Stuart Barton I July 11, 2017

**Summary**

VIX futures term structure is seen as a major driver of VIX ETP performance

Demand for long dated S&P 500 volatility products is often cited as accounting for the 'Volatility Risk Premium'

Vega convexity is often overlooked in the analysis of VIX ETPs

This article follows up on my earlier article Free Money? Explaining The VIX Term Structure available __here__. In that article I explained the primary driver behind the upward slope in the VIX futures term structure. I pointed out how VIX futures ultimately derive their price from the prices of S&P 500 options and how demand for billions of dollars of longer dated Vega exposure through products like variable annuities has served to inflate the price of longer dated volatility products. In this article I’ll advance the discussion one step further and explain how an upward sloping S&P 500 volatility term structure translates into an upward sloping VIX futures curve, and explain why VIX futures prices differ from implied S&P 500 option volatility. Each of these are important concepts to understand, help explain the source of VIX futures contango, and ultimately the source of decay and accrual in VIX Exchange Traded Products (ETPs) like the iPath S&P 500 VIX Short-Term Futures ETN (NYSEARCA:VXX) and VelocityShares Daily Inverse VIX Short-Term ETN (NASDAQ:XIV).

It is important to remember that the VIX Index and its futures don’t exist in a vacuum, but, as I’m sure most readers will be aware, are closely related to the prices of S&P 500 options through the volatility that the prices of those options imply. The VIX is priced from a portfolio of options defined by the CBOE, and the futures price from the market’s expectation of where the VIX Index will settle at expiration.

Connecting these VIX futures with S&P 500 options are the proxy hedges that can be made between VIX products and portfolios of S&P 500 options, and this relationship works to link these two products by arbitraging away profitable opportunities. In the simplest case this explains why an upward sloping S&P 500 curve translates into an upward sloping VIX futures curve. But there is more to know. The volatility that these two products reference is quite different and it is important to understand why. The clue is to be found in a term called ‘Vega Convexity’. Let me explain.

Vega is the (not so Greek) symbol used to express a financial product’s exposure to implied volatility. $1M of Vega would account for a $1M profit or loss for every one point change in implied volatility – say 15% to 16%. For many products this is a somewhat linear metric. For example, a volatility swap that pays out based on a formula multiplying changes in volatility by a fixed dollar amount, like our example above, may be $1M for every one vol point change. VIX futures demonstrate a somewhat similar profile to this.

The Vega on S&P 500 options however does not always act in this linear fashion, and Vega Convexity can account for a significant portion of an option’s exposure to implied volatility. For example, a 75% S&P 500 put option (i.e. 25% out of the money) might trade with an implied volatility of 20%, and, if we bought enough of these options to have a $1M of Vega exposure, we might reasonably expect the market value of our options to vary by $1M for every vol point change up or down. However, because of Vega Convexity, this linear relationship will not hold. Instead, if implied volatility were to rise, the Vega exposure on these options would also rise, perhaps increasing from their initial $1M of Vega to say $1.1M, and if implied volatility was to rise again, this exposure may also rise again. This increase in Vega exposure with changes in implied volatility is called Vega Convexity and accounts for an option’s non-linear payout with respect to volatility.

Vega Convexity comes about because the payoff from products like options and Variance Swaps (the topic of my next article) are linear in variance rather than volatility. For an option, this relationship can be seen in the ** d1** expression in the standard Black–Scholes formula for calculating the price of European options with variance being the square of standard deviation or σ in the equations below.

So this explains why options’ exposure to implied volatility is not linear, and why their payoffs exhibit a convexity in ‘volatility space’ - as the quants would put it.

This is an important concept to understand when considering the relationship between the implied volatility term structure of S&P 500 options and the VIX futures term structure. Although there is a constant ‘arbitrage’ between S&P 500 options and VIX futures, the difference in these two payoff types explains why the price and shape of the VIX futures curve often differs form the equivalent options term structure. Furthermore, understanding this difference offers us a conceptual tool to calculate by how much these two term-structures may differ. Perhaps an example will help consolidate things so far.

If a trader was to hedge a VIX future with a portfolio of options, the payoff from the VIX future would be linear while the payoff from the options hedge would be convex. If the trader was short the linear product (VIX future) and long the convex product (option portfolio) at an equal price in volatility terms, they would profit from volatility moves as the convex product outperformed the linear product as implied volatility rose, and underperforming the linear product as volatility fell. This is known as convexity trading.

So if profit from convexity can be captured, what is it worth, and how much more should a convex product be worth than a linear product like a VIX future? Well, the secret here is to consider what convexity might be worth over the life of the trade. If each time volatility spikes, and the dollar amount of Vega on a convex product increases, a trader might be able to sell that Vega, and then buy it back more cheaply if implied volatility subsequently fell. This is a fantastic sell high and buy low cycle that convexity traders love. For example, if you bought $1M of option Vega across a series of strikes and simultaneously sold $1M worth of VIX futures, each time volatility spiked the profit from the options would rise far more quickly (at a rate squared) than the loss on the VIX future (linear) and this might allow the trader to sell some Vega at the new higher implied volatility levels. Doing this repeatedly is known as convexity trading, or sometimes referred to as trading the volatility of volatility (vol of vol). More vol of vol allows the buy low sell high strategy to be executed more frequently and thus a long convexity position is also considered a long vol of vol position. While most traders use variance swaps rather than option portfolios, the outcome is much the same and this trade accounts for millions of dollars of Vega trading in the US markets each day.

The price of this convexity exposure (sometimes called Vomma) therefore depends on the market’s expectation of future vol of vol. If the vol of vol is expected to rise, the price of products with Vomma exposure will also rise, as traders become willing to pay up to gain exposure to that vol of vol. As expectations of future vol of vol fall the opposite applies and the price of Vomma falls. One good measure of vol of vol in the VVIX index, a kind of VIX of the VIX, deriving its value from the implied volatility of VIX Index options. This index typically prices between 70 and 120, and the last time I checked implied a volatility of the VIX (a vol of vol) of around 96.[1]

So how does this effect contango in the VIX futures market? Well, as explained so far, VIX futures should trade at a discount to the S&P 500 options term structure and that discount depends on implied vol of vol and time to maturity, with a higher implied vol of vol should account for a larger discount. Research delivered by a large hedge fund at the CBOE Risk Management Conference a few years back argued that VIX futures should trade on average at around a 10% discount to the equivalent volatility implied by the S&P 500 options market, and in VIX futures terms this might equate to about 1.5 vol points.[2] Colin Bennet in his book ‘Volatility Trading: Trading Volatility, Correlation, Term Structure and Skew’ has posited a similar discount.

But do VIX futures always trade at this discount to S&P 500 vol? Actually no. A check of S&P 500 option prices and the equivalent VIX Index price that they imply (complex but complete methodology from the CBOE in the footnote below)[3] reveals that this discount is frequently absent, and VIX futures could, at times, be considered overpriced.[4] Bennet argues this is due to a limited understanding of variance pricing among the retail clients, but is more likely related to the uncertainty in vol of vol.[5] While futures traders may introduce a vol of vol premium in the VIX futures market, option traders tend to only capture that part of the premium they believe is in excess of the risk the trade entails.

So a convexity premium can exist in the VIX futures market. For example, if a one month VIX future trades near parity with its associated options portfolio, it is likely that future is pricing at a premium. Furthermore, as time passes, and that future approaches maturity, that premium must shrink, as ultimately the future will settle against the VIX Index using the CBOE’s methodology mentioned earlier.

How does this underpricing of Vega convexity and resultant overpricing of VIX futures impact the decay on VIX ETPs? Well, because the convex products can theoretically never be worth less than the linear products, and because any premium in the linear product must decay before it expires, convexity decay is sometimes evident in VIX futures.

Furthermore, because VIX ETPs hold or track portfolios of VIX futures, Inverse VIX ETPs could benefit from periods when VIX futures are overvalued as a result of Vega convexity being undervalued.

This article was originally published on Seeking Alpha. For more Seeking Alpha articles by Stuart click __here__.

[1] Yahoo Finance, CBOE VIX Volatility index (^VVIX), July 7, 2017

[2] Kurella V. (2013) Variance and Convexity: A Practitioner’s Approach, 2013 CBOE Risk Management Conference. Bennett, C., & Gil, M. (2012) Volatility Trading: Trading Volatility, Correlation, Term Structure and Skew. Grupo Santander.

[3] CBOE, VIX White Paper. Available __here__.

[4] Bennett, C., & Gil, M. (2012) Volatility Trading: Trading Volatility, Correlation, Term Structure and Skew. Grupo Santander, pg 116

[5] Ibid.

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