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?