Living with the Higher Moments of the Normal Distribution
February 12, 2007
There was once a time, about a decade and a half ago, when only option traders used to talk about higher moments of a normal distribution. These days, almost every text on quantitative finance talks about higher moments and jargon about higher moments, especially the third and fourth moment, have even entered financial press.
The truth is, whether we are traders, punters, corporate lenders or ordinary mutual fund investors we are living with the higher moments every day. So what are these higher moments and how do they affect us? A complete answer to this question will result in a textbook, and we are no authors, but we'll try to present some simple and very concise explanations.
As you read below it will become apparent to you that odd moments of a normal distribution give us information about the symmetry of the distribution whereas the even moments tell us about the convexity of the distribution.
The question that proprietary traders often ask is that how often we will get a negative return from the asset. This question also dogs the speculator; he wants to make money as the market rallies or drops but in doing so, along the way, he often encounters market moves against him whereby his return is negative. Sometimes there could be a very large negative return, something that he may not have expected at all. He is actually, grappling with the third moment of the normal distribution, namely the skew . If a distribution has a skew-and all normal distributions in real life do, because they are not “normal”- then it means that there is a non-zero probability of realizing large negative returns. This is actually a negative skew . Skew is the shape, or the asymmetry in the distribution. A negative skew means that the left half of the normal distribution, i.e on the left side of the mean, is twisted in such a way that the probability of realizing negative returns is higher. Remember that in a theoretically correct, ideal normal distribution, positive returns and negative returns of equal magnitudes have more or less equal probabilities, which make the distribution symmetrical. Of course, a distribution can have positive skew as well, which will mean the existence of realizing a large positive return.
In common parlance, when we talk about skew we are trying to ascertain how often in our trading time span are we going to get negative returns as opposed to positive returns. Intuitively, a skew displays the correlation between the move of an asset and its volatility.
While ending the previous article we asked a question how uncertain is our uncertainty? That question will take us to the fourth moment of the distribution. Econometricians have a hideous sounding name for the fourth moment of a normal distribution. They call it Kurtosis and the phenomenon that causes a fourth moment to exist are called heteroskedacity. In simpler English it means varying volatility or more accurately, varying variance. It is difficult to tell whether the fourth moment entered the world of derivatives via standard econometric analysis of banks' research departments or that the traders always knew of the fourth moment. It was in their head all along. But fourth moment is something that every option trader, and some enterprising derivatives research analysts, can relate to. It signifies the volatility of volatility of an asset. Or one can say that the fourth moment is the gamma of that gamma, the second moment of the second moment. We have all heard of fat tails, once again the result of the existence of fat tails of a normal distribution. A changing volatility, i.e. a dispersion of the dispersion of the asset price can cause the tails of the normal distribution to become fatter than otherwise predicted, thereby increasing the risk. The product of the second moment with itself can sometimes become really scary. Some of the older and more experienced traders might have encountered very high volatility of volatility on a very small volatility of some assets (causing the distribution to become Pareto-Levy). Fourth moment is an indicator of convexity and if the fourth moment is negative it can cause many a sleepless nights.
Beyond the fourth moment things become a bit fuzzy but exotic option traders cannot escape them anyway. The fifth moment of a normal distribution measures the asymmetry sensitivity of the fourth moment. A barrier option trader whose position behaves in a more or less linear manner when the barrier is some distance away may suddenly be confronted with the prospect of markedly higher negative convexity as the market comes closer to the barrier. This is the work of the fifth moment.
The sixth moment of a normal distribution is generally associated with compound options (call on a call or a put on call, etc.). The seventh moment of a distribution can be really scary! But let's not even go there.
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