An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series

Statistical Papers volume 55, pages919–931(2014)Cite this article

Abstract

We introduce two types of ordinal pattern dependence between time series. Positive (resp. negative) ordinal pattern dependence can be seen as a non-paramatric and in particular non-linear counterpart to positive (resp. negative) correlation. We show in an explorative study that both types of this dependence show up in real world financial data.

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

Notes

  1. 1.

    We state below how to deal with the case \(x_n=x_{n+1}\).

References

  1. Bandt C (2005) Ordinal time series analysis. Ecol Model 182:229–238

    Article  Google Scholar 

  2. Bandi FM, Renò R (2009) Nonparametric leverage effects. Working paper, Università di Siena

  3. Barndorff-Nielsen OE, Shephard N (2002) Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J R Stat Soc B64:253–280

    MathSciNet  Article  Google Scholar 

  4. Black F (1976) Studies of stock market volatility changes. In: Proceedings of the business and economic statistics section. American Statistical Association, New York, pp 177–181

  5. Carr P, Wu L (2007) Stochastic skew in currency options. J Fin Econ 86(1):213–247

    Article  Google Scholar 

  6. Emmerich Cvan (2007) A square root process for modelling correlation. PhD thesis, Bergische Universität, Wuppertal

  7. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Fin Stud 6:327–343

    Google Scholar 

  8. Keller K, Sinn M, Emonds J (2007) Time series from the ordinal viewpoint. Stoch Dyn 2:247–272

    MathSciNet  Article  Google Scholar 

  9. Keller K, Sinn M (2005) Ordinal analysis of time series. Phys A 356:114–120

    Article  Google Scholar 

  10. Keller K, Sinn M (2011) Estimation of ordinal pattern probabilities in Gaussian processes with stationary increments. Comp Stat Data Anal 55:1781–1790

    MathSciNet  Article  Google Scholar 

  11. Lehmann EL (1966) Some concepts of dependence. Annu Math Stat 37:1137–1153

    Article  MATH  Google Scholar 

  12. Madan DB, Yor M (2011) The S &P 500 index as a sato process travelling at the speed of the VIX. Appl Math Finance 18(3):227–244

    MathSciNet  Article  MATH  Google Scholar 

  13. Nagel H, Schöbel R (1999) Volatility and GMM - Monte Carlo studies and empirical estimations. Stat Papers 49:297–321

    Article  Google Scholar 

  14. Romano M, Touzi N (1997) Contingent claims and market completeness in a stochastic volatility model. Math Fin 7:399–412

    Google Scholar 

  15. Sung SH (2012) Complete convergence of weighted sums under negatively dependent random variables. Stat Papers 53:73–82

    Article  MATH  Google Scholar 

  16. Veraart AED, Veraart AMV (2010) Stochastic volatility and stochastic leverage. Ann Finance 8(2–3):205–233

    Google Scholar 

  17. Whaley RE (1993) Derivatives on market volatility: hedging tools long overdue. J Deriv 1:71–84

    Article  Google Scholar 

  18. Whaley RE (2008) Understanding VIX. http://ssrn.com/abstract=1296743

  19. Wilmott P (2000) Quantitative finance, vol 1. Wiley, Hoboken

  20. Yu J (2005) On leverage in a stochastic volatility model. J Economet 127:165–178

    Article  Google Scholar 

  21. Zarei H, Jabbari H (2011) Complete convergence of weighted sums under negative dependence. Stat Papers 52:413–418

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The author wishes to thank two anonymous referees for their work. Their comments have helped to improve the paper. Furthermore he wishes to thank B. Funke (TU Dortmund) for carefully reading the manuscript and A. Dürre (TU Dortmund) for the implementation in R. The financial support of the DFG (German science Foundation) SFB 823: Statistical modeling of nonlinear dynamic processes (project C5) is gratefully acknowledged.

Author information

Affiliations

  1. Lehrstuhl IV, Fakultät für Mathematik, Technische Universität Dortmund, 44227,  Dortmund, Germany

    Alexander Schnurr

Authors
  1. Alexander Schnurr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Schnurr.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schnurr, A. An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series. Stat Papers 55, 919–931 (2014). https://doi.org/10.1007/s00362-013-0536-8

Download citation

Keywords

  • Ordinal patterns
  • Stationarity
  • Leverage effect
  • VIX
  • Model free data exploration
  • Econometrics

Mathematics Subject Classification

  • 62-07 (Primary)
  • 91G70
  • 91B84
  • 62M10 (Secondary)