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- American option (1)
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In the case of the equidistant discretization of the Airy differential equation (\discrete
Airy equation") the exact solution can be found explicitly. This fact is used
to derive a discrete transparent boundary condition (TBC) for a Schroedinger
equation with linear varying potential, which can be used in \parabolic equation"
simulations in (underwater) acoustics and for radar propagation in the troposphere.
We propose different strategies for the discrete TBC and show an efficient implementation.
Finally a stability proof for the resulting scheme is given. A numerical
example in the application to underwater acoustics shows the superiority of the new
discrete TBC.
In this work we derive an exact discrete artificial boundary condition
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 artificial boundary
condition on a purely discrete level.
Since the exact discrete artificial boundary condition
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 artificial boundary conditions
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 of the proposed method on several examples
and compare it to previously obtained discretized artificial boundary conditions.