In this master thesis we have examined the possibility of pricing multiple American options, on an underlying asset with discrete dividends, with a finite difference method. We have found a good and stable way to price one American option by solving the BSM PDE backwards, while also calculating the Greeks of the option with automatic differentiation. The list of Greeks for an option is quite extensive since we have been using a local volatility surface. We have also tried to find a way to price several American options simultaneously by solving a forward PDE. Unfortunately, we haven't found any previous work that we could use with our local volatility surface, while still keeping down the computational time. The closest we got was to calculate the value of a compound option in a forward mode, but in order to use this to value an American option, we needed to go through an iterative process which calculated a forward or backward European PDE in every step.