Using the Free-Boundary SABR ATM Correction for Implied Volatility

Summary

The document introduces an ATM implied-volatility correction in a free-boundary extension of SABR, motivated by pricing under negative rates. It describes a small-time heat-kernel approximation and gives a first-order correction for the at-the-money case with nonzero correlation. The expression depends on model parameters including correlation, volatility of volatility, the forward level, and transformed SABR parameters.

The text asks how to turn this correction into an ATM implied volatility but does not provide the derivation or implementation. It therefore serves as a starting point for studying the approximation, rather than a complete pricing recipe. Applying it requires the precise definitions and calibration of the transformed parameters, along with clarity about the time expansion and conventions used in the source paper. No numerical example, validation against prices, or discussion of approximation accuracy is supplied.

Key ideas

• The document presents a small-time approximation for SABR implied volatility based on a heat-kernel expansion.
• It gives a first-order correction for ATM volatility when correlation is nonzero.
• The correction depends on both standard SABR parameters and transformed parameters for the free-boundary construction.
• The document poses, but does not answer, how to use the correction to compute ATM implied volatility.

Tags

# The Free Boundary SABR: Natural Extension to Negative Rates


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.