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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?

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  • $\begingroup$ 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. $\endgroup$ – berkorbay Sep 20 '17 at 9:07
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The lower bounds are obvious since American options can be exercised at any time while European options can only be exercised at maturity.

The upper bounds are obtained from the property that an American option value is the Snell envelope of its discounted payoff, so that when the discounted payoff is a submartingale the American option should never be exercised early.

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  • $\begingroup$ 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? $\endgroup$ – A.Oreo Sep 20 '17 at 14:52
  • $\begingroup$ Why is American call < European call + 1/2 * discounted dividend not true? And the value of American put is not submartingale. $\endgroup$ – A.Oreo Sep 20 '17 at 15:08
  • $\begingroup$ Snell Envelope means the value under the optimal exercise policy, i.e. the best value that you can get by exercising intelligently. $\endgroup$ – noob2 Sep 20 '17 at 19:39
  • $\begingroup$ @noob2 so here American call is the Snell Envelope of European call? But, how about the put? American put is not submartingale. $\endgroup$ – A.Oreo Sep 21 '17 at 4:28
  • $\begingroup$ discounted (stock + reinvested dividends) is a martingale => discounted (stock + reinvested dividends minus strike) is a submartingale => the positive part of discounted (stock + reinvested dividends minus strike) is a submartingale (Jensen inequality) => the discounted payoff of an American call on (stock + reinvested dividends) is a submartingale. Therefore the American call on (stock + reinvested dividends) is in fact European. Now the European call on (stock + reinvested dividends) is worth less than the reinvested dividends + European call on stock, hence the first RHS inequality. $\endgroup$ – Antoine Conze Sep 21 '17 at 8:12

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