On the Continuous Limit of GARCH

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.