A General Gaussian Interest Rate Model Consistent with the Current Term Structure

Marco Di Francesco

doi:10.5402/2012/673607

Abstract

We describe an extension of Gaussian interest rate models studied in literature. In our model, the instantaneous spot rate is the sum of several correlated stochastic processes plus a deterministic function. We assume that each of these processes has a Gaussian distribution with time-dependent volatility. The deterministic function is given by an exact fitting to observed term structure. We test the model through various numeric experiments about the goodness of fit to European swaptions prices quoted in the market. We also show some critical issues on calibration of the model to the market data after the credit crisis of 2007.

Highlights

A short-rate model for the term structure of interest rates is based on the assumption of a specific dynamics for the instantaneous spot-rate process r for the definition of r, we refer for instance to the monographs 1–3
These models were the first approach to describe and explain the shape and the moves of the term structure of interest rates. These models are very convenient since the dynamics of the instantaneous spot rate drives all the term structure, in the sense that both rates and prices of bonds are defined as an expectation of a functional of r
In this paper we describe a general exogenous model in which the instantaneous spot rate r is the sum of correlated Gaussian stochastic processes with time-dependent volatility plus a deterministic function given by an exact fitting to the observed term structure

Summary

Introduction

A short-rate model for the term structure of interest rates is based on the assumption of a specific dynamics for the instantaneous spot-rate process r for the definition of r, we refer for instance to the monographs 1–3 These models were the first approach to describe and explain the shape and the moves of the term structure of interest rates. These models are very convenient since the dynamics of the instantaneous spot rate drives all the term structure, in the sense that both rates and prices of bonds are defined as an expectation of a functional of r. We describe all processes under the risk-neutral measure Q for the definition and the existence of Q, we refer to the monographs 1–4 .

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Conclusion