Boundary conditions for the single-factor term structure equation

TL;DR

This paper analyzes the boundary conditions for the single-factor term structure equation, establishing conditions for uniqueness of solutions and implications for numerical methods in modeling nonnegative short rates.

Contribution

It characterizes boundary behaviors in the term structure equation, providing conditions for solution uniqueness and practical insights for numerical implementation.

Findings

Boundary behavior determines solution uniqueness.

Attainable boundaries require boundary conditions for uniqueness.

Nonattainable boundaries do not need boundary conditions for uniqueness.

Abstract

We study the term structure equation for single-factor models that predict nonnegative short rates. In particular, we show that the price of a bond or a bond option is the unique classical solution to a parabolic differential equation with a certain boundary behavior for vanishing values of the short rate. If the boundary is attainable then this boundary behavior serves as a boundary condition and guarantees uniqueness of solutions. On the other hand, if the boundary is nonattainable then the boundary behavior is not needed to guarantee uniqueness but it is nevertheless very useful, for instance, from a numerical perspective.