The objective of this work is to assess and forecast the volatilities of prices on the Nigeria Stock Exchange. The ARCH family (ARCH, GARCH, TGARCH, EGARCH and PGARCH) and ARIMA models are used to assess and forecast volatilities in prices on the Nigeria stock market. The EGARCH model is found to be the most efficient for forecasting volatilities and has the capability to show the asymmetric effect. The assessment of volatilities in prices for 1985 to 2014 shows clustering, over the years. The forecasting performance shows the volatility in the Nigeria stock market to be on the increase for the next four years.