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I'm wondering what is the best practice for a consistent Forward Curve construction in the multicurve Framework (cf Bianchetti & Ametrano 2013):

Suppose for example that we have already bootstrapped a discount curve $C_{d}$ and we're building a $6M BOR$ forward curve: what instruments shall we use in the short end of the curve, say from $ON$ to $6M$, to be fully multicurve coherent ?

I've seen implementations where deposits are used, but then wouldn't this lead to "dirty" forwards on the short end since these instruments are not actually based on $6M BOR$ tenor?

Regards

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One possible solution is to build "synthetic" short term 6M IBOR deposits by extrapolating for $T < 6\text{M}$ from the 6M IBOR deposit and 1x7, 2x8, etc. 6M IBOR FRAs as I have seen done in various places, or better by extrapolating from the known 0x6, 1x7, 2x8, etc. OIS-6M IBOR basis as suggested in section 4.4.2 of the paper you are referring to.

In any case you could argue that when valuing a deal indexed on 6M IBOR, any fixing for the rate that covers a period $[T-6\text{M}, T]$ is already fixed when $T < 6\text{M}$ so the choice of short term extrapolation should not have a large impact when the 6M IBOR curve is used as a projection curve. It would have a small impact though when using the 6M IBOR curve as a discount curve, as would be the case say if you're valuing an uncollateralized deal under the assumption that unsecured funding is done at 6M IBOR.

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