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.

Power laws: Mandelbrotian is the new Gaussian?

Benoit Mandelbrot, the pioneer of fractals, died last year. I only found out months later. His death should have been announced as a great tragedy to the whole world. Mandelbrot I also only recently found out also made contributions to financial mathematics. I would like to discuss one aspect, which may not be original Mandelbrot, but which Taleb names after him.

The old

It has been well publicised that normal distributions do not work. At least they do not work for modelling financial markets. It is my impression that, despite knowing this, everybody continued to use them. Because the quants, the people who were supposed to ‘know’ said it was okay, the normal distribution became a kind of placebo.

These methods have, however, always had their critics. Taleb (who wrote Fooled by Randomness and The Black Swan) is one of them. The latest financial crisis proved them right. But still, the normal distribution is what everybody knows, and I think it will take a while before ‘better’ practices become widespread.

The new

One alternative to normal distributions that has been suggested, and which Taleb argues we should use, albeit with caution, is that of power laws. He calls such distributions Mandelbrotian, to challenge the conventional Gaussian. Power laws have, of course, been around for a very long time (they are hardly new). Zipf’s law is a specific example in the field of linguistics that was proposed in 1935 and the specific example we will be looking at was researched even earlier. I am not quite sure how much work on power laws has been done in terms of financial markets nor am I certain exactly what Mandelbrot’s contribution was.

Power laws are interesting creatures, showing themselves seemingly everywhere. They occur in nature where they give the frequency of different sizes of earthquakes, of the sizes of craters on the moon and the extinctions of species. These are some examples mentioned by John Gribbin in his book Deep Simplicity.

Things such as the distribution of city sizes and the disparity of wealth also seem to follow power law distributions. The ubiquity of this distribution not only in nature, but in human behaviour, makes it seem like some kind of universal law (which we cannot help but obey). Such an interpretation should be avoided of course. With interest in power laws high, you are more likely to hear about confirming evidence (this is what people will be looking for). Eventually we will find important examples that are not governed by power laws or we may find the power law distribution is a poorer estimate than some other law.

Definition

There is a general definition of a power law distribution, but I will merely define a special case, the Pareto distribution which has the distribution function

with support x > k and a > 0. There is a more general definition for which some of the properties I discuss below would only hold asymptotically. The Pareto distribution is also the source of the infamous 80-20 or Pareto law (not really a law). I have yet to establish exactly how this was derived but watch this space.

Why power laws are cool

Self-similarity

They display self-similarity. Self-similarity is a property of fractals (remember that Mandelbrot is famous for his work on fractals). Taleb calls these distributions scalable (as opposed to the normal distribution, which is not scalable). It follows from the following property:

So, for example, if a = 1, doubling x always lowers the probability of seeing a higher value by half. This also demonstrated by the fact that if you fix a new minimum wealth level you get a Pareto distribution with the same exponent:

More concretely, if X represented wealth levels, the inequality among the rich would be the same as that among the poor. So if your wealth is x_m, half the population will have twice that wealth, and half of them four times that and so on.

Infinite moments

For certain (common) cases they have infinite variance or even infinite mean. For a <= 2, the variance is infinite. For a <= 1, so is the mean. In these cases we will not be able to rely on the Central Limit Theorem (or the Law of Large numbers in the second case). This makes the maths harder and moves us irrevocably away from normality.

Fat tails

They have fat tails. This means that extreme values are more likely than for the normal distribution. The structure of the tails is given by the self similarity of the distribution as shown earlier. For the normal distribution doubling the wealth level would lower the probability of higher observations by much more than half and much more rapidly further up the tail we go.

  

Estimation difficulties

They are hard to fit. Determining the exponent to use is apparently not easy and the resulting distribution is very sensitive to use. Taleb gives a good illustration.

Can they help us?

The danger is that everyone will jump on the new bandwagon. I can see the headlines twenty years from now. Math whizzes cause greatest financial crash in history. Math models fail us again. These Mandelbrotian distributions are just another fraud. Taleb warns that we cannot model everything, that even appropriate mathematics is of but limited use. In any event we should use models with caution. I am very interested in how far we can take them. I am definitely not yet done with power laws. Expect more posts on this topic.

Some references

Gribbins links power laws and chaos theory (both linked to fractals):

• Gribbin, J., 2004. Deep Simplicity, New York: Random House, Inc.

 Taleb gives some useful examples and a mathless discussion:

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

 From Wikipedia¹: 

• Wikipedia, 2011a. Pareto distribution. Wikipedia. Available at: http://en.wikipedia.org/wiki/Pareto_distribution.
• Wikipedia, 2011b. Pareto principle. Wikipedia.
• Wikipedia, 2011c. Power law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Power_law_distribution .
• Wikipedia, 2011d. Zipf’s law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Zipf’s_law.

1 Yes I am going to shamelessly reference Wikipedia. It is useful and usually accurate. This is a blog, not an academic article. I am aiming for clarity of thought rather than accuracy of assumptions.