Adaptive partial differential equation methods for option pricing

This project investigates the application of finite difference schemes to option pricing problems. In particular, an adaptive mesh method is introduced to deal with difficulties that arise in the numerical approximation of PDE's in presence of discontinuities, such as a barrier. Compared to an equidistant mesh, this adaptive mesh method substantially increases the numerical accuracy with the same number of grid points. Several finite difference schemes for pricing American options are studied and compared both in one dimensional case and two dimensional case. The behaviors of the price and hedge factors of various types of barrier options and American barrier options are further studied in details.