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In the Black-Scholes model, the stock follows geometric Brownian motion $dS=\mu Sdt+\sigma SdW$, with constant volatility, and a risk-free account earns constant rate $r$. Frictionless continuous self-financing trading and no arbitrage lead to risk-neutral drift $r$. With $\tau =T-t$, risk-neutral lognormality givesThis is the Black-Scholes digital option formula, obtained by discounting $P(S_T>K)$, not the stock-weighted probability of an ordinary call. Differentiation gives the Delta hedgewhere $\Phi ,\phi$ are the standard normal distribution and density. At fixed $s\ne K$, the density term decays faster than ${\tau }^{-1/2}$ grows, so delta tends to zero. At $s=K$, $d_2\to 0$ and $\Delta \sim [K\sigma \sqrt{2\pi \tau }{]}^{-1}$ diverges. Thus a narrowing, increasingly large delta spike forms around the strike. The terminal payoff is discontinuous: frequent large hedge adjustments near the strike, transaction costs, jumps, discrete trading, and small model or price errors make practical replication difficult. A fixed path avoiding the strike and the at-the-strike limit need not behave alike.

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.