e study properties of a process
given by the
SDE
where
and
are some regular deterministic functions and
is a standard Brownian motion. We introduce a process
:
then
We substitute (*) and
obtain
or
We integrate the last relationship for
and
obtain
After multiplication by
we conclude
To explain relevance of the mean reverting equation let us consider an
equation
frequently used as a firstapproximation simplistic model for a forward curve
The
is the observation time and the
is the expiration time. The front end of the curve is most volatile. The
volatility decreases as
increases. This is a simple but realistic model of propagation of new
information through forward curve.
We integrate the above equation as
follows:
We introduce the "spot" price
and compute the SDE for the process
We
have
where the arrow marks application of the
operation.
Hence,
We conclude that the SDE for the
is mean reverting with the parameters
given
by
Note that
is the meanreversion parameter and the slope of the forward volatility curve.
