# 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?

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 $T$.