# CBOE S&P 3-month variance futures

CBOE S&P 500 Three-month Variance futures, one of the futures contracts based on “variance” and on “volatility” and listed on the CFE , was introduced for trading on May 18, 2004. It is based on based on the realized variance of the S&P 500 Composite Stock Price Index ("S&P 500"). The contract was delisted on October 12, 2012.

## Details

### Final Settlement Price

The final settlement value for CBOE S&P 500 Three-Month Variance futures contracts is based on a standardized calculation of the realized variance for the S&P 500. This calculation uses continuously compounded daily returns for a three-month period assuming a mean daily price return of zero. The calculated variance is then annualized assuming 252 business days per year. The final settlement value is this annualized, calculated variance multiplied by 10,000.

For purposes of calculating the settlement value, the three-month realized variance is calculated from a series of values of the S&P 500 beginning with the Special Opening Quotation ("SOQ") of the S&P 500 on the first day of the three-month period, and ending with the S&P 500 SOQ on the last day of the three-month period. All other values in the series are closing values of the S&P 500. For example, the final settlement value for a CBOE S&P 500 Three-Month Variance futures contract expiring on Friday, September 17, 2004, would be calculated using the S&P 500 SOQ on Friday, June 18, 2004, the closing prices of the S&P 500 from Monday, June 21, 2004 through Thursday, September 16, 2004, and the S&P 500 SOQ on Friday, September 17, 2004. (Though the previous example is dated, it nonetheless illustrates the principle well.)

### Variance Formula

Realized Variance

where:

• Ri = ln(Pi+1/Pi) - Daily return of the S&P 500 from Pi to Pi+1
• Pi+1 - The final value of the S&P 500 used to calculate the daily return.
• Pi - The initial value of the S&P 500 used to calculate the daily return.
• Ne = Number of expected S&P 500 values needed to calculate daily returns during the three-month period. The total number of daily returns expected during the three-month period is Ne - 1.
• Na = The actual number of S&P 500 values used to calculate daily returns during the three-month period. Generally, the actual number of S&P 500 values will equal the expected number of S&P 500 values (represented by Ne). However, if one or more "market disruption events" occurs during the three-month period, the actual number of S&P 500 values will be less than the expected number of S&P 500 values by an amount equal to the number of market disruption events that occurred during the three-month period. The total number of actual daily returns during the three-month period is Na - 1.

### Market Disruption Events

• (i) the occurrence or existence, on any trading day during the one-half hour period that ends at the Scheduled Close of Trading, of any suspension of, or limitation imposed on, trading on the primary exchange(s) of the companies comprising the S&P 500 in one or more securities that comprise 20 per cent or more of the level of the S&P 500; or
• (ii) if on any trading day the one or more primary exchange(s) determines to change the Scheduled Close of Trading by reducing the time for trading on such day, and either no public announcement of such reduction is made by such exchange or the public announcement of such change is made less than one hour prior to the Scheduled Close of Trading; or
• (iii) if on any trading day one or more primary exchange(s) fails to open and if in the case of either (i) or (ii) above, in the determination of CFE, such suspension, limitation, or reduction is deemed material. "Scheduled Close of Trading" means that time scheduled by each applicable exchange, as of the opening for trading in the applicable equity security, as the closing time for the trading of such equity security comprising the S&P 500 on the trading day.

Generally, if CFE determines that a market disruption event has occurred on a trading day, then the value of the S&P 500 on that day will be omitted from the series of values used to calculate three-month realized variance. For each such market disruption event, the value represented by Na in the standardized formula will be reduced by one.

If a market disruption event occurs on the Final Settlement Date, the final settlement value for CBOE S&P 500 Three-Month Variance futures will be determined in accordance with the Rules and By-Laws of The Options Clearing Corporation ("OCC"). These Rules and By-Laws list actions that may be taken if a final settlement value is unavailable or the normal settlement procedures cannot be utilized. Such actions include, but are not limited to, postponing the Final Settlement Date until the first succeeding trading day in which a market disruption event has not occurred. It is intended that the value of the S&P 500 on the final day in the three-month period, which is used in the calculation of the three-month realized variance for the CBOE S&P 500 Three-Month Variance futures contract, will equal the corresponding final settlement price for expiring series of S&P 500 options listed on the Chicago Board Options Exchange. Once the calculation period for three-month realized variance begins, the value represented by Ne will not change regardless of the number of market disruption events that occur during the three-month period, even if the Final Settlement Date is postponed. If the Final Settlement Date of the expiring futures contract is postponed, the calculation period for the next three-month realized variance will be shortened by the number of market disruption events that occurred at the beginning of the period. Likewise, the value represented by, Ne, will be reduced by the number of market disruption events that occurred at the beginning of the period. The first daily return of the shortened period for the next three-month realized variance will be calculated using the same procedure as described above (the initial value for the first daily return is the SOQ of the S&P 500 on the first day of the period and the final value for the first daily return is the closing value of the S&P 500 on the following trading day). For example, if the Final Settlement Date for the previous three-month realized variance is postponed to Tuesday, the initial value for the first daily return of the next three-month realized variance would be calculated using the SOQ of the S&P 500 on Tuesday morning and the closing value of the S&P 500 on Wednesday.

As soon as practical under the circumstances, CFE will notify Trading Privilege Holders of the existence of a market disruption event. Failure to provide such notice will have no effect on the determination by CFE that a market disruption event has occurred.

## References

CBOE S&P 500 3-Month Variance Futures. Chicago Board Options Exchange.