Q: how do you query the vol surface at a fitted strike but between 2 fitted maturities — Jun and next Dec, assuming today is Jan 1.

First Take sigma_J*sigma_J and sigma_D*sigma_D. Say we get Something like 20%^2 and 30%^2. Remember these are annualized sigmas. Suppose those maturities are 6 months and 24 months out. Raw variance values would be

Variance_J = (20%^2)* 6m/12m = .02

Variance_D = (30%^2)* 24m/12m = .18

Our assumption is that variance is linear with TTL, so let’s line up our raw variance values

6 months to expiry –> .02

15 months to expiry -> x

24 months to expiry -> .18

==> x = .10 (not annualized)

Annualized variance_x == x /(15/12) = .08

Annualized sigma_x = 28.28%

This estimate is better than a naïve linear interpolation like

6 -> 20%

15 -> ?????? — 25%

24 -> 30%