The (Mis)Behavior of Markets

by Benoît B. Mandelbrot

there is no before-and-after difference, then whatever dependence was originally present must have been negligible. Result: a measure for long-term dependence.

Or, in the terms of the Biblical metaphor, a Noah Effect produced by Joseph-style dependence.

That same pattern, of overshooting and crash, recurs—incessantly. Overshoot, then crash, Joseph Effect, then Noah Effect,

But that is little consolation to anyone living in real markets. They cannot forecast the next day’s weather with certainty—and so can never forecast when, exactly, the bubble will burst.

Speed it up during the boring parts, when there is little new information to digest.

with a normal price chart, and break it down into its two primitive components: one process that deforms time and another that generates a

By contrast, a multifractal has more than one scaling ratio in the same object—some parts of the object shrink quickly, others slowly.

In summary: The family starts with the parents. The father takes clock time and transforms it into trading time. The mother takes clock time and changes it into a price. Merged together, the baby takes the father’s trading-time and converts it into a price by the rules the mother provides. Last step: Use the new, baby generator to make a full fractal

The mother generator at top right is a Brownian motion, in conventional clock time—as

The wild price variability is settling down. These two bounds, below two hours

From the very beginning, in 1963, some economists had pointed out that the degree of wildness—the fatness of the tails—appeared to diminish as you looked at returns over longer and longer time-periods, from a day to a year to a decade. The common wisdom in economics was, and in some circles still is, that I may be right that

daily or weekly prices do not follow the standard model, but who cares? Most people, goes the argument, buy and hold for months, years, or decades—and in those time-scales, the conventional models work just fine. There is a fallacy in this, of course. Most people also do not contract HIV and then develop AIDS, but the few percent who do get it are very glad that the pharmaceutical industry has taken the time and expense to develop the necessary drugs to keep them alive longer. More importantly, the multifractal model successfully predicts what the data show: that at short time-frames prices vary wildly, and at longer time-frames they start to settle down.

But these papers miss the point. They assume that the “average” stock-market profit means something to a real person; in fact, it is the extremes of profit or loss that matter most. Just one out-of-the-average year of losing more than a third of capital—as happened with many stocks in 2002—would justifiably scare even the boldest investors away for a long while.

helps explain why so much of the world’s wealth remains in safe cash, rather than in anything riskier.

Japanese households keep 53 percent of their financial assets in cash, and barely 8 percent in shares

Unlike a broker, most investors do not care about “average” returns. For them, the rare, out-of-the-average catastrophes loom larger.

But if prices vary wildly, as I showed in the cotton market, the odds of ruin soar: They are on the order of one in ten or one in thirty. Considering

3. Market “Timing” Matters Greatly. Big Gains and Losses Concentrate into Small Packages of Time.

From 1986 to 2003, the dollar traced a long, bumpy descent against the Japanese yen. But nearly half that decline occurred on just ten out of those 4,695 trading days.

46 percent of the damage to dollar investors happened on 0.21 percent of the days. Similar statistics apply in other markets.

What matters is the particular, not the average. Some of the most successful investors are those who did, in fact, get the timing right. In the space of just two turbulent weeks in 1992, George Soros famously profited about $2 billion by betting against the British pound. Now,

A prudent investor would do as the Wall Street pros: Take a profit.

Prices Often Leap, Not Glide.

Continuity is a common human assumption.

Financial prices certainly jump, skip,

Time is different for every investor. Each time-scale you consider,

fractal analysis is that the same risk factors, the same formulae apply to a day as to a year, an hour as to a month. Only the magnitude differs, not the proportions.

found all the price variations followed the same statistical properties for days over a few decades and for months over eighty years.

There is, in finance, no barrier like that between the subatomic laws of quantum physics and the macroscopic laws of mechanics.

Time does not run in a straight line, like the markings on a wooden ruler. It stretches and shrinks, as if the ruler were made of balloon rubber.

In Pareto’s case, the scaling formula means that the odds of making more than ten billion once you make more than one billion are the same as those of making more than ten million once you make more than one million. With financial prices, scaling means that the odds of a massive price movement given a large one are akin to those of a large movement given a merely sizeable one. In both cases, the proportions are controlled by a scaling exponent,

Such is the confusion of scaling. It makes decisions difficult, prediction perilous, and bubbles a certainty.

It is an art to which the subjective judgment of the chartist matters more than the objective, replicable verdict of the numbers.

The persistent variety, especially those with an H exponent near 0.75, are especially curious, and these are the type common to many financial and economic data series.

But it does say that, when examining price charts, we should guard against jumping to conclusions that the invisible hand of Adam Smith is somehow guiding them. It is a bold investor who would try to forecast

Forecasting Prices May Be Perilous, but You Can Estimate the Odds of Future Volatility.

Markets are turbulent, deceptive, prone to bubbles, infested by false trends.

The fact that prices fell yesterday does not make them more likely to fall today. It remains possible for the absolute changes to be dependent: A 10 percent fall yesterday may well increase the odds of another 10 percent move today—but

If so, the correlation vanishes, in spite of the strong dependence.

Large price changes tend to be followed by more large changes, positive or negative. Small changes tend to be followed by more...

A fund manager or investor who cannot tolerate the risk of a large loss might, when the financial storm signs are up, simply trim his sails and avoid bold bets. And

You can measure the intensity and path of a hurricane, and you can calculate the odds of its landing; but, as anyone who lives on the U.S. Eastern Seaboard knows, you cannot predict with confidence exactly where it will land and how much damage it will

You cannot beat the market, says the standard market doctrine. Granted. But you can sidestep its worst punches.

Things sell below cost all the time. The price of a dress can drop 90 percent, simply by moving it from the shop window at the start of the season to the basement clearance rack at the end of the season.

London house prices more than doubled. So divorced from any idea of intrinsic value did property become that one developer

But the turbulent markets of the past few decades should have taught us, at the least, that value is a slippery concept, and one whose usefulness is vastly over-rated.

The prime mover in a financial market is not value or price, but price differences; not averaging, but arbitraging. People arbitrage between places or times. Between places: I had a friend who made his

A scalper buys a block of tickets today, and hopes to profit next month by reselling them dearly once the show is sold out.

Bouchaud’s method takes it as given that prices exhibit long-term dependence, have fat tails, and scale by a power law.