Consider a call option having a strike price $K$ and exercise time $t$; let $r$ be the nominal rate, $\sigma$ volatility and $S_0$ the underlying asset at $t = 0$. How to show that $C(t, S_0,K,\sigma,r) \geq (S_0 - e^{-rt}K)^{+}$ and $C(t, S_0,K,\sigma,r)\leq S_0$? I tried to use
$C(t, S_0,K,\sigma,r) = e^{-rt}\mathbb{E}[(S(t)-K)^{+}]$
But I could not get rid of the mean.
$C + Ke^{-rt}$ pays out at least $S(t)$, since $C + Ke^{-rt}$ provides enough to buy the stock and have and sell it for $S(t)$, so by no-arbitrage $C + Ke^{-rt} \ge S_0$.
If $C > S_0$ then you could make arbitrage profits of $C - S_0$ by selling the option and buying the stock. If the option is exercised you get the lower of $K$ and S(t), so overall you get a risk free gain of $C - S_0$ (immediately) + min(K,S) on expiry. By no-arbitrage, $C \le S_0$.