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 ?