Methods for Constructing Implied Volatility Indices
Summary
The document compares approaches to building an implied volatility index for an underlying asset. It explains that the earlier VXO approach used a small set of at-the-money options and Black–Scholes implied volatilities, while the later CBOE VIX methodology estimates model-free variance from a broad range of option strikes. The discussion notes that restricting the model-free calculation to only a few strikes is unsuitable, though a model-based index can be constructed with fewer options.
Alternatives mentioned include corridor variance swaps and state-price volatility indices. For sparse option chains, one response points to a three-option method for estimating a variance swap strike, which relates to the VIX in the continuous limit. The text also addresses whether the VIX uses delta: the current model-free method does not depend on an option pricing model or delta, though it can be viewed in relation to a gamma-hedged options portfolio. These approaches depend on option availability and liquidity; the cited proprietary delta- and vega-based method is not explained.
Key ideas
- The VIX methodology estimates model-free implied variance using a broad basket of options across strikes.
- A four-strike approximation is unsuitable for the model-free method, though sparse-strike model-based indices are possible.
- Corridor variance swaps and state-price volatility indices are alternative constructions.
- A three-option approach may estimate a variance swap strike when the option chain is sparse.
- The current VIX method does not rely on model-derived delta.
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Full text
# Calculating implied volatility index # Calculating implied volatility index What are common methods to compute implied volatility index? One could use VIX method on other underlying. It is also easy to limit the method to 4 atm strikes. Is this a good idea though? What are other approaches? PS IVolatility uses proprietary method that incorporates delta and Vega. Any insights into why and how? (I guess VIX method incorporates delta too) ## Answer by Martin Georg Haas (score 2, accepted) https://quant.stackexchange.com/a/57749 > 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. ## Answer by user34971 (score 1) https://quant.stackexchange.com/a/57720 > I was thinking of simply limiting set of options that go into computation to K0 strike+ 1 option on each side (cboe.com/micro/vix/vixwhite.pdf). Not sure if this is a good idea though. Hence the question If you have a full/complete options chain then naturally to calculate the VIX you should use the VIX formula, which should be interpreted as a definition. Assuming what you mean with "implied volatility index" is a VIX index but for general underliers, as many other definitions of an implied volatility index are possible, then: If you only have a few options, e.g. for an illiquid underlying, then you could use the method described in It takes three to smile. It describes how using only three options you can get an accurate estimate of (among others) the variance swap strike, and the VIX is (in the continuous limit) just the square root of the variance swap strike.
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