Calculating covariance from three variances

Question

I have been asked to look to refactor some code.

There is a line shown below:

$\text{implied covariance} = -\frac{(\text{var}_1 - \text{var}_2 - \text{var}_3)} {2}$,

where $\text{var}_1$ is the implied variance of AUDUSD, $\text{var}_2$ is the implied variance of USDCAD and $\text{var}_3$ is the implied variance AUDCAD

I understand that this is a calculation of covariance between AUDCAD.

However I don't understand the $\text{var}_1 - \text{var}_2 - \text{var}_3$ line. I thought the covariance between two variables was the variance of the two variables multiplied together divided by $n-1$.

Answer 1

I think what you are effectively looking at is

$$\ \begin{align} \log(S_{AUDCAD})&=\log(S_{AUDUSD})\pm\log(S_{USDCAD})\\ \Rightarrow z&=x\pm y \end{align} $$ Thus,

$$ \sigma_z^2=\mathrm{E}\left(\left(x\pm y\right)^2\right)- [\mathrm{E}(x\pm y)]^2 =\sigma_x^2+\sigma_y^2\pm 2\sigma_{xy} $$ Hence, $$ \tag{1} \sigma_{xy}=\frac{\sigma_z^2-\sigma_x^2-\sigma_y^2}{\pm 2} $$

Does that work for you?