Using SABR Volatility of Volatility to Summarize Smile Curvature
Summary
The document asks whether one scalar can summarize an implied volatility smile for a single maturity. It considers ATM curvature, estimated from nearby implied volatilities, and proposes SABR’s volatility-of-volatility parameter alpha as a possible alternative. The answer describes calibrating SABR to observed implied volatilities or option prices under Black–76, with the aim of obtaining parameters that represent the observed smile.
The discussion presents this as a candidate measure, not an established convention. It does not compare the measure with other summaries, provide empirical tests, or explain how alpha relates to curvature across different strikes or parameter settings. Its conclusion is that choosing a single-number representation is an empirical modeling question, so users should assess whether the chosen statistic captures the smile features relevant to their application.
Key ideas
- ATM second derivatives are one possible way to quantify local implied volatility smile curvature.
- SABR volatility of volatility, alpha, is proposed as another candidate summary.
- SABR parameters can be calibrated to observed implied volatilities or option prices.
- The discussion offers no evidence that one scalar is an accepted standard for representing a smile.
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# Expressing Volatility Smile as One Number
# Expressing Volatility Smile as One Number
Is there an accepted way in academia / industry to express the volatility smile as one number? (Not the full vol surface, but just the smile for a given option maturity: i.e. the implied vol as a function of strike).
The second derivative taken at the money is one of the ideas that came to my mind, i.e. if $f(K)$ is the implied vol and $K$ is the strike:
$$f^{''}(K)\approx\frac{f(K_{ATM}+h)-2f(K_{ATM})+f(K_{ATM}-h)}{h^2}$$ for some $h>0$. But that's just one of many obvious ways to summarize the "curvature" around the ATM point: is there an accepted way of expressing the smile as one number?
## Answer by KaiSqDist (score 2)
https://quant.stackexchange.com/a/79057
I looked into the vol of vol parameter in the SABR model and just wanted to share my thoughts on it as a prospective curvature measure. The SABR model is governed by the following stochastic processes:
$$ dF_t = \sigma_t(F_t)^{\beta}dW_t \quad d\sigma_t = \alpha \sigma_t dZ_t \quad dW_t dZ_t = \rho t $$
Some thoughts on properties of the SABR model and its feasibility in capturing the volatility skew/smile:
- The VoV measure mentioned seems to be $\alpha$ in the SABR model.
- Calibrate the SABR model parameters to the currently observed volatility skew or to the price of the options according to the Black-76 model - both calibration methods should produce the same parameter values.
In conclusion, this seems to be more of an empirical question of representation.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.