1. Consider a European option on some underlying stock with price S(t),0t T, and assume this stock pays no dividend. Assume the stock price follows the geometric Brownian motion, and the risk-free rate is r, compounded continuously. Besides paying h(S(T)) at maturity T, the option also pays qS(t)dt in the period (t,t+dt],0tT. Let f(t,S) be the price of the option in terms of the stock price S at time t. Try to use hedging argument to derive the Black-Scholes-Merton PDE f(t,S) satisfies