Propagator Models Of Price Impact

The simplest picture of market impact treats each trade as pushing the price by a fixed amount that stays pushed forever — buy some shares, the price permanently rises by a bit, done. Real markets don't behave that way: a trade's effect on price fades over subsequent minutes as liquidity providers refill the book and the temporary imbalance the trade created gets absorbed. Propagator models are built specifically to capture that fading.

Impact as an echo, not a step

In a propagator model, every trade sends out a small push to the price, and that push doesn't stay constant — it decays over time according to a decay kernel (the "propagator"), much like an echo that's loud right after the trade and fades as time passes. Total price impact at any moment is the sum of all decaying echoes from every trade made so far, some still loud, others nearly silent. Mathematically, if a trade of size $v$ at time $s$ contributes an impact of $v \cdot G(t-s)$ to the price at later time $t$, where $G$ is the decay kernel, total impact is the sum of these contributions across every trade. In plain English: impact isn't one number a schedule produces at the end — it's many separate pushes, each with its own decaying lifetime, overlapping and adding up at every point in time.

This decaying structure separates a propagator model from earlier "permanent impact" models, and it matters because it changes the optimal way to trade: if impact from early trades has already faded by the time later trades happen, front-loading a schedule is less costly than a permanent-impact model would suggest.

Worked example: two trades and their fading echoes

Suppose a simple exponential decay kernel: a trade's impact starts at some peak value and halves every ten minutes. A trader buys 10,000 shares at 10:00 a.m., creating a price push of 5 cents. By 10:10, that push has decayed to about 2.5 cents. Now a second 10,000-share trade happens at 10:10, adding its own fresh 5-cent push on top of what remains from the first. Total impact at 10:10 is the first trade's decayed 2.5 cents plus the second's fresh 5 cents, for 7.5 cents — not simply double the single-trade impact, since the two pushes sit at different points in their own decay curves. By 10:30, both have substantially faded, illustrating why spacing trades out can reduce peak combined impact compared to trading the same total size in one tight burst.

What this means in practice

Propagator models are the mathematical backbone of transient-impact models used to design real execution schedules, most notably the Obizhaeva-Wang model, which uses a specific exponential kernel to derive closed-form optimal trajectories. A key constraint on any valid decay kernel is the no-dynamic-arbitrage condition: the kernel can't decay in a way that lets a trader profit by buying and selling back and forth exploiting the impact pattern itself — see Gatheral's no-dynamic-arbitrage condition.