arXiv research

In the presence of a layer of metaprobabilities (from uncertainty concerning the parameters), the asymptotic tail exponent corresponds to the lowest possible tail exponent regardless of its probability. The problem explains "Black Swan" effects, i.e., why measurements tend to chronically underestimate tail contribution…

In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and genera…

In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can …

New bounds link generalization to stochastic optimizer's lower tail exponents.

We consider strictly stationary heavy tailed time series whose finite-dimensional exponent measures are concentrated on axes, and hence their extremal properties cannot be tackled using classical multivariate regular variation that is suitable for time series with extremal dependence. We recover relevant information ab…

There is convincing evidence showing that the probability distributions of stock returns in mature markets exhibit power-law tails and both the positive and negative tails conform to the inverse cubic law. It supports the possibility that the tail exponents are universal at least for mature markets in the sense that th…

Study on heavy tails in closing auction returns, explaining imbalance through limit order submission.

In this paper we tackle the problem of estimating the power-law tail exponent of income distributions by using the Hill's estimator. A subsample semi-parametric bootstrap procedure minimising the mean squared error is used to choose the power-law cutoff value optimally. This technique is applied to personal income data…

The higher-end tail of the wealth distribution in India is studied using recently published lists of the wealth of richest Indians between the years 2002-4. The resulting rank distribution seems to imply a power-law tail for the wealth distribution, with a Pareto exponent between 0.81 and 0.92 (depending on the year un…

The total duration of drawdowns is shown to provide a moment-free, unbiased, efficient and robust estimator of Sharpe ratios both for Gaussian and heavy-tailed price returns. We then use this quantity to infer an analytic expression of the bias of moment-based Sharpe ratio estimators as a function of the return distrib…

In a recent Nature paper, Gabaix et al. \cite{Gabaix03} presented a theory to explain the power law tail of price fluctuations. The main points of their theory are that volume fluctuations, which have a power law tail with exponent roughly -1.5, are modulated by the average market impact function, which describes the r…

Study free energy in spherical spin glasses, proving universality dichotomy.

We examine random variables in the power law/regularly varying class with stochastic tail exponent, the exponent $α$ having its own distribution. We show the effect of stochasticity of $α$ on the expectation and higher moments of the random variable. For instance, the moments of a right-tailed or right-asymmetric varia…

We introduce a simple generalization of rational bubble models which removes the fundamental problem discovered by [Lux and Sornette, 1999] that the distribution of returns is a power law with exponent less than 1, in contradiction with empirical data. The idea is that the price fluctuations associated with bubbles mus…

Detection of power-law behavior and studies of scaling exponents uncover the characteristics of complexity in many real world phenomena. The complexity of financial markets has always presented challenging issues and provided interesting findings, such as the inverse cubic law in the tails of stock price fluctuation di…

Direct measurements of Gini coefficients by conventional arithmetic calculations are a poor estimator, even if paradoxically, they include the entire population, as because of super-additivity they cannot lend themselves to comparisons between units of different size, and intertemporal analyses are vitiated by the popu…

The condition for stationary increments, not scaling, detemines long time pair autocorrelations. An incorrect assumption of stationary increments generates spurious stylized facts, fat tails and a Hurst exponent H_s=1/2, when the increments are nonstationary, as they are in FX markets. The nonstationarity arises from s…

We consider the evolution of scale-free networks according to preferential attachment schemes and show the conditions for which the exponent characterizing the degree distribution is bounded by upper and lower values. Our framework is an agent model, presented in the context of economic networks of trades, which shows …

New convergence bounds for online learning with heavy-tailed noise.

This paper addresses the statistical properties of time series driven by rational bubbles a la Blanchard and Watson (1982), corresponding to multiplicative maps, whose study has recently be revived recently in physics as a mechanism of intermittent dynamics generating power law distributions. Using insights on the beha…

Study shows flash crashes in finance are self-organized criticality events.

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual markets shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short m…

Proposes a new tail risk measure based on the most probable maximum risk event size.

Implementing a set of microeconomic criteria, we develop price dynamics equations using a function of demand/supply with key symmetry properties. The function of demand/supply can be linear or nonlinear. The type of function determines the nature of the tail of the distribution based on the randomness in the supply and…

Effects of randomness on non-integer power law tails in multiplicatively interacting stochastic processes are investigated theoretically. Generally, randomness causes decrease of the exponent of tails and the growth rate of processes. Explicit calculations are performed for two examples: uniformly distributed and two p…

On the framework of the Linear Farmer's Model, we approach the indeterminacy of agents' behaviour by associating with each agent an unconditional probability for her to be active at each time step. We show that Pareto tailed returns can appear even if value investors are the only strategies on the market and give a pro…

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not…

The multifractal behavior for tick data of prices is investigated in Korean financial market. Using the rescaled range analysis(R/S analysis), we show the multifractal nature of returns for the won-dollar exchange rate and the KOSPI. We also estimate the Hurst exponent and the generalized $q$th-order Hurst exponent in …

The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…

Detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency Deutsche Aktienindex data. The tail index ($α$), the Renyi exponents based on the box counting algorithm for the graph ($d_q$) and the generalized Hurst ex…

Study reveals that cryptocurrency price variations follow power-law distributions, influenced by age and market capitalization.

We assume the market price to diffuse in a hierarchical comb of barriers, the heights of which represent the importance of new information entering the market. We find fat tails with the desired exponent for the price change distribution, and effective multifractality for intermediate times.

We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric and asymmetric broad probability distributions of synthetic data are examined s…

It is shown phenomenologically that the fractional derivative $ξ=D^αu$ of order $α$ of a multifractal function has a power-law tail $\propto |ξ| ^{-p_\star}$ in its cumulative probability, for a suitable range of $α$'s. The exponent is determined by the condition $ζ_{p_\star} = αp_\star$, where $ζ_p$ is the exponent of…

A time series that represents daily values of the WIG index (the main index of Warsaw Stock Exchange) over last 5 years is examined. Non-Gaussian features of distributions of fluctuations, namely returns, over a time scale are considered. Some general properties like exponents of the long range correlation estimated by…

Starting from an exact relationship between news, threshold and price return distributions in the stationary state, I discuss the ability of the Ghoulmie-Cont-Nadal model of traders to produce fat-tailed price returns. Under normal conditions, this model is not able to transform Gaussian news into fat-tailed price retu…

In this note, we study the ultimate ruin probabilities of a real-valued L{é}vy process X with light-tailed negative jumps. It is well-known that, for such L{é}vy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to …

We investigate the use of the Hurst exponent, dynamically computed over a moving time-window, to evaluate the level of stability/instability of financial firms. Financial firms bailed-out as a consequence of the 2007-2010 credit crisis show a neat increase with time of the generalized Hurst exponent in the period prece…

We examine the performance of six estimators of the power-law cross-correlations -- the detrended cross-correlation analysis, the detrending moving-average cross-correlation analysis, the height cross-correlation analysis, the averaged periodogram estimator, the cross-periodogram estimator and the local cross-Whittle e…

We study the tick dynamical behavior of the yen-dollar exchange rate using the rescaled range analysis in financial market. It is found that the multifractal Hurst exponents with the short and long-run memory effects can be obtained from the yen-dollar exchange rate. This exists one crossover for the Hurst exponents at…

Personal income distributions in Japan are analyzed empirically and a simple stochastic model of the income process is proposed. Based on empirical facts, we propose a minimal two-factor model. Our model of personal income consists of an asset accumulation process and a wage process. We show that these simple processes…

We respond to the issues discussed by Farmer and Lillo (FL) related to our proposed approach to understanding the origin of power-law distributions in stock price fluctuations. First, we extend our previous analysis to 1000 US stocks and perform a new estimation of market impact that accounts for splitting of large ord…

In complex systems such as turbulent flows and financial markets, the dynamics in long and short time-lags, signaled by Gaussian and fat-tailed statistics, respectively, calls for a unified description. To address this issue we analyze a real dataset, namely, price fluctuations, in a wide range of temporal scales to em…

Are large scale research programs that include many projects more productive than smaller ones with fewer projects? This problem of economy of scale is particularly relevant for understanding recent mergers in particular in the pharmaceutical industry. We present a quantitative theory based on the characterization of d…

We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i…

We propose a family of models that enable predictive estimation of time-varying extreme event probabilities in heavy-tailed and nonlinearly dependent time series. The models are a white noise process with conditionally log-Laplace stochastic volatility. In contrast to other, similar stochastic volatility formalisms, th…

We consider the scaling behaviors for fluctuations of the number of Korean firms bankrupted in the period from August 1 2002 to October 28 2003. We observe a power law for the distribution of the number of the bankrupted firms. The Pareto exponent is close to unity. We also consider the daily increments of the number o…

One of the principal statistical features characterizing the activity in financial markets is the distribution of fluctuations in market indicators such as the index. While the developed stock markets, e.g., the New York Stock Exchange (NYSE) have been found to show heavy-tailed return distribution with a characteristi…