Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 29K / b / Solution

For $0\le t<T$, set $\tau =T-t$ andUnder the equivalent martingale measure, conditional log-normality givesThe risk-neutral pricing value of the digital call option is consequentlyAt $t=T$ this converges to the stated payoff away from $S_T=K$, with the payoff convention specifying the boundary value.

The delta hedge holds the derivative of the claim value with respect to the current stock price. Sincethe number of risky-asset units for $t<T$ isThis is the Black-Scholes digital option formula. The hedge becomes singular close to maturity near the strike, reflecting the discontinuity of the payoff.