• interest rate model
• multicurve model
• national bonds market model

In the classical interest rate models, have always been supposed the important no-arbitrage relationships, that relate the prices of the zero-coupon bonds and the LIBOR rates. These equalities allow, inside a model, to hedge perfectly forward-rate agreements in terms of zero-coupon bonds. As a consequence, forward rates of different tenors are related to each other by sharp constraints, that however might not hold in practice. For long time the gap between the models and the real market has been considered negligible, but starting from summer 2007, with the beginning of the credit crunch, the market quotes of zero-coupon bonds and of the LIBOR rates, started to violate the classical constraints in a macroscopic way. This because the counterparties began to consider the default of their creditors a realistic possibility, and lending money became a risky investment. Evidences of this fact can be seen for example in the historical data of the spread between the OIS swap rate with maturity one year and the EURIBOR swap rate with the same maturity. While before the crisis was nearly zero, consistently with the usual interest rate models, after the summer 2007 the situation drastically changed, and the OIS-IRS spread started growing, making the financial scenario totally inconsistent with the traditional models. This brought to the need to model separately the forward rates of the various tenor quoted in the market, since they couldn't no more obtained from the classical one-curve models. The result is a multi-curve model where each forward rate dynamic of a given tenor quoted in the market is described by a stochastic equation. The aim of the work is to present the main approaches proposed in literature for the construction of a multi-curve framework.
The first chapter is an introduction of the main definitions, the classical no-arbitrage formulas and the main interest rate derivatives, with a focus on the differences of the formulas in the case of the classical models and in the multi-curve models. Also are described the main techniques needed for bootstrapping the forward rates curve.
The second chapter is about a multi-curve model, adopting a short rate approach. Is supposed the existence of a risk-free short rate and of a risky short rate, and are chosen dynamics for the risk-free rate and for the spread between the two rates. Is obtained a formula for the FRA contracts, that can be used for the calibration.
In the third chapter are described two models, adopting an
Heath–Jarrow–Morton approach. For each tenor quoted in the market is supposed the existence of an instantaneous forward rate and are chosen HJM-dynamics for each one of them. In both models are deduced formulas for pricing swaptions, that can be used in the calibration.
In the fourth chapter is presented a model for the national bonds market. The shapes of the yield curves of the European states, while are similar for short maturities, present significant spreads for long maturities. This reflects the different reliability the investors has in the states, and the different liquidity risk attributed to the states by the market. This suggests, as in multi-curve frameworks, to produce a model, where each national bond dynamic is described by a different stochastic equation. Hence, as in the multi-curve short rate model, is supposed, for each state, the existence of a short rate, modelled as sum of a risk-free rate and a spread. The result is a multi-curve model for the bonds market. The last part also presents a possible pricing methodology for call and put options with underlying a national bond.