The Heston model is a partial differential equation which is used to price options and is a further developed version of the more famous Black-Scholes equation. Heston considers stochastic volatility which results in an extra variable and a more complex equation. This paper contains numerical solutions of the Heston model for the European and American option using finite difference and element methods. First the equation will be derived followed by numerical solutions for both European and American options with a finite difference method. The equation is then expressed in weak form and solved using a finite element method. Mathematical analysis concerning stability and uniqueness of the solution will be given for the finite element solver in the European case. All solvers are implemented using Matlab
