16 Market Makers
16.1 Introduction
Three questions structure this chapter. First, why does a bid-ask spread exist at all — why will a dealer simultaneously quote a lower price to buy and a higher price to sell the same asset? Second, how does an informed trader’s strategic demand choice interact with competitive market makers’ pricing rules to determine how efficiently private information is incorporated into prices? Third, if prices do reveal private information, what incentive remains for anyone to acquire that information in the first place? These questions sit at the heart of market microstructure, and the three canonical models examined here — Glosten-Milgrom, Kyle, and Grossman-Stiglitz — each isolates a different facet of the same underlying tension between informed and uninformed trading.
The first question matters because the bid-ask spread is the most visible cost of transacting in financial markets. A spread is not the result of dealer greed or market power alone; it arises even among perfectly competitive market makers as a rational response to the risk of trading against someone who knows more than they do. When an investor places a buy order, a competitive market maker cannot tell whether that investor is acting on a private signal that the asset is undervalued or merely has liquidity needs. Setting the ask equal to the expected value of the asset conditional on receiving a buy order — higher than the unconditional expectation — is the only way to avoid expected losses. The Glosten-Milgrom model formalizes this logic, showing that the spread is determined by the proportion of informed traders in the market and the magnitude of the information asymmetry, and that it shrinks as competition among market makers intensifies or as underlying asset volatility falls.
The second question takes the analysis a step further by asking how an informed trader should optimally exploit her informational advantage when she knows the market maker will adjust prices against her. The Kyle model addresses this: the insider trades strategically, spreading her demand across time to camouflage it within the noise generated by uninformed liquidity traders. In equilibrium the market maker sets a linear pricing rule whose slope — the Kyle lambda — measures market depth, the price impact of each additional unit of order flow. A higher fraction of noise trading relative to fundamental uncertainty produces a deeper market and a lower price impact, but never eliminates it entirely.
The third question strikes at the foundations of price informativeness. If prices were to perfectly aggregate and reveal all private information, the value of acquiring that information would vanish and no one would bother to collect it — yet without anyone collecting it, prices could not be informative. The Grossman-Stiglitz model resolves this paradox by showing that an equilibrium with fully revealing prices cannot exist when information acquisition is costly. Instead, prices are only partially informative: they reveal enough that the marginal informed trader is exactly indifferent between paying the cost of information and free-riding on the price signal. The chapter develops all three models in sequence, building from the simple binary-value setting of Glosten-Milgrom through the continuous Gaussian framework of Kyle to the competitive rational-expectations equilibrium of Grossman-Stiglitz.
When a retail trader places a “free” trade on an app, a market maker is on the other side, earning the bid–ask spread and paying the broker for the order. NBC News examined what lies “beneath the surface” of the 2021 GameStop frenzy — payment for order flow, and firms like Citadel Securities that handle a large share of retail volume. The spread those firms quote, and how it widens when they fear trading against someone better informed, is exactly the market-microstructure problem modeled in this chapter. Read it at NBC News.
16.2 The Glosten-Milgrom model of the spread
The Glosten-Milgrom model helps us to understand the motivation of liquidity providers when setting bid and ask prices.
Consider a set of identical, competitive market makers that do not know whether a stock will go up or down today. They must set prices at which they are willing to buy and sell shares of a particular stock. The price at which they will buy the stock is the bid (\(B\)) and the price at which they will sell the stock is the offer (or ask) (\(A\)).
Each market maker knows that the stock will be worth either \(V_0\) or \(V_1\) at the end of the day where \(V_0 < V_1\). Prior to the opening of trading, that market maker believes that the probability that the stock will be worth \(V_0\) is \(P(V_0) = 1-\pi\). Regardless of the future value of the stock, uninformed traders will arrive to both buy and sell the stock.
The market maker must decide where to set \(B\) and \(A\) in order to not lose money on average. If however, any market maker sets their spread wide enough to earn positive profits on average, competition will drive another market maker to come in and supply a better (narrower) bid/ask spread. The desire to not lose money, combined with competition means that in equilibrium, we will look for \(B = E(V|\mbox{next order is a sell})\) and \(A = E(V|\mbox{next order is a buy})\).
Suppose that amongst liquidity demanders, the fraction \(\mu\) are informed, meaning they know the value of the signal that has been received. Fraction \(1-\mu\) are uninformed. Each liquidity demander will demand exactly one share of the asset. The resolution of uncertainty is depicted in figure 1.
Thus, if an uninformed trader arrives at the market, the market maker will learn nothing about the value of the asset from trading with them. As such
\[E[V|\mbox{uninformed buy}] = E[V|\mbox{uninformed sell}] = E[V]\]
This implies that if traders were all uninformed, the spread on an asset would be zero. The bid \(B = E[V|\mbox{sell order}] = E[V]\) while the ask is \(A = E[V|\mbox{buy order}] = E[V]\).
However, an informed trader knows the true value of the asset. Therefore, if the market maker knew that the next trade to arrive is coming from an informed trader, then \(A = E[V|\mbox{buy order}] = V_{1}\) and \(B = E[V|\mbox{sell order}] = V_{0}\). The problem then for a market maker is that she doesn't know whether she is trading with an informed trader or an uninformed trader. As such, she must set the spread so that she doesn't lose money on average. That is, the money that she loses from trading with informed traders she recoups from trading with uninformed traders. We want to calculate \(E[V|\mbox{buy}]\). To do that, we first calculate \(P(\mbox{buy}) = \pi \mu + \pi(1-\mu)\frac{1}{2} + (1-\pi)(1-\mu)\frac{1}{2} = \pi \mu + \frac{1}{2}(1-\mu)\). (An informed trader buys only when the value is high, an event of probability \(\pi\mu\); an uninformed trader buys with probability \(\frac12\) regardless of value, contributing \(\frac12(1-\mu)\).) Bayes’ rule then gives the probability that a buyer is informed,
\[P(\mbox{In}|\mbox{buy}) = \frac{\pi\mu}{\pi\mu + \frac{1}{2}(1-\mu)}, \qquad P(\mbox{Un}|\mbox{buy}) = \frac{\frac{1}{2}(1-\mu)}{\pi\mu + \frac{1}{2}(1-\mu)},\]
and \(E[V|\mbox{buy},\mbox{In}] = V_1\) (an informed buyer knows the value is high), while \(E[V|\mbox{buy},\mbox{Un}] = E[V] = \pi V_1 + (1-\pi)V_0\). Then, to calculate the ask (similarly for the bid) we calculate
\[\begin{aligned} \begin{aligned} A & = E[V|\mbox{buy order}] = P(In|\mbox{buy})E[V|\mbox{buy}, \mbox{In}] + P(\mbox{Un}|\mbox{buy})E[V|\mbox{buy}, \mbox{Un}] \\ & = \frac{1}{\pi \mu + \frac{1}{2}(1-\mu)} \left (\pi\mu V_{1} + \frac{1-\mu}{2}\left [ \pi V_{1} + (1-\pi)V_{0} \right ] \right) \\ & = \frac{\pi(1+\mu)V_{1} + (1-\pi)(1-\mu)V_{0}}{1 + (2\pi - 1)\mu} \\ \end{aligned} \end{aligned}\]
The last line multiplies the numerator and denominator of the previous line by \(2\): the denominator becomes \(2\pi\mu + (1-\mu) = 1 + (2\pi-1)\mu\), and the numerator becomes \(2\pi\mu V_1 + (1-\mu)[\pi V_1 + (1-\pi)V_0] = \pi(1+\mu)V_1 + (1-\pi)(1-\mu)V_0\).
Whereas the bid is
\[\begin{aligned} \begin{aligned} B & = E[V|\mbox{sell order}] = P(\mbox{In}|\mbox{sell})E[V|\mbox{sell}, \mbox{In}] + P(Un|\mbox{sell})E[V|\mbox{sell}, \mbox{Un}] \\ & = \frac{1}{(1-\pi) \mu + \frac{1}{2}(1-\mu)} \left ((1-\pi)\mu V_{0} + \frac{1-\mu}{2}\left [ \pi V_{1} + (1-\pi)V_{0} \right ] \right) \\ & = \frac{\pi(1-\mu)V_{1} + (1-\pi)(1+\mu)V_{0}}{1 + (1-2\pi)\mu} \end{aligned} \end{aligned}\]
From this we can compute the spread \(A-B\). Note that the ask and bid have different denominators: writing \(D_A = 1 + (2\pi-1)\mu\) and \(D_B = 1 + (1-2\pi)\mu = 1 - (2\pi-1)\mu\), we have \(A = N_A/D_A\) and \(B = N_B/D_B\) with numerators \(N_A = \pi(1+\mu)V_1 + (1-\pi)(1-\mu)V_0\) and \(N_B = \pi(1-\mu)V_1 + (1-\pi)(1+\mu)V_0\). Combining over the common denominator \(D_A D_B = 1 - (2\pi-1)^2\mu^2\),
\[A - B = \frac{N_A D_B - N_B D_A}{1 - (2\pi-1)^2\mu^2}.\]
Expanding the numerator, the pieces symmetric in the two states cancel and one is left with \(N_A D_B - N_B D_A = 4\pi(1-\pi)\mu\,(V_1 - V_0)\), so
\[A - B = \frac{4\pi(1-\pi)\mu\,(V_{1} - V_{0})}{1 - (2\pi-1)^2\mu^2}.\]
If \(\pi = 0.5\) the denominator is \(1\) and \(2\pi-1 = 0\), so this becomes \(A-B = \mu(V_{1} - V_{0})\).
The Glosten-Milgrom model shows that there are several forces which narrow spreads and thus increase liquidity in limit order markets. These forces include
- Competition between market makers
- Decreases in the probability of informed trading, and
- Decreases in the volatility of the underlying asset.
16.3 The Kyle Model of Orderbook Shape
The participants in the Kyle model are similar to those in the Glosten-Milgrom model. We have uninformed traders, informed traders (or insiders in Kyle's language) and market makers. Uninformed traders have random demand that has mean 0 and a known variance. Let the demand for uninformed traders be given by \(u \sim N(0, \sigma^{2}_{u})\). Informed traders observe the future value of the asset \(v\) and select their demand \(x\) as a function of this observed \(v\) and the price set by the market makers. Assume that \(v \sim N(p_{0}, \Sigma_{0})\). A competitive set of market makers establish an order book described as a price that is a function of the number of orders that they observe \(y = x + u\). Since these market makers are competitive, they will in expectation earn zero profit. It is assumed that market makers cannot distinguish between informed and uninformed liquidity requests.
The expected profit to a market maker given the demand for liquidity \(y\) is \(E\pi = E(v - p)(-y)\) (recall that the market maker must take the opposite position of the liquidity demanders). If expected profits are to be zero for every \(y\), then in equilibrium \(p = E[v|y]\). In order to find an equilibrium in this model, we proceed by making an assumption about the behavior of market makers, which at the end we verify to be true. Let us assume that market makers set the price according to a linear function of the demand that they observe. Specifically, let \(P(y) = \mu + \lambda y\).
Now, we consider the problem of informed traders. The expected profit to informed traders is \(E\pi = E[(v-p(y))x|v]\). In order to maximize this profit, informed traders solve (given the price function of market makers.)
\[max_{x} E[\pi] = \max_{x} E[(v-\mu - \lambda(x + u))x|v]\]
Since \(Eu = 0\), the objective is \(E[\pi] = (v - \mu - \lambda x)x = vx - \mu x - \lambda x^2\). Differentiating with respect to \(x\) and setting the derivative to zero gives the first order condition
\[\frac{\partial E[\pi]}{\partial x} = v - \mu - 2\lambda x = 0.\]
Solving for \(x\) gives \(x = \frac{v - \mu}{2\lambda} = -\frac{\mu}{2\lambda} + \frac{1}{2\lambda}v\). Notice if the market makers use a linear pricing rule, then the informed traders will set their demand as a linear function of the signal that they receive.
Now consider the problem of market makers. Given that the demand that they observe is \(y = x + u\), they need to calculate \(E[v|x + u]\). Note that \(v\) and \(y\) are related since \(v\) determines \(x\) which is a part of \(y\). In fact, since \(y\) is a linear function of \(v\), \(y\) and \(v\) are jointly normally distributed. The demand \(y\) is \(y \sim N(-\frac{\mu}{2\lambda} + \frac{1}{\lambda}p_{0}, \frac{1}{\lambda^2}\Sigma_{0})\). The covariance between \(y\) and \(v\) is
\[\begin{aligned} \begin{aligned} E(y - Ey)(v - Ev) & = E(-\frac{\mu}{2\lambda} + \frac{1}{2\lambda}v + u +\frac{\mu}{2\lambda} + \frac{1}{2\lambda}p_{0} )(v- p_{0}) \\ & = E(\frac{1}{2\lambda}(v-p_{0}))(v-p_{0}) \\ & = \frac{1}{2\lambda}\Sigma_{0} \end{aligned} \end{aligned}\]
For two jointly normal random variables \((a,b)\), \(E[a|b] = Ea + \frac{cov(a,b)}{\sigma^{2}_{b}}(b - Eb)\). Applying this formula to \(y\) and \(v\) gives
\[E[v|y] = p_{0} + \frac{\frac{1}{2\lambda}\Sigma_{0}}{\frac{1}{4\lambda^2}\Sigma_{0} + \sigma^{2}_{u}}(y + \frac{\mu}{2\lambda} - \frac{1}{2\lambda}p_{0}).\]
Now, recall our original assumption that \(P(y) = \mu + \lambda y\). Since \(P(y) = E[v|y]\), we can compare coefficients and we notice that the coefficient on \(y\) implies that
\[\lambda = \frac{\frac{1}{2\lambda}\Sigma_{0}}{\frac{1}{4\lambda^2}\Sigma_{0} + \sigma^{2}_{u}}\]
To solve, cross-multiply by the denominator:
\[\lambda\left(\frac{1}{4\lambda^2}\Sigma_0 + \sigma^2_u\right) = \frac{1}{2\lambda}\Sigma_0 \quad\Longrightarrow\quad \frac{1}{4\lambda}\Sigma_0 + \lambda\sigma^2_u = \frac{1}{2\lambda}\Sigma_0.\]
Move the \(\Sigma_0\) terms to one side, \(\lambda\sigma^2_u = \frac{1}{2\lambda}\Sigma_0 - \frac{1}{4\lambda}\Sigma_0 = \frac{1}{4\lambda}\Sigma_0\), and multiply through by \(\lambda\):
\[\lambda^2\sigma^2_u = \frac{\Sigma_0}{4} \quad\Longrightarrow\quad \lambda^2 = \frac{\Sigma_0}{4\sigma^2_u}.\]
Taking the positive root (depth must be positive) produces the result that
\[\lambda = \frac{\sqrt{\Sigma_{0}}}{2\sigma_{u}}\]
Furthermore, it can be seen by inspection that \(\mu = p_{0}\) is a solution to that equation.
16.4 The Grossman-Stiglitz model of information and asset prices
In the Grossman-Stiglitz model, participants are similar to the Kyle model, except that they are risk averse (and have CARA utility) and markets are competitive. Consider a fraction a large number of traders. Fraction \(\lambda\) of them are given a signal that takes the form \(s = v + \epsilon\) where \(v\) is the true future value of the asset and \(\epsilon \sim N(0, \sigma^{2}_{\epsilon})\) is the noise term. The future value of the asset \(v \sim N(\mu_{0}, \sigma^{2}_{v})\). The risk-free asset has a net return of 0 and a normalized price of 1. Noise traders take the form of a noisy supply \(Z \sim N(0,\sigma^{2}_{z})\). We assume that \(Z, v\) and \(\epsilon\) are all independent of each other.
First, we consider the problem of the informed agents. As was shown in class, when traders have utility of the form \(U(w) = -e^{-\gamma w}\), and \(w\) is distributed normally, we can simplify their utility problem to become
\[\max_{\theta} \theta(E[v|I]-P) - \frac{\gamma}{2}\sigma^{2}(v|I)\]
where \(I\) represents the information that a trader has. This implies a demand function of
\[\theta(I) = \frac{E[v|I] - P}{\gamma \sigma^{2}(v|I)}\]
Using the demand functions for both informed and uninformed traders, equilibrium requires that
\[\lambda \frac{E[v|s] - P}{\gamma \sigma^{2}(v|s)} + (1-\lambda) \frac{E[v|P] - P}{\gamma \sigma^{2}(v|P)} = Z\]
To isolate \(P\), write the two demand weights as \(A_s = \frac{\lambda}{\gamma\sigma^2(v|s)}\) and \(A_p = \frac{1-\lambda}{\gamma\sigma^2(v|P)}\). The market-clearing condition is \(A_s(E[v|s] - P) + A_p(E[v|P] - P) = Z\). Distribute and collect the terms in \(P\):
\[A_s E[v|s] + A_p E[v|P] - (A_s + A_p)P = Z.\]
Moving the \(P\) term to one side and dividing by \(A_s + A_p\) gives
\[P = \frac{\lambda \frac{E[v|s]}{\gamma \sigma^{2}(v|s)} + (1-\lambda)\frac{E[v|P]}{\gamma \sigma^{2}(v|P)}}{\frac{\lambda}{\gamma\sigma^{2}(v|s)} + \frac{1-\lambda}{\gamma\sigma^{2}(v|P)}} - \frac{1}{ \frac{\lambda}{\gamma\sigma^{2}(v|s)} + \frac{1-\lambda}{\gamma\sigma^{2}(v|P)} } Z,\]
which is exactly the conjectured form \(P = a + bs - dZ\) once the conditional expectations below are substituted.
By the properties of updating for normal distributions, \(E[v|s] = \mu + \frac{\sigma^{2}_{v}}{\sigma^{2}_{v} + \sigma^{2}_{\epsilon}}s\) and given our assumption about the price function \(E[v|P] = \mu +\frac{b\sigma^{2}_{v}}{b^{2}(\sigma^{2}_{\epsilon}) + d^{2}\sigma^{2}_{z}}\left (P - a - b\mu \right)\).
Thus, the previous equation becomes \(KP = \lambda (\mu + Ms) + (1-\lambda)(\mu + N(P - a - b\mu))) - Z\) where
\[K = \frac{\lambda}{\gamma\sigma^{2}(v|s)} + \frac{1-\lambda}{\gamma\sigma^{2}(v|P)},\]
\[M = \frac{\sigma^{2}_{v}}{\sigma^{2}_{v} + \sigma^{2}_{\epsilon}}\]
and
\[N = \frac{b\sigma^{2}_{v}}{b^{2}(\sigma^{2}_{\epsilon}) + d^{2}\sigma^{2}_{z}}.\]
We notice that this has a linear form and hence can be solved for \(a\), \(b\) and \(d\). Concretely, substituting \(E[v|s] = \mu + Ms\) and \(E[v|P] = \mu + N(P - a - b\mu)\) into \(KP = \lambda(\mu + Ms) + (1-\lambda)(\mu + N(P-a-b\mu)) - Z\) and equating the coefficients of \(1\), \(s\), and \(Z\) on both sides of \(P = a + bs - dZ\) yields three equations in \(a\), \(b\), \(d\). Because \(N\) itself depends on \(b\) and \(d\) (see its definition above), this is a fixed-point system rather than a simple linear solve.
The thing to get out of this solution is that the price depends on the signal \(s\) but, because it is contaminated by the noisy supply \(Z\), does not allow someone who is only observing the price to completely infer the signal. This partial revelation is exactly what preserves a positive return to becoming informed, resolving the Grossman-Stiglitz paradox.
16.5 Application: A trading desk pricing the spread
Picture the trader at a market-making firm responsible for quoting bid and ask prices in the minutes before a Federal Reserve announcement. Every quote exposes the desk to the possibility that whoever trades against it knows something it does not, and the models in this chapter tell the trader exactly how to respond. The Glosten–Milgrom framework shows that the bid-ask spread is the desk’s defense against informed traders: the greater the chance that an incoming order carries private information, the wider the quotes must be to avoid being systematically picked off. As the announcement nears and the risk of informed trading rises, the desk widens its spread; once the news is public and uncertainty resolves, it tightens again. The spread that a retail investor experiences as a small, almost invisible cost is, on the other side of the trade, the precise output of this adverse-selection calculation.
Planet Money (NPR) — “Why Robinhood froze the buying of some stocks”: how payment for order flow, market makers like Citadel Securities, and the clearinghouse sit behind every “free” trade — the microstructure this chapter models.
16.6 Homework problems
16.6.1 Conceptual
MM-C1. Two stocks trade on the same exchange with identical fundamental values, but for stock X the market maker believes fraction \(\mu = 0.10\) of arriving traders are informed, while for stock Y she believes \(\mu = 0.40\). Using the logic of the Glosten-Milgrom model, explain which stock will have the wider bid-ask spread and why. In your answer, describe the specific mechanism by which the presence of informed traders forces a competitive, zero-profit market maker to quote \(A = E[V|\mbox{buy order}]\) above \(E[V]\) and \(B = E[V|\mbox{sell order}]\) below \(E[V]\).
MM-C2. A commentator argues that bid-ask spreads exist purely because market makers have monopoly power and use it to gouge traders. Drawing on the Glosten-Milgrom model, explain why a positive spread can arise even among perfectly competitive market makers who earn zero expected profit. What would the spread be in this model if every arriving trader were uninformed (\(\mu = 0\)), and what does that imply about the commentator’s claim?
MM-C3. Explain why, in the Glosten-Milgrom model, a competitive market maker sets the ask equal to \(A = E[V|\mbox{buy order}]\) rather than the unconditional expectation \(E[V]\). Walk through the role Bayes’ rule plays in computing \(P(\mbox{In}|\mbox{buy}) = \frac{\pi\mu}{\pi\mu + \frac{1}{2}(1-\mu)}\), and explain intuitively why observing a buy order is itself “bad news” for the market maker that pushes the quote above \(E[V]\).
MM-C4. Using the spread formula \(A - B = \frac{4\pi(1-\pi)\mu(V_1 - V_0)}{1 - (2\pi-1)^2\mu^2}\), explain how the spread responds to (i) the gap \(V_1 - V_0\) between the two possible asset values and (ii) the fraction \(\mu\) of informed traders. Which of the three spread-narrowing forces listed in the chapter — competition, less informed trading, lower volatility — does each of these two channels correspond to?
MM-C5. In the Kyle model the insider does not simply buy as many shares as possible when she learns that \(v\) is high. Explain why the insider optimally restrains her demand, and describe the role that the noise traders’ demand \(u \sim N(0,\sigma^2_u)\) plays in making it profitable for her to trade at all. What would happen to the informativeness of order flow if the noise-trader variance \(\sigma^2_u\) shrank toward zero?
MM-C6. The Kyle lambda \(\lambda = \frac{\sqrt{\Sigma_0}}{2\sigma_u}\) is described as a measure of “market depth” or price impact. Consider two assets: asset A has high fundamental uncertainty \(\Sigma_0\) and thin noise trading (low \(\sigma_u\)), while asset B has low \(\Sigma_0\) and heavy noise trading (high \(\sigma_u\)). Explain which asset has the deeper market, what “deeper” means for a trader submitting a large order, and why market makers rationally choose a steeper pricing rule for asset A.
MM-C7. The insider’s first-order condition in the Kyle model is \(v - \mu - 2\lambda x = 0\), giving the optimal order \(x = \frac{v - \mu}{2\lambda}\). Explain economically why the optimal order is proportional to the “surprise” \(v - \mu\) in the insider’s information and inversely proportional to \(2\lambda\). Why does a deeper market (smaller \(\lambda\)) induce the insider to trade more aggressively on the same signal, and how does the market maker’s linear pricing rule remain self-consistent in equilibrium?
MM-C8. The Grossman-Stiglitz model is often summarized by the claim that “prices cannot be perfectly informative if information is costly.” Explain the paradox this addresses: if the equilibrium price \(P = a + bs - dZ\) perfectly revealed the signal \(s\), why would no trader pay to become informed, and why does that in turn make it impossible for the price to be perfectly revealing? Describe the role of the noisy supply \(Z \sim N(0,\sigma^2_z)\) in resolving the paradox.
MM-C9. Suppose a regulator succeeds in eliminating all noise trading from a market described by the Grossman-Stiglitz model (so \(\sigma^2_z \to 0\)). Explain what happens to the ability of the price to reveal the informed traders’ signal, and use this to explain why the fraction \(\lambda\) of traders who choose to acquire the costly signal would collapse. Why does a healthy market, somewhat paradoxically, require some traders who trade for reasons unrelated to fundamental value?
MM-C10. Both Glosten-Milgrom and Kyle produce a wedge between the price a trader pays and the market maker’s unconditional expectation of value, yet the models attribute that wedge to different features of the trading environment. Compare the two: identify what plays the role of the “adverse-selection cost” in each, contrast the binary-value, single-share Glosten-Milgrom setting with the continuous Gaussian, strategic-quantity Kyle setting, and explain why the market maker earns zero expected profit in both despite systematically losing to informed traders.
MM-C11. The Grossman-Stiglitz demand function \(\theta(I) = \frac{E[v|I] - P}{\gamma \sigma^2(v|I)}\) shows how a CARA investor’s position depends on her information \(I\). Explain why informed traders (conditioning on the signal \(s\)) and uninformed traders (conditioning only on the price \(P\)) generally hold different positions, and how the term \(\sigma^2(v|I)\) in the denominator encodes the value of information. Why does a smaller posterior variance make an informed trader willing to take a larger position?
MM-C12. Textbooks distinguish “information-based” theories of the bid-ask spread (like Glosten-Milgrom) from “inventory-based” theories. Explain the core mechanism of each: in an information model, what generates the spread, and in an inventory model, why would a dealer who accumulates a large long or short position adjust her quotes? Give one empirical prediction that could help distinguish which mechanism dominates for a given asset.
16.6.2 Quantitative
MM-Q1. A stock will be worth either \(V_0 = 90\) or \(V_1 = 110\) at the end of the day. The market maker’s prior probability that the value is high is \(\pi = 0.5\), and fraction \(\mu = 0.3\) of arriving liquidity demanders are informed. Using the Glosten-Milgrom results, compute the ask \(A\), the bid \(B\), and the spread \(A - B\). (Since \(\pi = 0.5\) you may use the simplified spread formula \(A - B = \mu(V_1 - V_0)\) and verify it against the full ask and bid expressions.)
MM-Q2. A stock will be worth \(V_0 = 20\) or \(V_1 = 80\). The market maker’s prior on the high value is \(\pi = 0.7\), and fraction \(\mu = 0.5\) of arriving traders are informed. Compute \(E[V]\) and the ask \(A = \frac{\pi(1+\mu)V_1 + (1-\pi)(1-\mu)V_0}{1 + (2\pi-1)\mu}\). Verify that \(A > E[V]\) and explain in one sentence why the ask lies above the unconditional expectation.
MM-Q3. A stock will be worth \(V_0 = 40\) or \(V_1 = 60\). The market maker’s prior on the high value is \(\pi = 0.6\), and fraction \(\mu = 0.25\) of traders are informed. Compute the ask \(A = \frac{\pi(1+\mu)V_1 + (1-\pi)(1-\mu)V_0}{1 + (2\pi-1)\mu}\), the bid \(B = \frac{\pi(1-\mu)V_1 + (1-\pi)(1+\mu)V_0}{1 + (1-2\pi)\mu}\), and the spread \(A - B\). Then confirm your spread against the closed form \(A - B = \frac{4\pi(1-\pi)\mu(V_1 - V_0)}{1 - (2\pi-1)^2\mu^2}\).
MM-Q4. A stock will be worth \(V_0 = 100\) or \(V_1 = 140\), with \(\pi = 0.5\) and informed fraction \(\mu = 0.4\). Compute the spread using the simplified formula \(A - B = \mu(V_1 - V_0)\), then find \(A\) and \(B\) individually (recall that when \(\pi = 0.5\) the spread is symmetric about \(E[V]\)). Finally, state what the spread would be if the informed fraction rose to \(\mu = 0.8\), holding everything else fixed.
MM-Q5. In a Kyle-model market the fundamental value has variance \(\Sigma_0 = 16\) and the noise-trader demand has variance \(\sigma^2_u = 4\) (so \(\sigma_u = 2\)). Compute the equilibrium price-impact coefficient \(\lambda = \frac{\sqrt{\Sigma_0}}{2\sigma_u}\). Then, if total order flow observed by the market maker is \(y = 3\) and the prior mean is \(p_0 = 100\) (so \(\mu = p_0\)), compute the price \(P(y) = \mu + \lambda y\) that the market maker sets.
MM-Q6. In a Kyle market the fundamental variance is \(\Sigma_0 = 36\) and the noise-trader variance is \(\sigma^2_u = 9\) (so \(\sigma_u = 3\)). (a) Compute \(\lambda\). (b) Now suppose noise trading intensifies so that \(\sigma^2_u = 36\) (so \(\sigma_u = 6\)), with \(\Sigma_0\) unchanged; recompute \(\lambda\). (c) State whether the market became deeper or shallower and explain, in one sentence, what this means for the price impact of a large order.
MM-Q7. Consider a Kyle market with prior mean \(p_0 = 50\), fundamental variance \(\Sigma_0 = 25\), and noise variance \(\sigma^2_u = 100\). (a) Compute \(\lambda\). (b) An insider observes \(v = 60\). Using the insider’s optimal demand \(x = \frac{v - \mu}{2\lambda}\) with \(\mu = p_0\), compute her demand \(x\). (c) Now suppose the noise variance quadruples to \(\sigma^2_u = 400\) while \(\Sigma_0\) is unchanged; recompute \(\lambda\) and, for the same signal \(v = 60\), recompute \(x\), commenting on whether the insider trades more or less aggressively.
MM-Q8. A Kyle market has prior mean \(p_0 = 30\), fundamental variance \(\Sigma_0 = 64\), and noise variance \(\sigma^2_u = 16\) (so \(\sigma_u = 4\)). (a) Compute \(\lambda\) and set \(\mu = p_0\). (b) An insider observes \(v = 45\) and submits her optimal order \(x = \frac{v - \mu}{2\lambda}\); compute \(x\). (c) If the realized noise-trader demand is \(u = -2.5\), compute the total order flow \(y = x + u\) and the price \(P(y) = \mu + \lambda y\) the market maker sets, and state whether the insider bought at a price below the true value \(v = 45\).
MM-Q9. In a Grossman-Stiglitz setting the fundamental has variance \(\sigma^2_v = 9\) and the informed signal \(s = v + \epsilon\) has noise variance \(\sigma^2_\epsilon = 3\). Using \(E[v|s] = \mu + M s\) with \(M = \frac{\sigma^2_v}{\sigma^2_v + \sigma^2_\epsilon}\), compute the updating weight \(M\). If the prior mean is \(\mu = 20\) and an informed trader observes \(s = 26\), compute her conditional expectation \(E[v|s]\).
MM-Q10. In a Grossman-Stiglitz market the fundamental variance is \(\sigma^2_v = 16\) and the signal noise variance is \(\sigma^2_\epsilon = 16\). (a) Compute the updating weight \(M = \frac{\sigma^2_v}{\sigma^2_v + \sigma^2_\epsilon}\). (b) With prior mean \(\mu = 50\) and observed signal \(s = 70\), compute \(E[v|s] = \mu + M s\). (c) State, in one sentence, what happens to \(M\) as the signal becomes very noisy (\(\sigma^2_\epsilon \to \infty\)) and why that makes intuitive sense.
MM-Q11. An informed trader in the Grossman-Stiglitz model has CARA risk aversion \(\gamma = 2\). Conditional on her signal she believes \(E[v|s] = 25\) with conditional variance \(\sigma^2(v|s) = 5\), and the market price is \(P = 22\). Using the CARA demand \(\theta(I) = \frac{E[v|I] - P}{\gamma \sigma^2(v|I)}\), compute her optimal share demand \(\theta\). Then state, in one sentence, how her demand would change if her posterior uncertainty \(\sigma^2(v|s)\) doubled.
MM-Q12. Two Grossman-Stiglitz traders face the same price \(P = 24\) and the same risk aversion \(\gamma = 4\), and both hold the posterior belief \(E[v|I] = 30\) with conditional variance \(\sigma^2(v|I) = 2\). (a) Using \(\theta(I) = \frac{E[v|I] - P}{\gamma \sigma^2(v|I)}\), compute each trader’s demand \(\theta\). (b) A less risk-averse trader with \(\gamma = 1\) shares the same beliefs; compute her demand. (c) State, in one sentence, the general relationship between risk aversion and the size of the position an investor is willing to take on a given expected gain.