Introduction to financial mathematics

This is a course on Brownian motion, Ito calculus and its application to financial mathematics. The course is designed for students with a background in probability (BSc Mathematics level) and an exposure to bachelor level measure theory and stochastic processes. We start with a review of all the necessary background,  then move to the concept of Brownian motion and discuss some of its important properties like quadratic variation and the Markov property. We then discuss stochastic calculus, in particular the Ito-Doeblin formula and the derivation of the Black-Scholes-Merton formula for pricing European options. After making this link with financial mathematics, we introduce risk-neutral pricing and discuss Girsanov's Theorem and Fundamental Theorems of Asset Pricing. We (start and) end the course, with a discussion of financial options, option trading and the relevant mathematical concepts in practice.