f we are selling a derivative with the cash flow
at time 1 then we would like to hedge it with the traded assets. We are
assuming that the derivative
is not among components of
and is not a linear combination of such components. We require that for some
hedge
Hence, our asking price
is
Similarly, the bid
is
The random variable
is the price of riskless asset at time
.
Suppose that there is an opportunity to sell a contingent claim
for price
,
ask
.
Because there is not enough money to finance the hedge, such opportunity does
not change the
According to the result
(
Existence of incomplete
market pricing
), the last statement is equivalent to
Hence, we proved that
Therefore,
The opposite inequality
is derived by similar argument. Indeed, if we are going to offer
for
ask
then we are still not adding the acceptable opportunities and
Consequently,
Hence,


(Incomplete market ask)

Similarly,


(Incomplete market bid)

