20 April 2020

In this period, risk management is once again a hot topic. To manage risks, you need a model. It has been shown that traditional models fail to capture such risks. This is when the Mandelbrot name surges again. But what is it really about?

Most people know the name of the polish-born French mathematician Benoit Mandelbrot because
of the famous set^{1} that was named after him. But he also dedicated a lot of his time to the financial markets.

Like all of his work in mathematics, his focus was more on the structure of prices series than on their forecast. Although his work is not directly related to statistical robustness, he definitely paved the way for a more robust way to estimate financial risk.

He was a strong advocate for the representation of market prices as random phenomenon. Not necessarily because they are random in nature, but because they result from the combination of so many interactions (including human actions) that they are completely unpredictable.

The main benefit of such representation is to “provide estimates of the probability
of what the market might do and allow one to prepare for inevitable sea changes^{2}”.

To go one step further, he studied the price series of cotton futures and concluded that the common assumption that price changes follow a normal distribution was proven wrong. In fact, cotton future prices were following Lévy stable distributions. It may sound like a mere technicality, but the consequences are important. One of them is that it is much more difficult to compute probabilities of future price changes. The other is that massive price changes will appear more often than usually expected.

Mandelbrot introduced another important notion: self-similarity. It means that characteristics remain the same when changing the scale of representation. Self-similarity can be observed in nature. The best-known example is the Romanesco broccoli.

But self-similarity also exists in financial markets. Intra-day fluctuations of a security price will often graphically resemble weekly fluctuations of the same asset.

Such property can be used at the benefit of the (long term) investor by deducing long term probabilities from a short sample.

Mandelbrot’s work can help us evaluate risk in two ways:

- Unlikely (extreme) price changes will appear much more often than traditional risk models’ expectations, especially in the short term.
- We can learn about medium and long term risk by looking at shorter term (past) price changes.

Further readings:

- How Fractals Can Explain What's Wrong with Wall Street
- How the Mathematics of Fractals Can Help Predict Stock Markets Shifts
- Mandelbrot tips off the markets