If you clicked on this article looking for a critique of the movie Black Swan, starring Natalie Portman and Mila Kunis, I'm sorry to say that you're in the wrong place...but you should just stick around anyway!

The type of black swan in the lake today is of a much rarer variety: it is a statistical outlier in the markets that signals catastrophe. But where does it come from--and how can you profit from it?

The phrase "black swan" was made popular by Nassim Nicholas Taleb, a finance-extraordinaire turned writer, who sought to describe the nature of unusual events in financial markets. He gained notoriety after the crisis of 2008 when he spoke about the need to let systems break in order to be rebuilt and come back stronger (AKA- more resistant to future disasters). However, the concept of black swans has been around for much longer.

It is popular belief that prior to "Black Monday," traders did not concern themselves very much with the idea of a major crashes occurring on a 'regular basis' (that is, they generally said "if it crashes"). Afterwards, investors started to be wary of the possibility of the market going to hell in a hand-basket at any point in time, without a blaring sign (a shift to "when it crashes").

*(Before we move on, if you haven't read Alex Warfel's article on the VIX, you'll want to take a look since the VIX is a compliment to what we're about to discuss).*

Taking a step back for a moment, however, what does one really mean when they say "market crash"? Well, to take a page out of ye old statistics book (I'm sorry in advance), the stock market returns over a period can be plotted on a graph to show the general distribution over time (think normal distribution from your class). There is an average return (the mean) of the index over a period of time, as well as degrees to which the market deviates from that average (the standard deviation). The more standard deviations a return is from the mean, the less likely it is to happen (see below).

While the returns are thought to generally follow this distribution (cause apparently some are so bold as to claim that), there are occasions when returns at the tails (standard deviations of 2 or more) will be more prevalent. The possibility of this occurring is known as "tail risk," which is calculated using the price of out-of-the-money put and call options on the S&P.

As mentioned above, before Black Monday, people weren't really worried about the S&P falling too far below its current price, or even swinging too high above it in a short period of time, so prices of out-of-the-money put and call options where, to some degree, at parity (see the chart below). After the crash, the price of out-of-the-money put options increased due to a greater focus on the left-side tail risk of the market (AKA- the implied volatility curve was skewed towards puts).

For those of you who like to flex your math skills, the skewness of the S&P 500 can be found by the following equation, where $latex S$ is the coefficient of skewness, $latex R$ is the 30-day logarithmic return of the index, $latex\mu$ is the return. and $latex\sigma$ is the standard deviation.

$latex S=E[\big( \frac{R- \mu}{\sigma}\big)^3]$

Since this will give us a negative number in a very small range, we then take the value for $latex S$ and plug it into this equation:

$latex SKEW = 100-10*S$

This will give us the indexable value of SKEW that we all know and love. The value of SKEW ranges anywhere from 100 to 150, with lower values signaling that log-returns are normal, and outliers are not likely. As the value increases, the left-side tail grows "fatter" and the probability of an outlier occurring increases as well. Since it is on the left-side of the distribution, that outlier is likely to come in the form of a catastrophic market downturn.

According to the Chicago Board of Options Exchange (CBOE), the creators of the index, when $latex SKEW=115$, there is a 6% chance of a black swan, and at $latex SKEW=135$, there is a 12% chance. So, as an investor, how can you take advantage of your newly found knowledge of SKEW? One strategy is to use SKEW to gauge where the market is going, the other is to bet against the market.

If the market has priced in a high probability of a downturn (high SKEW), and you believe it, you could buy the high implied volatility options and write options with a low implied volatility (i.e.- write calls and buy puts, known as a "protective collar") If you think think the market is wrong, you would do the exact opposite, buying calls and selling puts (a regular "collar").

So go forth, find the SKEW and trade the heck out of it!