Asymmetric information in financial markets IMQF, Spring 2013

Homework problems

Asymmetric information in financial markets

1. Consider the Grossman and Stiglitz (1980) model where X ∼ N (µx, 1), Z ∼ N (µz, σ2z)

and agents can acquire a signal Y = X + ε with ε ∼ N (0, σ2ε ). Let the risk-aversion ofthe agents be τ = 1. Also, let λ denote the fraction of agents that is informed.

(a) What would be the equilibrium price? Conjecture price is of the form

Px = a+ bY − dZ

and solve for a, b, d (please provide values for all 3 coefficients).

(b) Determine the fraction of agents that would become informed if the cost of acquiringa signal was c.

2. Consider the Hellwig (1980) model where X ∼ N (µx, 1), Z ∼ N (µz, σ2z) and agents can

acquire i.i.d. signals of the form Yi = X + εi with εi ∼ N (0, σ2ε ). Let the risk-aversion ofthe agents be τ = 1. Also, let λ denote the fraction of agents that is informed.

(a) Solve for the equilibrium price as a function of µx, µz, λ and σ2ε . Conjecture priceis of the form

Px = a+ bX − dZ

(b) Define the ex-ante risk-premium on this stock to be η ≡ µx − E [Px]. How does ηdepend on µz, σ

2z , λ and σ2ε ?

(c) Determine the fraction of agents that would become informed if the cost of acquiringa signal was c. At an interior solution, that is for values of c such that λ ∈ (0, 1),how does the equilibrium risk-premium on the stock depend on σ2z? What is theintuition?

(d) Consider the following argument: “As the variance of the error term σ2ε goes downits quality increases. Therefore, if λ were determined endogeneously everyone wouldbecome infomed (i.e. λ would tend to 1) as σε tends to zero.” Do you agree? Whyor why not?

3. Consider the information seller in Admati and Pfleiderer (1986). Assume the risky assetsatisfies X ∼ N (0, 1), and aggregate supply is Z ∼ N (0, 1). Furthermore, assume thatthe information seller has perfect information, i.e. he gets to observe X. Let the risk-aversion of the agents be τ = 1.

(a) Consider selling newsletters with signals of the form Y = X + η, with η ∼ N (0, s)is the amount of noise that the seller adds to his information. Determine how manypeople should the information seller distribute his newsletter to, λ∗, and how muchnoise should he add s∗.

(b) Now assume that the information seller sells personalized newsletters with informa-tion Y = X + ηi, with ηi ∼ N (0, r) i.i.d. Fix the fraction of agents λ∗ from part(a), and solve for the optimal amount of noise that the monopolist seller show addto his signal r∗. Is the seller better or worse off adding personalized noise insteadof photocopied noise?

(c) If the monopolist seller could choose both λ and r, what should he set, i.e. howmany copies of the newsletter and how much noise should he add?

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Asymmetric information in financial markets IMQF, Spring 2013

4. Consider the Kyle (1985) model with m risk-neutral traders, each of whom observesa signal Yi = X + δ + εi, with X ∼ N (15, 1) denoting the payoff of the risky asset,δ ∼ N (0, σ2δ ) and εi ∼ N (0, σ2ε ). Let liquidity traders’ demand be Z ∼ N (0, 3). The setof random variables (X, δ, {εi}mi=1, Z) are independent.

(a) Solve for the equilibrium price. Hint: conjecture the equilibrium price is given byPx = α+ λω, where ω denotes the order flow.

(b) How does market depth, λ−1, depend on m and σ2ε ?

(c) How informative is the equilibrium as a function of m and σ2ε ? How aggresively dothe informed agents trade as a function of m and σ2ε ?

5. Consider the following variation of the one-period Kyle (1985) model. The terminal valueof the single risky asset is denoted by X, which is assumed to be normally distributedwith mean 0 and variance normalized to 1. There is a risk neutral insider who hasprivate information about the asset payoff X. Specifically, he observes a private signalYi = X + εi, where εi is normally distributed with mean 0 and variance σ2i . The orderplaced by noise traders is denoted by Z, where Z is normally distributed with mean0 and variance σ2z . Before the stock market opens, a public signal Yp, which both theinsider and the market maker observe, is released. The public signal is Yp = Yi + εp,where εp is normally distributed with mean 0 and variance σ2p. It is assumed that X,Z, εi, and εp are mutually independent. Let θi denote the market order placed by theinsider. There is a risk-neutral market maker who facilitates trading by absorbing theorder flow submitted by both the insider and the noise traders. He observes the publicsignal Yp and the total order flow θi + Z (but not its individual components). He sets aprice, P , conditional on his information set, so that he earns zero expected profits.

(a) What are the insider’s posterior beliefs about the payoff X after observing Yi andYp? What are the market maker’s posterior beliefs after observing Yp? For futurereference, let µi = E[X|Yi, Yp] and µp = E[X|Yp].

(b) Determine the distribution of µi conditional on the public signal Yp (note that Ypand µp are informationally equivalent). For future reference, let σ2i|P = var[µi|µp].

(c) Derive a linear equilibrium (i.e., an equilibrium in which the insider’s strategy andthe market maker’s pricing function are linear). Hint: You can save yourself a lotof work by conjecturing the right demand function θi and price function P .

(d) Calculate the ex-ante expected profit of the insider (before the realizations of Yi andYp are observed). Does the insider benefit from a more precise public signal (i.e.,from a lower σ2p)? Why or why not?

6. Consider the Kyle (1985) model with one single insider trader, who observes a signalY = X + ε, with X ∼ N (10, 1) denoting the payoff of the risky asset, and ε ∼ N (0, 7).Let liquidity traders’ demand be Z ∼ N (0, 13).

(a) We consider now the case of a risk-averse insider who has CARA preferences withrisk-aversion parameter τ . Solve for the equilibrium price for different values of τ(say τ ∈ [0, 20]).

(b) Solve the model when there are m > 1 risk-averse traders, with CARA preferencesand risk-aversion parameter τ , each receiving the same signal Y = X+ ε. How doesthe depth of the market vary with m?

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Asymmetric information in financial markets IMQF, Spring 2013

Hint: you may want to solve this problem numerically after characterizing the equilibriumanalytically.

7. You know the final payoff of an asset X, but you cannot trade on this asset (priorconviction from insider trading. . . ). You are deciding whether to give this information totwo of your friends, Asterix and Tintin, but you are not sure about whether you shouldgive it just to one of them or to both. Imagine that you want to maximize the sum of theex-ante certainty equivalent of wealth for both Asterix and Tintin. Recall the certaintyequivalent in these models is

Ui = − 1

τilog (−E[u(Wi)]) .

Asterix and Tintin will trade in a Kyle (1985) model with a risk-neutral market maker.Liquidity traders’ demand is given by Z ∼ N (0, σ2z). Both Asterix and Tintin haveCARA preferences with risk aversion τ .

(a) Imagine you gave both Asterix and Tintin your signal, i.e. you told them the valueof the asset X. Solve for the equilibria at the trading stage. What is the jointex-ante certainty equivalent for the two traders?

(b) Imagine you gave Asterix your signal, i.e. you told him the value of the asset X.You did not give any information to Tintin. Solve for the equilibria at the tradingstage. What is the ex-ante certainty equivalent for Asterix?

(c) Consider giving your friends signals of the form Yi = X + εi, where E[εi] = 0, butthe correlation structure of the εi’s is arbitrary. Namely, allow yourself to choosevar(εi) ∈ R2×2. Note that this subsumes the previous two cases (check).

Solve for the equilibria at the trading stage for an arbitrary information allocationcharacterized by var(εi) ∈ R2×2. What is the ex-ante certainty equivalent for bothtraders? What type of information allocation should you give to your friends?

Hint: you may want to consider corner solutions (where you give it to one but notthe other), and vary the risk-aversion parameter τ ∈ R+.

Hint: you may want to solve this problem numerically after characterizing the equilibriumanalytically.

8. Consider the Glosten and Milgrom (1985) model. Let X = 20 and X = 10 be equallylikely. Moreover let γ denote the probability of a liquidity trade buy order, and let µdenote the probability of informed trading.

(a) Solve for the equilibrium bid and ask prices. How does the bid-ask spread dependon γ? And on µ?

(b) Let γ = 0.4 and µ = 0.3. What are equilibrium values for the bid-ask spread? If abuy order comes through in the first trade, what quotes should the dealer offer inthe second period? If two buy orders come through in the first two trading rounds,what quotes should the deal post?

9. Consider the Easley and O’Hara (1987) model. Let γ denote the probability that aliquidity trader will put a small order, so that γ/2 is the probability that liquidity tradeis a small buy (same for a small sell), whereas (1−γ)/2 is the probability that a liquiditytrade is a large buy (and a large sell).

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Asymmetric information in financial markets IMQF, Spring 2013

Let the probability of occurrence of an information event be α = 0.8, and let the proba-bility of informed trading conditional on an information event be µ = 0.5. The two stockprices X = 20 and X = 10 are equally likely.

Solve for the equilibrium bid and ask prices (for small and large orders), as a function ofγ. How does the bid-ask spread depend on the parameter γ?

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