Arbitrage-Free Interpolation in Models of Market Observable Interest Rates

TL;DR

This paper introduces an arbitrage-free interpolation method for interest rate models that maintains Markovian properties and positivity of rates when extending from discrete to continuous tenors.

Contribution

It proposes a novel interpolation approach that preserves key properties of interest rate models during the extension process.

Findings

Preserves Markovian properties in continuous tenor extension

Guarantees positivity of interpolated interest rates

Maintains arbitrage-free conditions in the extended model

Abstract

Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing this extension; one which preserves the Markovian properties of the discrete tenor models and guarantees the positivity of all interpolated rates.