= Solution
{c}
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 $rSV_S-rV=rS-rS=0$. It also has the correct terminal value. Part <2/a> therefore gives
$$
S_t=e^{-r(T-t)}\mathbb E[S_T\mid\mathcal F_t],
\qquad
\boxed{\mathbb E[e^{-rT}S_T\mid\mathcal F_t]=e^{-rt}S_t.}
$$
Thus the discounted <stock> is a <martingale>. The explicit <lognormal distribution> in part <2/b> supplies the necessary finite <expectation>, so here this is a true martingale identity.
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