= Delta bound in an arithmetic stock model
{title2=$0\leq\Delta=\Phi(d)\leq1$}
= Gaussian call delta bound
{c}
{synonym}
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,
$$
0<\Delta<1.
$$
For a fixed spot $s\ne K$, the maturity limit is $\mathbf1_{\{s>K\}}$, 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 $\sigma\sqrt{T-t}$.
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