# Are there values of the strike price for which an American put and European put have the same no-arbitrage price?

Assuming the options do not pay dividends, is there a strike price that satisfies this?

• Trivially, at maturity (t=T) $P_E=P_A$, otherwise no, because there is a possibility of early exercise for the American Put. – noob2 Apr 6 '20 at 14:28
• If I am not mistaken, the possibility of early exercise imples $P_A \geq P_E$, but not $P_A > P_E$. There might be combinations of circumstances and contract parameters which guarantees early exercise would always be suboptimal. I don't know of any, but failing to provide a proof isn't a proof of failure. – Stéphane Apr 6 '20 at 16:02
• You are right (and I am wrong)... r=0 would do it. Also, even with positive r, a strike of zero would do it: you would never exercise. – noob2 Apr 6 '20 at 23:00