# If two lognormal distributions have correlation $\rho$, what is the correlation between the log of those distributions?

Suppose I have $$X,Y$$ which are lognormal with correlation, $$\rho$$, what is the correlation between $$log(X)$$ and $$log(Y)$$? I tried working it out analytically and I'm getting that you have to approximate is using a taylor series. Is that correct?

• Assuming you intend that $(\log X, \log Y)$ be bivariate normal, then the answer here should get you most of the way: stats.stackexchange.com/questions/132855/… ... you just need to write that covariance term on the right hand side in the equation at the end as a correlation times the two lognormal standard deviations, and simplify (there's some cancellation). Beware the difference between the notation there (where $\rho$ is the conventional log-scale correlation) and what you have above. Aug 19 '19 at 23:10
• There's also a related answer here: stats.stackexchange.com/questions/6853/… though it may be a little harder to see how to back it out from that one. Nevertheless it uses the same relationships as the first link. Aug 19 '19 at 23:11
• @Glen_b Thanks, basically, I just want to simulate this in python. Do you know if I can compute the correlation and then use this: stackoverflow.com/questions/31977175/…. I want this method in python (rdocumentation.org/packages/MethylCapSig/versions/1.0.1/topics/…). Aug 19 '19 at 23:37
• The answer to the question you link to describes the usual approach (generate MVN and then exponentiate). Aug 19 '19 at 23:43
• @Glen_b so would I need the formula you linked to get the correlation between the normals and then use that approach? Aug 19 '19 at 23:46