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Why SVI Uses Implied Variance and How It Relates to Total Variance

Article Quant Q&A · Author: jherek
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Summary

The discussion explains why Gatheral's SVI smile is parameterized in implied variance rather than volatility or total variance. Variance is a natural scale because the model is linear in variance in the far strike wings. This makes the asymptotic slopes, controlled by SVI parameters, directly useful for checking conditions associated with Lee's moment formula for extreme-strike implied volatility.

Total variance is closely related and can be represented by scaling the parameters with maturity, so the two formulations can be converted. The practical distinction is interpretability: total-variance parameters may become very small at short maturities. The thread also notes that traders may prefer SVI-JW, which describes the smile through at-the-money volatility, slopes, and curvature. These are representation and interpretation considerations; the discussion does not claim that one form is universally superior for calibration or trading.

Key ideas

  • SVI uses variance because its wings are linear in variance.
  • The asymptotic wing slopes make checks related to Lee's moment formula straightforward.
  • Variance and total variance parameterizations can be converted by maturity scaling.
  • Total-variance parameters may be harder to interpret at very short maturities.
  • SVI-JW provides a representation in terms of at-the-money volatility, slopes, and curvature.

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OptionsVolatilityDerivatives pricingStatistics
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# Why is the SVI parameterization in terms of variance?


# 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 $\sigma^2 T$ ?

## Answer by Fabien (score 3, accepted)

https://quant.stackexchange.com/a/49048

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 by raptor22 (score 3)

https://quant.stackexchange.com/a/49036

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.

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.