Few years ago I published two post trying to give simple explanations and intuition behind complicated formulas used for calculating vol indexes. However few of you emailed that some charts are missing from these older posts, and for technical reasons since I could not restore them, I decided to re-created new charts from scratch, and re-write the posts. In this post I will make many simplifications and sacrifice rigor to provide intuition behind calculations.

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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