If X = strike price, the upper bound for a American put option is P <= X, which makes sense. For a European put option, you must add a time value component to the upper bound [p <= X*exp(-rt)] since you have to wait until the expiration date to receive proceeds from the sale of the underlying...
The following passage is in the P1.T3.R19 Hull Financial Markets & Products Study Notes (Page 88):
Shouldn't the full price equal the present value of its cash flows PLUS accrued interest? (Passage states that AI is not zero.)
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