The LIBOR Forward Rate in a HJM-Lévy Framework

Abstract

An extended LIBOR forward rate model is derived through what we call the HJM-Lévy framework. The resulting model is a geometric Itô-Lévy process, of which the well known geometric Brownian motion with deterministic volatility is one of many special cases. One specific case of the LIBOR forward rate in the HJM-Lévy framework is a geometric Brownian motion with stochastic volatility. This special case is analyzed and implemented. Two caplet valuation formulas expressed by power series are derived for the model. One for the general geometric Itô-Lévy process, and one for the specific case of a geometric Brownian motion with stochastic volatility.