Can Math Beat Financial Markets?

Mathematical models help assess risk, but woe betide those who think math can predict stock market gains and losses...  financial markets are ruled not by Gaussian functions but by power laws.

자료(Source): Scientific American (page 1, page 2, page 3), By David Biello  | August 16, 2011

※ 발췌:

(...) Why do you argue that financial markets are ruled not by Gaussian functions but by power laws—relations in which the frequency of one event varies as a power of some attribute of that event and are generally more L-shape than bell shape?

For anything that is random and fluctuating, like a financial market, a Gaussian function is a wonderful way to make a histogram of the outcome. If the things that fluctuate are not correlated at all with one another, then it's demonstrable that a Gaussian function is the correct histogram.

The catch is: in a financial market, everything is correlated. The proof of that is that if the stock market were Gaussian, then you'd never have a flash crash. A Gaussian crash would be an event that goes out to maybe five standard deviations [that is, a rarity on par with one part in two million]. In markets, this is simply not true. There are events that are 100 standard deviations. Every economist knows for sure that these rare events occur and cannot be described by a Gaussian function. The question is: What are you going to do about it?

Power laws are simply way more accurate. If you don't know the risk, you are not going to make the right decision, and the economy is at risk from these big fluctuations. It's no surprise when they come. The only reason you have to wait awhile is because they are rare. Knowing that they will happen forces anyone prudent to have a plan for what to do if it happens.

The idea that it would be a power law that describes all the events, the tails and the middle is really a major contribution. It allows one to quantify risk. You can read off a plot of the law the numerical chance for a downturn of any given size. It's very small for something that is 100 standard deviations out but not so small for something that is 10 standard deviations out. In fact, the S&P 500 fluctuations—which if they were Gaussian, would pretty much be constrained to plus or minus five standard deviations—you find, in a 10-year period, the number of events that exceed five standard deviations is not just one, it's 64. And the number that exceeds 10 standard deviations is eight, and there was one event that exceeded 20 standard deviations. It looks like a power law, and that's what it is demonstrated to be when every trade of every stock is analyzed.

There are an awful lot of rare events and they're all ignored. This is not the best way I want my retirement funds invested. (...)