Modeling Variance of Variance: The Square-Root, the Affine, and the CEV GARCH Models ∗

Abstract

This paper develops a new econometric framework for investigating how the sensitivity of the financial market volatility to shocks varies with the volatility level. For this purpose, the paper first introduces the square-root (SQ) GARCH model for financial time series. It is an ARCH analogue of the continuous-time square-root stochastic volatility model popularly used in derivatives pricing and hedging. The variance of variance is a linear function of the conditional variance in the SQGARCH and of the square of it in the GARCH. After showing some implications of this difference, the paper introduces the constant-elasticity-of-variance (CEV) GARCH model, which allows more flexible fitting of variance-of-variance dynamics. The paper develops conditions for stationarity, the existence of finite moments, β-mixing, and other properties of the conditional variance process via the general state-space Markov chains approach. In particular, the paper generalizes the strict stationarity condition for the GARCH(1,1) and gives attention to a discrete-time analogue of the phenomenon known in the continuous-time finance literature as “volatility-induced stationarity,” which may occur with the integrated or mildly explosive CEVGARCH, shedding light on the stabilizing effect of the variance of variance. A diffusion limit for this model is also established. Several alternative models including the affine and the exponential CEV models are explored. The empirical estimates of the CEVGARCH model for the S&P 500 index and DM/US$ exchange rate time series suggest that the variance of variance grows faster than linearly with the conditional variance.