Exercise: Why no early exercise on non-dividend American calls

Tasks

1. Bound argument. Show that for an American call $V \ge S - Ke^{-r(T-t)}$ at any $(S, t)$. (Use the European put-call parity bound.)
2. Compare to immediate exercise value. The exercise value is $S - K$. Show that $S - Ke^{-r(T-t)} > S - K$ for any $r > 0, T - t > 0$. Conclude that holding strictly beats exercising.
3. Numerical verification. Implement the binomial tree for an American call (no dividends). Compare with the European call. Are they equal?
4. Add dividends. Now suppose the stock pays a continuous dividend yield $q$. Modify the binomial tree (use $u = e^{\sigma\sqrt{dt}}, p = (e^{(r-q)dt} - d)/(u - d)$). For $q = 0.05, r = 0.02$, is the American call still equal to the European? Explain.