Dragon-king of the outlier events

This is a black swan.

This is a dragon-king.

Which are you more terrified of?

In a new paper, author and physics professor Didier Sornette, presents the concept of “dragon-kings” — outlier events that exist within conventional power law distributions.

Power law distributions are kind of like bell-curve distributions, but their tails (where ‘extreme’ events occur) are much longer, so that extreme events are considered rather more frequent. Here for instance is a power law distribution, from Sornette’s paper, of the Dow Jones Industrial Average:

Sornette’s main assertion, then, is that while black swans, the outliers made famous by Nassim Nicholas Taleb, are generally considered unexpected since they are simply bigger than other events in the distribution, dragon-king events are a rather different beast altogether. Here’s his comment:

If events of large impacts are part of a population described by a power law distribution, the common wisdom is that there is no way to predict them because nothing distinguish them from their small siblings: their great sizes and impacts come out as surprises, beyond the realm of normal expectations. This is the view expounded for instance by Bak and co-workers in their formulation of the concept of self-organized criticality [14,15]. This is also the concept espoused by the “Black Swan Theory” [16], which views high-impact rare events as unpredictable . . .

The following examples suggest that, in a significant number of complex systems, extreme events are even “wilder” than predicted by the extrapolation of the power law distributions in their tail. Below, we document evidence for what can be termed genuine “outliers” or even better “kings” [17] or “dragons.”

On the financial side, dragon-kings work something like this, according to Sornette:

Fig.7 [of the DJIA above] shows the survival distribution of positive (continuous line) and negative daily returns (dotted line) of the Dow Jones Industrial Average index over the time interval from May 27, 1896 to May 31, 2000 [7]. No dragon-king is apparent and it seems that the distribution of large losses and large gains are pure asymptotic power laws [28].

But this is missing the forest for the tree! Our claim is that financial returns defined at fixed time scales, say at the hourly, daily, weekly or monthly time scales, are revealing only a part of the variability of financial time series, while a major risk component is gravely missing.

Since we are interested in characterizing the statistics of extreme events, to illustrate our claim, consider the simplified textbook example of a crash that occurs over a period of three days, as a sequence of three successive drops of 10% each summing up to a total loss of 30%. Now, a 10% market drop, that occurs over one day, can be seen in the data of the Nasdaq composite index to happen on average once every four years. Since there are approximately 250 trading days in a year, a 10% market drop is thus a 10-3 probability event. What is the probability for observing three such drops in a row? The answer is (10-3)3=10-9. Such one-in-one-billion event has a recurrence time of roughly 4 million years! Thus, it should never be observed in our short available time series. However, many crashes of such sizes or larger have occurred in the last decades all over the world.

What is wrong with the reasoning leading to the exceedingly small 10-9 probability for such a crash? It is the assumption of independence between the three losses! In contrast, our claim is that financial crashes are transient bursts of dependence between successive large losses. As such, they are missed by the standard one-point statistics consisting in decomposing the runs of losses into elementary daily returns. With some exaggeration to emphasize my message, I would say that by cutting the mammoth in pieces, we only observe mice.

We thus propose to analyze drawdowns (and their symmetrical drawups), because they are better adapted to capture the risk perception of investors, and therefore better reflect the realized market risks. Indeed, we demonstrate below that the distributions of drawdowns diagnose efficiently financial crashes, which are seen as dragon-kings, i.e., special events associated with specific bubble regimes that precede them.

Well, it sounds like a useful — albeit scary — distinction.

You can read the full paper here.

(H/T Paul Kedrosky)

Related links:
Predicting the credit crisis – FT Alphaville
The physics of the trading floor – Nature journal
Who controls the stock market, or, phycisists do it differently – FT Alphaville