Prophet Mandelbrot

I recently read a very entertaining book written by Benoit Mandelbrot and Richard Hudson, called The (Mis)behaviour of Markets. It is a popular science text, attempting to explain Mandelbrot’s fractal views on markets in simplified terms and with no maths. For years Mandelbrot has argued that conventional finance is wrong. This very fact was one of the lessons taken from the latest financial crisis (although it is yet to be seen if it will stick).

Foresight

What is interesting about the book is that it was written in 2004, before the latest financial crisis. Mandelbrot argues that conventional financial theory (based on normal distributions) could not be more wrong. Markets are wilder than these models could imagine. Continuing to use them may result in further financial crises (guess what, it did).

What is more surprising, perhaps, is that Mandelbrot has been saying this for over thirty years (and no one seems to have listened). Even before modern portfolio theory was developed, Mandelbrot argued against the mild view of risk posed by models based on the normal distribution. Like Taleb in 1987, Mandelbrot may have had some right to feeling smug given the events of 2007 onward.

What is wrong with conventional models?

Conventional financial models make a number of false assumptions that have been known for some time to be incorrect. However, in the absence of better models, they have continued to be used.

1. People are rational: people most certainly are not rational and do not always take into account all the information available. People tend to feel losses more heavily than gains which means we take different decisions when faced with choices framed in terms of losses as opposed to gains.
2. All investors are similar, apart from their appetite for risk: Investors are very different. Some are speculators in it only for a day, some are in it for the long run. Some believe in value investing, some are technical analysts. Return and variance are not the only things that matter to all investors (which is what the theory assumes). Some investors are big (able to influence prices), others are small (the theory assumes everyone is small).
3. Prices are continuous: This means they move smoothly. However, in reality it appears more likely that prices can jump erratically, moving from say 5 to 10 without hitting any number in between.
4. Prices changes (more accurately the logarithm of price changes) follows a normal distribution: in actual fact there are both far more boring pricing changes (very small) and wild changes (that is, fatter tails) than this model would predict.
5. Prices are independent over time: This means the price change yesterday does not impact it today. Mandelbrot argues that volatility tends to cluster with large price changes tending to be followed by more large price changes. He also argues that there is a long-term dependence in prices. The movement in prices today may still have an impact 100 years from now.
6. Volatility is constant: this is one critical assumption of the ubiquitous Black-Scholes formula for valuing option prices. This assumption is wildly wrong that quants have started modelling the intricate variation in the so-called implied volatility (if the model were correct there would be no such variation). This is ludicrous, in my opinion. Empirical evidence shows that volatility itself is very volatile.

A number of hacks have been made to work around some of the problems. For instance, the modelling of the volatility ‘skew’ in point 6 above. Complicated models such as GARCH and FIGARCH have been developed that allow for volatile volatility and long-term dependence. Mandelbrot argues that this is just tacking sticky tape onto a broken vase. Something entirely new is needed. His main premise for this seems to be that his models exhibit much greater parsimony (that is they need fewer parameters) – which is a way of saying they are more beautiful or elegant – and that they start with observations of actual market behaviour.

Mathematicians (and practitioners) love normal distributions and so tack on anything they can to make them work as it saves them the trouble of starting from scratch (It’s hard to admit that a hundred-year-old body of academic literature is largely defunct). Certainly I agree this is the wrong way to go about things. However, parsimony is also not the only measure of a model’s worth. Something things are just complicated (financial markets especially). Like Einstein we should not overcomplicate. Things should be as simple as possible, not simpler. Mandelbrot may well have given us a simpler, better foundation.

Fractal markets

Mandelbrot is most famous for his work on fractals (he coined the term ‘fractal’) and he applied it in many areas. Finance is one area to which it is naturally suited. However, it has not yet caught on, probably because the maths is harder and less well developed. I do not quite understand all the workings myself (not having gone through the math, yet), but the basic premise is that markets behave similarly on any scale (or most scales at least).

Consider a graph of the prices of a certain stock. The graph will look very similar, in terms of its swings, erratic movements and proportional price changes whatever period you look at, whether it be a year, a month or a day. That is, you can zoom in on one part of a price graph and get a miniature (statistical) replica of the whole graph (that is, it is equally “wiggly”). This can be seen in the following graphs of our old friend Berkshire Hathaway (from freestockcharts.com). Can you order them by length of period covered?

I

II

IIII

The first chart shows the daily price over about 2 days, the second the hourly price over a little more than 3 months and the last the daily price over a period of almost 2 years. Except for random variation, they are pretty much indistinguishable.

This would break down at very small time periods (over a minute, say, – prices may be constant) and over large time periods (the upward trend of stocks is likely to show more clearly and the progression may be smoother). It would also not, I would add, work for illiquid stocks where the (realised) price changes very infrequently (you can still think of the price moving in the fashion described, but only being observed when the stock is traded).

If you are a regular reader, you may also remember that in a previous post I discussed power-law distributions. Mandelbrot first suggested these might fit cotton prices, and since then many other price series. Power-laws display a scaling behaviour, which is a fractal property.

Anti-everything

The book attacks every paradigm of finance in existence today. It even says value investing, espoused by Warren Buffet (and Benjamin Graham before him), is mistaken. Technical analysis (which I have not heard many talking fondly of) is also debunked.

While I agree conventional finance has got things wrong and have my doubts about both the above paradigms (more so with technical analysis), Mandelbrot’s arguments could not convince me entirely. Mandelbrot’s view of technical analysis appeared to be a straw man (perhaps he needed to do so in order to make the book accessible) and I still have unanswered questions regarding technical analysis. However, it is a good beginning for my quest to understand the operations of the markets.

Final word

Though the book is not perfect, I would still recommend it to anyone in finance, to instil a sense of caution and of questioning. Too many people follow blindly what the ‘experts’ say. We still know very little about the markets (perhaps they are unknowable) and much work still needs to be done. I for one am rather excited that I might get to play a part.

Some references

Mandelbrot’s book:

• Mandelbrot, B., & Hudson, R. (2004). The (Mis)behaviour of Markets. London: Profile Books.

For Taleb's account of the crash in 1987:

• Taleb, N. N. (2007). The Black Swan. Penguin.