Communications of the
Korean Mathematical Society CKMS

We present a robust and accurate boundary condition for pricing financial options that is a hybrid combination of the payoff-consis\-tent extrapolation and the Dirichlet boundary conditions. The payoff-consistent extrapolation is an extrapolation which is based on the payoff profile. We apply the new hybrid boundary condition to the multi-dimensional Black--Scholes equations with a high correlation. Correlation terms in mixed derivatives make it more difficult to get stable numerical solutions. However, the proposed new boundary treatments guarantee the stability of the numerical solution with high correlation. To verify the excellence of the new boundary condition, we have several numerical tests such as higher dimensional problem and exotic option with nonlinear payoff. The numerical results demonstrate the robustness and accuracy of the proposed numerical scheme.

Keywords: multi-dimensional Black--Scholes equations, operator splitting method, extrapolation, linear boundary condition, high correlation

MSC numbers: Primary 91G60, 65N06