The Kyle (1985) model

Glosten–Milgrom’s informed trader has no decisions to make. He knows the value, so he buys or he sells, one unit, and the model is about what the maker infers. But a real informed trader faces the interesting question: how much should I trade?

Trade too little and you leave money on the table. Trade too much and you move the price against yourself — and worse, you reveal yourself, because the maker is watching the flow. Kyle (1985) is that trade-off solved exactly, and it produces the single most-used object in microstructure: λ, the price-impact coefficient.

The setup

One round of trading. Three players:

   the INFORMED TRADER   knows the true value v, drawn v ~ N(p₀, Σ₀).
                         Chooses a quantity x. Risk-neutral.

   the NOISE TRADERS     submit a random net quantity u ~ N(0, σ²ᵤ),
                         independent of v. They are not optimising;
                         they are the camouflage.

   the MARKET MAKER      sees ONLY the total order flow y = x + u —
                         never the split — and, being competitive
                         and risk-neutral, sets p = E[v | y].

The single most important line there is that the maker sees only y. It cannot tell a large informed buy from a run of lucky noise. That is the informed trader’s entire opportunity, and the reason he is willing to trade at all.

Note what’s switched off relative to M1.3: the maker is risk-neutral again and holds no inventory. Kyle is a pure-information model like Glosten–Milgrom — the new ingredient is strategy.

Conjecture, optimise, close the loop

The standard method: guess that both sides play linear strategies, solve each side’s problem taking the other as given, then demand the guesses be mutually consistent.

   informed trader:   x  =  β·(v − p₀)      trade proportional to your edge
   market maker:      p  =  p₀ + λ·y        price linear in total flow

Step 1 — the informed trader’s optimum. His profit is the edge he captures on the quantity he trades, at the price his own trading creates:

   π  =  (v − p)·x  =  (v − p₀ − λ(x + u))·x

   E[π | v]  =  (v − p₀)·x  −  λx²          (E[u] = 0)

   dE[π]/dx  =  (v − p₀) − 2λx  =  0

   ⟹   x*  =  (v − p₀) / (2λ)      so      β = 1/(2λ)

Already worth pausing on. The informed trader does not trade until the price reaches v — that would be x = (v − p₀)/λ. He trades exactly half that. The λx² term is his own market impact biting him, and the optimum splits the difference between edge and impact.

Step 2 — the maker’s zero-profit condition. Setting p = E[v | y] with everything jointly normal makes this a linear regression of v on y:

   λ  =  Cov(v, y) / Var(y)

   y = β(v − p₀) + u   ⟹   Cov(v,y) = βΣ₀ ,  Var(y) = β²Σ₀ + σ²ᵤ

   λ  =  βΣ₀ / (β²Σ₀ + σ²ᵤ)

Step 3 — the fixed point. Substitute β = 1/(2λ) and solve for λ:

   λ  =  (Σ₀/2λ) / (Σ₀/4λ² + σ²ᵤ)

   λ·(Σ₀/4λ² + σ²ᵤ)  =  Σ₀/2λ

   Σ₀/4λ + λσ²ᵤ      =  Σ₀/2λ

   λσ²ᵤ              =  Σ₀/4λ

   λ²                =  Σ₀ / (4σ²ᵤ)
   ─────────────────────────────────────────────────
        λ  =  ½·√(Σ₀ / σ²ᵤ)        β  =  √(σ²ᵤ / Σ₀)

Reading the result

Price impact is linear. p moves by λ per unit of net order flow, regardless of size. Not concave, not stepped — linear. Kyle’s λ is what people mean by “market impact” in a thousand execution papers, and it is why a single number can stand in for a market’s resilience.

Depth is 1/λ = 2·√(σ²ᵤ/Σ₀), a ratio of noise to uncertainty.

   more noise-trader volume σ²ᵤ   →   λ falls   →   DEEPER market
   more uncertainty about v Σ₀    →   λ rises   →   THINNER market

Both directions are intuitive once said aloud. Noise is camouflage: the more of it there is, the more the informed trader can hide, so the less any given order tells the maker, so the less the maker must move. Uncertainty is the opposite: when nobody knows what the asset is worth, every order is informative and the maker flinches.

That is the sentence to keep. Depth is not “how much money is resting in the book.” Depth is how hard it is to tell information from noise.

Exactly half the information reaches the price. Compute the maker’s residual uncertainty after seeing the flow:

   β² = σ²ᵤ/Σ₀   ⟹   β²Σ₀ = σ²ᵤ   ⟹   Var(y) = 2σ²ᵤ
   Cov(v,y) = βΣ₀ = √(σ²ᵤΣ₀)

   Var(v | y)  =  Σ₀ − Cov²/Var(y)  =  Σ₀ − σ²ᵤΣ₀/(2σ²ᵤ)  =  Σ₀/2

The informed trader trades precisely enough to halve the market’s variance about the true value, and keeps the other half. It isn’t a coincidence or a tuned parameter — it drops out of the optimisation. Information enters prices because someone profits by putting it there, and only ever partially, because full revelation would destroy the profit that motivated it.

A worked feel for λ

Suppose the market is unsure about a £100 stock to the tune of Σ₀ = 4 (so a standard deviation of £2), and noise flow has σ²ᵤ = 10⁶ (σᵤ = 1,000 shares).

   λ  =  ½·√(4 / 10⁶)  =  ½ × 0.002  =  0.001 £/share

   ⟹  a net 1,000-share buy moves the price  £1.00
   ⟹  depth 1/λ = 1,000 shares per £1 of price move
   ⟹  informed trader with v = £103 trades β(v−p₀)
        = √(10⁶/4) × 3 = 500 × 3 = 1,500 shares
        ...pushing the price to about £101.50, halfway to the truth

Now quadruple the noise, σᵤ = 2,000: λ halves to 0.0005, and the same informed trader doubles his size to 3,000 shares. More liquidity invites more informed trading — the two grow together, which is why deep markets are not naive markets.

What Kyle gives you, and what it doesn’t

It gives you the vocabulary that execution desks actually use: λ, market depth, linear impact, hiding in volume. It explains why big orders are sliced — impact is paid on each unit of net flow, so spreading a parent order across time (and across the noise arriving in that time) is worth real money. The multi-period version of the model makes this precise: the informed trader releases his information at a constant rate, trading so as to keep the market permanently unable to distinguish him.

What it doesn’t give you is the square-root law. Kyle says impact is linear in size; four decades of data say it goes roughly as √size. That gap between the cleanest model in the field and the most robust empirical regularity in the field is not a footnote — it is the subject of M1.5.

Source: Kyle (1985), “Continuous Auctions and Insider Trading”, Econometrica — §1 is the single-period model above and is about two pages of algebra; it repays doing by hand. O’Hara ch.4 for a slower walk-through, including the multi-period version where the informed trader’s optimal rate of information release is derived.