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# Take two - volatility, variance and higher levels thereof

## Take two - volatility, variance and higher levels thereof

**In our first article of our “VSTOXX® series” we highlighted VSTOXX® Futures and Options and the properties of “vol of vol”, stressing the differences to equity volatility in relation to skew and kurtosis. This second article takes a step back to present the different forms of volatility. We then take two steps forward to show how these tools can be applied to measure volatility of volatility, and review products along their merits and nuances. **

Rex Jones and Axel Vischer

In the fall of 2008, investors came to realize that many traditional diversifiers do not always deliver, as for example, commodities experienced severe price drops, as did the portfolios they were supposed to diversify. As correlations trended towards a level of 1 across previously uncorrelated assets, investors took particular note of market volatility and the returns achievable with related derivatives. Volatility gauges such as the VSTOXX® Index are negatively correlated to the EURO STOXX 50® Index with a correlation factor of about -0.75 on a 10-year basis. Also, this negative correlation is robust and intensifies in times of market stress. Academic research like “The Benefits of Volatility Derivatives in Equity Portfolio Management” by the EDHEC-Risk Institute shows how adding long volatility positions to equity portfolios stabilize returns. Back-testing and recent years of trading experience highlight the cost of equity market protection with index puts, and that volatility derivatives provide an efficient alternative investment tool for portfolio managers. In the last five years, volatility has emerged from a niche to a broader client base seeking uncorrelated returns in increasingly challenging markets.

Let us start out by looking at different types of volatility observable at market. Historical shows realized deviations while implied volatility derived from listed index options looks ahead how the market prices its expectation of volatility for the available options expiries. The figure below shows, for example, both realized and implied volatilities for the EURO STOXX 50® Index.

The VSTOXX® averages 3.3 percentage points higher than the realized volatility of the EURO STOXX 50® Index. Considering an average realized volatility of the EURO STOXX 50® Index of 19.6 percent between 2011 and 2013, the average volatility premium for the short-dated EURO STOXX 50® Index Options lies at 16.7 percent.

First things first

In table 1 below we take the next step and associate both implied and realized volatility with tradable products. The column labelled “Current “vol” products” lists products using the respective volatility either as an underlying or from which’s market prices volatility is derived from.

In particular, implied volatility is the price determinant for EURO STOXX 50® Index Options. Implied volatility is also the underlying of VSTOXX® Futures and Options that settle against a volatility forecast, the rolling 30-day implied volatility index derived from EURO STOXX 50® Index Options prices. Realized volatility, on the other hand, can be traded through products such as OTC variance & volatility swaps and listed Variance Futures.

Let us conclude this section on “current vol products” with an example of how these products are currently used in the market. A directional oriented investor is particularly interested in both the timing and execution in listed futures and options that settle against implied volatility – our VSTOXX® Futures and Options. The more recent market stress observed in October 2014 illustrates this challenge. Let us assume an investor entered a long volatility position in the first week of September as a tail risk hedge against an equity market downturn.

We see that implied volatility rose in the run up of market moving events, and that the VSTOXX® Index and VSTOXX® Futures prices then fell off sharply from these peaks as uncertainty became a fact. To maximize trading profits, investors in implied volatility products must actively manage positions and exit from strategies when they perceive an event to have transpired. Besides the roll costs associated with maintaining a long futures position, calendar spreads in VSTOXX® Futures can help to mitigate this timing risk as the spread between the front and second month VSTOXX® Futures widened and stayed wide. However, realized products offer investors additional exposure, at which daily observed volatility or variance levels are counted towards the payoffs of respective listed and OTC derivatives. They provide additional returns and performance in the aftermath of a market shakeup, as realized daily returns are also accrued when volatility is receding from peaks, but is still trading at elevated levels.

Two steps forward

After defining realized and implied volatility, as well as a first set of products designed to provide explicit exposure to these risk measures, we move on and investigate new potential products related to volatility of volatility. This means to combine the term “volatility of volatility” with labels like “realized” and “implied” ending up, for example, with something called “implied volatility of implied volatility” or “implied volatility of realized volatility”. What exactly are these measures and how could we use them to either manage our risk or gain new and interesting exposures? The right column of Table 1 provides an overview of the possible “combinations” as well as an indication which products could potentially be explored to trade them.

For cash and index derivatives trading, investors analyse implied volatility seeking price guidance and potential future direction. Likewise, as volatility investors, we focus on available market information derived from listed and OTC volatility trading instruments, as an efficient positioning in these products depends highly on prevailing market levels. This is best expressed in terms of “implied volatility of implied volatility”, the volatility implied out of VSTOXX® Options. We extracted this information in our first paper of this series in terms of at-the-money implied volatility and extend it here by applying the calculation methodology used for the actual VSTOXX to the proposed Volatility of VSTOXX® Index.^{1} In particular this requires a summation over the set of out-of-the-Money VSTOXX® option prices as outlined in the VSTOXX Index Guide.^{2 }

Next, there is realized volatility of implied volatility that is calculated by means of the historical volatility of the VSTOXX® Index. Figure 3 surprises at a first glance, as the implied volatility of implied volatility is not higher than the realized volatility of implied volatility.

This stands in contrast to the better known relationship in options trading that implied volatility is typically higher than realized volatility as also shown and discussed in Figure 1 for EURO STOXX® 50 Index Options, as sellers of implied volatility demand a premium to compensate them for the risks they take.

The selection of the correct underlying is required to better understand what is going on when comparing implied versus realized for "vol of vol". The Volatility of VSTOXX® Index is derived from the square root of implied variance from the first two VSTOXX® Index Options expiries. At this point, it is helpful to revisit the mechanics of the VSTOXX® Index in conjunction with the respective futures and options, as comparing implied volatility from the Volatility of VSTOXX® Index with the realized volatility of the VSTOXX® Index itself turns out not to be consistent.

VSTOXX® Futures and Options provide exposure to forward starting 30-day implied volatility. For example, on 1 December, the VSTOXX® Index is a rolling 30-day measure of implied volatility derived from December and the subsequent January expiries of the underlying EURO STOXX 50® Index Options. However, December VSTOXX® Options will settle against the 30-day VSTOXX® Index on their expiry day, i.e. at expiry their value will be based on January EURO STOXX 50® Index Options. Hence, we must consider the underlying of the first two VSTOXX® Options expiries used for calculation of the Volatility of VSTOXX® Index for a comparison of the implied versus realized volatilities of the VSTOXX®. This is not the VSTOXX® main index, but much more the price for the forward starting implied volatility for the 30-day time frame starting at the VSTOXX® Options expiration. Luckily, there is a market price for this: the prices of the VSTOXX® Futures for the matching expiration dates.

In figure 4, we see that the average spread between the Volatility of VSTOXX® Index and the realized volatility of front month VSTOXX® Futures is 5 percentage points, i.e. a volatility premium is also demanded by sellers of implied volatility of implied volatility. Interestingly, the volatility premium for long positions in VSTOXX® Options itself is merely 7.5 percent, half as high as the average volatility premium for the short-dated EURO STOXX 50® Index Options of 16.7 percent as seen in figure 1 for first dimension volatility.

By no means does the observed lower volatility premium of VSTOXX® Options impede systematic short volatility strategies expressed with VSTOXX® Options. While the lower volatility premium implies lower relative return potential versus a short volatility strategy in EURO STOXX 50® Index Options, the median implied and realized Volatility of VSTOXX® Index levels of 77 and 70 percent highlight the alpha generation potential during the observed timeframe. Also, the maximum and minimum deviations between the Volatility of VSTOXX® Index and the realized volatility of front month VSTOXX® Futures is strikingly higher than the premium implied versus realized for the EURO STOXX 50® Index volatility, marking maximum deviations of over 40 percent of implied versus realized "vol of vol".

As a final and more general observation, the VSTOXX® Index methodology gives more weight to options with downside strikes. In our previous article on the smile of the VSTOXX® we showed how the implied volatility skew for equity options takes on the form of a smile with implied volatility trading at higher levels for up- and down-side strikes versus the at-the-money implied volatility level. Because the VSTOXX® Options smile is actually more a “half-frown”, as downside VSTOXX® puts typically trade at far lower levels than upside VSTOXX® calls due to the jump risk of VSTOXX® on the upside, the Volatility of VSTOXX® Index lacks the relative higher implied volatility contribution from downside VSTOXX® puts. The Volatility of VSTOXX® Index provides an interesting reference point for VSTOXX® Options trading, and potential futures would offer investors pure exposure to forward implied volatility of the VSTOXX®.

Implied volatility of realized volatility is a very different product than the two “vol of vol” measures previously discussed. Currently it is only available in OTC markets in the form of variance swaptions and even there it is a niche market. Implied volatility of realized volatility of the EURO STOXX 50® Index could be extracted from potential options structured similarly to the recently listed Eurex Variance Futures, for example as options on these futures.

Finally, there is “realized volatility of realized volatility”, a measure which may seem too distant from the underlying EURO STOXX 50® Index, but it entails trading relevant information when considered together with the “implied volatility of realized volatility” as price determinant of OTC options on variance.

Summary

Volatility is a statistical measure derived from an underlying financial asset like an equity index. It also serves as the starting point for a range of higher order risk measures in the dimensions of implied and realized volatility. We highlighted established and growing products for the first dimension of volatility. We also discussed the characteristics of yet higher orders of volatility measures both as price determinant for existing bespoke volatility products, as well as their potential future utilization as next generation risk management and trading tools. Finally, one of the products discussed, the Volatility of VSTOXX® Index, is planned to be launched by STOXX® in the very near future.^{3}

The next article in this series will investigate implied correlation derived from EURO STOXX 50® Index option prices as well as the single name options on its constituent equities.*Eurex Exchange would like to thank the Program for Financial Mathematics of University of Chicago for its analysis. *

_{1 To be precise, the Volatility of VSTOXX® Index represents the square-root of implied variance of the square-root of implied variance.2 The VSTOXX Index Guide can be found in Chapter 7 of http://www.stoxx.com/download/indices/rulebooks/stoxx_strategy_guide.pdf . For the Volatility of VSTOXX® Index calculation one has to substitute Euro STOXX 50 option prices with VSTOXX option prices.3 The final index design of the Volatility of VSTOXX Index® uses VSTOXX® Options data only if there are more than 3 days left to expiration. The time series in this paper used VSTOXX® Options data with more than 2 days left to expiration.}

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