Dollar bills and Lévy flights

Florin Diacu’s book Mega Disasters: The Science of Predicting the Next Catastrophe (Princeton University Press, 2010) surveys a wide range of catastrophes—tsunamis, earthquakes, volcanic eruptions, whirlwinds (hurricanes, cyclones, and typhoons), rapid climate change, cosmic impacts, financial crashes, and pandemics. As a mathematician, he sets out to study what sorts of calamities can be predicted, especially since many dynamic systems exhibit the property of chaos. “To mathematicians,” he writes, “chaos is another name for high instability: similar starts don’t guarantee similar outcomes.” (p. xii)

Not surprisingly, Diacu concludes that few catastrophes can currently be predicted with any degree of accuracy or with much lead time. For now the best we can do in most cases is (1) to mitigate risk (for instance, imposing strict building codes in earthquake-prone areas or immunizing at-risk populations) and (2) to have procedures in place to manage the fallout from catastrophes.

As readers of this blog should know by now, I’m always on the lookout for new ways to frame familiar concepts, be they metaphors or models. Diacu has provided both an analogy and a statistical model for understanding the relationship between randomness and trend in financial markets.

The analogy comes from a 2006 study of how a potential virus would spread. The researchers decided to simulate the spread of the virus by tracking the movements of about half a million one-dollar bills in the U.S. They went to the well-known web site (by now over ten years old) Where's George. “What they found was not too surprising: the money moved chaotically at both the local and global levels. But it was interesting that they could characterize the dispersion in terms of what mathematicians call “Lévy flights,” named after the French mathematician Paul Pierre Lévy (1886-1971). These movements are characteristic to random walks with many short steps mixed with rare long jumps. In other words, the bills changed many hands in the same city before showing up in some other part of the United States.” (p. 160)

As usual, I’m late to the game: financial mathematics has long since incorporated modified versions of Lévy flights in its simulations because Lévy distributions allow for big jumps and have fat tails. But I figure I’m not alone in my tardiness, so here is an illustration of the difference between Brownian motion (left) and Lévy flights (right). Lévy flights, by the way, can also be characterized as random power law trajectories.



Diacu points to the fact that there is a close connection between Lévy flights and the Mandelbrot sets that Sornette used in his work on market catastrophes. (See my posts of October 23 and 26 on Sornette’s Why Stock Markets Crash, though you won’t find any mention of fractals there). One connection between the two is the absence of a characteristic scale; in fact, Lévy flights have been called scale-invariant fractals.

By now I’m in the deep end of the pool without my water wings. So let me climb out ungracefully and simply say that I find the Lévy flight model a fascinating way to combine the random walk theory of markets with the presence of trends.