What are common methods to compute implied volatility index?
One could use VIX method on other underlying.
Yes, the CBOE offers this for Apple, Google, Amazon, Goldman Sachs and IBM (see here). In my working paper here I use the CBOE VIX methodology on a sample of 268 individual equities in the same way. It also includes a comprehensive derivation of the CBOE model-free VIX method, option data structure and discussion about data requirements, if you're interested.
It is also easy to limit the method to 4 atm strikes. Is this a good idea though?
There are two methods for the CBOE VIX: up until 2003, it was based on 8 ATM-strike option contracts and used the Black&Scholes model to derive their implied volatilities (see e.g. this paper). The index is then quoted as a weighted, annualised standard deviation. This method was renamed and is quoted as "VXO" nowadays. However the B&S model makes assumptions that do not necessarily correspond to reality, which may introduce biases in the resulting implied volatility. Using only 4 strikes would be feasible using this model-based approach (see e.g. the New Zeeland index in this paper)
In 2003 the CBOE changed to a "model-free" approach to derive implied volatility, which got rid of most assumptions and thus the biases (see the VIX white paper). It is based on a "basket" of option contracts, which ideally contains a continuum of strikes from zero to infinity. In practice, the VIX is based on the SPX options (the whitepaper has 176 strikes in the example), which are suffiently liquid. Using the model-free methodology with only 4 strikes is infeasible.
What are other approaches?
- Corridor Variance Swaps (basically the CBOE VIX using a fixed range of strikes, see here)
- An approach called "State-Price Volatility Index" as in this paper
PS IVolatility uses proprietary method that incorporates delta and Vega. Any insights into why and how? (I guess VIX method incorporates delta too)
The current VIX method doesn't rely on an option pricing model and thus doesn't incorporate a delta. However, as the developers of the model-free implied volatility (at least one team of several authors who developed the method independently)
write in their paper, one could compare the VIX to a "gamma-hedged" portfolio of options.