Brownian Bridge™

1. In the Kyle (1985) single-period model, the equilibrium price impact coefficient is $\lambda = \frac{1}{2\sigma_u}\sqrt{\Sigma_0}$ , where $\sigma_u$ is the noise trader volume and $\Sigma_0$ is the prior variance of the asset value. What happens to $\lambda$ as $\sigma_u \to \infty$ (very noisy market), and what does this mean economically?

\lambda \to \infty

: a noisier market means any order flow is more likely to be informed, so the market maker raises the price more per unit of flow

\lambda \to 0

: a noisier market means the market maker cannot distinguish informed from uninformed flow, so the price reaction per unit of flow falls toward zero

\lambda

stays constant: noise trading and informed trading exactly offset each other

\lambda \to \Sigma_0^{1/2}

: the impact is determined solely by the fundamental value uncertainty

2. The square-root law of market impact states $\mathrm{Impact}(Q) \approx Y\sigma\sqrt{Q/V}$ , where $Q$ is the order size, $V$ is ADV, and $\sigma$ is daily volatility. A trader executes $Q = 0.01V$ (1% of ADV). If $Y = 1$ and $\sigma = 1\%$ /day, what is the expected price impact in basis points?

1 bps: the impact is linear in

Q/V

10 bps:

1\% \times \sqrt{0.01} = 1\% \times 0.1 = 0.1\% = 10

bps100 bps: equal to one day's volatility0.1 bps: the square root makes the impact much smaller than the linear approximation

3. Estimating Kyle's lambda from tick data by regressing mid-price changes on signed order flow can produce upward-biased estimates. The Lee-Ready algorithm is used to assign trade signs. Which of the following is the primary source of bias in this regression?

The Lee-Ready algorithm misclassifies approximately 50% of trades at the mid-price, creating noise that biases

\lambda

toward zeroBid-ask bounce: trades alternate between the bid and ask, creating negative autocorrelation in transaction price changes that inflates the apparent price sensitivity to order flowThe informed trader hides their trades in noise, making the signed flow uncorrelated with price changesMid-prices are quoted in discrete ticks, creating rounding errors that bias the regression slope

4. The permanent impact component of a trade measures the price move that persists after the trade is complete. A permanent impact close to zero implies that the trade was uninformative — the price reverts to its pre-trade level because no new information was revealed.

truefalse

5. A trader is considering whether to execute a large sell order using TWAP over 1 day or to execute it immediately. The square-root impact model gives $h(v) = \eta\sigma\sqrt{v/V}$ (temporary impact only; ignore permanent). For TWAP with constant rate $v_0 = Q/T$ , the total temporary impact cost is $\int_0^T h(v_0)\,dt = T \cdot h(Q/T)$ . For immediate execution at rate $v_0 \to \infty$ (a single market order of size $Q$ ), the cost is $h(Q/\Delta t)\cdot\Delta t$ for small $\Delta t$ . Which statement correctly compares these costs?

TWAP and immediate execution have the same total cost because

\int h(v)\,dt

is path-independentTWAP is cheaper than immediate execution: spreading the order reduces the impact rate at each instant, and since

h

is concave (

h(v) \propto \sqrt{v}

), TWAP's average cost is lower than the single-tranche costImmediate execution is always cheaper for concave impact functions because the cost per share decreases with quantityTWAP is always cheaper for any impact function, regardless of whether it is concave or convex

6. In the context of market impact estimation, what is a 'meta-order' and why is correctly identifying meta-order boundaries important for estimating the square-root law?

A meta-order is an order that spans multiple assets simultaneously; identifying it is important to measure cross-impactA meta-order is the sequence of child orders executed by an algorithm to fill a single parent order. Incorrect identification merges separate orders (overestimates

Q

) or splits one order (underestimates), both of which distort the estimated impact-vs-size relationshipA meta-order is an order submitted to multiple venues simultaneously; identifying it prevents double-counting the impactA meta-order is an iceberg order; identifying it is important to separate the displayed from the hidden liquidity

Quiz: Market Impact Estimation

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