The above link shows that there multiple ways to calculate implied volatility. My question is that for most of the common data sources like Bloomberg, Fidelity, etc, how is the implied volatility calculated?
https://www.cboe.com/micro/vix/vixwhite.pdf shows that for the VIX index, it "estimates expected volatility by averaging the weighted prices of SPX puts and calls over a wide range of strike prices". What does this mean mathematically? Can anyone shed more light on this?
Why is implied vol (and for that matter historical vol) correlated with bear markets and inversely correlated with bull markets? Since historical vol is the standard deviation of historical returns, why should the sign matter? Ie. if I add a point to a dataset that is far from the mean, it will increase variance regardless of the sign.