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?