VIX Option Selection, Strike-Range Errors, and Replication Size
Summary
The VIX calculation uses out-of-the-money SPX options with nonzero bids, so its option count changes as volatility and the available strike range change. The document asks how many options might be needed to replicate the index, but provides no practical estimate or count. Instead, it discusses research on how strike selection can affect the index and its interpretation as a fair measure of volatility.
The cited review describes a truncation issue associated with the rule that sets the minimum and maximum strikes. Proposed alternatives include using a fixed economic strike range to construct a corridor implied volatility measure, or extrapolating option prices into the tails. The latter approach is complicated by potentially mispriced deep out-of-the-money options. The document also reports a study’s findings about strike spacing and range: spacing errors were negligible below a stated standard-deviation threshold, and truncation errors were negligible beyond a stated distance from at-the-money. These are reported research results, not a current option-count estimate; no direct evidence is given on how many options are typically involved in live replication.
Key ideas
- The VIX includes out-of-the-money SPX options with nonzero bids, so the selected option count varies over time.
- A changing strike range can introduce truncation effects into the index calculation.
- A corridor volatility measure with an economically fixed range is one proposed alternative.
- Extrapolating prices into the tails is difficult when deep out-of-the-money options may be mispriced.
- The document reports that research links calculation error to strike spacing and range, but gives no live option-count estimate.
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Full text
# How many options would be required to dynamically replicate the VIX nowadays? # How many options would be required to dynamically replicate the VIX nowadays? The VIX is a portfolio of OTM options on the SPX with non-zero quotes. From CBOE white-paper: > Only SPX options quoted with non-zero bid prices are used in the VIX Index calculation. [...] As volatility rises and falls, the strike price range of options with non-zero bids tends to expand and contract. As a result, the number of options used in the VIX Index calculation may vary from month-to-month, day-to-day and possibly, even minute-to-minute. I know that the number of options changes all the time (with the weights), but I was wondering how big can this number get in practice. NOTE: I am not interested in the feasibility of such replication or to question any economic benefits in doing it. I am working on a research thesis and I could not find any sources that give an estimate of such quantity. ## Answer by Martin Georg Haas (score 6) https://quant.stackexchange.com/a/49731 Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2. > Andersen, Bondarenko & Gonzalez-Perez (2013) identify a new error source in VIX that generates a significant number of jumps in the volatility index unconnected with the underlying volatility process and that weakens the function of VIX as an annualized fair volatility index. This error source is related to the truncation error reported in Jiang & Tian (2005) but differs because it is generated by the CBOE rule that determines the minimum and maximum strikes considered in the volatility index formula (cutting-wings rule). After Andersen, Bondarenko & Gonzalez-Perez (2013) identify this additional error component, the CBOE added in the VIX white paper the following disclaimer: “as volatility rises and falls, the strike price range of options with nonzero bids tends to expand and contract. As a result, the number of options used in the VIX calculation may vary from month-to-month, day-to-day and possibly, even minuteto-minute.” Nevertheless, some adjustments or changes in the VIX formula should also be considered to reduce this deficiency. The literature basically suggests (i) to make the range of strikes economically invariant and compute the VIX as a Corridor Implied Volatility (CIV) index (see Andersen & Bondarenko (2007), Andersen & Bondarenko (2010), Andersen, Bondarenko & Gonzalez-Perez (2013)), or (ii) extrapolate option prices in the tails using implied volatility functions. Nevertheless, mispriced deep OTM options difficult the success of the extrapolation exercise. EDIT After finishing my Thesis about the subject I can provide one further interesting point: The Quality of the VIX calculation depends on the spacing of the strike prices, in terms of standard deviation of price (sd). Jiang & Tian (2005) (https://doi.org/10.1093/rfs/hhi027] find that the errors due to spacing are negligible below 0.35 sd. Regarding the range of strike prices, they find the errors due to so-called truncation of the range to be neglible beyond two sd above and below the at-the-money strike price.
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