# VIX Calculation

*The objective of this page is to explain the logic of VIX calculation and some of the underlying assumptions and parameters. Exact formulas are available in a short pdf named VIX White Paper on the official website of .*

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

## VIX Calculation: The Big Picture

VIX is interpreted as annualized implied volatility of a hypothetical option on S&P500 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. **There is a formula which directly derives variance from the whole set of prices of options with the same time to expiration.** Two different variances for two different times to expiration are then interpolated or extrapolated 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 the VIX calculation. - Calculate each option’s
**contribution**to the total variance of its expiration month. - Calculate the
**total variance**for the first month and the second month. - Calculate
**30-day variance**by interpolating or extrapolating 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
**VIX**.

*The rest of this page explains individual steps in more detail.*

## Options Included in VIX Calculation

### Expiration months included

The data used for VIX calculation are bid and ask quotes of **near term S&P500 options**. **Two monthly expirations** are used, but only those with **at least one week left to expiration**. For example, if the nearest expirations are in 4, 32, and 67 days, the front month (4 days to expiration) won’t be included, and the next two months (32 and 67 days) will be included in VIX calculation. This is to eliminate options in the last days before expiration, whose prices sometimes behave in strange ways.

### Strike prices included

**At the money and out of the money call and put options enter VIX calculation** and only options which have **non-zero bid** are included. This is to eliminate illiquid far out of the money options which can imply extreme values of volatility and 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 range and the 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

Expected variance of each expiration month is derived from a set of option prices and strikes, given 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 last business day before expiration – usually the third Friday of a month.

### Risk-free interest rate

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

## Contributions of Individual Options

The contribution of individual options to the calculation of total variance of an expiration month depends on the **option’s price**, the **strike price**, and the average **strike price increment** of neighbouring 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 the 30-day Variance from the Two Months

The 30-day variance is calculated by **interpolating or extrapolating the total variances of the two expiration months**. 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). If both expiration months have more than 30 days to expiration (e.g. 32 and 67 days), the first month’s weight is greater than 1 and the second month’s weight is negative. The sum of the weights is always 1.

## Calculating the VIX: Final Steps

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

The last step is to **multiply the result by 100**. While volatility usually is in percent, **the VIX is volatility times 100.** For example, if VIX is 22, it means that a hypothetical S&P500 option with 30 days to expiration has annualized implied volatility of 22%.

## New vs. Old VIX Calculation

**On 22 September 2003 CBOE changed the methodology of its Volatility Index calculation.** The VIX as we know it today is calculated using the new methodology. The old methodology index is still being calculated and published by CBOE under the ticker symbol VXO. The main differences are:

- The old methodology used S&P100 (OEX) option prices. The new methodology uses S&P500 (SPX) options.
- The old methodology used only at the money options. Under the new methodology, a wide range of strikes enters VIX calculation.
- The exact way how volatility is derived from option prices has changed. The old methodology used an option pricing model. The new methodology uses a direct formula.

Although the two methodologies of course produce different index values, the differences are not big and the two indexes (VIX and VXO) react to the same market conditions in a similar way. Daily VIX and VXO historical data is available (VXO starting from 1986, VIX from 1990).