4 - Predicting Financial Crashes?
Stock market crashes are momentous financial events that are fascinating to academics and practitioners alike. Within the efficient markets literature, only the revelation of a dramatic piece of information can cause a crash, yet in reality even the most thorough post-mortem analyses are typically inconclusive as to what this piece of information might have been. For traders, the fear of a crash is a perpetual source of stress, and the onset of the event itself always ruins the lives of some of them, not to mention the impact on the economy.
A few years ago, we advanced the hypothesis [Sornette et al., 1996] that stock market crashes are caused by the slow buildup of powerful "subterranean forces'' that come together in one critical instant. The use of the word "critical'' is not purely literary here: in mathematical terms, complex dynamical systems such as the stock market can go through so-called "critical'' points, defined as the explosion to infinity of a normally well-behaved quantity. As a matter of fact, as far as nonlinear dynamic systems go, the existence of critical points may be the rule rather than the exception. Given the puzzling and violent nature of stock market crashes, it is worth investigating whether there could possibly be a link.
In doing so, we have found three major points. First, it is entirely possible to build a dynamic model of the stock market exhibiting well-defined critical points that lies within the strict confines of rational expectations, a landmark of economic theory, and is also intuitively appealing. We stress the importance of using the framework of rational expectation in contrast to many other recent attempts. When you invest your money in the stock market, in general you do not do it at random but try somehow to optimize your strategy with your limited amount of information and knowledge. The usual criticism addressed to theories abandoning the rational behavior condition is that the universe of conceivable irrational behavior patterns is much larger than the set of rational patterns. Thus, it is sometimes claimed that allowing for irrationality opens a Pandora's box of ad hoc stories that have little out-of-sample predictive powers. To deserve consideration, a theory should be parsimonious, explain a range of anomalous patterns in different contexts, and generate new empirical implications.
Second, we find that the mathematical properties of a dynamic system going through a critical point are largely independent of the specific model posited, much more so in fact than "regular'' (non-critical) behavior, therefore our key predictions should be relatively robust to model misspecification.
Third, these predictions are strongly borne out in the U.S. stock market crashes of 1929 and 1987: indeed it is possible to identify clear signatures of near-critical behavior many years before the crashes and use them to "predict'' (out of sample) the date where the system will go critical, which happens to coincide very closely with the realized crash date. We also discovered in a systematic testing procedure a signature of near-critical behavior that culminated in a two weeks interval in May 1962 where the stock market declined by 12%. The fact that we "discovered'' the "slow crash'' of 1962 without prior knowledge of it just be trying to fit our theory is a reassuring sign about the integrity of the method. Analysis of more recent data showed a clear maturation towards a critical instability that can be tentatively associated to the turmoil of the US stock market at the end of october 1997. It may come as a surprise that the same theory is applied to epochs so much different in terms of speed of communications and connectivity as 1929 and 1997. It may be that what our theory addresses is the question: has human nature changed?
4.1 - Theory of Crashes with Rational Agents
Consider a single asset that pays no dividends and for simplicity, we ignore the interest rate, risk aversion, information asymmetry, and the market-clearing condition. In this dramatically stylized framework, rational expectations are simply equivalent to the familiar hypothesis that the present price of the asset is equal to its expectation over the future conditional on information revealed up to the present.
Suppose that we introduce an exogenous probability of crash. Then, simple calculations show that the higher the probability of a crash, the faster the price must increase (conditional on having no crash) in order to satisfy the no-arbitrage condition (i.e. that there are no "free lunch''). Intuitively, investors must be compensated by the chance of a higher return in order to be induced to hold an asset that might crash. This is the only effect that we wish to capture in this part of the model. This effect is fairly standard, and it was pointed out earlier in a closely related model of bubbles and crashes under rational expectations by Blanchard (1979, top of p. 389). It may go against the naive preconception that price is adversely affected by the probability of the crash, but our result is the only one consistent with rational expectations.
A few additional points deserve careful attention. First, the crash is modelled as an exogenous event: nobody knows exactly when it could happen, which is why rational traders cannot earn abnormal profits by anticipating it. Second, the probability of a crash itself is an exogenous variable that must come from outside this model. There is no feedback loop whereby prices would in turn affect either the arrival or the probability of a crash. This may not sound totally satisfactory, but it is hard to see how else to obtain crashes in a rational expectations model: if rational agents could somehow trigger the arrival of a crash they would choose never to do so, and if they could control the probability of a crash they would always choose it to be zero. In our model, the crash is a random event whose probability is driven by external forces, and once this probability is given it is rationally reflected into prices. If the other alternatives are to give up rational expectations or to give up studying crashes, we prefer to stick with our approach.
4.2 - The Crash
We now explain how to obtain a crash in terms of reasonable models of micro-level agent behavior. We start by a discussion in naive terms. A crash happens when a large group of agents place sell order simultaneously. This group of agents must create enough of an imbalance in the order book for market makers to be unable to absorb the other side without lowering prices substantially. One curious fact is that the agents in this group typically do not know each other. They did not convene a meeting and decide to provoke a crash. Nor do they take orders from a leader. In fact, most of the time, these agents disagree with one another, and submit roughly as many buy orders as sell orders (these are all the times when a crash does not happen). The key question is: by what mechanism did they suddenly manage to organize a coordinated sell-off?
We propose the following answer: all the traders in the world are organized into a network (of family, friends, colleagues, etc) and they influence each other locally through this network. Specifically, if I am directly connected with nearest neighbors, then there are only two forces that influence my opinion: (a) the opinions of these people; and (b) an idiosyncratic signal that I alone receive. Our working assumption here is that agents tend to imitate the opinions of their nearest neighbors, not contradict them. It is easy to see that force (a) will tend to create order, while force (b) will tend to create disorder. The main story that we are telling in here is the fight between order and disorder. As far as asset prices are concerned, a crash happens when order wins (everybody has the same opinion: selling), and normal times are when disorder wins (buyers and sellers disagree with each other and roughly balance each other out). We must stress that this is exactly the opposite of the popular characterization of crashes as times of chaos.
Our answer has the advantage that it does not require a global coordination mechanism: we will show that macro-level coordination can arise from micro-level imitation. Furthermore, it relies on a somewhat realistic model of how agents form opinions. It also makes it easier to accept that crashes can happen for no rational reason. If selling were a decision that everybody reached independently from one another just by reading the newspaper, either we would be able to identify unequivocally the triggering news after the fact (and for the crashes of 1929 and 1987 this was not the case), or we would have to assume that everybody becomes irrational in exactly the same way at exactly the same time (which is distasteful). By contrast, our reductionist model puts the blame for the crash simply on the tendency for agents to imitate their nearest neighbors. We do not ask why agents are influenced by their neighbors within a network: since it is a well- documented fact, we take it as a primitive assumption rather than as the conclusion of some other model of behavior. Presumably some justification for these imitative tendencies can be found in evolutionary psychology. Note, however, that there is no information in our model, therefore what determines the state of an agent is pure noise.
The output of the model is a quantity termed the susceptibility which measures the sensitivity of the average state to a perturbation. The susceptibility has a second interpretation as (a constant times) the variance of the average opinion around its expectation of zero caused by the random idiosyncratic shocks . Another related interpretation is that, if you consider two agents and you force the first one to be in a certain state, the impact that your intervention will have on the second agent will be proportional to the susceptibility. For these reasons, we believe that the susceptibility correctly measures the ability of the system of agents to agree on an opinion. If we interpret the two states in a manner relevant to asset pricing, it is precisely the emergence of this global synchronization from local imitation that can cause a crash. Thus, we will characterize the behavior of the susceptibility, and we will posit that the hazard rate of crash follows a similar process. We do not want to assume a one-to-one mapping between hazard rate and susceptibility because there are many other quantities that provide a measure of the degree of coordination of the overall system, such as the correlation length (i.e.~the distance at which imitation propagates) and the other moments of the fluctuations of the average opinion.
We argue that these properties are very robust to model misspecification. We claim that models of crash that combine the following features:
- A system of noise traders who are influenced by their neighbors; -Local imitation propagating spontaneously into global cooperation;
- Global coperation among noise traders causing crash;
- Prices related to the properties of this system;
- System parameters evolving slowly through time; would display the same characteristics as ours, namely prices following a power law in the neighborhood of some critical date, with either a real or complex critical exponent. What all models in this class would have in common is that the crash is most likely when the locally imitative system goes through a critical point.
In physics, critical points are widely considered to be the most interesting properties of complex systems. A system goes critical when local influences propagate over long distances and the average state of the system becomes exquisitely sensitive to a small perturbation. Another characteristic is that critical systems are self-similar across scales: in our example, at the critical point, an ocean of traders who are mostly bearish may have within it several islands of traders who are mostly bullish, each of which in turns surrounds lakes of bearish traders with islets of bullish traders; the progression continues all the way down to the smallest possible scale: a single trader [Wilson, 1979]. Intuitively speaking, critical self-similarity is why local imitation cascades through the scales into global coordination.
Because of scale invariance, the behavior of a system near its critical point must be represented by a power law (with real or complex critical exponent): it is the only family of functions that are homogenous, i.e.~they remain unchanged (up to scalar multiplication) when their argument gets rescaled by a constant. In general, physicists study critical points by forming equations to describe the behavior of the system across different scales, and by analyzing the mathematical properties of these equations. This is known as renormalization group theory (Wilson, 1979) as already discussed. Before renormalization group theory, the fact that a system's critical behavior had to be correctly described at all scales simultaneously prevented standard approximation methods from giving satisfactory results. But renormalization group theory turned this liability into an asset by building its solution precisely on the self-similarity of the system across scales. Let us add that, in spite of its conceptual elegance, this method is nonetheless mathematically challenging.
For our purposes, however, it is sufficient to keep in mind that the key idea proposed here is the following: the massive and unpredictable sell-off occuring during stock market crashes comes from local imitation cascading through the scales into global cooperation when a complex system approaches its critical point. Regardless of the particular way in which this idea is implemented, it will generate the same universal implications.
Strictly speaking, these equations are approximations valid only in the neighborhood of the critical point. We have proposed a more general formula with additional degrees of freedom to better capture behavior away from the critical point. The specific way in which these degrees of freedom are introduced is based on a finer analysis of the renormalization group theory that is equivalent to including the next term in a systematic expansion around the critical point and introduce a log-periodic component to the market price behavior.
4.3 - Extended Efficiency and Systemic Instability
Our main point is that the market anticipates the crash in a subtle self-organized and cooperative fashion, hence releasing precursory "fingerprints'' observable in the stock market prices. In other words, this implies that market prices contain information on impending crashes. If the traders were to learn how to decipher and use this information, they would act on it and on the knowledge that others act on it and the crashes would probably not happen. Our results suggest a weaker form of the "weak efficient market hypothesis'' [Fama, 1991], according to which the market prices contain, in addition to the information generally available to all, subtle informations formed by the global market that most or all individual traders have not yet learned to decipher and use. Instead of the usual interpretation of the efficient market hypothesis in which traders extract and incorporate consciously (by their action) all informations contained in the market prices, it may be that the market as a whole can exhibit an "emergent'' behavior not shared by any of its constituant. In other words, we have in mind the process of the emergence of intelligent behaviors at a macroscopic scale that individuals at the microscopic scale have not idea of. This process has been discussed in biology for instance in animal populations such as ant colonies or in connection with the emergence of conciousness [Anderson et al., 1988; Holland, 1992]. The usual efficient market hypothesis will be recovered in this context when the traders learn how to extract this novel collective information and act on it.
Most previous models proposed for crashes have pondered the possible mechanisms to explain the collapse of the price at very short time scales. Here in contrast, we propose that the underlying cause of the crash must be searched years before it in the progressive accelerating ascent of the market price, the speculative bubble, reflecting an increasing built-up of the market cooperativity. From that point of view, the specific manner by which prices collapsed is not of real importance since, according to the concept of the critical point, any small disturbance or process may have triggered the instability, once ripe. The intrinsic divergence of the sensitivity and the growing instability of the market close to a critical point might explain why attempts to unravel the local origin of the crash have been so diverse. Essentially all would work once the system is ripe. Our view is that the crash has an endegeneous origin and that exogeneous shocks only serve as triggering factors. We propose that the origin of the crash is much more subtle and is constructed progressively by the market as a whole. In this sense, this could be termed a systemic instability. This understanding offers ways to act to mitigate the build-up of conditions favorable to crashes.