An improvement of the douglas scheme for the Black-Scholes equation

Abstract

A well-known finite difference scheme for the valuation of options from the Black-Sholes equation is the Crank-Nicolson scheme. However, in the case of non-smoothpayoffs, the Crank-Nicolson scheme is known to produce unwanted oscillations forthe computed solution. As an alternative, Douglas scheme is generally recommendedfor better resolution of option price because it has fourth order accuracy in assetderivative. However, as noted by Shaw in his book, both these methods show“potentially nasty behavior when applied to simple option pricing”. We note that boththe Crank-Nicolson scheme and the Douglas scheme use a trapezoidal formula for timeintegration which is known to produce unwanted oscillations in the computed solution.This works since the trapezoidal formula is only A-stable and not L-stable. Chawlaand Evans proposed a new L-stable Simpson rule. We investigate the application ofthis L-stable third order rule for the time integration in the Black-Sholes equationafter it has been semi-discretized in the asset derivative by Numerov discretisation.By numerical experimentation with real option valuation problems, we compare theperformance of this new improved version of Douglas with both Crank-Nicolson andDouglas schemes. We also study the performance of this scheme for the valuation ofthe Greeks.