Market Impact Estimation

Setup

Why Market Impact Estimation Matters

Market impact is the price change caused by a trader's own activity. Correctly estimating market impact is essential for:

1. Transaction cost analysis (TCA): decomposing the implementation shortfall into pre-trade estimate and realised impact to assess execution quality.
2. Optimal execution: the Almgren-Chriss model requires the impact parameters $\eta$ (temporary) and $\gamma$ (permanent) as inputs; these must be estimated from data.
3. Portfolio construction: large-scale optimisers (BlackRock Aladdin, Axioma) incorporate impact costs as a non-linear transaction cost penalty; wrong impact estimates lead to over-trading.
4. Strategy capacity analysis: an alpha strategy with zero capacity above a certain AUM (assets under management) is unscalable; impact estimation defines this limit.

Conventions

• Price impact: the change in mid-price attributable to a trade of size $Q$ shares. Measured as a fraction of the pre-trade mid (in basis points) or in ticks.
• Participation rate: $\eta_{\mathrm{part}} = Q / V$, the fraction of average daily volume (ADV) traded. ADV is typically the 20- or 60-day average.
• Normalised quantity: $q = Q / (V \cdot \sigma_{\mathrm{daily}}^{1/2})$ in some models (Kyle normalisation).
• Convention: impact is measured as the mid-price move from the start of the trade to some point after completion. "Instantaneous" impact is measured at the moment of the last fill; "realised" or "total" impact is measured 15–30 minutes after completion (allowing temporary impact to decay).

Kyle's Lambda: The Linear Impact Model

The Kyle (1985) Model

Kyle (1985) studies a single-period model with one informed trader, one market maker, and uninformed noise traders. The informed trader submits order $x$ (unknown to the market maker); noise traders submit $u \sim \mathcal{N}(0, \sigma_u^2)$. The market maker observes total order flow $y = x + u$ and sets price $P = \mathbb{E}[V \mid y]$.

Assumptions:

• True value: $V \sim \mathcal{N}(\mu_0, \Sigma_0)$.
• Informed trader maximises expected profit: $\Pi = (V - P)x$, choosing $x$ optimally.
• Market maker is competitive: sets $P$ to be a martingale.

Equilibrium (Linear): In the unique linear equilibrium:

$P = \mu_0 + \lambda\, y, \qquad \lambda = \frac{1}{2\sigma_u}\sqrt{\Sigma_0},$

where $\lambda > 0$ is Kyle's lambda — the price sensitivity to order flow. The informed trader submits:

$x^* = \frac{V - \mu_0}{2\lambda} = \frac{\sigma_u}{\sqrt{\Sigma_0}}(V - \mu_0),$

which hides in the noise: $x^* \sim \mathcal{N}(0, \sigma_u^2)$, indistinguishable from noise trading.

Kyle's Lambda as a Price Impact Coefficient

In continuous time, Kyle (1985) and subsequent work show that the permanent price impact per unit of signed order flow is:

$\Delta P = \lambda \cdot Q_{\mathrm{signed}},$

where $Q_{\mathrm{signed}}$ is the net signed order flow (positive for buys, negative for sells) and $\lambda$ has units of $\text{USD}/\text{share}^2$ (or $\text{USD}/\text{share}$ per share of flow).

Estimation from tick data. Regress mid-price changes on signed order flow:

$\Delta M_{t} = \alpha + \lambda \cdot \mathrm{OF}_t + \varepsilon_t,$

where $\mathrm{OF}_t = \sum_{i: t_i \in \Delta t} q_i \cdot \mathrm{sign}(P_i - M_{i-1})$ is the signed order flow in the interval $[t-\Delta t, t]$ (the Lee-Ready rule assigns sign using the trade direction relative to the preceding mid). The slope $\lambda$ is Kyle's lambda for that stock and time period.

Units: for a stock with ADV = 10 million shares and daily volatility of 1%, a typical $\lambda \approx 10^{-7}$ to $10^{-6}$ $\text{USD}/\text{share}^2$, consistent with 1–10 bps impact per 1% of ADV traded.

The Square-Root Law

Empirical Evidence

A robust empirical finding across asset classes (equities, futures, FX) is that the expected price impact of a meta-order of size $Q$ follows a square-root law:

$\mathrm{Impact}(Q) \approx \sigma_{\mathrm{daily}} \cdot Y \cdot \sqrt{\frac{Q}{V}},$

where:

• $\sigma_{\mathrm{daily}}$: daily volatility of the asset.
• $V$: average daily volume (ADV).
• $Y$: a dimensionless constant, typically $Y \in [0.5, 1.5]$ (empirically $\approx 1$ for large-cap equities).
• $Q/V$: participation rate (fraction of ADV).

Key references: Almgren et al. (2005), Torre and Ferrari (1997), Grinold and Kahn (1999), Zarinelli et al. (2015).

The square-root law has several remarkable properties:

• Universal across markets: the same functional form fits equities, futures, and FX with different $Y$ but the same square-root dependence on quantity.
• Concave in $Q$: large orders have sublinear impact — doubling the order size less than doubles the price move. This is consistent with limit order book dynamics: large orders are filled across many price levels; the first shares hit the best ask, subsequent shares reach successively less liquid levels.
• Inconsistency with linear models: the linear model ($\lambda Q$) used in Kyle and Almgren-Chriss is an approximation for small $Q/V$. The empirically correct law is nonlinear.

Theoretical Justification

Gabaix et al. (2006) and Farmer et al. (2013) derive the square-root law from order book theory. The argument:

Consider a book with a power-law distribution of order sizes $P[q \geq x] \propto x^{-\alpha}$ with $\alpha \approx 3/2$. A meta-order of size $Q$ must consume all orders up to some depth $d$. The expected depth required is $\mathbb{E}[Q_{\mathrm{book}}(d)] \propto d^{1/(1-1/\alpha)} = d^{2}$ for $\alpha = 3/2$. Inverting: $d \propto Q^{1/2}$. The impact is proportional to $d$, giving $\mathrm{Impact} \propto Q^{1/2}$.