The “Crash of 1987” and its aftermath provided me with a sharp flash of awareness; one that I’ve never forgotten and one that I share with every trader who’s lived through something similar. In the derivatives business, we are surrounded by mathematical constructs, risk-metrics and financial theory solely predicated on the bell curve – or, normal distribution. Yet, once you live through a derivatives disaster, you know, deep inside, that derivatives behave in a very non-normal way.
Normal Distribution & its Moments
The bell curve (aka. normal distribution) has a unique curvature which is defined by a collection of all its individual “moments” which uniquely determines the distribution. Some “moments” describe the curves proportion while others its convexity.
The moments articulate the nature or shape of the probability distribution. Any distribution can be characterized by a number of features including: the mean, the variance, the skewness, the kurtosis, and so forth. And, whether you are an active options trader or a passive investor, you are forced to live with these higher moments each and every day.
So what are these moments, why are we living with them, and how do they affect our trading decisions?
The First Moment – the Mean
The first moment of a normal distribution seeks to determine the expected value of a variable from an average of previous values (i.e. finding the mean). Analysts, traders, and investors alike are all concerned with the problem of figuring out the likelihood or expectation of a certain outcome. An analyst, after weeks of research, places a value on a public company for a defined future period. A street-smart day trader may take the average of the last three days of XYZ and quickly conclude that XYZ will trade at such-and-such a price. In this case, both the analyst and the day trader are toying with the expected value of XYZ—the first moment of a normal distribution.
You aren’t predicting a price for a specific future date or series of dates; you are simply finding the average for past performance and using that to get an idea of where the share price might go next.
The Second Moment – Volatility
The second moment of a normal distribution estimates the variance or volatility of that asset over time. Options traders try to estimate the odds that a certain stock price will trade between, for example, $30 and $40 over the next 30 days. They seek to refine an estimated future share price into the variability of that price over time. The standard deviation—or implied volatility, in financial terms—is the second moment. The second moment attempts to capture the dispersion of values around the single expected value that you estimated in the first moment described above. Thus, the second moment captures the risk or volatility of a financial asset.
The second moment—the volatility—of the distribution is one of the most valuable notions in financial philosophy. Option values are based on this second moment. The second moment of the normal distribution is the “stress” in the orderliness which prompts asset prices to diverge from what everyone assumes them to be, thus stirring up risk in the system.
The Third Moment – Skewness of the Distribution
The third moment of a normal distribution is the skew or shape in the distribution. Options traders seek to make money by accurately predicting when the market will rally or drop. Yet, sometimes the market goes against the traders, causing them to lose money. Sometimes they confront sizeable negative returns, a situation they may not have anticipated at all. This is the skew. If a distribution has a skew—and all normal distributions in actuality do—it insinuates the ever-present possibility of large negative returns. This is actually negative skew.
Skew is the contour, or the unevenness, in a distribution, the dent in the bell curve. A negative skew suggests that the left half of the normal distribution (the left side of the mean) is twisted in such a way that the prospect of achieving negative returns is superior to that of achieving large positive returns. Recall that in a theoretically precise, ideal normal distribution, positive returns and negative returns of equivalent magnitudes have more or less equivalent probabilities. Hence the distribution is symmetrical, or balanced. Of course, a distribution can possess positive skew as well, which signifies the prospect of a large positive return.
When dealing with skew, traders strive to resolve how frequently in the trading time horizon they will obtain negative returns rather than positive returns. A skew demonstrates the relationship between the movement of an underlying asset and its volatility.
The Fourth Moment – Volatility of Volatility
The fourth moment of a normal distribution: Kurtosis or varying variance, the volatility of volatility. Mathematicians have a fearsome term for the fourth moment of a normal distribution—kurtosis. The fourth moment is a gauge of whether the distribution curve is tall and skinny or short and squat, measured up to the normal distribution of the same exact variance. It means varying volatility or, more precisely, varying variance. The fourth moment is something that all options traders can relate to, although they may not know its name or be able to provide details. It signifies volatility of the volatility of an underlying asset. What happens to the distribution of the curve when volatility changes? What occurs to the extreme downside skew if volatility changes?
A changing volatility can cause the tails of the normal distribution to become “fatter” or “skinnier” than otherwise predicted, thereby increasing the potential risk. This is the work of the fourth moment.