Why is the SVI parameterization in terms of variance?

Question

The Gatheral SVI parameterization reads $$\sigma^2 = a + b \left[\rho(k-m) + \sqrt{(k-m)^2+s^2}\right]\,.$$ Why is it expressed in terms of variance $\sigma^2$ and not directly in terms of volatility $\sigma$ or in terms of total variance $\sigma^2 T$ ?

Answer 1

One main characteristic of the SVI parameterization is to be linear in variance in the wings. It is a desirable property, since the criteria to obeys Lee's Moment Formula for Implied Volatility at Extreme Strikes translates then a simple condition on the asymptotic slopes, that is on $a$ and $b$.

And thus variance becomes the natural scale to find a parameterization. Now between total variance and variance, there is very little difference. The problem with expressing parameters in total variance is the interpretation of those: for very short maturities the numbers end up very small and it is difficult to make any sense of them.

Finally, for traders, other representations, such as SVI-JW (jump wings) detailed in Gatheral and Jacquier paper Arbitrage-free SVI volatility surfaces, with emphasis on at-the-money volatility, slopes and curvature is more natural.

Answer 2

Let $\tilde{a} = at$ and $\tilde{b} = bt$ and you can jump from a parametrization to another. In Gatheral and Jacquier's paper (Arbitrage-free SVI volatility surfaces) https://arxiv.org/pdf/1204.0646.pdf they parametrize total variance directly whereas in Zeliade's 2+3 optimization (Quasi-Explicit Calibration of Gatheral’s SVI model) http://www.zeliade.com/whitepapers/zwp-0005.pdf they parametrize for variance.