Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates

TL;DR

This paper applies perturbation methods to efficiently compute a key constant in the Schaefer-Schwartz interest rate model, improving bond price approximations by providing explicit formulas and demonstrating high accuracy with few terms.

Contribution

It introduces a perturbation approach to calculate the nonlinear constant in the model, enabling closed-form coefficients and improved approximation accuracy.

Findings

Perturbation series coefficients are derived in closed form.

Few terms in the series suffice for high accuracy.

Method enhances bond price approximation in the model.

Abstract

We deal with the interest rate model proposed by Schaefer and Schwartz, which models the long rate and the spread, defined as the difference between the short and the long rates. The approximate analytical formula for the bond prices suggested by the authors requires a computation of a certain constant, defined via a nonlinear equation and an integral of a solution to a system of ordinary differential equations. In this paper we use perturbation methods to compute this constant. Coefficients of its expansion are given in a closed form and can be constructed to arbitrary order. However, our numerical results show that a very good accuracy is achieved already after using a small number of terms.