Volatility smile
The smile is fitted before differentiation. Second derivatives amplify quote noise, so the bandwidth below directly controls how stable the density estimate is.
Raw vs smoothed IV — Oct 16, 2026
Call side
Model inputs
0.055
Gaussian kernel bandwidth in log-moneyness
4.30%
Continuous discount rate r
1.30%
Continuous yield q
ATM smoothed IV
15.15%
Strike 580
Downside wing IV
25.63%
Strike 480
Upside wing IV
14.59%
Strike 640
Fit RMSE
1.68%
Root mean squared IV residual vs raw quotes
Refitted call curve
Smoothed IVs converted back to Black-Scholes call prices; this curve is what gets differentiated twice.