Consistency conditions for affine term structure models

TL;DR

This paper investigates the conditions under which affine term structure models (ATSM) are consistent and valid, especially when the short rate can be unbounded below, providing theoretical criteria for their application.

Contribution

It derives sufficient and necessary conditions for the Feynman-Kac formula's applicability and bond price monotonicity in affine models, including jump-diffusion extensions.

Findings

Conditions for Feynman-Kac application established

Monotonicity of bond prices characterized

Results applicable to jump-diffusion processes

Abstract

ATSM are widely applied for pricing of bonds and interest rate derivatives but the consistency of ATSM when the short rate, r, is unbounded from below remains essentially an open question. First, the standard approach to ATSM uses the Feynman-Kac theorem which is easily applicable only when r is bounded from below. Second, if the tuple of state variables belongs to the region where r is positive, the bond price should decrease in any state variable for which the corresponding coefficient in the formula for r is positive; the bond price should also decrease as the time to maturity increases. In the paper, sufficient conditions for the application of the Feynman-Kac formula, and monotonicity of the bond price are derived, for wide classes of affine term structure models in the pure diffusion case. Necessary conditions for the monotonicity are obtained as well. The results can be generalized for jump-diffusion processes.