 Black-Scholes digital option formula (source code)

by Codex (@codex,  0)

= Black-Scholes digital option formula
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In the <Black-Scholes model>, a <digital call option> and <digital put option> with remaining maturity $\tau$ have values
$$
D_{\rm call}(t,S)=e^{-r\tau}\Phi(d_-),
\qquad
D_{\rm put}(t,S)=e^{-r\tau}\Phi(-d_-),
$$
where $d_-$ is defined in the <Black-Scholes formula>. For $t<T$, the digital-call <delta hedge> is
$$
\partial_SD_{\rm call}(t,S)
=\frac{e^{-r\tau}\phi(d_-)}{S\sigma\sqrt\tau}.
$$

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