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Preprint 8-2008
A fast, stable and accurate numerical method for the Black-Scholes equation of American options
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Author(s) : Matthias Ehrhardt , Ronald Mickens
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 8-2008
- 35A35 Theoretical approximation to solutions
- 65N99 None of the above, but in this section
- 91B26 Market models
We derive an exact discrete artificial boundary condition (ABC) for the Crank-Nicolson scheme for solving the Black-Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.
Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.
Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.
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