# 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.

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.