Scaling, stability and distribution of the high-frequency returns of the IBEX35 index

Abstract: In this paper we perform a statistical analysis of the high-frequency returns of the IBEX35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data using maximum likelihood estimation and different goodness-of-fit tests rejects the L\'evy-stable law as a plausible underlying probabilistic model. The analysis shows that the Normal Inverse Gaussian distribution provides an overall fit for the data better than any of the other subclasses of the family of the Generalized Hyperbolic distributions and certainly much better than the L\'evy-stable laws. Furthermore, the right (resp. left) tail of the distribution seems to follow a power-law with exponent \alpha=4.60 (resp. \alpha =4.28). Finally, we present evidence that the observed stability is due to temporal correlations or non-stationarities of the data.