Extrapolating Option Implied Volatility with SVI

Summary

The document explains how the Stochastic Volatility Inspired (SVI) parameterization can represent implied total variance across option strikes. Its formula describes a smile using five parameters: a level, slope scale, skew, horizontal shift, and curvature scale. With several market quotes near at-the-money, these parameters can be calibrated and the fitted curve used to estimate implied volatility at distant, out-of-the-money strikes.

The evidence is a brief conceptual answer; it supplies no calibration example, quote data, error analysis, or comparison with other extrapolation methods. The approach depends on whether the fitted parameterization is a credible representation of the volatility surface beyond the observed strikes. Sparse near-the-money observations alone do not establish that the extrapolated tails are reliable, so the document presents a way to produce estimates rather than a guarantee of accuracy.

Key ideas

• SVI models implied total variance as a function of log-moneyness using five parameters.
• Market quotes can be used to calibrate the SVI parameters.
• A calibrated curve provides implied volatility estimates for strikes beyond the observed quotes.
• Extrapolation quality depends on whether SVI represents the volatility surface in the unobserved region.

Tags

# Extrapolating SVI


# Extrapolating SVI












In his paper Gatheral presents the following parametrization of the implied total variance $w(k,T) = \sigma_{BS}(k,T)^2T$

$$ w(k) = a + b\{\rho (k-m) + \sqrt{(k-m)^2 + \sigma^2} \}.$$

Assuming that we only have a few market prices e.g. 6 or 7 which are close to at-the-money. I wanted to know if there are any common techniques to extrapolate the implied volatility for Strikes that are far out-of-the-money.

## Answer by Gordon (score 0, accepted)

https://quant.stackexchange.com/a/22065

The parametrization, such as this and the SABR volatility, is for an easier looking up of the volatility on a volatility surface. When you have 6 or 7 market quotes, you can calibrate the parameters $a$, $b$, $\rho$, $m$, and $\sigma$. Once this is done and assuming that this parametrization is a bona fide representation of the volatility surface, you are then able to look up implied volatilities for deep out-of-the-money strikes.

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.