Fixed-payout binary call and its link to vanillas

A European binary option pays a fixed cash amount $A$ at $T$ if $S_T > X$ and nothing otherwise. Under Black-Scholes, what is it worth, and how does it relate to the price of an ordinary call?

$A e^{-rT} N(d_2)$. It is the discounted risk-neutral probability of finishing above $X$, times $A$. Since $\partial C/\partial X = -e^{-rT}N(d_2)$, the binary equals $-A\,\partial C/\partial X$: it is the limit of a tight call spread, long $A/\varepsilon$ calls at $X - \varepsilon$ and short $A/\varepsilon$ at $X$.