Modeling Nelson–Siegel Yield Curve Using Bayesian Approach

  • Sourish DasEmail author
Part of the New Economic Windows book series (NEW)


Yield curve modeling is an essential problem in finance. In this work, we explore the use of Bayesian statistical methods in conjunction with Nelson–Siegel model. We present the hierarchical Bayesian model for the parameters of the Nelson–Siegel yield function. We implement the MAP estimates via BFGS algorithm in rstan. The Bayesian analysis relies on the Monte Carlo simulation method. We perform the Hamiltonian Monte Carlo (HMC), using the rstan package. As a by-product of the HMC, we can simulate the Monte Carlo price of a Bond, and it helps us to identify if the bond is over-valued or under-valued. We demonstrate the process with an experiment and US Treasury’s yield curve data. One of the interesting observation of the experiment is that there is a strong negative correlation between the price and long-term effect of yield. However, the relationship between the short-term interest rate effect and the value of the bond is weakly positive. This is because posterior analysis shows that the short-term effect and the long-term effect are negatively correlated.


  1. 1.
    Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  2. 2.
    Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer, New York (1985)CrossRefGoogle Scholar
  3. 3.
    Betancourt, M., Girolami, M.: Hamiltonian Monte Carlo for hierarchical models. Technical report (2013).
  4. 4.
    Carlin, B.P., Louis, T.A.: Bayesian Methods for Data Analysis, 3rd edn. CRC Press, New York (2008)zbMATHGoogle Scholar
  5. 5.
    Chen, Y., Niu, L.: Adaptive dynamic nelson-siegel term structure model with applications. J. Econ. 180(1), 98–115 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Das, S., Dey, D.: On bayesian analysis of generalized linear models using jacobian technique. Am. Stat. 60, (2006).
  7. 7.
    Das, S., Dey, D.: On dynamic generalized linear models with applications. Methodol. Comput. Appl. Probab. 15, (2013).
  8. 8.
    Das, S., Dey, D.K.: On bayesian inference for generalized multivariate gamma distribution. Stat. Probab. Lett. 80, 1492–1499 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diebold, F., Li, C.: Forecasting the term structure of government bond yields. J. Econ. 130(1), 337–364 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hoffman, M.D., Gelman, A.: The no-u-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res. 15, 1593–1623 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Joao, F.C., Laurin., M.P., Marcelo, S.P.: “Bayesian Inference Applied to Dynamic Nelson-Siegel Model with Stochastic Volatility” Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, 30(1), (2010)Google Scholar
  12. 12.
    Meinhold, R.J., Singpurwalla, N.D.: Understanding the kalman filter. Am. Stat. 37(2), 123–127 (1983)Google Scholar
  13. 13.
    Mrcio, P.L., Luiz, K.H.: Bayesian extensions to diebold-li term structure model. Int. Rev. Financ. Anal. 19, 342–350 (2010)CrossRefGoogle Scholar
  14. 14.
    Nelson, C.R., Siegel, A.F.: Parsimonious modeling of yield curve. J. Bus. 60(4), 473–489 (1987)CrossRefGoogle Scholar
  15. 15.
    Nielsen, B.: Bond yield curve holds predictive powers treasury rates (2017).
  16. 16.
    Nikolaus, H., Fuyu, Y.: Bayesian inference in a stochastic volatility Nelson–Siegel model. Comput. Stat. Data Anal. 56, 3774–3792 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2005)Google Scholar
  18. 18.
    Sambasivan, R., Das, S.: A statistical machine learning approach to yield curve forecasting. IEEE proceedings ICCIDS-2017 (2017)Google Scholar
  19. 19.
    Sambasivan, R., Das, S.: Fast gaussian process regression for big data. Big Data Res. (2018). Scholar
  20. 20.
    Spencer Hays, H.S., Huang, J.Z.: Functional dynamic factor models with applications to yield curve forecasting. Ann. Appl. Stat. 6(3), 870–894 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Team., S.D.: Stan modeling language users guide and reference manual (2016).
  22. 22.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Chennai Mathematical InstituteIndia and Commonwealth Rutherford Fellow, University of SouthamptonSouthamptonUK

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