The 2008/9 financial crisis intensified the search for realistic return models, that capture real market movements. The assumed underlying statistical distribution of financial returns plays a crucial role in the evaluation of risk measures, and pricing of financial instruments. In this dissertation, we discuss an empirical study on the evaluation of the traditional portfolio risk measures, and option pricing under the hybrid Brownian motion model, developed by Shaw and Schofield. Under this model, we derive probability density functions that have a fat-tailed property, such that “25-sigma” or worse events are more probable. We then estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using four equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE Top 40 index. We apply the historical method and Variance-Covariance method (VC) in the valuation of VaR. Under the VC method, we adopt the GARCH(1,1) model to deal with the volatility clustering phenomenon. We backtest the VaR results and discuss our findings for each probability density function. Furthermore, we apply the hybrid model to price European style options. We compare the pricing performance of the hybrid model to the classical Black-Scholes model. / Dissertation (MSc)--University of Pretoria, 2017. / National Research Fund (NRF), University of Pretoria Postgraduate bursary and the General Studentship bursary / Mathematics and Applied Mathematics / MSc / Unrestricted