How is implied volatility derived?

Question

How to compute Implied Volatility Calculation?

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.

Answer 1

For How VIX works you can read this wonderful blog : http://onlyvix.blogspot.com/2011/09/intuitive-understanding-of-vix-formula.html

It provide wonderful non mathematical explanation of the how vix is actually computed.

Now comes to your last answer why vix is inversely related to market movement ?

In simple words, if market is more volatile then investor would expect higher return from the market. To get the higher return, traders will continue to sell the stock until its price is consistent with the perceived risk. Lets take simple example : if current price of stock is $ 100 and investor expect price to be 110 after one year. Here expected return is 10 percent which is consistent with the perceived risk of lets say 20%. If this risk increased then investor would expect higher return from the same stock. Assume, risk increased to 25% and investor expect now 12% return. So now the fair price is 110/1.12 = 98.21. So stock is overprice and investor continue to sell it until price reaches 98.21 which is consistent with revised expected return of 12%.

Reverse happen when volatility fall.

The above explanation is simply based on risk return theory of stock price. I hope it would help you to understand why VIX is inversely related to market movement.