I was reading the paper: https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf
In the equation $(5)$, he is defining the variance of the spread as:
$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\sigma_1 \sigma_2 S_1 S_2 \rho$$
whereas I have always seen it defined as:
$$\sigma_1^2 + \sigma_2^2 - 2\sigma_1\sigma_2\rho$$
This is for 2 correlated GBM and the spread is $S_1 - S_2$.
What am I missing?