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onlyaxel
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Joined: July 12th, 2010, 12:22 pm

Cholesky factorisation+negative semi-definite matrix

September 1st, 2010, 4:29 pm

Hi everybody,Does anyone know how to simulate correlated brownian motions (paths followed by stocks) when the variance covariance matrix is negative semi definite, so we can't do the Cholesky factorisation. Is it a good idea to use the eigenvalue cleaning (I am sure how to use it when there is complex eigenvalues in addition to negative eigenvalues), or is there another method to regularize the matrix.Thanks in advance
 
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eh
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Joined: March 2nd, 2010, 9:26 am

Cholesky factorisation+negative semi-definite matrix

September 2nd, 2010, 7:15 am

Quotethe variance covariance matrix is negative semi definiteThis is a contradiction. By definition a covariance matrix is positive semi-definite.
 
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Costeanu
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Joined: December 29th, 2008, 5:33 pm

Cholesky factorisation+negative semi-definite matrix

September 2nd, 2010, 2:11 pm

Hi onlyaxel,If by negative semi-definite, you really mean negative semi-definite, there is nothing you can do. If however, you mean a matrix which is not postive semi-definite, then you are probably talking about a corrupted correlation matrix. This can happen sometimes, for example when you calculate the matrix from time series some of which may miss some time-points. For a corrupted correlation matrix there is a procedure to "fix" it, i.e. to find the "nearest" correlation matrix. The two properties of a correlation matrix are: - non-negative eigenvalues- 1 on the diagonalYou do the following two steps repeatedly: 1. remove non-negative eigenvalues -> diagonalize your matrix, and floor your eigenvalues at 0, and then change back to the original base2. replace whatever you find on the diagonal with 1 After each step you find a matrix that has only one of the two properties, but after enough iterations you will get one that has both of them within numerical accuracy. If your corrupted correlation matrix is small enough, this procedure converges fairly rapidly, maybe in a hudred steps or so. If the matrix is large, you need a faster algorithm. For such algorithm, consult work done by Highman (just google "nearest correlation matrix"). Best,V.
 
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onlyaxel
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Joined: July 12th, 2010, 12:22 pm

Cholesky factorisation+negative semi-definite matrix

September 2nd, 2010, 8:16 pm

Thank you very much to all of you and especially Costeanu. I thought correlation matrix just needed to have only 1 on the diagonal and to be symmetric and so my simulations failed always for n>8. But now with your explanations, It is clear Thx again