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Thesis (M.Sc. (Statistics)) -- University of Limpopo. 2014 / A number of previous research studies have investigated volatility and financial risks in the ermeging markets. This dissertation investigates stock returns volatility and financial risks in the Johannesburg Stock Exchange (JSE). The investigation is con- ducted in modelling volatility using Autoregressive Moving Average-Generalised Au- toregressive Conditional Heteroskedastic (ARMA-GARCH)-type models. Daily data of the log returns at the JSE over the period 08 January, 2002 to 30 December, 2011 is used. The results suggest that daily returns can be characterised by an ARMA (1, 0) process. Empirical results show that ARMA (1, 0)-GARCH (1, 1) model achieves the most accurate volatility forecast. Modelling tail behaviour of rare and extreme events is an important issue in the risk management of a financial portfolio. Extreme Value Theory (EVT) is applied to quantify upper extreme returns. Generalised Ex- treme Value (GEV) distribution is used to model the behaviour of extreme returns. Empirical results show that the Weibull distribution can be used to model stock re- turns on the JSE. In using the Generalised Pareto Distribution (GPD), the modelling framework used accommodates ARMA and GARCH models. The GPD is applied to ARMA-GARCH filtered returns series and the model is referred to as the ARMA- GARCH-GPD model. The threshold value is estimated using Pareto quantile plot while peak-over-threshold approach is used to model the upper extreme returns. In general, the ARMA-GARCH-GPD model produces more accurate estimates of ex- treme returns than the ARMA-GARCH model. The out of sample forecast indicates that the ARMA (1, 3)-GARCH (1, 1) model provides the most accurate results.