Wall Street's wild swings last week helped skew both retirement portfolios and mathematical models of the financial markets. After all, a standard Gaussian function—a bell curve—would predict that such extreme dips and rises would be exceedingly rare and not prone to following one after the other on succeeding days.

Gaussian functions might be able to describe the distribution of grades in a big college class, with most students getting, say, B–/C+, and enable you to predict how many students will get A's or fail. But evidently, they do a poor job at explaining steep fluctuations in stock prices, although some economists and modelers think they are the best tool available to describe financial markets.

So can any math accurately describe market behavior and enable you to beat it? To find out, Scientific American spoke with statistical physicist H. Eugene Stanley of Boston University, a proponent of applying the approaches and concepts of physics to economics.

[An edited transcript of the interview follows.]

Can mathematical models beat markets?
They haven't yet. Science is about empirical fact. There is no question that optimistic people think they can beat the market, but they don't do it consistently with mathematical models. No model can consistently predict the future. It can't possibly be.

So what can math predict?
What you can do is predict the risk of a given event. The risk just means the chance that something bad will happen, for example. That you can do with increasing accuracy because we have more and more data. It's like insurance companies: they cannot tell you when you are going to die, but they can predict the risk that you will die given the right information. You can do the same thing with stocks. If you lose less, you get ahead of those who lose more.

Why do economists and "quants"—those who use quantitative analysis to make financial trades—have such faith in their mathematical models then?
If they're just to reduce risk, then they're very valuable. If you're worried, for example, about the segment of the Chinese economy that deals with steel, you make a model of what that whole market is all about and then you see if we did this what would likely happen. They're right some of the time. It's better than nothing.

But when they have excessive faith in these models, it's not justified. Math starts with assumptions; the real world does not work that way. Economics, which calls itself a science, too often doesn't start with looking at empirical facts in any great detail. Fifteen years ago even the idea of looking at huge amounts of data did not exist. With a limited amount of data, the chance of a rare event is very low, which gave some economists a false sense of security that long-tail events did not exist.

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.

Are algorithm-based trading programs causing these fluctuations, like in the "flash crash" in 2010, when the Dow Jones Industrial Index momentarily dropped roughly 1,000 points in minutes?
There is no question that a huge percentage of trades are done electronically by algorithms. Of course, the flash crash was triggered by that. But we had problems before [algorithm-based trading programs]. We've had lots of crashes.

The speed of a flash crash is vastly greater than the speed of crashes before. Things move fast because everybody knows everything all the time. In that sense, it's not the algorithm-based trading that moves the markets, it's that the information is so instantaneous.

Does this understanding of financial markets suggest anything about how to invest, like when to buy or sell?
It can't predict the future. The key thing is that it tells you not to listen to those who tell you now is the time to buy or sell if their advice is based on something wrong, as it sometimes is.

Is this all a result of the interlinked global financial system?
I believe so. The finances of every country are interlinked to the finances of every other country and, because they are interlinked, if one key players goes down then the other players know things aren't going to be as good. A useful analogy is coupled networks, which are far more susceptible to a cascade of failures than uncoupled networks.

Is there a remedy for preventing markets from affecting one another?
It would be nice if there were, but the system is going to be more fragile because it is so interlinked. Maybe something like a circuit breaker, something that slows trading down? There are two answers to that: one is that [a circuit breaker] is good and allows people to cool off, to realize the economy is still there. The opposite view is that we just pick up where we left off.

For something like the flash crash, you might try it out. If you lose five to 10 percent in the space of an hour, then you stop the market. Of course, stopping the market at a low point could do the opposite and convince everyone something bad is happening.

Can anything predict the market?
Let me tell you a story: two to the power of 10 is 1,024. One way to predict the market is to call up 1,024 trading places and tell half of them by week's end the market will be up and the other half that the market will be down. At the end of the week, forget about the half that knows you were wrong. Keep doing that for 10 weeks and, at the end, you will have called the market correctly for one person who will think you are a genius.

The economy is a very complex system—like the weather—that we understand bits of. You sure as heck can't decide on a Monday whether the weather will be nice on the coming weekend. No one can predict where the market will be at the end of the week.