Lognormality of rates and term structure models

Abstract

A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting instantaneous rates are well defined (they do not explode) and remain positive, contrary to those derived in [2]. They are also bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann in [22] and [23] for Markovian lognormal short rates to (non-Markovian) lognormal forward rates. We show also existence of invariant measures for the proposed term structure dynamics