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Lithuanian Mathematical Journal

, Volume 49, Issue 1, pp 5–25 | Cite as

Strong consistency of maximum likelihood estimators for a discrete-time random field HJM-type interest rate model

  • E. FülöpEmail author
  • G. Pap
Article
  • 33 Downloads

Abstract

We consider a discrete time Heath–Jarrow–Morton-type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. Strong consistency of maximum likelihood estimators is proved for stable and unstable no-arbitrage models containing a simple stochastic discounting factor.

Keywords

strong consistency of maximum likelihood estimators from nonindependent samples Heath–Jarrow–Mortontype forward interest rate model geometric spatial autoregression field no-arbitrage models stochastic discounting factors 

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References

  1. 1.
    L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986.zbMATHGoogle Scholar
  2. 2.
    E. Fülöp and G. Pap, Asymptotically optimal tests for a discrete time random field HJM type interest rate model, Acta Sci. Math. (Szeged), 73(3–4):637–661, 2007.MathSciNetGoogle Scholar
  3. 3.
    E. Fülöp and G. Pap, Note on strong consistency of maximum likelihood estimators for dependent observations, in E. Kovács, P. Olajos, and T. Tómács (Eds.), Proc. of the 7th International Conference on Applied Informatics, Eger, Hungary, 2007, Volume 1, B.V.B. Nyomda és Kiadó Kft, Eger, 2008, pp. 223–228.Google Scholar
  4. 4.
    J. Gáll, Some Problems in Discrete Time Financial Market Models, Ph.D. dissertation, University of Debrecen, Hungary, 2008.Google Scholar
  5. 5.
    J. Gáll, G. Pap, and W. Peeters, Random field forward interest rate models, market price of risk and their statistics, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53(2):233–242, 2007.CrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Gáll, G. Pap, and M. van Zuijlen, Maximum likelihood estimator of the volatility of forward rates driven by geometric spatial AR sheet, J. Appl. Math., 2004(4):293–309, 2004.zbMATHCrossRefGoogle Scholar
  7. 7.
    J. Gáll, G. Pap, and M. van Zuijlen, Forward interest rate curves in discrete time settings driven by random fields, Comput. Math. Appl., 51(3-4):387–396, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Gáll, G. Pap, and M. van Zuijlen, Joint ML estimation of all parameters in a discrete time random field HJM type interest rate model, Report No. 0606, Radboud University Nijmegen, The Netherlands, 2006.Google Scholar
  9. 9.
    R.S. Goldstein, The term structure of interest rates as a random field, Rev. Fin. Stud., 13(2):365–384, 2000.CrossRefGoogle Scholar
  10. 10.
    D. Heath, R.A. Jarrow, and A. Morton, Bond pricing and the term structure of interest rates: A discrete time approximation, J. Fin. Quant. Anal., 25(4):419–440, 1990.CrossRefGoogle Scholar
  11. 11.
    D. Heath, R.A. Jarrow, and A. Morton, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica, 60(1):77–105, 1992.zbMATHCrossRefGoogle Scholar
  12. 12.
    R.D.H. Heijmans and J.R. Magnus, Asymptotic normality of maximum likelihood estimators obtained from normally distributed but dependent observations, Econometric Theory, 2(3):374–412, 1986.MathSciNetGoogle Scholar
  13. 13.
    R.I. Jennrich, Asymptotic properties of non-linear least squares estimators, Ann. Math. Statist., 40(2):633–643, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D.P. Kennedy, The term structure of interest rates as a Gaussian random field, Math. Finance, 4(3):247–258, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    W. Peeters, Volatility estimation for different structures of random field interest rate models in discrete time, Publ. Math. Debrecen, 72(3), 2008.Google Scholar
  16. 16.
    A. Pelsser, Efficient Methods for Valuing Interest Rate Derivatives, Springer-Verlag, London, 2000.zbMATHGoogle Scholar
  17. 17.
    P. Santa-Clara and D. Sornette, The dynamics of the forward interest rate curve with stochastic string shocks, Rev. Fin. Stud., 14(1):149–185, 2001.CrossRefGoogle Scholar
  18. 18.
    A.W. van der Vaart, Asymptotic Statistics, Cambridge University Press, Cambridge, 1998.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Probability Theory, Faculty of InformaticsUniversity of DebrecenDebrecenHungary

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