Value of a one-dollar touch claim at a higher barrier

A stock currently trades at $1 and follows geometric Brownian motion; the risk-free rate is zero. A perpetual contract pays $1 at the first moment the stock price hits a level $H>1$ (and nothing if it never does). What is the contract worth today?

$1/H$. With $r=0$ the stock is a martingale under the risk-neutral measure. Stopping it at the hitting time of $H$ (it cannot go negative, so optional stopping applies) gives $1=H\cdot\mathbb{P}(\text{hit } H)$, hence the hit probability, and the claim's value, is $1/H$. Equivalently, holding $1/H$ shares replicates it.