Why American Puts Can Be Worth More Than European Puts
Summary
The document corrects the misconception that American and European options have equal values whenever the underlying stock pays no dividends. It uses a two-period binomial example for an American put and shows that, at an intermediate down node, exercising immediately gives a greater payoff than holding the option for another period. The reason is that early exercise can be optimal for puts, even without dividends.
The answer contrasts puts with calls: with nonnegative interest rates and no dividends, early exercise is not optimal for an American call, so its value matches the corresponding European call. The discussion is a conceptual correction supported by the example’s exercise-versus-continuation comparison. It does not derive a general early-exercise boundary or cover negative rates, dividends, or alternative pricing models.
Key ideas
- An American put can be worth more than its European counterpart even when the stock pays no dividends.
- At a binomial-tree node, compare immediate exercise value with the discounted expected value of continuation.
- Early exercise is optimal when the immediate put payoff exceeds the continuation value.
- For nonnegative interest rates and nondividend-paying stocks, American calls have the same value as European calls.
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Full text
# European option and American option are equivalent in this case?
# 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!
## Answer by Mark Joshi (score 3, accepted)
https://quant.stackexchange.com/a/18169
European puts need not agree with American ones. The equality is true for call options when there is non-negative interest rates and non-positive dividends.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.