e
are assuming that the price of some underlying asset is given
by
and we are considering a derivative
that may jump in value depending on some Cox process
The
drift
,
volatility
,
intensity of the Cox process
and the size of the jump
are all functions of
.
So this is a Markovian situation with the
being the state.
We form a
portfolio
where
is another derivative. In reality we would, of course, sell the derivative
and hedge it with the underlying
and more liquid derivative
:
However, the result would be the same and our unnatural way keeps expressions
symmetrical.
We calculate the
differential
and choose the
and
to remove both sources of
randomness:
We solve for the
For such
we
have
Note that the expression on the LHS only depends on the function
while the expression on the RHS only depends on
.
The functions
and
are not related. Therefore, both parts of the equation do not, in fact, depend
on
or
.
We conclude that there is a function
depending on the state, such
that
The probabilistic meaning of the above equation is provided in the section
(
Backward_equation_for_jump_diffusion
).
