Tag Archives: Benoit Mandelbrot

Remembering Benoît Mandelbrot



Benoît Mandelbrot
the discover/inventor of fractals has died at 85. His work has been hugely influential in areas as diverse as computer graphics, finance and ecology.

In computer graphics fractals have been used to produce more realistic landscapes and vegetation, in finance his work as inspired people such as Nassim Taleb and others to think about the distribution of events, and in ecology fractals have been extensively used to understand the scaling of landscapes.

Mandelbort described his own career as a fractal:

“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”

There have been obituaries in the New York Times Benoît Mandelbrot, Novel Mathematician, Dies at 85, The Telegraph (UK), the Guardian, NPR, and The Atlantic.

Below are some links to his work:

Some Mandelbrot obituaries and appreciations have been published in The Telegraph (UK), The New York Times and The Atlantic.

Mandlebrot is probably most famous for the Mandelbrot set seen above and many version of which are seen below. R code to generate a mandelbrot set is here.

Economist on fat tails and finance

A special report on the future of finance in The Economist Fallible mathematical models: In Plato’s cave:

… although the normal distribution closely matches the real world in the middle of the curve, where most of the gains or losses lie, it does not work well at the extreme edges, or “tails”. In markets extreme events are surprisingly common—their tails are “fat”. Benoît Mandelbrot, the mathematician who invented fractal theory, calculated that if the Dow Jones Industrial Average followed a normal distribution, it should have moved by more than 3.4% on 58 days between 1916 and 2003; in fact it did so 1,001 times. It should have moved by more than 4.5% on six days; it did so on 366. It should have moved by more than 7% only once in every 300,000 years; in the 20th century it did so 48 times.

In Mr Mandelbrot’s terms the market should have been “mildly” unstable. Instead it was “wildly” unstable. Financial markets are plagued not by “black swans”—seemingly inconceivable events that come up very occasionally—but by vicious snow-white swans that come along a lot more often than expected.

This puts VAR in a quandary. On the one hand, you cannot observe the tails of the VAR curve by studying extreme events, because extreme events are rare by definition. On the other you cannot deduce very much about the frequency of rare extreme events from the shape of the curve in the middle. Mathematically, the two are almost decoupled.

The drawback of failing to measure the tail beyond 99% is that it could leave out some reasonably common but devastating losses. VAR, in other words, is good at predicting small day-to-day losses in the heart of the distribution, but hopeless at predicting severe losses that are much rarer—arguably those that should worry you most.

When David Viniar, chief financial officer of Goldman Sachs, told the Financial Times in 2007 that the bank had seen “25-standard-deviation moves several days in a row”, he was saying that the markets were at the extreme tail of their distribution. The centre of their models did not begin to predict that the tails would move so violently. He meant to show how unstable the markets were. But he also showed how wrong the models were.

Modern finance may well be making the tails fatter, says Daron Acemoglu, an economist at MIT. When you trade away all sorts of specific risk, in foreign exchange, interest rates and so forth, you make your portfolio seem safer. But you are in fact swapping everyday risk for the exceptional risk that the worst will happen and your insurer will fail—as AIG did. Even as the predictable centre of the distribution appears less risky, the unobserved tail risk has grown. Your traders and managers will look as if they are earning good returns on lower risk when part of the true risk is hidden. They will want to be paid for their skill when in fact their risk-weighted returns may have fallen.

Turbulence and Finance: Taleb and Mandelbrot

On PBS’s NewsHour, Paul Solman interviewed Nassim Nicholas Taleb and Benoit Mandelbrot about how strongly coupled systems can produce unpredictable turbulence.  They strike very resilience oriented themes – narrow over-optimization leading to a loss of resilience.

PAUL SOLMAN: In the [Black Swan], Taleb wrote, “The increased concentration among banks seems to have the effect of making financial crises less likely. But when they happen, they are more global in scale and hit us very hard. True, we now have fewer failures, but, when they occur, I shiver at the thought.”

NASSIM NICHOLAS TALEB: The banking system, the way we have it, is a monstrous giant built on feet of clay. And if that topples, we’re gone.

Never in the history of the world have we faced so much complexity combined with so much incompetence and understanding of its properties.

PAUL SOLMAN: But there’s been complexity before. There has been overextension of credit before. We’ve had crashes in American history many times before. We’re a resilient system. Won’t we pull out of it?

NASSIM NICHOLAS TALEB: Let me tell you why it’s not like before. Look at what’s happening. The world is getting so fragile that a small shortage of oil — small — can lead to the price going from $25 to $150.

PAUL SOLMAN: A barrel.

NASSIM NICHOLAS TALEB: A barrel. A small excess demand in an agricultural product can lead to an explosion in price.

We live in a world that is way too complicated for our traditional economic structure. It’s not as resilient as it used to be. We don’t have slack. It’s over-optimized.

PAUL SOLMAN: What do you mean by “over-optimized”?

NASSIM NICHOLAS TALEB: Let me tell you what is happening in the ecology of the banking system. They’re swelling to large banks, OK, because it’s vastly more optimal to have one large bank than 10 small banks. It’s more efficient.

PAUL SOLMAN: Well, we’ve certainly seen the consolidation of the industry.

NASSIM NICHOLAS TALEB: Exactly. And that consolidation is what’s putting us at risk, because we are — when one bank, large bank makes a mistake, OK, it’s 10 times worse than a small bank making a mistake.

PAUL SOLMAN: So, getting back to your fundamental work and insight, this is a system that can become turbulent or is inherently turbulent, that doesn’t have enough of a buffer, and that’s the danger?BENOIT MANDELBROT: That is not well-understood. In fact, that is misunderstood for which tools have been developed which assume that changes are always very small.

If one of them comes, nothing bad happens. If several of them come together, very bad things have happened. And the theory does not take account of that, and the theory doesn’t take account of very large and sudden changes in anything.

The theory thinks that things move slowly, gradually, and can be corrected as they change, whereas, in fact, they may change extremely brutally.

NASSIM NICHOLAS TALEB: Now you understand why I’m worried. I hope I’m wrong. I wake up every morning — actually, I don’t wake up every morning now. I start to wake up at night the last couple of weeks hoping that I’m wrong, begging to be wrong.

I think that we may be experiencing something that is vastly worse than we think it is.

PAUL SOLMAN: And we think it’s pretty bad.

NASSIM NICHOLAS TALEB: It’s worse. Of all the books you read on globalization, they talk about efficiency, all that stuff. They don’t get the point. The network effect of that globalization, OK, means that a shock in the system can have much larger consequences.

via Global Guerrillas