The topic of Future/FRA adjustment has already been addressed on a theoretical point view, roughly we need a rate model to calculate the covariance between the money market account of the discount rate, and the floating rate. However, in practice, let s take the EUR case, the discount curve is ESTR, and the floating rate is 3m euribor. Several instruments can give you an idea of the FRAs at the IMM dates, indeed, for the first year, one can construct the ESTR curve out of ecb swaps, and the first ESTr vs 3m euribor IMM FRAs are available. Also, the 1Y imm swap rate is recovered using ester vs3m imm fras. Now, by using the futures prices, you can get the convexity adjustments. Despite recovering IMM swaps in live pricing, the convexity adjustments are far off compared to any rate model calibrated to futures options, or even convexity given by any broker. There are an infinite numbers of convex adjustments that can recover IMM swaps, but how can one construct a 3m curve consistent with IMM swaps, Ester vs 3m euribor imm fras, while the implied convexity looks decent.
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$\begingroup$ 3M IMM swaps, i.e. 3M FRAs, 3M IMM ESTR swaps, the futures markets, and the relevant convexity markets all have consistent prices. If they dont one or the other is more attractive to trade to a EUR derivatives trader: these will be arbitraged away (considering bro costs).. The convexity price is made up of two components; theoretical convexity implied from vol, and supply/demand. Since the supply/demand is a latent variable the mkt price of convexity may or may not agree with your theoretical vol price. That's OK though becuase all the other components and trading activity are consistent $\endgroup$– Attack68 ♦Commented Feb 7 at 11:28
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