Hawkes Model Research Articles

Dynamic optimal execution in a mixed-market-impact Hawkes price model

We study a linear price impact model, including other liquidity takers, whose flow of orders is driven by a Hawkes process. The optimal execution problem is solved explicitly in this context, and the closed-form optimal strategy describes in particular how one should react to the orders of other traders. This result enables us to discuss the viability of the market. It is shown that Poissonian arrivals of orders lead to quite robust price manipulation strategies in the sense of Huberman and Stanzl (Econometrica, 72:1247–1275, 2004). Instead, a particular set of conditions on the Hawkes model balances the self-excitation of the order flow with the resilience of the price, excludes price manipulation strategies, and gives some market stability.

Exploiting Spillovers to Forecast Crashes

We develop Hawkes models in which events are triggered through self as well as cross-excitation. We examine whether incorporating cross-excitation improves the forecasts of extremes in asset returns compared to only self-excitation. The models are applied to US stocks, bonds and dollar exchange rates. In-sample, a Lagrange Multiplier test indicates the existence of cross-excitation for these series. Out-of-sample, we find that the models that include spillover effects forecast crashes and the Value-at-Risk significantly more accurately than the models without.

Density matrix of black hole radiation

Hawking's model of black hole evaporation is not unitary and leads to a mixed density matrix for the emitted radiation, while the Page model describes a unitary evaporation process in which the density matrix evolves from an almost thermal state to a pure state. We compare a recently proposed model of semiclassical black hole evaporation to the two established models. In particular, we study the density matrix of the outgoing radiation and determine how the magnitude of the off-diagonal corrections differs for the three frameworks. For Hawking's model, we find power-law corrections to the two-point functions that induce exponentially suppressed corrections to the off-diagonal elements of the full density matrix. This verifies that the Hawking result is correct to all orders in perturbation theory and also allows one to express the full density matrix in terms of the single-particle density matrix. We then consider the semiclassical theory for which the corrections, being non-perturbative from an effective field-theory perspective, are much less suppressed and grow monotonically in time. In this case, the R\'enyi entropy for the outgoing radiation is shown to grow linearly at early times; but this growth slows down and the entropy eventually starts to decrease at the Page time. In addition to comparing models, we emphasize the distinction between the state of the radiation emitted from a black hole, which is highly quantum, and that of the radiation emitted from a typical classical black body at the same temperature.

Numerical Simulation of Flow-Induced Noise in High Pressure Reducing Valve.

The main objective of this paper is to study the characteristics of flow-induced noise in high pressure reducing valve (HPRV) and to provide some guidance for noise control. Based on computational fluid dynamics (CFD), numerical method was used to compute flow field. Ffowcs Williams and Hawkings Model was applied to obtain acoustic signals. The unsteady flow field shows that noise sources are located at the bottom of plug for valve without perforated plate, and noise sources are behind the plate for valve with perforated plate. Noise directivity analysis and spectrum characteristics indicate that the perforated plate could help to reduce noise effectively. Inlet pressure has great effects on sound pressure level (SPL). The higher inlet pressure will lead to larger SPL at high frequency. When the maximum Ma is close to 1, SPL at low frequency becomes very high.

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.

Modelling systemic price cojumps with Hawkes factor models

Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating portfolios of highly liquid stocks, we find that there are a large number of high-frequency cojumps. We show that the dynamics of these jumps is described neither by a multivariate Poisson nor by a multivariate Hawkes model. We introduce a Hawkes one-factor model which is able to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets.

Modeling foreign exchange market activity around macroeconomic news: Hawkes-process approach.

We present a Hawkes-model approach to the foreign exchange market in which the high-frequency price dynamics is affected by a self-exciting mechanism and an exogenous component, generated by the pre-announced arrival of macroeconomic news. By focusing on time windows around the news announcement, we find that the model is able to capture the increase of trading activity after the news, both when the news has a sizable effect on volatility and when this effect is negligible, either because the news in not important or because the announcement is in line with the forecast by analysts. We extend the model by considering noncausal effects, due to the fact that the existence of the news (but not its content) is known by the market before the announcement.

Specification Testing in Hawkes Models

We propose various specification tests for Hawkes models based on the Lagrange Multiplier (LM) principle. Hawkes models can be used to model the occurrence of extreme events in financial markets. Our specific testing focus is on extending a univariate model to a multivariate model, that is, we examine whether there is a conditional dependence between extreme events in markets. Simulations show that the test has good size and power, in particular for sample sizes that are typically encountered in practice. Applying the specification test for dependence to US stocks, bonds and exchange rate data, we find strong evidence for cross-excitation within segments as well as between segments. Therefore, we recommend that univariate Hawkes models be extended to account for the cross-triggering phenomenon.

Hawk models, hawk mimics, and antipredator behavior of prey

Prey typically respond to potential predators by taking flight. This results in an optimal flight initiation distance (FID) at which the risk of remaining and the cost of flight are equal. Thus, FID is strongly negatively correlated with susceptibility to predation by the sparrowhawk Accipiter nisus across species of small European birds. However, not everything that looks like a hawk is in fact a hawk. Aggressive mimicry arises from the resemblance between a dangerous model such as a predatory hawk and an innocuous mimic such as a cuckoo that makes errors made by individuals that encounter models and mimics potentially dangerous and life threatening. A prime example of such aggressive mimicry is the hawk-like appearance of common cuckoos Cuculus canorus and other cuckoos. If mimicry is efficient, we should expect that species of small birds that are prey of hawks and hosts of cuckoos react just as strongly to a cuckoo as to the presence of a dangerous model like a sparrowhawk. We used FID of small birds as a measure of the reaction of prey to models and mimics, predicting that FID would be negatively related to susceptibility to predation and positively to risk of cuckoo parasitism. Both susceptibility to sparrowhawk predation and rate of brood parasitism by the common cuckoo independently explained variation in FID, consistent with the expectation that both hawks and cuckoos have imposed significant selection pressures on FID and that species of small birds perceive cuckoos as true mimics of hawks.

Quadratic Hawkes Processes for Financial Prices

We introduce and establish the main properties of QHawkes (Quadratic Hawkes) models. QHawkes models generalize the Hawkes price models introduced in E. Bacry et al. (2014), by allowing all feedback effects in the jump intensity that are linear and quadratic in past returns. A non-parametric fit on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. Our model exhibits two main properties, that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily be at the critical point. Finally, we provide numerical simulations of our calibrated QHawkes model, which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.

Effective measure of endogeneity for the Autoregressive Conditional Duration point processes via mapping to the self-excited Hawkes process

In order to disentangle the internal dynamics from exogenous factors within the Autoregressive Conditional Duration (ACD) model, we present an effective measure of endogeneity. Inspired from the Hawkes model, this measure is defined as the average number of direct offspring for each observed event. Having an explicit meaning allowing one to define a distance to criticality, this “branching ratio” parameter also characterizes the average fraction of events that are triggered due to internal feedback mechanisms within the total population. We provide a direct comparison of the Hawkes and ACD models based on numerical simulations and show that our effective measure of endogeneity for the ACD can be mapped onto the “branching ratio” of the Hawkes model.

Hawkes model for price and trades high-frequency dynamics

We introduce a multivariate Hawkes process that accounts for the dynamics of market prices through the impact of market order arrivals at microstructural level. Our model is a point process mainly characterized by four kernels associated with, respectively, the trade arrival self-excitation, the price changes mean reversion, the impact of trade arrivals on price variations and the feedback of price changes on trading activity. It allows one to account for both stylized facts of market price microstructure (including random time arrival of price moves, discrete price grid, high-frequency mean reversion, correlation functions behaviour at various time scales) and the stylized facts of market impact (mainly the concave-square-root-like/relaxation characteristic shape of the market impact of a meta-order). Moreover, it allows one to estimate the entire market impact profile from anonymous market data. We show that these kernels can be empirically estimated from the empirical conditional mean intensities. We provide numerical examples, application to real data and comparisons to former approaches.

Joint tail and vocal alarm signals of gray squirrels (Sciurus carolinensis)

Threat-specific vocalizations have been observed in primates and ground squirrels, but their contemporaneous usage with visible signals has not been experimentally analyzed for association with threat type. Here we examine the eastern gray squirrel, an arboreal squirrel that uses both vocal and tail signals as alarms. Squirrels were presented with cat and hawk models simulating natural terrestrial or aerial predator attacks and also with control objects that do not resemble predators but approach in a similar manner. Individuals responded with tail signals (twitches and flags) and vocalizations (kuks, quaas and moans), but only flags and moans are associated with predator type. Moans were elicited primarily by aerial stimuli and flags by terrestrial stimuli. Eastern gray squirrels use an alarm-signaling system in which signals in each modality potentially are associated with particular attributes of a threat or may be general alarms. Terrestrially-approaching stimuli yielded vocal and tail alarm signals regardless of whether the stimulus resembled a predator. With aerially-approaching stimuli, however, quaas were used more often when the stimulus resembled an aerial predator than when it did not. An approaching object’s physical appearance may therefore affect squirrels’ responses to aerial, but not terrestrial, objects. When the stimuli resembled real predators approaching in the natural manner (terrestrially or aerially), both tail flags and vocal moans were associated with predator type, so we also considered moans and flags together. The presence and absence of moans and flags in an alarm signaling bout yields a higher statistical index of predictive association as to whether the threat is aerial or terrestrial than does either component alone.

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.

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

We propose a novel index of short-term endogeneity (or reflexivity) derived by calibrating the Hawkes self-excited conditional Poisson model on empirical time series of trades. The Hawkes model accounts simultaneously for the co-existence and interplay between the exogenous impact of news and the endogenous mechanism by which past trading activity may influence future trading activity. Technically known in the mathematical literature on branching processes as the branching ratio, the reflexivity index is quantified for several commodity futures markets (corn, oil, soybean, sugar, and wheat) and also for a benchmark equity futures market (E-mini S&P 500). Specifically, the reflexivity index is 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. We find an overall increase of the level of short-term endogeneity since the mid-2000s to October 2012, with a typical value nowadays around 0.6–0.7, implying that at least 60–70 per cent of commodity price changes are now due to self-generated activities rather than novel information. Our robustness tests show that the branching ratio provides a ‘pure’ measure of endogeneity that is independent of the rate of activity, order size, volume or volatility. We complement our analysis by relating the endogeneity dynamics of these futures markets to their price dynamics, particularly around the commodity bubble that developed since 2006 and culminated in mid-2008. While our index does not have a long-term memory, interestingly, we find that it can still provide some interesting insights when the mechanisms working at longer time scales cascade down to shorter terms.

Comparing the Hawkes and Trigger Process Models for Aftershock Sequences Following the 2005 Kashmir Earthquake

In an earlier study (Van Lieshout and Stein in Math Gesoci 44(3):309–326, 2012) we postulated the existence of two major earthquakes in the 2005 Kashmir disaster instead of a single one, based upon the pattern of aftershocks. In this study, we explore this hypothesis further by fitting several spatial point pattern models. In particular, we discuss the Hawkes and the trigger process models for earthquake aftershock sequences following the Kashmir catastrophe in 2005. The minimum contrast method is used for estimation of the parameters. The study shows that the trigger model fits better than the Hawkes model. The most likely number of main shocks is rounded to 2 generating the almost 200 aftershocks, whereas the Hawkes model would estimate a parent process of approximately 18 parents with on average about 10 descendants. We conclude that the spatial pattern of aftershocks can best be understood as a mixture of two bivariate normal distributions centered around two major shocks and estimate the parameters.

Modelling Systemic Cojumps with Hawkes Factor Models

Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating a set of 20 high cap stocks traded at the Italian Stock Exchange, we find that there is a large number of high frequency cojumps. We show that the dynamics of these jumps is described neither by a multivariate Poisson nor by a multivariate Hawkes model. We introduce a Hawkes one factor model which is able to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets.

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 of the application of the self-excited Hawkes process to high-frequency financial data and carefully analyze a set of effects that lead to significant biases in the estimation of the 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 a few 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 (n = 1) while the true value is actually n = 0. 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: (i) the impact of overnight trading in the analysis of long-term trends, (ii) intraday seasonality and detrending of the data and (ii) vulnerability of the analysis to day-to-day nonstationarity 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, Bercot and Bouchaud (2013) that financial market have been continuously functioning at or close to criticality (n = 1) cannot be supported. In contrast, our previous results on E-mini S&P 500 Futures Contracts and on major commodity future contracts are upheld.

Study on Noise Generation in a High Speed Centrifugal Fan

This study presents the investigation of aerodynamics and aeroacoustics of centrifugal fan using commercial computational fluid dynamics (CFD) code. The unsteady turbulent flow of the fan is simulated with Detached Eddy Simulation (DES) and the acoustics sources are computed based on the pressure fluctuations. The Ffowcs Williams and Hawking model is used to predict the tonal noises in aeroacoustics simulation.

Motif statistics and spike correlations in neuronal networks

Motifs are patterns of subgraphs that are the building blocks of complex networks. Recent experiments have characterized the frequencies with which different motifs occur in biological neural networks, and found remarkable deviations from what we would expect if the networks were randomly connected [1]. Here, we study the impact of such patterns of connectivity on the level of correlated, or synchronized, spiking activity among pairs of cells. Correlations in spiking activity have been shown to strongly impact the neural coding of information. We use a linear, stochastic model of recurrent networks. A cell’s time-dependent firing rate is perturbed from its baseline level by convolution of a response kernel and the input signal from presynaptic neurons. Each neuron generates spikes as an inhomogeneous Poisson process. Previous studies have shown that such models can capture pairwise correlations in integrate and fire networks [2,3], and they are closely related to spike response and Hawkes models [4,6]. For this model, there is an explicit expression for pairwise correlation in terms of the connectivity matrix. By expanding this expression in a series, one can relate each term to a different motif (with a different number of connections). Through a resumming technique, we show that the average correlation across the network can be closely approximated using the frequencies of only first and second order motifs. These are the diverging motif—two cells both receiving projections from another cell, its counterpart the converging motif—two cells projecting to a common cell, and the chain motif—three cells linked by two consecutive projections. Specifically, we show that the prevalence of diverging and chain motifs tends to increase correlation, while the converging motif makes no contribution to the average correlation. Moreover, we numerically show that variance of correlations across the network is largely determined by the frequency of the chain motif (see Figure ​Figure1)1) alone. Finally, we demonstrate potential effect of motif statistics on neural coding by showing how motif frequencies impact linear Fisher information. In particular, we find that the linear Fisher information is only affected by converging motif frequency (more information with given more converging motifs). Figure 1 Each dot represent one network sample plotted against its chain and diverging motif frequencies. Color shows the standard deviation of correlations in the network. Inset is the same plot with respect to diverging and converging motifs.