Consider a derivative that pays off
at time
, where
is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that is price at time
(
) has the form
where
is the stock price at time
and
is a function only of
and
.
(a) Derive and ordinary differential equation satisfied by
. I am guessing that I should use Black-Scholes here?
(b) What is the boundary condition for the differential equation for
?
(c) Show that
where
is the risk free interest rate and
is the stock price volatility. This need part (a) right?