Looking at the use of fractals in finance

The offshoot of the belief in the bell curve distribution is that we underestimated the risks in the financial markets.

Mon, Oct 25, 2010
The Business Times

By Teh Hooi Ling
Senior Correspondent

I RECEIVED an email message from my university professor on Monday with the subject head: Obituary - Benoit Mandelbrot.

My professor wrote: 'Dear Hooi Ling,

'There is no Nobel Prize for mathematics. This is a great man and I learnt so much from him even though I have never met him. He deserves more than a Nobel Prize and his work, which (an) author like Nassim Taleb popularized in his book Black Swan, is still breaking new path. His work on fractal mathematics (nonparametric, semiparametric, etc) breaks the monopoly of the field of Euclidean mathematics, which all of us were taught from young.

'Read and write about him if you can and I hope one day, his story will be told by a movie.'

I can't remember if my professor had mentioned Mr Mandelbrot in his lectures before, but the first time I wrote about him was when I attended a presentation by Jeremy Grantham, a founding partner of Grantham, Mayo, van Otterloo (GMO), a Boston-based privately owned global investment manager, back in 2005.

A legend

Mr Grantham is a legend himself, having been described as one of the most influential people on Wall Street.

In his presentation then, Mr Grantham pooh-poohed efficient market theory, which claims that the current price of a stock reflects all publicly available information. His two favourite quotes were from English writer G K Chesterton and mathematician Benoit Mandelbrot.

Mr Mandelbrot said: 'The whole world of economics is enormously more complex than the world of physics. And therefore the teaching of business schools, including Yale's, is unrealistic. Even though economics is a very old subject, it has not truly come to grips with the main difficulty, which is the inordinate practical importance of a few extreme events.'

That, of course, was the premise of Mr Taleb's book - The Black Swan: The Impact of the Highly Improbable - which I'm a huge fan of.

Mr Mandelbrot, 85 when he passed away last week, was described by The Financial Times as 'one of the most original and influential mathematicians of the 20th century'. To The New York Times, he was 'a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields'. And for The Independent, he was 'the man whose mathematical method revolutionised our understanding of everything from economics to cauliflowers and coastlines'.

Peter Tyson, editor in chief of NOVA Online, a science website, succinctly described Mr Mandelbrot's contribution as follows: 'Mr Mandelbrot invented the term fractal to describe the 'roughness' he saw all around him in nature - the jagged shape of a cloud, the rugged indentations of a coastline. Classical Euclidean mathematics, the kind we learn in school, serves well for the human-made world of straight lines, circles, and squares. But nature's non-linear shapes were generally considered unmeasurable - until Mandelbrot developed fractal geometry.'

In Mr Mandelbrot's own words: 'In the whole of science, the whole of mathematics, smoothness was everything. What I did was open up roughness for investigation,' he said in Hunting the Hidden Dimension.

Suddenly, something as ragged as a coastline could come under mathematical scrutiny. While he couldn't actually measure a coastline, Mr Mandelbrot found he could measure its roughness. It required rethinking one of the basic concepts in math - dimension.

In Mr Mandelbrot's view, another dimension exists between two and three dimensions. It's a fractal dimension, and the rougher something is, the higher its fractal dimension. This roughness, this fractal dimension, he discovered, could be measured quite well using fractal geometry.

Fractals - he noted - have another distinctive quality: self-similarity. Look at the branches of a tree. Big branches give rise to smaller branches that give rise to yet smaller branches, yet all look essentially the same; they're just at different scales. They're self-similar.

Natural selection, as he pointed out in his classic 1982 book The Fractal Geometry of Nature, has favoured fractal self-similarity over and over throughout evolutionary history. From the self-similar stalks on a head of broccoli, to the ever-smaller branching of blood vessels in the human body, fractals are everywhere in nature.

And thanks to Mr Mandelbrot, they're everywhere in the human world now, too, noted Mr Tyson. The antenna in your cell phone? Fractal. Those noise barriers along highways? Fractal. Even some of Jackson Pollock's art pieces are fractals.

Fractal geometry

Since Mr Mandelbrot introduced fractal geometry to the world in the 1970s, his new math has informed fields as diverse as biology and physics, ecology and engineering, medicine and cosmology.

In finance, fractals are used to challenge the efficient market theory. It has long been held and taught in finance courses that market prices reflect all publicly available information and that from day to day, the price movements of financial assets are random. This implies that returns should follow the 'bell curve' distribution often found in the natural world. As such, many models like the modern portfolio theory and the Black-Scholes model for pricing options are developed based on the premise of a normal distribution for price movements.

But as Mr Mandelbrot and Mr Taleb noted in an article to The Financial Times in 2006, measures of uncertainty using the bell curve, or the Gaussian model, simply disregard the possibility of sharp jumps or discontinuities.

According to the bell curve, 99.7 per cent of observations would fall within three standard deviations of the mean. But in the financial markets, outlying events occur much more frequently than the bell curve suggests.

Underestimating risks

So the offshoot of the belief in the bell curve distribution is that we underestimated the risks in the financial markets - and that's one of the reasons for the fiasco the world found itself in in 2007/8.

But the thing is, price movements in the financial markets do approximate the bell curve to a large degree. For example, I calculated the daily movements of OCBC Bank way back from 1975 until this week. In the 9,341 observations, the average daily movement is 0.053 per cent. The standard deviation (SD) is 1.69 per cent.

So according to the bell curve, 68 per cent of the daily movements should fall within one SD of the average, or mean. In fact, 78.8 per cent of OCBC's daily price movements fell within one SD of its mean.

About 94.5 per cent are within two SDs, when the bell curve says 95 per cent should be within that range.

And here's where the outliers or the fat tails come in. According to the bell curve, almost all the observations or 99.7 per cent of them, should fall within three SDs of the mean. In OCBC's case, only 98.5 per cent of them did.

On the few extreme days, OCBC's share price actually went some 10 SDs from the mean, which according to the bell curve should almost never occur.

The bell curve says that out of the past 35 years, there should only be 28 days when OCBC's price movement is more than three SDs from its mean daily movement. In actual fact, there were 142 days when it moved more than that.

Some market commentators contested the importance of fractal or the chaos theory in the field of finance. The bell curve works most of the time, they said. So it serves a purpose. Without it, investors would be 'flying blind'.

But the counter argument, of course, is that the comfort afforded by the bell curve has made a lot of people blind to the high impact, outlier risk - which, when they occur, can wipe out an entity completely, be it an individual, a bank, or as seen in 2008, even the entire financial system.

Hence Mr Mandelbrot's contribution can hardly be overstated.

The writer is a CFA charterholder

This article was first published in The Business Times.