uppose a stock
follows the
SDE
in the risk neutral world. The
is a deterministic function of time. The
is the moment of observation. We aim to express the volatility
as a function of
and its derivatives with respect to strike.
We use the
representation
in terms of distribution density
We calculate the
derivative
and substitute the equation
(
Forward_Kolmogorov
):
We evaluate each integral via integration by parts and with help of results
(
Distribution density via Call
).
Summary
(Differentiating
call with respect to maturity
1)Assume
then
We now switch to a process of the
form
where the function
is still given by
Thus we cannot simply do a
change
of variables to reduce to the previous case. We repeat our
calculations.
The equation (
Forward_Kolmogorov
) now has
the
form
We integrate by parts and use the formulas
(
Distribution density via
Call
):
Summary
(Differentiating
call with respect to maturity 2)
Assume
then
