Skip to main content
SpringerLink
Log in

Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This study aims to investigate the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime switching. An essential difference shown in this study between CIR processes with and without regime switching is that the stationary distribution of those with regime switching may be heavy-tailed. We first provide sharp criteria to justify the existence of a stationary distribution for the CIR process with regime switching, which is applied to study the long-term returns of interest rates. Then, we provide a criterion to identify whether this distribution is heavy-tailed. Our results provide theoretical evidence of the existence of regime switching for interest-rate models based on empirical evidence of a heavy-tailed distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ang A, Bekaert G. Short rate nonlinearities and regime switches. J Econom Dynam Control, 2002, 26: 1243–1274

    Article  Google Scholar 

  2. Ang A, Bekaert G. International asset allocation with regime shifts. Rev Financ Stud, 2002, 15: 1137–1187

    Article  Google Scholar 

  3. Bao J, Yuan C G. Long-term behavior of stochastic interest rate models with jumps and memory. Insurance Math Econom, 2013, 53: 266–272

    Article  MathSciNet  Google Scholar 

  4. Bardet J, Guerin H, Malrieu F. Long time behavior of diffusions with Markov switching. ALEA Lat Am J Probab Math Stat, 2010, 7: 151–170

    MathSciNet  MATH  Google Scholar 

  5. Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. Classics Series. Philadelphia: SIAM Press, 1994

    Book  Google Scholar 

  6. Brown S J, Dybvig P H. The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates. J Finance, 1986, 41: 617–630

    Article  Google Scholar 

  7. Cox J C, Ingersoll J E, Ross S A. A theory of the term structure of interest rates. Econometrica, 1985, 53: 385–407

    Article  MathSciNet  Google Scholar 

  8. Deelstra G, Delbaen F. Long-term returns in stochastic interest rate models. Insurance Math Econom, 1995, 17: 163–169

    Article  MathSciNet  Google Scholar 

  9. de Saporta B, Yao J F. Tail of a linear diffusion with Markov switching. Ann Appl Probab, 2005, 15: 992–1018

    Article  MathSciNet  Google Scholar 

  10. Dothan U L. On the term structure of interest rates. J Financ Econ, 1978, 6: 59–69

    Article  Google Scholar 

  11. Duffie D, Kan R. A yield-factor model of interest rates. Math Finance, 1996, 6: 379–406

    Article  Google Scholar 

  12. Elliott R J, Siu T K. Pricing regime-switching risk in an HJM interest rate environment. Quant Finance, 2016, 16: 1791–1800

    Article  MathSciNet  Google Scholar 

  13. Foss S, Korshunov D, Zachary S. An Introduction to Heavy-Tailed and Subexponential Distributions. New York: Springer, 2011

    Book  Google Scholar 

  14. Garcia R, Perron P. An analysis of the real interest rate under regime shifts. Rev Econom Statist, 1996, 78: 111–125

    Article  Google Scholar 

  15. Gray S F. Modeling the conditional distribution of interest rates as a regime switching process. J Financ Econ, 1996, 42: 27–62

    Article  Google Scholar 

  16. Hamilton J D. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 1989, 57: 357–384

    Article  MathSciNet  Google Scholar 

  17. Hull J, White A. Pricing interest rate derivatives. Rev Financ Stud, 1990, 3: 573–592

    Article  Google Scholar 

  18. Jeanblanc M, Yor M, Chesney M. Mathematical Methods for Financial Markets. London: Springer, 2009

    Book  Google Scholar 

  19. Longstaff F A, Schwartz E. Interest rate volatility and the term structure: A two-factor general equilibrium model. J Finance, 1992, 47: 1259–1282

    Article  Google Scholar 

  20. Shao J. Criteria for transience and recurrence of regime-switching diffusion processes. Electron J Probab, 2015, 20: 1–15

    Article  MathSciNet  Google Scholar 

  21. Shao J. Ergodicity of regime-switching diffusions in Wasserstein distances. Stochastic Process Appl, 2015, 125: 739–758

    Article  MathSciNet  Google Scholar 

  22. Shao J. Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space. SIAM J Control Optim, 2015, 53: 2462–2479

    Article  MathSciNet  Google Scholar 

  23. Shao J, Xi F. Stability and recurrence of regime-switching diffusion processes. SIAM J Control Optim, 2014, 52: 3496–3516

    Article  MathSciNet  Google Scholar 

  24. Shreve S. Stochastic Calculus for Finance. II: Continuous-Time Models. New York: Springer-Verlag, 2004

    Book  Google Scholar 

  25. Vasicek O A. An equilibrium characterization of the term structure. J Financ Econ, 1977, 5: 177–188

    Article  Google Scholar 

  26. Yin G, Zhu C. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modeling and Applied Probability, vol. 63. New York: Springer, 2010

  27. Zhang Z, Tong J, Hu L. Long-term behavior of stochastic interest rate models with Markov switching. Insurance Math Econom, 2016, 70: 320–326

    Article  MathSciNet  Google Scholar 

  28. Zhao J. Long time behaviour of stochastic interest rate models. Insurance Math Econom, 2009, 44: 459–463

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771327, 11431014 and 11831014).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Tongtong Hou

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Jinghai Shao

Authors
  1. Tongtong Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinghai Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinghai Shao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, T., Shao, J. Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching. Sci. China Math. 63, 1169–1180 (2020). https://doi.org/10.1007/s11425-017-9392-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9392-5

Keywords

MSC(2010)

Search

Navigation