Implied Correlation: Var(a/c) Formula Explained

[volplayer]

Hi,

Haven't studied math for a while and thought I'd ask you for help. It regards implied correlation based on implied volatility for FX options.

(b/c)=(a/c)/(a/b)

Var(b/c) = Var(a/c)+Var(a/b) - 2*Sigma(a/c)*Sigma(a/b)*Corr(a/c,a/b)

When breaking out the Corr(a/c,a/b) from the formula, we get the following:

Corr(a/c,a/b) = (Var(a/c)+Var(a/b)-Var(b/c)) / (2*Sigma(a/c)*Sigma(a/b))

Now let's break out (a/c)

(a/c) = (b/c) * (a/b)

Now I have understood that the Corr(b/c,a/b) formula is the following

Corr(b/c,a/b) = (Var(a/c)-Var(b/c)-Var(a/b)) / (2*Sigma(b/c)*Sigma(a/b))

Does this mean the Var(a/c) formula is like the following

Var(a/c) = Var(b/c)+Var(a/b) + 2*Sigma(b/c)*Sigma(a/b)*Corr(b/c,a/b) ?

I.e. you have a PLUS instead of a MINUS infront of the 2*Sigma*Sigma*Corr part?


Happy if someone could answer this.

 

[fresh_42]

The minus sign is correct. What we have here is just a version of the law of cosines although your notation as fractions is very confusing. Try to determine your random variables in a proper way first.