= Solution
In the <Black-Scholes model>, the stock follows geometric Brownian motion $dS=\mu S\,dt+\sigma S\,dW$, 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 gives
$$
\boxed{V(t,s)=e^{-r\tau}\Phi(d_2),\qquad
d_2=\frac{\log(s/K)+(r-\sigma^2/2)\tau}{\sigma\sqrt\tau}.}
$$
This 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 hedge>
$$
\boxed{\Delta(t,s)=\frac{e^{-r\tau}\varphi(d_2)}{s\sigma\sqrt\tau},}
$$
where $\Phi,\varphi$ 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 $s=K$, $d_2\to0$ 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.
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