Self-Exciting Jumps: Hawkes Processes

Jumps come in clusters. You have seen it on every screen you have ever watched: the market is quiet for hours, then one violent print arrives, and within minutes there are three more. The crash is not a single event but a burst. This is the central empirical fact that today's lesson is built to honor, because most of the machinery you have met so far quietly assumes the opposite.

Start with the model that gets it wrong, so the fix has something to push against. The simplest model of random arrivals is the Poisson process. In a Poisson process, events arrive at a constant average rate, and — this is the crucial part — the past tells you nothing about the future. The fact that a jump just happened does not change the chance of the next one. Arrivals are memoryless. For modeling the buses on a sleepy route, that is fine. For modeling jumps in a stressed market, it is badly, expensively wrong, because in markets one jump makes the next jump more likely. A large sell order trips stops, the stops trigger more selling, liquidity providers widen, and the move feeds on itself. The events are not independent. They are contagious.

So you need a process whose rate of arrivals is not constant but reacts to its own history. That is exactly what a Hawkes process is: a point process that excites itself. Every time an event fires, it temporarily lifts the probability that another event fires soon after, and that bump fades away as time passes without further events. When you string these bumps together, quiet periods and violent bursts emerge naturally from a single mechanism, rather than being painted on by hand.

Let us make that precise, because at Tier 400 intuition has to cash out as a formula you can estimate. The object that governs a point process is its conditional intensity, written lambda-of-t. Think of the intensity as the instantaneous arrival rate: over a tiny sliver of time, the probability of an event is the intensity times the width of that sliver. In a Poisson process the intensity is a flat constant. In a Hawkes process the intensity is alive — it jumps up at every event and decays between events.

The standard Hawkes specification, and the one SVQ uses, writes the intensity as a baseline plus a sum of decaying excitations. In words: the intensity at time t equals a constant baseline, call it mu, plus a contribution from every past jump, where each past jump adds an immediate kick of size alpha that then decays exponentially at rate beta as time moves on. Formally,

lambda-of-t equals mu plus the sum, over every past event time t-sub-i, of alpha times e to the minus beta times the quantity t minus t-sub-i.

Read each symbol as a piece of market behavior, because each one is doing real work. Mu, the baseline, is the background rate of jumps when nothing has happened recently — the market's ambient nervousness with no fresh shock in memory. Alpha is the size of the self-excitation: the instant a jump lands, the intensity leaps upward by alpha, encoding how strongly one jump begets another. Beta is the decay rate, and it controls memory. A large beta means the excitement from a jump dissipates almost at once, so clusters are tight and short-lived. A small beta means a jump's influence lingers, so a shock keeps the process agitated for a long time. The characteristic memory length is one over beta — that is roughly how long, on average, a single jump's echo persists.

Now the number that matters most for your judgment. Divide alpha by beta and you get the branching ratio, often written n. It has a beautiful interpretation: it is the expected number of direct children each jump spawns — the average count of new jumps caused by one jump through self-excitation. If n is well below one, the process is stable: each shock dies out, its offspring fewer than itself, and bursts are finite. As n climbs toward one, each jump nearly reproduces itself, clusters grow long, and the process edges toward criticality. At n equal to or above one the math diverges — the intensity explodes and the model is no longer stationary. Empirically, many stressed markets sit unnervingly close to one, which is precisely why a single shock can cascade.

This is where the institute's throughline reappears. A clustered jump is the loudest "evidence" a market ever produces, and your job is to ask whether that evidence is genuine or an artifact of the process exciting itself. The Hawkes decomposition answers it directly: it splits any observed burst into the part that is exogenous — driven by mu, by genuine outside news — and the part that is endogenous — driven by alpha and the market reacting to its own moves. A high branching ratio means most of what you are seeing is the market talking to itself, not new information arriving. That distinction is the difference between a signal worth trading and a feedback loop about to unwind.