# 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?) • I believe you may be referring to the "New VIX calculation method" aka the "variance swap method" which indeed does take all options at a certain maturity into account, though it is not simply an averaging of Black Scholes IV's, but something original. There is a lot written on that, including on quant.stackexchange. An overview: math.nyu.edu/research/carrp/papers/pdf/JODCarrWu.pdf – noob2 Nov 6 '15 at 14:10
• this is also called "model free implied volatility" method – noob2 Nov 6 '15 at 16:28
• It may be called "model-free", but it isn't. It relies on the hypothesis that the underlying's price follows an Itô process under the risk-neutral measure; or, in other words, it postulates an absence of jumps in the price process. Bit problematic in 2008-09. – ocstl Feb 24 '16 at 23:48

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

• fitting a vol surface can help to make variance swap prices. If one is meticulous about making an arb free surface (including calendar arbs), the wings can be well controlled even when strikes are somewhat sparse. Then choose the expiry that is preferred and do the usual var swap pricing with the appropriate slice (smile) of the vol surface. – FinanceGuyThatCantCode Mar 29 '17 at 13:13

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.