Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions

Abstract

After presenting historical bubbles and crashes, this chapter distills a number of pertinent mechanisms behind stock market crashes. One source of the sometimes irrational behavior of investors can be traced to their psychology, which motivates a brief discussion of behavioral economics. The collective behavior of all traders determines the probability distribution function of the daily returns from the stock market, which shows distinctly fat tails. This motivates us to cover power laws, fractals, random walks with increments drawn from fat-tailed distributions, and Levy-stable distributions as their limiting case. This context is used to derive the central limit theorem and question an underlying assumption of the Black-Scholes theory; being based on a (Gaussian) Wiener process. After a short review of extreme-value theory, we introduce Sornette’s theory of finite-time divergencies, sometimes visible in time series of stock markets.