Eur. Phys. J. B 3, 125-137 (1998)
https://doi.org/10.1007/s100510050291

“String” formulation of the dynamics of the forward interest rate curve

LPMC (CNRS UMR 6622) , Université des Sciences, Parc Valrose, 06108 Nice Cedex 2, France and Institute of Geophysics and Planetary Physics, and Department of Earth and Space Sciences, UCLA, Los Angeles, CA 90095-1567, USA

Corresponding author: ^a sornette@naxos.unice.fr.

Received: 30 January 1998
Revised: 12 February 1998
Accepted: 16 February 1998
Published online: 15 May 1998

Abstract

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportunities. This condition takes a form similar to a fluctuation-dissipation theorem, albeit on the same quantity (the forward rate), linking the bias to the covariance of variation fluctuations. We provide the general structure of the models that obey this constraint in the framework of stochastic partial (possibly non-linear) differential equations. We derive the general solution for the pricing and hedging of interest rate derivatives within this framework, albeit for the linear case (we also provide in the appendix a simple and intuitive derivation of the standard European option problem). We also show how the “string” formulation simplifies into a standard N-factor model under a Galerkin approximation.