Variance of a spread for options on spreads

Question

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?

Answer 1

I think the variance of the instantaneous shifts in the spread is meant:

$V \left[ dX \right]=V \left[ dS_1-dS_2 \right]$

And the individual variances (in the conditional and local sense) are:

$V \left[ dS_1 \right]= \sigma_1^2 S_1^2dt$

$V \left[ dS_2 \right]= \sigma_2^2 S_2^2dt$

And the covariance term is, assuming the two Brownians are correlated:

$C\left[ dS_1 , dS_2\right]=\rho \sigma_1 \sigma_2 S_1 S_2dt$

Now if plug these into your formula, you get the equation 5.