Consider an American put and a European put with the same st

Consider an American put and a European put with the same strike price K and expiration T. In this problem, we investigate the relationship between the price P of the American put and the price p of the European put. We already know that P - p > 0 since an American put is always worth at least as much as a European put. Consider a portfolio where you sell one American put and buy one European put and invest the proceeds P - p. Note that the initial value of the portfolio is zero. Show that if r = 0, then the portfolio is guaranteed to be worth at least P - p at time T. Deduce that P = p.

Solution

At t = 0 the portfolio value will be zero as at that P = p

As the time increases the value of the american put will increase more than the value of european put so if the person sells the american put and buy european and invest remaining then again after investing he can earn more and then again invest and P-p wil be the least profit.