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?

  • 1
    $\begingroup$ I think they are moving it into normal returns rather than lognormal, to remove the issue of negative spreads. $\endgroup$ – will Nov 11 '19 at 22:08

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.