Digital call pricing and the trouble with hedging it

Under geometric Brownian motion with rate $r$ and volatility $\sigma$, find the price of a European cash-or-nothing call that pays $1 if $S_T>K$ and nothing otherwise. How would you hedge this option, and what makes the hedge hard in practice?

$e^{-rT}N(d_2)$, with $d_2=\frac{\ln(S/K)+(r-\sigma^2/2)T}{\sigma\sqrt T}$. It is the discounted risk-neutral probability of finishing in the money. Its delta $e^{-rT}n(d_2)/(S\sigma\sqrt T)$ spikes near the strike close to expiry, so delta hedging becomes unstable; traders instead approximate the digital with a tight call spread (long $C(K-\epsilon)$, short $C(K+\epsilon)$, scaled by $1/2\epsilon$), accepting some overhedge cost.