We develop a finite difference scheme for pricing barrier options, which has the second order accuracy in time. The barrier structure of the option is one of the most common feature in financial derivatives, and it is appeared as knock-in, knock-out or early redemption conditions. It is well known that the conventional Crank-Nicolson scheme could produce worse result than the implicit scheme for pricing the barrier option due to the numerical jump on the barrier levels.
We consider the barrier level as an interface of decomposed domains, so that we could impose the Poincare-Steklov operator on the barrier level. The advantage of our scheme is that we do not solve the partial differential equations on the barrier level and the Crank-Nicolson type time discretization could be blended. We tested our algorithm to several benchmark problems including real world problems such as the equity linked securities(ELS).