Stock Returns and Roughness Extreme Variations: A New Model for Monitoring 2008 Market Crash and 2015 Flash Crash

We use Student’s t-copula to study the extreme variations in the bivariate kinematic time series of log–return and log–roughness of the S&P 500 index during two market crashes, the financial crisis in 2008 and the flash crash on Monday August 24, 2015. The stable and small values of the tail dependence index observed for some months preceding the market crash of 2008 indicate that the joint distribution of daily return and roughness was close to a normal one. The volatility of the tail and degree of freedom indices as determined by Student’s t-copula falls down substantially after the stock market crash of 2008. The number of degrees of freedom in the empirically observed distributions falls while the tail coefficient of the copula increases, indicating the long memory effect of the market crash of 2008. A significant change in the tail and degree of freedom indices associated with the intraday price of S&P 500 index is observed before, during, and after the flash crash on August 24, 2015. The long memory effect of the stock market flash crash of August 2015 is indicated by the number of degrees of freedom in the empirically observed distributions fall while the tail coefficient of the joint distribution increases after the flash crash. The small and stable value of degrees of freedom preceding the flash crash provides evidence that the joint distribution for intraday data of return and roughness is heavy-tailed. Time-varying long-range dependence in mean and volatility as well as the Chow and Bai-Perron tests indicate non-stability of the stock market in this period

Towards Finding Efficient Tools for Measuring the Tail Index and Intensity of Long-range Dependent Network Traffic

Many researchers have discussed the effects of heavy-tailedness in network traffic patterns and shown that Internet traffic flows exhibit characteristics of self-similarity that can be explained by the heavy-tailedness of the various distributions involved. Self-similarity and heavy-tailedness are of great importance for network capacity planning purposes in which researchers are interested in developing analytical methods for analysing traffic characteristics. Designers of computing and telecommunication systems are increasingly interested in employing heavy-tailed distributions to generate workloads for use in simulation - although simulations employing such workloads may show unusual characteristics. Congested Internet situations, where TCP/IP buffers start to fill, show long-range dependent (LRD) self-similar chaotic behaviour. Such chaotic behaviour has been found to be present in Internet traffic by many researchers. In this context, the 'Hurst exponent', H, is used as a measure of the degree of long-range dependence. Having a reliable estimator can yield a good insight into traffic behaviour and may eventually lead to improved traffic engineering. In this paper, we describe some of the most useful mechanisms for estimating the tail index of Internet traffic, particularly for distributions having the power law observed in different contexts, and also the performance of the estimators for measuring the intensity of LRD traffic in terms of their accuracy and reliability

Analytic model for MPEG-4 and H.263 encoded video traces

Summary form only given. We present the analysis of statistical distributions of 1-hour long publicly available empirical samples of MPEG-4 and H.263 encoded video traces. Each video has been encoded to different quality levels. The results of our analysis show that: i) although the autocorrelation of the studied traces asserts long-range dependence, however, the distribution of the same traces is not heavy-tailed; ii) for different quality levels, the traces belong to the same statistical distribution; iii) most of the studied traces show long-tail behavior which can be modeled using lognormal distribution function

Analytic model for MPEG-4 and H.263 encoded video traces

Summary form only given. We present the analysis of statistical distributions of 1-hour long publicly available empirical samples of MPEG-4 and H.263 encoded video traces. Each video has been encoded to different quality levels. The results of our analysis show that: i) although the autocorrelation of the studied traces asserts long-range dependence, however, the distribution of the same traces is not heavy-tailed; ii) for different quality levels, the traces belong to the same statistical distribution; iii) most of the studied traces show long-tail behavior which can be modeled using lognormal distribution function

Heavy-Tailed Features and Empirical Analysis of the Limit Order Book Volume Profiles in Futures Markets

This paper poses a few fundamental questions regarding the attributes of the volume profile of a Limit Order Books stochastic structure by taking into consideration aspects of intraday and interday statistical features, the impact of different exchange features and the impact of market participants in different asset sectors. This paper aims to address the following questions: 1. Is there statistical evidence that heavy-tailed sub-exponential volume profiles occur at different levels of the Limit Order Book on the bid and ask and if so does this happen on intra or interday time scales ? 2.In futures exchanges, are heavy tail features exchange (CBOT, CME, EUREX, SGX and COMEX) or asset class (government bonds, equities and precious metals) dependent and do they happen on ultra-high (<1sec) or mid-range (1sec -10min) high frequency data? 3.Does the presence of stochastic heavy-tailed volume profile features evolve in a manner that would inform or be indicative of market participant behaviors, such as high frequency algorithmic trading, quote stuffing and price discovery intra-daily? 4. Is there statistical evidence for a need to consider dynamic behavior of the parameters of models for Limit Order Book volume profiles on an intra-daily time scale ? Progress on aspects of each question is obtained via statistically rigorous results to verify the empirical findings for an unprecedentedly large set of futures market LOB data. The data comprises several exchanges, several futures asset classes and all trading days of 2010, using market depth (Type II) order book data to 5 levels on the bid and ask

Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models

Long-Range Dependence (LRD) and heavy-tailed distributions are ubiquitous in natural and socio-economic data. Such data can be self-similar whereby both LRD and heavy-tailed distributions contribute to the self-similarity as measured by the Hurst exponent. Some methods widely used in the physical sciences separately estimate these two parameters, which can lead to estimation bias. Those which do simultaneous estimation are based on frequentist methods such as Whittle’s approximate maximum likelihood estimator. Here we present a new and systematic Bayesian framework for the simultaneous inference of the LRD and heavy-tailed distribution parameters of a parametric ARFIMA model with non-Gaussian innovations. As innovations we use the α-stable and t-distributions which have power law tails. Our algorithm also provides parameter uncertainty estimates. We test our algorithm using synthetic data, and also data from the Geostationary Operational Environmental Satellite system (GOES) solar X-ray time series. These tests show that our algorithm is able to accurately and robustly estimate the LRD and heavy-tailed distribution parameters