 Solution

Take the terminal financial payoff $f(S)=S$. The function $V(S,t)=S$ satisfies the Black-Scholes equation: its time derivative and second stock derivative vanish, while $rS{V}_S-rV=rS-rS=0$. It also has the correct terminal value. Part a therefore givesThus the discounted stock is a martingale. The explicit lognormal distribution in part b supplies the necessary finite expectation, so here this is a true martingale identity.