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(By XIBOR I intend any EURIBOR or LIBOR rate. By RFR I intend SOFR for the USD and ESTR (€STR) for EUR.)

I am mainly focused on the EUR rates market (but also a bit on the USD market) and looking for a model able to price EURIBOR swaps, caps, floors, swaptions and EURIBOR exotics (callable or not) like TARNs, range accruals etc, as well as ESTR/SOFR linked (OIS) swaps and SOFR swaptions (until their EUR counterparts appear). To sum up, I need a model pricing vanilla XIBOR derivatives, XIBOR exotics (callable or not) as well as well as vanilla RFR derivatives (OIS swaps and swaptions, caps floors, RFR futures).

I came across Mercurio's and Lyashenko's FMM model and it's extended version (see their two SSRN papers) where they define backward-looking forward rates $R_j (t)$ and forward-looking forward rates $F_j (t)$ which, to quote their first paper, "are set at the beginning of their application period. For instance, a forward-looking rate at time $T_{j−1}$ with maturity $T_j$ can be defined similarly to an OIS swap rate".

First, unless I missed some big hypothesis of a constant spread between the forward-looking forward rate based on the RFR and the XIBOR rate (or more directly between the RFR the XIBOR rate), I don't see what link this $F_j (t)$ could have with a XIBOR. Second, this $F_j (t)$ defined similarly to an OIS swap rate conflicts with the introduction of their first paper where it's written that that the classic interest rate modeling framework can be naturally extended to describe the evolution of both the forward-looking (IBOR-like) and backward-looking (setting-inarrears) term rates using the same stochastic process.

For sure I am missing something (probably linked to some collateral) somewhere but I can't see what, or misunderstanding the "XIBOR-like" signification.

For sake of completeness, the SSRN links for the two papers :

Looking Forward to Backward-Looking Rates: A Modeling Framework for Term Rates Replacing LIBOR

and

Looking Forward to Backward-Looking Rates: Completing the Generalized Forward Market Model

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