Statistical Methods and Applications

, Volume 18, Issue 1, pp 87–107 | Cite as

An online estimation scheme for a Hull–White model with HMM-driven parameters

  • Christina Erlwein
  • Rogemar MamonEmail author
Original Article


This paper considers the implementation of a mean-reverting interest rate model with Markov-modulated parameters. Hidden Markov model filtering techniques in Elliott (1994, Automatica, 30:1399–1408) and Elliott et al. (1995, Hidden Markov Models: Estimation and Control. Springer, New York) are employed to obtain optimal estimates of the model parameters via recursive filters of auxiliary quantities of the observation process. Algorithms are developed and implemented on a financial dataset of 30-day Canadian Treasury bill yields. We also provide standard errors for the model parameter estimates. Our analysis shows that within the dataset and period studied, a model with two regimes is sufficient to describe the interest rate dynamics on the basis of very small prediction errors and the Akaike information criterion.


Regime-switching Markov model Interest rate dynamics Mean-reversion Filtering Optimal parameter estimation 


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  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki BF(eds) Second International Symposium on Information Theory. Academiai Kiado, Budapest, pp 267–281Google Scholar
  2. Albert J, Chib S (1993) Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts. J Business Econ Statist 11: 1–15CrossRefGoogle Scholar
  3. Armstrong J, Collopy F (1992) Error measures for generalizing about forecasting methods: empirical comparisons. Int J Forecasting 8: 69–80CrossRefGoogle Scholar
  4. Bansal R, Zhou H (2002) Term structure of interest rate with regime shifts. J Finan 57: 1997–2043CrossRefGoogle Scholar
  5. Bicego M, Murino V, Figueiredo M (2003) A sequential pruning strategy for the selection of the number of states in hidden Markov models. Pattern Recognit Lett 24: 1395–1395zbMATHCrossRefGoogle Scholar
  6. Black F, Karasinski P (1991) Bond and option pricing when short rates are Log normal. Financial Analysts J 47: 52–59CrossRefGoogle Scholar
  7. Brennan MJ, Xia Y (2002) Dynamic asset allocation under inflation. J Finan 57: 1201–1238CrossRefGoogle Scholar
  8. Brigo D, Mercurio F (2006) Interest rate models—theory and practice with smile, inflation and credit, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  9. Cox J, Ingersoll J, Ross S (1985) A theory of the term structure of interest rates. Econometrica 53: 363–363zbMATHCrossRefMathSciNetGoogle Scholar
  10. Driffill J, Kenc T, Sola M (2003) An empirical examination of term structure models with regime shifts. Computing in economics and finance 65. Society for Computational EconomicsGoogle Scholar
  11. Duffie D, Kan R (1996) A yield-factor model of interest rates. Math Finan 64: 379–406CrossRefGoogle Scholar
  12. Elliott R (1994) Exact adaptive filters for Markov chains observed in Gaussian noise. Automatica 30: 1399–1408zbMATHCrossRefGoogle Scholar
  13. Elliott R, Aggoun L, Moore J (1995) Hidden Markov models: estimation and control. Springer, New YorkzbMATHGoogle Scholar
  14. Elliott R, Krishnamurthy V (1999) New finite-dimensional filters for parameter estimation of discrete-time linear Gaussian models. IEEE Trans Autom Control 44: 938–951zbMATHCrossRefMathSciNetGoogle Scholar
  15. Elliott R, Fischer P, Platen E (1999) Filtering and parameter estimation for a mean reverting interest rate model. Can Appl Math Q 7: 381–400zbMATHMathSciNetGoogle Scholar
  16. Elliott R, Hunter W, Jamieson B (2001) Financial signal processing: a self calibrating model. Int J Theor Appl Finan 4: 567–584CrossRefMathSciNetGoogle Scholar
  17. Elliott R, Mamon R (2002) An interst rate model with a Markovian mean reverting level. Quantitat Finan 2: 454–458CrossRefMathSciNetGoogle Scholar
  18. Elliott R, Sick G, Stein M (2003) Modelling electricity price risk. Working paper. University of CalgaryGoogle Scholar
  19. Evans M (2003) Real risk, inflation risk and the term structure. Econ J 113: 345–389CrossRefGoogle Scholar
  20. Fama E, Gibbons M (1984) A comparison of inflation forecasts. J Monetary Econ 13: 327–348CrossRefGoogle Scholar
  21. Garcia R, Perron P (1996) An analysis of the real interest rate under regime shifts. Rev Econ Statist 78: 111–125CrossRefGoogle Scholar
  22. Garthwaite P, Jolliffe I, Jones B (2002) Statistical inference, 2nd edn. Oxford Science Publications, OxfordzbMATHGoogle Scholar
  23. Gray S (1996) Modelling the conditional distribution of interest rates as a regime-switching process. J Finan Econ 42: 27–62CrossRefGoogle Scholar
  24. Hamilton J (1988) Rational expectations econometric analysis of changes in regime: an investigation of the term structure of interest rate. J Econ Dyn Control 12: 385–423zbMATHCrossRefMathSciNetGoogle Scholar
  25. Hamilton J (1990) Analysis of time series subject to change in regime. J Econom 45: 39–70zbMATHCrossRefMathSciNetGoogle Scholar
  26. Hansen B (1992) The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. J Appl Econom 7: S61–S82CrossRefGoogle Scholar
  27. Hansen B (1996) Erratum—the likelihood ratio test under nonstandard contions: testing the Markov switching model of GNP. J Appl Econom 11: 195–198CrossRefGoogle Scholar
  28. Hardy M (2001) A regime-switching model of long-term stock returns. North Am Actuarial J 5(2): 41–53zbMATHMathSciNetGoogle Scholar
  29. Heath D, Jarrow R, Morton A (1992) Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60: 77–105zbMATHCrossRefGoogle Scholar
  30. Hull J, White A (1990) Pricing interest rate derivative securities. Rev Finan Stud 3: 573–592CrossRefGoogle Scholar
  31. Kim C (1994) Dynamic linear models with Markov-switching. J Econom 60: 1–22zbMATHCrossRefGoogle Scholar
  32. Landén C (2000) Bond pricing in a hidden Markov model of the short rate. Finan Stochastics 4: 371–389zbMATHCrossRefGoogle Scholar
  33. Longstaff F, Schwartz E (1992) Interest volatility and the term structure: a two-factor general equilibrium model. J Finan 47: 1259–1282CrossRefGoogle Scholar
  34. Lucia J, Schwartz E (2002) Electricity prices and power derivatives: evidence from the Nordic Power Exchange. Rev Derivatives Res 5: 5–50zbMATHCrossRefGoogle Scholar
  35. Mamon R (2000) Market models of interest rate dynamics with a joint short rate/HJM approach. Ph.D. thesis, University of Alberta, CanadaGoogle Scholar
  36. Naik V, Lee M (1997) Yield curve dynamics with discrete shifts in economic regimes: theory and estimation. Working paper. University of British Columbia, CanadaGoogle Scholar
  37. Nicolato E, Venardos E (2003) Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math Finan 13: 445–466zbMATHCrossRefMathSciNetGoogle Scholar
  38. Otranto E, Gallo G (2002) A nonparametric Bayesian approach to detect the number of regimes in Markov switching models. Econom Rev 21: 477–496zbMATHCrossRefMathSciNetGoogle Scholar
  39. Pelsser A (2000) Efficient methods for valuing interest rate derivatives. Springer, LondonzbMATHGoogle Scholar
  40. Psaradakis Z, Spagnolo N (2003) On the determination of the number of regimes in Markov-switching autoregressive models. J Time Ser Anal 24: 237–252zbMATHCrossRefMathSciNetGoogle Scholar
  41. Smith D (2002) Markov-switching and stochastic volatility diffusion models of short-term interest rates. J Business Econ Statist 20: 183–197CrossRefGoogle Scholar
  42. Wu C (1983) On the convergence properties of the EM algorithm. Ann Statist 11: 95–103zbMATHCrossRefMathSciNetGoogle Scholar
  43. Vasicek O (1977) An equilibrium characterization of the term structure. J Finan Econ 5: 177–177CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.CARISMA School of Information Systems, Computing and MathematicsBrunel UniversityUxbridgeUK
  2. 2.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada

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