Fitting a Polynomial to Extrapolate Implied Volatility by Moneyness and Tenor
Summary
The document asks how to estimate implied volatility at moneyness and tenor combinations outside the observed volatility surface. It proposes a quadratic regression using moneyness and tenor as explanatory variables, then clarifies how to organize the observations: for each date, use the surface values across moneyness levels and tenors as the response vector, paired with the corresponding moneyness and tenor inputs. Repeating this fit across dates produces date-specific coefficients that can be applied to evaluate a chosen combination.
The response explains the data layout and evaluation step, but gives no quantitative comparison, validation, or error estimates. The proposed polynomial form and its behavior beyond observed moneyness or tenor ranges are not assessed. Consequently, the method is a basic extrapolation recipe whose reliability outside the sampled surface remains uncertain; a fit alone does not establish that implied volatility follows a quadratic relationship.
Key ideas
- Fit the surface separately for each date using observations across its moneyness and tenor grid.
- Pair each observed volatility with the matching moneyness and tenor inputs in the regression.
- Use the coefficient set for the desired date to evaluate volatility at a nonstandard point.
- The document provides no evidence that the polynomial extrapolates accurately beyond observed values.
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Full text
# Extrapolate Implied Volatility Surface # Extrapolate Implied Volatility Surface 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]1 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! ## Answer by StableSong (score 0) https://quant.stackexchange.com/a/52891 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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