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Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations


  • Ruijun Bu
  • Ludovic Giet
  • Kaddour Hadri
  • Michel Lubrano


We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail:, Oxford University Press.

Suggested Citation

  • Ruijun Bu & Ludovic Giet & Kaddour Hadri & Michel Lubrano, 2011. "Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 9(1), pages 198-236, Winter.
  • Handle: RePEc:oup:jfinec:v:9:y:2011:i:1:p:198-236

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    References listed on IDEAS

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    Cited by:

    1. Bu, Ruijun & Jawadi, Fredj & Li, Yuyi, 2017. "An empirical comparison of transformed diffusion models for VIX and VIX futures," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 46(C), pages 116-127.
    2. Wang, Chou-Wen & Yang, Sharon S. & Huang, Hong-Chih, 2015. "Modeling multi-country mortality dependence and its application in pricing survivor index swaps—A dynamic copula approach," Insurance: Mathematics and Economics, Elsevier, vol. 63(C), pages 30-39.
    3. Oleg L. Kritski & Vladimir F. Zalmezh, 2017. "Asymptotics for Greeks under the constant elasticity of variance model," Papers 1707.04149,, revised Jul 2017.
    4. David Zimmer, 2015. "Asymmetric dependence in house prices: evidence from USA and international data," Empirical Economics, Springer, vol. 49(1), pages 161-183, August.
    5. Bu, Ruijun & Cheng, Jie & Hadri, Kaddour, 2016. "Reducible diffusions with time-varying transformations with application to short-term interest rates," Economic Modelling, Elsevier, vol. 52(PA), pages 266-277.
    6. Bu Ruijun & Cheng Jie & Hadri Kaddour, 2017. "Specification analysis in regime-switching continuous-time diffusion models for market volatility," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 21(1), pages 65-80, February.
    7. repec:ids:ijbder:v:3:y:2017:i:2:p:153-175 is not listed on IDEAS
    8. Choi, Hwan-sik, 2016. "Information theory for maximum likelihood estimation of diffusion models," Journal of Econometrics, Elsevier, vol. 191(1), pages 110-128.
    9. Zhu, Wenjun & Wang, Chou-Wen & Tan, Ken Seng, 2016. "Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests," Journal of Banking & Finance, Elsevier, vol. 69(C), pages 20-36.
    10. Eraker, Bjørn & Wang, Jiakou, 2015. "A non-linear dynamic model of the variance risk premium," Journal of Econometrics, Elsevier, vol. 187(2), pages 547-556.
    11. Huang, MeiChi & Wu, Chih-Chiang & Liu, Shih-Min & Wu, Chang-Che, 2016. "Facts or fates of investors' losses during crises? Evidence from REIT-stock volatility and tail dependence structures," International Review of Economics & Finance, Elsevier, vol. 42(C), pages 54-71.

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates


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