Option pricing using numerical methods for PDEs

Now a days mathematics can be used for many different purposes or topics, and every day new fields to be applied are found. One of this fields, which is becoming more and more popular, is financial mathematics. This thesis has as a target get an approach to financial mathematics, in this case option pricing. In finance an \emph{option} is a \emph{derivative}, which price has to be fixed. Therefore the main goal of this thesis is to study two different models for option pricing. In the latest history many different people have studied and created different models to compute the price of these options. However, they are difficult to understand because the theory behind the price of these options includes many different branches of mathematics, such as: statistics, probability, stochastic processes, partial differential equations, numerical calculus, etc Due to the complexity, neither of the models studied will be derived, that is not the objective. They will be just assumed, having as a target make a deep study on the partial differential equation governing the models, and solving them using numerical methods The first model that is introduced is the Black and Scholes Model, presented in 1957 by two mathematicians. In this case the price depends only on two variables, time and price of underlying asset. Assuming the partial differential equation, some theoretical results are going to be obtained. Afterwards, the job will be converting the model, i.e. the partial differential equation into a numerical problem, first by bounding the domain and building boundary conditions, and finally using finite differences(numerical method) for solving it. The second and more complex model is the Heston model, introduced in 1993. We will basically proceed as the previous one. However, in this case the model depends on three variables (time, price of underlying asset and volatility), therefore the finite differences approximation is going to be tougher. In this case the focus will be more on how to solve the problem, that is how to convert it on a numerical problem. As before, bounding the domain, studying the boundary conditions and finally applying finite differences. As the end of the work, the two models will be implemented in \emph{matlab} and simulated with different parameters to interpret if the results obtained are as expected