Volatility, persistence, and survival in financial markets

Persistence in active turbulence

Active fluids such as bacterial swarms, self-propelled colloids, and cell tissues can all display complex spatiotemporal vortices that are reminiscent of inertial turbulence. This emergent behavior, despite the overdamped nature of these systems, is the hallmark of active turbulence. In this Letter, using a generalized hydrodynamic model, we present a study of the persistence problem in active turbulence. We report that the persistence time of passive tracers inside the coherent vortices follows a Weibull probability density whose shape and scale are decided by the strength of activity-contrary to inertial turbulence that displays power-law statistics in this region. In the turbulent background, the persistence time is exponentially distributed that is remindful of inertial turbulence. Finally we show that the driver of persistence inside the coherent vortices is the temporal decorrelation of the topological field, whereas it is the vortex turnover time in the turbulent background.

Stocks and cryptocurrencies: Antifragile or robust? A novel antifragility measure of the stock and cryptocurrency markets

In contrast with robust systems that resist noise or fragile systems that break with noise, antifragility is defined as a property of complex systems that benefit from noise or disorder. Here we define and test a simple measure of antifragility for complex dynamical systems. In this work we use our antifragility measure to analyze real data from return prices in the stock and cryptocurrency markets. Our definition of antifragility is the product of the return price and a perturbation. We explore different types of perturbations that typically arise from within the system. Our results suggest that for both the stock market and the cryptocurrency market, the tendency among the 'top performers' is to be robust rather than antifragile. It would be important to explore other possible definitions of antifragility to understand its role in financial markets and in complex dynamical systems in general.

Persistence in Brownian motion of an ellipsoidal particle in two dimensions

We investigate the persistence probability p(t) of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed its sign in the given time interval. We explicitly consider two cases-diffusion of a free particle and that of a harmonically trapped particle. The latter is particularly relevant in experiments that use trapping and tracking techniques to measure the displacements. We provide analytical expressions of p(t) for both the scenarios and show that in the absence of the shape asymmetry, the results reduce to the case of an isotropic particle. The analytical expressions of p(t) are further validated against numerical simulation of the underlying overdamped dynamics. We also illustrate that p(t) can be a measure to determine the shape asymmetry of a colloid and the translational and rotational diffusivities can be estimated from the measured persistence probability. The advantage of this method is that it does not require the tracking of the orientation of the particle.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process

The problem of zero crossings is of great historical prevalence and promises extensive application. The challenge is to establish precisely how the autocorrelation function or power spectrum of a one-dimensional continuous random process determines the density function of the intervals between the zero crossings of that process. This paper investigates the case where periodicities are incorporated into the autocorrelation function of a smooth process. Numerical simulations, and statistics about the number of crossings in a fixed interval, reveal that in this case the zero crossings segue between a random and deterministic point process depending on the relative time scales of the periodic and nonperiodic components of the autocorrelation function. By considering the Laplace transform of the density function, we show that incorporating correlation between successive intervals is essential to obtaining accurate results for the interval variance. The same method enables prediction of the density function tail in some regions, and we suggest approaches for extending this to cover all regions. In an ever-more complex world, the potential applications for this scale of regularity in a random process are far reaching and powerful.

Persistence of a Brownian particle in a time-dependent potential

We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times, leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential decay and an algebraic decay. Analytical calculations and numerical simulations show that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is determined by the exponent of the decay. Finally, using the two-time correlation function for the position of the Brownian particle, we construct the persistence probability for the Brownian walker in such a scenario.

Fluctuations in the zeros of differentiable Gaussian processes

The stochastic point processes formed by the zero crossings or extremal points of differentiable, stationary Gaussian processes are studied as a function of their autocorrelation function. The properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function, their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin. The distribution for the number of zeros occurring in an asymptotically large interval are approximately negative-binomial or binomial depending upon whether the relative variance or Fano factor is greater or less than unity. The correlation properties of the zeros are such that they are repelled from each other or are "antibunched" if the autocorrelation function of the Gaussian process is characterized by a single scale size, but occur in clusters if more than one characteristic scale size is present. The intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves. When bunching occurs the interval density becomes bimodal, indicating the interval sizes within and between the clusters. The interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the Gaussian process at large delay times. Poisson distributed fluctuations of the zeros occur only exceptionally but never form a Poisson process.

Dynamic scaling approach to study time series fluctuations

We propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface-growth problem. In this framework, the fluctuation sampling time interval plays the role of time variable, whereas the physical time is treated as the analog of spatial variable. In this way we found that the fluctuations of many real-world time series satisfy the analog of the Family-Viscek dynamic scaling ansatz. This finding permits us to use the powerful tools of kinetic roughening theory to classify, model, and forecast the fluctuations of real-world time series.

Extreme times for volatility processes

Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.

Scaling and memory of intraday volatility return intervals in stock markets

We study the return interval tau between price volatilities that are above a certain threshold q for 31 intraday data sets, including the Standard and Poor's 500 index and the 30 stocks that form the Dow Jones Industrial index. For different threshold q, the probability density function Pq(tau)scales with the mean interval tau as [Formula: see text], similar to that found in daily volatilities. Since the intraday records have significantly more data points compared to the daily records, we could probe for much higher thresholds and still obtain good statistics. We find that the scaling function f(x)is consistent for all 31 intraday data sets in various time resolutions, and the function is well-approximated by the stretched exponential, f(x) similar to e(-ax)(gamma), with gamma=0.38+/-0.05 and a=3.9+/-0.5, which indicates the existence of correlations. We analyze the conditional probability distribution Pq(tau/tau0) for tau following a certain interval tau0, and find Pq(tau/tau0) depends on tau0, which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval (tau/tau0) increases with tau0, consistent with the memory found for Pq(tau/tau0). Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by Pq(tau/tau0), have long-term correlations with correlation exponents similar to that of volatility records.