Lévy processes with application

Specialeforsvar ved Ahmad Alwali

Titel: Lévy processes with appplication

Abstract: This thesis deals with the application of Lévy processes within mathematical finance. In the first section we introduce and discuss Lévy processes where the focus will be on two characteristic theorems: Lévy-Khintchine and Lévy-Itô decomposition. Furthermore, we clarify how Lévy processes can be implemented into financial models. These models are based on the Lévy processes: double-exponential jump-diffusion Kou model and variance-gamma model. The Kou model is an expansion of the Black-Scholes model where we have added the compound Poisson process. By subordination theory of Lévy processes, a Brownian motion subordinates from a variance-gamma process. At last we simulate pricing of assets using tge Kou model and variance-gamma model respectively. By the Kou model, the parameter $p$ tends to go upward, which could mean good news that effects the market positively. As for the parameter $1-p$ is going down-ward, which effects the market negatively. Using the variance-gamma model we have the parameters $\theta$ and $\nu$ which are used to control the skewness and kurtosis 

Vejleder:  Jesper Lund Pedersen
Censor:   Bjarne Astrup Jensen, CBS