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Your $P_I(t,T)$ is the formula for the so-called "pseudo" discount curve. It can be used to compute relevant LIBOR forward rates and LIBOR zero rates. The "true" discount curve is of course the OIS discount curve, which can be built independently of the LIBOR curve.


You wrote Given this, what does the value of 1M LIBOR curve at 1Y point represent? It is a real number X such that: The following deal can be agreed today in the swap market: You will pay me the amount X (fixed in advance) one year from now, and in return I agree to pay you one year from now the amount Y equal to the 1 Month Libor Rate published at that ...


Libor includes risk. It is riskier to make a 6m loan than two 3m loan. So the 6m Libor curve is not the same as the 3m one. Ther difference is the basis spread. When using a short rate model, you are modelling one curve. As a first approximation, you can deduce the other curves by adding a deterministic basis spread.

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