A comparison between finite differenceand binomial methods for solvingAmerican single-stock options

Abstract: In this thesis, we compare four different finite-difference solvers with a binomial solver for pricing American options, with a special emphasis on achievable accuracy under computational time constraints. The three finite-difference solvers are: an operator splitting method suggested by S. Ikonen and J. Toivanen, a boundary projection method suggested by M. Brennan and E. Schwartz, projected successive overrelaxation and second order accurate operator splitting method known as Peaceman-Rachford. The binomial method is a modified variant employing an analytical final step as suggested by M. Broadie and J. Detemple. The model problem is an American put option, and we empirically examine the effects of the relevant numerical parameters on the quality of the solutions. For the finite-difference methods we utilize both a Crank-Nicolson discretization and a fully implicit second-order-in-time discretization. We conclude that the operator splitting method suggested by S. Ikonen and J. Toivanen is the Alternating Direction Implicit algorithm known as the Douglas-Rachford algorithm. We also conclude that the accuracy of the Peaceman- Rachford algorithm degrades to first order for the American option problem. Of the finite-difference methods tried, the Douglas-Rachford algorithm has the highest performance in terms of accuracy under computational time constraints. We conclude that it does, however, not outperform the modified binomial model