Hawkes Processes for Order Flow

If you watch a market's tape, orders do not trickle in at a steady rate. They come in bursts — a flurry of trades, then quiet, then another flurry. A plain The Poisson Process, which assumes events arrive at a constant average rate with no memory, completely misses this. A Hawkes process fixes it with one idea: every event makes the next one more likely for a little while. Activity feeds on itself, so bursts are built in.

This "self-excitement" is exactly what real order flow looks like. One aggressive trade triggers hedging trades, which trip stop orders, which set off market-making adjustments, which spark more trades. Hawkes processes turn that cascade into a clean, estimable model — and one number falls out that tells you how much of the market's activity is the market reacting to itself rather than to outside news.

The self-exciting intensity

In a Poisson process the arrival rate is a constant $\mu$. In a Hawkes process the rate — called the intensity $\lambda(t)$, the instantaneous chance per unit time of the next event — jumps up after every event and then fades:

$$\lambda(t) = \mu + \sum_{t_i < t} \alpha\, e^{-\beta (t - t_i)} .$$

Reading it: $\mu$ is the baseline rate from outside causes (genuine news), the same as Poisson. The sum runs over every past event time $t_i$; each one adds a bump of size $\alpha$ that decays away at speed $\beta$. So $\alpha$ is how big a kick each event gives to the arrival rate, and $\beta$ is how fast that kick fades. Right after a flurry, $\lambda(t)$ is high — more events likely — which is precisely the clustering we see.

The one number that matters: the branching ratio

Think of it like an epidemic. Each event is a "parent" that spawns some number of "child" events before its influence dies out. The average number of children per event is the branching ratio:

$$n = \frac{\alpha}{\beta} .$$

This is just the total size of one bump (its area) — the height $\alpha$ spread over its decay time $1/\beta$. If $n \ge 1$ each event spawns at least one child on average and the process explodes; the market is only stable (stationary) when $n < 1$. And $n$ has a gorgeous interpretation: it is the fraction of activity that is endogenous, self-generated by the market reacting to itself. Empirically, estimated branching ratios for liquid markets sit disturbingly close to 1 (often 0.7–0.95), meaning the overwhelming majority of trading is the market echoing its own moves, not responding to fresh news.

Worked example

Suppose you fit a Hawkes model to a stock's trade arrivals and get a baseline $\mu = 0.5$ trades per second, a kick $\alpha = 8$ per second, and a decay $\beta = 10$ per second. Then:

• Branching ratio: $n = \alpha/\beta = 8/10 = 0.8$. Below 1, so the process is stable — good.
• Cluster size. One outside-driven trade spawns on average $n = 0.8$ children, each of which spawns $0.8$ grandchildren, and so on. The total number of trades set off by a single exogenous one is the geometric sum $1 + 0.8 + 0.8^2 + \dots = 1/(1 - 0.8) = 5$. So each genuinely new trade drags five trades in total through the network.
• Average trade rate. The long-run intensity is the baseline amplified by that same factor: $\bar\lambda = \mu/(1 - n) = 0.5/0.2 = 2.5$ trades per second. Only $\mu = 0.5$ of that is truly exogenous; the other 2.0 is the market reacting to itself.

Now watch what happens as the market gets more reflexive. If tighter feedback pushes $n$ to $0.95$, the cluster size jumps to $1/(1-0.95) = 20$ and the trade rate to $0.5/0.05 = 10$ per second — the same trickle of real news, but four times the activity and much longer-lived bursts. This sensitivity near $n = 1$ is why crowded, high-feedback markets can flip from calm to frantic on a tiny nudge.

Hawkes processes are the natural upgrade from Poisson whenever "one event begets another." They underpin modern models of order-book dynamics, of how Market Impact decays (the impact kernel and the Hawkes kernel are close relatives), and of the high-frequency feedback loops that make markets bursty. When someone says order flow "clusters," the Hawkes process is the math that says it precisely.