Partial differential equations (PDEs) frequently result when solving the valuation of financial derivatives, in particular options, that may arise due to the models, including the Black Scholes framework. Where to consider the solution of these PDEs numerically, which is often preferable over an analytical solution that is often not available in case of more complex instruments, this paper discusses the application of finite difference methods (FDMs). We talk about explicit, implicit and Crank Nicolson schemes involving European and American option pricing. Their implementation, stability and convergence are examined. The results show that finite difference offers quick and versatile means of valuation of options especially when the boundary conditions or payoff is not standard.