Multi-curve Pricing: Short Notes

Issue:
After Mid 2007, a lot of the good-o textbook wisdom failed to hold, mainly in two aspects:

1. Some rates use to match one another almost exactly (i.e. zero spread), for example deposit rate vs overnight swap rate; now the spread is much larger.
2. Swaps with different settlement frequencies have very large rate spread, i.e. the size of the swap rate depends on the fixing.

Proposed remedies:

1. Use separate curves for discounting and forwarding
2. Treat LIBOR's with different fixings as independent underlyings

Questions:
Even in the good-o textbook context, shall we expect LIBOR's with different tenors to have zero spread? Or is it just that tenor used to be irrelevant in the old narrative?

Reference:
Henrard 2009, "Irony in Discounting: The Crisis"
Bianchetti 2009, "Two Curves, One Price"
Mercurio 2009, "Interest Rates and the Credit Crunch: New Formulas and Market Models"