A class of locally state-dependent models for forward curves

TL;DR

This paper introduces a novel class of locally state-dependent stochastic models for forward curves in finance, ensuring well-posedness and linking to existing models for practical application.

Contribution

It develops a Hilbert-space based framework with locally state-dependent coefficients, guaranteeing existence, uniqueness, and positivity of solutions, and connects to established models for fixed delivery times.

Findings

Model captures entire forward curve dynamics with a single SPDE.

Certain projections are Markovian and satisfy 1D SDEs.

Conditions for positivity and well-posedness are established.

Abstract

We present a dynamic model for forward curves within the Heath-Jarrow-Morton framework under the Musiela parametrization. The forward curves take values in a function space H, and their dynamics follows a stochastic partial differential equation with state-dependent coefficients. In particular, the coefficients are defined through point-wise operating maps on H, resulting in a locally state-dependent structure. We first explore conditions under which these point-wise operators are well defined on H. Next, we determine conditions to ensure that the resulting coefficient functions satisfy local growth and Lipschitz properties, so to guarantee the existence and uniqueness of mild solutions. The proposed model captures the behavior of the entire forward curve through a single equation, yet retains remarkable simplicity. Notably, we demonstrate that certain one-dimensional projections of the model are Markovian and satisfy a one-dimensional stochastic differential equation. This connects our Hilbert-space approach to well established models for forward contracts with fixed delivery times, for which existing formulas and numerical techniques can be applied. This link allows us to examine also conditions for maintaining positivity of the solutions. As concrete examples, we analyze Hilbert-space valued variants of an exponential model and of a constant elasticity of variance model.