Brownian Bridge™

1. In a limit order book with price-time priority, trader A posts a limit buy for 100 shares at $50.10. Five microseconds later, trader B posts a limit buy for 200 shares at$ 50.10. A sell market order for 150 shares then arrives. How are the shares allocated?

Trader A receives 75 shares and trader B receives 75 shares (pro-rata)Trader A receives 100 shares and trader B receives 50 shares (price-time priority)Trader B receives all 150 shares (larger order has priority)Trader A receives all 150 shares (earlier order has priority regardless of fill)

2. The microprice is $\hat{M} = P_{\mathrm{bid}} \cdot \frac{Q_{\mathrm{ask}}}{Q_{\mathrm{bid}} + Q_{\mathrm{ask}}} + P_{\mathrm{ask}} \cdot \frac{Q_{\mathrm{bid}}}{Q_{\mathrm{bid}} + Q_{\mathrm{ask}}}$ . If the best bid is $100.00 (500 shares) and the best ask is$ 100.01 (100 shares), what is the microprice and what does it signal?

$100.005 — balanced book, no directional signal$100.008 — the microprice is closer to the ask, signalling upward price pressure$100.002 — the microprice is closer to the bid, signalling downward price pressure$100.010 — the microprice equals the ask because the ask is thinner

3. In the Glosten-Milgrom model, a fraction $\pi$ of traders are informed (they know the true asset value) and $1-\pi$ are uninformed. At the break-even spread, a market maker sets the ask to equal the expected value of the asset conditional on receiving a market buy. With two value states $V_L$ and $V_H$ and equal prior probability, the equilibrium spread is:

s = (V_H - V_L)/2

, independent of

\pi

s = \pi(V_H - V_L)

, proportional to both the informed fraction and the value uncertainty

s = \pi(V_H - V_L)/2

, since only half the informed traders buy

s = (1-\pi)(V_H - V_L)

, since uninformed traders determine the spread

4. A Fill-or-Kill (FOK) order and an Immediate-or-Cancel (IOC) order both execute immediately; the difference is that FOK allows partial fills while IOC requires a complete fill.

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5. Order flow imbalance (OFI) over a time window measures net changes at the best bid and ask. Empirically, Cont, Kukanov, and Stoikov (2014) found that OFI predicts mid-price changes with a linear relationship $\Delta M \approx \beta \cdot \mathrm{OFI}$ . Why does OFI outperform trade direction (i.e., net signed trade flow) as a price impact predictor at short horizons?

OFI incorporates both passive (limit order) and aggressive (market order) activity at the touch, while trade direction only captures market ordersOFI is easier to compute from Level-1 data than trade directionOFI has a longer memory and therefore captures more of the price impactOFI eliminates the adverse selection component, leaving only the informational component

6. A market maker's limit sell at the ask is filled. Subsequently, the mid-price rises by 2 ticks. This represents a loss for the market maker due to adverse selection. In what sense is a resting limit order a 'free option' granted to the market?

It gives the market maker the right (but not obligation) to sell at the askIt gives any market participant the right to buy from the market maker at the posted price, regardless of subsequent price movesIt gives the exchange the option to match the order at a better priceIt gives the market maker an option on the spread income

Quiz: Limit Order Book Mechanics

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