# Why is the SVI parameterization in terms of variance?

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 $$\sigma^2 T$$ ?

• If you look at Gatheral's paper then it is expressed in terms of total variance, were you looking at Zeliade's paper? – raptor22 Oct 2 '19 at 15:09
• Which paper are you referring to? It is in terms of variance in his initial presentation in Madrid "A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives". – jherek Oct 3 '19 at 8:25

## 2 Answers

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.

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.

• The question is not whether you can jump from one to the other, it is more why the choice of variance vs. vol or total variance? – jherek Oct 3 '19 at 8:25
• Well the only difference should be in the interpretation of these two parameters. Increasing $a$ translates either the variance (= total variance by $at$) or total variance (=variance by $a/t$) and same for $b$ with the slope of put/call wings (see p. 5 of the linked Gatheral paper). It seems clear that working with total variance is more natural for deriving arbitrage conditions. However I guess that it is easier to interpret the former parametrization. If you look at p. 6/7 they introduce another paramtrization (SVI-JW) of the variance that is more interpretable. – raptor22 Oct 3 '19 at 8:42
• But in essence it is really the same thing and does not really change anything practically. Using on or the other is imho only a matter of notations. (plus you can compare parameters accross tenors with the variance parametrization). – raptor22 Oct 3 '19 at 8:46