Every serious execution decision - horizon, urgency, strategy choice - leans on an estimate of what the order will cost before a single unit trades. Market impact models provide that estimate. This part builds the standard model intuition, shows how it plugs into scheduling, and covers the calibration traps that make impact modeling as much craft as science.

Temporary vs. Permanent Impact

Impact has two economically distinct components, distinguishable by what happens after you stop trading (the reversion diagnostic of Part 12):

• Temporary impact is the premium paid for demanding liquidity faster than it naturally replenishes - pressure that decays once the demand stops. It depends on how fast you trade.

• Permanent impact is the lasting price shift from the information your trading reveals: determined one-sided flow updates everyone’s beliefs. It depends mostly on how much you trade, not how fast.

The split matters because only temporary impact rewards patience. Stretching the horizon dilutes the liquidity premium but does little to the information content - you cannot schedule your way out of permanent impact.

The Square-Root Law

The most robust empirical regularity in execution research: for an order of size Q in an asset with daily volume V and daily volatility σ, expected impact behaves approximately as

impact ≈ c × σ × √(Q / V)

with the constant c of order one, remarkably stable across asset classes - equities, futures, FX, and (with wider error bars) crypto. Three practical readings:

• Concavity: impact grows with the square root of size - the fourth doubling of an order hurts less per unit than the first. Costs rise steeply at first, then flatten.

• Volatility scaling: the same percentage-of-volume trade costs several times more in a volatile alt than in a calm large-cap. Volatility is the price of liquidity.

• Participation as the lever: at fixed size, what the trader actually controls is the rate - which is why participation caps (Part 6) are the standard risk control.

Around this backbone, production models add a temporary-impact term that grows with participation rate, a decay profile describing how pressure relaxes, and adjustments for spread, depth, and time of day.

From Model to Schedule

Plug an impact model and a risk model into the optimizer of Part 10 and each candidate horizon yields a point on a curve: fast schedules with high expected cost and low variance, slow schedules with low cost and high variance. The result is the execution version of an efficient frontier. The trader’s risk aversion picks the point; the model determines the frontier’s shape. Pre-trade, the same model produces the cost estimate that anchors TCA (Part 12) - and a sanity check: an order whose estimated cost exceeds its expected alpha shouldn’t be traded at that size at all. Feeding impact estimates back into portfolio construction is one of the highest-value uses of the whole modeling stack.

Calibration: Why This Is Hard

• Endogeneity. Trading decisions correlate with the very price moves being measured - desks trade urgently precisely when prices are already running. Naive regressions of price change on trading attribute the alpha to impact, badly overestimating it. Clean calibration needs randomized experiments (deliberately varying urgency on comparable orders) or careful controls.

• Sparse extremes. The orders you most need the model for - the very largest - are exactly the ones you have least data on, because desks avoid trading them carelessly. Extrapolating the square-root curve into that tail is an act of informed faith.

• Regime drift. Impact constants move with volatility regimes, venue landscapes, and fee changes. Models need re-estimation on rolling windows, and their error bars deserve as much attention as their point estimates.

• Crypto specifics. Reported volume is unreliable on loosely surveilled venues (inflating V flatters the estimate), so depth-based measures - visible book depth within a band around mid, across venues - often anchor better. And because books are thin relative to flow, the temporary/permanent mix skews temporary: crypto prices bend far on pressure and snap back hard, which patient execution can exploit (Part 14).

Using the Model Honestly

Treat impact estimates as distributions, not prices: a good pre-trade report says “35 bps ± 20” and states the participation assumption behind it. Track realized-versus-predicted across many orders (the TCA loop), and be most suspicious of the model exactly where it is most consequential - large, urgent, illiquid orders. The desks that get this right don’t have better formulas; they have better feedback loops.

One market structure stresses every assumption in this part at once: fragmented venues, unreliable volume, thin books, no consolidated tape. The series closes there - Part 14: Execution in Crypto Markets.