When you get test results from an annual physical, the word “normal” is music to your ears. But “normal” also implies there’s a “not normal,” with all sorts of attendant judgments and the specter of medical TV dramas. Maybe that’s why traders become traders. We see the world differently. We’re not normal—in the best sense. To us, “strangle” is something we trade. And “net liquidating value” has nothing to do with happy hour.

So, no surprise here: theoretical assumptions that underlie much of market and options theory—e.g., market returns are normally distributed—aren’t always right. Traders get that. Going deeper, traders never forget that our assumptions are constantly challenged.

When you hear “normal distribution,” think bell curve. And when you hear “returns,” think percentage returns of a stock or index over some period like a day, week, or month.

Options theory (e.g., Black-Scholes and other options pricing formulas) assume that market returns are normally distributed, because the normal distribution is “mathematically tractable” (semi-easy to compute), and that distribution is a good representation of what we see in the markets. Options theory assumes most daily stock and index returns are more frequently closer to 0% than they are to -20% or +20%. Those +/- 20% changes are possible, but they tend to be less frequent than changes of 1%, for example.

So, theory assumes market returns (not market prices, which are assumed to have a lognormal distribution) are normal. But traders like us know the markets don’t always behave with predictable happy endings. We see stocks take off in rallies, careen and crash, gap higher and lower, or have disruptive series of consecutive up-or-down days that make it tough for anyone to describe any of this as a normal distribution.

## The New Normal

Enter kurtosis, the term used to describe how data can deviate from the normal distribution. The red line in Figure 1 is a normal distribution without any “excess kurtosis.” The blue line has less kurtosis, and the green line has more kurtosis. Although the data is always most frequent around the middle, the green distribution has more data frequency in the “wings.”

Kurtosis describes data that has a fatter tail or taller peak in a normal distribution. And market data can exhibit those fatter tails, which are larger percentage-price changes— more frequently than a normal distribution would suggest. For example, if the normal theoretical distribution suggests that five out of 1,000 price changes will be greater than -10%, but in real market data you see eight out of 1,000 price changes greater than -10%, you can say the market data has “excess kurtosis.” When the distribution has “fat tails,” you call it “leptokurtic.” Try saying that 10 times fast.

You can test data (e.g., stock returns) for kurtosis. If you’re one who gets turned on by math formulas, test percentage price change data using the kurtosis formula. If it’s too dense for you, there are other ways.

Kurtosis [X] = [E [(X - µ )^{4} ] / σ^{4}] - 3

Just so you know what the symbols are,

µ =E [X]

σ = E[( X -µ)^{2} ]

Kurtosis is the fourth “moment” of the normal distribution (the first being its mean, the second its standard deviation, the third its skewness). It is by nature backward looking because it uses historical returns.

That’s why calculating the kurtosis of stock or index returns isn’t nearly as helpful as seeing the market’s interpretation of kurtosis—the implied volatility (IV) skew. Really, then, kurtosis is just a name for what we observe in the markets. We have bigger price changes than we expect (think 1987 and 2008 crashes), and naturally make certain adjustments to our strategies and the way we look at options prices. For example, some institutions buy far out-of-the-money (OTM) puts as a hedge for a crash they don’t think will happen, but could.

Kurtosis may be foreign to you. But you do know that lurking big price changes surprise everyone, while you see institutions buying OTM options as hedges. In response, market makers increase the price of those OTM options, which in turn increases the options’ implied volatility (“vol”) or IV. Then you may see this flashing across your screen: “I’m sorry, the test for kurtosis suggests your distribution isn’t normal.” You’re skewed.

## What Now?

In a Black-Scholes normal distribution world, a single vol input should accurately price all the options on a stock or index. But market makers, with their intuitive knowledge of kurtosis, set the prices of the OTM options higher than the theoretical value of the single vol Black-Scholes model.

Consider this: If the at-the-money (ATM) IV for a $100 stock is 25%, and you use that 25% to price the options with 60 days to expiration at the 90 strike, the 90 put would have a theoretical value of 0.72. But say that put’s market price is 1.20. That would make its implied vol 30%. That higher implied vol is a signal of how likely the market considers a large potential price change for the stock. If the market doesn’t anticipate larger price changes—up or down—the further OTM options have lower values and lower implied vols. If the market anticipates larger price changes, like it might around earnings or a news event, the further OTM options have higher values and higher implied vols. Kurtosis is why we see implied vol skew.

In fact, the story goes that implied vol skew was born during the ’87 crash. Before that, market makers were cool with pricing a stock’s or index’s options with a single vol. Why? Most of those guys weren’t around in 1929, and hadn’t seen the market drop that much in a single day. Then Black Monday happened, wiped out a whole bunch of traders, and taught the survivors about the potentially higher frequency of large price drops. Since then, OTM options have had higher implied vols because traders understand that a crash could happen at any time and naturally price those options higher.

To see the impact of kurtosis on options in action, go to the **Trade** tab on your thinkorswim^{®} platform by TD Ameritrade and look at equidistant OTM calls and puts in the same expiration (Figure 2). For example, if the stock is $100, look at the 90 puts and 110 calls. If you see the 90 puts trading for 1.10, and the 110 calls trading for 1.00, that suggests the market anticipates the stock is somewhat more likely to have a 10-point drop than a 10-point rally. The market expects the distribution of returns could have a slightly fatter tail on the downside.

Alternatively, if the 90 puts were $1.50, and the 110 calls $1, this suggests the market’s fear of a $10 drop is higher than the expectation the stock could rise $10. In this case, the market expects the kurtosis—deviation from the normal distribution—to potentially be much larger. The downside tail would be expected to be fatter than the upside tail.

Just because OTM options are often priced to reflect what the market anticipates doesn’t mean that bad news always, or necessarily, happens. On the other hand, if options prices indicate the market is calm and no large price changes are expected, just remember the market often has a mind of its own. Stay alert. Normal is often a mirage.

So, the expectation of kurtosis in the distribution of stock returns and indices can boost the relative theoretical values of OTM options. And that can help guide your choice of strategy. As a trader, you might be tempted to sell those naked options short, believing this “kurtosis” thing won’t happen to you. If you make that mistake, you could lose your entire account value. Look at 2008. That was a big drop that placed historical market data miles from a nice, clean normal distribution.

That’s why using defined-risk verticals when you see higher implied vols (that is, bigger expected kurtosis) could be a smarter choice, albeit with a higher commission. If you’re bullish, for example, and you see much higher implied vol for OTM puts, a short put vertical that’s short an OTM put and long a further OTM put, can still take advantage of the elevated put prices. But its defined-risk nature—max risk being the difference between the strikes minus the credit received—means that even if the market crashes, the loss, while bad, is not necessarily devastating.

## Keep 'Em Small

So, you begin to understand that larger, unforeseen (aka “black swan”) price changes could wipe you out. But suppose you still want to sell naked options. Consider selling further OTM options to give the stock or index more room to drop or rise before it passes the breakeven point of the short options. Yes, you’ll have a smaller potential profit selling a further OTM option, regardless of its implied vol. But that extra amount OTM could give you some cushion. Remember, though, it only takes one big price change to cause catastrophic losses on short naked options. Keep these positions small and monitor them closely.

Kurtosis sounds like a scary virus, but it’s just fancy market geek-speak for something that, at times, could be quite revealing regarding market data. Don’t worry too much about theories and formulas and the numerical kurtosis. Focus on seeing hard evidence before you put on an options trade, and adjust your strategy thoughtfully. Your doc may not give you amazing news, but you likely have recourse. As always, don’t doubt the messenger.