uppose the economy has one risky asset
given by the SDE
under the original probability measure. There is possibility of short time
borrowing at some deterministic riskless rate
.
We want to determine the present
price of a derivative paying
at maturity
.
Let
be the time
price of the derivative. To apply the Ito formula we need to know the
functional dependence of
.
If we make the simplifying assumption
then
because the filtration
of the model is generated by
:
and the
is Markovian. Suppose that at the time moment
we are short one unit of derivative and long
units of the underlying asset
The value of the position is
At the infinitesimally close moment
the value of the portfolio
becomes
Hence, the infinitesimal change in portfolio's value
is
by
(
Ito_formula
),
The choice
makes the value
instantaneously deterministic. Hence, it has to perform as the money market
account (MMA) during the small interval
:
where
and
Therefore,
We compare the last result with the proposition
(
Backward Kolmogorov for
discounted payoff
) and
conclude
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