Consider a derivative that pays off S at time T, where S7 is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that its price at time t (t)

where S is the stock price at time t and h is a function only of t and T.

(a) By substituting into the Black-Scholes-Merton partial differential equation, derive an ordinary differential equation satisfied by h(t,T).

(b) What is the boundary condition for the differential equation for h(t, T)?

(c) Show that h(t,T) = 0.50n(n-1)+r(n1)](T1).AppendixLO1