Security prices are often represented as solutions to parabolic partial differential equations (PDEs) arising out of the stochastic calculus. This course focuses on the study of these equations, from both a theoretical and a numerical point of view. Several finite difference methods are presented, and their performance is compared with other methods, e.g., binomial methods. Applications include the Black-Scoles formula for American options as the solution of a PDE as well as some exotic option pricing.