Analysis of intra-day fluctuations in the Mexican financial market index
Léster Alfonso, Danahe E. Garcia-Ramirez, Ricardo Mansilla, César A. Terrero-Escalante
AAnalysis of intra-day fluctuations in the Mexicanfinancial market index -L´ester Alfonso a , Danahe E. Garcia-Ramirez b , Ricardo Mansilla c , C´esar A.Terrero-Escalante d ∗ a Universidad Aut´onoma de la Ciudad de M´exico,C.P. 09790, M´exico D.F., M´exico b Departamento de Astronom´ıa,DCNE-CGT, Universidad de Guanajuato,C.P. 36023, Guanajuato, M´exico. c Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades,Universidad Nacional Aut´onoma de M´exico,Ciudad Universitaria, C.P. 04510, M´exico D.F., M´exico. d Facultad de Ciencias, Universidad de Colima,Bernal D´ıaz del Castillo 340, Col. Villas San Sebasti´an,C.P. 28045, Colima, Colima, M´exico.
Abstract
In this paper, a statistical analysis of high frequency fluctuations of the IPC,the Mexican Stock Market Index, is presented. A sample of tick–to–tickdata covering the period from January 1999 to December 2002 was analyzed,as well as several other sets obtained using temporal aggregation. Our re-sults indicates that the highest frequency is not useful to understand theMexican market because almost two thirds of the information correspondsto inactivity. For the frequency where fluctuations start to be relevant, theIPC data does not follows any α -stable distribution, including the Gaussian,perhaps because of the presence of autocorrelations. For a long range oflower-frequencies, but still in the intra-day regime, fluctuations can be de-scribed as a truncated L´evy flight, while for frequencies above two-days, aGaussian distribution yields the best fit. Thought these results are consis-tent with other previously reported for several markets, there are significantdifferences in the details of the corresponding descriptions. ∗ Author’s names are arranged alphabetically. Corresponding author: [email protected]
Preprint submitted to Physica A February 14, 2020 a r X i v : . [ q -f i n . S T ] F e b eywords: stock markets, high frequency fluctuations distribution, tailbehavior, autocorrelations
1. Introduction
Is a fundamental assumption of the economic theory of markets that thefinancial markets are efficient in the sense of the community of economicagents being able to discount all the possibilities of arbitrage, incorporatingwith it all the relevant information for the prices formation, so that it isimpossible to design an (always) winning strategy for investment[1, 2]. Mathematically, this is described as a process, where a future increase ordecrease of the current price is the result of a random event. Underlying sucha random walk is a binomial distribution, which after a very large number ofsteps (price changes), converges to a normal, Gaussian, distribution. In turn,normal distributions are common in the description of systems in equilibrium,a stable state, small fluctuations around which decay exponentially with time.However, it seems that the financial market cannot actually be modeledas this kind of system; data for most financial indexes around the worldare statistically described by probability distributions that exhibit a largeskewness or kurtosis, relative to that of a normal distribution. It may meansthat strong perturbations are not, de facto, necessary for the market enteringa critical state; recurrent significant deviations of economic variables fromtheir average values could be just the cumulative and unavoidable resultof many small-scale processes and interactions occurring within the marketsystem, continuously subjected to a net external action like, for instance,publicly available announcements of annual earnings, stock splits, companiesprofits forecasts, new securities or, even more, the punctual action of agentshaving preferential access to restricted or confidential information.In this regard, for years now particularly good fits to financial data havebeen obtained using L´evy-stable distributions [3]. Nevertheless, it is obviousthat, strictly speaking, the market can neither be such a random process.First, it is unreasonable to expect data from any real (as opposed to hypo-thetical) process to have an infinite variance. Secondly, it is reasonable to For more details on the background and definitions of the concepts we use in thispaper see book [2]. α -stable L´evy distribution cannot be rejected at the 5% sig-nificance level. This implies that the daily data can safely being consideredas independent and identically distributed, characterized by an infinite vari-ance. Our aim here is to study how this conclusion changes when tick-to-tickdata is analyzed.The paper is organized as follows. In the next section we provide a briefreview of L´evy distributions, in particular of the stable and L´evy truncateddistributions which we use for fitting the IPC fluctuations. The descriptionof this data, as well as the sets derived from them and used for our analysisis presented in section 3. Next in sections 4 and 5 we proceed to present theanalysis of the probability distribution and serial dependence of the actualfluctuations of the IPC data. We devote the last section to discuss our resultsand state the main conclusions.
2. L´evy distributions
As it was mentioned in the introduction, L´evy distributions are amongthe more frequently used to fit data from complex processes. Particularly, ithas been found to be an excellent fit to the distribution of stock returns aswell as of other financial time series. In this section we will briefly reviewthe definition and properties of the stable and truncated L´evy distributions.
While studying the behavior of sums of independent random variablesPaul L´evy [3] introduced a skew distribution specified by scale γ , exponent4 , skewness parameter β and a location parameter µ . Since the analyti-cal form of the Levy stable distribution is known only for a few cases, theyare generally specified by their characteristic function. The most popularparameterization is defined by Samorodnitsky and Taqqu [13] with the char-acteristic function: φ ( t ) = (cid:40) exp (cid:0) − γ | t | (cid:2) iβ π sign(t) ln( | t | ) + i µ t (cid:3)(cid:1) , if α = 1 . exp (cid:0) − γ α | t | α (cid:2) − iβ tan (cid:0) πα (cid:1) sign( t ) + iµt (cid:3)(cid:1) , otherwise . (1)where sign ( t ) stands for the sign of t . Then, the probability density functionis calculated from it with the inverse Fourier transform in the form: f ( x ; α, β, γ, µ ) = 12 π (cid:90) + ∞−∞ φ ( t )e − itx dt . (2)L´evy distributions are characterized by the property of being stable underconvolution, i.e, the sum of two independent and identically L´evy-distributedrandom variables, is also L´evy distributed with the same stability index α .The stability parameter α lies in the interval (0 , α represents asharp peak but heavy tails which asymptotically decay as power laws withexponent − ( α + 1). For the normal distribution α = 2. For symmetricdistribution (like the normal distribution), the skewness parameter β = 0.The skewness parameter must lie in the range [ − , β = +1 , − γ lies in the interval (0 , ∞ ),while the location parameter µ is in ( −∞ , + ∞ ).The asymptotic behavior of the L´evy distributions is described by theexpression f ( x ; α ) ≈ | x | − − α . (3)Hence, the variance of the Levy stable distributions is infinite for all α < As mentioned in the previous subsection, α -stable Levy distributions haveinfinite variance, hence, they have power-law tails that decay too slowly.Therefore, the fit of empirical data by L´evy stable distributions will usuallyoverestimate the probability of extreme events. In particular, real pricesfluctuations have finite variance, so their distribution decays slower than aGaussian, but faster than a Levy-stable distribution, with the tails betterdescribed by an exponential law than by a power law. The truncated L´evy5ight (TLF) was proposed by Mantegna and Stanley [9] to overcome thisproblem, and can be defined as a stochastic process with finite variance andscaling relations in a large, but finite interval. They defined the truncatedL´evy flight distribution as: P ( x ) = x > lcP L ( x ) if − l ≤ x ≤ l x < − l , (4)where P L ( x ) is a symmetric L´evy distribution. As the TLF has a finitevariance, with sequential convolution it will converge to a Gaussian process,but the convergence is very slowly, as was demonstrated by Mantegna andStanley [9]. However, the cutoff in the tail given by (4) is abrupt. Thisproblem was solved by Koponen [10], who introduced an infinitely divisibleTLF with an exponential cutoff with the characteristic function:log φ ( t ) = − c α cos( πα ) (cid:20)(cid:0) t + λ (cid:1) α cos (cid:18) α arctan | t | λ − λ α (cid:19)(cid:21) , (5)where c is the scaling factor, α is the stability index, and λ is the cutoffparameter. The L´evy α -stable law is restored by setting to zero the cutoffparameter. For small values of x , the truncated Levy density described bythe characteristic function (5), behaves like a L´evy-stable law of index α [14]. It was used by Matacz [15] to describe the behavior of the AustralianAll Ordinaries Index.
3. Data sets from IPC values
For our analysis we used the IPC value over the period January 1999-December 2002 which comprises 4321427 transactions. Taking into accountthat the Mexican trading day is of six and a half hours, this give us a meantime between transactions of 5 . N = 1164256 elements with a mean value over the period of6209 . . . Y k , are now irregularly sampled, with a mean time between fluctuationsof 19.3 seconds. This set is plotted in figure 1.6 igure 1: IPC actual fluctuations. The horizontal red line corresponds to the mean valueover the period. The actual fluctuations are then taken as the difference of the naturallogarithm of the Y k , S k = ln Y k +1 − ln Y k , for k = 1 , , . . . , N . (6)For this study we also used sets corresponding to the convolution of thedensity distributions of high-frequency data { S k } , i.e., sets obtained aftersumming for different values of N conv , S N conv j = j × N conv (cid:88) k =1+( j − × N conv S k , for j = 1 , , . . . , ¯ N , (7)where ¯ N is the multiple of N conv closer to N .For completeness we analyzed also a set of closing values of the IPC forthe same period downloaded from the Yahoo Finance website.7 . Probability distribution of IPC fluctuations We started by analyzing the non-convoluted data. The correspondingdistribution is presented in figure 2, where a logarithmic scale is used forthe vertical axis and the horizontal axis has been rescaled dividing by thestandard deviation. The plots of best fits to a Gaussian (narrow blue curvebelow the data points) and a L´evy (wide red curve above the data points)distributions are also shown. It can be observed that the probability distri-
Figure 2: Distributions for N conv = 0. In the horizontal axis Z stands for the fluctuationsdata or the values of the corresponding fits. bution function for this data does not correspond to a normal distribution,but it neither does to a L´evy one. This is confirmed by the results of theKolmogorov-Smirnov test shown in table 1 for the best fit of data to an α -stable distribution.From this table we can also observe that as N conv reaches a value around70, the test cannot longer reject the hypothesis of the probability densityfunction for this data being an α -stable distribution. As it is shown in fig.3,L´evy scaling holds over a long range of values of N conv . For instance, infig.4 is presented the data and the best fits for a normal and an α -stabledistribution for N conv = 100.The values of α keep steadily increasing as N conv is also increased. Asa matter of fact, as it can be seen in table 1 and it is represented in Fig.5,the value of the stable coefficient slowly converges to 2, while the convolutioninvolves larger blocks of data. That is, for instance, the value of α for N conv =2700. For this case the corresponding statistics are presented in the last row For the analysis of stable distributions, we used the library developed by J. Nolan [16]. able 1: Results of the Kolmogorov-Smirnov Goodness of Fit Test (K-S test). N conv α β γ δ K-S Statistics p-value Reject H ? p = 0 .
050 1 . − . . . . . . . . − . . . . . . − . . . . . . − . . . . . . − . . . . . . − . . . . . . − . . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . . . . − . . . . − . . . . . . − . . . . . . − . . . . . . − . . . . . . − . . . . . . . . . . . igure 3: Results of the Kolmogorov-Smirnov Goodness of Fit Test versus N conv .Figure 4: Distributions for N conv = 100. In the horizontal axis Z stands for the fluctua-tions data or the values of the corresponding fits.Figure 5: The stable coefficient α as function of N conv .
10f table 1 and the distributions are plotted in Fig.6. As we can see, for
Figure 6: Distributions for N conv = 2700. In the horizontal axis Z stands for the fluctua-tions data or the values of the corresponding fits. N conv = 2700 a normal distribution and the corresponding L´evy distribution(with α = 2) give both a very good fit to the convoluted data. This is astrong evidence that, indeed, the variance of the data is finite. α = 2Note in figure 5 that α ≈ N conv around 2000 too. Also in table 1 itcan be seen that the convergence to α = 2 is not just slow, as has been notedpreviously [9], but it is also non-uniform. We believe that this is an effectof the finite number of elements in the sample. To verify that, we simulateddata by truncating sets generated using the stable library by Nolan [16] andfollowing expression (4). For the three sets we used the same parameters, butthe length of the series are one, two and five millions elements respectively.In fig.7 is plotted how the stable coefficient α for each original set (withouttruncation) evolves with convolution. It is noticeable the high quality ofthe simulated data, since in each case α converges to a value significantlydifferent from 2 and, the larger the size of the set, the closer this value getsto the α used for generating the set.The truncation was done by erasing out the elements of a given set withabsolute value greater than n std × σ , where σ is the standard deviation of thecorresponding data. It implies that, the smaller n std , the fewer the elementsremaining in the truncated set. There is a range of n std when the simulationworks. For high n std (i.e., small truncation), the length of the series is notlarge enough for observing the convergence to α = 2, thought it seems toconverge to a value significantly different from the value used for generating11 igure 7: The stable coefficient α as function of N conv , without truncation. the set, i.e., the case without convolution presented in fig.7. An exampleis given in fig.8. On the other hand, for low n std (i.e., large truncation), Figure 8: The stable coefficient α as function of N conv for n std = 50. the Kolmogorov-Smirnov test rejects that the data correspond to a stabledistribution. Neither it is a normal distribution, but it converges very fast to α = 2. An example is given now in fig.9. Therefore, reasonable truncationcan be performed, such that it can be obtained a series that for low N conv Figure 9: The stable coefficient α as function of N conv for n std = 10. the test cannot reject them to be stable-distributed, but with convolutionconverges to α = 2. The corresponding example is given in fig.10. Figure 10: The stable coefficient α as function of N conv for n std = 35. From these simulations it can be observed that the convergence is notonly slow, but also non uniform. However, the larger the amount of elementsin a given set, the smaller the size of the irregularities. If considering the13hole population, (a condition for the generalization of the Central LimitTheorem), the convergence can be expected to be uniform.
Since our sample is finite and this affect the estimation of the parameter α ,we further analyzed the transition from L´evy to Gaussian regime by followingthe procedure proposed in Ref.[17]. We study the behavior of the excesskurtosis for convoluted samples, from N conv = 0 to N conv = 3000. The excesskurtosis k ≡ (cid:104) ( S i − µ ) (cid:105)(cid:104) ( S i − µ ) (cid:105) − , (8)gives a statistical measure of the heaviness of the tail of a distribution withmean value µ . A normal process shows zero excess kurtosis for the popula-tion, while it is positive for leptokurtic distributions like L´evy-stable distri-butions.The results we obtained are presented in fig. 11. It can be seen that Figure 11: Excess kurtosis for the samples as a function of N conv . The circle encloses theL´evy to Gaussian crossover. N conv between 2700 and 3000. This confirms the result discussed above for the fitof the convoluted data to a stable L´evy distribution (see table 1), that gives avalue of α = 2 for N conv = 2700. This way, the crossover time can be set equalto N c = 2700. Recalling that, during the analyzed period (from January 1999to December 2002), the average time between successive fluctuations is closeto 20 seconds and that for the Mexican market one trading day is equal to 6.5hours, we find that the Levy-Gaussian crossover is approximately 2.3 tradingdays.
5. Serial dependence in the IPC data
From the previous section we concluded that for convolutions below N conv =70 data does not fit neither a Gaussian nor any L´evy distribution. A com-mon hypothesis for both cases is the data being independent and identicallydistributed. In figures 12 and 13 we present the results for the analysis ofautocorrelation for the set { S k } . In principle, the vertical axis would cover Figure 12: Autocorrelations for N conv = 0 (left) and N conv = 10 (right) values from − δ , i.e., denote the correlation between S k and S k + δ . The blue dashed lines in these figures indicate approximate limitsof correlation coefficients expected under a null hypothesis of uncorrelateddata. We successfully tested these limits using the simulations of truncated15 igure 13: Autocorrelations for N conv = 50 (left) and N conv = 150 (right) L´evy distributions described in subsection 4.1. As it can be observed the fluc-tuations exhibit positive autocorrelations, which are correspondingly dilutedafter convoluting the series. It suggests a mild serial dependence betweenfluctuations within an interval of about 48 minutes.In figures 14 are shown the autocorrelations for the series of absolute
Figure 14: Autocorrelations for the absolute value of the log returns for N conv = 0 (left)and N conv = 150 (right) values of S k for N conv = 0 (left) and N conv = 150 (right). This is a measureof volatility and it exhibits long range serial dependence, a fact consistentwith findings reported for other markets [2, 7].16inally, we present in figure 15 the autocorrelations for the data and Figure 15: Autocorrelations for the log returns of the dayly closing (left) and the corre-sponding absolute value (right). absolute value of the data of the daily closing set. As it can be observedthese last results are totally consistent with those obtained for the tick–to–tick set.
6. Discussion
Our results suggest that the statistical description of the Mexican mar-ket strongly depends on the time scale of interest. For the analyzed period,around 73% of the IPC tick–to–tick data, sampled every 5 . . S & P
500 index, sampled at a 1 min time scale, and also sometimes reported forvarious asset returns [7]. The autocorrelation in the IPC are diluted by con-volution, i.e., the aggregation of elements of the set of fluctuations in blocksof length N conv . This is characteristic of random walks with short memory.Conversely, in the case of a deterministic process with noise, even if auto-correlations are initially hidden, they surface and get more noticeable with17onvolutions. Therefore, this serial dependence seems to reflect less the in-ternal mechanics of the market (due to the law of supply and demand) thanthe (complex, noisy) external action on it.It is worthy to note that sets obtained by convolution of the IPC tick–to–tick data can be safely used to describe the activity at lower frequencies.We tested this by comparing, for instance, the fit for N conv = 1200, whichcorresponds to one trading day, with the daily data from Yahoo finance forthe same period (from January 1999 to December 2002). From table 1 we seethat α = 1 . . S & P
500 (during the six yearperiod from January 1984 to December 1989), estimated the crossover time tobe of the order of one month; Matacz [15], for the Australian All Ordinariesshare market index for the period 1993–1997, found that the crossover timeis approximately 19 trading days, and in Cuoto Miranda and Riera [17],a crossover of approximately 20 days was found for the Sao Paulo StockExchange Index in Brazil (IBOVESPA), during the 15 years period 19862000.The differences outlined here with respect to the stylized facts found forother markets seems to indicate that in the given period the behavior of theMexican market was atypical: the effects induced by external factors (forinstance, different kinds of expected announcements and unexpected actionof privileged agents) led the dynamics for periods of about an hour. As thiseffect vanishes, significant (as compared to normal) fluctuations of the indexfrom its average value were likely within an interval of a couple of tradingdays, but if analyzed over longer intervals, data describes a usual randomwalk, i.e., with the size of the fluctuations decaying exponentially.Nevertheless, there are reasons for expecting the statistical descriptionof the IPC to also depend on the period analyzed, and the variation fromperiod to period not just being given by the presence of critical events. Asmentioned before, in Ref.[8] we analyzed the daily closure data for the IPCcovering the period from 04 / / / / α = 1 .
64, which is significantly lower than the value of 1 . / / S & P
500 about a decade earlier.
7. Acknowledgments
This research was supported by the Sistema Nacional de Investigadores(M´exico). It was also partially funded by a grant from the Consejo Nacionalde Ciencia y Tecnologa of Mexico (SEP–CONACyT) CB-284482. D. G-R. thanks the support of the Facultad de Ciencias, Universidad de Colima,where most of her contribution was done.
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