Benoit Mandelbrot's Contributions to Price Analysis

[RichardCox 10]

Before his death, Benoit Mandelbrot was one of the most celebrated mathematicians in the world, and most of his work was based on the belief that the wider view of price behavior in financial markets is primitive and in need of large revisions. Mandelbrot's work sought to update these “medieval” views of market behavior so that economists could update the ways asset price changes are understood on a fundamental level. Mandelbrot's legacy as one of the main contributors to the field of technical analysis is firmly established but there are still some traders that view Mandelbrot's work as being disconnected from the real workings of asset markets.

 

Here, we will look at some of Mandelbrot's main contributions to the study of price behavior. The work started roughly 50 years ago and involved the creation of a new type of mathematics. Rather than focusing on accounting or simple geometry, Mandelbrot equations dealt with mathematical shapes rough and complex, rather than smooth and simplistic. Additionally, the pattern repetition that Mandelbrot found was largely uncommon and unexpected relative to most of the similarly focused work that had come before him.

 

Mandelbrot's equations would later be referred to as Fractals and the wider applications for these equations have been seen in environments that are far-removed from the world of finance (being used to describe clouds and coastal formations, to forecast the ways rocks and metals will wear over time, and to generate computer graphics programs.

 

Mandelbrot’s Approach to Finance Markets

 

Focusing on how Mandelbrot's work relates to a financial context, specific examples can be seen in the ways Mandelbrot criticized a large majority of his colleagues attempted to predict the changes in price activity. Mandelbrot’s initial assertions were that market participants should be looking to describe market behavior before attempting to forecast or predict it. While this might sound disappointing to some looking for the “holy grail” of price forecasting, for Mandelbrot, this was an essential step in the process because accurate predictions would be impossible without first having accurate descriptions of what exactly is happening in market environments.

 

Mandelbrot’s Critiques of Prevailing Ideas

 

Since Mandelbrot largely disagreed with most of the price forecasting approaches around him,it should be noted that Mandelbrot's main contention was with the idea that price activity is determined by dual opposing positions (the decision to either push prices higher, or to drive them lower). This idea he saw as an overly simplistic view of market activity, reducing asset valuations to nothing more than the toss of a coin: Heads, markets buy and push prices higher, Tails, markets sell and drive prices lower. Before Mandelbrot, this idea reigned as an explanation of how prices remain generally supported: Buyers and Sellers constantly bump into each other and keep prices suspended with real valuation changes occurring slowly.

 

Mandelbrot argued, however, that this view does not adequately account for the occurrence of major swings in market volatility. Imagine a stock that falls from $50 to 5 cents. Surely, for Mandelbrot, these suspension ideas (and the binary Coin Toss view) contain some inefficiencies and cannot be relied upon to forecast all possible price outcomes. According to Mandelbrot, large increases in price activity should never happen when viewing markets in a binary, suspension fashion. But, of course, we know that these situations occur all the time and Mandelbrot main focus was to explain why this is the case.

 

Changing the Prevailing Approach

 

One of Mandelbrot’s suggestions was to modify the binary (Coin Toss) suspension model to that the equations matched the real world occurrences. When looking at the financial markets, this modification meant applying Fractals because of the ways they can plot and describe the rough and seemingly unlevel nature of price activity. Uneven changes in price activity (or any observable environment) will make it difficult to formulate equations, and for Mandelbrot, this is the reason simple geometry proved insufficient when describing these environments.

 

This is the reason Mandelbrot’s work with Fractal geometry became a primary focus. With this form of mathematics, he felt he could write equations that mirrored market behavior but he still found himself confronting problems with the ability to forecast future price behavior (especially in cases where major swings in price were seen). While many of Mandelbrot’s critics agreed with the idea that traditional statistics were insufficient when looking to forecast wild variations in price, some contended that when looking at longer time frames the market actually does smooth out in ways less rough than Mandelbrot was suggesting. This debate led many to favor Mandelbrot’s models over short time frames, while preferring to use more traditional models as durational periods are extended.

 

Theories Gauging Market Performance

 

In his 1982 book The Fractal Geometry of Nature, Mandelbrot looked more broadly, attempting to find evidence continual reappearance of fractals within the universe, and this is the work that ultimately granted him the notoriety he would later receive. The appearance of Fractals in many of the patterns we encounter in daily life (in clouds, plants, geographical formations), can be described by Mandelbrot’s mathematics. This gained him attention from practitioners of Chaos Theory and computer graphics visionaries and both of these fields would later draw largely from Mandelbrot’s Fractals equations.

Mandelbrot’s equations for price forecasting evolved later in his life.

 

The changing approach to price analysis allowed for new ways to measuring market performance, and in modeling the drastic price swings with a greater level of precision. Mandelbrot's formulas are now viewed as a means for measuring and assessing the broader climate seen in the market over a given period. For Mandelbrot’s theories, consequences must be tested a large number of times and show high levels of accuracy in order to be applicable to asset trading.

 

Mandelbrot's central contention is that Coin Tossing approaches are overly simplistic and almost suggest that the trading outcome of one day is equal to those seen on every other day. Anyone with any real experience in the markets knows that this just isn’t the case. Trading with Mandelbrot’s theories typically involves looking at the number of price changes and then isolating the number of important trading days to small intervals. For Mandelbrot, it is this significant minority that is the important period, and the other trading days can be avoided altogether - as the ability to forecast movements is less likely to result in substantial gains.

 

A Summary of Mandelbrot’s Fractals

 

Benoit Mandelbrot’s Fractal theory has been studied for decades and is based on the idea that financial markets (and science as a whole) is based on our sensations. Most of these sensations are commonly understood: When we see something, we recognize this as the sensation of optics. When we hear something, we recognize this as the sensation of acoustics. When temperatures rise, we develop theories to explain the sensation of heat. For Mandelbrot, one of the most fundamental sensations can be described as “roughness.”

 

Imagine primitive man agitated by roughness in the same way that humans become agitated by heat. Roughness was difficult for primitive man to overcome, and this, for Mandelbrot required the use of Fractal geometry to remedy. Mandelbrot argues that the ability to measure the non-basic sensations (such as Roughness) came relatively recently. These advances came with ability to make accurate physical measurements, as we essentially turn sensations into abstract numbers and equations. The patterns that mark Fractals repeat in a similar fashion, and with this repetition Mandelbrot suggests that we can gain an accurate description of “rough” and irregular constructions - like a cloud formation, coastline, plant structure - or the financial markets. For Mandelbrot, it is this description that is the first step to efficient price forecasting.

[clmacdougall 11]

I read Mandelbrots " The Misbehavior of Markets" and enjoyed it, but I did not necessarily agree with or understand it fully. Could you take a moment to visually show me what Mandelbrots fractal theory would inspire in a trade setup. I would love to see how you would produce that on a chart.

 

Regards

[BlueHorseshoe 10]

  clmacdougall said:

I read Mandelbrots " The Misbehavior of Markets" and enjoyed it, but I did not necessarily agree with or understand it fully. Could you take a moment to visually show me what Mandelbrots fractal theory would inspire in a trade setup. I would love to see how you would produce that on a chart.

 

Regards

 

Hello,

 

Mandelbrot's fractal theory might cause you to assess probabilities using power laws of exponentiation rather than assuming a Gaussian distribution of outcomes (as one would expect flipping a coin as the article describes). You would price an Option completely differently to the prevalent Merton-Scholes model, for instance. You wouldn't rely on linear regression methods and concepts like GARCH. You would anticipate the occurrence of 'outliers' with far greater frequency than that predicted by a bell curve approach. And you wouldn't work with an artificially imposed two sigma threshold such as that imposed by Bollinger Bands.

 

Hope that helps.

 

BlueHorseshoe

[RichardCox 10]

Hi clmac, there will be trading setups with Fractals in a future article. I will post a link in this thread.

[suby 10]

Great Thread Richard!

 

I look forward to the link

 

Happy to see some posters on TL tackle markets from an academic point of view rather than the junk thats promoted/misconstrued in regards to technical analysis