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)?


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.


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