# VIX Calculation Explained

This page explains the logic of VIX calculation and some of the underlying assumptions and parameters. Exact formulas are available in VIX White Paper by CBOE.

If you are not familiar with the VIX index, you may first want to see a more basic explanation: What is VIX?

## Basic Logic of VIX Calculation

The VIX is interpreted as annualized implied volatility of a hypothetical option on the S&P500 stock index with 30 days to expiration, based on the prices of near-term S&P500 options traded on CBOE.

Contrary to what many people believe, the VIX is not calculated using Black-Scholes or any other option pricing model.

Instead, there is a formula which directly derives variance from prices of different options with the same time to expiration. Two different variances for two different expirations are then interpolated to get 30-day variance. This variance is then transformed into standard deviation (by taking the square root) and multiplied by 100.

## VIX Calculation Step by Step

• Select the options to be included in VIX calculation – a range of call and put strikes in two consecutive expirations around the target 30-day mark.
• Calculate each option's contribution to the total variance of its expiration.
• Sum up the contributions to get total variances of the first and second expiration.
• Calculate 30-day variance by interpolating the two variances, depending on the time to expiration of each.
• Take the square root to get volatility as standard deviation.
• Multiply the volatility (standard deviation) by 100.
• The result is the VIX index value.

## Options Included in VIX Calculation

### Expirations Included

The data used for VIX calculation are bid and ask quotes of short term S&P500 options.

Because the target time horizon for the VIX index is 30 days, two consecutive expirations with more than 23 days and less than 37 days are used. These can include the standard monthly expirations as well as weekly S&P 500 options.

The two expirations are referred to as "near-term" and "next-term".

As soon as the near-term options get less than 24 days to expiration, they are no longer used. The previously next-term expiration becomes the new near-term expiration and the next available longer expiration is added as the new next-term. This rollover happens every week.

### Strike Prices Included

The set of options for each expiration includes both call and put options, but only at the money and out of the money strikes. In the money options are not included.

Furthermore, only options which have non-zero bid are used. This is to eliminate illiquid far out of the money options which can imply extreme values of volatility and would therefore distort the final VIX value.

The selection of strikes goes from the at the money strike up (for calls) and down (for puts), until two consecutive strikes with zero bid price are found in each direction. No other options beyond such two consecutive zero bid strikes are included.

As a result, the strike range and total number of options included in VIX calculation vary over time, in line with changes in S&P500 index value and changes in quotes on individual S&P500 options.

Only S&P500 option quotes directly from CBOE are used.

## Parameters Used in VIX Calculation

For each option in the set described above, expected variance is calculated from the option's price (premium) and strike relative to spot S&P500 index value. Other inputs entering the calculation are time to expiration and risk-free interest rate.

### Time to Expiration

The time to expiration for a particular option is calculated very precisely in minutes. The end of the period is the moment when the exercise-settlement value is being determined, which is the open (8:30 am Chicago time) on the settlement day for monthly S&P500 options (usually the third Friday of a month) and close of trading (3:00 pm) for weekly options.

### Risk-Free Interest Rate

The interest rate used in the VIX calculation is the bond-equivalent yield of US T-bills which mature closest to the particular option expiration. Therefore, two different interest rates may be used for the two different expirations which enter VIX calculation.

## Contributions of Individual Options

The contribution of individual options to the calculation of total variance of an expiration depends on the option's price, the strike price, and the average strike price increment of neighboring strikes. In general, at the money options influence the final result the most and the contributions decrease as you go further out of the money.

## Getting 30-day Variance from the Two Expirations

The 30-day variance is calculated by interpolating the total variances of the two expirations. The weights of the two variances depend on how close or far each expiration is from the desired 30-day mark (the closer, the greater weight).

The two weights must add up to 1.

Until October 2014 when only monthly expirations were used, if both expiration months had more than 30 days left (e.g. 32 and 67 days), the first month's weight was greater than 1 and the second month's weight was negative.

## Calculating the VIX from 30-day Variance

Having calculated the 30-day variance, we need to take the square root to transform variance into standard deviation (which is the traditional way how volatility is quoted and the VIX is no exception).

The last step is to multiply the result by 100. While volatility is usually quoted in percent, the VIX is volatility times 100.

For example, if the VIX index is 22, it means that a hypothetical S&P500 option with 30 days to expiration has annualized implied volatility of 22%.

## Old VIX Calculation Methods

### 22 September 2003 – 5 October 2014

Until October 2014, the VIX calculation used monthly options only. The rule was two nearest monthly expirations with at least one week left to expiration. For example, if the nearest expirations were in 4, 32, and 67 days, the front month (4 days to expiration) wouldn't be included, and the next two months (32 and 67 days) would be used in VIX calculation. This was to eliminate options in the last days before expiration, whose prices sometimes behave in strange ways.

Once weekly S&P 500 options became liquid enough, it was logical for CBOE to start using them from 6 October 2014. This made the window around the 30 days target narrower and the calculation more precise.

The old version of the VIX using monthly options only is still being calculated and available under the symbol VIXMO.

### Before 22 September 2003

Until September 2003 the VIX was calculated in an entirely different way, even using a different underlying:

• S&P100 (OEX) option prices were used. The current method uses S&P500 (SPX) options.
• Only at-the-money options were included. Under the current method, a wide range of at-the-money and out-of-the-money strikes enters VIX calculation.
• The exact way how volatility was derived from option prices was different. The old method used an option pricing model, while the current method uses a direct formula.

The pre-2003 method index was calculated and published by CBOE under the symbol VXO until 23 September 2021.

Obviously, the two methods produce different index values, although the differences are relatively small and the two indices (VIX and VXO) react to the same market conditions in a similar way.

Daily historical data is available starting from 1990 for the VIX and from 1986 for VXO. Therefore, VXO data covers the interesting events of October 1987, when all-time VXO highs were reached.

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