Discrete-scale invariance and complex dimensions

Sornette, Didier

doi:10.1016/s0370-1573(97)00076-8

Cited by 607 publications

(659 citation statements)

References 152 publications

(221 reference statements)

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“…This is in line with Sornette's (1998) hypothesis that a normally functioning market shows a high weighted average value of regularity exponents, combined with a narrow spectrum, indicating fairly accurate global persistence.…”

Section: Multifractal Patterns Around Stock Market Crashessupporting

confidence: 90%

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

Los

Yalamova

2004

SSRN Journal

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The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or "randomness" in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or "drawdowns." In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, however, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5 < H < 0.7, these spectra tend to be skwed towards anti-persistence in the returns.

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Section: Multifractal Patterns Around Stock Market Crashessupporting

confidence: 90%

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

Los

Yalamova

2004

SSRN Journal

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“…In the minimization of Q(A, B, C n ; Φ) given by (18), the linear parameters A, B and C n are determined accordingly to Eq. (20) as a function of the nonlinear parameters Φ.…”

Section: Fitting Proceduresmentioning

confidence: 99%

Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction

Zhou

Sornette

2003

Physica A: Statistical Mechanics and its Applications

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We propose a straightforward extension of our previously proposed log-periodic power law model of the "anti-bubble" regime of the USA market since the summer of 2000, in terms of the renormalization group framework to model critical points. Using a previous work by Gluzman and Sornette (2002) on the classification of the class of Weierstrass-like functions, we show that the five crashes that occurred since August 2000 can be accurately modelled by this approach, in a fully consistent way with no additional parameters. Our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish "anti-bubble" regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes. We develop retrospective predictions, that confirm the existence of significant arbitrage opportunities for a trader using our model. Finally, we offer a prediction for the unknown future of the US S&P500 index extending over 2003 and 2004, that refines the previous prediction of Sornette and Zhou (2002).

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“…The idea is to generalise the real exponent z to a complex exponent β + iω, such that a power law is changed into (t c − t) β+iω , whose real part is (t c − t) β cos (ω ln(t c − t)) [48]. The cosine will decorate the average power law behaviour with so-called log-periodic oscillations, the name steming from the fact the oscillations are periodic in ln(t c − t) and not in t. As we shall see, these log-periodic oscillations can account for a large part of the observed variability around the power law.…”

Section: Generalisation To Power Laws With Complex Exponents: Log-permentioning

confidence: 99%

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

Johansen

Sornette

2001

Physica A: Statistical Mechanics and its Applications

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Contrary to common belief, both the Earth's 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 occuring 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 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 wellknown finite-size effects and friction and suggests that we have already entered the transition region to a new regime. In theoretical support, a multivariate analysis coupling population, capital, R&D and technology shows that a dramatic acceleration in the population 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. IntroductionBoth the world economy as well as the human population have grown at a tremendous pace especially during the last two centuries. It is estimated that 2000 years ago the population of the world was approximately 300 million and for a long time the world population did not grow significantly, since periods of growth were followed by periods of decline. It took more than 1600 years for the world population to double to 600 million and since then the growth has accelerated. It reached 1 billion in 1804 (204 years later), 2 billion in 1927 (123 years later), 3 billion in 1960 (33 years later), 4 billion in 1974 (14 years later), 5 billion in 1987 (13 years later) and 6 billion in 1999 (12 years later). This rapidly accelerating growth has raised sincere worries about its sustainability as well as concerns that we humans as a result might cause severe and irreversible damage to eco-systems, global weather systems etc [6,22]. At, what one may say the other extreme, the optimists expect that the innovative spirit of mankind will be able to solve the problems associated with a continuing increase in the growth rate [15,46]. Specifically, they believe that the world economic development will continue as a successive unfolding of revolutions, e.g., the Internet, bio-technological and other yet unknown innovations, replacing the prior agricultural, industrial, medical and information revolutions of the past. Irrespective of the interpretation, the important point is the presence of an acceleration in the growth rate. Here, it is first shown that, contrary to common belief, both the Earth human population as well as its economic output have grown faster that exponential for most of the known history and most strikingly so in the last centuries. Furthermore, we will show that both the population growth rate and the economic growth rate are consistent with a spontaneous singula...

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Discrete-scale invariance and complex dimensions

Cited by 607 publications

References 152 publications

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction

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

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