The art of fitting financial time series with Levy stable distributions

This paper illustrates a procedure for fitting financial data with $\alpha$-stable distributions. After using all the available methods to evaluate the distribution parameters, one can qualitatively selectthe best estimate and run some goodness-of-fit tests on this estimate in order to quantitatively assess its quality. It turns out that, for one of the two investigated data sets (DJIA from 2000 to present), an $\alpha$-stable fit of log-returns is reasonably good. However, for the other data set (MIB30 from 2000 to present), the fit is not as good as in the previous case. The issue of goodness-of-fit tests is critically discussed.