Hawkes Process

Table of Contents

1. Summary

A Hawkes process is a point process in which every event raises the rate of further events, and the raise fades with time. A burst of events therefore breeds more events, and a quiet spell stays quiet until something starts it. Real tick streams behave like this. A Poisson process cannot reproduce it, because a Poisson process forgets every event at once. Hawkes (1971) introduced the process. Bacry, Mastromatteo and Muzy (2015) review its use in finance. This note covers the univariate process with an exponential kernel: one stream of events, one fading memory.

2. Layperson's mental model

Think of a room where people talk. When the room is silent, someone speaks now and then, at a low steady rate. Each remark prompts replies, and each reply prompts more, so a remark can start a conversation. The conversation dies away when the replies stop coming. The low steady rate is the baseline \(\mu\). How many replies one remark brings is the branching ratio \(n\). How fast the prompting fades is the decay \(\beta\). 

// the rate just before the next event, a gap after the last one
double rate = baseline + (rate_after_last_event - baseline) * std::exp(-decay * gap);
// the rate jumps by alpha at each event
rate += alpha;

3. Detail

3.1. The intensity

With event times \(t_1 < t_2 < \dots\) the conditional intensity is

\begin{equation} \lambda(t) = \mu + \sum_{t_k < t} \phi(t - t_k), \qquad \phi(x) = \alpha\, e^{-\beta x} \end{equation}

\(\mu > 0\) is the baseline rate. The kernel \(\phi\) is the extra rate that an event adds, \(x\) time units later. With the exponential kernel the jump at an event is \(\alpha\) and the decay is \(\beta\).

3.2. The branching ratio and stability

The kernel integrates to the branching ratio

\begin{equation} n = \int_0^\infty \phi(x)\,dx = \frac{\alpha}{\beta} \end{equation}

\(n\) is the mean number of events that one event causes directly. The process is stable, with a finite long-run rate, only when \(n < 1\). For \(n \ge 1\) the count grows without bound, as a family tree that never dies out. For \(n < 1\):

  • The long-run mean rate is \(\bar\lambda = \mu / (1 - n)\).
  • The long-window Fano factor, the variance of the count over its mean, is \(1/(1-n)^2\). At \(n = 0\) it is one, the Poisson value. At \(n = 0.8\) it is 25.
  • The mean number of events in the family that one outside event starts is \(1/(1-n)\).

Filimonov and Sornette (2012) call \(n\) the endogeneity or reflexivity ratio: the share of events that earlier events caused rather than outside news. They fit E-mini S&P 500 futures from 1998 to 2010 and report that the share of price changes that outside news caused fell from about 70% in 1998 to under 30% after 2007, so the endogenous share rose to about 70% or more. Hardiman, Bercot and Bouchaud (2013) argue that a kernel with a power-law tail gives an \(n\) close to one.

3.3. The cluster view

Hawkes and Oakes (1974) showed that the process is a family tree. Outside events, called immigrants, arrive as a Poisson process of rate \(\mu\). Each event, an immigrant or not, has a Poisson number of children with mean \(n\), and each child comes after its parent by a gap drawn from the kernel scaled to a density, which is \(\text{Exp}(\beta)\) here. The cluster view gives the three results above and it is the plain way to explain the process to a reader who has not met intensities.

3.4. A different parametrisation

The triple \((\mu, \alpha, \beta)\) is the usual one, and the control panel of the approach note names it. It mixes a rate that a person can measure with two that are hard to feel. The triple \((\bar\lambda, n, \beta)\) is easier to set:

\begin{equation} \mu = \bar\lambda\,(1 - n), \qquad \alpha = n\,\beta \end{equation}

\(\bar\lambda\) is the long-run rate, which the person already knows from a ticks-per-hour setting. \(n\) is how clustered the stream is, from 0 to just under

  1. \(1/\beta\) is how long a burst lasts. Changing \(n\) then changes the clustering

and leaves the load unchanged.

3.5. The memory is one number

With the exponential kernel the whole history reduces to the current intensity. Between events, \(\lambda(t) - \mu\) decays by \(e^{-\beta\,\Delta}\) over a gap \(\Delta\), and each event adds \(\alpha\). So the intensity, and nothing else, is the state of the process, and it is a Markov process in that state. A simulator keeps one number and never reads back the list of events. Other kernels, such as the power law, lack this property and cost more for each event.

3.6. Two exact ways to simulate

Thinning (Ogata 1981), the general method, works for any kernel. Between events, the exponential-kernel intensity only falls, so its value just after the last event bounds it. Draw a candidate gap from an exponential law at that bound. Read the true intensity there. Keep the candidate with probability true over bound, otherwise move on from the candidate with the lower bound. Rejected candidates cost a draw each and carry no event.

Direct inversion (Dassios and Zhao 2013) needs no rejection for the exponential kernel. The next event is the earlier of two competing arrivals: a baseline arrival and an arrival from the fading excess. Let \(\lambda^+\) be the intensity just after the last event.

  1. Draw the baseline gap \(S_1 = -\ln(U_1)/\mu\).
  2. Compute \(D = 1 + \beta\,\ln(U_2) / (\lambda^+ - \mu)\).
  3. If \(D > 0\), the excess gap is \(S_2 = -\ln(D)/\beta\). If \(D \le 0\) the excess never produces an event, and \(S_2\) is infinite.
  4. The gap is \(\min(S_1, S_2)\).
  5. The new \(\lambda^+\) is \(\mu + (\lambda^+ - \mu)\,e^{-\beta\,\text{gap}} + \alpha\).

Both methods draw from the same law. Direct inversion draws two uniforms for each event, always. It does not extend to a power-law kernel, and thinning does.

3.7. Checking a simulated stream

Two checks need no data. The mean rate over a long run should be \(\mu/(1-n)\), and the Fano factor of counts over a window much longer than \(1/\beta\) should be \(1/(1-n)^2\). A sharper check is the time-rescaling test of the point process note: the rescaled gaps must follow \(\text{Exp}(1)\).

3.8. The multivariate form

A Hawkes process can have many streams, each one exciting the others through a matrix of kernels. That models, for example, a trade tick that excites a quote tick, or one currency pair that excites another. This note, and the implementation in ORE Studio, treat one stream. The reason is in tick.

4. In the code

service::hawkes_arrival_process, in ores.analytics.quant/service/processes/hawkes_arrival_process.hpp, draws gaps by direct inversion. It is built by process_factory::make_arrival_process from a mode, a rate in ticks per hour, a branching ratio and a decay; the factory converts the rate to arrivals per second and the class turns the observable triple into the baseline \(\mu\) and the jump \(\alpha\). The constructor refuses a branching ratio of one or more. The tests check the long-run mean rate, the Fano factor and the time rescaling of the rescaled gaps.

5. See also

5.1. Further reading

  • Hawkes, A. G. (1971). "Spectra of some self-exciting and mutually exciting point processes." Biometrika, 58(1), 83-90. The original paper.
  • Hawkes, A. G., & Oakes, D. (1974). "A cluster process representation of a self-exciting process." Journal of Applied Probability, 11(3), 493-503.
  • Ogata, Y. (1981). "On Lewis' simulation method for point processes." IEEE Transactions on Information Theory, 27(1), 23-31.
  • Dassios, A., & Zhao, H. (2013). "Exact simulation of Hawkes process with exponentially decaying intensity." Electronic Communications in Probability, 18(62), 1-13. https://doi.org/10.1214/ECP.v18-2717.
  • Filimonov, V., & Sornette, D. (2012). "Quantifying reflexivity in financial markets: Toward a prediction of flash crashes." Physical Review E, 85(5),
    1. https://doi.org/10.1103/PhysRevE.85.056108. arXiv:1201.3572.
  • Hardiman, S. J., Bercot, N., & Bouchaud, J.-P. (2013). "Critical reflexivity in financial markets: a Hawkes process analysis." European Physical Journal B (2013). https://doi.org/10.1140/epjb/e2013-40107-3. arXiv:1302.1405. Fits the E-mini S&P 500 future and finds a kernel that integrates to about one in every period from 1998 to 2011.
  • Bacry, E., Mastromatteo, I., & Muzy, J.-F. (2015). "Hawkes processes in finance." Market Microstructure and Liquidity, 1(1), 1550005. https://doi.org/10.1142/S2382626615500057. arXiv:1502.04592.
  • Bacry, E., Bompaire, M., Gaïffas, S., & Poulsen, S. (2017). "tick: a Python library for statistical learning, with a particular emphasis on time-dependent modelling." arXiv:1707.03003. The journal version, with Philip Deegan as a fifth author and the title "tick: a Python Library for Statistical Learning, with an emphasis on Hawkes Processes and Time-Dependent Models", is Journal of Machine Learning Research 18(214), 1-5 (2018).
  • Wikipedia: Hawkes process.

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