Brownian Bridge™

1. In the Glosten-Milgrom model, after observing a market buy order, the dealer updates their belief $\mu_t = \Pr[V = V_H]$ upward. What is the economic mechanism by which this causes a permanent price impact?

The market buy order increases demand for the stock, pushing prices up mechanicallyThe buy order is evidence that an informed trader may have acted, raising the dealer's estimate of the true asset value; the dealer revises their mid-quote upward accordinglyThe buy order reduces the available ask liquidity, widening the spread and increasing the midThe dealer raises their ask to avoid further adverse selection from the same informed trader

2. In the Avellaneda-Stoikov model, the reservation price is $r(t,q) = S_t - q\gamma\sigma^2(T-t)$ . A market maker is short 20 shares ( $q = -20$ ), with $\gamma = 0.01$ , $\sigma = 0.25$ , and $T-t = 0.5$ (half a year). What is the reservation price relative to mid, and how does this affect the market maker's quotes?

r = S - 0.025

; the market maker quotes below mid, trying to sell

r = S + 0.025

; the market maker quotes above mid, trying to buy to reduce the short

r = S

; the short position has no effect on reservation price

r = S + 0.0125

; the effect is halved because the position is short not long

3. The Avellaneda-Stoikov optimal half-spread contains two terms: $\frac{1}{\gamma}\ln(1 + \gamma/k)$ and $\frac{1}{2}\gamma\sigma^2(T-t)$ . What does each term represent, and what happens to the total spread as $t \to T$ (end of the trading session)?

Both terms grow as

t \to T

, so the market maker widens spreads near the closeThe first term is the adverse selection component and grows; the second is inventory risk and shrinks. Total spread grows near the closeThe first term is a time-invariant spread floor (pure profit margin / adverse selection). The second is inventory risk that vanishes as

t \to T

, so the total spread narrows to the floor near the closeBoth terms vanish as

t \to T

, so the market maker quotes zero spread at close

4. In the Avellaneda-Stoikov model, if the market maker has zero inventory ( $q=0$ ), the optimal bid and ask quotes are symmetric around the mid-price $S_t$ .

truefalse

5. The Glosten-Milgrom model predicts that the bid-ask spread should narrow as more trades occur. What is the intuition, and does this prediction match empirical observations in equity markets?

More trades increase the dealer's inventory, widening the spread — this contradicts the GM predictionMore trades reveal information, reducing the dealer's uncertainty

\mu_t(1-\mu_t)

about the true value and narrowing the adverse selection component of the spread. Empirically, spreads do narrow during the session as price discovery occurs, consistent with GMMore trades reduce order arrival rates, and lower arrival rates narrow the spread by reducing adverse selection exposureThe GM model predicts a constant spread, independent of the number of trades

6. In the Avellaneda-Stoikov model, the order arrival intensity is $\lambda(\delta) = Ae^{-k\delta}$ . What is the optimal posted half-spread that maximises the expected income per unit time from one side of the book (ignoring inventory risk), treating the problem as a single-period revenue maximisation?

\delta^* = 0

— the market maker should always quote at the mid to maximise fill rate

\delta^* = 1/k

— obtained by maximising

\delta \cdot \lambda(\delta) = A\delta e^{-k\delta}

with respect to

\delta

\delta^* = k

— the spread equals the elasticity parameter

\delta^* = A/k

— proportional to the baseline arrival rate

Quiz: Adverse Selection and Inventory Models

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