Long-term memory stability in the Italian stock market

The standard hypothesis that stock returns are log-Normally distributed, i.e. that the stochastic processes generating such returns are independently and identically distributed, has not been settled in an increasing number of empirical works. In particular, many authors, analizing financial time series by different quantitative tools, found the presence of long-term memory. In this work it is assumed that the stochastic processes generating the stock returns are fractional Brownian motions, instead of (classical) standard Brownian motions, i.e. Normally distributed stochastic processes able to generate long-term dependent, instead of independent, and identically distributed stock returns. These stochastic processes are characterized by the so-called "exponent of Hurst" H, which is a kind of measure of the long-term dependence; in particular, if 0<0.5 there is negative long-term dependence, if H=0.5 there is independence and if 0.5<1 there is positive dependence. In some previous works, the author estimated the value of the Hurst exponent for some time series of stock returns coming from the Italian financial market, finding evidence of positive long-term dependence. In this work, the author continues this research in the Italian stock market, analyzing the stability of the long-term memory over time and thus the dynamic behaviour of this long-term memory, instead of just the static one. This analysis is carried out by using a modified version of the "forward-rolling" and the "backward-rolling" techniques. Note that the proposed analysis, studying the dynamics behaviour of the long-term dependence, is able to give some indications with regard to the useful (i.e., H=0.5) or not useful (i.e. H≠0.5) application of the classical financial models (like the Black and Scholes one) based on the hypothesis of independence.