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Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates

Author

Listed:
  • Li Chen

    (Princeton University)

  • H. Vincent Poor

    (Princeton University)

Abstract

In this paper, a class of regular quadratic Gaussian processes is defined to characterize quadratic term structure models (QTSMs) in a general Markovian setting. The primary motivation for this definition is to provide a more general model for the quadratic term structure of the forward curve, while maintaining the analytical tractability of the traditional QTSMs. It is demonstrated that the tractability of QTSMs does not necessarily rely on the Ornstein-Uhlenbeck state processes used in their traditional definition. Rather, the crucial element that provides analytical solutions for the prices of zero-coupon bonds and their options is a so-called quadratic Gaussian property as defined in this paper. In order to retain this property for a general Markov process, it is shown that, under the regularity conditions, no jumps are allowed in the infinitesimal generator of the process. It is further shown that the coefficient functions defined in the quadratic Gaussian property can be determined by multi-variate Riccati equations with a unique admissible parameter set. The implications of this result for modeling the term structure of risk-free rates and defaultable rates are discussed.

Suggested Citation

  • Li Chen & H. Vincent Poor, 2003. "Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates," Finance 0303008, EconWPA.
  • Handle: RePEc:wpa:wuwpfi:0303008
    Note: Type of Document - pdf; prepared on IBM PC - PC-TEX/UNIX Sparc TeX; to print on HP/PostScript/Franciscan monk; pages: 20; figures: included/request from author/draw your own. We never published this piece and now we would like to reduce our mailing and xerox cost by posting it.
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    References listed on IDEAS

    as
    1. Leippold, Markus & Wu, Liuren, 2002. "Asset Pricing under the Quadratic Class," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(02), pages 271-295, June.
    2. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    3. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    4. Dong-Hyun Ahn & Robert F. Dittmar, 2002. "Quadratic Term Structure Models: Theory and Evidence," Review of Financial Studies, Society for Financial Studies, vol. 15(1), pages 243-288, March.
    5. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance 0207014, EconWPA.
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    Citations

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    Cited by:

    1. Gourieroux, Christian & Sufana, Razvan, 2011. "Discrete time Wishart term structure models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(6), pages 815-824, June.
    2. Realdon, Marco, 2006. "Quadratic term structure models in discrete time," Finance Research Letters, Elsevier, vol. 3(4), pages 277-289, December.
    3. Likuan Qin & Vadim Linetsky, 2014. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing," Papers 1411.3075, arXiv.org, revised Sep 2015.
    4. Gibson, Rajna & Lhabitant, Francois-Serge & Talay, Denis, 2010. "Modeling the Term Structure of Interest Rates: A Review of the Literature," Foundations and Trends(R) in Finance, now publishers, vol. 5(1–2), pages 1-156, December.
    5. Si Cheng & Michael R. Tehranchi, 2015. "Polynomial term structure models," Papers 1504.03238, arXiv.org, revised Aug 2016.
    6. Gourieroux, C. & Monfort, A., 2008. "Quadratic stochastic intensity and prospective mortality tables," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 174-184, August.
    7. Satoshi Yamashita & Toshinao Yoshiba, 2011. "Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process," IMES Discussion Paper Series 11-E-20, Institute for Monetary and Economic Studies, Bank of Japan.
    8. Li Chen & Damir Filipovic, 2003. "Modeling Credit Risk by Affine Processes," Finance 0303006, EconWPA.
    9. Realdon, Marco, 2009. ""Extended Black" term structure models," International Review of Financial Analysis, Elsevier, vol. 18(5), pages 232-238, December.
    10. Li Chen & H. Vincent Poor, 2003. "Credit Risk Modeling and the Term Structure of Credit Spreads," Finance 0312009, EconWPA.
    11. Zorana Grbac & Laura Meneghello & Wolfgang J. Runggaldier, 2015. "Derivative pricing for a multi-curve extension of the Gaussian, exponentially quadratic short rate model," Papers 1512.03259, arXiv.org, revised Jun 2016.
    12. Peng Cheng & Olivier Scaillet, 2007. "Linear-Quadratic Jump-Diffusion Modeling," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 575-598.
    13. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.

    More about this item

    Keywords

    Quadratic term structure models; option pricing; defaultable rates; time-homogenous Markov processes;

    JEL classification:

    • C39 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Other

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