# European option and American option are equivalent in this case?

This is Question No.11 from 2007 May MFE Exam.

For a two-period binomial model for stock prices, you are given:

(1) Each period is 6 months.

(2) The current price for a nondividend paying stock is $$70.00$$

(3) $$u=1.181$$, $$d=0.890$$

(4) The continuously compounded risk-free interest rate is $$5\%$$.

Calculate the current price of a one-year American put option on the stock with strike price of $$80.00$$.

I supposed that for a nondividend paying stock, the price of American put option should be the same as the price of the corresponding European option. Following that thought, I constructed the binomial tree and my calculation is that

$$24.553\times e^{-0.05} \times (1-0.465)^2+6.4237\times e^{-0.05}\times 2 \times (1-0.465)\times 0.465$$

But I was reading the answer, and apparently when calculating the payoff at node $$P_d$$, the answer suggests it is optimal to early exercise the option.

$$P_d=\max (K-S_d,e^{-rh} [P_{ud}p+P_{dd}(1-p)])=\max(80-62.30, e^{-0.05*0.5}[6.42\times 0.465+24.55\times(1-0.465)])=\max(17.70,15.72)$$.

Is there anything wrong with the question? Or did I miss something? Thank you very much!