After reading Antifragile and The Black Swan, I wanted to hear a bit more about randomness straight from the source, and Mandelbrot was the inspiration for both of these books.

Mandelbrot is an odd duck. He's a bit of an outcast from both the Financial and Mathematics communities (finance types didn’t like him because he focuses on obscure applied mathematics, mathematicians think finance types are focused too much on money and not enough on pure math).

Mandelbrot is most famous for coining the idea of Fractals; self-similar shapes which don't lose resolution as you zoom in or out. Good examples would be:

  • coastlines (which have a similar shape at 5,000 ft resolution and 10ft resolution)

  • mountain ranges (peaks look like little boulder and piles of rocks from far away)

  • ferns (which have their own intricate branching structure when zoomed up close)

He devoted his life to studying this "roughness" in shapes and in nature. In addition to looking at 2d and 3d space, Mandelbrot argued that this "roughness" coefficient should be an extra dimension to look at various patterns, even in 2d space.

The book takes this thinking and applies it to economics. Mandelbrot's big argument is that the way we look at markets today is mostly wrong.

We tend to think of markets as "random walks". The price from one day to the next will follow some normal bell curve distribution, and either increase or decrease, but not wildly divulge from the mean. There's a whole host of formulae to describe the returns of markets in this way: Sharpe’s CAPM, Black-Scholes, etc.

Mandelbrot instead argues that all of these are wrong, because they don't empirically match the way that markets work. Instead, markets tend to cascade and jump wildly in spurts, much more than a random walk might predict. He argues that you instead of estimating in terms of random walks, one should look at the rate of change or the delta in markets. Within those deltas, you will find both fractal behavior and power laws related to it.

From looking empirically at Markets, it’s important instead to take in several ideas into account.

  • market moves do not follow a bell curve — instead jumps will follow a power law. when looking at the progression of cotton prices, there will be a handful of outsized jumps that happen much more frequently than one might expect. the way to model these should not be gaussian randomness, but instead using power law coefficients.

  • prices leap, not glide — it’s easy to think of market prices as shifting continuously. after all, we tend to graph stocks as these constantly moving continuous lines. in reality, markets jump based upon the bid-ask spread, so any model based upon a continuous price is almost inherently flawed. in algorithmic trading times, we should expect this effect to get even bigger.

  • analysis should be done not by time, but by trading volume — one of Mandelbrot’s biggest points is that we tend to think of stock prices when charted against time. instead, we should really be thinking more about them as a function of the number of trades. times when trading volume is high, will looked like a zoomed out version of times when trading volume is low. it’s all fractal!

Overall, the book taught me quite a bit about how markets actually behave, when viewed from an empirical sense. If you like Taleb, or reading about finance in general I think you’d enjoy it.

Additionally, this has underscored my conviction that visualizing data can lead to deep levels of insight. Mandelbrot got his start by simply looking at shapes, and understanding their repeating, self-referential patterns. I think more fields could (and should) do the same. Data visualization is powerful!