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 const...

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Bibliographic Details
Published inMathematica Slovaca Vol. 68; no. 3; pp. 617 - 624
Main Author Stehlíková, Beáta
Format Journal Article
LanguageEnglish
Czech
French
German
Russian
Published Heidelberg De Gruyter 26.06.2018
Walter de Gruyter GmbH
Subjects
34E05
65H05
91G30
asymptotic expansion
Asymptotic methods
bond price
Closed form solutions
Computation
Differential equations
Exact solutions
interest rate spread
Interest rates
interest rates model
long-term rate
Mathematical models
Nonlinear analysis
nonlinear equation
Nonlinear equations
Ordinary differential equations
Parameter identification
Perturbation methods
Pricing policies
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Summary: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. A quantity entering the nonlinear equation is expressed in a closed form, but it contains infinite sums and evaluations of special functions. In this paper we use perturbation methods to compute the constant of interest as an asymptotic serie with coefficients given in closed form and expressed using elementary functions. A quick computation of the bond prices, which our approach allows, is essential for example in calibration of the model by means of fitting the observed yields, where the theoretical bond prices need to be recalculated for every observed date and maturity, as well as every combination of parameters considered. The first step of our derivation is identification of a small parameter in the problem, since it is not immediately clear. We verify our choice by numerical experiments using the values of parameters from the literature.
ISSN:0139-9918
1337-2211
DOI:10.1515/ms-2017-0129