Financial modelling with stochastic processes

The celebrated Black-Scholes model for describing stock prices and how to price and hedge options is fundamental in financial mathematics. The accompanying Black-Scholes formula also serves as a benchmark for describing risks of assets via the implied volatility. The Black-Scholes model has some obvious inaccuracies. In this course some of the important weaknesses of the Black-Scholes model is pointed out both what regards statistical properties and theoretical aspects on risks by options. The focus in this course is financial modeling beyond the Black-Scholes world. Then there is non-uniqueness of option pricing and perfect hedging is impossible. The market is incomplete: there is an inherent risk with option trading. In this course an important point is to handle the incompleteness. The main approach here is modeling via jump process, in particular Lévy processes. A fundament of the course is on building and simulating Lévy processes relevant in financial modeling. Models with stochastic volatility are also included in the course.