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
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 dis-
tribution. 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 ]/ σ4] - 3
Just so you know what the symbols are,
Kurtosis is the fourth “moment” of the
normal distribution (the first being its mean,
That’s why calculating the kurtosis of
stock or index returns isn’t nearly as help-
ful 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 cer-
tain 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.
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
FIGURE 1: More or less kurtosis? The green
distribution curve, or data with a fatter tail, has more
kurtosis. The taller peak, as seen in the blue
distribution curve, has less kurtosis.
For illustrative purposes only.