Chaos and coherence: a new framework for interest–rate modelling

TLDR

The paper introduces elementary axioms for stochastic finance that use the state‑price density to determine the stochastic dynamics of the interest‑rate term structure. By treating the state‑price density as a potential, the authors identify an asymptotic random variable whose conditional variance equals the density, then apply Wiener chaos expansion to this variable to parametrize discount‑bond dynamics and derive analytic expressions for bond prices, volatility, short rate, and risk premium, with extensions to FX and general assets. They establish a natural mapping from admissible term‑structure trajectories to the symmetric Fock space, showing that any stochastic bond‑price system corresponds to an incoherent element of that space.

Abstract

A set of elementary axioms for stochastic finance is presented wherein a prominent role is played by the state–price density, which in turn determines the stochastic dynamics of the interest–rate term structure. The fact that the state–price density is a potential implies the existence of an asymptotic random variable X ∞ with the property that its conditional variance is the state–price density. The Wiener chaos expansion technique can then be applied to X ∞ , thus enabling us to ‘parametrize’ the dynamics of the discount–bond system in terms of the deterministic coefficients of the chaos expansion. Using this method, we find that there is a natural map from the space of all admissible term–structure trajectories to the symmetric Fock space F naturally associated with the space of square-integrable random variables on the underlying probability space. An element of F is either coherent or incoherent, and a stochastic bond–price system is necessarily represented by an incoherent element of F . Making use of the linearity of F we derive simple analytic formulae for the bond–price system, the volatility structure, the short rate, and the risk premium associated with an arbitrary admissible term–structure model. Extensions to foreign–exchange markets and general asset systems are also developed.