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If one imposes a form $r(t) = \text{...}$ on the cc. short rate, and aims to fit the short end of a SOFR (or another modern RFR) using futures, how would one best go about this within a "curve-fitting" paradigm e.g. least-squares on fitting error a la fitting a bond curve by minimizing error of $\sum_b (\sum_i c_{i,b} \exp(-\int r(s)ds))$ (where b is a bond, c a cashflow).

I have seen this done by converting a 3m compounded SOFR quote into a forward quote, how does that sound in principle (convexity adjustment aside)?

It should be noted that the form $r(t)$ would be chosen to answer a specific question (it could be piecewise flat around meeting dates, it could be a smooth hump around the cycle) — i.e. the question is how to best do it for an arbitrary but well-behaved function.

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