Author(s) :
Matthias Ehrhardt
,
Ronald Mickens
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 8-2008
MSC 2000
- 35A35 Theoretical approximation to solutions
-
65N99 None of the above, but in this section
-
91B26 Market models
Abstract :
In this work we improve the algorithm of Han and Wu
(SIAM J. Numer. Anal. 41 (2003), 2081-2095) for American Options
with respect to stability, accuracy and order of computational effort.
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.
Keywords :
Black-Scholes equation, computational finance, option pricing, finite difference method, artificial boundary condition, free boundary problem, American option
Notes :
accepted: International Journal of Theoretical and Applied Finance (IJTAF)