Vladimir Filimonov

Apparent Criticality and Calibration Issues in the Hawkes Self-Excited Point Process Model: Application to High-Frequency Financial Data

We present a careful analysis of possible issues on the application of the self-excited Hawkes process to high-frequency financial data. We analyse a set of effects leading to significant biases in the estimation of the ‘criticality index’ n that quantifies the degree of endogeneity of how much past events trigger future events. We report the following model biases: (i) evidence of strong upward biases on the estimation of n when using power law memory kernels in the presence of outliers, (ii) strong effects on n resulting from the form of the regularization part of the power law kernel, (iii) strong edge effects on the estimated n when using power law kernels and (iv) the need for an exhaustive search of the absolute maximum of the log-likelihood function due to its complicated shape. Moreover, we demonstrate that the calibration of the Hawkes process on mixtures of pure Poisson process with changes of regime leads to completely spurious apparent critical values for the branching ratio (), while the true value is actually . More generally, regime shifts on the parameters of the Hawkes model and/or on the generating process itself are shown to systematically lead to a significant upward bias in the estimation of the branching ratio. We demonstrate the importance of the preparation of the high-frequency financial data, in particular: (a) the impact of overnight trading in the analysis of long-term trends, (b) intraday seasonality and detrending of the data and (c) vulnerability of the analysis to day-to-day non-stationarity and regime shifts. Special care is given to the decrease of quality of the timestamps of tick data due to latency and grouping of messages to packets by the stock exchange. Altogether, our careful exploration of the caveats of the calibration of the Hawkes process stresses the need for considering all the above issues before any conclusion can be sustained. In this respect, because the above effects are plaguing their analyses, the claim by Hardiman et al. [Eur. Phys. J. B – Cond. Matter Comp. Syst., 2013, 86, 442] that financial market has been continuously functioning at or close to criticality () cannot be supported. In contrast, our previous results on E-mini S&P 500 Futures Contracts and on major commodity future contracts are upheld.

A Stable and Robust Calibration Scheme of the Log-Periodic Power Law Model

We present a simple transformation of the formulation of the log-periodic power law formula of the Johansen-Ledoit-Sornette model of financial bubbles that reduces it to a function of only three nonlinear parameters. The transformation significantly decreases the complexity of the fitting procedure and improves its stability tremendously because the modified cost function is now characterized by good smooth properties with in general a single minimum in the case where the model is appropriate to the empirical data. We complement the approach with an additional subordination procedure that slaves two of the nonlinear parameters to what can be considered to be the most crucial nonlinear parameter, the critical time tc defined as the end of the bubble and the most probable time for a crash to occur. This further decreases the complexity of the search and provides an intuitive representation of the results of the calibration. With our proposed methodology, metaheuristic searches are not longer necessary and one can resort solely to rigorous controlled local search algorithms, leading to dramatic increase in efficiency. Empirical tests on the Shanghai Composite index (SSE) from January 2007 to March 2008 illustrate our findings.

Quantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets

We propose a “reflexivity” index that quantifies the relative importance of short-term endogeneity for several commodity futures markets (corn, oil, soybean, sugar, and wheat) and a benchmark equity futures market (E-mini S&P 500), from mid-2000s to October 2012. Our reflexivity index is defined as the average ratio of the number of price moves that are due to endogenous interactions to the total number of all price changes, which also include exogenous events. It is obtained by calibrating the Hawkes self-excited conditional Poisson model on time series of price changes. The Hawkes model accounts simultaneously for the co-existence and interplay between the exogenous impact of news and the endogenous mechanism by which past price changes may influence future price changes. Our robustness tests show that our index provides a ‘pure’ measure of endogeneity that is independent of the rate of activity, order size, volume or volatility. We find an overall increase of the reflexivity index since the mid-2000s to October 2012, which implies that at least 60–70 percent of commodity price changes are now due to self-generated activities rather than novel information, compared to 20–30 percent earlier. While our reflexivity index is defined on short-time windows (10–30 min) and thus does not capture long-term memory, we discover striking coincidence between its dynamics and that of the price hikes and abrupt falls that developed since 2006 and culminated in early 2009.

Power Law Scaling and 'Dragon-Kings' in Distributions of Intraday Financial Drawdowns

We investigate the distributions of e-drawdowns and e-drawups of the most liquid futures financial contracts of the world at time scales of 30 seconds. The e-drawdowns (resp. e-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 e-drawdowns and e-drawups exhibit power law tails, albeit with exponents significantly larger than those for the return distributions. This paradoxical result can be attributed to (i) the existence of significant transient dependence between returns and (ii) the presence of large outliers (dragon-kings) characterizing the extreme tail of the drawdown/drawup distributions deviating from the power law. The study of the tail dependence between the sizes, speeds and durations of drawdown/drawup indicates a clear relationship between size and speed but none between size and duration. This implies that the most extreme drawdown/drawup tend to occur fast and are dominated by a few very large returns. We discuss both the endogenous and exogenous origins of these extreme events.

The Hawkes process with renewal immigration & its estimation with an EM algorithm

In its original form, the self-excited Hawkes process is a cluster process where immigrants follow a Poisson process, and each immigrant may form a cluster of multi-generational offspring. The Hawkes process is generalized by replacing the Poisson immigration process with a renewal process. This generalization makes direct MLE impossible. Thus, two EM algorithms are introduced: The first extends the existing EM algorithm for the Hawkes process to consider renewal immigration. It treats the entire branching structure-which points are immigrants, and which point is the parent of each offspring-as missing data. The second algorithm reduces the amount of missing data, considering only if a point is an immigrant or not as missing data. This significantly reduces computational complexity and memory requirements, enabling estimation on larger datasets. Both algorithms are found to perform well in simulation studies. A case study shows that the Hawkes process with renewal immigration is superior to the standard Hawkes process for the modeling of high-frequency price fluctuations. Further, it is demonstrated that misspecification of the immigration process can bias estimation of the branching ratio, which quantifies the degree of self-excitation.

Estimation of the Hawkes Process with Renewal Immigration Using the EM Algorithm

We introduce the Hawkes process with renewal immigration and make its statistical estimation possible with two Expectation Maximization (EM) algorithms. The standard Hawkes process introduces immigrant points via a Poisson process, and each immigrant has a subsequent cluster of associated offspring of multiple generations. We generalize the immigration to come from a Renewal process; introducing dependence between neighbouring clusters, and allowing for over/under dispersion in cluster locations. This complicates evaluation of the likelihood since one needs to know which subset of the observed points are immigrants. Two EM algorithms enable estimation here: The first is an extension of an existing algorithm that treats the entire branching structure - which points are immigrants, and which point is the parent of each offspring - as missing data. The second considers only if a point is an immigrant or not as missing data and can be implemented with linear time complexity. Both algorithms are found to be consistent in simulation studies. Further, we show that misspecifying the immigration process introduces significant bias into model estimation - especially the branching ratio, which quantifies the strength of self excitation. Thus, this extended model provides a valuable alternative model in practice.

Views to a war: systematic differences in media and military reporting of the war in Iraq

The quantitative study of violent conflict and its mechanisms has in recent years greatly benefited from the availability of detailed event data. With a number of highly visible studies both in the natural sciences and in political science using such data to shed light on the complex mechanisms underlying violent conflict, researchers have recently raised issues of systematic (reporting) biases. While many sources of bias are qualitatively known, biases in event data are usually not studied with quantitative methods. In this study we focus on a unique case - the conflict in Iraq - that is covered by two independently collected datasets: Iraq Body Count (IBC) reports of civilian casualties and Significant Action (SIGACT) military data. We systematically identify a number of key quantitative differences between the event reporting in the two datasets and demonstrate that even for subsets where both datasets are most consistent at an aggregate level, the daily time series and timing signatures of events differ significantly. This suggests that at any level of analysis the choice of dataset may substantially affect any inferences drawn, with attendant consequences for a number of recent studies of the conflict in Iraq. We further outline how the insights gained from our analysis of conflict event data have broader implications for studies using similar data on other social processes.

Spurious trend switching phenomena in financial markets

The observations of power laws in the time to extrema of volatility, volume and intertrade times, from milliseconds to years reported by Preis et al. (2010, 2011), are shown to result straightforwardly from the selection of biased statistical subsets of realizations in otherwise featureless processes such as random walks. The bias stems from the selection of price peaks that imposes a condition on the statistics of price change and of trade volumes that skew their distributions. For the intertrade times, the extrema and power laws results from the format of transaction data.

Most Efficient Homogeneous Volatility Estimators

We present a new theory of homogeneous volatility (and variance) estimators for arbitrary stochastic processes. The main tool of our theory is the parsimonious encoding of all the information contained in the OHLC prices for a given time interval by the joint distributions of the high-minusopen, low-minus-open and close-minus-open values, whose analytical expression is derived exactly for Wiener processes with drift. The efficiency of the new proposed estimators is favorably compared with that of the Garman-Klass, Roger-Satchell and maximum likelihood estimators.

On the Modeling of Financial Time Series

This paper discusses issues related to modeling of financial time series. We discuss so-called empirical “stylized facts” of real price time-series and the evolution of financial models from trivial random walk introduced by Louis Bachelier in 1900 to modern multifractal models, that nowadays are the most parsimonious and flexible models of stochastic volatility. We focus on a particular model of Multifractal Random Walk (MRW), which is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. The paper presents a method of numerical simulation of realizations of MRW using the Circulant Embedding Method and discuss methods of its calibration.