I know that it is possible to simulate two correlated GBM in e.g. Matlab (Generating Correlated Asset Paths in MATLAB) based on cholesky decomposition. However, they take as input the correlation matrix, which from my understanding is just the Pearson correlation coefficient. However, if I look at correlation between two time series, cross-correlation is the correct measure. This also results in a lag. Is it possible to incorporate the lag into the simulation of the correlated GBM?
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If I am understanding your question correctly, maybe you can simulate two correlated GBM's and then apply the lag manually afterward.
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$\begingroup$ So in that case I suppose I should use the correlation coefficient from the maximum lag to simulate the two correlated GBMs? $\endgroup$– WillartDec 30 '20 at 8:26