Faster Than Exponential: Can You See a Crash Coming?

8 min read Keywords:  Sornette, log-periodic power law, bubbles, finite-time singularity, dragon kings, black swans, crash prediction, critical points
A small boat pitched almost vertical on a black sea while the crew struggle with the sail and one figure is sick over the side
The Storm on the Sea of Galilee, Rembrandt van Rijn, 1633. Isabella Stewart Gardner Museum, Boston (public domain).

The last two essays were pessimistic about prediction. Crashes are endogenous, the biggest moves have no special cause, and hunting for the grain that triggered the avalanche is hunting for something that was never special. Take that seriously and the only sane response is defense: carry slack, cut leverage, expect the cascade.

Didier Sornette spent a career arguing that this gives up too early. Not about every crash, but about a particular kind: the ones at the end of a bubble. His claim is that a bubble has a shape, that the shape is visible while it is forming, and that the shape lets you say something, with all the usual hedges, about when it will break. This is the optimistic case in this series, and it is worth taking seriously, including where it falls down.

#I. Two kinds of growth

Start with the difference between fast and dangerous.

Exponential growth is what compound interest does. A fixed percentage each period, so the thing doubles on a regular schedule. It looks dramatic, but the growth rate is constant. A savings account, a healthy company, a population with steady birth rates: all exponential, all sustainable in the sense that nothing about the rate is speeding up.

A bubble is something else. In a bubble the growth rate itself grows. Prices rise, and the rise pulls in buyers who push prices up faster, which pulls in more buyers still. The thing that is increasing is not just the price but the speed of the price. This is faster-than-exponential growth, and it has a strange property: it does not head toward infinity in the far future, it heads toward infinity at a specific finite time. Mathematicians call that a finite-time singularity. In a market it cannot literally happen, so what the singularity really marks is the moment the trend has to break.

Two panels. Left, on an ordinary scale, an exponential curve and a faster-than-exponential curve start together; the faster one bends upward and shoots toward a vertical wall at a fixed time. Right, on a log scale, the exponential is a straight line while the faster-than-exponential one still curves upward.

Exponential growth has a constant rate, so on a log scale it is a straight line. Faster-than-exponential growth keeps curving up even on a log scale, and it runs into a wall at a finite time. That wall is the danger.

The log-scale panel on the right is the quick test. Steady compounding is a straight line there. A bubble is not. It keeps bending upward, because the rate keeps rising, and that upward bend is the fingerprint of the positive feedback that cannot last.

#II. The wobble that gives it away

Here is where Sornette goes past everyone else. He argues that the climb toward the singularity is not smooth. It carries oscillations, and the oscillations speed up as the critical time approaches, squeezing closer and closer together near the end.

A price path that rises with growing speed while oscillating, the swings getting faster and tighter as it nears a dashed vertical line marked as the predicted crash time, after which it drops.

The bubble accelerates, and the wobbles on the way up get faster as the end nears. Sornette’s model fits this shape to estimate the crash time. The wobbles are the part that is supposed to let you read the clock.

The technical name is a log-periodic power law, and the intuition behind the wobble is that the herding has a kind of rhythm, a discrete echo that gets compressed in time as the system tightens toward its critical point. You do not need the machinery. The claim is simply that a forming bubble has both an accelerating trend and an accelerating wobble, and that fitting both at once pins down the critical time better than the trend alone. Sornette built an entire research program around this, the Financial Crisis Observatory at ETH Zurich, dedicated to fitting these shapes to live markets and logging the forecasts in advance.

#III. Dragon kings and black swans

This is a direct quarrel with Taleb, and Sornette picked it on purpose.

Taleb’s black swan is an event so far out in the tail that it is effectively unpredictable. In the language of the earlier essays, it sits on the power-law line: rare, large, but the same kind of thing as a small event, just a more extreme draw from the same distribution. Nothing about it is special except its size, so nothing lets you see it coming.

Sornette’s answer is the dragon king. His claim is that the very largest events in some systems are not on the line. They are a separate population, generated by a different mechanism, the runaway feedback of a bubble reaching its critical point. And because they have their own mechanism, they leave their own signature, which means they are partly foreseeable in a way a black swan is not.

A log-log plot of event size against how often it happens. Most points fall on a straight power-law line and are labeled black swans, just rare draws. A few large points sit well above the extrapolated line and are labeled dragon kings, with their own mechanism.

A black swan sits on the line: a rare draw from the same law. A dragon king sits above the line: too big and too frequent to belong to that law, the mark of a separate mechanism. Sornette’s bet is that dragon kings, unlike black swans, give warning.

So the disagreement is sharp and clean. Taleb says the giant events are unforeseeable and you should build robustness. Sornette says a subset of giant events announce themselves and you can act on the warning. They cannot both be fully right, and the fight is more useful than either position alone.

#IV. How much to believe

I find the picture compelling and I would not trade on it, and those two things are not in tension.

The honest reading is that Sornette is on strong ground about the mechanism and much weaker ground about the forecast. That a bubble is faster-than-exponential feedback heading for a break is convincing and well evidenced. That you can read the date of the break off the wobbles is far shakier. The critical time the model returns is not a date, it is a wide, fuzzy probability, and Sornette himself frames the crash as a random event whose odds rise as you approach, not an appointment. The fits look gorgeous after the fact. The live, pre-registered record is thinner and contested, and it suffers the usual selection problem, where the hits get remembered and the misses get explained. A bubble can also deflate without a crash, sliding back down instead of snapping, so the singularity is a hazard that can be avoided, not a destiny.

None of that kills the idea. It just sets the right expectation. The signature is real. Whether the signature is tradeable is a different and much harder claim, and Sornette is on much firmer footing for the first than the second.

#V. What it is good for

Used correctly, this is not a market-timing tool. It is a regime detector, and that is still worth a great deal.

If a market is climbing faster than exponentially, with the trend bending up even on a log scale, you are not in a normal market. You are in a positive-feedback regime that, by its own internal logic, has to end, and probably end sharply. You cannot say when. You can say that the distribution of what happens next has gone lopsided, that the crash hazard is elevated, and that this is the time to act like the previous essays told you to: trim, hedge, hold the insurance you bought when it was cheap.

In other words, the right way to use Sornette is to let his diagnosis tell you when to behave like Taleb. The next essay puts a number on how reflexive markets actually are, the one after that asks what the calm itself can tell you, and the last one presses Taleb’s case that the one quantity you would need to trust here, how close the market is to its critical point, is exactly the quantity you cannot.

#Further reading

The bubble-as-critical-point model:

Dragon kings versus black swans: