . and b = 1 . for S&P 500, and k = − . and b = 1 . forNASDAQ, respectively. The open circle in each plot of Fig. 5 presents the valuesof F and ∆ α for the real data. −3 ∆ α rnd F r nd (a) ShuffledOriginal −4 (b) ∆ α rnd F r nd −4 (c) ∆ α rnd F r nd −3 (d) ∆ α rnd F r nd Fig. 5. Scatter plots of the dependence of the shuffled F rnd and the corresponding ∆ α rnd .(a) HSI, (b) SZSC, (c) S&P 500, and (d) NASDAQ. Two striking facts emerge from Fig. 5. First, the 4000 points of (∆ α rnd , F rnd ) col-lapse on a same linear line since the values of k and b are identical for the fourplots. Second, the four points of (∆ α, F ) for the four real data sets also locate on8he same line. For other trading days, we have observed similar phenomena, whichput further evidence on our conclusion that the real and reshuffled time series haveundistinguishable scaling behaviors.The values of ∆ α and F for each original time series are compared with the aver-ages h ∆ α rnd i and h F rnd i of the 1000 corresponding shuffled data sets. The resultsfor the four indexes are illustrated in Fig. 6. The solid line is the main diagonal y = x . We find that ∆ α ≈ h ∆ α rnd i and F ≈ h F rnd i for all cases, which impliedthat the multifractal spectra of the shuffled data are very close to that of the real dataand the f ( α ) curves of real index data can be completely interpreted by the randomfluctuations of the original data sets. We stress that there are no extreme values inthe intraday index prices so that one can not attribute the observed multifractalityto tail fatness that is absent in the present case. Hence, the multifractal property inhigh-frequency stock market indexes obtained by partition function method is notstatistically significant. It is just an illusion. −5 −4 −3 −2 −1 −5 −4 −3 −2 −1 (a) 〈∆α rnd 〉 , 〈 F rnd 〉 ∆ α , F ∆α ∼ 〈∆α rnd 〉 F ∼ 〈 F rnd 〉 −6 −4 −2 −6 −4 −2 (b) 〈∆α rnd 〉 , 〈 F rnd 〉 ∆ α , F −6 −4 −2 −6 −4 −2 (c) 〈∆α rnd 〉 , 〈 F rnd 〉 ∆ α , F −4 −3 −2 −1 −4 −3 −2 −1 (d) 〈∆α rnd 〉 , 〈 F rnd 〉 ∆ α , F Fig. 6. Comparison of F and ∆ α obtained from the shuffled data and the real data. (a) HSI,(b) SZSC, (c) S&P 500, and (d) NASDAQ. In the presence case, to test the presence of multifractality amounts to testingwhether the local singularity exponent α = 1 , or ∆ α = 0 . As a last step, weimpose a very strict null hypothesis to investigate whether the f ( α ) spectrum iswider than those produced by chance. The null hypothesis is the following: H : ∆ α ∆ α rnd . (8)We can compute the p -value, which is the probability that the null hypothesis is9rue. The smaller the p -value, the stronger the evidence against the null hypothe-sis and favors the alternative hypothesis that the presence of of multifractality isstatistically significant. The false probability is estimated by p = Pr(∆ α ∆ α rnd ) . (9)Under the conventional significance level of . , the multifractal phenomenon isstatistically significant if and only if p . . While p > . , the null hypothe-sis cannot be rejected. A similar null hypothesis can be described as follows: H : F > F rnd , (10)where the false probability is p = Pr( F > F rnd ) . (11)Using the conventional significance level of . , the multifractal phenomenon isstatistically significant if and only if p . .For the four examples shown in Fig. 5, we find that p = 1 and p = 1 for HSI, p = 0 . and p = 0 . for SZSC, p = 0 . and p = 0 . for S&P 500, and p = 0 and p = 0 for NASDAQ. Obviously, we can not distinguish the real datafrom the shuffled data beside NASDAQ for the chosen trading days. We also findthat p ≈ p for all the trading days. More generally, Table 1 shows the statisticaltests for the all the each trading days. About half of the trading days can not passthe statistical inference, indicating that multifractality is absent in the those tradingseries. Table 1Statistical tests for the presence of multifractal nature in the four indexes investigated.
Indexes HSI SZSC S&P 500 NASDAQPercentage of p . p . We have investigated the multifractal features in intraday minutely high-frequencystock market indexes (including HSI, SZSC, S&P 500, and NASDAQ) for indi-vidual trading days. The resultant scaling functions τ ( q ) have been confirmed tobe linear and the singularities α are close to 1 so that ∆ α is close to 0. Thisanalysis implies that there is no multifractality in the indexes. Further evidencebased on bootstrapping technique shows that that the scaling behavior of the shuf-fled data is undistinguishable from that of the raw data. Specifically, we find that,101) almost all points (∆ α, F ) of the raw data sets locate on the same straight line F rnd = − α rnd + 1 extracted from the points (∆ α rnd , F rnd ) of the shuffled data;(2) for each time series, ∆ α ≈ h ∆ α rnd i and F ≈ h F rnd i ; and (3) the two ratherstrict null hypotheses cannot be rejected for about half of the time series. There isthus no doubt that the reported multifractal nature in the indexes of HSI and SZSC[42, 43, 44] is not a fact but a fiction. This conclusion is further verified by twoindexes (S&P 500 and NASDAQ) in a developed stock market. We believe that ouranalysis and conclusion apply for other market indexes or common stock priceswhen one concerns intraday stock prices or indexes rather than their returns.In addition, we cast doubts on the efforts to use this illusionary multifractal featureto forecast the stock market [43] and to define a risk index for risk management[44]. However, to be more conservative, we do not deny the potential usefulnessof those techniques proposed based on some nonexistent properties. The idea touse multifractal nature to predict or to manage risks in stock markets should beinvestigated based on the returns or other alternative financial quantities. After all,one cannot build a palace on a sand beach. Acknowledgments:
We are indebted to Prof. Bing-Hong Wang for providing the HSI data and fruitfuldiscussion. This work was partly supported by the National Natural Science Foun-dation of China (Grant No. 70501011), the Fok Ying Tong Education Foundation(Grant No. 101086), and the Shanghai Rising-Star Program (No. 06QA14015).
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