Skip to content

Propagator Model (Transient Impact Kernel)

Algorithm description

The propagator model (also "transient impact model") represents the mid-price return at time \(t\) as a linear superposition of the decaying impact of all past signed order-flow events. Each event leaves a "footprint" that fades over time according to a kernel \(G\).

\[r_t = \sum_{t' \leq t} G(t - t') \cdot \varepsilon_{t'} + \eta_t\]

where: - \(\varepsilon_{t'} \in \{-1, +1\}\) (or signed volume) is the signed trade / order-flow event at time \(t'\). - \(G(\tau) \geq 0\) is the response function / propagator: how much price impact from an event at lag \(\tau\) remains at the current instant. - \(\eta_t\) is an uncorrelated residual.

For a meta-order executed over a window, the expected price displacement decomposes as:

\[\mathbb{E}[p(t+\tau) - p(t)] = \theta(Q) \cdot I(\tau) + \alpha \cdot H(\tau)\]

where \(\theta(Q) = \epsilon Y_0 \sigma (Q/V)^\delta\) is the instantaneous (square-root) impact, \(I(\tau)\) is the mechanical impact-decay kernel with \(I(0) = 1\), \(I(\infty) \approx 0\), and \(\alpha \cdot H(\tau)\) is a (possibly zero) predictor-induced term for informed trades.

Typical functional forms

Form Formula Used in
Power-law decay \(G(\tau) \sim \tau^{-\beta}\), \(\beta \in [0.2, 0.5]\) BFL 2008, Noble-Rosenbaum-Souilmi 2026
Stretched exponential \(G(\tau) \sim e^{-(\tau/\tau_0)^\gamma}\) fits some FX and futures data
Exponential \(G(\tau) = A e^{-\tau/\tau_0}\) tractable; too fast for equity meta-orders
Multivariate Hawkes kernel \(G_{ij}(\tau)\) matrix of kernels Bacry et al.; see hawkes-process

The paper of record (bouchaud-farmer-lillo-propagator) argues for slow power-law decay as the best empirical fit on equities.

Why it works (the long-memory reconciliation)

The empirical puzzle the propagator model solves:

  • Signed order flow is long-memory — autocorrelation \(\mathbb{E}[\varepsilon_t \varepsilon_{t+\tau}] \sim \tau^{-\gamma}\) with \(\gamma < 1\) on most liquid equities, driven by meta-order splitting over days.
  • Prices are near-martingales — returns have negligible autocorrelation at short horizons.

A fixed permanent impact model (\(G(\tau) = \text{const}\)) would imply predictable returns, contradicting martingality. A purely temporary impact model (\(G\) with short memory) cannot explain why large metaorders move prices over days.

The propagator framework reconciles both: \(G\) decays slowly enough to let long-memory flow accumulate into persistent price moves, yet fast enough that returns remain unpredictable. The two power-law exponents must satisfy a self-consistency relation: \(\beta + \gamma/2 \approx 1\) (Bouchaud et al.).

Deconvolution

The raw measured impact of meta-orders is biased upward at long lags because correlated subsequent trades from the same investor keep reinforcing the original direction. brokmann-slow-decay-impact shows that after deconvolving the investor's own autocorrelated order flow, the mechanical propagator \(I(\tau)\) decays essentially all the way to zero — the commonly-reported "2/3 plateau" is an artefact, not a structural feature.

Deconvolution proceeds via an OU-predictor toy model that yields an analytic convolution relating \(I_\text{raw}\) to the true \(I\) and the signal autocorrelation; inversion gives the true kernel.

Stability / self-consistency

For the propagator to produce finite, stationary returns:

\[\int_0^\infty |G(\tau)|^2 \, d\tau < \infty\]

A fat-tailed \(G\) with \(\beta < 1/2\) can diverge. In multivariate (multi-asset, multi-event-type) extensions, the spectral radius of the kernel matrix must stay below 1 (same stability condition as for Hawkes processes).

Computational complexity

  • Naive simulation: \(O(n^2)\) for \(n\) events (each event sums over all past).
  • With exponential kernel: \(O(n)\) via recursive updates.
  • With sum-of-exponentials: \(O(Kn)\) for \(K\) components — common approximation for power-law kernels.
  • Fitting / deconvolution: iterative Wiener-filter-style inversion; or MLE under a specific kernel family.

Applications in this wiki

Paper Role of propagator
bouchaud-farmer-lillo-propagator Introduces the framework; surveys empirics
eisler-bouchaud-kockelkoren-order-book-events Generalises propagator to six event types (MO/LO/CA × at-best/inside-spread). See event-type-impact-decomposition.
models-for-all-order-book-events Fully dynamic extension — history-dependent bare impacts for small-tick stocks via linear AR model on gaps.
brokmann-slow-decay-impact Empirical calibration via deconvolution on CFM meta-orders
reality-gap-lob-simulation Embeds a power-law decay kernel into an extended queue-reactive simulator to reproduce concave impact during execution and partial reversion after
order-flow-filtration Uses Hawkes kernel norms — the multivariate generalisation — as diagnostic of OBI→return excitation
csi300-ou-levy-ofi OU-Lévy process is an alternative framing of the same "transient response to a shock" idea

When to use / when not to use

Use when: - Modelling metaorder execution cost over minutes to days. - Backtesting strategies where self-impact matters (propagator tells you the cost of your own trading). - Reconciling long-memory order flow with near-martingale prices.

Avoid or adapt when: - Sub-second horizons — individual event impact is more naturally modelled via queue dynamics (e.g., queue-reactive-model). - Non-stationary regimes (crashes, auctions) — \(G\) is state-dependent; constant-kernel assumption breaks. - Highly concave/non-linear interactions at large sizes — the linear superposition assumption becomes poor.

Relationship to other impact models

Model Permanent? Transient? Nonlinearity
Fixed permanent (Kyle) yes no any
Propagator (BFL) no (decays to 0) yes linear superposition
History-dependent permanent yes yes interpolates both
Square-root meta-order law phenomenological concave in \(Q\) emergent from propagator + long-memory

The square-root law of meta-order impact \(I(Q) \sim \sigma(Q/V)^{1/2}\) is an emergent consequence of the propagator + long-memory flow — it is not assumed; it falls out.

Implementations

  • Hawkes libraries (tick.hawkes, hawkeslib) implement multivariate propagator kernels.
  • Custom MLE / Wiener-filter deconvolution in NumPy/SciPy suffices for exponential and power-law kernels.
  • LOBFrame (deep-lob-forecasting) does not model propagator impact explicitly — a gap that a hybrid LOBFrame+propagator simulator would close.

Connections