# Abstracts

**Consistent yield curve modelling**

*Philipp Harms (ETH Zurich, Switzerland)*

Thursday June 5, 11:30-12:00 | session 7.4 | Interest Rates | room K

We present a class of HJM models, which share numerical tractability with factor models, but allow for consistent re-calibration by today’s yield curve. By consistency, we mean that one and the same model is used for simulation, calibration, and estimation of the yield curve. From a mathematical point of view, a rich enough set of increment processes is described, whose concatenation converges to a limit process.

**A Tractable Multi-Factor Dynamic Term-Structure Model for Risk Management**

*Roland Seydel (German Finance Agency, Germany)*

Thursday June 5, 12:00-12:30 | session 7.4 | Interest Rates | room K

We present an affine arbitrage-free dynamic term-structure model based on a representation of instantaneous forward rates as sum of exponentials. The model, which is Gaussian and belongs to the class of Heath-Jarrow-Morton-type models, is intuitively appealing as a suitable linear combination of the stochastic factors can be interpreted as stochastic evolution of stable principal components of the yield curve. Focusing on applications, we derive general principal components in such an affine-linear model, calibrate the model to government bond prices, and derive simple formulas to price caps and floors.

**Analytical RMBS pricing formulas consistent with observed term structures of interest rates and prepayment rates**

*Yukio Muromachi (Tokyo Metropolitan University, Japan)*

Thursday June 5, 12:30-13:00 | session 7.4 | Interest Rates | room K

Recently, Residential Mortgage-Backed Securities (RMBSs) have become more and more important for long-term investments. They have some different features from ordinary bonds, for example, ``negative convexity' to the downward movements of the interest rates. In order to evaluate their features appropriately, the prepayment risk and interest rate risk must be considered based on the available data of their term structures. And, for the risk management purpose, quick pricing is needed because of repeated pricing under various kinds of economic scenarios.

We derive semi-analytical pricing formulas of RMBS under stochastic interest rates and prepayment rates. As the interest rate model, we use the Quadratic Gaussian++ (QG++) model because it generates positive future interest rates and can reconstruct the initial term structure. Recently, the model has been paid much attention to because it is suitable for describing low interest rate environments, however, the theoretical prices of swaptions are not so consistent with the market prices. In order to overcome the problem, we introduce the QG++ model with time-dependent parameters, so that the differences between theoretical and market prices can be reduced drastically, and the estimated values become more reasonable as model parameters. While we assume that the prepayment rates consist of the interest-rate sensitive term and the baseline (interest-rate insensitive) term, and consider the term structures of not only the baseline prepayment rates but also the interest-rate sensitivities of prepayment rates. We use the QG++ model as the baseline prepayment rates, and propose a positive interest-rate sensitive term model. As a result, we can construct a very quick RMBS pricing model, in which the stochastic interest rates and prepayment rates are always positive, and the observed term structures of initial interest rates, swaption prices, prepayment rates and interest-rate sensitivities can be reflected.

Based on the observed data of interest rates and prepayment rates in Japan, we show some numerical examples of our model and compare the theoretical prices with the market ones.

The authors would like to offer special thanks to AFAS, Inc. (www.afasinc.co.jp).