Delta bound in an arithmetic stock model ($0\le \Delta =\Phi (d)\le 1$)

For positive volatility $\sigma >0$ and positive remaining maturity, the call price in an arithmetic stock model with interest has option delta $\Delta =\Phi (d)$, where $d$ is the standardized discounted moneyness. Since a finite argument of the standard normal distribution function gives a value strictly between zero and one,For a fixed spot $s\ne K$, the maturity limit is $1_{\{s>K\}}$, the derivative of the call financial payoff away from its kink. The payoff derivative is undefined at $s=K$. Under the nondegenerate Gaussian terminal law, the event $S_T=K$ has probability zero; it is this strike event that is exceptional. At zero interest use the same formula with normal scale $\sigma \sqrt{T-t}$.