Physica A 388 (2009) 59–62

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Physica A

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Quantifying price fluctuations in the Brazilian stock marketB.M. Tabak a,b,∗, M.Y. Takami a, D.O. Cajueiro c, A. Petitinga ca Banco Central do Brasil, Brazilb Universidade Católica de Brasilia, Brazilc Universidade de Brasilia, Brazil

a r t i c l e i n f o

Article history:Received 30 October 2007Received in revised form 16 September2008Available online 30 September 2008

Keywords:Power law distributionStock marketEmerging marketsEconophysics

a b s t r a c t

This paper investigates price fluctuations in the Brazilian stock market. We employ arecently developed methodology to test whether the Brazilian stock price returns presenta power law distribution and find that we cannot reject such behavior. Empirical resultsfor sub-partitions of the time series suggests that for most of the time the power lawis not rejected, but that in some cases the data set does not conform with a power lawdistribution.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Recent research in econophysics has shown that financialmarkets can be considered as complex systems inwhich a largenumber of agents interact with each other. This has stimulated a large body of research on unveiling the characteristics ofprice fluctuations in financial markets [1].The study of asset price fluctuations is at the core of the financial literature as most important models such as the Black

and Scholes option pricing model and Capital Asset pricing models assume specific dynamics for asset prices [2] which areat odds with recent findings in the econophysics literature [1].Empirical evidence has shown that price fluctuations follow complex patterns and present power law distributions [3–

10]. An important finding, which is a stylized fact in the econophysics literature, is that many stock markets exhibit powerlawswith α ' 3 [3].1 However, most of the literature has estimated tail indices based on the log–log linear regression of theempirical cumulative distribution function and the Hill estimate [12]. These techniques usually overestimate the exponentin finite samples [13,14]. A recent paper [4] has suggested a new approach that seeks to overcome some of these problemsin the estimation of tail exponents. We will employ this approach to formally test for power law behavior in the Brazilianstock market, employing a large intraday data set.2The contribution of this paper is twofold. First,we testwhether the power lawdistributionholds for different sub-samples

for the Brazilian stock market. Second, we incorporate data on the recent sub-prime crisis and test whether power lawexponents collapse to specific thresholds as is suggested in Refs. [15,16].

∗ Corresponding address: Banco Central do Brasil, SBS Quadra 3 Bloco B Ed. Sede, 9, 70074-900 Brasilia, DF, Brazil. Tel.: +55 61 3414 245.E-mail address: Benjamin.tabak@bcb.gov.br (B.M. Tabak).1 This issue has been extensively debated in the literature with results suggesting that in some cases the power law may not be an adequate

representation. For example, Matia et al. [11] find that the distribution of daily returns in the Indian stock market deviates from the power law and decaysexponentially.2 The Brazilian stock market is an important example of emerging stock market behavior, due to its large liquidity and size.

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.09.028

60 B.M. Tabak et al. / Physica A 388 (2009) 59–62

Fig. 1. (a) Plot of the IBOVESPA - 5 min intraday observations. (b) Plot of the IBOVESPA - 5 min intraday returns.

The remainder of the paper is structured as follows. The next section describes the methodology that is employed toestimate the power law exponents and to test the hypothesis that the power law describes the fluctuations of the Brazilianstock market. Section 3 discusses the empirical results, while Section 4 concludes the paper.

2. Methodology

A variable x follows a power law distribution if

p(x) = Ax−α. (1)

The scaling parameter α usually belongs to the interval [2, 3]. However, there are some exceptions.In empirical applications researchers have estimated the scaling parameter α using a least-squares linear regression on

the logarithm of the histogram. In general, if one takes the log on both sides of Eq. (1), then one can see that the power lawdistribution obeys

ln p(x) = ln A− α ln x, (2)

which suggests that it follows a straight line on a doubly logarithmic plot.Recent literature has shown that the least-squares estimator of the scaling parameter has serious biases and suggested

employing a maximum likelihood (MLE) method for the estimation [4].The MLE estimator of α is given by

α̂ = 1+n

n∑i=1ln xixmin

, (3)

where n, xi and xmin are the number of observations, the observed value of variable X and the lower bound to the power lawbehavior, with xi and xmin being positive.To test the power law hypothesis, [4] suggests using the Kolmogorov–Smirnov (KS) statistic. This statistic may be used

to assess the goodness of fit of the power law to the data. This is done by generating synthetic data and calculating the KSstatistic for each fit and calculating a p-value as the fraction of the KS statistics for the synthetic data whose value exceedsthe KS statistic for the time series under analysis. A small p-value suggests that the power law distribution may be ruled outas an alternative for modeling the distribution of empirical data.

3. Empirical tests

We employ intraday data for the Sao Paulo Stock Exchange Index (IBOVESPA) from August 2, 2006 to August 29, 2007.We employ both five- and fifteen-minute returns in our analysis. Therefore, we have 20,664 and 7056 observations in theformer and latter cases, respectively.Fig. 1a presents the IBOVESPA for 5 min prices, while Fig. 1b presents 5 min returns. We use only intraday returns to

exclude discontinuity jumps between two consecutive days due to overnight effects. The recent fall in the index was due tothe subprime crisis that hit the US in July 2007.We estimate α for 5 and 15 min returns and find evidence of coefficients equal to 3.51 and 3.72, respectively (Table 1).

These results contrast with the work of Ref. [17], which uses daily data to study the IBOVESPA and finds a scaling indexof 1.66 for a three-decade period. Therefore, our results suggest that these exponents may depend on the frequency withwhich the data set is being analyzed. If we use high frequency data the power law behavior is evident. Our results are inline with evidence presented in Ref. [18], which shows that α increases with 1t . These results are consistent with largerkurtosis and larger deviation from normality in higher frequency data.

B.M. Tabak et al. / Physica A 388 (2009) 59–62 61

Table 1Estimates and statistics of power laws for Brazilian returns.

1t α KS (p-value) xmin

5 3.51 0.67 0.002015 3.72 0.35 0.0048

This table shows the MLE estimates of α and the p-value of the KS statistic for null hypothesis that the time series follows a power law behavior. We study5 and 15 min returns.

Fig. 2. (a) Empirical CDF of the IBOVESPA returns (5 min intraday observations). (b) Plot of the empirical density function (using a kernel density).

Fig. 3. (a) Plot of the ‘‘rolling sample’’ approach (without overlapping) for α estimated using the maximum likelihood. We employ 5 min returns for 10trading days. (b) Plot of the ‘‘rolling sample’’ approach (without overlapping) for the p-value of the Kolmogorov–Smirnov statistic estimated using themaximum likelihood. We employ 5 min returns for 10 trading days.

Fig. 2(a), (b) present the empirical cumulative distribution function and the kernel density function for the IBOVESPA. Asone can notice the data set has large fat tails. The excess kurtosis is equal to 26.69, whereas the skewness is − 0.72.We also test whether we can reject the power law distribution to characterize the data. The KS suggests that the power

law seems to provide a good fit to the data.Recent research suggests that asset price dynamics change over time, which could imply that although we cannot reject

the power law distribution for the entire sample it could be rejected in different sub-samples (see Refs. [19,20]). This couldbe motivated by the fact that in specific time periods such as crisis periods, traders may employ different trading strategies;benchmarks may also change, which implies different asset price dynamics.Ref. [10] shows that the Indian stock market shows long tails consistent with power law behavior, with exponents close

to 3 at different time scales (1 min and 1 day). Ref. [9] shows that evidence of power law behavior is robust to the choiceof the time period under consideration. Therefore, in order to check the robustness of the results we also test whether thepower law holds for different sub-periods.We employ five-minute returns and a moving window of fixed length (10 days) to estimate the power laws, with

approximately 820 observations. In our estimations we roll the sample using non-overlapping windows. Results arepresented in Fig. 3(a), (b).The first important result is that for most of the periods the power law behavior is not ruled out as can be seen from the

high p-values. However, in approximately 16% of cases the p-values are below the 5% threshold and we are able to reject thepower law behavior. This suggests that the dynamics of stock prices may change abruptly and that perhaps there is morecomplicated dynamics, especially within crisis periods.An interesting finding is that rejection of the power law coefficient is related to falls in the α parameter as the Spearman

rank correlation is around 70%. Furthermore, we have implemented the same tests for the negative tail of the distributionand results remain qualitatively the same.

62 B.M. Tabak et al. / Physica A 388 (2009) 59–62

It is important to notice that we have included the period in which the subprime crisis hit the US stock market, whichhad a huge impact on the Brazilian stock market. From July 23, 2007 to August 16, 2007 the IBOVESPA fell 18.96% due to theimpact of the subprime crisis. The subprime crisis is due to the sharp rise in foreclosures in the subprime mortgage marketin the US.3 The α exponent converges to 3 in the middle of the first semester of 2007, prior to the crisis. Recent papers [15,16,21–23] have shown that when bubbles arise, the α exponents tend to specific values (critical) depending on the specificmarkets being evaluated. Therefore, our results seem to reinforce this recent literature. Some caveats apply to our results.We have only one crisis periodwithin our sample period, which limits any further statistical analysis. Further research couldexploit whether we can find such patterns for a variety of stock markets prior to crisis.

4. Conclusions

This paper tests whether Brazilian intraday stock returns follow a power law distribution. Empirical results suggest thatone cannot reject the power law distribution, although when studying sub-samples the power law is rejected 16% of thetime (using non-overlapping observations). Furthermore, α exponents converged to 3 prior to the sub-prime crisis, a periodin which the Brazilian stock index lost 20% of its value. These results together suggest that asset price dynamics may changeover time and that power law exponents may have informational content regarding the future evolution of stock markets.Further research should exploit whether this is the case for a variety of emerging and developed markets.

Acknowledgements

The authors are grateful the comments made by anonymous reviewers and the editor H.E. Stanley that have helpedus improve the paper. Benjamin M. Tabak and Daniel O. Cajueiro gratefully acknowledge financial support from CNPQFoundation. The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the BancoCentral do Brasil.

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3 This implies increasing interest rates, which have had a negative impact on adjustable rate mortgages, and homeowners found themselves unable tomeet their financial commitments while lenders found themselves with huge losses.