How the Normal Distribution Keeps Crashing The Global Financial System

The only normal people are the ones you don’t know very well.
— Alfred Adler

What is normal? Normal is conforming to a standard; usual, typical, expected. And what could be more normal than the humble normal distribution?

The normal distribution, also called Gaussian distribution after German mathematician Carl Friedrich Gauss (the Duke of Mathematicians) is everywhere, from the social sciences to the natural sciences to machine learning.

So it was only natural for Charles S. Peirce, writing in the late 19th century, to call it normal, referring to its widespread use and usefulness.

When the normal distribution is not explicitly used, it implicitly inhabits the core of many scientific models: significance tests in psychology, such as T-tests, make Gaussian assumptions about model residuals. Many financial models, such as the Black-Scholes model, crucially depend on the assumption of “normality”.

Normal distributions, crashing markets, and an old man that, according to DALL-E, is supposed to be Gauss.

But what is normal in this context, really? As we will see later, assuming normality in places where things are far from the normal, such as the global financial system, can lead to disastrous outcomes.

Still, the normal distribution is too widespread to easily get replaced. Its popularity rests on several pillars, some of them intuitively clear, some less apparent:

• Its nice mathematical properties stand out to everyone learning about it for the first time. Exponential functions are relatively easy to handle for the aspiring high school mathematician, given that their derivative is again an exponential function. In Bayesian statistics, it comes in handy that the conjugate prior of a normal distribution is again a normal distribution.
• The normal distribution sits at the heart of the central limit theorem. This important theorem that I have written a whole article about states that when adding large enough samples even from highly skewed distributions will lead to a distribution of the sample means which is normal. Toss a coin fifty times, and the number of heads will be roughly centered around the expected value of 25 heads. Repeat the fifty coin tosses hundreds of times, and you will end up with a nice Gaussian distribution over the…