This article proposes an extension to the CGARCH model in order to capture the characteristics of short-run and long-run asymmetry and persistence, and examine their effects in modeling and forecasting the conditional volatility of the stock markets from the region of Latin America during the period from 2 January 1992 to 31 December 2014. In the sample analysis, the estimation results of the CGARCH-class model family reveal the presence of short-run and long-run significant asymmetric effects and long-run persistency in the structure of stock price return volatility. The empirical results also show that the use of symmetric and asymmetric loss functions and the statistical test of Hansen ( 2005) are sound alternatives for evaluating the predictive ability of the asymmetric CGARCH models. In addition, the inclusion of long- * Corresponding author.