LectureNote: MarketMicrostructure

Transcription

1 LectureNote: MarketMicrostructure Albert S. Kyle University of Maryland Finance Theory Group Summer School Washington University, St. Louis August 17, 2017

2 Overview Importance of adverse selection in financial market trading. Model of Treynor (1971): Market maker, informed trader, liquidity trader. Dealer markets and organized exchanges. Summary of issues in literature: Grossman and Stiglitz(1980) and related papers. One-period model of Kyle(1985). Continuous model of Kyle(1985). Pete Kyle University of Maryland p. 1/32

3 Treynor Model of Adverse Selection Dealer trades against on order which might come from an informed trader or a liquidity(noise) trader. CommonPrior: DealerbelievesvalueisV = V H = 110or V L = 90withequalprobabilities. Order is from informed trader with probability π = 0.10, fromnoisetraderwithprobability 1 π = InformedtradersobserversvalueofV H = 110orV L = 90 perfectly. Buys one contract if high, sells one contract if low. Noise trader buys of sells one contract randomly with equal probabilities. Whatarebidandaskpricesatwhichdealerbreakseven? Pete Kyle University of Maryland p. 2/32

4 Solution to Treynor Model Conditional on buy order arriving, market maker calculates P ASK = π V H +(1 π) E[V] = (0.10) (110)+(0.90) (100) (1) = 101 Conditional on sell order arriving, market maker calculates P BID = π V L +(1 π) E[V] = (0.10) (90)+(0.90) (100) (2) = 99 Bid-ask spread is 2 dollars as a result of adverse selection. Can turn into dynamic model easily. Pete Kyle University of Maryland p. 3/32

5 Dealer Markets and Organized Exchanges Dealer markets: Customers cannot post limit orders. Organized exchanges: respect time and price priority. Dealer markets are less anonymous: Dealers know identity of customer. Customer commits to size of order. Dealer markets have non-competitive arrangements: opaque inside markets for dealers only, implicit collusion over spreads. Literature assumes search is important in dealer markets. PK disagrees. Do customers search for dealers or vice versa? Dealer relationships are expensive for customers. PK thinks equilibrium is tournaments. Pete Kyle University of Maryland p. 4/32

6 History Organized exchanges set up as cartels with monopolistic fixed commissions, supported by exclusive dealing and extensive self-regulation. Regulators banned fixed commissions in 1970s. Dealer market allows more rent extraction than centralized exchange. Opaque prices. Costly dealer relationships. Regulators opened up dealer markets in 1990s. Large tick size and high fees allowed organized exchanges (NYSE) market makers(specialists) to extract rents on organized exchanges. Pete Kyle University of Maryland p. 5/32

7 Markets Today Regulatorscutticksizeto$0.01in2001. Regulation NMS(National Market System) in U.S. and Mi- FiD in EU broke monopoly franchise of organized exchange. Market fragmentation: competing exchanges with low fees. Institutional investors less likely to execute large blocks non-anonymously with dealers. Traders execute their own orders anonymously in fragmented markets with algorithmic trading High frequency traders with fast algorithms have replaced human market makers. Equities, government securities, foreign exchange, commodity futures, corporate bonds changing rapidly. Pete Kyle University of Maryland p. 6/32

8 Some Papers Grossman and Stiglitz(1980) Diamond and Verrecchia(1981) Hellwig(1980) Milgrom and Stokey(1982) Kyle(1989) Glosten and Milgrom(1985) Kyle(1985) Kyle, Obizhaeva, Wang(2017) KyleandLee(2017):Twopapers Glosten(1994) Pete Kyle University of Maryland p. 7/32

9 Technical Themes and Issues in the Papers Exponential utility and normally distributed random variables( CARA-normal assumptions) imply quadratic optimization with linear solutions. Rational expectations equilibrium: Traders learn from prices. Noisy rational expectations: Prices do not fully reveal the private information of informed traders. Need noise trading or overconfidence to generate trading in the presence of adverse selection. Competitive rational expectations: Unrealistic idea that traders think they do not affect the price. Imperfect competition: Consistent with linearity. Monopoly power over both information and price. Pete Kyle University of Maryland p. 8/32

10 Some Economic Issues Relationship to efficient markets hypothesis: Are returns predictable given public information? Information content of prices: How much information do price reveal about fundamental value? Understanding market liquidity: Is liquidity supplied by intermediaries? Or do traders supply liquidity to one another? Incentives to acquire costly private information: What is the profitability of trading on costly private information? Competition and Monopoly Power: How important is strategic trading in financial markets? Pete Kyle University of Maryland p. 9/32

11 Grossman and Stiglitz(1980) Competitive rational expectations equilibrium. Identical informed traders with same noisy signal. Identical uninformed traders(market makers). Exogenous noise traders. Exponential utility and normal random variables imply linear solution. Price is noisy signal of value(mixed with noise). Imperfect learning from prices. Solution is algebraically complicated because of asymmetry between informed traders and uninformed traders. Pete Kyle University of Maryland p. 10/32

12 Diamond and Verrecchia(1982) Exponential utility, normally distributed random variables, competitive rational expectations equilibrium. Symmetrically informed traders, no uninformed traders or market makers. Trade motivated by uncorrelated endowment shocks of same variance. Informed traders have different private information of same precision. Equilibrium is much less algebraically complicated than Grossman and Stiglitz model. Pete Kyle University of Maryland p. 11/32

13 Smart Money and Noise Trading Simplest possible model with simplest notation. One Period Model: Informed investors trade with noise traders: σv 2 =Priorvarianceofliquidationvalue Dollars2 Shares 2 1 A =Agg.riskaversion(smartinvestors) Dollars σu 2 =Varianceofnoisetrading Shares2 τ =Precisonofinformation 1 Dimensionless Dimensional Analysis: Buckingham π Theorem (3) Usedimensionalparameters σ 2 V and Aforscaling. Use dimensionless parameters for solution: τ = Information, θ := Aσ V σ U = Noise (4) Pete Kyle University of Maryland p. 12/32

14 Setup Assume Z V, Z U, Z I arenid(0,1)anddimensionless. V = σ V Z V =LiquidationValue U = σ U Z U =NoiseTrading I = τ 1/2 Z V +(1 τ) 1/2 Z I =Information Timeline: (5) Time 0:Informedtradersobserve I. Time 1:Tradeatprice P. Time 2:LiquidationV realize,returnsv P. Pete Kyle University of Maryland p. 13/32

15 CARA Normal Framework With normally distributed V, solving max x is equivalent to solving E[ exp( A(V P)x] (6) max x E[V P]x 1 2 Avar[V P]x2. (7) Pete Kyle University of Maryland p. 14/32

16 Solution Calculate expectations: E[Z V I] = τ 1/2 I, var[z V I] = 1 τ (8) CARA-normal demands and market clearing imply Demand = E[V P I,P] A var[v I,P] =U =Supply (9) Solvefor P: P σ V = τ 1/2 I A(1 τ)σ 2 V U σ V = τ Z V +τ 1/2 (1 τ) 1/2 Z I (1 τ)θ Z U (10) Pete Kyle University of Maryland p. 15/32

17 Sharpe Ratios Sharpe ratio and expected squared Sharpe ratio for informed traders: SR I = E[V P I] (var[v P I]) 1/2 = ( τ ) 1/2 I, (11) 1 τ E [ (SR I ) 2] = τ 1 τ =Signal (12) Noise Note:Indynamicmodels,theSharperatiohasatimedimension, which is ignored here. Pete Kyle University of Maryland p. 16/32

18 Statistical Identification Mean and variance from perspective of economist : E[V P P] = (1 τ)2 θ 2 τ +(1 τ) 2 θ2 P 0as θ 0, (13) var[v P P] σ V = τ(1 τ)+(1 τ)2 θ 2 τ +(1 τ) 2 θ 2 1 τ as θ 0. (14) var[p] σ V = τ +(1 τ) 2 θ 2 τ as θ 0. (15) var[v P] σ V = 1 τ +(1 τ) 2 θ 2 1 τ as θ 0. (16) Conclusion: τ and θ statistically identified from repeated realizationsof P andv. Noise leads to excess volatility and mean reversion. Pete Kyle University of Maryland p. 17/32

19 Homework: Verify equations in previous pages. Calculate Sharpe ratio and expected squared Sharpe for economist. Pete Kyle University of Maryland p. 18/32

20 Where does research go from here? Market Order Model : Mimics a dealer market by maintaining a distinction between dealers and other traders. Discussed next as Kyle (1985): Model uses restriction on non-dealers from placing price-contingent orders. Does not use search. Limit Order Model : Equilibrium in demand schedules treats all traders symmetrically with single-price auction which protects time and price priority of orders. Important to consider strategic order submission. Single-period model: Kyle(1989), Kyle and Lee(2017). Continuous-time model with smooth trading: Kyle, Obizhaeva, Wang(2017) Pete Kyle University of Maryland p. 19/32

21 What generates trade? What prevents trade from collapsing? Completely inelastic noise trading: Willing to suffer large losses. Kyle(1985), Kyle(1989). Endogenous hedging incentives large enough to overcome adverse selection. Kyle and Lee(2017) Overconfident informed traders who trade too much based on their private signals. Kyle, Obizhaeva, Wang(2017). Insight: Much of intuition from single-period model carries over to dynamic model. Pete Kyle University of Maryland p. 20/32

22 Kyle(1985): Assumptions Monopolistic informed trader: Observes private signal. Takes account of impact on price. Noise trading: Exogenous normal distributions. Market makers: Risk neutral perfect competitors(reduced form). Implies market efficiency : Prices follow martingale. Note:Noriskaversioninmodel! Equilibrium defined by optimal trading strategy for informed trader and pricing rule for market makers. One period, multi-period, and continuous-time models. Market-order model since informed trader conditions on information, not price. Pete Kyle University of Maryland p. 21/32

23 One-Period Model Liquidation value V has common prior distribution V N(P 0,σV 2)(dollarspershare). Informed trader observes liquidation value V. Chooses quantity X(V) to maximize expected profits. NoisetraderstradeexogenousquantityU N(0,σU 2). Marketmakerssetprice P asfunctionof orderflow Y = X +U suchthat P(Y) = E[V X +U =Y]. Look for equilibrium with linear price function P(Y) = P 0 +λ Y. (17) Pete Kyle University of Maryland p. 22/32

24 Solution to One-Period Model Informed traders choose X to maximize Profit = E[(V P 0 λ (X +U)) X V] (18) First-order condition implies Solutionforλis V P 0 2 λ X = 0. (19) X = β (V P 0 ), where β = 1 2 λ. (20) λ = Model solution is cov[β V +U,V] var[β V +U] = β σ 2 V β 2 σ 2 V +σ2 U (21) β = σ U σ V, λ = 1 2 σv σ U. (22) Pete Kyle University of Maryland p. 23/32

25 Solution Properties Market impact costs arise endogenously in equilibrium. Noise traders losses equal informed trader s profits. Market makers offer fixed, break-even supply schedule. Modelsolutioncaneasilybeguesseduptoconstantsusing dimensional consistency. Priceisnoisy: Onlyonehalfofinformedtrader sprivate information is incorporated into prices. Equilibrium is unique within class of models offering linearpriceasfunctionoforderflow. Pete Kyle University of Maryland p. 24/32

26 Kyle(1985): Continuous Model Assumptions Tradingininterval t [0,T]. LiquidationvalueV N(P 0,T σ V ). Noise traders exogenous inventory process follows Brownianmotion du(t) = σ U db(t). Informed trader trades dx(t) to maximize profits, taking into account price impact in present and future. Orderflowis dy(t) = dx(t)+du(t). Marketmakerssetpricesothat P(t) = E[V PastOrderFlow = {Y(t): <s< t}]. (23) Pete Kyle University of Maryland p. 25/32

27 Continuous-time Model Solution Supply schedule is a fixed linear schedule offering instantaneous liquidity : P(t) = P 0 +λ Y(t), λ = σ V σ U. (24) Informed trader moves price linearly toward V: dx(t) dt = β(t) (V P(t))dt, β(t) = 1 T t σu σ V. (25) Errorvarianceofmarketmakersis var[v PastOrderFlow] = (T t) σ 2 V. (26) Pete Kyle University of Maryland p. 26/32

28 Properties of Continuous Solution Derivative 1/(T t)replaces 1/2inone-periodmodel. Informed trader trades like perfectly discriminating monopolist, moving gradually along supply schedule of market makers. Informed trader does not intentionally camouflage his trading, but price is noisy estimate of value because informed trade is hidden in noise trading. ReturnsvolatilityisconstantσV 2(dollarspersharepersquare root of time) Noise traders trade too aggressively. Could reduce transactions costs by one half by smoothing out trading over an arbitrarily short period of time! Pete Kyle University of Maryland p. 27/32

29 Kyle and Lee(2017) = Symmetric Generalization of Kyle 1989 Numberoftraderswithineachgroup M Numberofgroupsoftraders N Fundamentalvolatility σ V Residualuncertainty σ Y Riskaversion A Privateinformationτ I Endowmentshock σ S Exogenousnoisetrading Σ Z Pete Kyle University of Maryland p. 28/32

30 Measuring Information and Competition Equilibrium in demand schedules Traders learn from prices Informational efficiency ϕ τ := Traders trade strategically σ 2 V var{v i l,s l,p} = 1+τ I +(N 1)τ I ϕ. (27) Competition χ χ := x l x PT l = Aσ 2 V 2λ +Aσ 2 V 1 τ +σ 2 Y 1 τ +σ 2 Y. (28) Pete Kyle University of Maryland p. 29/32

31 Equilibrium Existence condition: ϕ< ϕ soc := Informational efficiency: 1 ϕ 1 =(Aσ Vσ S ) 2 ( 1+σ 2 τ Y τ ) 2 I + (Aσ VΣ Z ) 2 τ I M 2 (N 1) MN 2 MN 2+N. (29) ( ) 2 ( MN 1 MN 2 ϕ soc ϕ soc ϕ ) 2 ( 1+σ 2 Y τ ) 2. (30) Competition and Information: ϕ soc ϕ χ = ϕ soc ϕ + 2(1+(N 1)ϕ). (31) (MN 2+N) Pete Kyle University of Maryland p. 30/32

32 Take-aways Information is in the price. Competition is in the quantity. ϕandthepriceareindependentof M when Σ 2 Z = 0. Thereisaninverserelationshipbetweenϕandχ. Perfectcompetitionrequires M orϕ 0 Grossman-Stiglitz paradox goes away because prices are never fully revealing in a perfectly competitive market. Thepuzzleistheremaybenotradeevenwhenthereare ex-ante gains from trade: No-trade theorem. Ifnotrade,pricesarenotfullyrevealingunless M Vanishingnoise(Σ 2 Z 0)limitiswell-defined. Pete Kyle University of Maryland p. 31/32

33 Conclusion Are there better mechanisms than single-price auctions for generating both better risk sharing and more informative prices simultaneously? Can such a better mechanism be found in the standard CARA normal setting? QUESTIONS? Pete Kyle University of Maryland p. 32/32