Anders Johansen

CRASHES AS CRITICAL POINTS

We study a rational expectation model of bubbles and crashes. The model has two components: (1) our key assumption is that a crash may be caused by local self-reinforcing imitation between noise traders. If the tendency for noise traders to imitate their nearest neighbors increases up to a certain point called the “critical” point, all noise traders may place the same order (sell) at the same time, thus causing a crash. The interplay between the progressive strengthening of imitation and the ubiquity of noise is characterized by the hazard rate, i.e. the probability per unit time that the crash will happen in the next instant if it has not happened yet. (2) Since the crash is not a certain deterministic out come of the bubble, it remains rational for traders to remain invested provided they are compensated by a higher rate of growth of the bubble for taking the risk of a crash. Our model distinguishes between the end of the bubble and the time of the crash: the rational expectation constraint has the specific implication that the date of the crash must be random. The theoretical death of the bubble is not the time of the crash because the crash could happen at any time before, even though this is not very likely. The death of the bubble is the most probable time for the crash. There also exists a finite probability of the attaining the end of the bubble without crash. Our model has specific predictions about the presence of certain critical log-periodic patterns in pre-crash prices, associated with the deterministic components of the bubble mechanism. We provide empirical evidence showing that these patterns were indeed present before the crashes of 1929, 1962 and 1987 on Wall Street and the 1997 crash on the Hong Kong Stock Exchange. These results are compared with the statistical tests on synthetic data.

Predicting Financial Crashes Using Discrete Scale Invariance

We present a synthesis of all the available empirical evidence in the light of recent theoretical developments for the existence of characteristic log-periodic signatures of growing bubbles in a variety of markets including 8 unrelated crashes from 1929 to 1998 on stock markets as diverse as the US, Hong-Kong or the Russian market and on currencies. To our knowledge, no major financial crash preceded by an extended bubble has occurred in the past 2 decades without exhibiting such log-periodic signatures.

Stock Market Crashes, Precursors and Replicas

We present an analysis of the time behavior of the S&P 500 (Standard and Poors) New York stock exchange index before and after the October 1987 market crash and identify precursory patterns as well as aftershock signatures and characteristic oscillations of relaxation. Combined, they all suggest a picture of a kind of dynamical critical point, with characteristic log-periodic signatures, similar to what has been found recently for earthquakes. These observations are confirmed on other smaller crashes, and strengthen the view of the stockmarket as an example of a self-organizing cooperative system.

Large financial crashes

We propose that large stock market crashes are analogous to critical points studied in statistical physics with log-periodic correction to scaling. We extend our previous renormalization group model of stock market prices prior to and after crashes (D. Sornette, A. Johansen, J.P. Bouchaud, J. Phys. I France 6 (1996) 167) by including the first non-linear correction. This predicts the existence of a log-frequency shift over time in the log-periodic oscillations prior to a crash. This is tested on the two largest historical crashes of the century, the October 1929 and October 1987 crashes, by fitting the stock market index over an interval of 8 yr prior to the crashes. The good quality of the fits, as well as the consistency of the parameter values obtained from the two crashes, promote the theory that crashes have their origin in the collective “crowd” behavior of many interacting agents.

Finite-time singularity in the dynamics of the world population, economic and financial indices

Contrary to common belief, both the Earths human population and its economic output have grown faster than exponential, i.e., in a super-Malthusian mode, for most of the known history. These growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052±10 signaling an abrupt transition to a new regime. The degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in 1970, i.e., about 80 years before the predicted singular time, corresponding to approximately 4% of the studied time interval over which the acceleration is documented. This rounding-off of the finite-time singularity is probably due to a combination of well-known finite-size effects and friction and suggests that we have already entered the transition region to a new regime. As theoretical support, a multivariate analysis coupling population, capital, R&D and technology shows that a dramatic acceleration in the population growth during most of the timespan can occur even though the isolated dynamics do not exhibit it. Possible scenarios for the cross-over and the new regime are discussed.

The NASDAQ Crash of April 2000: Yet Another Example of Log-Periodicity in a Speculative Bubble Ending in a Crash

Abstract:The Nasdaq Composite fell another % on Friday the 14th of April 2000 signaling the end of a remarkable speculative high-tech bubble starting in spring 1997. The closing of the Nasdaq Composite at 3321 corresponds to a total loss of over 35% since its all-time high of 5133 on the 10th of March 2000. Similarities to the speculative bubble preceding the infamous crash of October 1929 are quite striking: the belief in what was coined a “New Economy” both in 1929 and presently made share-prices of companies with three digits price-earning ratios soar. Furthermore, we show that the largest draw downs of the Nasdaq are outliers with a confidence level better than 99% and that these two speculative bubbles, as well as others, both nicely fit into the quantitative framework proposed by the authors in a series of recent papers.

Stock market crashes are outliers

Abstract:We call attention against what seems to be a widely held misconception according to which large crashes are the largest events of distributions of price variations with fat tails. We demonstrate on the Dow Jones Industrial Average that with high probability the three largest crashes in this century are outliers. This result supports the suggestion that large crashes result from specific amplification processes that might lead to observable pre-cursory signatures.

Financial ``Anti-Bubbles'': Log-Periodicity in Gold and Nikkei collapses

We propose that the herding behavior of traders leads not only to speculative bubbles with accelerating over-valuations of financial markets possibly followed by crashes, but also to anti-bubbles with decelerating market devaluations following all-time highs. For this, we propose a simple market dynamics model in which the demand decreases slowly with barriers that progressively quench in, leading to a power law decay of the market price characterized by decelerating log-periodic oscillations. We document this behavior of the Japanese Nikkei stock index from 1990 to present and of the gold future prices after 1980, both after their all-time highs. We perform simultaneously parametric and nonparametric analyses that are fully consistent with each other. We extend the parametric approach to the next order of perturbation, comparing the log-periodic fits with one, two and three log-frequencies, the latter providing a prediction for the general trend in the coming years. The nonparametric power spectrum analysis shows the existence of log-periodicity with high statistical significance, with a preferred scale ratio of λ≈3.5 for the Nikkei index and λ≈1.9 for the Gold future prices, comparable to the values obtained for speculative bubbles leading to crashes.

A HIERARCHICAL MODEL OF FINANCIAL CRASHES

We follow up our previous conjecture that large stock market crashes are analogous to critical points in statistical physics. The term “critical” refers to regimes of cooperative behavior, such as magnetism at the Curie temperature and liquid–gas transitions, and is characterized by the singular mathematical behavior of relevant observables. To illustrate the concept of criticality, we present a simple hierarchical model of traders exhibiting “crowd” behavior and show that it has a well-defined critical point, whose mathematical signature is a power law dependence of the price, modulated by log-periodic structures, as recently found in market data by several independent groups.

Probing human response times

In a recent preprint (Dialog in e-mail traffic, preprint cond-mat/0304433), the temporal dynamics of an e-mail network has been investigated by Eckmann, Moses and Sergi. Specifically, the time period between an e-mail message and its reply were recorded. It will be shown here that their data agrees quantitatively with the frame work proposed to explain a recent experiment on the response of “internauts” to a news publication (Physica A 296(3–4) (2001) 539) despite differences in communication channels, topics, time-scale and socio-economic characteristics of the two population. This suggest a generalized response time distribution ∼t−1 for human populations in the absence of deadlines with important implications for psychological and social studies as well the study of dynamical networks.