Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 6 / ii / Solution

The call spread in the hint has payoffIt is between zero and one and converges to $1_{\{s>K\}}$ as $ϵ\downarrow 0$. In the Black-Scholes model with positive volatility and $t<T$, the conditional terminal stock distribution has no atom at $K$. The strict and non-strict indicator therefore have the same discounted expectation. The dominated convergence theorem in the risk-neutral pricing formula givesEquivalently this is $e^{-r(T-t)}\Phi (d_2)$. The no-atom condition explains the endpoint subtlety: the displayed spread has value zero at $s=K$, whereas the printed digital pays one there. At maturity the strike derivative can be one-sided or undefined, so the smooth pricing identity is understood before expiry.