Volatility Products such as VIX Futures

VIX futures and Nikkei 225 VI Futures come up when estimating the cost of hedging against a sharp fall in stocks and when explaining the profit or loss of rolling from one futures contract to the next. How the final settlement value is set is used in calculating settlement dates and checking the amounts paid and received.

The VIX as a 30-day expected volatility

The VIX is an index calculated from the prices of S&P 500 Index options. It measures how much the market expects the US stock market to move over the next 30 days¹. Option prices carry the market's view of volatility, the size of price moves. The volatility backed out of the price of a single option is called implied volatility, and the VIX condenses the view priced into many options into a single number. Cboe began calculating the VIX in 1993 and in 2003 switched to the current method, which uses S&P 500 Index options².

The calculation uses put options with strike prices below a cutoff ($K_0$ in the formula below) and call options with strike prices above it, one option per strike. At the cutoff strike, both the put and the call are used. Apart from the cutoff, all of these options are out of the money, meaning none would pay off if exercised right now. Prices are the midpoints between the bid and the ask, and the calculation covers options across a wide range of strikes³. For the nearest contract month (the near term) and the one after it (the next term), the following variance is calculated.

$$\sigma^2 = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2}\, e^{RT} Q(K_i) - \frac{1}{T}\left(\frac{F}{K_0} - 1\right)^2$$

The symbols in the formula are as follows⁴.

• $T$ is the time to expiration in years, counted in minutes and divided by 365 days.
• $K_i$ is the $i$-th strike price.
• $\Delta K_i$ is half the distance between the strikes on either side. For the strikes at either end, it is the distance to the neighboring strike.
• $Q(K_i)$ is the midpoint price of that option.
• $R$ is the risk-free interest rate to expiration.
• $F$ is the forward level derived from call and put prices.
• $K_0$ is the strike closest to $F$ at or below it.

Each option contributes in proportion to its price and $\Delta K_i$, and in inverse proportion to the square of its strike. These are the same weights used to replicate the payoff of a variance swap with a portfolio of options⁵.

The near-term and next-term variances are interpolated by their days to expiration into a 30-day variance, and the VIX is its square root multiplied by 100⁶. Because $T$ is measured in years, the VIX is an annualized volatility expressed in percent. For example, a VIX of 20 means the market expects the S&P 500 Index to move at an annual rate of about 20% over the next 30 days. During regular trading hours, Cboe calculates and disseminates the VIX every 15 seconds⁷.

Deep diveCalculating the 30-day VIX from the near-term and next-term variances

Let $T_1$ and $\sigma_1^2$ be the time to expiration in years and the variance for the near term, and $T_2$ and $\sigma_2^2$ those for the next term. Let $M_{T_1}$ and $M_{T_2}$ be the number of minutes from now until the near-term and next-term expirations, $M_{30}$ the number of minutes in 30 days, and $M_{365}$ the number of minutes in 365 days. The VIX is calculated as follows⁶.

$$\text{VIX} = 100 \times \sqrt{\left\{ T_1 \sigma_1^2 \frac{M_{T_2} - M_{30}}{M_{T_2} - M_{T_1}} + T_2 \sigma_2^2 \frac{M_{30} - M_{T_1}}{M_{T_2} - M_{T_1}} \right\} \times \frac{M_{365}}{M_{30}}}$$

The expression in braces takes each contract month's variance multiplied by its time in years (the variance that builds up until expiration) and interpolates along a straight line to the 30-day point. Multiplying by the ratio of 365 days to 30 days turns the result back into an annual rate.

Final settlement of VIX futures

VIX futures are futures contracts that settle by paying or receiving the difference between the traded price and the VIX value set on the final settlement date. Cboe launched VIX futures on March 24, 2004, on Cboe Futures Exchange (CFE)². The main terms are as follows⁸.

The final settlement value is calculated differently from the VIX that is disseminated every day. The daily VIX interpolates between two contract months, but the final settlement value uses options from only one contract month, the one that expires exactly 30 days after the final settlement date (for monthly contracts, the third Friday of the following month). It also uses prices set in a special opening auction that Cboe holds that morning, not the midpoints of bid and ask quotes⁹.

Consider a hypothetical example in which you buy one VIX futures contract at 20.00. If the final settlement value is 25.00, the buyer receives (25.00 − 20.00) × USD 1,000, or USD 5,000, and the seller pays it. If the final settlement value is 16.00, the buyer pays (16.00 − 20.00) × USD 1,000, or USD 4,000, and the seller receives it.

Figure description

This figure shows, in points, the profit and loss at final settlement for a party that bought one VIX futures contract at 20.00. The horizontal axis is the final settlement value and the vertical axis is the profit or loss. The profit or loss is a straight line that rises by the same amount as the final settlement value rises. It is 0 when the final settlement value is 20.00, a profit above that, and a loss below it. At 25.00 the profit is 5 points (USD 5,000), and at 16.00 the loss is 4 points (USD 4,000). The seller's profit or loss is this figure flipped upside down.

Besides the monthly contracts, there are also contracts with a final settlement date every week, and their ticker symbols carry the week number. The final settlement value of a weekly contract is calculated from S&P 500 Index options that settle in the afternoon (SPXW)⁸.

Exercise and settlement of VIX options

VIX options are options on the VIX. Trading in them began on Cboe Options Exchange in February 2006². They are European options that can be exercised only on the expiration date, which is usually a Wednesday. The expiration date is set in the same way as the final settlement date of VIX futures, and the value used to settle exercised options (the Special Opening Quotation) is also the same¹⁰⁹.

When an option is exercised, the parties pay or receive USD 100 times the difference between the exercise settlement value and the strike price, in cash on the business day after expiration¹⁰. Consider a hypothetical example in which you hold a VIX call option with a strike price of 25. If the settlement value is 30, the buyer receives (30 − 25) × USD 100, or USD 500, and the seller pays it. The buyer's profit or loss is this USD 500 minus the premium paid at the time of the trade. If the value is 25 or lower, the option is not exercised and the buyer receives nothing. In that case, the buyer loses the premium, and the seller keeps it.

Nikkei 225 VI and Nikkei 225 VI Futures

The Nikkei Stock Average Volatility Index (Nikkei 225 VI) is an index calculated by Nikkei Inc. that shows how much the market expects the Nikkei Stock Average to move over the next 30 days. It is calculated from the prices of Nikkei 225 Options and Nikkei 225 Futures on Osaka Exchange, using a method widely used to estimate the fair level of payments on a variance swap (the fair variance). It uses the near-term and next-term options and interpolates between them so that the time to expiration is 30 days¹¹. Nikkei Inc. began calculating and publishing the index in November 2010, and in January 2012 switched to real-time calculation every 15 seconds¹². How Nikkei 225 Options work is covered in Equity Index Options.

Because the index is an annual volatility, dividing it by about 16, the square root of the number of business days in a year (about 245), gives a rough daily volatility¹³. For example, an index value of 32 corresponds to a daily volatility of about 2%.

Nikkei 225 VI Futures are futures on the Nikkei 225 VI, and trading in them began on Osaka Exchange on February 27, 2012¹⁴. They compare with VIX futures as follows⁸¹⁴.

Nikkei 225 VI Futures have eight consecutive nearby contract months. The final settlement value is called the final settlement price, or special quotation (SQ)¹⁴.

Consider a hypothetical example in which you buy one unit of Nikkei 225 VI Futures at 22.00. If the SQ is 25.00, the profit is (25.00 − 22.00) × 10,000 yen, or 30,000 yen. If the SQ is 20.00, the loss is (20.00 − 22.00) × 10,000 yen, or 20,000 yen.

Futures term structure and the profit or loss from rolling

Lining up VIX futures prices in order of time to expiration gives the term structure of volatility¹⁵. According to a 2017 notice from the Financial Industry Regulatory Authority (FINRA) in the United States, VIX futures prices had tended to be higher the later the expiry, a shape called contango¹⁶. For Nikkei 225 VI Futures too, the term structure tends to slope upward in normal times, with futures priced above the index and tending to fall toward it as expiration approaches. By contrast, in turbulent markets the term structure tends to slope downward, with prices lower the later the expiry (backwardation)¹⁷¹⁸.

Futures on a volatility index cover the volatility over the 30 days from the day the final settlement value is set¹⁹. Today's VIX looks at the 30 days from today, while a futures contract with final settlement two months from now looks at the 30 days starting two months from now.

Consider a hypothetical example in which the term structure is shown below.

Suppose you buy the futures expiring in 2 months at 18.00, and a month later the term structure has exactly the same shape. The contract you bought is now the futures expiring in 1 month, so its price has fallen to 17.00, a loss of 1 point (USD 1,000). Even though the VIX stays at 15.00, simply holding the futures makes their price fall. Conversely, if stocks fall sharply during the month and this contract rises to 25.00, the profit is 7 points (USD 7,000).

To keep holding futures with a constant time to expiry, you repeatedly sell the nearer contract and buy a later one. This is called rolling. In a market in contango, a fall in price like the one in the previous example repeats with each switch, so the profit or loss from continued rolling (the roll yield) is negative¹⁸¹⁶. In a market in backwardation, it is positive instead¹⁶.

Protection against a sharp fall in stocks and its cost

The VIX often rises sharply when equity indices fall sharply²¹. For this reason, buying VIX futures or Nikkei 225 VI Futures can serve as a hedge against a sharp fall in stocks. On the other hand, if no sharp fall comes, the futures price tends to decline gradually, as described in the previous section, and that decline is the cost of the hedge¹⁷.

Consider a hypothetical example in which an investor who holds USD 1 million of US stocks buys 10 VIX futures contracts expiring in 2 months at 18.00. The results a month later are as follows.

In this example, when stocks fall sharply, the profit on the futures makes up for part of the loss on the stocks. When nothing happens, the 10,000-dollar decline in the futures is the cost of one month of protection. The VIX does not always rise when stocks fall sharply, however. VIX futures can move differently from the VIX, and in general their moves are smaller than those of the VIX²¹. So how much of the loss on stocks the profit on the futures can offset depends on the market at the time.

Risks

With futures such as Nikkei 225 VI Futures, the trading amount is larger than the margin deposited, so price moves can cause losses greater than the margin deposited²⁴. How margin works is covered in Futures Margin and Daily Settlement.

Sellers can suffer large losses when stocks fall sharply, because volatility indices jump within a short time when the market falls. On March 14, 2011, for example, the Nikkei Stock Average fell about 6% from the previous business day, March 11, while the Nikkei 225 VI rose about 74%, from 23.44 to 40.72²⁵. Osaka Exchange cautions investors without sufficient assets or experience to avoid selling Nikkei 225 VI Futures²⁵.

ETPs that are easily mistaken for tracking the VIX also call for caution. Many US volatility ETPs track not the VIX itself but a futures index that rolls VIX futures to keep a constant time to expiry. In an October 2017 notice, FINRA pointed out that because of the negative roll yield, these ETPs tend to lose value over long periods, and that some have lost more than 90% of their value since launch²¹. As of fiscal 2016, an ETF linked to a VIX short-term futures index was also listed on the Tokyo Stock Exchange. This index represents the return from selling the nearest futures and buying the next contract month every day, keeping the weighted average time to expiry at one month. As a result, it moves differently from the VIX²⁶.

Notes

1. Cboe, "Volatility Index Methodology: Cboe Volatility Index" (the "Methodology"), section 1.1 "Index Objective" (page 3). The VIX is defined as a measure of the market's expectation of 30-day forward looking volatility of the US equity market. ↩

2. Cboe, the Methodology, section 1 "Introduction" (page 3). It covers the introduction in 1993, the 2003 change to a method using S&P 500 Index options, the launch of VIX futures on March 24, 2004, and the launch of VIX options in February 2006. ↩ ↩² ↩³

3. Cboe, "Cboe Volatility Index Mathematics Methodology" (the "Mathematics"), section 3(a) (ii) (page 7) and (iii) (pages 7–8). The composition and the use of midpoint prices are in the Methodology, section 3.1, Step 3 (page 6). ↩

4. Cboe, the Mathematics, section 3(a) "Single Term," formula (1) and the table of symbols (pages 6–7), and section 3(a) (i) "Time to Expiration" (page 7). The source defines $Q(K_0)$ as the average of the put and call prices at strike $K_0$. ↩

5. Cboe, the Mathematics, section 1 "Introduction" (page 4) and section 3(a) (iv) "Calculating Volatility" (page 8). ↩

6. Cboe, the Mathematics, section 3(b) (page 8), and the Methodology, the table in section 2.1 (page 4). The source's $M_{CM}$ is written as $M_{30}$. ↩ ↩²

7. Cboe, the Methodology, section 4 "Calculation and Dissemination" (page 8). Regular trading hours (RTH) are 9:31 a.m. to 4:15 p.m. US Eastern Time. ↩

8. Cboe, "VIX Futures Contract Specifications," the items on contract size, tick size, final settlement date, final settlement value, settlement, ticker symbols, and contract expirations. ↩ ↩² ↩³ ↩⁴

9. Cboe, the Methodology, section 3.4 "Calculation of Final Settlement Value for VIX Derivatives" (pages 7–8). ↩ ↩² ↩³

10. Cboe, "VIX Options Product Specifications," "Exercise Style," "Expiration Date," and "Settlement of Option Exercise." ↩ ↩²

11. Nikkei Inc., "「日経平均ボラティリティー・インデックス」リアルタイム算出要領" (Real-time calculation rules for the Nikkei Stock Average Volatility Index, in Japanese; January 27, 2025 edition), "１：考え方" (Approach), and "２：指数の計算" (Index calculation), items "（１）基本事項" (Basics) and "（３）算式" (Formula) (pages 2–4). The 30 days are interpolated as 2,592,000 seconds. ↩

12. Nikkei Inc., "「日経平均ボラティリティー・インデックス」リアルタイム算出要領" (Real-time calculation rules for the Nikkei Stock Average Volatility Index, in Japanese), the cover (page 1) and "２：指数の計算" (Index calculation), item "（１）基本事項" (Basics) (page 2). ↩

13. Osaka Exchange, "日経平均ボラティリティー・インデックス先物取引の利用のコツ" (Tips for using Nikkei Stock Average Volatility Index futures, in Japanese; 4th edition, November 2024; the "Tips"), note 2 to Article 3 (page 4). The source puts the number of business days in a year at about 245, and its square root at 15.65. ↩

14. Osaka Exchange, "制度概要（日経平均VI先物）" (in Japanese; English page: "Contract Specifications," Nikkei 225 VI Futures, https://www.jpx.co.jp/english/derivatives/products/vi/225-vi-futures/01.html), the rows "取引開始日" (Opening Date), "限月取引" (Contract Months), "取引最終日" (Last Trading Day), "SQ日" (SQ Day), "取引単位" (Contract Unit), "呼値の単位" (Tick Size), "最終清算数値（SQ値）" (Final Settlement Price), "証拠金" (Margin), and "決済方法" (Settlement method). ↩ ↩² ↩³ ↩⁴ ↩⁵

15. Osaka Exchange, the Tips, Article 7 (page 8). Using the Nikkei 225 VI and Nikkei 225 VI Futures as examples, the source calls a chart with time on the horizontal axis and volatility on the vertical axis "ボラティリティーの期間構造" (the term structure of volatility). ↩

16. FINRA, "Regulatory Notice 17-32" (October 16, 2017), "Background and Discussion" and endnote 4. The source says VIX futures prices have tended to rise as expiries go further out, and calls a market with a negative roll yield "contango" and the opposite "backwardation." ↩ ↩² ↩³

17. Osaka Exchange, the Tips, Article 6 (page 7). ↩ ↩²

18. Osaka Exchange, the Tips, Article 8 (page 9). The source calls the upward-sloping shape "アップワードスローピング" (upward sloping) and the downward-sloping shape "ダウンワードスローピング" (downward sloping). ↩ ↩²

19. Osaka Exchange, the Tips, Article 4 (page 5). The source describes Nikkei 225 VI Futures as "a product that forecasts the size of market moves over the 30 days from the futures' expiration date" (our translation). ↩

20. Osaka Exchange, the Tips, Article 10 (page 11). The source shows trading volume by contract month from April to October 2016 and the order book on October 21, 2016. ↩

21. FINRA, "Regulatory Notice 17-32," "Summary" and "Background and Discussion." ↩ ↩² ↩³

22. FINRA, "Regulatory Notice 17-32," endnote 7. As an example from an enforcement action issued at the same time, the source describes brokers who recommended ETPs linked to VIX futures to customers under the mistaken belief that they could be used as a long-term hedge in a market downturn. ↩

23. FINRA, "Regulatory Notice 17-32," "Summary" and "Sales Practice Obligations Relating to Volatility-Linked ETPs." ↩

24. Osaka Exchange, the Tips, the disclaimer (page 16). ↩

25. Osaka Exchange, the Tips, Article 13 (page 15). The values for March 11 and 14, 2011, are the closing values given in the source. ↩ ↩²

26. Osaka Exchange, the Tips, Article 11 (pages 12–13). Together with trading value for the first half of fiscal 2016, the source lists an ETF linked to the S&P 500 VIX Short-Term Futures Index (total return). ↩