Universal behavior of the interoccurrence times between losses in financial markets: Independence of the time resolution

Abstract

We consider representative financial records (stocks and indices) on time scales between one minute and one day, as well as historical monthly data sets, and show that the distribution P(Q)(r) of the interoccurrence times r between losses below a negative threshold -Q, for fixed mean interoccurrence times R(Q) in multiples of the corresponding time resolutions, can be described on all time scales by the same q exponentials, P(Q)(r)∝1/{[1+(q-1)βr](1/(q-1))}. We propose that the asset- and time-scale-independent analytic form of P(Q)(r) can be regarded as an additional stylized fact of the financial markets and represents a nontrivial test for market models. We analyze the distribution P(Q)(r) as well as the autocorrelation C(Q)(s) of the interoccurrence times for three market models: (i) multiplicative random cascades, (ii) multifractal random walks, and (iii) the generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. We find that only one of the considered models, the multifractal random walk model, approximately reproduces the q-exponential form of P(Q)(r) and the power-law decay of C(Q)(s).

Nonlinear q-Generalizations of Quantum Equations: Homogeneous and Nonhomogeneous Cases—An Overview

“…The authors of [92] verified that, by considering the q-plane-wave solution of Equation (6), one finds the balance equation with R( x, t) given by Equation (39), although the velocity field above yields:…”

“…This q-generalized thermostatistical theory has been useful in the study of a considerable number of physical systems, including long-range-interacting many-body Hamiltonian systems [11][12][13][14], low-dimensional dynamical systems [15][16][17][18][19], cold atoms [20][21][22], plasmas [23,24], trapped atoms [25], spin-glasses [26], power-law anomalous diffusion [27,28] and granular matter [29], high-energy particle collisions [30][31][32][33], black holes and cosmology [34,35], chemistry [36], economics [37][38][39], earthquakes [40], biology [41,42], solar wind [43,44], anomalous diffusion and central limit theorems [45][46][47][48][49][50][51][52][53], quantum entangled and non-entangled systems [54,55], quantum chaos [56], astronomic...…”

Statistical Collapse of Excessive Market Losses

“…Empirical market data on excessive profits and losses [5][6][7][8] define excessive profits as those greater than some positive fixed threshold Q and excessive losses as those below some negative threshold −Q. The mean interevent time † between losses versus Q has been used in this paper as an aggregated basic variable.…”

“…Interevent times constitute a universal stochastic measurement of market activity on time-scales that range from one minute to one month [5,6]. The mean interevent time can be used as a control variable that produces a universal description of empirical data collapse [7], i.e., the distribution of interevent times for a fixed mean interevent time is a universal statistical quantity unaffected by time-scale, type of market, asset, or index.…”

Coupled uncertainty provided by a multifractal random walker

“…To this end, we study the criticality in coupled systems by implementing the bivariate multifractal random walk method. However, it could be instructive to read the applications of this method in other disciplines [37][38][39][40][41]. This paper is organized as follows: In section II method for analysis will be explained.…”