Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 68, Issue 3

Issues

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

Beáta Stehlíková
  • Department of Applied Mathematics and Statistics Faculty of Mathematics, Physics and Informatics Mlynská dolina, SK–842 48 Bratislava Slovakia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-05-18 | DOI: https://doi.org/10.1515/ms-2017-0129

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. 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.

MSC 2010: 34E05; 65H05; 91G30

Keywords: interest rates model; long-term rate; interest rate spread; bond price; nonlinear equation; asymptotic expansion

This work was supported by by VEGA 1/0251/16 grant.

References

  • [1]

    Ahrens, R.: Predicting recessions with interest rate spreads: a multicountry regime-switching analysis, J. Int. Money Financ. 21 (2002), 519–537.CrossrefGoogle Scholar

  • [2]

    Babbs, S. H.—Nowman, K. B.: Kalman filtering of generalized Vasicek term structure models, J. Financ. Quant. Anal. 34 (1999), 115–130.CrossrefGoogle Scholar

  • [3]

    Brigo, D.—Mercurio, F.: Interest Rate Models, Theory and Practice: With Smile, Inflation and Credit, Springer, 2007.Google Scholar

  • [4]

    Christiansen, Ch.: Multivariate term structure models with level and heteroskedasticity effects, J. Bank. Financ. 29 (2005), 1037–1057.CrossrefGoogle Scholar

  • [5]

    Christiansen, Ch.: Predicting severe simultaneous recessions using yield spreads as leading indicators, J. Int. Money Financ. 32 (2013), 1032–1043.Web of ScienceCrossrefGoogle Scholar

  • [6]

    Choi, Y.—Wirjanto, T. S.: An analytic approximation formula for pricing zero-coupon bonds, Financ. Res. Lett. 4 (2007), 116–126.CrossrefGoogle Scholar

  • [7]

    Corzo, T. S.—Schwartz, E. S.: Convergence within the EU: Evidence from interest rates, Econ. Notes 29 (2000), 243–266.CrossrefGoogle Scholar

  • [8]

    Fouque, J.-P.—Papanicolaou, G.—Sircar, R.—Solna, K.: Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, 2011.Google Scholar

  • [9]

    Funahashi, H.—Fukui, T.: A unified approximation formula for zero-coupon bond prices, Working paper, 2014, available at SSRN: http://ssrn.com/abstract=2472503.

  • [10]

    Gómez-Valle, L.—Martinéz-Rodriguéz, J.: Improving the term structure of interest rates: Two-factor models, Int. J. Financ. Econ. 15 (2010), 275–287.Web of ScienceGoogle Scholar

  • [11]

    Hinch, E. J.: Perturbation Methods, Cambridge University Press, 1991.Google Scholar

  • [12]

    Holmes, M. H.: Introduction to Perturbation Methods, Springer, 2013.Google Scholar

  • [13]

    Honda, T.—Tamaki, K.—Shiohama, T.: Higher order asymptotic bond price valuation for interest rates with non-Gaussian dependent innovations, Financ. Res. Lett. 7 (2009), 60–69.Web of ScienceGoogle Scholar

  • [14]

    Kwok, Y.-K.: Mathematical Models of Financial Derivatives, Springer, 2008.Google Scholar

  • [15]

    Schaefer, S. M.—Schwartz, E. S.: A two-factor model of the term structure: An approximate analytical solution, J. Financ. Quant. Anal. 19 (1984), 413–424.CrossrefGoogle Scholar

  • [16]

    Stehlíková, B.: A simple analytic approximation formula for the bond price in the Chan-Karolyi-Longstaff-Sanders model, Int. J. Numer. Anal. Mod. -B 4 (2013), 224–234.Google Scholar

  • [17]

    Stehlíková,, B.—Capriotti, L.: An effective approximation for zero-coupon bonds and Arrow-Debreu prices in the Black-Karasinski model, Int. J. Theor. Appl. Financ. 17(2014), 1450037.CrossrefGoogle Scholar

  • [18]

    Stehlíková, B.—Ševčovič, D.: Approximate formulae for pricing zero-coupon bonds and their asymptotic analysis, Int. J. Numer. Anal. Mod. 6 (2009), 274–283.Google Scholar

  • [19]

    Ševčovič, D.—Urbánová Csajková, A.: On a two-phase minmax method for parameter estimation of the Cox, Ingersoll, and Ross interest rate model, Cent. Eur. J. Oper. Res. 13 (2005), 169–188.Google Scholar

  • [20]

    Zíková, Z.—Stehlíková, B.: Convergence model of interest rates of CKLS type, Kybernetika 3 (2012), 567–586.Google Scholar

  • [21]

    EMMI – European Money Markets Institute. Euribor Rates. http://www.emmi-benchmarks.eu/euribor-org/euribor-rates.html

  • [22]

    Long-term interest rates.https://www.ecb.europa.eu/stats/money/long/html/index.en.html

About the article


Received: 2016-08-24

Accepted: 2016-10-26

Published Online: 2018-05-18

Published in Print: 2018-06-26


Citation Information: Mathematica Slovaca, Volume 68, Issue 3, Pages 617–624, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0129.

Export Citation

© 2018 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in