On the Statistical Equivalence at Suitable Frequencies of GARCH and Stochastic Volatility Models with the Corresponding Diffusion Model

Continuous-time models play a central role in the modern theoretical finance literature, while discrete-time models are often used in the empirical finance literature. The continuous-time models are diffusions governed by stochastic differential equations. Most of the discrete-time models are autoregressive conditionally heteroscedastic (ARCH) models and stochastic volatility (SV) models. The discrete-time models are often regarded as discrete approximations of diffusions because the discrete-time processes weakly converge to the diffusions. It is known that SV models and multiplicative GARCH models share the same diffusion limits in a weak-convergence sense. Here we investigate a much stronger convergence notion. We show that SV models are asymptotically equivalent to their diffusion limits at the basic frequency of their construction, while multiplicative GARCH models match to the diffusion limits only for observations singled-out at frequencies lower than the square root of the basic frequency of construction. These results also reveal that the structure of the multiplicative GARCH models at frequencies lower than the square root of the basic frequency no longer obey the GARCH framework at the observed frequencies. Instead they behave there like the SV models. AMS 1991 Subject Classification. Primary: 62B15. Secondary: 90A09, 90A20, 90A16, 62M99.