In this paper, we introduce Brownian motion, and some of its drawbacks in connection to the financial modeling. We then introduce geometric Brownian motion as the basis for European call option pricing as we navigate our way through the Black-Scholes-Merton equation. Lévy Processes round out the background information of the paper as we discuss Poisson and compound Poisson processes and the pricing of European call options using the stochastic calculus of jump processes. Ornstein-Uhlenbeck processes are then constructed. Finally we review and analyze the Barndorff-Nielsen and Shepard model. We provide its application to price European call options using the fast Fourier transform and the direct integration method.