Hawkes Processes: Self-Exciting Events

A Poisson process has a constant arrival rate — events don't care about the past. Markets aren't like that: a trade makes the next trade more likely, an order triggers a cascade. A Hawkes process captures this with an intensity that jumps up at every event and decays between them.

The self-exciting intensity

The conditional intensity (instantaneous arrival rate given the history) is

\lambda(t) = \mu + \sum_{t_i < t} \phi(t - t_i),

$\mu > 0$ — the baseline rate (exogenous events),
$\phi(\cdot) \ge 0$ — the excitation kernel: how much each past event lifts the rate.

The standard choice is an exponential kernel:

\phi(t) = \alpha\,e^{-\beta t}, \qquad \alpha, \beta > 0 .

Every event adds $\alpha$ to the intensity, which then decays at rate $\beta$ . So bursts beget bursts — and then quiet returns.

Stability and the branching ratio

Each event spawns, on average,

n = \frac{\alpha}{\beta} = \int_0^\infty \phi(t)\,dt

"offspring" events. This branching ratio must satisfy $n < 1$ for the process to be stable (otherwise activity explodes). Read $n$ as the fraction of activity that is endogenous — self-generated by the market rather than driven by outside news. Empirically $n$ is high (often $0.7$ – $0.95$ ) in liquid markets: most activity is the market reacting to itself.

Why quants care

Hawkes processes reproduce volatility clustering and trade clustering that Poisson models can't. The next lessons plug this intensity into a jump-diffusion price and into the order book.

Free preview of the Hawkes & Microstructure course.

Based on Ibrahim Lanre Adedimeji’s “Hawkes Processes: A Stochastic Gem.”

The self-exciting intensity

The conditional intensity (instantaneous arrival rate given the history) is

\lambda(t) = \mu + \sum_{t_i < t} \phi(t - t_i),

\mu > 0

— the baseline rate (exogenous events),

\phi(\cdot) \ge 0

— the excitation kernel: how much each past event lifts the rate.

The standard choice is an exponential kernel:

\phi(t) = \alpha\,e^{-\beta t}, \qquad \alpha, \beta > 0 .

Every event adds

\alpha

to the intensity, which then decays at rate

\beta

. So bursts beget bursts — and then quiet returns.

Stability and the branching ratio

Each event spawns, on average,

n = \frac{\alpha}{\beta} = \int_0^\infty \phi(t)\,dt

"offspring" events. This branching ratio must satisfy

n < 1

for the process to be stable (otherwise activity explodes). Read

n

as the fraction of activity that is endogenous — self-generated by the market rather than driven by outside news. Empirically

n

is high (often

0.7

–

0.95

) in liquid markets: most activity is the market reacting to itself.

Why quants care

Hawkes processes reproduce volatility clustering and trade clustering that Poisson models can't. The next lessons plug this intensity into a jump-diffusion price and into the order book.

Free preview of the Hawkes & Microstructure course.

Based on Ibrahim Lanre Adedimeji’s “Hawkes Processes: A Stochastic Gem.”