How to Implement Market Models Using VBA

CHAPTER 6

Yield Curve Models

6.1 SHORT RATE MODELS

6.1.1 Introduction

Basically, short rate dynamics yield the following expression:

or, more generally:

Depending on whether w_t 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 Q^t, any asset denominated in zero-bonds has the following martingale property:

On the other hand, under ...