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The correct delta hedging of a derivative $V$ in a model where volatility $\sigma$ is a function of the underlier $S$ requires a stock holding of an amount $$ \frac{dV}{dS}=\frac{\partial V}{\partial S}+\frac{\partial V}{\partial \sigma}\frac{\partial \sigma}{\partial S}. $$ If I understand correctly, this article makes the point that by thinking about volatility dynamics in terms of a volatility level and a shape curve, it is possible to deduce the $\partial \sigma/\partial S$ from the Skew Stickiness Ratio (SSR) defined by Bergomi (2004) and Bergomi (2016) as $$ \mathcal{R}=\frac{1}{\mathcal{S}}\frac{d\sigma_F}{d\log S}, $$ where $\sigma_F$ is the at the forward volatility for a certain maturity and $\mathcal{S}=\partial\sigma_F/\partial K$ is the volatility skew for the same maturity.
Can anybody explain me how would it be possible? $\partial\sigma/\partial S=\mathcal{S R}$ does not hold due to the logarithmic transformation of $S$. In addition, we know from different perspectives that under the sticky-strike assumption $d\sigma/d S=0$ while $\mathcal{R}=1$. Can anybody help me make sense of this?

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