Longstaff and Schwartz models for American Options

This thesis consists of two parts. In the first part the Longstaff and Schwartz least squares method (a Monte-Carlo method) for pricing American type options is investigated. The method is based on valuation through a comparison between the value of early exercise and the conditional expected value of continued possession. The result is regressed on a set of basis functions. In this paper the functions are simple polynomials. The results is compared to the results from a finite difference scheme and proves to be reliable. In the second part of the thesis a finite element model is implemented to solve a two factor partial differential equation model, known as the Longstaff and Schwartz two factor equilibrium model. The implementation is first tested against an analytical solution for a simple bond and is then used to price a floorlet (as an example). The model is then expanded with a successive over relaxation solver that can handle obstacle problems to price American type derivatives. An American put option on a bond is priced and compared with the value of a European put option on the same bond and with the same maturity. Implementation of the finite element method 16 3.2.1 Boundary conditions 16 3.2.2 The analytical solution for Bonds and European derivatives 18 3.2.3 Numerical solution using FEM 19 3.2.4 The grid 21 3.2.5 Implementation of the model and simple validations 22 3.2.6 Comparison between analytical and FEM models for the price of a bond 23 3.2.7 Limitations of the implemented model 25 3.3 Results 26 3.3.1 Floors and floorlets 26 3.3.2 American and European put option on a bond 27 4 Discussion 29 Appendix A 30 Bibliography 31 4