Improved risk estimation in multifractal records: Application to the value at risk in finance

The dynamics of the aggressive order during a crisis

We investigate the dynamics of aggressive order in the financial market to further understand volatility. To analyze aggressive order, market orders in the order book are scrutinized. The market orders have different degrees of aggressiveness; therefore, we categorize market orders into four types: types Zero, One, A, and B, of which type B is the most aggressive. To examine the dynamics and impacts of each type of order, we use both macro- and micro-level approaches. From the macroscopic perspective, the burstiness and memory of type B is highly correlated with volatility. When traders face a financial crisis, they place bursty aggressive orders, and the orders are more predictable than usual. From the microscopic perspective, we additionally focus on the influence of the orders, particularly the price impact and resilience. The aggressive order has a greater impact than others, even when the price change of the aggressive order is smaller. Moreover, the aggressive order delivers more information on price because the aggressive order has a higher price impact than the execution cost.

Forecasting extreme atmospheric events with a recurrence-interval-analysis-based autoregressive conditional duration model

With most city dwellers in China subjected to air pollution, forecasting extreme air pollution spells is of paramount significance in both scheduling outdoor activities and ameliorating air pollution. In this paper, we integrate the autoregressive conditional duration model (ACD) with the recurrence interval analysis (RIA) and also extend the ACD model to a spatially autoregressive conditional duration (SACD) model by adding a spatially reviewed term to quantitatively explain and predict extreme air pollution recurrence intervals. Using the hourly data of six pollutants and the air quality index (AQI) during 2013–2016 collected from 12 national air quality monitoring stations in Beijing as our test samples, we attest that the spatially reviewed recurrence intervals have some general explanatory power over the recurrence intervals in the neighbouring air quality monitoring stations. We also conduct a one-step forecast using the RIA-ACD(1,1) and RIA-SACD(1,1,1) models and find that 90% of the predicted recurrence intervals are smaller than 72 hours, which justifies the predictive power of the proposed models. When applied to more time lags and neighbouring stations, the models are found to yield results that are consistent with reality, which evinces the feasibility of predicting extreme air pollution events through a recurrence-interval-analysis-based autoregressive conditional duration model. Moreover, the addition of a spatial term has proved effective in enhancing the predictive power.

Bayesian model selection for complex dynamic systems

Time series generated by complex systems like financial markets and the earth’s atmosphere often represent superstatistical random walks: on short time scales, the data follow a simple low-level model, but the model parameters are not constant and can fluctuate on longer time scales according to a high-level model. While the low-level model is often dictated by the type of the data, the high-level model, which describes how the parameters change, is unknown in most cases. Here we present a computationally efficient method to infer the time course of the parameter variations from time-series with short-range correlations. Importantly, this method evaluates the model evidence to objectively select between competing high-level models. We apply this method to detect anomalous price movements in financial markets, characterize cancer cell invasiveness, identify historical policies relevant for working safety in coal mines, and compare different climate change scenarios to forecast global warming.

Systematic changes in stock market prices or in the migration behaviour of cancer cells may be hidden behind random fluctuations. Here, Mark et al. describe an empirical approach to identify when and how such real-world systems undergo systematic changes.

Universality of market superstatistics

We use a key concept of the continuous-time random walk formalism, i.e., continuous and fluctuating interevent times in which mutual dependence is taken into account, to model market fluctuation data when traders experience excessive (or superthreshold) losses or excessive (or superthreshold) profits. We analytically derive a class of "superstatistics" that accurately model empirical market activity data supplied by Bogachev, Ludescher, Tsallis, and Bunde that exhibit transition thresholds. We measure the interevent times between excessive losses and excessive profits and use the mean interevent discrete (or step) time as a control variable to derive a universal description of empirical data collapse. Our dominant superstatistic value is a power-law corrected by the lower incomplete gamma function, which asymptotically tends toward robustness but initially gives an exponential. We find that the scaling shape exponent that drives our superstatistics subordinates itself and a "superscaling" configuration emerges. Thanks to the Weibull copula function, our approach reproduces the empirically proven dependence between successive interevent times. We also use the approach to calculate a dynamic risk function and hence the dynamic VaR, which is significant in financial risk analysis. Our results indicate that there is a functional (but not literal) balance between excessive profits and excessive losses that can be described using the same body of superstatistics but different calibration values and driving parameters. We also extend our original approach to cover empirical seismic activity data (e.g., given by Corral), the interevent times of which range from minutes to years. Superpositioned superstatistics is another class of superstatistics that protects power-law behavior both for short- and long-time behaviors. These behaviors describe well the collapse of seismic activity data and capture so-called volatility clustering phenomena.

Symplectic geometry spectrum regression for prediction of noisy time series

We present the symplectic geometry spectrum regression (SGSR) technique as well as a regularized method based on SGSR for prediction of nonlinear time series. The main tool of analysis is the symplectic geometry spectrum analysis, which decomposes a time series into the sum of a small number of independent and interpretable components. The key to successful regularization is to damp higher order symplectic geometry spectrum components. The effectiveness of SGSR and its superiority over local approximation using ordinary least squares are demonstrated through prediction of two noisy synthetic chaotic time series (Lorenz and Rössler series), and then tested for prediction of three real-world data sets (Mississippi River flow data and electromyographic and mechanomyographic signal recorded from human body).

Predictability of critical transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However, especially in the presence of noise, it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance.

Detecting and interpreting distortions in hierarchical organization of complex time series

Hierarchical organization is a cornerstone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectra are symmetric while those extracted from empirical data are often asymmetric. Using selected time series representing such diverse phenomena as price changes and intertransaction times in financial markets, sentence length variability in narrative texts, Missouri River discharge, and sunspot number variability as examples, we show that the resulting singularity spectra appear strongly asymmetric, more often left sided but in some cases also right sided. We present a unified view on the origin of such effects and indicate that they may be crucially informative for identifying the composition of the time series. One particularly intriguing case of this latter kind of asymmetry is detected in the daily reported sunspot number variability. This signals that either the commonly used famous Wolf formula distorts the real dynamics in expressing the largest sunspot numbers or, if not, that their dynamics is governed by a somewhat different mechanism.

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

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).

Detrended cross-correlation analysis consistently extended to multifractality

We propose an algorithm, multifractal cross-correlation analysis (MFCCA), which constitutes a consistent extension of the detrended cross-correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods, like multifractal extension, have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter λ(q). This relation provides information about the character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from the stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

Performance of multifractal detrended fluctuation analysis on short time series

The performance of the multifractal detrended analysis on short time series is evaluated for synthetic samples of several mono- and multifractal models. The reconstruction of the generalized Hurst exponents is used to determine the range of applicability of the method and the precision of its results as a function of the decreasing length of the series. As an application the series of the daily exchange rate between the U.S. dollar and the euro is studied.

Scaling properties and universality of first-passage-time probabilities in financial markets

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 its 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 related quantity, the survival probability, shows, away from the center and tails of the distribution, a hyperbolic t(-1/2) decay typical of a Markovian dynamics, albeit the existence of memory in markets. Modifications of the Weibull and Student distributions are good candidates for the phenomenological description of first-passage time properties under certain regimes. The scaling strategies shown may be useful for risk control and algorithmic trading.

Financial factor influence on scaling and memory of trading volume in stock market

We study the daily trading volume volatility of 17,197 stocks in the US stock markets during the period 1989-2008 and analyze the time return intervals τ between volume volatilities above a given threshold q. For different thresholds q, the probability density function P(q)(τ) scales with mean interval 〈τ〉 as P(q)(τ)=〈τ〉(-1)f(τ/〈τ〉), and the tails of the scaling function can be well approximated by a power law f(x)∼x(-γ). We also study the relation between the form of the distribution function P(q)(τ) and several financial factors: stock lifetime, market capitalization, volume, and trading value. We find a systematic tendency of P(q)(τ) associated with these factors, suggesting a multiscaling feature in the volume return intervals. We analyze the conditional probability P(q)(τ|τ(0)) for τ following a certain interval τ(0), and find that P(q)(τ|τ(0)) depends on τ(0) such that immediately following a short (long) return interval a second short (long) return interval tends to occur. We also find indications that there is a long-term correlation in the daily volume volatility. We compare our results to those found earlier for price volatility.

Recurrence interval analysis of trading volumes

We study the statistical properties of the recurrence intervals τ between successive trading volumes exceeding a certain threshold q. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, the weighted KS statistic and the Cramér-von Mises criterion. The measurements of the conditional probability distribution and the detrended fluctuation function show that both short-term and long-term memory effects exist in the recurrence intervals between trading volumes. We further study the relationship between trading volumes and price returns based on the recurrence interval analysis method. It is found that large trading volumes are more likely to occur following large price returns, and the comovement between trading volumes and price returns is more pronounced for large trading volumes.