American Call Spread early exercise

Question

It is generally agreed that it is not optimal to exercise the call option, $C_t = (S_t-K)^+$ early. My question is this, can anyone help in proving or disproving if the American call spread, $C_t = (S_t-K_1)^+ - (S_t-K_2)^+ $ can have an early exercise?

Answer 1

For a call spread with $K_2>K_1$, it is optimal to exercise once the underlying hits $K_2$, because it is of the maximum payoff for the spread.

It not exercised, there is a chance for the underlying $S_t$ to move back below $K_2$, resulting in less payoff down the road.