Fractal finance

Can the infinite complexity of the financial markets be described by the simple rules of mathematicians?

January 4, 2011

How risky is the stockmarket? How rough is its ride? The traditional way to answer that question is to look at a representative period of history – say the last few years or decades – and use standard statistical methods to compute the average fluctuation over a given period. That will give you an idea of the maximum you would expect to lose during the next day or month or year.

One drawback with this approach, is that it assumes the future will resemble the past. But there is a deeper problem as well; for if you try to measure the typical price fluctuation, over different “representative” time periods, then you won’t get a consistent answer.

Imagine you have a physical object – say, a rock – and you pass it round a group of people, and everyone takes turns measuring its size, and they all get different results. Risk is a bit like that. Risk assessment techniques such as Value at Risk (VaR) are a bit like that too. No wonder financiers worry about these models.

Mathematicians have known for some time that measurement is not a straightforward process – especially when the object being measured is not straight. If you measure a smooth curved surface, like the circumference of a round table, using a straight ruler, then the accuracy of the answer will depend on the length of the ruler. A short ruler will do a better job of following the curve, so gives a better result than a long ruler.

The length of the ruler defines a scale of measurement, and for a small enough scale, the surface will appear straight – just as the Earth seems flat to people walking on its surface. The result will therefore converge on the correct answer. However, if the surface is not smooth, then things get more complicated. In fact, the answers can be all over the map.

In the 1920s, the English scientist Lewis Fry Richardson noticed that, on maps, countries had very different impressions of the length of their shared borders. For example, Spain thought its border with Portugal was 987km, while Portugal thought it was 1214km.

This problem has not gone away with improved technology. According to my internet search, the length of the British coastline is either 12,429 km (the CIA World Factbook), or 17,820 km (the UK Ordnance Survey).
The reason for the different answers is again that they are based on different scales. Small countries tend, it seems, to use finer scale maps. But the finer the scale, the more nooks and crannies the map picks up, and the longer the coastline seems. So, unlike with smooth curves, the answer no longer converges on a single answer.

In fact it turns out that things like coastlines, or natural boundaries between countries, or indeed many phenomena in nature, are not just curved, or a little unstraight – they are infinitely crooked. No matter how far you zoom in, the kinks don’t go away. The measured length just gets longer and longer, and never converges to a single answer. The boundaries are so complex, that in a well-defined mathematical sense, their dimension is not that of a line at all, but somewhere between a line and a two-dimensional plane.

In the 1970s, the mathematician Benoit Mandelbrot (who died in October 2010) coined the term fractal – from the Latin word fractus for broken – to describe such objects. Perhaps the most famous fractal figure is his eponymous Mandelbrot set. The border of this object has a fractal dimension of 2, the same as the plane.

Mandelbrot’s fractal theory was motivated by his study of financial data. Like coastlines, prices do not vary in a smooth, continuous fashion, but are a collection of zigs and zags. Their roughness doesn’t go away when you zoom in. Trying to measure the average price change is like trying to measure the length of the British coastline – there’s no consistent or meaningful way to do it.

So why is it that financial data have these fractal properties? One clue is that such fractal statistics are typical of complex systems, operating at a state known as self-organised criticality – or, more graphically, the “edge of chaos.” If left to their own devices, many processes, such as those which shape a landscape, naturally evolve towards that state. The economy is no exception.

While fractals are ubiquitous in nature, not all systems are equally rough or broken. There are cases where a little smoothness is useful. A plot of the human heart beat, for example, has fractal qualities, but an overly rough or erratic pulse – as measured by fractal dimension – is a symptom of a heart condition known as atrial fibrillation. A graph of brain waves also follow a fractal pattern, but in epileptics a sudden increase in the fractal dimension can herald the onset of a seizure.

This points to a couple of interesting ideas. One is that tracking changes in the fractal properties of markets might give us some insight into their health. A project known as the Financial Crisis Observatory, headed by Didier Sornette from the Swiss Federal Institute of Technology (ETH) in Zurich, uses such techniques to search for precursors of financial seizures.

But instead of trying to predict the next crisis, another approach is to lower the chances of it happening in the first place. After all, the financial system is something we have designed ourselves, so there should be some way of smoothing out its fluctuations – just as our bodies keep a rein on heart beats and brain waves.

This might sound a little optimistic, since financial crashes have been around for as long as finance. But in other areas of science and engineering, we build in safeguards and regulations that make operation safer and smoother.

After all, one thing you never hear from a nuclear engineer is “We’re operating at the edge of chaos!” Maybe one day that will be true of financial engineers too.

David Orrell is a mathematician and author. His most recent book is Economyths: Ten Ways That Economics Gets It Wrong.

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