How Markets Aggregate Implied Volatility Across Options
Summary
An underlying’s quoted implied volatility is a summary measure, not the volatility of one universal option. The document describes two common approaches: use an at-the-money option’s implied volatility, often interpolated to a 30-day maturity, or estimate a variance-swap level from a strip of calls and puts. The VIX methodology is presented as a reference for the latter; in theory, the strike weighting varies inversely with the square of strike price.
A variance-swap replication ideally uses a broad range of strikes, but actual markets provide only a limited liquid set, so practical implementations must make choices about available prices and option style. For a historical series, one suggested simple approach is to calculate an at-the-money straddle’s implied volatility each day and track it alongside the underlying. The document does not prescribe one universal method, and the result depends on the measure, maturity, option selection, and data quality.
Key ideas
- An underlying-level implied volatility depends on the aggregation method and option set.
- A variance-swap estimate combines calls and puts across strikes with heavier weighting at lower strikes.
- Practical variance replication is constrained by the strikes with sufficiently liquid prices.
- Many vendors use at-the-money implied volatility interpolated to a standard maturity.
- A historical series can be built by recalculating at-the-money implied volatility for each day.
Tags
Full text
# Calculating historical implied volatility # Calculating historical implied volatility I know that each individual option has it's own implied volatility, but how do you go about calculating the overall implied volatility for an underlying? For example when someone sais the IV of a certain underlying is 40%, they are not referring to a specific option/strike. They mean that the option market as a whole is implying a volatility of 40%. How is that 40% calculated? Im guessing it is something along the lines of calculating the IV for every option available and taking some sort of average? Secondly, how do you go about calculating the historical IV over a given time period. For example in most options trading platforms (eg: TWS, ThinkOrSwim, etc) you can pull up a chart of a specific underlying along with it's IV over a given time period. How would you go about recreating that? Again I presume you do something like: - One day at a time, get the closing price for every active option - Calculate the IV for all the options at every strike - Perform some sort of average - Move to the next day It seems impractical to calculate the IV of every single active option. Is it perhaps only done using the front month? (and if so, does that include weeklies and monthlies?) ## Answer by RiskyScientist (score 2) https://quant.stackexchange.com/a/27799 One way to do this would be to try to replicate the VIX calculation, which is calculated as the square root of a 30 day variance swap level. A variance swap can be replicated (in theory) using standard European calls and puts (you would need to convert American style stock option prices to European style prices using option models). The weighting scheme is inversely proportional to the square of the strike price, and in theory uses all option prices from zero to infinite strike. Of course we don't have this many option prices, and the actual liquid set of option prices is much smaller than the set that gets closing prices on the exchange, so one needs to make pragmatic choices. There are many papers on how to practically replicate variance swaps. Here is one good paper by JPM: JPM Var Swap Paper ## Answer by onlyvix.blogspot.com (score 1) https://quant.stackexchange.com/a/24504 Most software vendors use ATM implied volatility (usually interpolated to 30 days) which is how the old VIX index was calculated. AFAIK no vendor provides new VIX/MFIV as default - it is simply not as robust. ## Answer by glaucoOptions (score 1) https://quant.stackexchange.com/a/27792 For historical volatility I actually like this article: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf it provides several of the better known methods for calculating historical vol, which of course could be done manually. Just being aware of the upsides and downsides of each method. As for implied vol, yes as onlyvix has said it's generally ATM that they use. You could just take the closest atm straddle for each day and calculate it via some sort of root finding method such as the Newton Method. Then keep track of each day vs where the underlying was. There are also several online resources that have this done for you as well.
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