# How does delta adjustment relate to skew stickiness ratio (SSR)?

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?