*The (Mis)behavior of Markets*Benoit Mandelbrot, writing with Wall Street Journal editor Richard Hudson, argues that financial markets are fractal and that this is critically important to understanding financial risk. It takes a historical approach and is pitched at a popular audience, using no equations at all and slowly introducing the key concepts.

Part one is a brief history of traditional financial theory and risk analysis, going back to Bachelier. This covers basic probability, normal curves, and the Efficient Market Hypothesis, as well as applications by Markowitz, Sharpe, and Black and Scholes to asset valuation, portfolio construction, and the pricing of risk. A chapter "The Case Against the Modern Theory of Finance" considers the problems with all of this, both in its assumptions about investors and price changes, and in contrary evidence.

Part two presents "The New Way", explaining fractals and their applications to markets. It begins with market turbulence and the basic concept of a fractal — "a pattern or object whose parts echo the whole, only scaled" — using a simple graphical generator to generate stock charts. Continuing the geometrical approach, it explains "roughness" and the concept of fractal dimension; this is illustrated with a number of geometrical examples including the famous Mandelbrot Set.

Mandelbrot then describes his own experiences trying to understand historical cotton prices, in the process explaining the basics of power laws and scaling and the "fat tail" Cauchy distribution.

"Gaussian or not, scaling or not: Does it matter? Yes. First, it shows that prices can and do gyrate wildly. The market is very risky — far more risky than if you blithely assume that prices meander around a polite Gaussian average."

"data that scale can produce surprising patterns — patterns that, if you glance at them, you would swear are periodic, predictable, and bankable. Anyone studying the cotton price records could easily imagine he was seeing 'corrections,' 'resistance levels,' and the other signals that a technical analyst seeks to buy, sell, or hold. It is fool's gold."

In the early part of the 20th century the hydrological engineer Harold Edwin Hurst found that the range in level of the annual Nile floods increases with longer time periods faster than the expected square-root power. Mandelbrot generalises this long-term dependence using fractal mathematics.

Unexpected variability can be due to wild price swings (the Noah Effect) or to long-term dependence (the Joseph Effect). Mandelbrot considers how these work in markets and their interaction in investment bubbles.

Basic fractal models can be refined using the concept of a multifractal, where there are different scaling ratios in the one object. Mandelbrot argues that this allows flexibility while being less ad hoc and more elegant than the "GARCH" approach, which sticks with traditional normal curves and Brownian prices but allows the parameters to vary with volatility.

Part three presents some brief practical advice in the form of ten "heresies of finance": markets are turbulent; and riskier than standard theories imagine; timing matters greatly; prices can leap rather than glide and discontinuities matter; time is flexible; all markets work alike; markets are uncertain and bubbles are inevitable; markets are deceptive; forecasting prices is hard, but one can "estimate the odds of future volatility"; and "value" has limited value. This is interesting but also quite general, and probably of little direct use to investors. As a final chapter on outstanding problems says, "fractal investment analysis has more questions than answers", with obvious open problems in stock analysis, portfolio construction, valuing options, and managing risk.

I'm not convinced by everything Mandelbrot writes, but I think he's correct in his key arguments: fractal mathematics is the appropriate choice for modelling many aspects of markets and this changes the way we should approach risk. (One of my concerns is the extent to which financial markets are embedded in real world economies which are driven at least in part by non-financial considerations.)

*The (Mis)behavior of Markets* presents an important application of
some important mathematics, in a way that will be broadly accessible.
It seems to find itself catalogued and stocked in bookshops under
"business and finance", but I think it will appeal more as popular
science than as investment advice.

Mandelbrot and Hudson do a good job presenting complex concepts to a lay audience. All the equations and unnecessary technical details are consigned to endnotes, but the core ideas are still there. The presentation may be slow for those with both mathematical training and finance experience, but even they should find some novel material, even if it's only in the history and background.

February 2010

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