Consider a derivative that pays off 
at time T, where ST is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that its price at time 
has the form
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
where r is the risk-free interest rate and σ is the stock price volatility.