- Review of Measure Theory.
- Review of Probability Theory.
- Review of Stochastic Processes.
- Infinitely Divisible Distributions.
- Lévy Processes.
- Martingales and Lévy Processes.
- Poisson Random Measures.
- The Lévy-It? Decomposition.
- Stochastic Integration with respect to Lévy Processes.
- Exponential Martingales and Change of Measure.
- The Black-Scholes Model.
- Exponential Lévy Models.
- Pricing in Exponential Lévy Models.
- Hedging in Exponential Lévy Models.
If time permits we will also cover the following topics:
15. Simulation of Lévy Processes.
16. Risk measures.
In this course, we will not follow a particular book. I will try to upload the notes of the topic covered each week one week in advance. Therefore, there is no need to buy a book to follow the lectures.
However, most of the material I will present (but not all) can be found in the following two references:
[A] David Applebaum. Lévy Processes and Stochastic Calculus. Second Edition. (2009) Cambridge University Press.
[CT] Rama Cont & Peter Tankov. Financial Modelling with Jump Processes. (2004) Chapman & Hall/CRC
The book [A] focus its attention to present the more mathematical aspects of the course. In particular, the properties of Lévy processes and stochastic integration with respect to martingale valued measures.
The book [CT] also covers most of the mathematical aspects of the course but, in addition, it presents many financial applications.
If you want to buy a book and you are more interested in the financial applications of Lévy processes I recommend you to buy [CT].