or most of financial
problems the payoff function does not vanish for both of the infinities.
Hence, if one
uses
at the lattice boundaries then such boundaries have to be far away to make
such positions of the state variable improbable. Such problem is even more
apparent if American feature is involved.
One alternative is to
use


(Second derivative localization)

on the boundary because it is true for most of the practical payoffs. However,
such condition, when substituted into the scheme, destroys the normality of
the problem leading to the difficulties described in the section
(
Stability of general
evolution equation
). By using the formula
(
Second derivative
localization
) we introduce a Jordanian block of at least second order that
creates unstable behavior near the boundary. When propagating to the center
the error competes with the slight stability of the scheme presumably obtained
excluding the difficulty on the boundary.
For some contracts one may
use
for a known constant
and then transform the problem to the homogenous
condition
by changing the function
.
This way normality is
preserved:
However, such change will introduce a RHS component into the equation. If the
original problem has a blow up at spacial infinity then the difficulty is
likely to migrate into the RHS. The same problem occurs when using
for a known linear function
and the lattice boundary
.
