On the Continuous Limit of GARCH

ICMA Centre Discussion Paper No. DP2005-13

20 Pages Posted: 27 Jul 2004

See all articles by Carol Alexander

Carol Alexander

University of Sussex Business School

Emese Lazar

University of Reading - ICMA Centre

Date Written: October 2005

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 continuous time limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes, yet it has been the point of much debate between econometricians. The seminal work of Nelson (1990) derived the GARCH(1,1) limit as a stochastic volatility process, uncorrelated with the price process. But then a subsequent paper of Corradi (2000) that derives the limit as a deterministic volatility process and several other contradictory papers followed. We reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to consider its limit. Instead it is legitimate to use the weak definition of GARCH that is aggregating in time. This model differs from strong GARCH by defining the discrete time process on the best linear predictor of the squared errors, rather than the conditional variance itself. We prove that its continuous limit is a stochastic volatility model with 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. Under certain assumptions our limit model reduces to Nelson's GARCH diffusion.

Keywords: GARCH diffusion, stochastic volatility, time aggregation, continuous limit

JEL Classification: C32, G13

Suggested Citation

Alexander, Carol and Lazar, Emese, On the Continuous Limit of GARCH (October 2005). ICMA Centre Discussion Paper No. DP2005-13. Available at SSRN: https://ssrn.com/abstract=569081 or http://dx.doi.org/10.2139/ssrn.569081

Carol Alexander (Contact Author)

University of Sussex Business School ( email )

Falmer, Brighton BN1 9SL
United Kingdom

HOME PAGE: http://www.carolalexander.org

Emese Lazar

University of Reading - ICMA Centre ( email )

Whiteknights Park
P.O. Box 242
Reading RG6 6BA
United Kingdom
+44 (0)1183 786675 (Phone)
+44 (0)1189 314741 (Fax)

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