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Can we interchangeably use Cholesky decomposition of covariance and correlation matrix to generate simulations? If not, in which situations do we use one or the other and why? Thanks in advance.

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The usual approach is to decompose the correlation matrix. See e.g. here sitmo.com/article/generating-correlated-random-numbers Given the relationship between correlation and covariance $\rho_{XY}=\frac{cov(X,Y)}{\sigma_X \sigma_Y}$ you can always turn a covariance matrix into the correlation one. –  Probilitator Mar 23 at 9:25

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You can use the either, as both necessarily are symmetric positive definite; covariance is a personal preference. It's really just a matter of scaling, as $\mathcal{N}(0,\Sigma)$ is distributionally $\sqrt{\Sigma} \mathcal{N}(0,1) $.

Correlation would require additional scaling (i.e. multiplication of every $\mathcal{N}(0,\rho)$ element by its respective volatility, and therefore requires more operations).

Glasserman (p. 72-74) also uses the covariance matrix for his introduction to Cholesky factorization, so I suspect it is not unusual, however I have also seen correlation (e.g. example of @Probilitator).

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+1 for pointing to Glasserman's approach –  Probilitator Mar 23 at 13:52
    
Yes, ideally, we should be able to use either of those to generate the random samples. But are there any practical implications? Like while I was discussing the same thing with someone who works as a market risk consultant, he told me that some of the implementations only calculate correlation matrix once a month and calculate daily covariance matrix based on daily variances and the correlation matrix. I guess this approach would save some computation time but is this a correct approach? –  user3212376 Mar 27 at 7:20

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