Suppose I am long spread option with underlying : rate A - rate B. The vega on the option would be positive. But if I want to compute the option vega with respect to individual rates, can I use the below formula?
sprdVar = (v1)^2 – 2 * rho * v1 * v2 + (v2)^2
where sprdVar = spread variance, i.e. sprdVol^2 v1, v2 = vol of the assetA/B (in normal vol)
d(sprdVar)/d(v1) = 2*(v1) – 2 * rho * v2
So for the spread vols to be negative wrt asset A, v1- rho*v2 < 0 . Will this be correct or the actual rate levels will also play a role in determining the option vega sensitivity from individual rates? Thanks