2
$\begingroup$

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 $\sigma^2 T$ ?

$\endgroup$
  • 1
    $\begingroup$ If you look at Gatheral's paper then it is expressed in terms of total variance, were you looking at Zeliade's paper? $\endgroup$ – raptor22 Oct 2 at 15:09
  • $\begingroup$ 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". $\endgroup$ – jherek Oct 3 at 8:25
3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – jherek Oct 3 at 8:25
  • $\begingroup$ 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. $\endgroup$ – raptor22 Oct 3 at 8:42
  • $\begingroup$ 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). $\endgroup$ – raptor22 Oct 3 at 8:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.