Explicit local volatility formula for Cheyette-type interest rate models

TL;DR

This paper derives an explicit analytical formula for local volatility in Cheyette interest rate models, extending the Dupire framework and aiding model calibration.

Contribution

It introduces a novel explicit formula for local volatility in Cheyette models, derived from an extension of the Dupire approach.

Findings

Provides an explicit formula for local volatility in Cheyette models.

The formula extends to multi-factor models.

Facilitates practical model calibration.

Abstract

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.