The first section of the thesis presents the classic Black-Scholes formula, derived by solving partial differential equations and doing probabilistic calculations. The details of these calculations are normally omitted from textbooks on the subject, so it felt like a good idea to include them here, as a reference.
Section two uses Girsanov's Theorem to find the equivalent martingalemeasure for the Black-Scholes market model, and with the help of thismeasure presents an alternative way to arrive at the Black-Scholes formula.
In section three, we investigate what happens to the option price if the parameters of the model, especially the volatility, are changed. Through a series of MATLAB simulations, culminating in an animated movie, it is demonstrated that the classic method of calculating the greek vega should be approached with a great deal of caution.
The fourth section of the thesis leaves behind the safety of continuity and introduces Itô-Lévy processes and a suitable version of the Itô formula to go along with them. Inspired by the financial crisis, particular attention is given to the Poisson process, which is introduced into our market model in an attempt to simulate the possibility of sudden (discontinuous) market falls.
Section five starts off with a discussion on EMMs and how they relate to the notion of market completeness. An equivalent martingale measure for the (discontinuous) market model we used in section four is calculated, and later on used to find the option price, similarly to what was done for the(continuous) Black-Scholes market in section two.