Sign of a binary option's sensitivity to vol

If volatility rises, what happens to the price of a European digital that pays a fixed sum when the stock ends above the strike? Explain both intuitively and with a formal derivation.

It can go either way: vega is negative in the money and positive out of the money. With price $e^{-rT}N(d_2)$ and $\partial d_2/\partial\sigma = -d_1/\sigma$, vega $= -e^{-rT}\phi(d_2)\,d_1/\sigma$. Intuitively, extra volatility threatens a digital that is already likely to pay but gives an unlikely one a better chance; vega is zero where $d_1 = 0$, slightly below the strike.