 Solution

The barrier-triggered put exercises at the particular stopping time ${\tau }_B\wedge T$. The American put option optimizes over all allowed exercise times, soIt can be strictly smaller, but can never exceed the American value. Equality is possible if immediate exercise is already triggered and optimal; a fixed barrier is not generally the optimal finite-maturity exercise rule.

The comparison with the European put option can go in either direction. By the early exercise premium for a fixed barrier,Thus an early payment gains interest but sacrifices remaining put time value; the expression inside braces need not have a fixed sign for arbitrary barriers.

For a concrete general construction take current spot $S\ge K$, $r>0$, $\sigma >0$, and $t<T$. Set $\gamma =2r/{\sigma }^2$ and $b_{\infty }=K\gamma /(1+\gamma )$, the perpetual put option exercise boundary. For $0<B<b_{\infty }$, the perpetual value at $B$ equals $K-B$ and strictly dominates the finite-maturity European value. The premium formula and positive probability of hitting $B$ therefore give $E_B>P$. Strictness follows from the perpetual verification argument: before a fixed positive expiry, its discounted value loses drift $rK$ whenever the stock is below $b_{\infty }$, with positive expected occupation time when started at $B<b_{\infty }$.

Conversely, for $B<K$ sufficiently close to $K$, every possible payment of the exploding put is at most $K-B$, whether paid at touch or on surviving to expiry. Hence $0\le E_B\le K-B\to 0$, while $P(S,t;K)>0$. This gives $E_B<P$ for such barriers, and also strict inequality below the American value.

These examples establish both possibilities in general, not a claim that every fixed initial state admits both. If $S<b_{\infty }$ and only alive barriers $B<S$ are allowed, all those barriers are in the same perpetual stopping region and the premium comparison is nonnegative.