## DOUBLE THE FUN WITH CBOE's VVIX^{SM} Index

#### Introduction

CBOE is expanding its suite of volatility benchmarks with a new index called the VVIX Index, the VVIX for short. The VVIX is a volatility of volatility measure in that it represents the expected volatility of the 30-day forward price of the CBOE Volatility Index (the VIX®). It is this expected volatility that drives the price of VIX® nearby options.

The VVIX is designed to guide and inform the growing number of investors in VIX-based products. The VVIX and its term structure convey:

- The expected volatility that determines VIX® option prices.
- The expected volatility of the VIX® itself to a nearby horizon.
- The mean and standard distribution of settlement values of VIX® futures and options.

Different points on the VVIX term structure price portfolios of VIX® options (VVIX portfolio) to different expirations. A position in a VVIX portfolio replicates the volatility of VIX® forward prices. VVIX portfolio prices have usually been at a premium relative to future realized volatility. The discount is a volatility risk premium. For nearby expirations, these prices have also tended to surge at the same time as VIX®. These features suggest several trading opportunities:

- Buying a VVIX portfolio returns the difference between realized and expected volatility less the volatility risk premium. Conversely selling a VVIX portfolio returns the difference between expected and realized volatility plus the volatility risk.
- To the extent that volatility expectations are unbiased, consistently selling a VVIX portfolio captures the volatility risk premium.
- Buying a short-dated VVIX portfolio can cushion losses from extreme increases in VIX® futures prices.

#### 1. Historical Behavior of VVIX June 2006 - February 2012

**Chart 1a. The VIX® and VVIX, Time Series**

**Chart 1b. The VVIX versus the VIX® Sorted in Ascending Order **

Chart 1a. is a graph of the time series of the VVIX and VIX® between June 2006 and February 2012; Chart 1b. shows the variation of the VVIX when the VIX® is sorted in ascending order. Several features are apparent from these charts and from an analysis of index returns:

**The range of values of the VVIX is at a significantly higher level than that of the VIX®.**

The VVIX ranges between 60 and 145 around an average of 86. The VIX ranges between 10 and 81 around an average of 24. Also note that the range of variation of the VVIX tends to widen at higher values of the VIX.

**Except at high values of VIX, there is little correlation between variations of the VIX® and VVIX.**

The VVIX and VIX® both reached local peaks in October 2008, during the credit crisis of 2008 and in May 2010, the "flash crash" month. In general however, the relationship between their variations is weaker than the relationship between the VIX® and the S&P 500.

**Since the flash crash of May 2010, the VVIX has rarely dropped below 80.**

This suggests that a new volatility regime came about after the flash crash. Market participants appear to have become more tentative about the future value of the VIX®.

**The following additional features of VVIX characterize its term structure and its tendency to mean revert.**

**The VVIX tends to revert to its historical mean. **

Since the VVIX is a rate variable, it is mean reverting. Table 1 shows the historical value of the VVIX 30, 60 and 90 days later, starting from different initial intervals. For example, when the initial value of the VVIX is between 30 and 50, its average value 30 days later is 78. The mean reversion is centered about the historical mean of VVIX, approximately 85 and the fixed point is the interval from 80 to 90 volatility points. Starting from this interval, the VVIX ends in the same interval. Starting from an initial value below its historical mean, the VVIX tends to end in a higher range in subsequent months. Conversely, from an initial value above its historical mean, the VVIX tends to revert to a lower range.

**Table 1 Mean Reversion of VVIX**

The term structure of the VVIX is downward sloping

Based on past history, the term structure of the VVIX is downward sloping. This is normal because a forward price is a forecast, and a short term forecast tends to be more volatile than a long-term forecast.

**Chart 2 Term Structure of the VVIX **

#### 2. Applications of the VVIX

Volatility Premium Capture

The VVIX and its term structure price the volatility of VIX® futures prices. This is useful information for trading that volatility. VVIX embeds a risk premium that can be captured by selling the VVIX portfolio on a consistent basis. The risk premium is illustrated in the bottom panel of Chart 3 where there is a positive spread between the daily value of the December 2010 VVIX and the volatility subsequently realized until the December 2010 expiration.

The top panel of Chart 3 shows that the risk premium is offset in periods when the market significantly underestimates future realized volatility, as it did in September and October 2008, the onset of the credit crisis. An investor who sensed that VVIX undervalued future volatility prior to that period could have profited by buying the VVIX portfolio.

**Chart 3 VVIX vs. Realized Volatility of VIX® Futures Price**

Points on the VVIX Term Structure yield estimates of the fair value of the VIX® at expiration dates

At date t, VVIX _{t,T} is the point on the VVIX term structure corresponding to expiration date T. VVIX _{t,T} generates an alternative estimate of the final settlement value of VIX futures expiring at T. This estimate is an approximate "fair value" for VIX® futures which is based on VIX® option prices.

Fair value_{t} =

The derivation of the fair value is based on a result of B. Dupire described in Appendix 2.

The first term under the square root sign, E_{t}[VIX_{T} ^{2}], is the expected variance of the S&P 500 from T to T+30. It is estimated from S&P 500 option (SPX) strips that replicate spot variances to different dates. The second term is a multiple of VVIX_{t,T} .

Any deviation between the fair value and the price of VIX futures is a potential signal to buy or sell VIX futures. It also suggests the following quasi-arbitrage: a spread between the portfolio of SPX options that replicates the forward S&P 500 variance VIX_{T}^{2} and a position in VIX® futures that approximately replicates the same forward variance.

The fair value of VIX May 2009 VIX® futures, the VIX® futures price, and the final settlement value are shown in Chart 4.

**Chart 4. Fair Value of May 2009 VIX® Futures **

Points on the VVIX Term Structure proxy for the standard deviation of the VIX® at expiration dates

The standard deviation expresses the degree of confidence the market has in forecasts of VIX. For any expiration of VIX® futures, the standard deviation is obtained by multiplying the value of the corresponding VVIX with the price of VIX® futures. VVIX is deannualized and expressed in decimals. The derivation is described in more detail in the Appendix.

Paths that are one standard deviation from the mean are shown in Chart 5 for December 2008 and December 2009 VIX® futures. Uncertainty about the expiration value of December 2008 VIX futures started to increase in September 2008 - the onset of the credit crisis - and still prevailed close to expiration. The picture for December 2009 is closer to the norm. Uncertainty about the expiration value decreases over the life of the futures contract.

**Chart 5 Estimated Standard Deviation of VIX®, Dec 2008 and Dec 2009 VIX® **

Options involve risk and are not suitable for all investors. Prior to buying or selling an option, a person must receive a copy of Characteristics and Risks of Standardized Options. Copies are available from your broker,by calling 1-888-OPTIONS, or from The Options Clearing Corporation at www.theocc.com. Supporting documentation for any claims, comparisons, statistics or other technical data in this document is available by visiting www.cboe.com or contacting CBOE at www.cboe.com/Contact. The information in this document is provided solely for informational purposes is not intended and should not be construed to constitute investment advice or recommendations to purchase or sell securities or futures. Investors should consult their tax advisor as to how taxes affect the outcome of contemplated options transactions. Past performance is not indicative of future results. This document contains index performance data based on back-testing, i.e., calculations of how the index might have performed prior to launch. Backtested performance information is purely hypothetical and is provided in this document solely for informational purposes. Backtested performance does not represent actual performance, and should not be interpreted as an indication of actual performance. The methodologies of the CBOE volatility indexes are owned by Chicago Board Options Exchange, Incorporated (CBOE) and may be covered by one or more patents or pending patent applications. CBOE®, Chicago Board Options Exchange®, CBOE Volatility Index® and VIX® are registered trademarks and SPX and VVIX are service marks of CBOE. Standard & Poor's, S&P® and S&P 500® are trademarks of Standard & Poor's Financial Services, LLC and have been licensed for use by CBOE.