A Fourth Order L-stable Method for the Black-Scholes Model with Barrier Options

Abstract

In this paper we develop a fourth order L-stable method for solving the Black-Scholes equation with barrier options. The method is based on a rational approximation to the matrix exponential function possessing real and distinct poles which allows the implementation of the algorithm as Backward Euler-like solves on concurrent processors. Due to barriers or nonsmooth payoffs which cause discontinuities in the solution, standard .A-stable methods, such as Crank-Nicolson, are prone to produce large and spurious oscillations in the numerical solution which would lead to poor estimates of options. The proposed higher order method does not suffer these drawbacks due to its strong stability properties at infinity. © Springer-Verlag 2003.