Abstract: This article demonstrates theoretically that without imposing any structure on the underlying forcing process, the model-free CBOE volatility index (VIX) does not measure market expectation of volatility but that of a linear moment-combination. Particularly, VIX undervalues (overvalues) volatility when market return is expected to be negatively (positively) skewed. Alternatively, we develop a model-free generalized volatility index (GVIX). GVIX is generic because no diffusion assumption is necessary, and VIX is only a special case of the GVIX. Empirically, VIX generally understates the true volatility, and the estimation errors considerably enlarge during volatile markets.