Stuart Barton I August 10, 2017

#### **Summary**

VarSwaps make up a significant portion of the equity volatility market.

Retail investors trading ETFs should understand what underlies the volatility products they are investing in.

Knowing the dynamics behind the volatility market could help investors make more informed decisions.

A net short bias in retail volatility products may introduce a short convexity position in the wholesale volatility market.

Exchange traded volatility products like ETFs and ETNs continue to grow in popularity, and this rapid expansion into a relatively complex asset class has left many investors confused about some of the fundamental concepts needed to reasonably understand the market that underlies these popular retail products.

Equity volatility trading has been around in one form or another since 1973, the year Black and Scholes published their seminal article 'The pricing of options and corporate liabilities' and the year the Chicago Board of Trade established the Chicago Board Options Exchange, and listed the first standardized, exchange-traded stock options in the United States (Source: Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy, 81(3), 637-654).

By delta hedging those listed equity options, traders were able to extract the volatility risk component from the option and realize the volatility implied in their prices. Owners of options could profit handsomely through daily stock hedging (delta hedging) of their options during volatile periods, but risked realizing very little on quiet, less volatility days.

While hugely popular in the 1980s, 90s and 2000s, the problem with this type of volatility trading is that stocks (or index futures) need to be constantly traded to realize an option's implied volatility - selling stock as stock prices rise, and buying it back as stock prices fall. A potentially profitable activity, but ultimately a very time consuming one as well.

Enter the Volatility swap. In the early 1990s banks began offering their customers pure volatility bets - the ability to buy or sell the realized volatility of a stock or index over a fixed period of say six months or one year. In 1999 Derman et al. published the first widely accepted rigorous description of these Volatility swaps and also introduced many to the concept of a Variance Swap or VarSwap [Source: Demeterfi, K., Derman, E., Kamal, M., & Zou, J. (1999). More than you ever wanted to know about volatility swaps. Goldman Sachs quantitative strategies research notes, 41].

Simply put, a volatility swap is a contract between two parties where the seller promises to pay the buyer the realized volatility over the 'strike' price at the swaps' expiry. If the strike was 12% volatility, and the buyer owned say $100,000 per volatility point, and over the 1 year life of the product the stock (or index) realized 15% volatility, the seller would pay the buyer (15 - 12) x $100,000 = $300,000. This could be described as making three vol points on $100k of Vega.

VarSwaps differ from Volatility Swaps in that their payout is in variance units rather than volatility units. For everyone trying to remember their intro to statistics class, equity volatility is a measurement of annualized Standard deviation (σ), and variance is the square of that Standard Deviation (σ^2).

So why would anyone be interested in variance instead of volatility? Well this is perhaps one of the most important facts about the volatility market that many retail traders forget - the wholesale volatility market is dominated by trading options, and the payout of an option is linear in variance space not in volatility space. I explain this in a little more detail in a previous article available __here.__

So why does this matter to the retail volatility trader? Well, banks, hedge funds, and large market makers are far more willing to make tight prices in products they can efficiently hedge, and VarSwaps can be hedged more efficiently with options because their payouts are both in variance.

But how are these VarSwaps traded and what do retail volatility traders need to know about them? Firstly, VarSwaps are often confusingly traded with reference to volatility (rather than variance) and Vega (exposures to volatility) rather than exposure to variance. This is confusing because traders often say they bought $1m of S&P 500 VarSwap at 12 vol, when what they have actually bought is the square root of $1m exposure to the square of 12% volatility. Following me so far?

Perhaps a simple example will help. A trader believes that the realized volatility of the S&P 500 Index will be 14% over the next year, while the 1 year S&P 500 VarSwap is priced at just 12% in the market. Should he buy it? Well possibly, but first it is important to understand what he would be buying if he bought that VarSwap.

First of all, VarSwaps are swaps, that is to say, like Volatility Swaps, VarSwaps are priced according to their strike, so a swap priced in the market at 12% (in vol terms) represents the strike a trader can get exposure at for a present consideration of zero dollars paid today. This is often more accurately described as the volatility strike of the zero cost VarSwap.

So what does this 12% represent? Despite what it says on the sticker the swap is actually not for a volatility product struck at 12% (0.12), but rather for a variance product struck at 0.0144. And this odd market practice is important to understand. Because the variance product is traded in variance (σ2) and not in standard deviation (σ), as volatility rises the payout of the VarSwap rises to the square of volatility. Formulaically this payout can be expressed as the following:

That is to say, the payout at maturity will equal the difference in the actual realized variance over the life of the product and its agreed strike multiplied by the swap's variance notional.

And how much variance notional must be traded for an equivalent exposure of $1M of Vega? Well, the variance notional is simply calculated as the following:

To those who have read my previous article, it should be clear that VarSwaps are strongly convex - that is to say their payout changes not linearly with movements in volatility but instead with the square of movements in volatility. Sebastien Bossu demonstrates this well in his diagram below [Source: Bossu, S. (2016) Introduction to Variance Swaps. Available here]. As you might imagine things can get very exciting during very large volatility movements.

And why should the retail investor care about VarSwaps? Well, a great number of retail products like ETFs and ETNs are based on indexes of rolling VIX futures, and a lot of these futures are hedged by the professional market with options and VarSwaps. You may have heard people describing VIX futures for instance as volatility based forward starting VarSwaps.

Now, if end users of these products - that is to say people buying or selling them without hedging, impose a misbalance on the professional market, it is likely then that professional traders will have a spread hedge on between strongly convex products (like VarSwaps) and more linear products (like VIX futures) and this spread could bring about some interesting characteristics in the volatility market.

For example, if end users - as the press leads me to believe at the moment - are net short retail volatility products - professional traders are likely long the other side of those traders, while at the same time hedging this long volatility position with short positions in options and VarSwaps. If this is the case the professional market would be considered net short convexity.

This is important to consider, because if the professional market is short convexity the market may exhibit some interesting activity if volatility were ever to move upward quickly. For example, if a large bank with a net long ETF/ETN position has hedged with Variance Swaps, and volatility spikes, the long Vega position of the ETFs and ETN would remain relatively linear, while the short Vega position in VarSwaps would explode into life, possibly leaving that large bank beyond its agreed short Vega risk limits, and even possibly forcing it to buy more Vega in a spiking vol market.

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