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On The Continuous Limit of GARCH

Author

Listed:
  • Carol Alexandra

    (ICMA Centre, University of Reading)

  • Emese Lazar

    (ICMA Centre, University of Reading)

Abstract

GARCH processes constitute the major area of time series variance analysis hence the limit of these processes is of considerable interest for continuous time volatility modelling. The limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes yet it has been the subject of much debate. The seminal work of Nelson (1990) derived this limit as a stochastic volatility process that is uncorrelated with the price process but a subsequent paper of Corradi (2000) derived the limit as a deterministic volatility process and several other contradictory papers followed. In this paper we reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to examine its limit. Instead it is legitimate to use the weak definition of GARCH that is time aggregating. We prove that its continuous limit is a stochastic volatility model that reduces to Nelson's GARCH diffusion only under certain assumptions. In general, the weak GARCH limit has correlated Brownian motions in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density.

Suggested Citation

  • Carol Alexandra & Emese Lazar, 2005. "On The Continuous Limit of GARCH," ICMA Centre Discussion Papers in Finance icma-dp2005-13, Henley Business School, University of Reading.
    • Handle: RePEc:rdg:icmadp:icma-dp2005-13
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    File URL: http://www.icmacentre.ac.uk/pdf/discussion/DP2005-13.pdf
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    Citations

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    Cited by:

    1. Hafner, Christian M. & Laurent, Sebastien & Violante, Francesco, 2017. "Weak Diffusion Limits Of Dynamic Conditional Correlation Models," Econometric Theory, Cambridge University Press, vol. 33(3), pages 691-716, June.
    2. Buccheri, Giuseppe & Corsi, Fulvio & Flandoli, Franco & Livieri, Giulia, 2021. "The continuous-time limit of score-driven volatility models," Journal of Econometrics, Elsevier, vol. 221(2), pages 655-675.
    3. Kallsen, Jan & Vesenmayer, Bernhard, 2009. "COGARCH as a continuous-time limit of GARCH(1,1)," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 74-98, January.
    4. Bali, Turan G. & Wu, Liuren, 2006. "A comprehensive analysis of the short-term interest-rate dynamics," Journal of Banking & Finance, Elsevier, vol. 30(4), pages 1269-1290, April.
    5. Rodrigo Alfaro & Natán Golberger, 2013. "The Impact of Persistence in Volatility over the Probability of Default," Working Papers Central Bank of Chile 689, Central Bank of Chile.
    6. Badescu, Alexandru & Elliott, Robert J. & Ortega, Juan-Pablo, 2015. "Non-Gaussian GARCH option pricing models and their diffusion limits," European Journal of Operational Research, Elsevier, vol. 247(3), pages 820-830.

    More about this item

    Keywords

    GARCH; stochastic volatility; time agtregation; continuous limit;
    All these keywords.

    JEL classification:

    • 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
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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