A Lévy process is a stochastic process that has stationary and independent increments. Log returns of financial assets tend to portray stochastic behaviours possessing distributions with heavy tails, high peaks and negative skewness which justifies the adoption of Lévy processes on modeling these phenomena. In this dissertation we consider two Lévy processes linked to orthogonal polynomials which are the Meixner process and Brownian motion. We build two option pricing models based on these Lévy processes. Both models make use of the Fourier transform methods and their efficiency is judged by the size of the error measures that calculate the distance between the market and model prices. The two models are compared to each other in terms of efficiency, simplicity in application and completeness. We use data from S&P500 index and JSE indices to determine the performances of the models in both liquid (US) and illiquid (SA) markets.