No arbitrage global parametrization for the eSSVI volatility surface

The log‐moment formula for implied volatility

“…Extensions of this framework can be found in Guo et al. (2016) and detailed calibration methodology in Hendriks and Martini (2019), Martini and Mingone (2021), Martini and Mingone (2022), and Mingone (2022). Expanding it as x tends to $$-\infty$$ yields $$$$\begin{equation*} \mathrm{I}_{\mathrm{SSVI}}(x) = \sqrt {\frac{\theta (1-\rho )\varphi (\theta )}{2}}\sqrt {|x|} + \frac{c}{\sqrt {|x|}} + \mathcal {O}{\left(|x|^{-3/2}\right)}, \end{equation*}$$$$where the constant c depends explicitly on θ and $$\varphi (\theta )$$, but we omit the details for clarity.…”