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Extreme Events

Extreme events are intuitively associated with natural disasters, industrial accidents, or market crashes. Yet how are extreme events defined? How can we measure their prevalence?

Fig. 1: Illustration of Seesaw outcomes with a crash. In the game, the y-axis would be rescaled (automatically) to make sure all outcomes are visible.

The easiest way to explain extreme events is to contrast them with non-extreme ones that are “normally distributed”. This distribution, also called Gaussian, or bell curve, is known to many from school. It is completely determined by its mean (for simplicity set to zero in Fig. 2) and its width (called the standard deviation). Body heights, for example, are nearly normally distributed: The probability for somebody to be three times the average height is effectively zero. In a study on heights, we could discard such an outlier as a “measurement error”. Adding random events - for example die rolls - is another example that normally leads to a normal distribution.

Fig. 2: There are many phenomena that repeatedly exhibit extreme events that shouldn't occur for a normal distribution.

Extreme events are different. Earthquakes, for example, often differ by several orders of magnitude (factors of 10): most of them go by unnoticed; yet the probability that the energy released during an earthquake is surpassed by a factor of 100 or 1000 during a human lifetime cannot be dismissed. One has to prepare particularly for those rare events - as long as this is even possible.

Likewise, price changes at stock markets are not normally distributed. At the first glance, the Dow Jones (Fig. 3 top) exhibits roughly exponential growth. Looking closer, however, one notices fluctuations around this average behaviour. To visualise these movements, one uses the “log return” (Fig. 3 botton). It measures the ratio between two subsequent prices (Example log returns: no change: r = 0; 1% change: r ≈ 0.01; 10% change: r ≈ 1.1; ...).

Dow Jones
Fig. 3: The Dow Jones Industrial Average is the weighted average of the stock prices of large US-American companies. Top: its value (price, p) reflects important events, e.g. a: The economic crisis of the 1930s; b: September 11. 2001; c: The economic crisis after 2008. Bottom: price changes are measured using the log return r = ln( p(r) / p(t-1) ). The most extreme crash (so far) was d: October 19, 1987. Data source:

Regarding the changes of the Dow Jones (or of other prices), one can see two prominent deviations with respect to independent, normally distributed events. First, there are clusters in the amplitudes of the fluctuations. If the market is turbulent today, it will likely also be turbulent in a year. Hence, there are hidden correlations acting between prices over long periods of time. Second, large price jumps protrude like needles. Hence one cannot say that crisis and jumps are exceptions: they reoccur with varying magnitudes and their distribution has stayed virtually the same since centuries. Therefore, one might hypothesize the existence of a fundamental principle behind the aforementioned phenomena.

(Neo-) classical equilibrium models in economics and game-theory cannot explain how these phenomena emerge: they just attribute them to external factors that affect the market but don't react to it. Similar approaches are not exclusive to models for financial markets - macroeconomic models are also of this type. Particularly in financial markets, however, we can observe today how high-frequency trading generates such price-jumps within fractions of a second. Yet there is no reason to believe that relevant external events can influence the market at such high frequencies.

In The Seesaw Game, a comparison with the Dow Jones is displayed for particularly large fluctuations. Even if it looks like an error, the outcome likely wasn't uncommon for real markets! Please find further information using the navigation list.