He told me that he saw this in a book by Practical Econometrics by Carol Alexander.
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$\begingroup$ This is most easily seen by examining the contrapositive: when two random variables have a correlation of $\pm 1,$ then one is a constant plus some nonzero multiple of the other (almost surely). Since two Lognormal distributions with different geometric standard deviations do not enjoy any such relationship, they cannot possibly achieve such extreme correlations. One way to find the achievable extremes uses copula theory. $\endgroup$– whuber ♦Commented Sep 21, 2021 at 21:25
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1$\begingroup$ Several questions on site ask about the correlations between bivariate lognormals, and there are answers that detail the calculation of $\rho$. e.g. stats.stackexchange.com/questions/41734/… $\endgroup$– Glen_bCommented Sep 21, 2021 at 23:22
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$\begingroup$ Try searching for lognormal correlation which turns up multiple relevant posts $\endgroup$– Glen_bCommented Sep 21, 2021 at 23:29
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