Chapter 9
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 extremevalue theory, we introduce Sornette’s theory of finite-time divergencies, sometimes
visible in time series of stock markets.
Much of the financial and economic theory discussed in previous chapters is based
on the notion of markets in equilibrium, a concept heralded as the “efficient market
hypothesis.” This hypothesis is probably a good approximation to the truth much of
the time, but once in a while markets enter a phase where they grow very rapidly—a
bubble—and later contract even more rapidly—a crash. On the top panel in Fig. 9.1
we show the historic evolution of the Dow Jones Index [1] from 1900 until 2010 on
a logarithmic scale. It rises from about 50 in 1900 to about 10000 in 2010. The curve
is neither smooth nor following an exponential growth. Instead, there are periods of
steep rises, examples are in the late 1920s, mid 1980s, the 1990s, and the first years of
the new millennium. During these times the market was in a bubble that subsequently
deflated in a more or less violent crash, visible as a significant drop in the index. The
lower panel shows the day-to-day variation of the index (I (i + 1) − I (i))/I (i). We
observe that crashes are accompanied by increased volatility or wildly fluctuating
day-to-day variations. Crashes are identified by large single-day drops, such as the
30% loss during the crash in October 1987. These patterns of bubbles and crashes are
difficult to reconcile with the idea of a market in equilibrium. To better understand
this behavior, we will therefore discuss historical examples of crashes, followed by
a presentation of the likely mechanisms causing them. Finally, we spend some time
to characterize the mathematics of crashes.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_9
113
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 extremevalue theory, we introduce Sornette’s theory of finite-time divergencies, sometimes
visible in time series of stock markets.
Much of the financial and economic theory discussed in previous chapters is based
on the notion of markets in equilibrium, a concept heralded as the “efficient market
hypothesis.” This hypothesis is probably a good approximation to the truth much of
the time, but once in a while markets enter a phase where they grow very rapidly—a
bubble—and later contract even more rapidly—a crash. On the top panel in Fig. 9.1
we show the historic evolution of the Dow Jones Index [1] from 1900 until 2010 on
a logarithmic scale. It rises from about 50 in 1900 to about 10000 in 2010. The curve
is neither smooth nor following an exponential growth. Instead, there are periods of
steep rises, examples are in the late 1920s, mid 1980s, the 1990s, and the first years of
the new millennium. During these times the market was in a bubble that subsequently
deflated in a more or less violent crash, visible as a significant drop in the index. The
lower panel shows the day-to-day variation of the index (I (i + 1) − I (i))/I (i). We
observe that crashes are accompanied by increased volatility or wildly fluctuating
day-to-day variations. Crashes are identified by large single-day drops, such as the
30% loss during the crash in October 1987. These patterns of bubbles and crashes are
difficult to reconcile with the idea of a market in equilibrium. To better understand
this behavior, we will therefore discuss historical examples of crashes, followed by
a presentation of the likely mechanisms causing them. Finally, we spend some time
to characterize the mathematics of crashes.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_9
113