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
by
where the drift
,
volatility
,
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
and
depending on
.
We form a
portfolio
where
is the trading strategy. We calculate the
differential
and choose the
and
to remove both sources of
randomness:
We solve for the
For such
we
have
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.
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