The aim of this thesis is to analyse the volatility of 11 sectorial stock return data of S&P 500 Index during the 2008 global financial crisis and the recent COVID-19 global pandemic. S&P 500 is a large stock market index that tracks the performance of 500 companies that are some of the largest in the world. The proposed method to conduct the thesis is by choosing the appropriate GARCH model. To do this, BIC (Bayesian Information Criteria) is used to find the best fitted model for each sector and each crisis periods. The use of GARCH family models comes in the form of examining for volatility clustering, which is a phenomenon in financial time series where large changes are followed by large changes, meaning there are periods of high volatility and periods of low volatility, i.e., volatility persistence. Asymmetry effect is the tendency of negative shocks to decrease stock value thereby increasing leverage and cause an increase in volatility, this is also referred to as the leverage effect. Results give evidence to big impact on the stock market. It shows higher volatility persistence during recession times, significant asymmetric effect, and leverage effect in most sectors. EGARCH and TGARCH models are superior to GARCH (1,1) model in estimating volatility during a crisis event. The thesis can be beneficial for economic and financial policy makers, researchers, analysts in all sectors, investors, both private and retail, in times of economic recession so that making financial decisions and reacting to such circumstances may lead to a healthier economy.