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2
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Is the delta of a binary option the same as the delta for a regular European option?
Hence,
$$C_{\text{binary}}(S,K) = \lim_{\epsilon \to 0}\frac{C(S, K - \epsilon) - C(S,K)}{\epsilon} = \frac{\partial C}{\partial K}(S,K),$$
and we see the distinction between the delta of the binary … and vanilla call options:
$$\Delta_{\text{binary}} = \frac{\partial C_{\text{binary}}}{\partial S} = \frac{\partial^2 C}{\partial S \partial K} \neq \frac{\partial C}{ \partial S}= \Delta$$ …