Barrier digital call ($e^{-rT}\mathbb{Q}({\sup }_{s\le T}{S}_s\ge c)$)

A cash-at-maturity upper-barrier digital option pays a fixed amount if the stock touches an upper barrier by maturity. In the Black-Scholes model, write $a=\log (c/S_0)/\sigma$ and $b=-(r-{\sigma }^2/2)/\sigma$. Its unit-payoff price is $e^{-rT}$ times the linear-boundary Brownian first-passage distribution at $T$. This differs from paying at the hitting time, when the discount factor is random, and from an ordinary terminal-price digital option.