Why American Option Values Do Not Fall with Longer Maturities
Summary
The document explains why an American call or put, with the other contract and market assumptions held fixed, is worth at least as much when its maturity is extended. The central argument is based on exercise flexibility: any stopping time available before the shorter expiry remains available under the longer expiry, while the holder also gains the option to exercise later. Since the longer contract permits every strategy available to the shorter one, its optimal expected discounted payoff cannot be lower.
The formal explanation expresses price as the supremum of expected discounted exercise payoffs over admissible stopping times and notes that the shorter maturity’s set is contained in the longer one’s set. This establishes nondecreasing value under the stated comparison. It does not imply a strict increase, and the conclusion assumes the contracts are otherwise comparable, including the underlying payoff and relevant model inputs. A second answer invokes time value and early exercise flexibility but offers less formal support.
Key ideas
- An American option’s value is the best expected discounted payoff over permitted exercise times.
- Extending maturity preserves all stopping times available under the shorter contract.
- The longer maturity adds possible exercise times, so its value cannot be lower under otherwise identical assumptions.
- The result establishes nondecreasing value, not necessarily a strict increase.
Tags
Full text
# American call and put prices, increasing in maturity
# American call and put prices, increasing in maturity
Show that American call and put prices are increasing in maturity $T$.
Does this mean I need to show that as $T$ increases than the American call and put prices increase as well? If so, how do I go about showing this?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/22756
For American options, the longer the maturity, the more choices for the optimal exercises time, then the option value is bigger. For example, consider maturities $T_1$ and $T_2$, for the same option except for different maturities. Any optimal exercise time within $[0, T_1]$ is a possible exercise time within $[0, T_2]$, with a better time possibly falls in $(T_1, T_2]$.
Formally, the value of an American option, with maturity $T$, is given by \begin{align*} \sup_{\tau\in \mathcal{T}_{[0,T]}}E\Big(e^{-r\tau}\max\big(\phi(S_{\tau}-K, 0) \big)\Big), \end{align*} where $\mathcal{T}_{[0,T]}$ is the set of stopping times with values in $[0, T]$. Here, $\phi=1$ for a call and $-1$ for a put. Note that, for $0 < T_1 < T_2$, $\mathcal{T}_{[0, T_1]} \subset \mathcal{T}_{[0,T_2]}$. Therefore, \begin{align*} \sup_{\tau\in \mathcal{T}_{[0,T_1]}}E\Big(e^{-r\tau}\max\big(\phi(S_{\tau}-K, 0) \big)\Big) \le \sup_{\tau\in \mathcal{T}_{[0,T_2]}}E\Big(e^{-r\tau}\max\big(\phi(S_{\tau}-K, 0) \big)\Big). \end{align*} That is, American call and put prices are increasing in maturity $T$.
## Answer by Will Gu (score 0)
https://quant.stackexchange.com/a/22761
american options are at least as expensive as their european counterparts. So it's enough to argue that european options increase in value as time to expiry prolongs, given other metrics remain the same. This is because of the "time value" of options.
On the other hand, longer time give you more opportunity to early exercise, which adds in zero or positive addition values. Thus, the options have more value.
Two factors come together, the american options increase in value as time to expiry increases.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.