Smile with the Gaussian term structure model

TL;DR

This paper introduces an affine extension of the Gaussian term structure model, enhancing its ability to model covariances with new numerical tools for pricing and simulation of interest rate derivatives.

Contribution

The paper develops an affine extension of the LGM with semidefinite covariance processes and provides practical numerical methods for pricing and Monte Carlo simulation.

Findings

The model captures the covariance structure with an affine process.

An expansion method for caplet and swaption prices improves accuracy.

A second order scheme enhances Monte Carlo simulation for exotic options.

Abstract

We propose an affine extension of the Linear Gaussian term structure Model (LGM) such that the instantaneous covariation of the factors is given by an affine process on semidefinite positive matrices. First, we set up the model and present some important properties concerning the Laplace transform of the factors and the ergodicity of the model. Then, we present two main numerical tools to implement the model in practice. First, we obtain an expansion of caplets and swaptions prices around the LGM. Such a fast and accurate approximation is useful for assessing the model behavior on the implied volatility smile. Second, we provide a second order scheme for the weak error, which enables to calculate exotic options by a Monte-Carlo algorithm.