Price impact in practice

M1 — Market microstructure

Kyle gave us a clean, derivable answer: impact is linear in order flow, at rate λ. It is the most-cited result in the field, and it is empirically wrong — in an interesting way.

This lesson is the collision between the two cleanest things in microstructure: the best model and the most robust measurement. Neither survives intact.

First, split the impact in two

Before measuring anything, separate two effects that a price chart hopelessly conflates. You buy a large parent order over an hour. The price rises. You stop. Some of the rise stays; some of it drains away.

   price
     │                    ╭─────╮
     │                ╭───╯      ╰────╮
     │            ╭───╯                ╰──────────────  ← permanent
     │        ╭───╯                                        component
     │  ──────╯
     │
     └──────┬──────────────────┬─────────────────────▶ time
         you start          you stop

     the hump you push up while trading  =  TEMPORARY impact
     the level it settles at afterwards  =  PERMANENT impact

The two have completely different causes, and you already have both models:

   PERMANENT  ← Glosten–Milgrom (M1.2).  The market inferred information
                from your flow and genuinely revised its belief. It does
                not revert, because nobody thinks they were wrong.

   TEMPORARY  ← Ho–Stoll (M1.3).  You consumed resting liquidity and left
                makers holding inventory they didn’t want, so they skewed
                their quotes. As they work back to flat, it decays.

This is why M1.3 insisted on the distinction. Permanent impact is the price of being informative; temporary impact is the price of being in a hurry. You can reduce the second by trading patiently. You cannot reduce the first at all — if your trading genuinely predicts the future, the market is right to move against you, and that cost is inseparable from having alpha in the first place.

The square-root law

Now measure it. Across equities, futures, FX, options, across decades, across venues, one relationship keeps showing up:

              ΔP  ≈  Y · σ · √( Q / V )

     ΔP  price move caused by the order (in the same units as σ)
     σ   the asset’s daily volatility
     Q   the size of your parent order
     V   the asset’s average daily volume
     Y   a constant of order ½ to 1, mildly venue- and asset-dependent

Note what it does not contain: any parameter fitted per stock beyond the two you already know (σ and V), and any hint of Kyle’s linearity. The dependence on size is concave — the marginal cost of the next share falls as your order grows.

Some numbers, for a stock with 2% daily volatility, taking Y = 1:

   you trade… (% of ADV)     √(Q/V)      price move    cost as % of ADV traded
   ──────────────────────────────────────────────────────────────────────────
        0.1%                  0.032         0.06%             tiny
        1%                    0.100         0.20%             noticeable
        10%                   0.316         0.63%             expensive
        50%                   0.707         1.41%             you are the market
       100%                   1.000         2.00%             a full day’s σ

The concavity is the whole story. Going from 1% to 10% of volume — ten times the size — only triples the price move. That is very good news if you are large, and it is the reason large funds exist at all.

But the cost scales as Q^1.5

Here is the trap, and it catches people who have just learned the square-root law and drawn the wrong conclusion from its concavity.

You pay the impact on every share you trade, so:

   cost  ≈  ΔP × Q  ∝  √Q × Q  =  Q^1.5

Ten times the size is only three times the price move, but thirty times the total cost. Impact per share is concave; the bill is convex. Two things follow immediately:

Every strategy has a capacity. Your expected alpha scales like Q — twice the position, twice the profit. Your cost scales like Q^1.5. There is therefore a size at which the two cross and the strategy stops making money, no matter how good the signal. This is not a failure of execution; it is arithmetic, and it is why a strategy that prints money on £10m can be worthless at £1bn.

Slicing is worth real money. If impact depends on your participation rate rather than your absolute size, then splitting a parent order into child orders spread over time — and letting fresh liquidity and fresh noise traders arrive between them — genuinely reduces the bill. That is what the entire execution-algorithm industry sells: VWAP, TWAP, implementation-shortfall and participation-rate algos are all variations on how slowly dare I go.

And there’s the trade-off that makes execution a real problem rather than a solved one: trade fast and pay impact; trade slow and bear risk that the price moves for unrelated reasons before you’re done. The optimal schedule balances the two — that’s the Almgren–Chriss framework, and it is the direct commercial descendant of everything in M1.

Why the square root? (honestly: unsettled)

There is no consensus derivation. Three serious families of explanation:

Latent liquidity. The visible book is a tiny fraction of real trading intent; most liquidity is latent, in the minds of traders who reveal it only as the price approaches. Under plausible assumptions about how that latent density is shaped near the current price, √ falls out.
Fair pricing / no-arbitrage in impact. If impact were linear, a trader could in some setups profitably round-trip against their own footprint; requiring that this be impossible pins the exponent near ½.
Propagator models. Model impact as a decaying kernel applied to each child order. Order flow is famously long-memory (it autocorrelates for hours), and the decay must exactly offset that persistence or prices would be predictable — which forces a concave aggregate.

They are not mutually exclusive and none is airtight. Be comfortable saying the law is an extremely robust empirical regularity with contested foundations — that is the honest state of the field, and pretending otherwise is a tell.

Reconciling with Kyle

Kyle isn’t refuted so much as local. Its linear λ describes a single auction with a fixed information set and Gaussian noise; the square-root law describes a parent order worked over hours against a market whose liquidity replenishes and whose participants adapt. Kyle is the right object for “what does one unit of surprise flow do right now”; the square-root law is the right object for “what will this £50m cost me by close.”

Both survive because they answer different questions. Reaching for the wrong one is the actual error.

Source: Bouchaud, Bonart, Donier & Gould, Trades, Quotes and Prices (2018) — the definitive modern treatment, and ch.11–12 cover impact properly. For the original empirics, Almgren et al. (2005); for the execution-scheduling problem, Almgren & Chriss (2000), “Optimal execution of portfolio transactions”. Bouchaud’s “Why is the order flow so persistent?” is the readable way into propagators.