Modelling the term structure of interest rates \'{a} la Heath-Jarrow-Morton but with non Gaussian fluctuations

TL;DR

This paper extends the Heath-Jarrow-Morton interest rate model by incorporating Paretian (heavy-tailed) fluctuations, deriving new stochastic differential equations, and analyzing the implications for bond pricing and jump processes.

Contribution

It introduces a generalized HJM model with Paretian noise, derives a new Itô's lemma for such variables, and explores the large-time behavior of Paretian jump processes.

Findings

Paretian fluctuations prevent the elimination of drift in discounted bond prices.

Large-time jump distribution exhibits a unique decay rac{K}{(\u221a{ ext{ln}(k t)})^2} in Fourier space.

The model connects to fractional diffusion equations, highlighting non-Gaussian market dynamics.

Abstract

We consider a generalization of the Heath Jarrow Morton model for the term structure of interest rates where the forward rate is driven by Paretian fluctuations. We derive a generalization of It\^{o}'s lemma for the calculation of a differential of a Paretian stochastic variable and use it to derive a Stochastic Differential Equation for the discounted bond price. We show that it is not possible to choose the parameters of the model to ensure absence of drift of the discounted bond price. Then we consider a Continuous Time Random Walk with jumps driven by Paretian random variables and we derive the large time scaling limit of the jump probability distribution function (pdf). We show that under certain conditions defined in text the large time scaling limit of the jump pdf in the Fourier domain is \tilde{omega}_t(k,t) \sim \exp{-K/(\ln(k t))^2} and is different from the case of a random walk with Gaussian fluctuations. We also derive the master equation for the jump pdf and discuss the relation of the master equation to Distributed Order Fractional Diffusion Equations.