The Misbehavior of Markets by Benoit Mandelbrot

Prices in financial markets are not predictable and do not follow any meaningful patterns. Risk, however, does follow patterns that can be modeled and used to better understand the market.

Many financial models are flawed in two ways: the assumption that price changes are statistically independent and normally distributed (aka “Brownian motion”). Price changes echo through time like ripples on a lake, and cluster in time much like wind turbulence. They also follow a power law distribution which results in wild, not mild, fluctuations with time.

Market fluctuations of a security can be described by two terms. The first, H, is an exponent of price dependence (memory). The second, Alpha, characterizes the volatility of the security.

Traditional market wisdom holds that prices follow the capital asset pricing model (CAPM), which states that a security’s returns vary as a function of the firm’s beta, as well as the risk-free rate and the market risk premium. There are three well known anomalies to this theory, however. The first is the P/E effect, which states that firms with lower P/E ratios tend to outperform those having higher ratios of price to earnings. The second is the “small firms in January effect,” which observes that stocks, particularly those of smaller companies, tend to rally in January. The third is the market to book effect, which states that firms with higher book values as a proportion of their market value tend to outperform those with lower ratios. In a 1992 paper, Fama and French showed that these effects explain firm performance more accurately than the CAPM.

Fractal geometry can be used to model markets and market risk more adequately than traditional normally distribution based models. A fractal exists when an object or series can be broken down into smaller parts, each an echo of the whole. Think of a head of cauliflower, heap of dirt, or a rugged coastline.

Fractals are a study of roughness. A more rough and rugged surface has a higher fractal dimension, which leads to greater levels of scaling with additional granularity.

Construction of a fractal starts with a single geometric object, such as a straight line (the black line in the diagram below).  This is called the initiator. The next part is the generator, from which the fractal will be made. The generator is a specific pattern that is formed within the initiator. In the case of a straight line, a generator might be a zigzag (the blue line). To create the fractal, sequentially replace each segment that looks like an initiator with a generator (the green lines).

Not all fractals are self-similar, I.e. are identical with further scaling. Some simply self-affine, meaning the generator is stretched out or distorted in some instances. Some are multifractals, which scale in many ways at different points.

In financial markets, prices move in fits and spurts rather than linear increments. As a result, normal time can be converted into “trading time” to model this multifractal complexity. To do this, create a fractal model to convert normal time to trading time, and another fractal model to convert trading time into price. The resulting models can be used to relate normal time with price.