When Benoit Mandelbrot passed away last year, I thought it would be nice to re-read his (mis)Behaviour of the Markets, to symbolically “pay tribute” to this visionary maverick. I really enjoyed the book first time round and it still reads very well. It is more of “vulgarisation” book, telling the story of how Mandelbrot developed his theory of fractals (it is an easy and quick read: not a single equation in the main text) and how the models can have a relevance (or even provide a new paradigm) in the financial markets.
The book is really divided into two main parts: first the classical modern finance theory, later opposed to Mandelbrot’s fractal view of the Markets, risk, ruin and reward – where he introduces his two main model components: H: the exponent of price dependence and α: the parameter characterizing volatility.
A History of Modern Finance Theory
Mandelbrot traces the origins of Modern Finance Theory back to little-known French mathematician: Louis Bachelier, who, in 1900, published his Théorie de la Spéculation thesis, mostly ignored at the time. The theory introduced its key model: the random walk or brownian motion, which forms a large part of Modern Finance Theory’s foundations. It is not until the 1960’s that Bachelier’s ideas would catch up, when translated to English and republished. Fama’s Efficient Market Hypothesis simply represents a broader version of Bachelier’s work, which “would be developed into a great edifice of modern economics and finance (and five Nobel Memorial Medals in economic science)”.
Mandelbrot first presents the stepping stones of Modern Finance before arguing that there are basic flaws in the theory:
The principal building blocks with which the modern house of finance is constructed all sit on the theoretical foundations laid by Bachelier a century ago.
This book argues that the foundation needs re-pouring, before any more repairs are done to the building. To understand why this matters, let us first look more closely at the structure as it exists today
The presentation of Modern Finance is an interesting chronological re-telling of how the theory shaped itself throughout its various developments.
It starts with Harry Markowitz, who applied Bachelier’s theory to develop his Modern Portfolio Theory using the Mean Variance model. This is often considered as the start of financial engineering. William Sharpe then simplified some of Markowitz’s work with the Capital Asset Pricing Model. Black and Scholes followed suit by contributing their famous eponymous option pricing model.
The book describes the discoveries and explains the main concepts more in detail, which makes for an interesting recap.
But then, Mandelbrot starts to deconstruct these theories:
The whole edifice hung together – provided you assume Bachelier and his latter-day disciples are correct. Variance and standard deviation are good proxies for risk, as Markowitz posited – provided the bell curve correctly describes how prices move. Sharpe’s beta and cost-of-capital estimates make sense – provided Markowitz is right and, in turn, Bachelier is right. And Black-Scholes is right – again, provided you assume the bell curve is relevant and that prices move continuously. Taken together, this intellectual edifice is an extraordinary testament to human ingenuity. But the whole is no stronger than its weakest member
The rest of the first part of the book forms Mandelbrot’s case against the Modern Theory of Finance. In it, he explains that some of the assumptions in the models are wrong, borrowing from behavioural finance principles, to mention that investors are not rational for example. The easier assumption to refute using simple facts, is that of price changes following a brownian motion. The presence of fat-tails in market returns distribution or the P/E effect are such evidence contradicting the theory. In here, Mandelbrot even takes the example of a simple moving average strategy being profitable:
A by-now substantial body of economics research suggests that there is, indeed, money to be made in such a “trend following” strategy.
Fractals Applied to the Markets
Mandelbrot is the inventor of fractals. He actually coined the term and founded a new branch of mathematics based on fractal geometry.
My life’s work has been to develop a new mathematical tool to add to man’s survival kit. I call it fractal and multifractal geometry. It is the study of roughness, of the irregular and jagged. I coined its name in 1975. Fractal is from fractious, past participle of frangere, to break, as I was reminded by one of my son’s Latin dictionaries. The same root survives in many common words, including fraction and fragment. I developed these ideas over many decades of intellectual wanderings – pulling together many stray, forgotten, under-explored, and seemingly unrelated artifacts and issues of the mathematical past, extending them in every direction, and creating a new, coherent body of mathematics. Fractal geometry has come to be viewed as “natural”. It is used today for an improbably diverse set of tasks: compressing digital images over the internet, measuring meta-structures, analysing brain waves in an EEG machine, designing ultra-small radio antennae, making better optical cables, and studying the anatomy of lung bronchia.
After an introduction to fractals in general, the second part dives into the two main tenets of Mandelbrot’s theory.
The first one is fractal scaling, basically based on the fact that market price changes do not follow a gaussian distribution but instead a power-law distribution, in which the tails drop off much slower than in the usually assumed bell curve (ie. fat tails), giving infinite variance and explaining why extreme price movements are much more frequent than anticipated by the “classic” models – something which is arguably a good reason for trend following to work.
Mandelbrot describes how he came to that conclusion starting with his study of cotton prices in 1961 while working as an IBM researcher, and coming across similar concepts by George Zipf, Vilfred Pareto or Paul Levy.
The power law distributions are characterised by their parameter α describing how fast the tails drop off (ie linking intensity to frequency).
The second main concept is that of long memory or long range dependence, characterized by the Hurst exponent H. Presenting some studies in the Nile river hydrology, Mandelbrot establishes the concept of trend persistence in natural phenomenon: periods of floods or droughts tend to come in streaks: they exhibit more serial correlation and for longer than one would expect. Applying similar calculations to market prices shows that financial instruments display more (trend persistence) or less (anti-persistence) long-term memory than the normal case (when H = 0.5).
To measure these two effects, I developed new statistical tools. Some focus on α, the index mentioned earlier. A low-α market would be risky, prone to wild price swings. A market with a higher α differs less from the classic coin-tossing market. Other of my statistical test focus on H, the Hurst coefficient for long-range dependence described earlier. An H of one half implies each price change is independent of the last. A larger H suggests the data are “persistent”, trending in the same direction. A smaller H implies “anti-persistence”, a tendency to double back on themselves.
To separate the two effects, measured by H and α, I developed a statistical test called rescaled range analysis or R/S. It is of a type known by statisticians as “non-parametric”, tests that make no simplifying assumptions about how the data are organised.
Now, as fate would have it, under some circumstances these two effects are so closely interrelated that H is simply equal to 1/α. Take the coin-tossing case: its H is one half and its α is two. Mathematically the relation between the two effects is quite profound; it presents what mathematicians call a dual relationship.
Mandelbrot then quickly presents his “current best model” of how a market works, the fractional Brownian motion of multifractal time, and how to use it to generate graphically synthetic market data exhibiting desired H and α.
Disappointment in the Conclusion?
The book ends without a direct practical application of the fractal concepts to trading or managing money, which can be disappointing for some readers. Mandelbrot is not shy of admitting that his work is still in-progress and to be developed by further generations. Indeed, his models could be compared to those of Bachelier, which took decades to begin having a practical application in finance. Nevertheless, the new concepts are interesting and might give some food for thought for further research. I did try a while ago to implement and use the rescaled range analysis test for trading without much success. I’d be interested to hear other readers’ experience using fractal concepts in trading…
UPDATE: A reader kindly pointed me to another review/summary pdf of the book which can be found there