The purpose of this thesis is to explore stochastic volatility models to price American and European options. The two methods used are both based on a quadrinomial tree, but the first uses an Ornstein-Uhlenbeck process and the Monte Carlo method with a quadrinomial recombining tree and the second uses the Heston model and a tree-based approach that combines a grid and bilinear interpolation to estimate the option price. The thesis is split into four chapters. In the first chapter, it gives an overview of options, option pricing models, and numerical methods. The second chapter discusses the quadrinomial recombining tree, and the third presents the tree-based approach that uses a grid and bilinear interpolation. Finally the fourth, presents the results of both methods and then compares their performance and flexibility.