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## AT-THE-MONEY IMPLIED VOLATILITY

Studies of the volatility surface have focused on at-the-money implied volatility time behaviour in terms of decay rates being slower than $1/T$.  Empirically, however, even power law $T^{-\alpha}$ may not be quite right.  The graph below shows an example from the stock USO, volatility surface of 2015-04-06.  The black line is log(implied volatility) and the green line is the fit of a cubic polynomial.  If only the linear term is significant, then a power law is sufficient.  However, the red line is the best fit by a linear function (R^2=0.716) and the green line which seems quite accurate (R^2=0.9915) is a cubic polynomial fit.  If the linear fit were sufficient we would have decay $ATMIV(T) = T^{-0.19}$ which is what would result from a power law long memory effect.  The cubic fit is $log(ATMIV) \sim -0.99 -0.22 T -0.04 T^3 -0.0017 T^2$.  The quadratic term is insignificant:

> summary(model)

Call:
lm(formula = x ~ poly(log(t), 3))

Residuals:
1 2 3 4 5 6 7
-0.0005498 0.0060277 -0.0103279 -0.0034048 0.0088524 0.0021038 -0.0030762
8
0.0003747

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.992963 0.002780 -357.196 3.69e-10 ***
poly(log(t), 3)1 -0.220855 0.007863 -28.089 9.56e-06 ***
poly(log(t), 3)2 -0.001724 0.007863 -0.219 0.83720
poly(log(t), 3)3 -0.040101 0.007863 -5.100 0.00698 **

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.007863 on 4 degrees of freedom
Multiple R-squared: 0.9951, Adjusted R-squared: 0.9915
F-statistic: 271.7 on 3 and 4 DF, p-value: 4.465e-05