e seek an
asymptotic expansion for solution
of the
problem
for integrable function
We integrate the equation
in
from
to
:
and then we substitute the same expression into the
RHS:
and simplify for
,
We now provide alternative derivation of the same result.
According to the section (
Backward
Kolmogorov equation
), the solution
of the problem
may be represented as
where
is the standard Brownian motion. We perform the following standard
calculations.
We make the change of
variable
then
We arrive to the situation of the proposition
(
Asymptotic of
integral with Gaussian kernel
). We introduce the
notation
and arrive
to
The proposition
(
Asymptotic of
integral with Gaussian kernel
) requires regular
power
series for the function
.
We do not have those in case of the call payoff. However, we can consider a
domain of
away from the singularity or restrict attention to a regular
or approximate a call payoff with a smooth function. In either of such
situations, we have uniformly converging Taylor
series:
in some bounded domain. We apply the proposition
(
Asymptotic of
integral with Gaussian kernel
) and
find
thus
and we confirm our calculation.
