 Cash-at-hit digital call

A barrier claim paying a fixed cash amount at the first time the stock reaches a higher level, provided this happens before expiry. In the Black-Scholes model, take $a=\log (c/S_0)/\sigma$ and $b=({\sigma }^2/2-\rho )/\sigma$. Its unit-cash price is ${\mathbb{E}}_Q[e^{-\rho T_{a,b}}{1}_{\{T_{a,b}\le T\}}]$, given by truncated discounted Brownian first passage. The effective square root $\sqrt{b^2+2\rho }=|\rho +{\sigma }^2/2|/\sigma$ is real for every real interest rate.