Information Percolation in Segmented Markets Darrell Duffie, - PowerPoint PPT Presentation

Information Percolation in Segmented Markets Darrell Duffie, Gustavo Manso, Semyon Malamud Stanford University, U.C. Berkeley, EPFL Probability, Control, and Finance In Honor of Ioannis Karatzas Columbia University, June, 2012 Duffie-Manso-Malamud Information Percolation 1

Figure: An over-the-counter market. Duffie-Manso-Malamud Information Percolation 2

Cusip: 592646-AX-1 5% 4% Markup 3% 2% 1% 0% -1% 1 2 3 4 5 6 7 8 9 10 Day Figure: Transaction price dispersion in muni market. Source: Green, Hollifield, and Sch¨ urhoff (2007). See, also, Goldstein and Hotchkiss (2007). Duffie-Manso-Malamud Information Percolation 3

Figure: Daily trade in the federal funds Market. Source: Bech and Atalay (2012). Duffie-Manso-Malamud Information Percolation 4

Information Transmission in Markets Informational Role of Prices: Hayek (1945), Grossman (1976), Grossman and Stiglitz (1981). ◮ Centralized exchanges: • Wilson (1977), Townsend (1978), Milgrom (1981), Vives (1993), Pesendorfer and Swinkels (1997), and Reny and Perry (2006). ◮ Over-the-counter markets: • Wolinsky (1990), Blouin and Serrano (2002), Golosov, Lorenzoni, and Tsyvinski (2009). • Duffie and Manso (2007), Duffie, Giroux, and Manso (2008), Duffie, Malamud, and Manso (2010). Duffie-Manso-Malamud Information Percolation 5

Figure: Many OTC markets are dealer-intermediated. Duffie-Manso-Malamud Information Percolation 6

Model Primitives ◮ Agents: a non-atomic measure space ( G, G , γ ) . ◮ Uncertainty: a probability space (Ω , F , P ) . ◮ An asset has a random payoff X with outcomes H and L . ◮ Agent i is initially endowed with a finite set S i = { s 1 , . . . , s n } of { 0 , 1 } -signals. ◮ Agents have disjoint sets of signals. ◮ The measurable subsets of Ω × G are enriched from the product σ -algebra enough to allow signals to be essentially pairwise X -conditionally independent, and to allow Fubini, and thus the exact law of large numbers (ELLN). (Sun, JET, 2006). Duffie-Manso-Malamud Information Percolation 7

Information Types After observing signals S = { s 1 , . . . , s n } , the logarithm of the likelihood ratio between states X = H and X = L is by Bayes’ rule: n log P ( X = H | s 1 , . . . , s n ) P ( X = L | s 1 , . . . , s n ) = log P ( X = H ) log p i ( s i | H ) � P ( X = L ) + p i ( s i | L ) , i =1 where p i ( s | k ) = P ( s i = s | X = k ) . We say that the “type” θ associated with this set of signals is n log p i ( s i | H ) � θ = p i ( s i | L ) . i =1 Duffie-Manso-Malamud Information Percolation 8

ELLN for Cross-Sectional Type Density ◮ The ELLN implies that, on the event { X = H } , the fraction of agents whose initial type is no larger than some given number y is almost surely � � F H ( y ) = 1 { θ α ≤ y } dγ ( α ) = P ( θ α ≤ y | X = H ) dγ ( α ) , G G where θ α is the initial type of agent α . ◮ On the event { X = L } , the cross-sectional distribution function F L of types is likewise defined and characterized. ◮ We suppose that F H and F L have densities, denoted g H ( · , 0) and g L ( · , 0) respectively. ◮ We write g ( x, 0) for the random variable whose outcome is g H ( x, 0) on the event { X = H } and g L ( x, 0) on the event { X = L } . Duffie-Manso-Malamud Information Percolation 9

Information is Additive in Type Proposition Let S = { s 1 , . . . , s n } and R = { r 1 , . . . , r m } be disjoint sets of signals, with associated types θ and φ . If two agents with types θ and φ reveal their types to each other, then both agents achieve the posterior type θ + φ . This follows from Bayes’ rule, by which log P ( X = H | S, R, θ + φ ) log P ( H = H ) = P ( X = L ) + θ + φ, P ( X = L | S, R, θ + φ ) log P ( X = H | θ + φ ) = P ( X = L | θ + φ ) Duffie-Manso-Malamud Information Percolation 10

Dynamics of Cross-Sectional Density of Types Each period, each agent is matched, with probability λ , to a randomly chosen agent (uniformly distributed). They share their posteriors on X . Duffie and Sun (AAP 2007, JET 2012): With essential-pairwise-independent random matching of agents, � + ∞ g ( x, t + 1) = (1 − λ ) g ( x, t ) + λg ( y, t ) g ( x − y, t ) dy, x ∈ R , a . s . −∞ which can be written more compactly as g ( t + 1) = (1 − λ ) g ( t ) + λg ( t ) ∗ g ( t ) , where ∗ denotes convolution. Duffie-Manso-Malamud Information Percolation 11

Solution of Cross-Sectional Distribution Types ◮ The Fourier transform of g ( · , t ) is � + ∞ 1 e − izx g ( x, t ) dx. ˆ g ( z, t ) = √ 2 π −∞ ◮ From (11), for each z in R , d g 2 ( z, t ) , dt ˆ g ( z, t ) = − λ ˆ g ( z, t ) + λ ˆ (1) ◮ Thus, the differential equation for the transform is solved by ˆ g ( z, 0) ˆ g ( z, t ) = g ( z, 0) . (2) e λt (1 − ˆ g ( z, 0)) + ˆ Duffie-Manso-Malamud Information Percolation 12

Solution of Cross-Sectional Distribution Types Proposition The unique solution of the dynamic equation (11) for the cross-sectional type density is the Wild sum � e − λt (1 − e − λt ) n − 1 g ∗ n ( θ, 0) , g ( θ, t ) = (3) n ≥ 1 where g ∗ n ( · , 0) is the n -fold convolution of g ( · , 0) with itself. The solution (3) is justified by noting that the Fourier transform ˆ g ( z, t ) can be expanded from (2) as � e − λt (1 − e − λt ) n − 1 ˆ g ( z, 0) n , g ( z, t ) = ˆ n ≥ 1 which is the transform of the proposed solution for g ( · , t ) . Duffie-Manso-Malamud Information Percolation 13

Numerical Example ◮ Let λ = 1 and P ( X = H ) = 1 / 2 . ◮ Agent α initially observes s α , with P ( s α = 1 | X = H ) + P ( s α = 1 | X = L ) = 1 . ◮ P ( s α = 1 | X = H ) has a cross-sectional distribution over investors that is uniform over the interval [1 / 2 , 1] . ◮ On the event { X = H } of a high outcome, this initial allocation of signals induces an initial cross-sectional density of f ( p ) = 2 p for the likelihood P ( X = H | s α ) of a high state. Duffie-Manso-Malamud Information Percolation 14

On the event { X = H } , the evolution of the cross-sectional population density of posterior probabilities of the event { X = H } . 4 t = 4 . 0 t = 3 . 0 t = 2 . 0 t = 1 . 0 3.5 t = 0 . 0 3 Population density 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Conditional probability of high outcome. Duffie, Stanford Information Percolation in Large Markets 9

Multi-Agent Meetings The Boltzmann equation for the cross-sectional distribution µ t of types is d dtµ t = − λ µ t + λ µ ∗ m . t We obtain the ODE, d µ m dt ˆ µ t = − λ ˆ µ t + λ ˆ t , whose solution satisfies µ m − 1 ˆ µ m − 1 0 ˆ = . (4) t µ m − 1 µ m − 1 e ( m − 1) λt (1 − ˆ ) + ˆ 0 0 Duffie-Manso-Malamud Information Percolation 16

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 17

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 18

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 19

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 20

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 21

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 22

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Duffie-Manso-Malamud Information Percolation 23

Other Extensions ◮ Privately gathered information. ◮ Public information releases (such as tweets or transaction announcements). • Duffie, Malamud, and Manso (2010). ◮ Endogenous search intensity • Duffie, Malamud, and Manso (2009). Duffie-Manso-Malamud Information Percolation 24

A Segmented OTC Market ◮ Agents of class i ∈ { 1 , . . . , M } have matching probability λ i . ◮ Upon meeting, the probability that a class- j agent is selected as a counterparty is κ ij . ◮ At some time T , the economy ends, X is revealed, and the utility realized by an agent of class i for each additional unit of the asset is U i = v i 1 { X = L } + v H 1 { X = H } , for strictly positive v H and v i < v H . Duffie-Manso-Malamud Information Percolation 25

Trade by Seller’s Price Double Auction ◮ If v i = v j , there is no trade (Milgrom and Stokey, 1982; Serrano-Padial, 2008). ◮ Upon a meeting with gains from trade, say v i < v j , the counterparties participate in a seller’s price double auction. ◮ That is, if the buyer’s bid β exceeds the seller’s ask σ , trade occurs at the price σ . ◮ The class of one’s counterparty is common knowledge. Duffie-Manso-Malamud Information Percolation 26