The fact that the American-style options can be exercised at any time its holder thinks as the more convenient, reflects the aspects of our problem: the valuation strategy of the American options, taking into account that they are worth at least as their European counterparties, due to the "early exercise" feature that is embedded in the American options. It is this early exercise feature that makes the valuation of the American options very complicated. The mathematical solution must be unique. The book has the following structure: first, we are conducting a literature review, focusing more on the main methods that we are going to analyse. Following, we present numerical results for both American calls and puts, obtained using each and every valuation method for different sets of parameters and continuous dividend yields. Afterwards, we will interpret the case where there are discrete dividends paid by the stock and investigates the convergence (or divergence) of the results obtained with the alternative methods. Finally, the aggregate concluding remarks are contained, based on the results generated in the previous chapters.