# Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$

Are there any robust / non-parametric results for pure stochastic volatility models, in terms of bounds or preferably accurate approximation, for the implied volatility skew $$\partial IV(k) / \partial k$$ when the correlation between the stochastic instantaneous volatility and the asset approaches $$\pm 1$$?

What I mean with "robust" is that if the SV process is $$dS_t = \sigma_t S_t dZ_t$$ $$d\sigma_t = a (t,\sigma_t) dt + b(t,\sigma_t) dW_t$$ $$dW_t dZ_t = \rho dt$$ whatever $$a$$ and $$b$$ are, the bound/approximation on the skew should not depend on the particular form of $$a$$ and $$b$$, hence can the bound/approximation be expressed in terms of Black-Merton-Scholes quantities only (as these BMS quantities are directly observable in the market)?

EDIT:

I can derive that, if $$\Sigma_{d_2}$$ denotes the implied volatility where both the BMS vanna and volga of an option is zero, and $$\Sigma_{d_1}$$ is the implied volatility where only the BMS volga of a vanilla option is zero, then $$\rho E_t \left[ \bar{\sigma} \int_t^T \sigma_u dW_u \right] \approx \Sigma_{d_1} - \Sigma_{d_2}$$ with $$\bar{\sigma} = \sqrt{ \frac{1}{T-t} \int_t^T \sigma_u^2 du}$$ So if I know the value of $$\Sigma_{d_1} - \Sigma_{d_2}$$ as $$\lvert \rho \rvert \rightarrow 1$$, then I can back out $$E_t \left[ \bar{\sigma} \int_t^T \sigma_u dW_u \right]$$. And given the observable value of $$\Sigma_{d_1} - \Sigma_{d_2}$$ I can then subsequently back out $$\rho$$.

Hence my question. Giving it a bounty.