  The crisis that affected financial markets in the last years leaded market
practitioners to revise well known basic concepts like the ones of discount
factors and forward rates. A single yield curve is not sufficient any longer to
describe the market of interest rate products. On the other hand, using
different yield curves at the same time requires a reformulation of most of the
basic assumptions made in interest rate models. In this paper we discuss market
evidences that led to the introduction of a series of different yield curves.
We then define a HJM framework based on a multi-curve approach, presenting also
a bootstrapping algorithm used to fit these different yield curves to market
prices of plain-vanilla contracts such as basic Interest Rate Swaps (IRS) and
Forward Rate Agreements (FRA). We then show how our approach can be used in
practice when pricing other interest rate products, such as forward starting
IRS, plain-vanilla European Swaptions, Constant Maturity Swaps (CMS) and CMS
spread options, with the final goal to investigate whether the market is
actually using a multi-curve approach or not. We finally present some numerical
examples for a simple formulation of the framework which embeds by construction
the multi-curve structure; once the model is calibrated to market prices of
plain-vanilla options, it can be used via a Monte Carlo simulation to price
more complicated exotic options.
