Dr. Lothar Banz
Master of Science, Diplom-Ökonom

PhD project: hp-TDG Methods for Financial Option Pricing in particular American Put Options

Over the last decades options became an important feature of the financial market. They exist in many different forms and differ mainly in the imposed rights and conditions. These options can be used to hedge risk or as an leverage to amplify speculative yields. Therewith an accurate pricing which prevents arbitrage or guaranteed losses becomes increasingly important. The most common methods are binomial trees, Monte Carlo simulations and finite difference schemes but they suffer from significant theoretical and practical drawbacks.
In this project the time-discontinuous Galerkin hp-finite element method is applied to the Black-Scholes partial differential equation for American put options. Two competing but equivalent weak formulations are considered, a variational inequality and a mixed formulation in which the Lagrange multiplier space is spaned by biothogonal basis functions. This project focuses especially on

• hp-TDG for parabolic obstacle problems
• biorthogonal basis functions
• semi-smooth Newton
• a priori and a posterior error estimators

PhD project: hp-TDG Methods for Frictional Elasto-dynamic Contact Problems in Linear Elasticity

High order methods are very efficient to obtain high accuracy with only moderate degrees of freedom and large element sizes. Hence they are well suited for large time scale simulations as they frequently appear in biotribology.
In this project the hp-FE time discontinuous Galerkin method is applied to elasto-dynamic contact problems with friction. The hyperbolic problem is written as a system involving only first order time derivatives and the weak contact condition is resolved using a biothogonal Lagrange multiplier.This project focuses especially on

• hp-TDG for hyperbolic obstacle problems
• biorthogonal basis functions
• semi-smooth Newton
• a priori and a posterior error estimators
• existance and uniqueness