Multiple Yield Curve Modelling
Yinglin Zhang
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 fi-nancial 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 mar-ket 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 devel-opment.
The thesis is composed by five chapters.
In the first Chapter we recall shortly the classical interest rate theory. We be-gin 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 mathe-matical issues and general pricing formulas of the main interest rate derivatives (both linear and non linear), like IRS, OIS, FRA, basis swap, caps/floors, swap-tions 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 liq-uidity exposure were used to construct the so called discounting curve. This reference yield curve is used as the discount factor (i.e. numeraire for martin-gale 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 instru-ments. 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 Marko-vian 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 Vasiček-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 mea-sure. 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.
