Expressing Volatility Smile as One Number

Question

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 1

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.