An extensive family of univariate models of autoregressive conditional heteroskedasticity is applied to Peru's daily stock market returns for the period January 3, 1992 to March 30, 2012 with four different specifications related to the distribution of the disturbance term. This concerns capturing the asymmetries of the behavior of the volatility, as well as the presence of heavy tails in these time series. Using different statistical tests and different criteria, the results show that: (i) the FIGARCH (1,1)-t is the best model among all symmetric models while the FIEGARCH (1,1)-Sk is selected from the class of asymmetrical models. Also, the model FIAPARCH (1,1)-t is selected from the class of asymmetric power models; (ii) the three models capture well the behavior of the conditional volatility; (iii) however, the empirical distribution of the standardized residuals shows that the behavior of the tails is not well captured by either model; (iv) the three models suggest the presence of long memory with estimates of the fractional parameter close to the region of nonstationarity.