• finanza matematica
• multiple yield curve

The last crisis of 2007 has affected massively the financial market with the
raising of the credit crunch. Credit and liquidity risk even among the biggest financial
institutions manifests in significant spreads which were negligible before
the crisis, like Eonia-Euribor spread, OIS-IRS spread, basis swap spread etc.
The classical interest rate theory, which was sufficient enough to explain market
dynamics pre-crisis, now becomes problematic, since the usual no-arbitrage
relationships began to be violated in a macroscopic way.
Many new models are being developed concerning this problem. The main
idea is to model risky interbank rates of different tenors as separated assets, as
in general risk increases with tenor length. Such approach is called multi-curve
approach, in contrast with the classical single-curve approach where all market
issues are described by a single yield curve.

In this thesis we give an introduction and some simple examples of multiple
yield curve modelling methods, since the argument is still in continuous development.

The thesis is composed by five chapters.

In the first Chapter we recall shortly the classical interest rate theory. We begin
with the First Fundamental Theory which is the basis of every market model.
After that, we give the basic definition of bonds and interest rates as mathematical
issues and general pricing formulas of the main interest rate derivatives
(both linear and non linear), like IRS, OIS, FRA, basis swap, caps/floors, swaptions
etc. We give also a synthetic view of principal pre-crisis bond market
models which will be extended in the multi-curve context.

The second Chapter presents the problem formulation of multiple yield curve
modelling.
Overnight rates (e.g. Eonia, OIS etc.) which have reduced credit and liquidity
exposure were used to construct the so called discounting curve. This
reference yield curve is used as the discount factor (i.e. numeraire for martingale
measure) to value future cash flows, it completes the risk-free picture where
the classical theory continues to hold.
Interbank rates, considered risky, now must be modelled separately. Their
yield curves of different tenors are called forwarding curves. Our aim is to
construct a coherent market model consisting of both discounting curve and
forwarding curves.
In this Chapter we will also show how to use an interest rate model in
practice.
1. Bootstrapping technique gives a method of constructing initial yield curves,
both discounting and forwarding.
2. Calibration is the most important step in which the market parameters
are adapted to market data.
3. Calibrated model is then used to price future cash flows of market instruments.
If tractable formula is not available, numerical methods must be
used. The most used method is Monte Carlo simulation which is however
computationally slow and expensive; a more sophisticate method requires
the discretization of associated PDEs when original SDEs have a Markovian
setting, such as Fokker-Planck equations whose strong solution gives
the correspond SDE solution’s density function.

A first simple multi-curve model is shown in Chapter three. It is an extended
short rate model which uses a Vasicek-type factor structure for modelling both
risk-free short rate and short rate spreads between the risk-free one and the risky
one. This modelling approach leads to a generalization of affine term structure,
and the calculation of an adjustment factor between pre-crisis and post-crisis
FRA prices. However, this model has several problems, like the assumption of
negative values with non zero probability and the difficulty to apply calibration
procedure.

Chapter four gives a more elaborated modelling method. It is inspired both
by the classial HJM framework and Libor model, i.e. it models both risk-free
instantaneous forward rate and risky forward Libor rates under T-forward measure.
Particular volatility assumption ensures that SDE solutions are Markovian
processes; moreover, the entire model dynamics can be driven by a finite family
of Markov processes, thus the original model becomes a generalization of shifted
multi-factor Hull-White model. Unlike the previous model, this model leads to
a real Black’s formula for caps and an approximated one for swaptions, which
simplified enormously the calibration problem.

Finally, in the last Chapter a numerical example is given for a simplified
case of HJM-type model presented in Chapter four. Initial yield curve graphs
and calibrated model parameters are obtained by concrete market data.