Stuart Barton, PhD, CFA I August 27, 2019

Today, volatility can be traded like any other asset. VIX linked Exchange Traded Products (ETPs) and their underlying futures have become some of the most liquid instruments traded in the United States – almost $1bn a day of the popular Barclays iPath Series B S&P 500 VIX Short Term Futures ETN (VXX) trade each day for example. But unlike simpler assets, volatility and the products that are derived from them, are far more difficult to price.

Rigorous attempts at valuing volatility products stretches back almost 50 years with the publication in 1973 of ‘The Pricing of Options and Corporate Liabilities’ by Fischer Black and Myron Scholes and the model we now call the Black–Scholes options pricing model.

But a lot has changed in those 50 years. From a simple model to systematically value put and call options, today we rely on a series of complex models that Dr. Emanuel Derman described earlier this month as allowing ‘…traders to treat the volatility parameter in the model as an asset and trade it..’ __here__. I couldn’t agree more.

Most volatility traders will be familiar with Emanuel’s work. A fellow South African and Alumnus of the University of Cape Town, he is now professor of finance at Colombia University and has over the last 30 years developed and transformed our understanding of volatility from a mathematical curiosity to a growing asset class. Emanuel is credited together with Iraj Kani, for example, with introducing the world to the concept of ‘Volatility Smile’ in 1994 in their paper ‘The Volatility Smile and Its Implied Tree’ __here__. Since then, volatility ‘Smile’ and ‘Skew’ have been central in valuing volatility-based products.

Put simply, the Black–Scholes options pricing model ‘assumes that a stock's future return volatility is constant, independent of the strike and time to expiration of any option on that stock.’ Emanuel observed that this was how implied volatilities were modeled before the stock market crash of 1987. After that everything changed.

What Emanuel described in that paper was a new reality, one where traders valued the possibility of volatility rising in a falling stock market and priced options with this in mind. Rather than all options on a particular stock or index implying the same volatility, each strike implied a different volatility, with lower strike options implying a higher volatility than At The Money options (ATM). Emanuel’s observations seem obvious now, and volatility skew can be seen in all equity markets, including the S&P 500.

So if options of different strikes are priced at different implied volatilities, what does it really mean when we describe volatility as rising or falling? If ATM volatility is 20 and the 10% Out of The Money (OTM) puts (i.e the 90% strike puts) imply a 25 volatility, if spot prices fall by 10% from the 100% to 90% and ATM volatility rises from 20 to 25, has implied volatility risen? Well simply put, no!

This introduces an interesting facet of VIX linked products. The VIX index is calculated from a series of implied volatilities of S&P 500 options based around today’s ATM strike. If that spot price of the S&P 500 index moves (and thus the ATM strike moves), the options that the VIX Index references changes, and the VIX value may rise or fall accordingly. This change in the VIX is therefore the result of a change in the S&P 500’s spot price and not the result of any change in actual implied volatility.

Let’s look at an example. Say the VIX is pricing at 15 while the S&P 500 spot price is 2,800. If there is 1 volatility point of skew in the S&P 500 options market (i.e. 99% options are priced 1 volatility point above ATM options) and the S&P 500 falls from 2,800 to 2,772, the VIX will likely rise by 1 point to 16, even when volatility in the S&P 500 options market remains absolutely stable. This behavior is often called the ‘Sticky Strike effect’ and is often overlooked by less experienced volatility traders.

So how should this influence volatility trading during significant regime changes like political change or a bear market? Well, in 1999 Emanuel published a further paper ‘Regimes of Volatility’ in which he described seven regimes under which different volatility assumptions could prove useful. In summary he describes three volatility models that he believes can be used to predict volatility changes during different market conditions: the sticky strike model, the sticky delta model, and the sticky tree model.

The sticky strike model is relatively intuitive, and I have made a rudimentary attempt at describing it above – effectively if implied volatility remains constant, it is assumed that the implied volatility of any option strike remains constant while the spot price of the underlying rises and falls. This is the simplest assumption we can make.

The sticky delta model takes this simple approach one step further by assuming that instead of implied volatilities of a particular strike remaining constant, the implied volatilities of options with a particular delta remain constant. This is seen as important because strike prices of options are arbitrary, and the delta of an option may be a more important facet for mapping implied volatility. An ATM option is likely to be 50 delta no matter where spot prices move and thus in a stable volatility regime, ups and downs in spot price will result in a more consistent ATM implied volatility and thus VIX index values. If the market functioned like this, the VIX index would be a far more consistent indicator of implied volatility than it actually is.

So how should we expect implied volatility to behave in a more volatile or bearish market? Emanuel describes this under his ‘jumpy index, sticky implied tree’ regime. He characterizes this regime by an index entering a period of high volatility with ‘appreciable downward jumps’ and recommends applying his sticky tree model to predict the evolution of implied volatility over time. Under this model the implied volatility of individual strikes varies inversely with the index level, and at a rate twice that implied by the sticky strike approach! A worrying assertion if true. But what does this mean for traders using VIX linked products?

For one it may imply the VIX is undervalued during periods of volatility. While the lower strike OTM options used to price the VIX already imply volatility above the present ATM volatility through the implied skew, Emanuel’s sticky tree model argues these options are still significantly underpriced. His argument can therefore be extended to assert that during periods of volatility, the VIX index, and the products that track it are also undervalued. This may explain the recent interest in tail risk strategies – strategies that buy up small out of the money options in preparation for very large moves. These options may seem expensive today, but if Trump’s knee jerk policy making persists, Dr. Emanuel Derman’s ‘jumpy index, sticky tree’ analysis may explain why these options may prove extremely good value for money.

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