In this part of the book we look in detail at what a number of significant Gaussian affine models tell us about expectations, risk premia and the behaviour of the yield curve in general. Some of the models are well known; some are introduced here for the first time.

Our intention is not to present an encyclopaedia of models. Rather, we want to discuss the strengths and the shortcomings of a handful of models, each of which is a salient representative of an interesting class. In a way, each model we present in this part of the book tries to answer the questions left open by the Vasicek approach that we have examined in detail in Parts I to II of this book.

As explained in the Introduction, we do not consider affine models as (just) mathematical entities endowed with formal properties (axioms and theorems). Rather, we look at them as restricted-view windows on the financial reality we would like to understand. Therefore the calibration of a model, and its congruence with the empirical evidence about how yields behave, take centre stage in this part of the book.

Broadly speaking, we will look at two classes of models – which, for lack of a better description, we will refer to as the ‘financial-story’ and the ‘statistically motivated’ models. With the first class, one starts from a simplified picture of how the world works (a ‘financial story’), and one translates as best one can this picture into a set of relationships among the chosen variables: a simple model. One then makes the model speak (ie, one extracts predictions from the model), and compares the model predictions with reality. If the match is acceptable, this is corroborating evidence that the financial story we started with was not entirely fanciful, and we can use the model for predictions about quantities that we would like to know, but cannot observe – such as for instance, the risk premium.