I have a moneynessratio-tenor volatility surface and want to extrapolate the implied volatility for moneynessratios > 150%.

The volatility surface was downloaded for different points in time, so I basically have one matrix for every moneyness ratio which holds per column a different (standard) tenor.

These matrices were merged together into one big matrix to be able to process the data.

This table looks as follows:

![Date 50.7 50.14 .... 60.7
2019-02-01 0.67 0.68 0.97
2019-02-02 0.43 0.26 0.26
2019-02-03 0.69 0.66 0.13]

To extrapolate I want to use the quadradic polynom of the form IV = MN² + MN + TN² + TN + alpha with MN = Moneynessratio, TN = Tenor, IV = Implied Volatility, alpha = Intercept.

MN is a matrix that contains the moneynessratios which is the same for every day (e.g. 50%) TN is a matrix that contains the tenor which is the same for every day (e.g. 7).

If I run now a regression, I regress the 1st column of every explanatory variable on the 1st column of the indepenendt variable and get two betas (for every column of the explanatory variables one beta).

After the regression is done, I have a two matrices of betas.

How can I now use them to extrapolate for a non-standard moneyness ratio (e.g. 168 %) and a non standard tenor (e.g. 9 days)?

Thank you very much for your help!

  • $\begingroup$ I would advise against extrapolating a volatility surface with a polynomial, especially one of that form. Pade approximants where you can fix the asymptotic behaviour are a much safer bet. $\endgroup$ – will Aug 27 '20 at 15:07

The solution was to use the observation per day across the moneyness levels and tenors. In my provided table would line two be the the y-vector in the regression and the values moneyness (see values before "." in header) levels the entries for the explanatory variable and the tenors (values after "." in header) the entries for the tenor variables.

Then this regression is repeated per row in the vola surface which results in a beta per date.

You just have to grab the beta according to your desired evaluation date and multiply the observations with it.


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