\begin{align}
\begin{split}
\partial_k \hat{I}(0,k^{*}) = \frac{-\partial_kBS(0,X_0,k^*,I^E(0,k^*))-\mathbb{E}\left(e^{k^*-X_T}1_{X_T\geq k^*}\right)}{\partial_{\sigma}BS(0,X_0,k^*,I(0,k^*))}.
\end{split}
\label{skew_estimator}
\end{align}
