A new paper from the ETH Zurich defines bubbles as episodes of unsustainable and quickening asset price growth with accelerating corrections and rebounds. In order to recognize such patterns it is critical to focus on the broader picture and correct time scale, rather than concurrent detail. Bubbles arise from innovations, valuation uncertainty and various positive feedback mechanisms that make prices spiral away from equilibrium. A critical state is often indicated by asset prices growing faster than exponentially.

“Financial bubbles: mechanisms and diagnostics”, Didier Sornette and Peter Cauwels,
ETH Zurich, April 2014

The below paragraphs are excerpts from the paper. Cursive lines and emphasis have been added.

What is an asset price bubble?

We define a bubble as a period of unsustainable growth, when the price of an asset increases ever more quickly, in a series of accelerating phases of corrections and rebounds. More technically, during a bubble phase, the price follows a faster-than-exponential power law growth process, often accompanied by log-periodic oscillations. This dynamic ends abruptly in a change of regime that may be a crash or a substantial correction. Because they leave such specific traces, bubbles may be recognized in advance: that is, before they burst.”

“Our research suggests that bubbles occur on all possible time scales. This means that they can form, develop and mature over a period of an hour, but also over a period of a century. As we do not know the time base of the bubble, the key to hunting for bubbles is to scan the data using different window sizes.”

Are asset price bubbles predictable?

“By its nature, trading is carried out by interacting players connected in a hierarchical network structure who affect one another continuously. As such, financial markets are open, adaptive, out-of-equilibrium systems that are subject to nonlinear dynamics, created particularly but not only by imitation and herd behaviour. Such structures are extremely hard to model, and it is practically impossible to understand their behaviour in full detail. In fact, this assertion can be formalised by algorithmic information theory…stating that, in a nutshell, most dynamic systems are intrinsically unpredictable.”

“However, if we take a step back and look at the broader, lower-resolution picture, predictability creeps back in and one can often be surprised to discover unexpected emerging macroscopic properties, like financial bubbles… To understand the complexity around us, and especially in economics and finance, we need to step back from traditional concepts such as utility maximisation or equilibrium in closed systems, and use innovative ways of thinking to discover sometimes surprisingly simple rules.”

“In general, economic models are designed so as to ensure the existence of an equilibrium, or at least a solution at any point in time. Here, we take another view by accepting that the market can have different phases with distinct dynamics, depending on whether the pricing mechanism follows a sustainable or a non-sustainable process. As a matter of fact, the non-existence of a solution is the key point in our methodology for predicting the end of a bubble.”

How do bubbles develop?

“When pricing stocks [according to a dividend discount model], estimates of different factors such as dividends, returns and growth expectations are needed, small differences in these estimates…may result in large deviations in the expected price…By voting with his or her wallet, each investor contributes to the collective knowledge, and the aggregate information is revealed in the asset price…Markets experience transient phases in which they disconnect in specific, dangerous ways from the fundamental value. These are situations in which investors are following the herd, and pushing a price up on an unsustainable growth trajectory.”

“A bubble starts with a new opportunity or expectation. This can be a ground-breaking new technology, the access to a new market, or a significant technical trading event such as the breaking of a support line. In any case, there must be a good story about terrific future prospects. Smart money flows in at the early stage, which leads to a first wave of price appreciation. Attracted by the prospect of extrapolated higher returns, more investors follow. At some point, demand goes up as the price increases, and the price goes up as the demand increases. This is a positive feedback mechanism, which fuels a spiralling growth away from equilibrium. As when pulling the plug out of the bathtub, the equilibrium has been broken and there is no longer any serious price determination at the intersection of supply and demand.”

“There are no positive feedback mechanisms in the standard financial models, which assume that the growth of asset prices is essentially a stochastic proportional process, fuelled by the mechanism of compound returns or interest rates. This means that, apart from its volatility, a stock price is supposed to grow, on average, exponentially at a constant rate of return. When positive feedback is involved, the dynamics change drastically. Now, the growth rate is no longer constant, but starts growing itself, which makes the price follow a (faster-than-exponential) hyperbolic course until, at some point, the growth rate becomes so large that the price hits a wall and the model breaks down.”

“There are many positive feedback mechanisms. They can be classified into two broad classes: (i) technical/rational and (ii) behavioural.

  • In the first class we find, for instance, option hedging and portfolio insurance techniques. Option hedging is the simple implementation of the Black-Scholes option theory, in which the risk associated with selling, say, a call option is eliminated by buying more of the underlying asset if the price has gone up recently and vice-versa.
  • Positive-feedback behavioural mechanisms include imitation and herd behaviour. It can, in fact, be proved that imitation turns out to be the optimum investment strategy in conditions of serious uncertainty, since the price, which results from the aggregate polling of decisions, can be proxied by the aggregate sentiments of one’s ‘friends’.”

When does a bubble burst?

“By nature, a financial bubble is an unsustainable process in which the system is gradually pushed towards criticality. In a critical system, small events can have huge impacts. There is no point in arguing about local causes and consequences when the system has reached a critical state, as it is the criticality that matters.”

“In principle, by verifying mathematically whether the price follows a hyperbolic course instead of an exponential one, it can be determined whether a positive feedback mechanism, which characterises the bubble phase of an asset, is at work. If this is the case, the price trajectory cannot be sustainable and a critical point will be reached at which a change in the market will occur. That is the point where the risk of a crash or a major correction is highest.”

“The risk of a major correction, or even a crash, becomes substantial when a bubble develops towards maturity, and that it is therefore very important to find evidence of bubbles and to follow their development from as early a stage as possible.”