Fractional term structure models: No-arbitrage and consistency

TL;DR

This paper introduces fractional Brownian motion-driven HJM interest rate models, establishing no-arbitrage conditions under transaction costs and analyzing the compatibility of Nelson-Siegel family with fractional dynamics.

Contribution

It extends HJM models to fractional Brownian motion, providing no-arbitrage drift conditions and characterizing invariant manifolds for fractional models.

Findings

Fractional HJM models are arbitrage-free under certain conditions.

Nelson-Siegel family is incompatible with fractional HJM dynamics.

Finite-dimensional invariant manifolds are characterized for fractional models.

Abstract

In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White models. It turns out that similar to the Brownian case such a family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family.