There are two formulas that are used to calculate vol indexes: the theoretical variance swap formula

and its practical discrete reformulation:

While these formulas may look complicated it is actually really simple if viewed in a chart. Below are 3 charts that will illustrate step by step what the formula means, and I hope will provide intuitive understanding behind the formula. For the charts below I used simulated prices of calls and puts on a hypothetical stock with time to expiration = 0.1 and 20% annualized volatility, so we should expect the volatility index in this theoretical example to be about 20. I plot prices of calls and puts vs strike.

Let's consider just OTM options:

The higher the perceived risk, the higher volatility will cause higher options prices, and "taller" price curves on the chart. Conversely, lower prices = lower curves. The intersections of these two curves looks like a curved pyramid. Higher prices, higher vol would create taller pyramid.

The old VIX index and any other ATM / ATMF based index has a very simple meaning - it is proportional to the height of the pyramid - red line on the chart - which is just ATM price of a call or put. Approximate formula for ATM option ( call or put ) is 0.4 * volatility * index price * √ time to expiration . The reverse of the formula is volatility = height of the pyramid / 0.4 / index price / √ time to expiration . In our case height is 2.53, and volatility = 2.53 / 0.4 / 100 / √ 0.1 = 0.2 or 20% , exactly the vol we were expecting. So, in summary -

**ATM vol is height of the curve**.
Current VIX formula has a slightly different meaning - which involves an extra step. First, prices are re-weighted, or multiplied by 2/√ time to expiration / strike

^{2}. Then the area of the new pyramid is volatility squared.
In our case the area is 0.04, or 20% squared, just like we expected.

**In summary - old VIX / ATM vol is the height of the pyramid, and current VIX ( or VSTOXX, or other vol indexes ) are the area of the pyramid.**

Send me an email if you want a copy of excel spreadsheet with calculations.

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