Why VIX Option Weights Scale Inversely with Strike Squared

Summary

The answer explains the inverse-square strike weighting in the option strip used to calculate the VIX by connecting it to variance swap replication. The construction aims to hold dollar gamma approximately constant across strikes. Since dollar gamma scales with the square of the underlying price, options at lower underlying levels contribute less dollar gamma unless their weights compensate for that difference.

Weighting option prices by the inverse square of strike is presented as the mechanism that supports this replication objective, including the greater weight assigned to lower strikes. The explanation responds to the concern that the weighting appears to favor out-of-the-money puts over calls. It is brief and relies on the variance-swap and gamma intuition; it does not derive the full VIX formula, discuss discretization or market conventions, or quantify how the weighting behaves in practice.

Key ideas

• The VIX option strip is connected to replication of variance swap exposure.
• The replication aims for constant dollar gamma across strikes.
• Dollar gamma scales with the square of the underlying price level.
• Inverse-square strike weighting compensates for the lower dollar gamma associated with lower levels.

Tags

# VIX Calculation - weighting of strikes


# VIX Calculation - weighting of strikes












The formula for calculating the VIX from SPX options is given on page 4 of this document:

https://www.cboe.com/micro/vix/vixwhite.pdf

My question is, why is the option price at each strike weighted by the inverse of the strike price squared? This would seem to overweight OTM put options (with low strikes) and underweight OTM call options (with high strikes).

## Answer by mbison (score 1, accepted)

https://quant.stackexchange.com/a/19536

As the other comments already suggest this topic has been discussed many times and the references / links that are provided are far more detailed than my quick solution below.

In the construction / replication of a variance swap one tries to achieve a CONSTANT dollar gamma. This is done by buying a strip of calls and puts, weighted by 1/K^2 as you suggested. Now why did I all-cap the word CONSTANT? because the dollar gamma = $\gamma S^2$. As you can see the black scholes gamma is scaled by the squared level of the stock price. So to get a constant dollar gamma you need to have more of the lower strike options else they won't generate enough dollar gamma because the level of the stock is too low.

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.