IceQuants

I was talking to a friend of mine the other day, about, financial science of course, and the question came up: “I know that by transforming a vector of uncorrelated normals with the cholesky-factoring the correlation matrix will result in a vector of correlated normals… but why does that work?”. I was going to rig up the argument in a hurry, in a bar, on the back of a napkin, but I couldn’t recall it. I promised my friend I’d give him a very simple one-liner argument and the intention here is to fulfill the promise.

First, recall that the correlation matrix of a vector of unit normals is given by

(Notice by the way, that there is an easy fix to the stupid 038 appearing in the matrix, fix explained here but for some reason I don’t have write access to the file /wp-content/plugins/latexrender/latex.php so we’ll have to wait for this to be legible.)

Anyway, that was just to recall what the correlation matrix was. Now assume we Cholesky factorize it as follows:

and consider the outcome of an uncorrelated vector normal transformed by the cholesky factor:

We know that the this must be a unit vector normal, the only thing left to determine is its correlation matrix:

But the cholesky matrix is deterministic and comes out of the expectation, and using the uncorrelated property of we get:

And since the only thing which prevented the last two equations to fit in one line is the unusually narrow margins here, the argument has been presented as a one-liner.

This entry was posted on Wednesday, January 31st, 2007 at 10:41 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.