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The presence of skew causes a correlation between volatility and spot. This correlation produces a negative shadow delta for all forward starting products (forward starting options have a theoretical delta of zero).

How does this produce a negative shadow delta? The exact mechanism is not clear to me.

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Basically, the author is saying that the delta of an option,

$dC/dS = \frac{\partial C}{\partial S} + \frac{\partial C}{\partial v}\frac{\partial v}{\partial S}$,

where the $\frac{\partial C}{\partial S}$ is the delta assuming constant volatility, the $\frac{\partial C}{\partial v}$ is the vega of the option, and the $\frac{\partial v}{\partial S}$ describes how the implied volatility of the option moves as the spot price moves. This second term is the "shadow delta" being referred to.

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  • $\begingroup$ The use of a total derivative was what I was missing here $\endgroup$ – Permian Feb 19 '17 at 17:06

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