American option pricing with an adaptive penalization strategy using the sparse grid combination technique combined with the Parareal-algorithm

My name is Anna Clevenhaus and I am a second year Phd-Student in the Applied Mathematics and Numerical Simulation (AMNA) group of the University of Wuppertal. My interest in numerical simulations and analysis started during my bachelor degree when I studied mathematics and chemistry. Afterwards I started with the international curriculum of Computer Simulation in Science with the focus on Financial Mathematics. For my master thesis, I studied European option pricing with the Heston model extended with a stochastic correlation process, where alternating direction implicit schemes have been used to solve the resulting partial differential equation.

Why I became a PhD student in the AMNA Group
During my master courses I worked in a tax consultant office. Back then I had the possibility to directly compare working in a research field and gaining practise. That’s why the choice between a PhD, where I am able to learn and study more about financial mathematics, and working in the industry was easy. I applied for a PhD-position and was happy to get the PhD-position in the AMNA working group being already familiar to me due to my master thesis. One positive side effect was that due to my Phd topic I got the chance to focus furthermore on option pricing and additionally be able to connect my interest in high performance computing with financial modelling.

What I will do during my Phd
As I have considered European option pricing in my master thesis, now my research focuses on American option pricing. American option pricing problems result in free boundary problems. One can consider front-fixing or penalty methods for solving those problems. Another approach is to rewrite the problem to a Linear complementary problem to avoid a explicit computation of the free boundary value. In my research I focus on an improvement for penalty terms due to time dependence. For the grid structure I use a sparse grid setup including additional restrictions to the resolution caused by the underlying models. In the sparse grid approach, several grids are connected due to the sparse grid combination technique. As underlying models the Heston model as well as extensions with additional stochastic processes will be studied. In a final step a time parallelization will be considered, more precisely the Parareal-algorithm.