When the SABR ATM Volatility Approximation Breaks Down
Summary
This discussion examines the commonly used SABR at-the-money volatility approximation that keeps only the leading term, alpha divided by the forward price raised to one minus beta. The fuller Hagan expression includes correction terms involving expiry, the volatility of volatility, and the correlation and beta parameters. The replies explain that the leading term is justified when those corrections are small, rather than as a universal identity.
Long expiries and large volatility of volatility can make the omitted terms material, while the approximation is often reasonable when the expansion terms remain small. One reply notes that the ATM expression can instead be handled as a cubic equation, so the shortcut is mainly useful for rough scale intuition or mental calculations. The discussion concerns Hagan’s expansion; the expansion itself may differ from the theoretical SABR model’s actual ATM volatility, especially with long maturities or high volatility of volatility.
Key ideas
- The leading-term SABR ATM volatility approximation is valid only when correction terms are small.
- Long expiries and high volatility of volatility can make the omitted terms significant.
- Correlation and volatility-of-volatility parameters affect the size of the expansion correction.
- The Hagan ATM expression can be treated as a cubic equation rather than approximated.
- Hagan’s expansion may diverge from the theoretical SABR ATM volatility in challenging regimes.
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Full text
# SABR ATM volatility
# 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
## Answer by Xiaotian Deng (score 3)
https://quant.stackexchange.com/a/41955
In short , this claim does not hold under all circumstances.
There are a few ways to break down such approximation.
- The options under consideration have very long expiry, i.e. $T$ is very large
- As expiration date approaches, the volatility smile becomes more pronounced, i.e. $v$ becomes relatively large.
- 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.
## Answer by jherek (score 0)
https://quant.stackexchange.com/a/47335
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.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.