If data for multiple stock prices has a specific correlation matrix, is the correlation matrix preserved when those prices are converted to multivariate log-differenced returns?

  • $\begingroup$ Hi: No. The return is (approximately ) the first difference of the log price and there's no theory that says that taking the log and differencing preserves correlations. In fact, the reason why such a transformation is used is to induce approximate stationarity. Prices themselves are not stationary so it's not a good idea to use them when doing econometric type analysis. $\endgroup$ – mark leeds Sep 30 at 2:35
  • $\begingroup$ i ask because i know how to generate correlated GBM prices, and i know how to generate correlated random (non-GBM) returns, but as you can see, the latter, if converted to prices, will not display desired GBM behavior $\endgroup$ – develarist Sep 30 at 3:32
  • $\begingroup$ The relationship between price and log-return correlation for GBM is derived in quant.stackexchange.com/a/45881/8153 $\endgroup$ – RRL Sep 30 at 5:02
  • $\begingroup$ good i saw that when u first put it here, but the question is about targeting those moments during generation of the gbm, not their relationship $\endgroup$ – develarist Sep 30 at 5:05

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