# Infinite Horizon Barrier Option Paradoxe [duplicate]

I've came across this question which is puzzling me. Imagine that interest rates are zero and that you observe a stock $S_t$ whose value today $S_0$ is equal to 1\$. We consider the derivative that pays 1\$ at the time $\tau$ where the stock crosses a fixed barrier $B > 1\$$. We will suppose that the risk-neutral probability of the event$\mathbb{Q} \left(\exists t \geq 0, \quad S_t\geq B\right)=1$Let us now consider the following pricing approaches : • Martingale Approach : Using this approach, we know that the value of the derivative today is the expected (discounted, but rates are 0.0) payoff under the risk-neutral probability. The payoff is exactly equal to the indicator function$1_{\exists t \geq 0,S_t\geq B}$So the price, is simply the probability of that event which is 1. • Replication Approach Imagine that one wants to replicate the option using the underlying, he can buy$1/B$of the stock today. At the time where the stock value is equal to$B$, the portfolio is worth$1/B * B = 1$which is the payoff of the option.$\implies$A portfolio of$1/B$shares today replicates the payoff of the option at each state of the world. By no arbitrage the value of the option should be$1/B\$

The two approaches yield different pricing results, yet I cannot find the logical flow behind one method or the other. Can you please help ?

Thanks!