Why American Puts Can Be Worth More Than European Puts
Summary
The document asks why an American put can have a different value from a European put when both are considered under the Black–Scholes framework. It contrasts the pricing inequality and payoff constraint for an American option with the familiar result that, absent dividends, an American call is worth the same as its European counterpart. The central issue is early exercise: an American holder may exercise before expiry, while a European holder cannot.
The answer focuses on a deeply in-the-money put when interest rates are positive. In that situation, the European put’s value can fall below its immediate exercise payoff because receiving the strike sooner has value; exercising an American put captures that payoff and prevents an arbitrage opportunity. The explanation is qualitative and does not derive the pricing boundary or quantify the value difference. Its intuition depends on positive rates and the stated moneyness conditions.
Key ideas
- An American option’s value must be at least its immediate exercise payoff.
- A European put can be worth less than immediate exercise value when it is deeply in the money and rates are positive.
- Early exercise lets an American put holder receive the strike sooner.
- The no-arbitrage condition rules out an American put price below its exercise payoff.
- The explanation is qualitative and does not calculate the early exercise boundary.
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Full text
# Difference in pricing of American call and put
# Difference in pricing of American call and put
In Paul Wilmotts quantitative finance books he says that the the value of an American option satisfies the following
$$ \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2 \frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV \leq 0 \quad \quad (1) $$ and by the no arbitrage condition we have
$$V(S,t) \geq P(S,t) \quad \quad (2)$$ where P is the time dependent payoff. Now to quote the justification he gives for the price of American and European call options to be same is that
"If we substitute the Black–Scholes European call solution, in the absence of dividends, into the inequality (1) then it is clearly satisfied; it actually satisfies the equality. If we substitute the expression into the constraint (2) with P (S, t) = max(S − E, 0) then this too is satisfied. The conclusion is that the value of an American call option is the same as the value of a European call option when the underlying pays no dividends "
My question is why cant the same be said about American put options .As far as I understand the European put also satisfies (1) with equality .
## Answer by siou0107 (score 3, accepted)
https://quant.stackexchange.com/a/67845
For a very ITM put (i.e. $S \ll E$), when interest rates are positive, you can have $V \left(S, t\right) < \left(E - S\right)^+$ (see figure 2.8 p. 34 in the book). That would correspond roughly to the situation where, as time passes, the loss of time value is virtually null while the discounting effect (present value increases as time goes by) gives a positive theta to your option.
On an American option, this cannot be true as it would give rise to arbitrage opportunities (buy the put and exercise it immediately).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.