Normal SABR Implied Volatility and Its Strike Approximation

Summary

The document examines the normal implied volatility formula for SABR when beta is zero, corresponding to a Bachelier model for the forward. It contrasts a formula using the forward-minus-strike difference with a further approximation that substitutes a logarithmic expression involving the forward and strike. The question asks why this substitution is made and whether it is intended to handle cases where prices or strikes have different signs.

The author reports that the two forms behave differently in implementation: the logarithmic version fits deep out-of-the-money strikes poorly and behaves badly near zero, while the difference-based expression can support calibration to negative strikes. These are reported observations rather than a derivation or systematic comparison. The document provides no resolution of when the approximation is appropriate, so it is best read as a modeling question highlighting that algebraic approximations can affect calibration and boundary behavior.

Key ideas

• The beta-zero SABR case gives a normal implied volatility formula for options on a forward.
• The document compares a forward-minus-strike input with a logarithmic approximation involving both values.
• The logarithmic form is reported to fit deep out-of-the-money strikes poorly.
• The author also reports unstable behavior near zero and notes that the difference form permits negative strikes.
• The document raises the purpose and limitations of the approximation but does not resolve them.

Tags

# Hagan's implied vol formula for Normal SABR


# Hagan's implied vol formula for Normal SABR












I'm sure there's an obvious answer to this question so apologies. But reading through this seminal paper on the SABR model, the author's provide an explicit formula for the (normal) implied vol of a $\tau$-expiry, strike $K$ option, $\sigma_N(K)$ (ibid. equations A67a,b). For the case $\beta=0$ (i.e. for a Bachelier distribution of the forward $f$) this formula simplifies to \begin{equation} \sigma_N(K)=\alpha\frac{\zeta}{D(\zeta)}\left(1+\frac{2-3\rho^2}{24}\nu^2\tau\right)\tag{1} \end{equation} where $\alpha,\nu,\rho$ are the remaining SABR parameters, \begin{equation}\zeta=\frac{\nu}{\alpha}(f-K) \tag{2}\end{equation} and \begin{equation}D(\zeta)=\ln\left(\frac{\sqrt{1-2\rho\zeta+\zeta^2}-\rho+\zeta}{1-\rho}\right).\end{equation} Now, the authors "further simplify" $(1)$ by recasting $\zeta$ (through a series expansion of the term $(f-K)$) as \begin{equation}\zeta=\frac{\nu}{\alpha}\sqrt{fK}\ln\left(\frac{f}{K}\right)\tag{3}\end{equation} with $D(\zeta)$ as before (ibid. equations A69a,b). It escapes me as to exactly why this is done: is it purely to eliminate $K$ and $f$ having opposite signs and setting $K \in (0,\infty]$? Not only does taking $\zeta$ as given by $(2)$ allow for smile calibration to negative strikes, taking it in the form $(3)$ (implemented on a computer) gives poor fitting for deep OTM strikes and nasty behavior for strikes close to zero.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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