= Early exercise premium for a fixed barrier
{title2=$E_B-P=\mathbb E[e^{-r(\tau_B-t)}(K-B-P(B,\tau_B;K))\mathbf1_{\{\tau_B\leq T\}}]$}
For an alive <barrier-triggered put>, the difference from the corresponding <European put option> is the displayed discounted conditional expectation, evaluated under the <risk-neutral measure>. To prove it, condition the European payoff on the first hitting time or expiry and use the <Strong Markov property>. At a hit, continuation of the European claim is worth $P(B,\tau_B;K)$, whereas the fixed exercise rule pays $K-B$ immediately. On survival to expiry the two payoffs coincide. The relevant stopped quantities are bounded, so the conditional-expectation argument is justified.
Back to article page
