Modeling Correlated Geometric Brownian Motion Returns

Summary

The document asks how to simulate multivariate geometric Brownian motion returns with a target correlation matrix, using a Cholesky factorization. It focuses on modeling returns directly rather than simulating correlated prices and then deriving returns. The author reports that the latter approach did not preserve the intended correlation in the resulting returns, motivating the request for a source that applies Cholesky to the GBM return process.

No derivation, implementation, reference, or empirical demonstration is provided, so the proposed modeling concern is not resolved here. The observation about correlation changing under price-to-return transformation is the author’s reported experience, not a general result established by the document. Readers would need further analysis to determine the appropriate process and assumptions for their application.

Key ideas

• The document asks how to use Cholesky factorization to generate correlated GBM returns with a specified correlation matrix.
• It raises concern that correlations specified for prices may not carry over to returns derived from those prices.
• The author provides no derivation or source that resolves the modeling question.

Tags

# Source on multivariate correlated geometric Brownian motion returns, not prices


# Source on multivariate correlated geometric Brownian motion returns, not prices












Can anyone provide a source that formulates how to generate multivariate geometric Brownian motion returns using the Cholesky method with target correlation matrix, instead of correlated GBM prices?

If instead, correlated GBM prices are started, and then transformed to returns, I found that the correlation matrix of the prices following this route does not carry over to the returns, and is lost. so it would be better to model correlated GBM returns immediately, not prices. But where is a source that formulates the cholesky method with multivariate GBM (not just `randn`)?

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