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From Merton (1973) the following boundary condition is valid for an American put.

G(S,t:E) >= Max [0,E-S]

I dont understand how the Rational put price can be greater than the potential payoff? You may pay for something that will always yield you less right?

Could you think of this as - European put payoff sets the lower floor of and american put?

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    $\begingroup$ You are confusing the current intrinsic value $\max \{ E - S_t, 0 \}$, which is what you get when you exercise immediately, with the max. potential payoff $E$. Given that you have the option to exercise at any time, the option price can never be lower than the current intrinsic value. I am voting to close the question for being too basic. $\endgroup$ – LocalVolatility Jan 20 '18 at 12:29
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When you exercise at any moment prior to expiry, you obtain the price at payoff, i.e. the same as you would get when you hold it till expiry and the price stayed constant.

What makes your pur more valuable than the price at expiry is the right to hold it further -- that is your free choice to exercise it now or hold it for some more time. At expiry this right vanishes.

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