The magic of an Allaisian appraisal: implied and historical volatility revisited the VIX: once bitten, twice shy?

Many contributions have dealt with the relation between implied and historical volatility in reference to the S&P100 index and on mostly limited samples of data. A large part of this literature finds that implied volatility defined directly from option prices is a biased estimator of future realized volatility, although some dissent has been expressed Christensen and Prabhala (1998). We investigate the issue on the larger market of the S&P500, using the VIX index as the measure of implied volatility and on a much larger sample (314 months), extending from January 1990 to December 2016. Our results are in line with most of the literature inasmuch as they invalidate the efficient market hypothesis. More originally, however, we use a time series analysis derived from Maurice Allais's "lost" work on monetary theory and show that the VIX incorporates a subtle version of perceived and memorized past data – the "missing link" in relating implied to realized volatility - rather than reflecting any kind of "rational expectation" of future realized volatility. Incidentally, we show that the VIX seems to have been over-valued until the middle of the first decade of our century and to be since then averagely under-valued. Amazingly enough, this trend of affairs seems to be steadily confirmed by the financial market, which calls for additional research, even if we offer two possible explanations.