Comparing the distributions of the WIG20 and S&P500 index

Our aim is to compare the distribution of two stock indices: the Polish WIG20 and the American S&P500. We consider the Closing price of these indices (X) in the years 1994– 2006. The observed values x t (denoting the closing price at day t) were transformed to z t = ln(x t+1 − ln(x t) appropriately. The derived variable Z will be called daily return. We have 2 goals: 1. to compare the daily dynamics of WIG20 and S&P, 2. to model mathematically the distribution of daily returns of these stocks. Motivation for goal no. 1. The two stocks are existing in two different economy systems. WIG20 is relatively young; S&P has long tradition. Is the behavior of the two series-with elapsing time-similar? Motivation for goal no. 2. The attempt of modelling statistically or mathematically the distribution of the variable Z (stock returns) has a long history. Already in 1900, Bachelier proposed the first model for the stochastic process of returns (quoted after Gopikrishnan and al., 1999). Bachelier proposed the model of an uncorrelated random walk with independent , (i.i.d) random variables. However, it seems that this is only a rough approximation of the true model governing the phenomenon. Some researchers (see, e.g. Kon, 1984, Gopikr-ishnan and al., 1999, Brabazon and O'Neil, and quite a lot of other researchers) argue, that the distribution of returns, when observed in short time intervals (days, or smaller time intervals) is not normal. This is stated by calculating the kurtosis of the distribution. Also, it is believed, that the variable Z has heavy tails and is leptokurtic. Other supposed alternatives are: Student's t distribution, mixture of Gaussians or Students' t. Usually, to our knowledge, only univariate Student's t was considered. We do it considering bi-variate Student's t distribution and apply in that case Mardia's tests for normality of distributions. We got a lot of interesting results. 1. The autocorrelation functions of the observed variable X look similar for both indices, although that of the Polish WIG20 is decidedly more ragged (see draft figure 1). 2. The hypothesis on the Gaussianity of the distributions should be rejected. The departure from Gaussianity is caused rather by a higher concentration near the mean than by heavy tails (see figures 2 and 4). Formally, the kurtosis for WIG20 is higher than that for SP500 (Figure 3). 3. We stated also a problem about calculating the kurtosis from samples …