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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.

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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.

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  • $\begingroup$ Thanks for your answer. The issue I have with the parametrization is the high number of parameters. Having 6 observations to fit 5 parameters, should be fine in theory. However it seems to be quite delicate in practice due to overfitting, especially when the observations are all located around the same area. What would you do if you have even less parameters, say less than 5 ? $\endgroup$
    – Jonkie
    Dec 3 '15 at 7:13
  • $\begingroup$ If you have less data, certain parameters may need to be pre-specified. $\endgroup$
    – Gordon
    Dec 3 '15 at 13:41
  • $\begingroup$ Are there any common pre specifications of the parameters for equities ? $\endgroup$
    – Jonkie
    Dec 5 '15 at 12:08
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    $\begingroup$ For deep OTM strikes it is common to use extrapolation based on a direct parametric form for options prices. See the RBS paper quarchome.org/risktailspaper_v5.pdf $\endgroup$
    – Antoine Conze
    Jan 2 '16 at 13:32
  • $\begingroup$ @AntoineConze the method provided in the paper from RBS requires the user to pre-specify the weights of the tails. After having worked out both the recommended and the additional choice (footnote 7) for the extrapolation functions. I am wondering if there is a general consensus on setting the for a large number of equities. $\endgroup$
    – Jonkie
    Jan 4 '16 at 8:18

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