On volatility variation in ARCH(1) and GARCH(1;1) continuous limits

Abstract

The variance of many time series in biology and finance is not a constant function withrespect to time. Different time-discrete statistical models have been proposed but recently timecontinuouslimit models have been developed. This article is concerned with the analysis of volatility variation by employing the continuous limit of a time-discrete statistical models ARCH (AutoRegressive Conditionnal Heteroscedasticity) and GARCH (Generalized AutoRegressive Conditionnal
Heteroscedasticity). Specifically, under some technical assumptions, we prove the uniqueness of the time-continuous processes ARCH(1)-M and GARCH(1;1)-M and we derive the related stationary probability distribution functions. Moreover by employing numerical simulations we show that the volatility variation is higher in the time-discrete GARCH(1;1) than ARCH(1). The results are of great interest in the financial markets.