SABR ATM volatility

The ATM implied volatility is important in SABR when calibrating the model. Let's consider the ATM vol (for a european call option): $$\sigma = \frac{\alpha}{f^{1-\beta}} \left[ 1+ \left(\frac{(1-\beta)^2}{24}\frac{\alpha^2}{f^{2-2\beta}}+\frac{1}{4}\frac{\rho \beta v}{f^{1-\beta}}+\frac{1}{24} (2-3\rho^2)v^2 \right)T \right]$$ where $v$ is rest is obvious and same notation is used in Hagans original paper.

However, it commonly mentioned in literature that this volality can be estimated by first term: $$\sigma \approx \frac{\alpha}{f^{1-\beta}}$$

How does one proof/show that claim?

https://www.next-finance.net/IMG/pdf/pdf_SABR.pdf

• It is pretty obviuos when you look at the terms and some become really small. I can also demonstrate it graphically. But I still want a better argument that this hold for all values of the parameters
– Kim
Sep 14, 2018 at 6:41

In short , this claim does not hold under all circumstances.

There are a few ways to break down such approximation.

1. The options under consideration have very long expiry, i.e. $$T$$ is very large

2. As expiration date approaches, the volatility smile becomes more pronounced, i.e. $$v$$ becomes relatively large.

3. Under extreme market condition, the magnitude of $$\rho$$ and $$v$$ become significant resulting a non-trivial expansion term.

However, in most of the cases, the expansion terms are way smaller than 1, thus I would argue such approximation is legit for most of time.

As @XiaotianDeng mentioned, the simple at-the-money approximation you mention does not always hold: it works only if you assume that $$\alpha^2 T, \nu^2 T$$ are small, typically $$o(1)$$. I wanted to add that there is really no need for such an approximation, except, possibly, to do calculations in your head, or for understanding the scale of $$\alpha$$ against $$\sigma_{atm}$$.

The at-the-money volatility is the solution of a cubic equation, as per your first equation, and this can be solved exactly via Cardano's formula.

Last point, this is really the SABR Hagan expansion ATM volatility, which is used in practice. The actual theoretical SABR model ATM vol may differ in case of large vol of vol or/and long maturities.