## AT-THE-MONEY IMPLIED VOLATILITY

June 9, 2017 by zulfahmed

Studies of the volatility surface have focused on at-the-money implied volatility time behaviour in terms of decay rates being slower than . Empirically, however, even power law 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 which is what would result from a power law long memory effect. The cubic fit is . 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

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