How does the volatility of volatility depend on volatility?

We investigate the state dependence of the variance of the instantaneous variance of the S&amp
P 500 index empirically. Time-series analysis of realized variance over a 20-year period shows strong evidence of an elasticity of variance of the variance parameter close to that of a log-normal model, albeit with an empirical autocorrelation function that one-factor diffusion models fail to capture at horizons above a few weeks. When studying option market behavior (in-sample pricing as well as out-of-sample pricing and hedging over the period 2004&ndash
2019), messages are mixed, but systematic, model-wise. The log-normal but drift-free SABR (stochastic-alpha-beta-rho) model performs best for short-term options (times-to-expiry of three months and below), the Heston model&mdash
in which variance is stationary but not log-normal&mdash
is superior for long-term options, and a mixture of the two models does not lead to improvements.