# The Free Boundary SABR: Natural Extension to Negative Rates

In the paper by Antonov, Konikov and Spector An alternative approximation for the SABR model is presented. I'm interested to implement the formula for the ATM swaptions implied volatilities in the non-zero correlation case, that is based on mimicking the heat kernel expansion for small-time options. $$dF_t=\sigma F_tdW_t \\ \text{volatility expansion} \sigma=\sigma_0+\sigma_1 T \\ \sigma_0=\gamma\frac{|\ln \frac{K}{F_0}|}{s_{\text{min}}} \\ \frac{\sigma_1}{\sigma_0}=\frac{\ln(K^\beta\sqrt{\nu_0\nu_{\text{min}}})-A_{\text{min}}-\ln\sigma_0-\frac{1}{2}\ln(KF_0)}{\frac{s_{\text{min}}^2}{\gamma^2}}$$ Proceeding in this direction, they derive the SABR ZC using new specified parameters $$\tilde{\gamma}$$ and $$\tilde{\beta}$$, to arrive at the first ATM correction expressed by: $$\frac{\tilde{v}_0^{(1)}}{\tilde{v}_0^{(0)}}\bigg|_{K=F_0}=\frac{1}{12}\left(1-\frac{\tilde{\gamma}^2}{\gamma^2}-\frac{3}{2}\rho^2\right)\gamma^2+\frac{1}{4}\beta\rho\nu_0\gamma F_0^{\beta-1}.$$ How can one use this ATM correction to compute the ATM implied volatility?