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 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.GARCH, stochastic volatility, time agtregation, continuous limit