e introduce the forward rate
connected to the defaultless bond price
by the
relationship
where the
is the observation time and
is the maturity of the bond. We are given a reference filtration
and the SDE
in the real world, where the
and
are
-adapted
vector and matrix valued processes.
Our intention is to compute an SDE for the
by direct differentiation of (*), to produce a risk neutral measure from
Girsanov's theorem and a 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
have
Hence,
We introduce
for some
-adapted process
and
continue
By existence of the risk neutral measure
(
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
(**):
Summary
If the riskless bond and the forward rate are given by the real world SDEs (*)
and (**) then in the risk neutral world the bond and forward rates evolve
according
to
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|
(Bond SDE)
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