A Highly Accurate Adaptive Finite Difference Solver For The Black-Scholes Equation

In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black-Scholes equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence in space is two, both for low-order and high-order standard FD schemes. To obtain a method that gives higher accuracy, we use an extra grid in a limited space- and time-domain. This new method is called FD6G2. The FD6G2 method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy, the adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method.