A class of locally state-dependent models for forward curves
Nils Detering, Silvia Lavagnini
TL;DR
This work develops a Hilbert-space–valued forward-curve model in the Filipović space under the Musiela parametrization, where drift and diffusion are state dependent through point-wise maps and multiplicative operators. By establishing conditions on the point-wise map $\Psi$ and the corresponding diffusion, the authors prove (local) existence and uniqueness of mild solutions and study positivity, as well as invertibility for measure changes. A key contribution is showing that fixed-delivery forwards induce one-dimensional SDEs, enabling tractable pricing and calibration while preserving arbitrage-consistency across maturities. The paper also provides two concrete specifications—the exponential (geometric) model and a CEV-style model—to illustrate the framework and its connections to classical models. Together, these results yield a flexible, scalable approach to modeling entire forward curves with state-dependent dynamics and practical avenues for calibration and risk management.
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.