# Inside Volatility Trading: New Year, New Probabilities, New Risk

Kevin Davitt
January 19, 2022

A new year tends to bring hope, expectations, and for some, the intent to make changes. In capital markets before the calendars flip to January there are countless efforts at prognostication. You can find a synopsis of some of the big banks’ forecasts for the S&P 500 Index in 2022 here, or here, or here.

There will never be consensus because (as we point out often) the future is always uncertain. In general, investors and the research teams at banks tend to expect the future to look like the recent past. There’s also a tendency to overweight the probability of recent – potentially rare – events repeating themselves.

This availability bias is an investigation into where probability and expectations diverge. Probability is quantitatively driven and there are discernable answers. Expectations are subjective but may be informed by a given probability as well as your risk exposure. Let’s explore how option markets are a valuable reflection of expectations, probabilities, and risk.

## Probabilities

Probability is the likelihood of a specific event/outcome occurring. It’s typically reflected in percentages (0% - 100%). Examples often reference rolling a six-sided die. The probability of rolling a six is 1/6 = 0.166 or 16.6%.

More complicated math is involved when calculating something like the probability of winning Powerball. You need to use the Combination Formula.

## Combination Formula

Where: nCr = number of combinations

n = total number of objects in the set

r = number of choosing objects from the set

The “!” symbol is a factorial, where a number is multiplied by all the numbers before it

The probability of winning Powerball (you must match five balls drawn from a drum containing balls numbered 1 – 69 and match one ball drawn from a second drum with balls numbered 1 – 26) works out to 1/292,201,338.

Combinations assume that outcome order doesn’t matter. We do not need to see the Powerball numbers appear in the same order that we selected them in; we simply need to see that all of our numbers match the numbers generated by the Powerball machine. You need to use a Permutation Formula if the order does matter and here are some practical examples.

Translation: You should not expect to win Powerball because the probability of winning is exceptionally low.

Curious to know more about probabilities? Check out this edition’s Simply Put!

## Annual Returns

Let’s turn our attention to the broad market. The visual below buckets annual S&P 500* (Total Return Index) performance by deciles. The data covers 1926 – 2021 and the highlighted number illustrates the frequency (in years) when S&P 500 Index returns fell into each performance bucket.

For example, the S&P 500 Total Return Index gained 40% or more during a calendar year five times.

## S&P 500 Index Annual Return Performance

Source: Cboe Options Institute

We can establish expectations for equity markets based on history, but expectation is not the same as probability. This calls to mind the ubiquitous disclaimer on any financial media: “past performance is not indicative of future results.”

In a game of chance, probabilities do not change. In capital markets, change is a constant. Consequently, there are unique opportunities as well as different risks.

We know based on historical data that from 1926 – 2021 the large cap equity index (S&P 500 Index®) generally returned between -20% and +20%. In 54 (green highlighted portion) of the past 95 years, annual performance has fallen within that range. However, that does not mean there’s a (54/95 = 0.5684) 56.84% probability that that pattern is followed in 2022.

There’s no “right answer” because the future remains uncertain.

Our expectations may be appropriately informed by history, but you can’t calculate probabilities the same way you do with games of chance. In the markets there are limitless potential outcomes.

## Historical S&P 500 Index Outcomes  (1926 – 2021)

• Mean annual return +11.9%
• Min/Max annual return: -43.8% & +52.6%

Stock market performance over the past five years has been exceptional (mean +19.7%). Since 2008, the S&P 500 Index has averaged a positive annual return of 16.6% with only one down year. It’s easy to calculate averages and make assumptions, but every year is an independent event. Past performance is not indicative of future results.

## Alternative

So, what’s the alternative?

Well, the options market is replete with valuable information. You need to know where to look and how to interpret the data. Below is a snapshot of the Standard December 2022 SPX options with the index measuring 4700 (January 6, 2022).

## Delta Chart

Source: Cboe LiveVol & Options Institute

What does this table tell us about probabilities and risks?

The option Greek delta can serve multiple roles. Delta represents how much an options theoretical value should change for each one point move in the underlying. Delta is also an estimate of the likelihood of the underlying expiring above/below a given strike.

For example, with the S&P 500 Index measuring 4700 the implied probability of the S&P 500 Index expiring above 5800 was 7.1%. That probability (delta) is sensitive to the existing implied volatility assumptions, which change over time. All else equal, a higher volatility assumption would yield a higher probability (and vice versa). The SPX Index measuring 5800 would represent a 23% rally (change from spot) before December 16, 2022.

What’s the implied probability of a 23% decline in spot by the same point in time?

In other words, what’s the delta of the 3600 put?

15%.

Based on this information, the implied probability of a 23% advance is just over 7% whereas a decline of 23% has a 15% implied probability. Why such divergence? Based on the first chart, it looks like S&P 500 Index returns are positively skewed on an annual basis. There are more outcomes on the right side of the median (0% return). Does the market currently reflect significant concern about inflation or Omicron?

The short answer is no, which is typical for the index options market. Let’s understand why.

## Vol Skew

Based on history, we know that equity markets tend to move up gradually. We also know that they can reprice risk and sell off quickly. There are countless examples of that dynamic. In the October edition of Inside Volatility, we focused on “Black Monday” and the emergence of index option skew.

During the early portion of the pandemic, it took about a month for the S&P 500 Index to lose 33% of its value. Then it took five months for the index to make new highs. That was a highly unusual recovery. On average, it takes two years for the S&P 500 Index to make new highs following a bear market.

On a related note, implied volatility (IV) for index options tends to decline when the market moves higher. IVs typically increase when the market sells off. The long-term negative correlation between the S&P 500 Index and the VIX Index® is a great example.

## SPX Index & VIX Index Long-Term Negative Correlation

Source: Bloomberg

The visuals highlight the negative S&P 500 Index correlation to the VIX Index. On a daily basis, over the past five years the correlation has ranged between -0.38 and -0.95. If you look at the weekly correlation over the past decade the range vacillates between -0.54 and -0.94. Correlations change over time, but typically when the S&P 500 Index moves higher, the VIX Index declines (and vice versa) (although that relationship is not always maintained).

Building off that point, if the S&P 500 Index is up 23% in December 2022, chances are implied volatility will be lower than current measures. Conversely, if the S&P 500 Index is trading closer to 3600 (down 23%), implied volatilities will likely be substantially higher.

Option greeks, including delta, are impacted by volatility assumptions. The imbalanced probability we just highlighted, where the 23% out-of-the-money (OTM) call and 23% OTM puts have a very different delta, is skew at work. This dynamic is most pronounced at “the wings,” the extreme ends of the distribution curve.

If we compare the up/down 11% strikes (4200 put and 5200 call) the deltas are only slightly different. Looking back at our annual performance by decile visual, history says the odds of up <10% are equal to down <10% for the year.

The other key factor for index options is the fact that collectively market risk is tilted to the downside. Investors of all types are asset owners. Shareholders benefit from equity index values increasing. Based on last year’s performance and data from S&P Global, roughly \$16 trillion (USD) in assets are indexed to the S&P 500 Index. Assuming linear exposure, if the broad market declines by 10%, about \$1.6 trillion in wealth is lost.

As such, there’s a natural desire for protection. There’s also a willingness on the part of some market participants to forgo some upside to finance protection. In short, there’s consistent demand for left tail (downside) protection. There’s also a steady supply of right tail (OTM calls) options used to offset the cost of “portfolio insurance.”

That combination skews the supply and demand dynamic for OTM downside relative to upside. Ergo, the volatility implied by equidistant S&P 500 Index calls and puts is different.

## Expectations & Perspective

“Life is like topography, Hobbes. There are summits of happiness and success, flat stretches of boring routine and valleys of frustration and failure.” – Calvin & Hobbes

My first favorite “author” was Bill Watterson. I grew up looking forward to the weekend for several reasons, but the Sunday comics and a new Calvin & Hobbes strip were certainly included. Bill Watterson was able to convey powerful truths about the world through the eyes of six-year-old boy. Calvin believed in shaping one’s own reality by changing either one’s own perspective or the perspectives of those around. He knew that learning was complicated and sometimes very difficult. The big picture lessons I took from Calvin & Hobbes are ones I see reflected in the markets to this day.

Expectations for economic growth and potential S&P 500 Index returns are lower when compared to years past. The probability of multiple rate hikes (based on current Fed Fund futures prices) is high.

Similarly, an options delta, which is a proxy for probability, is impacted by the option’s implied volatility. “Probability” in the context of the markets is different than pure mathematical calculations because the path and potential outcomes are infinite.

That reality gave rise to continued growth in options. 2021 was another blockbuster year for equity, ETF, and Index options. More than 9.8 billion total options were traded and cleared by the OCC. More and more people have chosen to learn about and use options to shape and define potential outcomes over a specific time frame and to gauge collective risk.

This year will likely bring some summits of success as well as valleys of frustration and failure.

Here’s to 2022. Will it look like the recent past? Time will tell.

*S&P 500 Index launched in 1957: Composite Index 1926 – 1956

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