31 research outputs found
Long-Time Fluctuations in a Dynamical Model of Stock Market Indices
Financial time series typically exhibit strong fluctuations that cannot be
described by a Gaussian distribution. In recent empirical studies of stock
market indices it was examined whether the distribution P(r) of returns r(tau)
after some time tau can be described by a (truncated) Levy-stable distribution
L_{alpha}(r) with some index 0 < alpha <= 2. While the Levy distribution cannot
be expressed in a closed form, one can identify its parameters by testing the
dependence of the central peak height on tau as well as the power-law decay of
the tails. In an earlier study [Mantegna and Stanley, Nature 376, 46 (1995)] it
was found that the behavior of the central peak of P(r) for the Standard & Poor
500 index is consistent with the Levy distribution with alpha=1.4. In a more
recent study [Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999)] it was found
that the tails of P(r) exhibit a power-law decay with an exponent alpha ~= 3,
thus deviating from the Levy distribution. In this paper we study the
distribution of returns in a generic model that describes the dynamics of stock
market indices. For the distributions P(r) generated by this model, we observe
that the scaling of the central peak is consistent with a Levy distribution
while the tails exhibit a power-law distribution with an exponent alpha > 2,
namely beyond the range of Levy-stable distributions. Our results are in
agreement with both empirical studies and reconcile the apparent disagreement
between their results
The origin of power-law distributions in self-organized criticality
The origin of power-law distributions in self-organized criticality is
investigated by treating the variation of the number of active sites in the
system as a stochastic process. An avalanche is then regarded as a first-return
random walk process in a one-dimensional lattice. Power law distributions of
the lifetime and spatial size are found when the random walk is unbiased with
equal probability to move in opposite directions. This shows that power-law
distributions in self-organized criticality may be caused by the balance of
competitive interactions. At the mean time, the mean spatial size for
avalanches with the same lifetime is found to increase in a power law with the
lifetime.Comment: 4 pages in RevTeX, 3 eps figures. To appear in J.Phys.G. To appear in
J. Phys.
Volatility Effects on the Escape Time in Financial Market Models
We shortly review the statistical properties of the escape times, or hitting
times, for stock price returns by using different models which describe the
stock market evolution. We compare the probability function (PF) of these
escape times with that obtained from real market data. Afterwards we analyze in
detail the effect both of noise and different initial conditions on the escape
time in a market model with stochastic volatility and a cubic nonlinearity. For
this model we compare the PF of the stock price returns, the PF of the
volatility and the return correlation with the same statistical characteristics
obtained from real market data.Comment: 12 pages, 9 figures, to appear in Int. J. of Bifurcation and Chaos,
200
Mesoscopic modelling of financial markets
We derive a mesoscopic description of the behavior of a simple financial
market where the agents can create their own portfolio between two investment
alternatives: a stock and a bond. The model is derived starting from the
Levy-Levy-Solomon microscopic model (Econ. Lett., 45, (1994), 103--111) using
the methods of kinetic theory and consists of a linear Boltzmann equation for
the wealth distribution of the agents coupled with an equation for the price of
the stock. From this model, under a suitable scaling, we derive a Fokker-Planck
equation and show that the equation admits a self-similar lognormal behavior.
Several numerical examples are also reported to validate our analysis
Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series
Notwithstanding the significant efforts to develop estimators of long-range
correlations (LRC) and to compare their performance, no clear consensus exists
on what is the best method and under which conditions. In addition, synthetic
tests suggest that the performance of LRC estimators varies when using
different generators of LRC time series. Here, we compare the performances of
four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis
(DFA), Backward Detrending Moving Average (BDMA), and centred Detrending Moving
Average (CDMA)]. We use three different generators [Fractional Gaussian Noises,
and two ways of generating Fractional Brownian Motions]. We find that CDMA has
the best performance and DFA is only slightly worse in some situations, while
FA performs the worst. In addition, CDMA and DFA are less sensitive to the
scaling range than FA. Hence, CDMA and DFA remain "The Methods of Choice" in
determining the Hurst index of time series.Comment: 6 pages (including 3 figures) + 3 supplementary figure
Crackling Noise
Crackling noise arises when a system responds to changing external conditions
through discrete, impulsive events spanning a broad range of sizes. A wide
variety of physical systems exhibiting crackling noise have been studied, from
earthquakes on faults to paper crumpling. Because these systems exhibit regular
behavior over many decades of sizes, their behavior is likely independent of
microscopic and macroscopic details, and progress can be made by the use of
very simple models. The fact that simple models and real systems can share the
same behavior on a wide range of scales is called universality. We illustrate
these ideas using results for our model of crackling noise in magnets,
explaining the use of the renormalization group and scaling collapses. This
field is still developing: we describe a number of continuing challenges
The Forbes 400, the Pareto power-law and efficient markets
Statistical regularities at
the top end of the wealth distribution in the United States are
examined using the Forbes 400 lists of richest Americans,
published between 1988 and 2003.
It is found that the wealths are distributed according to a power-law
(Pareto) distribution.
This result is explained using a
simple stochastic model
of multiple investors that incorporates the
efficient market hypothesis
as well as the multiplicative nature of financial market fluctuations