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I'm currently reading about how to value an american option and I have a few questions about it. Would be very grateful if anyone can spare the time and answer them.

$1)$ Since an american put and a european put both have the same payoff and they have identical cash flow, don't they have the same value at any point by the Law of One Price?

$2)$ The source I'm using says that you need the condition of "holder chooses the early exercise strategy in order to maximize the option’s value" in order to price it. What does this condition mean? From what I've read is it that once you reach some "optimal" value, you the exercise the option if the value decreases? If yes, then how is this "optimal" value determined?

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1) No, they do NOT have the same Payoff:

  • European Option: $[K-S(T)]^+$ (i.e. only at time T)
  • American Option: $[K-S(t)]^+$ (at some point t, possibly prior to time T)
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  • $\begingroup$ But doesn't this contradict the Law of One price? They have the same cash flow on $(t,T]$ and the same price at the end. $\endgroup$ – asdf Apr 18 '18 at 13:57
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    $\begingroup$ They do not have the same cash flow: Assume it would be optimal to exercise the american option prior to maturity at t=2. Since you can only exercise the European option at Maturity t=T, you will never achieve the same cash flow. $\endgroup$ – Vanity Apr 18 '18 at 14:04
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2) This boils down to quite a lot of theory about "optimal stopping times / Snell Envelop and so on" and i would advise you to read about it in one of the standard textbooks on martingale pricing. Economically this would mean that the payoff basically is defined as the payoff you get by exercising the option early optimally, by choosing time t such that the payoff is maximised.

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