In this paper we develop a high-order adaptive finite difference space-discretization for the Black-Scholes (B-S) equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence is two, both for low-order and high-order standard finite difference (FD) schemes. To obtain a sixth-order scheme we use an extra grid in a limited space- and time-domain. The new sixth-order 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 in several dimensions the adaptive FD6G2-method is superior to both standard and adaptive second-order FD-methods.
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