The objective of the course is to introduce the theory of mathematical finance and the mathematical tools on which this is based. The focus chosen is on the pricing of financial assets via arbitrage theory. We will concentrate on discrete time models, e.g. Cox-Ross-Rubinstein and multinomial models, and classical assets as European call and put options.

In the above market models we will also study optimal portfolio problems, that is we study how to obtain a strategy maximizing the expected utility of the final wealth. For this we will concentrate on complete markets.

The mathematical tools presented in the course belong to the theory of probability and stochastic processes. They include the concepts of probability measures, conditional expectations, and martingales. The concepts, methods, and models discussed have a value by themselves and can be applied beyond the focus of this course. They also constitute the base for follow-up courses at master level.

Topics

- Elements of probability theory and stochastic processes: probability measures, conditional expectations, convergences, filtrations, martingales, change of measure
- Financial assets: derivatives of European type
- Discrete time models: Cox-Ross-Rubinstein and multinomial models as preparation to the classical Black-Scholes model
- Pricing methods: concept on arbitrage, risk-neutral evaluation, Snell envelope
- Optimal portfolio problems via martingale methods