Skip to main content
SpringerLink
Log in

Volatility modeling with COGARCH(1,1) driven by Meixner-Lévy process: an application to Tokyo stock exchange market (Nikkei225)

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, the authors propose a new outlook to model Tokyo stock exchange market’s volatility (Nikkei225) in the concept of continuous-time GARCH(1,1) driven by Meixner Lévy Process for the first time. GARCH(1,1) model is estimated as an appropriate discrete-time model for the timeseries, then Meixner Lévy process driven continuous-time model is developed to analyze the volatility characteristics of standardized log returns of Nikkei225. Pseudo Maximum Likelihood (PML), is employed as the parameter estimation process for the Meixner-COGARCH(1,1) model. The data covers the period from 2005 to 2013. The empirical results show that continuous-time GARCH(1,1) driven by a Meixner Lévy process successfully captures volatility clustering and heavy-tail behaviour of Nikkei225.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ando T (2006) Pricing Nikkei 225 options using nonlinear and non-Gaussian stochastic volatility model with leverage effect. J Japan Stat Soc 36(2):173–197

  2. Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). J R Stat Soc Ser B 63:167–241

    Article  MathSciNet  MATH  Google Scholar 

  3. Björk T (2009) Arbitrage theory in continuous time, 3rd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  4. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654

    Article  MathSciNet  MATH  Google Scholar 

  5. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327

    Article  MathSciNet  MATH  Google Scholar 

  6. Boźejko M, Lytvynov E (2009) Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization. arXiv:0812.0895v2

  7. Brockwell PJ (2004) Representations of continuous-time ARMA processes. J Appl Probab 41A:375–382

    Article  MathSciNet  MATH  Google Scholar 

  8. Brockwell PJ, Marquardt T (2005) Lévy driven and fractionally Integrated ARMA process with continuous tome parameter. Stat Sin 15(2):477–494

    MATH  Google Scholar 

  9. Brockwell PJ, Chadraa E, Lindner A (2006) Continuous-time GARCH processes. Ann Appl Probab 16(2):790–826

    Article  MathSciNet  MATH  Google Scholar 

  10. Buchmann, B, Müller, G. (2012). Limit experiments of GARCH. Bernoulli 18(1):64–99

  11. Corradi V (2000) Reconsidering the continuous time limit of the GARCH (1,1) process. J Econ 96:145–153

  12. Einstein A (1905) Über die von molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann Phys 322(8):549–560

    Article  MATH  Google Scholar 

  13. Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50:987–1008

    Article  MathSciNet  MATH  Google Scholar 

  14. Fama E (1963) Mandelbrot and the stable paretian hypothesis. J Bus 36(4):420–429

    Article  Google Scholar 

  15. Fischer MJ (2014) Generalized hyperbolic secant distributions with applications to finance. Springer, Berlin

    Book  MATH  Google Scholar 

  16. Grant PM (2008) Option pricing: a GARCH model with Lévy process innovations. Dissertation, University of Johannesburg

  17. Grigelionis B (1999) Processes of Meixner type. Lith Math J 39(1):33–41

    Article  MathSciNet  MATH  Google Scholar 

  18. Haug S, Klppelberg C, Lindner A, Zapp M (2007) Method of moment estimation in the COGARCH(1,1) model. Econo J 10(2):320–341

    Article  MathSciNet  MATH  Google Scholar 

  19. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6–2:327–343

    Article  MATH  Google Scholar 

  20. Ishida I, Watanabe T (2009) Modeling and forecasting the volatility of the Nikkei 225 realized volatility using the ARFIMA-GARCH model In Global COE Hi-Stat discussion paper series No. 32. Institute of Economic Research, Hitotsubashi University

  21. Jarque CM, Bera AK (1980) Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econ Lett 7:313–318

    MathSciNet  Google Scholar 

  22. Jing BY, Kong XB, Liu Z (2012) Modeling high-frequency financial data by pure-jump processes. Ann Stat 40:759–784

    Article  MathSciNet  MATH  Google Scholar 

  23. Kallsen J, Vesenmayer B (2009) COGARCH as a continuous-time limit of GARCH(1,1). Stoch Process Appl 119(1):74–98

    Article  MathSciNet  MATH  Google Scholar 

  24. Klüppelberg C, Lindner A, Maller R (2004) A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour. J Appl Probab 41(3):601–622

    Article  MathSciNet  MATH  Google Scholar 

  25. Kong XB, Liu Z, Jing BY (2015) Testing for pure-jump processes for high-frequency data. Ann Stat 43:847–877

    Article  MathSciNet  MATH  Google Scholar 

  26. Le Cam L (1964) Sufficiency and approximate sufficiency. Ann Math Stat 35:1419–1455

  27. Mandelbrot B (1963) The variation of certain speculative prices. J Bus 36(4):394–419

    Article  Google Scholar 

  28. Maller RA, Müller G, Szimayer A (2008) GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli 14:519–542

    Article  MathSciNet  MATH  Google Scholar 

  29. Merton R (1973) The theory of rational option pricing. Bell J Econ Manag Sci 4:141–183

    Article  MathSciNet  MATH  Google Scholar 

  30. Mittnik S, Paolella MS, Rachev ST (1998) Unconditional and conditional distribution models for the Nikkei index. Asia Pac Finan Mark 5(2):17–34

    Article  MATH  Google Scholar 

  31. Müller G (2010) MCMC estimation of the COGARCH(1,1) model. J Finan Econ 8(4):481–510

    Google Scholar 

  32. Nelson DB (1990) ARCH models as diffusion approximations. J. Econ 45:7–38

    Article  MathSciNet  MATH  Google Scholar 

  33. Said SE, Dickey DA (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71:599–607

    Article  MathSciNet  MATH  Google Scholar 

  34. Samuelson PA (1965) Rational theory of warrant pricing. Ind Manag Rev 6(2):1331

    Google Scholar 

  35. Schoutens W, Teugels JL (1998) Lévy processes, polynomials and martingales. Commun Stat Stoch Models 14(1 and 2):335–349

  36. Schoutens W (2001) Meixner process in finance. EU Random Report

  37. Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, New York

    Book  Google Scholar 

  38. Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52(3–4):591–611

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang Y (2002) Asymptotic non-equivalence of GARCH models and diffusions. Ann Stat 30:754–783

    Article  MATH  Google Scholar 

  40. Zivot E, Andrews DWK (1992) Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. J Bus Econ Stat 10(3):251–270

Download references

Author information

Authors and Affiliations

  1. Department of International Finance, Yeditepe University, Istanbul, Turkey

    Yavuz Yıldırım & Gazanfer Ünal

Authors
  1. Yavuz Yıldırım
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gazanfer Ünal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gazanfer Ünal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yıldırım, Y., Ünal, G. Volatility modeling with COGARCH(1,1) driven by Meixner-Lévy process: an application to Tokyo stock exchange market (Nikkei225). Int. J. Dynam. Control 6, 582–588 (2018). https://doi.org/10.1007/s40435-017-0351-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-017-0351-5

Keywords

Search

Navigation