Quantitative Microstructure Self-Exciting Point Processes Alan Hawkes (1971)

Hawkes (1971) High-Frequency Process Lab

Simulate Alan Hawkes' (1971) self-exciting point process via Ogata's (1981) modified thinning algorithm. Analyze branching ratios, endogenous algorithmic cascades, memory half-life, and cubic variance explosion near flash crash criticality.

Theoretical Foundations: Self-Exciting Point Processes & Endogenous Reflexivity
1. Conditional Intensity with Exponential Memory

The conditional arrival hazard rate $\lambda(t)$ combines exogenous baseline immigration $\mu$ with memory of all past event arrival times $t_i < t$:

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

Each order arrival immediately increases conditional intensity by jump magnitude $\alpha > 0$, decaying exponentially at mean-reversion speed $\beta > 0$ with memory half-life $t_{1/2} = \frac{\ln 2}{\beta}$.

2. Branching Ratio & Flash Crash Criticality

The branching ratio $\eta = \frac{\alpha}{\beta}$ determines the average number of endogenous offspring spawned per event. The unconditional expected event rate and asymptotic count variance are:

$$\bar{\lambda} = \frac{\mu}{1 - \eta}, \quad \text{Var}(N(t)) \propto \frac{t}{(1 - \eta)^3} \quad (\eta < 1)$$

As $\eta \to 1$, order flow variance explodes cubically ($1/(1-\eta)^3$), revealing how electronic markets near criticality suffer endogenous liquidity flash crashes from minor exogenous tremors.

Model Parameters

High-Frequency Algorithmic Cascades: High branching ratio (η = 0.84), rapid memory decay (half-life = 0.14s), and intense endogenous order clustering.

Branching Ratio η 0.840 Near-Critical
Memory Half-Life $t_{1/2}$ 0.14 s Decay Speed
Endogenous Reflexivity 82.5% Feedback %
Variance Multiplier 244.1x Instability
Total Order Events 89 orders
Empirical Rate 2.97 /s
Theoretical Rate $\bar{\lambda}$ 3.12 /s
Log-Likelihood $\ln L$ 142.3
Dual 5×5 Microstructure Sensitivity Grids
Sensitivity Grid 1: Excitation Jump α vs Decay β → Branching Ratio η

Shows the transition across subcritical (η < 0.85), near-critical (η ≥ 0.85), and explosive supercritical regimes (η ≥ 1.0):

Sensitivity Grid 2: Baseline μ vs Branching η → Expected Rate $\bar{\lambda} = \frac{\mu}{1 - \eta}$

Quantifies how endogenous self-excitation magnifies exogenous order flow volume across market regimes:

Market Microstructure Mastery: Hawkes Self-Exciting Processes
Related Market Microstructure & Quantitative Finance Simulators
Engle-Russell ACD Model Lab

Autoregressive Conditional Duration model for irregular transaction arrival intervals and high-frequency trade clustering.

Explore ACD Lab
Ho-Stoll (1981) Market Making Lab

Optimal dealer pricing, inventory risk management, CARA reservation prices, and asymmetric bid-ask quote shading.

Explore Ho-Stoll Lab
Kyle-Obizhaeva Invariance Lab

Market microstructure invariance: 3/2 power law trade sizing, transactional business time, and execution slippage.

Explore Invariance Lab
Bouchaud Propagator Lab

Transient market impact propagator, long-memory order signs, bare power-law kernels, and critical diffusivity balance.

Explore Propagator Lab
Frequently Asked Questions: Hawkes Point Processes & High-Frequency Reflexivity

Alan Hawkes' (1971) self-exciting point process is a stochastic model where the occurrence of an event momentarily increases the conditional intensity or arrival hazard rate of future events. In market microstructure, it captures how transactions, quote revisions, and order cancellations trigger recursive cascades of algorithmic responses.

The branching ratio $\eta = \alpha / \beta$ represents the expected number of secondary or child events generated by a single parent event. When $\eta < 1$, the process is stationary and mean-reverting. When $\eta$ approaches 1, the market approaches criticality, where virtually all trades are endogenous algorithmic feedback loops.

The Hawkes process uses an exponential memory kernel $\alpha e^{-\beta(t - t_i)}$. Each transaction increases instantaneous arrival intensity by $\alpha$, which then relaxes exponentially back toward baseline $\mu$ at decay rate $\beta$, naturally reproducing empirical high-frequency volatility clustering and fat-tailed trade durations.

As established by Bacry et al. (2013) and Bouchaud et al., the asymptotic order flow count variance scales cubically as $1 / (1 - \eta)^3$. As $\eta \to 1$, infinitesimal exogenous shocks trigger massive self-fulfilling feedback avalanches, demonstrating the physical mechanism behind flash crashes.

Empirical calibration across global equity and futures order books (Hardiman et al., 2013; Filimonov & Sornette, 2012) showed that the branching ratio has climbed from approximately 0.3 in the 1990s to between 0.70 and 0.95 in modern markets, proving that over 70% to 90% of trading activity is endogenous algorithmic reaction rather than new macroeconomic information.