This paper proposes a closed form solution for pricing an American put option on a non-dividend paying stock. An American put option grants its holder rights, but not obligation to sell a stock in a fixed price at any time up until maturity. In the past decades, there is no closed form solution for pricing American options although many people made great efforts. In this paper, an optimally early exercise strategy of an American put option on a non-dividend paying stock is set up. That is, an American put option should be early-exercised when the maximum option premium of early exercise is no less than the value of its European counterpart; otherwise, it should not be early-exercised. Based on this strategy, a series of lemmas is proposed and a closed form formula is drawn. Also, this paper shows that Merton (1973)’s formula does not do a good job for pricing perpetual American put options and shows the price of a perpetual American put option on a non-dividend paying stock is equal to the strike price.