# Intuitively understand boundaries of American Call and Put

Denote American Call/Put $C_{am}/P_{am},$ European Call/Put $C_v/P_v,$ with constant risk-free interest rate $r,$ dividend yield rate $D,$ strike $K,$ maturity $T.$

1.We have the well know inequalities: $$C_v\leq C_{am} \leq C_v + S_t(1 - e^{-D(T-t)})$$ $$P_v\leq P_{am} \leq P_v + K(1 - e^{-r(T-t)})$$ Surely, we can build the portfolios to proof the inequalities, but is there any intuitive ways to demonstrate above inequalities? Or when does the = hold, for none-zero $D$ and $r$ since the proof by portfolio method is not clear to see the conditions of =.

For example, $S_t(1 - e^{-D(T-t)})$ is actually the sum of discounted dividend, that means

American call will never be larger than the European call adding the dividend of its underlying asset.

And we also have

it's always optimal to exercise American call immediately before the ex-dividend etc.

Maybe the conditions of = can effectively solve the problem.

2.Moreover, for the low boundary(value of forward) of European call $$\max(e^{-D(T- t)}S_t - e^{-r(T- t)}K,0)\leq C_v(t),$$ some book said it can be regard as the American call, I can not understand this statement?

• Dividend rule is valid for only regular/planned dividends. If the cash dividend is extraordinary, the contract adjusts itself. Though, it is mostly at CBOE's discretion. – berkorbay Sep 20 '17 at 9:07

• Here Snell envelope of its discounted payoff means the expectation of discounted payoff namely the value European call? And you want to show (European call + discounted dividend) is submartingale and dominants the (European call), then it should dominant the American call? How about the put? Sorry could you show some detail? – A.Oreo Sep 20 '17 at 14:52
• Why is American call < European call + 1/2 * discounted dividend not true? And the value of American put is not submartingale. – A.Oreo Sep 20 '17 at 15:08