A unified view of LIBOR models

TL;DR

This paper introduces a comprehensive framework for LIBOR rate modeling using semimartingales, unifying various existing models and providing conditions for arbitrage-free and martingale properties.

Contribution

It offers a unified approach encompassing multiple LIBOR models, with verifiable conditions for arbitrage-freeness and martingale behavior, and structural insights into model properties.

Findings

Framework covers models driven by Brownian motion and jump processes

Conditions for arbitrage-free and martingale LIBOR rates derived

Only exponential affine models preserve structure under measure changes

Abstract

We provide a unified framework for modeling LIBOR rates using general semimartingales as driving processes and generic functional forms to describe the evolution of the dynamics. We derive sufficient conditions for the model to be arbitrage-free which are easily verifiable, and for the LIBOR rates to be true martingales under the respective forward measures. We discuss when the conditions are also necessary and comment on further desirable properties such as those leading to analytical tractability and positivity of rates. This framework allows to consider several popular models in the literature, such as LIBOR market models driven by Brownian motion or jump processes, the L\'evy forward price model as well as the affine LIBOR model, under one umbrella. Moreover, we derive structural results about LIBOR models and show, in particular, that only models where the forward price is an exponentially affine function of the driving process preserve their structure under different forward measures.