In this paper, a heavy-tailed distribution approach is considered in order to explore the behavior of actual financial time series. We show that this kind of distribution allows to properly fit the empirical distribution of the stocks from S&P500 index. In addition to that, we explain in detail why the underlying distribution of the random process under study should be taken into account before using its self-similarity exponent as a reliable tool to state whether that financial series displays long-range dependence or not. Finally, we show that, under this model, no stocks from S&P500 index show persistent memory, whereas some of them do present anti-persistent memory and most of them present no memory at all.