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A Random Walk Gets More "Wiggly"
on July 11, 2014
Mandelbrot sets out to demolish most of the theoretical bases of financial theory that led to several of the financial crises in the last several decades, foremost of which is that the random motions in prices of commodities and stocks can be assumed to be normally distributed. This sounds like an esoteric sort of argument, but anyone who wishes to win in any game of chance must have some solid notion of how to deal with risk. If one uses the standard model employing the normal (Gaussian) distribution, one will always underestimate the probability of rare events. This can lead to ruin, sometimes on a small scale. As an example, Mandelbrot talks about the rise and fall of the mother of all hedge funds - Long Term Capital Management which took a measly $3.6 billion bailout in the late 1990's because it underestimated risk. But it can happen on a much larger scale as in the crash of 2008 when many large financial institutions in the US held leveraged positions in mortgage security debt instruments. Long story short, everyone underestimated the risk of the unexpected happening, and it nearly crashed western civilization. The cost of that mistake will be measured in the $trillions.
Mandelbrot goes through the models that set up the whole thing: Bachelier, Sharpe, Black-Scholes, and standard portfolio theory. He briefly discusses their power. It's a great, if somewhat sketchy overview of what tools financiers and bankers often use. But in each case, lurking in the background are the assumptions of normality in price movements, and of statistical independence between time periods and between different asset classes.
There is no question that Mandelbrot proves that cotton price fluctuations are badly described by the normal distribution. The quantitative and qualitative information he brings to other asset classes is much less robust. He gives us very good arguments as to why other classes behave as does cotton; but It is hard to say that he brings the same level of quantitative rigor to these. For those of us who want the argument to end with everyone believing the fractal story, it's a bit of a disappointment. What he does do, though, is to describe the Cauchy distribution function which, with some slight generalizations can produce distribution functions that will accurately characterize time series price data whose variation obeys power-laws in the tails of the distribution. The upshot is that anyone with a solid understanding of college level statistics could go on to derive their own Black-Scholes formula.
His publisher appears to have set two rules: 1) no math of any sort in the body of the book, and 2) only simple algebraic equations in the notes. These prohibitions have several consequences. One is that the book is quite readable to anyone, even someone who has not finished eighth grade algebra. A reader can get a vague sense for what Mandelbrot is saying without the math. The flip side is that people who have finished eighth grade algebra may find the arguments hand-wavy when they could be much more solid. Anyone who has a solid background in statistics is likely to be able to fill in the gaps much better, but they will find the arguments fall far short of the kind of proof that one would expect in a 300 page book written by a world-famous mathematician. The people who have studied Black-Scholes, understand its derivation, and use it everyday will likely want a little bit more data and a lot more math before they kill the beast that writes their paychecks. Specifically, they will want a replacement method, which Mandelbrot only hints at.
I found the text here to be a little bit discursive and somewhat repetitive. I often enjoyed his anecdotes, but I did find myself skipping paragraphs, pages, and even chapters. I bought the book knowing that markets have fractal behavior, and hoping to be able to make my own mathematical models based on information in this book. It did allow me to make the intuitive connection between power-law behavior and fractal behavior. And I believe the book has gotten me to the point where I can do all the steps required to price risk and characterize random motions in the prices of assets; although I think a six page monograph that admitted mathematical notation would have been more than sufficient.