Options are financial contracts whose value depends on the value of their underlying asset (they are financial derivatives). The European call (put) option is a contract which gives its owner the right, but not the obligation of buying (selling) the underlying asset on the exercise date \(T\) at an agreed-upon price \(K\) (the strike price). The definition for American options is analogous, except that the asset can be bought (sold) at any point in time \(t\) up until the exercise date \(T\). Exotic options are those whose definition is different from standard options. Often they have (one or more) non-standard underlying assets, they have knock-out barriers which determine payoff, their value depends on the asset price in every point in time (Asian options), etc. The main problem when trading options is determining their value (price) when they are issued and throughout their life. There are many ways to tackle this problem; mostly from the stochastic and numerical perspectives. The goal of this thesis was to cover one of the numerical approaches - using finite-element methods. The first topic we cover in the first chapter are definitions of the terms and variables we use throughout the later chapters. We define what the options’ value actually is (what the payoff function looks like). From the no-arbitrage condition we derive the a priori bounds for the options’ value. We conclude the value of an American option is always going to be greater or equal than that of an European option. From the callput parity we get the bounds for call and put options. After that, important market parameters are introduced and we discuss how the value of an option depends on them. We then show the geometrical interpretation of an American options’ value. At the end of the chapter the Black-Scholes-Merton market model is introduced: its assumptions and the Black-Scholes partial differential equation. The second chapter deals with finite-element methods in general. First we show how to conduct the discretisation and define the approximation using the weighted residuals principle. After that we list some of the most popular basis and weighting functions. We choose the Galerkin method, where the weighting functions are equal to the basis functions, to solve the problem. For the basis functions we choose hat functions and then use this approach to use finite-element methods on an example. The resulting method is then applied to standard options in the third chapter. In order to use the method on American options, we familiarise ourselves with the obstacle problem in the fourth chapter. We try to find the numerical solution of the problems’ alternative definition, the linear complementarity problem. We carry on the solving by introducing the variational form of the obstacle problem. After the theoretical results, we get a problem which we then solve using finite-element methods. Using this method we proceed by developing algorithms with applications to American options. In the fifth chapter we sketch out an application to exotic options. We reserve the sixth chapter for error estimates. Strong and weak solutions are defined and we get the error estimate using Céa’s lemma. In the last chapter, we pick a few examples to showcase the usage of our MATLAB implementation of the algorithm for valuing American options. The first appendix is provided as a supplement or a reminder on the SOR numerical method, while the second appendix contains the MATLAB implementation of the algorithm for valuing American options.