Analysis of intra-day fluctuations in the Mexican financial market index

-Léster Alfonso^a, Danahe E. Garcia-Ramirez^b, Ricardo Mansilla^c, César A. Terrero-Escalante^d ^a Universidad Autónoma de la Ciudad de México,
C.P. 09790, México D.F., México
^b Departamento de Astronomía,
DCNE-CGT, Universidad de Guanajuato,
C.P. 36023, Guanajuato, México.
^c Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades,
Universidad Nacional Autónoma de México,
Ciudad Universitaria, C.P. 04510, México D.F., México.
^dFacultad de Ciencias, Universidad de Colima,
Bernal Díaz del Castillo 340, Col. Villas San Sebastián,
C.P. 28045, Colima, Colima, México.

(August 18, 2025)

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–tick data covering the period from January 1999 to December 2002 was analyzed, as well as several other sets obtained using temporal aggregation. Our results indicates that the highest frequency is not useful to understand the Mexican market because almost two thirds of the information corresponds to inactivity. For the frequency where fluctuations start to be relevant, the IPC data does not follows any $\alpha$ -stable distribution, including the Gaussian, perhaps because of the presence of autocorrelations. For a long range of lower-frequencies, but still in the intra-day regime, fluctuations can be described as a truncated Lévy flight, while for frequencies above two-days, a Gaussian distribution yields the best fit. Thought these results are consistent with other previously reported for several markets, there are significant differences in the details of the corresponding descriptions.

keywords:

stock markets , high frequency fluctuations distribution , tail behavior , autocorrelations

MSC:

91B82 , 60E07 , 62D05

^†^†journal: Physica A

1 Introduction

Is a fundamental assumption of the economic theory of markets that the financial markets are efficient in the sense of the community of economic agents being able to discount all the possibilities of arbitrage, incorporating with it all the relevant information for the prices formation, so that it is impossible to design an (always) winning strategy for investment[1, 2]. ¹¹1For more details on the background and definitions of the concepts we use in this paper see book [2]. Mathematically, this is described as a process, where a future increase or decrease of the current price is the result of a random event. Underlying such a random walk is a binomial distribution, which after a very large number of steps (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 modeled as this kind of system; data for most financial indexes around the world are statistically described by probability distributions that exhibit a large skewness or kurtosis, relative to that of a normal distribution. It may means that strong perturbations are not, de facto, necessary for the market entering a critical state; recurrent significant deviations of economic variables from their average values could be just the cumulative and unavoidable result of many small-scale processes and interactions occurring within the market system, continuously subjected to a net external action like, for instance, publicly available announcements of annual earnings, stock splits, companies profits forecasts, new securities or, even more, the punctual action of agents having preferential access to restricted or confidential information.

In this regard, for years now particularly good fits to financial data have been obtained using Lévy-stable distributions [3]. Nevertheless, it is obvious that, strictly speaking, the market can neither be such a random process. First, it is unreasonable to expect data from any real (as opposed to hypothetical) process to have an infinite variance. Secondly, it is reasonable to expect data from financial transactions at a given time to have some memory of the previous transactions, i.e., the elements of the corresponding time series should not be really independent each from the others. This is easily noted, for instance, during steeped variations of the stock returns due to panic or euphoria.

Commonly there are two causes of autocorrelation in this kind of data, irregular sampling and causality determined by market microstructure. In agreement with the ‘efficient market hypothesis’, it is intuitive not to expect persistent serial dependence in price changes, otherwise it would be used to influence the market. Consistently with this, autocorrelation is often neutralized after homogenization of the corresponding series by time averaging of subsamples. This typically results in sets with information correlated with those of daily or lower frequency data. It is for these last sets that fitting to stable distributions have already yielded remarkably good results [4, 5, 6, 7, 8]. More precisely, it has been verified that the actual density distribution behind the data could be a so-called Lévy-truncated [9, 10] which, after sequential convolution corresponding to the sum of the values over increasing time intervals, also converges to a Gaussian distribution. This convergence is ultra-slow, therefore the truncation still allows for a relatively high probability of extreme fluctuations which, as it was already noted, seems to be a distinctive feature of most financial markets.

Despite the success in describing markets activity as a Lévy (truncated) flight, it is widely acknowledged that the description will not be complete unless high-frequency data is also brought into the study. On one hand, most market models are based on a variety of hypothesis regarding the long-memory features of volatility which are difficult to extract from daily or lower-frequency data, but can be observed in intraday data. For instance, as it was already mentioned, crisis in finances may not necessarily be linked only to strong perturbations over several days, but also to the cumulative effect of intraday weaker perturbations. On the other hand, the quality of risk analysis of investment depends on the accuracy of measurement of ex post volatility and of forecast evaluation. There is evidence of the improvement in both directions thanks to the availability of high-frequency data from liquid financial markets such as the foreign-exchange, bond or equity-index markets afford [11]. Last, but not least, the analysis of the impact of sample size on the probability estimation of extreme events yields that a large volume of data is needed in order to determine the actual stable distribution corresponding to a given asymptotic scaling[12, 8]. This is the kind of volume that commonly characterizes the sets of high frequency data.

It must be pointed out that market microstructure and short term interactions become relevant while analyzing intra-day fluctuations and, since they depend on local socio-economic factors, the stylized facts of the market as a complex system can be difficult to extract. Therefore, it is very important to study different realizations of high frequency financial data in order to discriminate local and universal properties of the market dynamics. It is with this intention that in this paper we report a first step into analyzing the tick-to-tick data of the Indice de Precios y Cotizaciones (IPC), the main benchmark stock index in Mexico. It has been previously shown in Ref.[8] that it can be dismissed that a normal distribution relies under the data of daily closure values of the IPC. On the other hand, the null hypothesis that it comes from $\alpha$ -stable Lévy distribution cannot be rejected at the 5% significance level. This implies that the daily data can safely being considered as independent and identically distributed, characterized by an infinite variance. Our aim here is to study how this conclusion changes when tick-to-tick data is analyzed.

The paper is organized as follows. In the next section we provide a brief review of Lévy distributions, in particular of the stable and Lévy truncated distributions which we use for fitting the IPC fluctuations. The description of this data, as well as the sets derived from them and used for our analysis is presented in section 3. Next in sections 4 and 5 we proceed to present the analysis of the probability distribution and serial dependence of the actual fluctuations of the IPC data. We devote the last section to discuss our results and state the main conclusions.

2 Lévy distributions

As it was mentioned in the introduction, Lévy distributions are among the more frequently used to fit data from complex processes. Particularly, it has been found to be an excellent fit to the distribution of stock returns as well as of other financial time series. In this section we will briefly review the definition and properties of the stable and truncated Lévy distributions.

2.1 Stable distributions

While studying the behavior of sums of independent random variables Paul Lévy [3] introduced a skew distribution specified by scale $\gamma$ , exponent $\alpha$ , skewness parameter $\beta$ and a location parameter $\mu$ . Since the analytical form of the Levy stable distribution is known only for a few cases, they are generally specified by their characteristic function. The most popular parameterization is defined by Samorodnitsky and Taqqu [13] with the characteristic function:

\phi(t)=\begin{cases}\exp{\left(-\gamma|t|\left[1+i\beta\frac{2}{\pi}\rm{sign}(t)\ln(|t|)+i\mu t\right]\right)},&\text{if $\alpha=1$}.\\ \exp{\left(-\gamma^{\alpha}|t|^{\alpha}\left[1-i\beta\tan\left(\frac{\pi\alpha}{2}\right){\rm sign}(t)+i\mu t\right]\right)},&\text{otherwise}.\end{cases}

(1)

where $sign(t)$ stands for the sign of $t$ . Then, the probability density function is calculated from it with the inverse Fourier transform in the form:

f(x;\alpha,\beta,\gamma,\mu)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}\phi(t){\rm e}^{-itx}dt\,.

(2)

Lévy distributions are characterized by the property of being stable under convolution, i.e, the sum of two independent and identically Lévy-distributed random variables, is also Lévy distributed with the same stability index $\alpha$ . The stability parameter $\alpha$ lies in the interval $(0,2]$ . Small $\alpha$ represents a sharp peak but heavy tails which asymptotically decay as power laws with exponent $-(\alpha+1)$ . For the normal distribution $\alpha=2$ . For symmetric distribution (like the normal distribution), the skewness parameter $\beta=0$ . The skewness parameter must lie in the range $[-1,1]$ . When $\beta=+1,-1$ , one tail vanishes completely. The parameter $\gamma$ lies in the interval $(0,\infty)$ , while the location parameter $\mu$ is in $(-\infty,+\infty)$ .

The asymptotic behavior of the Lévy distributions is described by the expression

f(x;\alpha)\approx|x|^{-1-\alpha}\,.

(3)

Hence, the variance of the Levy stable distributions is infinite for all $\alpha<2$ .

2.2 Truncated Lévy distributions

As mentioned in the previous subsection, $\alpha$ -stable Levy distributions have infinite variance, hence, they have power-law tails that decay too slowly. Therefore, the fit of empirical data by Lévy stable distributions will usually overestimate the probability of extreme events. In particular, real prices fluctuations have finite variance, so their distribution decays slower than a Gaussian, but faster than a Levy-stable distribution, with the tails better described by an exponential law than by a power law. The truncated Lévy flight (TLF) was proposed by Mantegna and Stanley [9] to overcome this problem, and can be defined as a stochastic process with finite variance and scaling relations in a large, but finite interval. They defined the truncated Lévy flight distribution as:

P(x)=\left\{\begin{array}[]{lll}0&\mbox{if }x>l\\ cP_{L}(x)&\mbox{if }-l\leq x\leq l\\ 0&\mbox{if }x<-l\,,\end{array}\right.

(4)

where $P_{L}(x)$ is a symmetric Lévy distribution. As the TLF has a finite variance, with sequential convolution it will converge to a Gaussian process, but the convergence is very slowly, as was demonstrated by Mantegna and Stanley [9]. However, the cutoff in the tail given by (4) is abrupt. This problem was solved by Koponen [10], who introduced an infinitely divisible TLF with an exponential cutoff with the characteristic function:

\log\phi(t)=-\frac{c^{\alpha}}{\cos(\frac{\pi\alpha}{2})}\left[\left(t^{2}+\lambda^{2}\right)^{\frac{\alpha}{2}}\cos\left(\alpha\arctan\frac{|t|}{\lambda}-\lambda^{\alpha}\right)\right]\,,

(5)

where $c$ is the scaling factor, $\alpha$ is the stability index, and $\lambda$ is the cutoff parameter. The Lévy $\alpha$ -stable law is restored by setting to zero the cutoff parameter. For small values of $x$ , the truncated Levy density described by the characteristic function (5), behaves like a Lévy-stable law of index $\alpha$ [14]. It was used by Matacz [15] to describe the behavior of the Australian All 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 account that the Mexican trading day is of six and a half hours, this give us a mean time between transactions of $5.2$ seconds. With the aim of analyzing the statistics of nontrivial index fluctuations, the repeated values (the intervals where no change in the returns was recorded) were removed and the set was reduced to $N=1164256$ elements with a mean value over the period of $6209.392$ , a variance of $744334.3$ and an excess of kurtosis of $1537.442$ . These ticks, $Y_{k}$ , are now irregularly sampled, with a mean time between fluctuations of 19.3 seconds. This set is plotted in figure 1.

Refer to caption — Figure 1: IPC actual fluctuations. The horizontal red line corresponds to the mean value over the period.

The actual fluctuations are then taken as the difference of the natural logarithm of the $Y_{k}$ ,

S_{k}=\ln Y_{k+1}-\ln Y_{k}\,,\quad{\rm for}\,k=1,2,\dots,N\,.

(6)

For this study we also used sets corresponding to the convolution of the density distributions of high-frequency data $\{S_{k}\}$ , i.e., sets obtained after summing for different values of $N_{conv}$ ,

S_{j}^{N_{conv}}=\sum_{k=1+(j-1)\times N_{conv}}^{j\times N_{conv}}S_{k}\,,\quad{\rm for}\,j=1,2,\dots,\bar{N}\,,

(7)

where $\bar{N}$ is the multiple of $N_{conv}$ closer to $N$ .

For completeness we analyzed also a set of closing values of the IPC for the same period downloaded from the Yahoo Finance website.

4 Probability distribution of IPC fluctuations

We started by analyzing the non-convoluted data. The corresponding distribution is presented in figure 2, where a logarithmic scale is used for the vertical axis and the horizontal axis has been rescaled dividing by the standard deviation. The plots of best fits to a Gaussian (narrow blue curve below the data points) and a Lévy (wide red curve above the data points) distributions are also shown.²²2For the analysis of stable distributions, we used the library developed by J. Nolan [16].

It can be observed that the probability distribution function for this data does not correspond to a normal distribution, but it neither does to a Lévy one. This is confirmed by the results of the Kolmogorov-Smirnov test shown in table 4 for the best fit of data to an $\alpha$ -stable distribution.

$N_{conv}$	$\alpha$	$\beta$	$\gamma$	$\delta$	K-S Statistics	p-value
0	$1.2565$	$-0.0024$	$0.3796$	$0.0014$	$0.0179$	$0.0$	Yes
10	$1.5788$	$0.0056$	$2.3138$	$-0.0051$	$0.0097$	$0.0$	Yes
20	$1.6107$	$0.0089$	$3.8497$	$-0.0114$	$0.0076$	$0.0026$	Yes
30	$1.6296$	$0.0072$	$5.2420$	$-0.0075$	$0.0081$	$0.0115$	Yes
40	$1.6539$	$0.0140$	$6.6326$	$-0.03553$	$0.0099$	$0.0064$	Yes
50	$1.6523$	$0.2332$	$7.8216$	$-0.0497$	$0.0108$	$0.0084$	Yes
60	$1.6594$	$0.0145$	$8.9391$	$-0.0492$	$0.0100$	$0.0404$	Yes
70	$1.6647$	$-0.0053$	$10.0793$	$0.0359$	$0.0078$	$0.2602$	No
80	$1.6682$	$0.0010$	$11.1893$	$0.0339$	$0.0089$	$0.2022$	No
90	$1.6688$	$0.0141$	$12.1705$	$0.0103$	$0.0102$	$0.1365$	No
100	$1.6845$	$0.0029$	$13.2942$	$0.0329$	$0.0099$	$0.2053$	No
110	$1.6916$	$0.0080$	$14.2635$	$-0.0547$	$0.0123$	$0.0819$	No
120	$1.6887$	$0.0049$	$15.2096$	$-0.0066$	$0.0114$	$0.1607$	No
130	$1.6826$	$-0.0291$	$16.0270$	$0.2303$	$0.0131$	$0.0903$	No
140	$1.6874$	$-0.0211$	$16.9094$	$0.2623$	$0.0115$	$0.2163$	No
150	$1.6909$	$-0.0027$	$17.8762$	$0.1131$	$0.0107$	$0.3344$	No
1200	$1.8693$	$-0.3670$	$71.8964$	$6.8822$	$0.0150$	$0.5328$	No
2500	$1.9065$	$-0.6018$	$108.4828$	$12.0903$	$0.0301$	$0.7860$	No
2700	$2.0000$	$0.7973$	$122.6070$	$5.0104$	$0.0309$	$0.7985$	No

Analysis of intra-day fluctuations in the Mexican financial market index

Abstract

keywords:

MSC:

1 Introduction

2 Lévy distributions

2.1 Stable distributions

2.2 Truncated Lévy distributions

3 Data sets from IPC values

4 Probability distribution of IPC fluctuations

4.1 Convergence to $\alpha=2$

4.2 Lévy to Gaussian crossover

5 Serial dependence in the IPC data

6 Discussion

7 Acknowledgments

References

Analysis of intra-day fluctuations in the Mexican financial market index

Abstract

keywords:

MSC:

1 Introduction

2 Lévy distributions

2.1 Stable distributions

2.2 Truncated Lévy distributions

3 Data sets from IPC values

4 Probability distribution of IPC fluctuations

4.1 Convergence to α=2\alpha=2

4.2 Lévy to Gaussian crossover

5 Serial dependence in the IPC data

6 Discussion

7 Acknowledgments

References

4.1 Convergence to $\alpha=2$