Discrete-Time Interest Rate Modelling

TL;DR

This paper develops an axiomatic framework for discrete-time interest rate models using a pricing kernel approach, linking arbitrage-free pricing with equilibrium economics and deriving explicit asset pricing formulas.

Contribution

It introduces a novel axiomatic scheme for discrete-time interest rate modeling that ensures arbitrage-free conditions and links to economic principles.

Findings

Derived a general expression for asset prices including dividends and retained earnings.

Established conditions for the existence of a previsible money-market account.

Provided explicit examples of discrete-time interest rate models.

Abstract

This paper presents an axiomatic scheme for interest rate models in discrete time. We take a pricing kernel approach, which builds in the arbitrage-free property and provides a link to equilibrium economics. We require that the pricing kernel be consistent with a pair of axioms, one giving the inter-temporal relations for dividend-paying assets, and the other ensuring the existence of a money-market asset. We show that the existence of a positive-return asset implies the existence of a previsible money-market account. A general expression for the price process of a limited-liability asset is derived. This expression includes two terms, one being the discounted risk-adjusted value of the dividend stream, the other characterising retained earnings. The vanishing of the latter is given by a transversality condition. We show (under the assumed axioms) that, in the case of a limited-liability asset with no permanently-retained earnings, the price process is given by the ratio of a pair of potentials. Explicit examples of discrete-time models are provided.