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How to Implement Market Models Using VBA by François Goossens

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CHAPTER 6

Yield Curve Models

6.1 SHORT RATE MODELS

6.1.1 Introduction

Basically, short rate dynamics yield the following expression:

image

or, more generally:

image

Depending on whether wt is a single Brownian or a vector, the model is said to be one-factor or multi-factor. Also, the diffusion process can involve several correlated variables (e.g., the Gaussian additive model).

In the literature, it is common to read that models are classified into to two main categories:

  • Equilibrium models, derived from general assumptions based on economic considerations that aim to model the short rate and draw the longer term rates from it;
  • Arbitrage-free models, which are only concerned about fitting the parameters of the models with liquid market data.

In this section, we will address successively one-factor, two-factor, and forward curve models, without reference to any other criterion. Obviously, one chapter cannot cover the matter exhaustively, but the models presented give a global overview of the main features of interest rate modeling theory: further on, these models will be used for pricing some interest rate exotics.

The role of the forward measure By definition, under the forward measure Qt, any asset denominated in zero-bonds has the following martingale property:

On the other hand, under ...

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