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

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.