Under what conditions will both European and American put options worth the same?

It is well-known that on a non-dividend paying stock, it is suboptimal to exercise an American call option earlier. In other words, both European and American call options on the same non-dividend paying stock worth the same. This can be proven using the Put-Call Parity.

However, I am not sure about European and American put options. I know that due to the ability that an American option can be exercised any time prior to maturity, it should worth at least as much as European put option. I am interested to know the conditions that give equality. In particular,

Question: Under what conditions will both European and American put options worth the same?

My motivation behind asking this question is so that I can ensure that my binomial tree implementation to price American put option is correct.

When dividend is zero, my binomial output the same price for both European and American call options, which is a good sign. However, I have no alternative to check whethee the tree gives correct price for American put option or not.

• Well, trivial examples would be zero strike ($K=0$) and at maturity ($t=T$). Also, the early exercise premium equals zero if $r=0$. – Kevin Jun 13 '20 at 5:08
• Perhaps you could explain your second statement? – Idonknow Jun 13 '20 at 10:38
• Early exercise is about receving your cash early and not waiting until the option expires. If there's no interest rate (no discounting, no time value of money), then there's no point of early exercise. More mathematically, look at the early exercise premium derived in Carr, Jarrow, Myneni (1992). It clearly equals zero if $r=0$. – Kevin Jun 13 '20 at 11:19
• To test your code you need an example where American and European prices are different. For an interesting test case try a put option that is deeply in the money (S very low compared to K) and a very high interest rate (double digit). This is a case where the American should be exercised early. – noob2 Jun 14 '20 at 8:03
• ... i.e. the american should have a price = Intrinsic Value, while the European price should be lower than this. – noob2 Jun 14 '20 at 15:41