Non-Parametric Analyses of Log-Periodic Precursors to Financial Crashes

We apply two non-parametric methods to test further the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The analysis using the so-called (H,q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the ln(t_c-t) variable, where t_c is the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency $f = 1.02 \pm 0.05$ corresponding to the scaling ratio $\lambda = 2.67 \pm 0.12$. These values are in very good agreement with those obtained in past works with different parametric techniques.Comment: Latex document 13 pages + 58 eps figure

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

We propose that imitation between traders and their herding behaviour not only lead 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 decorated by decelerating log-periodic oscillations. We document this behaviour on the Japanese Nikkei stock index from 1990 to present and on the Gold future prices after 1980, both after their all-time highs. We perform simultaneously a parametric and non-parametric analysis 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 one providing a prediction for the general trend in the coming years. The non-parametric power spectrum analysis shows the existence of log-periodicity with high statistical significance, with a prefered scale ratio of $\lambda \approx 3.5$ for the Nikkei index $\lambda \approx 1.9$ for the Gold future prices, comparable to the values obtained for speculative bubbles leading to crashes.Comment: 14 pages with 4 figure

Significance of log-periodic precursors to financial crashes

We clarify the status of log-periodicity associated with speculative bubbles preceding financial crashes. In particular, we address Feigenbaum's [2001] criticism and show how it can be rebuked. Feigenbaum's main result is as follows: ``the hypothesis that the log-periodic component is present in the data cannot be rejected at the 95% confidence level when using all the data prior to the 1987 crash; however, it can be rejected by removing the last year of data.'' (e.g., by removing 15% of the data closest to the critical point). We stress that it is naive to analyze a critical point phenomenon, i.e., a power law divergence, reliably by removing the most important part of the data closest to the critical point. We also present the history of log-periodicity in the present context explaining its essential features and why it may be important. We offer an extension of the rational expectation bubble model for general and arbitrary risk-aversion within the general stochastic discount factor theory. We suggest guidelines for using log-periodicity and explain how to develop and interpret statistical tests of log-periodicity. We discuss the issue of prediction based on our results and the evidence of outliers in the distribution of drawdowns. New statistical tests demonstrate that the 1% to 10% quantile of the largest events of the population of drawdowns of the Nasdaq composite index and of the Dow Jones Industrial Average index belong to a distribution significantly different from the rest of the population. This suggests that very large drawdowns result from an amplification mechanism that may make them more predictable than smaller market moves.Comment: Latex document of 38 pages including 16 eps figures and 3 tables, in press in Quantitative Financ

Stock market crashes are outliers

We call attention against what seems to 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 index that with high probability the three largest crashes in this century are outliers. This result supports suggestion that large crashes result from specific amplification processes that might lead to observable pre-cursory signatures.Comment: 8 pages, 3 figures (accepted in European Physical Journal B

Stochastics theory of log-periodic patterns

We introduce an analytical model based on birth-death clustering processes to help understanding the empirical log-periodic corrections to power-law scaling and the finite-time singularity as reported in several domains including rupture, earthquakes, world population and financial systems. In our stochastics theory log-periodicities are a consequence of transient clusters induced by an entropy-like term that may reflect the amount of cooperative information carried by the state of a large system of different species. The clustering completion rates for the system are assumed to be given by a simple linear death process. The singularity at t_{o} is derived in terms of birth-death clustering coefficients.Comment: LaTeX, 1 ps figure - To appear J. Phys. A: Math & Ge

A Critical Behaviour of Anomalous Currents, Electric-Magnetic Universality and CFT_4

We discuss several aspects of superconformal field theories in four dimensions (CFT_4), in the context of electric-magnetic duality. We analyse the behaviour of anomalous currents under RG flow to a conformal fixed point in N=1, D=4 supersymmetric gauge theories. We prove that the anomalous dimension of the Konishi current is related to the slope of the beta function at the critical point. We extend the duality map to the (nonchiral) Konishi current. As a byproduct we compute the slope of the beta function in the strong coupling regime. We note that the OPE of $T_{\mu\nu}$ with itself does not close, but mixes with a special additional operator $\Sigma$ which in general is the Konishi current. We discuss the implications of this fact in generic interacting conformal theories. In particular, a SCFT_4 seems to be naturally equipped with a privileged off-critical deformation $\Sigma$ and this allows us to argue that electric-magnetic duality can be extended to a neighborhood of the critical point. We also stress that in SCFT_4 there are two central charges, c and c', associated with the stress tensor and $\Sigma$, respectively; c and c' allow us to count both the vector multiplet and the matter multiplet effective degrees of freedom of the theory.Comment: harvmac tex, 28 pages, 3 figures. Version to be published in Nucl. Phys.

Stock mechanics: a general theory and method of energy conservation with applications on DJIA

A new method, based on the original theory of conservation of sum of kinetic and potential energy defined for prices is proposed and applied on Dow Jones Industrials Average (DJIA). The general trends averaged over months or years gave a roughly conserved total energy, with three different potential energies, i.e. positive definite quadratic, negative definite quadratic and linear potential energy for exponential rises (and falls), sinusoidal oscillations and parabolic trajectories, respectively. Corresponding expressions for force (impact) are also given. Keywords:Comment: 14 pages, 3 figures, scehudled for IJMPC 17/ issue

Three-body properties of low-lying 12^{12}Be resonances

We compute the three-body structure of the lowest resonances of $^{12}$Be considered as two neutrons around an inert $^{10}$Be core. This is an extension of the bound state calculations of $^{12}$Be into the continuum spectrum. We investigate the lowest resonances of angular momenta and parities, $0^{\pm}$, $1^{-}$ and $2^{+}$. Surprisingly enough, they all are naturally occurring in the three-body model. We calculate bulk structure dominated by small distance properties as well as decays determined by the asymptotic large-distance structure. Both $0^{+}$ and $2^{+}$ have two-body $^{10}$Be-neutron d-wave structure, while $1^{-}$ has an even mixture of $p$ and d-waves. The corresponding relative neutron-neutron partial waves are distributed among $s$, $p$, and d-waves. The branching ratios show different mixtures of one-neutron emission, three-body direct, and sequential decays. We argue for spin and parities, $0^{+}$, $1^{-}$ and $2^{+}$, to the resonances at 0.89, 2.03, 5.13, respectively. The computed structures are in agreement with existing reaction measurements.Comment: To be published in Physical Review

Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories

For quantum field theories that flow between ultraviolet and infrared fixed points, central functions, defined from two-point correlators of the stress tensor and conserved currents, interpolate between central charges of the UV and IR critical theories. We develop techniques that allow one to calculate the flows of the central charges and that of the Euler trace anomaly coefficient in a general N=1 supersymmetric gauge theory. Exact, explicit formulas for $SU(N_c)$ gauge theories in the conformal window are given and analysed. The Euler anomaly coefficient always satisfies the inequality $$. This is new evidence in strongly coupled theories that this quantity satisfies a four-dimensional analogue of the $c$-theorem, supporting the idea of irreversibility of the RG flow. Various other implications are discussed.Comment: latex, 27 page

Bayesian model comparison for compartmental models with applications in positron emission tomography

We develop strategies for Bayesian modelling as well as model comparison, averaging and selection for compartmental models with particular emphasis on those that occur in the analysis of positron emission tomography (PET) data. Both modelling and computational issues are considered. Biophysically inspired informative priors are developed for the problem at hand, and by comparison with default vague priors it is shown that the proposed modelling is not overly sensitive to prior specification. It is also shown that an additive normal error structure does not describe measured PET data well, despite being very widely used, and that within a simple Bayesian framework simultaneous parameter estimation and model comparison can be performed with a more general noise model. The proposed approach is compared with standard techniques using both simulated and real data. In addition to good, robust estimation performance, the proposed technique provides, automatically, a characterisation of the uncertainty in the resulting estimates which can be considerable in applications such as PET