e present a negative result, to show how easy it is to have a situation when
the delta hedging argument does not deliver a pricing equation.
We are assuming that a price of underlying asset is given
where the drift
intensity of the Poisson process
and the size of the jump
are all functions of
So this is a Markovian situation with the
being the state.
Suppose we have two traded derivatives
We form a
is the trading strategy. We calculate the
and choose the
to remove both sources of
We solve for the
We see that, unfortunately, the functions do not separate. One may attempt to
apply the risk neutral approach formally to try and see what kind of PDE
should be expected and then attempt to separate the functions. It seems that
this does not work. Apparently, this means that there is a significant room
for difference of opinions about the price. Such opinions are dictated by the
way of replication of the derivative.