A highly accurate adaptive finite difference solver for the Black–Scholes equation

Abstract

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.

Keywords:

• adaptive method
• option pricing
• high-order approximation
• finite difference method
• accurate

2000 AMS Subject Classification :

• 65N06
• 65N12
• 65N50

Acknowledgements

Jonas Persson was funded by FMB. The Swedish Graduate school in Mathematics and Computing. Lina von Sydow was supported by The Swedish Research Council under Contract No. 621-2003-5280.