How the VIX Approximates Variance Swap Volatility

Summary

The document explains the conceptual origin of the VIX calculation by connecting it to the pricing of volatility swaps. The cited foundational work on volatility swaps is presented as the detailed source for understanding why the index uses its option-based formula, with a blog post suggested for a more intuitive introduction.

Operationally, the VIX approximates the relevant volatility-swap calculation using a discrete set of listed options rather than a continuous range of strikes. It excludes some extremely out-of-the-money options whose prices may be unrepresentative, and it interpolates or extrapolates between expirations to estimate a constant 30-day horizon. These are practical approximations used to turn the theoretical relationship into an index calculation. The document is a brief pointer rather than a derivation: it provides no equations or empirical comparison of the approximations, and details of the formula and its assumptions require consulting the cited source material.

Key ideas

• The VIX calculation is linked to the theory of volatility swaps.
• Listed options approximate a continuous-strike calculation through a discrete sum.
• Some extremely out-of-the-money options are excluded because their prices may be unreliable.
• Interpolation or extrapolation helps target a 30-day maturity when exact expirations are unavailable.
• The document points readers to source material instead of deriving the formula.

Tags

# Why is the VIX computed that way?


# Why is the VIX computed that way?












The VIX as a clear definition as defined in this paper I am interested to know why they came up with this formula.

I smell some reasonably complicated explanation here so any pointer to a paper would be fine with me. Thanks

## Answer by sashkello (score 7, accepted)

https://quant.stackexchange.com/a/9203

In that white paper itself they quote where it came from: “More than you ever wanted to know about volatility swaps” by Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou, Goldman Sachs Quantitative Strategies Research Notes, March 1999. This is a classic article which you should definitely read if you are trading volatility. While there might be a clearer explanation somewhere, this is the original and quite comprehensive work.

For some intuitive ideas behind this, also see this post in OnlyVIX blog.

The calculation of VIX itself is taking this Volatility Swaps idea and approximating it with discrete set of options (sum instead of integral), throwing away some ridiculously OTM options (prices of which are not representative at all). Since there is not always an expiration 30 days ahead, they approximate this point by interpolating (or extrapolating). These are merely some technical things they have to do to apply the formula.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.