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I use a LMM model from a well known vendor, using a SOFR swap curve and SOFR swaptions. The calibration set include many/all of the ATM swaptions from 1m-1y to 30y-30y and I get a very good fit for volatility quotes. The next thing I am trying to incorporate is correlations between short rates and long rates. The model I use offers the capability to calibrate to various correlation matrices at various future observation points like 1y,2y,5y etc... One matrix must be provided for each observation horizon. However, the correlations that the model can calibrate to is effectively a cross-sectional (across paths) of forward CMS rates at each forward horizon. This is a bit unusual, because it is much more common to observe absolute changes in rate levels for correlation (and volatility) purposes. In addition, it introduces spurious elements as at times, shorter CMS rates can be 1.5% versus 4.5% for 30y rates. The vendor has no obvious way to modify the calibration algorithm, so my question is how to use historical CMS data to compute the correlations that their model fits to. In particular, if I have say a data set of 10 years of daily data, or 2,500 points, how can I capture the different future observation horizons in my historical analysis? All my data sets are spot historical data sets. So how can I create a correlation matrix of outright rates (as mentioned above, I'd prefer changes thereof) after 1y, then 2y then say 5y based on a data set of spot rates?

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