Many historical financial return series exhibit substantially more large price changes than an independent Gaussian model calibrated to their ordinary fluctuations predicts.

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average, of no change. In fact, the bell curve fits reality very poorly. From 1916 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve. The far edges flare too high: too many big changes. Theory suggests that over that time, there should be fifty-eight days when the Dow moved more than 3.4 percent; in fact, there were 1,001. Theory predicts six days of index swings beyond 4.5 percent; in fact, there were 366. And index swings of more than 7 percent should come once every 300,000 years; in fact, the twentieth century saw forty-eight such days. Truly, a calamitous era that insists on flaunting all predictions. Or, perhaps, our assumptions are wrong. ## The Power of Power Laws

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Second, contrary to orthodoxy, price changes are very far from following the bell curve. If they did, you should be able to run any market price's records through a computer, analyze the changes, and watch them fall into the approximate normality assumed by Bachelier's random walk. They should cluster about the mean, or average, of no change. In fact, the bell curve fits reality very poorly. From 1916 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve. The far edges flare too high, too many big changes. Theory suggests that over that time, there should be 58 days when the Dow moved more than 3.4%. In fact, there were 1,001. Theory predicts six days of index swings beyond 4.5%. In fact, there were 366. And index swings of more than 7% should come once every 300,000 years. In fact, the 20th century saw 48 such days. Truly, a calamitous era that insists on flaunting all predictions.