``String'' formulation of the Dynamics of the Forward Interest Rate Curve

TL;DR

This paper introduces a novel string-based formulation for modeling the forward interest rate curve, deriving conditions for arbitrage-free dynamics, and providing solutions for derivative pricing and hedging within this framework.

Contribution

It presents a new string-based approach to interest rate modeling, deriving arbitrage-free PDE conditions and linking the model to standard N-factor models via Galerkin approximation.

Findings

Derived arbitrage-free PDE conditions similar to fluctuation-dissipation theorem.

Provided a general solution for interest rate derivative pricing and hedging.

Showed how the string model reduces to standard N-factor models.

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.