uppose the 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
( Forward_Kolmogorov): 
We evaluate each integral via integration by parts and with help of results
( Distribution density via Call).
 
Summary
Assume
then
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