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The Continuous Limit of GARCH Processess


  • Carol Alexandra

    () (ICMA Centre, University of Reading)

  • Emese Lazar

    () (ICMA Centre, University of Reading)


Contrary to popular belief, the diffusion limit of a GARCH variance process is not a diffusion model unless one makes a very specific assumption that cannot be generalized. In fact, the normal GARCH(1,1) prices of European call and puts are identical to the Black-Scholes prices based on the average of a deterministic variance process. In the case of GARCH models with several normal components – and these are more realistic representations of option prices and returns behaviour – the continuous limit is a stochastic model with uncertainty over which deterministic local volatility governs the return. The risk neutral model prices of European options are weighted averages of Black-Scholes prices based on the integrated forward variances in each state. An interesting area to be considered for application of this model is path dependent options. Since both marginal and transition price densities are lognormal mixtures the mixture GARCH option pricing model is not equivalent to the mixture option pricing models that have previously been discussed by several authors.

Suggested Citation

  • Carol Alexandra & Emese Lazar, 2005. "The Continuous Limit of GARCH Processess," ICMA Centre Discussion Papers in Finance icma-dp2004-09, Henley Business School, Reading University, revised Jul 2004.
  • Handle: RePEc:rdg:icmadp:icma-dp2004-09

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    References listed on IDEAS

    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Garcia, Rene & Luger, Richard & Renault, Eric, 2003. "Empirical assessment of an intertemporal option pricing model with latent variables," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 49-83.
    3. Emese Lazar & Carol Alexander, 2006. "Normal mixture GARCH(1,1): applications to exchange rate modelling," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(3), pages 307-336.
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    More about this item


    GARCH diffusion; normal mixture; stochastic volatility; time aggregation;

    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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