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 embrace both regimes. By means of Kramers-Moyal (KM) coefficients evaluated from empirical time series, we obtain the evolution equation for the probability density function (PDF) of price returns. We also present consistent asymptotic solutions for the timescale dependent equation that emerges from the empirical analysis. From these solutions, new relationships connecting PDF characteristics, such as tail exponents, to parameters of KM coefficients arise. The results reveal a dynamical path that leads from Gaussian to fat-tailed statistics, furnishing insights on other complex systems where akin crossover is observed.
In the existing financial literature, entropy based ideas have been proposed in portfolio optimization, in model calibration for options pricing as well as in ascertaining a pricing measure in incomplete markets. The abstracted problem corresponds to finding a probability measure that minimizes the relative entropy (also called $I$-divergence) with respect to a known measure while it satisfies certain moment constraints on functions of underlying assets. In this paper, we show that under $I$-divergence, the optimal solution may not exist when the underlying assets have fat tailed distributions…
I report a new statistical distribution formulated to confront the infamous, long-standing, computational/modeling challenge presented by highly skewed and/or leptokurtic ("fat- or heavy-tailed") data. The distribution is straightforward, flexible and effective. Even when working with far fewer data points than are routinely required, it models non-Gaussian data samples, from peak center through far tails, within the context of a single probability density function (PDF) that is valid over an extremely broad range of dispersions and probability densities. The distribution is a precision tool t…
In this paper we introduce an efficient fat-tail measurement framework that is based on the conditional second moments. We construct a goodness-of-fit statistic that has a direct interpretation and can be used to assess the impact of fat-tails on central data conditional dispersion. Next, we show how to use this framework to construct a powerful normality test. In particular, we compare our methodology to various popular normality tests, including the Jarque--Bera test that is based on third and fourth moments, and show that in many cases our framework outperforms all others, both on simulated…
We empirically analyze a large sample of firm sales growth expectations. We find that the relationship between forecast errors and lagged revision is non-linear. Forecasters underreact to typical (positive or negative) news about future sales, but overreact to very significant news. To account for this non-linearity, we propose a simple framework, where (1) sales growth dynamics have a fat-tailed high frequency component and (2) forecasters use a simple linear rule. This framework qualitatively fits several additional features of data on sales growth dynamics, forecast errors, and stock return…
Financial time series have been investigated to follow fat-tailed distributions. Further, an empirical probability distribution sometimes shows cut-off shapes on its tails. To describe this stylized fact, we incorporate the cut-off effect in superstatistics. Then we confirm that the presented stochastic model is capable of describing the statistical properties of real financial time series. In addition, we present an option pricing formula with respect to superstatistics.
We construct a two-tailed peak-over-threshold Hawkes model that captures asymmetric self- and cross-excitation in and between left- and right-tail extreme values within a time series. We demonstrate its applicability by investigating extreme gains and losses within the daily log-returns of the S&P 500 equity index. We find that the arrivals of extreme losses and gains are described by a common conditional intensity to which losses contribute twice as much as gains. However, the contribution of the former decays almost five times more quickly than that of the latter. We attribute these asymmetr…
We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails -- ubiquitous in the statistics of social, economic, and ecological systems. Our main goal is to provide a series of exact results on the dynamics and asymptotic behaviour of increasingly complex versions of a basic multiplicative process with resets, including discrete and continuous-time variants and several degrees of randomness in the parameters that control the process. In particular, we show how the power-law distributions are built up as…
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness e…
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯人工智能实验室）；City University of Hong Kong（香港城市大学）
We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns. Our model captures well both the time-varying characteristic and the asymmetrical heavy-tail property of financial time series. It combines the merits of a popular sequential neural network model, i.e., LSTM, with a novel parametric quantile function that we construct to represent the conditional distribution of asset returns. Our model also captures individually the serial dependences of higher moments, rather than just the volatility. Across a wide range of asset classes, t…
We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called "value investing", i.e. systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are normally distributed and uncorrelated across time. All this changes when the funds are allowed to leverage, i.e. borrow from a bank, to purchase more assets than their wealth would otherwise permit. During good times competition drives investors to funds that use more leverage, because they have higher pr…
In the post-crisis era, financial regulators and policymakers are increasingly interested in data-driven tools to measure systemic risk and to identify systemically important firms. Granger Causality (GC) based techniques to build networks among financial firms using time series of their stock returns have received significant attention in recent years. Existing GC network methods model conditional means, and do not distinguish between connectivity in lower and upper tails of the return distribution - an aspect crucial for systemic risk analysis. We propose statistical methods that measure con…
We investigate the financial market dynamics by introducing a heterogeneous agent-based opinion formation model. In this work, we organize the individuals in a financial market by their trading strategy, namely noise traders and fundamentalists. The opinion of a local majority compels the market exchanging behavior of noise traders, whereas the global behavior of the market influences the fundamentalist agents' decisions. We introduce a noise parameter $q$ to represent a level of anxiety and perceived uncertainty regarding the market behavior, enabling the possibility for an adrift financial a…
2T-POT Hawkes model for left- and right-tail conditional quantile forecasts of financial log-returns: out-of-sample comparison of conditional EVT models
We present some stylized facts exhibited by the time series of returns of the Mexican Stock Exchange Index (IPC) and compare them to a sample of both developed (USA, UK and Japan) and emerging markets (Brazil and India). The period of study is 1997-2011. The stylized facts are related mostly to the probability distribution func- tion and the autocorrelation function (e.g. fat tails, non-normality, volatility cluster- ing, among others). We find that positive skewness for returns in Mexico and Brazil, but not in the rest, suggest investment opportunities. Evidence of nonlinearity is also docume…
Inference over tails is usually performed by fitting an appropriate limiting distribution over observations that exceed a fixed threshold. However, the choice of such threshold is critical and can affect the inferential results. Extreme value mixture models have been defined to estimate the threshold using the full dataset and to give accurate tail estimates. Such models assume that the tail behavior is constant for all observations. However, the extreme behavior of financial returns often changes considerably in time and such changes occur by sudden shocks of the market. Here we extend the ex…
In this paper, we use the generalized Hurst exponent approach to study the multi- scaling behavior of different financial time series. We show that this approach is robust and powerful in detecting different types of multiscaling. We observe a puzzling phenomenon where an apparent increase in multifractality is measured in time series generated from shuffled returns, where all time-correlations are destroyed, while the return distributions are conserved. This effect is robust and it is reproduced in several real financial data including stock market indices, exchange rates and interest rates. …
We establish new results for estimation and inference in financial durations models, where events are observed over a given time span, such as a trading day, or a week. For the classical autoregressive conditional duration (ACD) models by Engle and Russell (1998, Econometrica 66, 1127-1162), we show that the large sample behavior of likelihood estimators is highly sensitive to the tail behavior of the financial durations. In particular, even under stationarity, asymptotic normality breaks down for tail indices smaller than one or, equivalently, when the clustering behaviour of the observed eve…
Volatility is a key measure of risk in financial analysis. The high volatility of one financial asset today could affect the volatility of another asset tomorrow. These lagged effects among volatilities - which we call volatility spillovers - are studied using the Vector AutoRegressive (VAR) model. We account for the possible fat-tailed distribution of the VAR model errors using a VAR model with errors following a multivariate Student t-distribution with unknown degrees of freedom. Moreover, we study volatility spillovers among a large number of assets. To this end, we use penalized estimation…
We investigate the distributions of epsilon-drawdowns and epsilon-drawups of the most liquid futures financial contracts of the world at time scales of 30 seconds. The epsilon-drawdowns (resp. epsilon- drawups) generalise the notion of runs of negative (resp. positive) returns so as to capture the risks to which investors are arguably the most concerned with. Similarly to the distribution of returns, we find that the distributions of epsilon-drawdowns and epsilon-drawups exhibit power law tails, albeit with exponents significantly larger than those for the return distributions. This paradoxica…
One of the standardized features of financial data is that log-returns are uncorrelated, but absolute log-returns or their squares namely the fluctuating volatility are correlated and is characterized by heavy tailed in the sense that some moment of the absolute log-returns is infinite and typically non-Gaussian [20]. And this last characteristic change accordantly to different timescales. We propose to model this long-memory phenomenon by superstatistical dynamics and provide a Bayesian Inference methodology drawing on Metropolis-Hasting random walk sampling to determine which superstatistics…
Financial markets provide an ideal frame for the study of crossing or first-passage time events of non-Gaussian correlated dynamics mainly because large data sets are available. Tick-by-tick data of six futures markets are herein considered resulting in fat tailed first-passage time probabilities. The scaling of the return with the standard deviation collapses the probabilities of all markets examined, and also for different time horizons, into single curves, suggesting that first-passage statistics is market independent (at least for high-frequency data). On the other hand, a very closely rel…
