e introduce forward rate
connected to defaultless bond price
by the
relationship
where
is observation time and
is maturity of the bond. We are given a reference filtration
and the real world SDE
where
and
are
-adapted
vector and matrix valued processes.
Our intention is to compute an SDE for
by direct differentiation of (*), to produce the risk neutral measure from
Girsanov's theorem and the requirement that
would drift with riskless rate
in the risk neutral world and, finally, to compute the risk neutral world
version of the (**).
We introduce the convenience
notations
and
for any function of two variables
.
We
calculate
We introduce
for some
-adapted process
and
continue
By existence of the risk neutral measure, see
(
Risk neutral Brownian motion
),
there has to be a
such
that
We differentiate the above relationship with respect to the variable
:
and substitute (***) and (****) into
(**):
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