Kurtosis= What?

When we have a stock, we like to look at the distribution of its returns to figure out the likelihood of certain outcomes. The 4 main things this distribution tells us is the average, or the mean, the standard deviation, 0r how spread out the data is, skew, or which way the data is “biased”, and kurtosis.

Kurt-toe-what?

In technical terms, kurtosis is the measure of tailedness of random variables. It accounts for how much of the difference from the average is in large swings, or small common differences. If we were looking at a normal distribution, it would be little movements around the peak, or the large movements out to the side.

So kurtosis measures the “height” of the distribution compared to the “spreadout-ness”. Kurtosis is about large shifts versus small shifts, not the direction of the move (which is measured by skew).

A random variable with zero kurtosis is mesokurtic, a negative kurtosis is playtykurtic and one with positive kurtosis is leptokurtic. As far as modeling for option prices, we primarily deal with mesokurtic and leptokurtic. A high kurtosis means that extreme variations form expected value are very common.

kurtosis_2

So leptokurtic, or what we think that market follows, means that these extreme moves are more likely to occur. Mesokurtic would mean that extreme moves often just as happen as we think they would. Platykurtic, which we don’t really use, means that extreme moves happen even less than we think that they would.

BUT the large shifts are more common than we think they would be, and the small shifts are less common, leading us to a leptokurtic distribution. Taking the Black-Sholes model, we assume that the returns on stocks are normally distributed (which requires a zero kurtosis) but the market begs to differ.

People don’t always follow the assumptions that the structure of models rely on all the time (especially not Black-Sholes). They go with their own beliefs, which sometimes means “buying high and selling low” rather than “selling high and buying low”, the former of which the Black-Sholes model assumes. Because of this (and other even more complicated reasons) Black-Sholes assumes that volatility should be the same at each strike point in the stock. So XYZ priced at $40 should have the same volatility at $39, $41, and $50.

But this isn’t true.

There’s a thing called a volatility smile that shows what the market believes. This shows that investors believe that large shifts are more likely compared to small shifts, and that returns aren’t normally distributed- rather, they are leptokurtic, or positive.

volatilitysmile2

So, stock returns are heavy tailed. It’s leptokurtic.

These heavy tails mean that the underlying is going to swing big eventually. Big outlier moves happen more frequently than we think they should.

There are some big shifts in the market. So it would make sense to stay close to the stock price with your options, and stay away from the volatility smile curves (the inflection points). Or, you can go really far out, to the dimples of the smile, and take some premium there.

Another thing to remember is our market has negative skew. When it goes down, it goes down BIG. So when we deal with our positive kurtosis in the marketplace, where the big moves happen more often than we believe they should, these big moves tend to be to the downside, due to the negative skew. The market has a positive upward drift over time, but tends to skew to the negative side for the big moves.

A way to hedge yourself from this swing risk is to stay small. Carry some negative deltas to combat the “surprise” big moves. Pay attention to the numbers that the stock is giving you.

Who knew kurtosis could be so rewarding?

Disclaimer: These views are not investment advice, and should not be interpreted as such. These views are my own, and do not represent my employer. Trading has risk. Big risk. Make sure that you can balance your risk/reward, and trade small, and trade often.

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