Market-making with Search and Information Frictions

Transcription

1 Market-making with Search and Information Frictions Benjamin Lester Philadelphia Fed Venky Venkateswaran NYU Stern Ali Shourideh Carnegie Mellon University Ariel Zetlin-Jones Carnegie Mellon University Sep 2017 Disclaimer: The views expressed here do not necessarily reflect the views of the Federal Reserve Bank of Philadelphia. 1 / 29

2 Liquidity in Financial Markets A common metric: Bid-ask spreads More generally, non-linear pricing or the price impact of trade Two prominent theories to generate/interpret spreads: 1 Asymmetric information: investors know more about asset than dealers See, e.g., Glosten-Milgrom 2 Search frictions: dealers have market power, investors trade infrequently See, e.g., Duffie-Garleanu-Pedersen 2 / 29

3 Asymmetric Information + Search Large literatures studying each friction in isolation Theoretical: study effects of changes to severity of each friction Empirical: attempts to quantify each friction Important questions: are theoretical predictions robust when both frictions present? can we empirically disentangle the two? Hard to answer without a unified framework... 3 / 29

4 This paper Develop a tractable framework of a dynamic asset market with: 1 search frictions and market power 2 asymmetric information where dealers learn over time from market-wide trading activity 4 / 29

5 This paper Develop a tractable framework of a dynamic asset market with: 1 search frictions and market power 2 asymmetric information where dealers learn over time from market-wide trading activity Show that their interaction novel predictions Result: Reducing search frictions can lead to wider bid-ask spread (without asymmetric information) more competition tighter spreads (with asymmetric information) slower learning wider spreads Result: More information can lead to wider bid-ask spread (without market power) less adverse selection tighter spreads (with market power) higher markups wider spreads 4 / 29

6 Literature Asymmetric information without search frictions Large traders, strategic revelation of information: Kyle(1985), Back & Baruch (2012),... Large markets, aggregation: Grossman & Stiglitz(1980), Glosten & Milgrom(1985),... This paper: Large markets, aggregation with search, market power Decentralized trading with asymmetric information Common value (idiosyncratic): Inderst(2005), Camargo & Lester(2014), Lauermann & Wolinsky(2016)... Common value (agg.): Wolinsky(1990), Blouin & Serrano(2001), Duffie & Manso(2007)... This paper: Learning about aggregate states from market-wide activity Market-making under full information Duffie, Garleanu & Pedersen(2005), Lagos & Rocheteau(2009)... This paper: Private information, adverse selection (common values) Market-making with asymmetric information Private value: Gehrig(1990), Spulber(1996) This paper: Common values, Learning 5 / 29

7 the economic environment 6 / 29

8 Agents and Assets Discrete time, infinite horizon A market for a single asset, quality is either l or h A continuum of traders can hold q {0, 1} units of the asset measure Nq e of new traders with q assets enter each period all traders exit market with probability 1 δ in each period traders have private info about asset quality + their own preferences A continuum of dealers can hold unrestricted positions (long or short) don t know asset quality, but learn from trading activity 7 / 29

9 Preferences Given asset quality j {l, h}, trader i who owns an asset receives: flow payoff ω t + ε i,t per period terminal payoff c j upon exit, with c h > c l with ω t F (ω) = market-wide liquidity shock, mean zero, iid over time ε i,t G(ε) = idiosyncratic liquidity shock, mean zero, iid over time For each unit he holds, dealer receives: payoff v j, with v h > v l no liquidity shocks 8 / 29

10 Search, Trading and Information In every period, each trader gets an opportunity to trade w.p. π. Conditional on trading, she meets 1 a single dealer w.p. α m (monopolist) she meets more than 1 dealer w.p. α c 1 α m (competitive) Dealers observe number of competing dealers but not asset quality/trader preferences offer to buy (sell) at bid (ask) price, denoted B k t (A k t ) k {c, m} trader accepts or rejects. if she rejects, no trade occurs in that period. 9 / 29

11 Search, Trading and Information In every period, each trader gets an opportunity to trade w.p. π. Conditional on trading, she meets 1 a single dealer w.p. α m (monopolist) she meets more than 1 dealer w.p. α c 1 α m (competitive) Dealers observe number of competing dealers but not asset quality/trader preferences offer to buy (sell) at bid (ask) price, denoted B k t (A k t ) k {c, m} trader accepts or rejects. if she rejects, no trade occurs in that period. After trades occur in each period, dealers observe total trading volume Confounds quality (common value) and market-wide liquidity (private value) volume is a noisy signal about asset quality 9 / 29

12 Search, Trading and Information In every period, each trader gets an opportunity to trade w.p. π. Conditional on trading, she meets 1 a single dealer w.p. α m (monopolist) she meets more than 1 dealer w.p. α c 1 α m (competitive) Dealers observe number of competing dealers but not asset quality/trader preferences offer to buy (sell) at bid (ask) price, denoted B k t (A k t ) k {c, m} trader accepts or rejects. if she rejects, no trade occurs in that period. After trades occur in each period, dealers observe total trading volume Confounds quality (common value) and market-wide liquidity (private value) volume is a noisy signal about asset quality Dealers are informationally small and all have common beliefs Beliefs summarized by µ t Prob t(j = h) Interpretation via Interdealer Market 9 / 29

13 optimal behavior and equilibrium 10 / 29

14 Traders Optimal Behavior Let W q j,t value of owning q {0, 1} units of quality j {l, h} asset at t R j,t = W 1 j,t W 0 j,t reservation value at t when quality is j {l, h} 11 / 29

15 Traders Optimal Behavior Let W q j,t value of owning q {0, 1} units of quality j {l, h} asset at t R j,t = W 1 j,t W 0 j,t reservation value at t when quality is j {l, h} Given bid and ask prices (B k t, A k t ) and shocks (ε i,t, ω t), Owner should sell if ε i,t sufficiently small, hold otherwise: B k t + W 0 j,t ε i,t + ω t + W 1 j,t Non-owner should buy if if ε i,t sufficiently large, do nothing otherwise: A k t + W 1 j,t + ε i,t + ω t W 0 j,t 11 / 29

16 Traders Optimal Behavior Owner i sells in a k meeting iff ɛ i,t ɛ k j,t B k t R j,t ω t Non-owner i buys in a k meeting iff ɛ i,t ɛ k j,t A k t R j,t ω t Reservation values satisfy where R j,t = (1 δ)c j + δe [R j,t+1 ] + δπe Ω j,t+1 }{{} Net option value Ω j,t+1 = α k max{bk t+1 R j,t+2 ω t+1 ε i,t+1, 0} max{ A k t+1 k=c,m }{{} + R j,t+2 + ω t + ε i,t+1, 0} }{{} option to sell option to buy 12 / 29

17 Aggregate Positions N q j,t = measure of traders holding q {0, 1} units of asset when quality is j {l, h} N 1 j,t+1 = δ N1 t 1 π + π 1 α k G(ε k j,t ) + N 0 t π 1 α k G(ε k j,t ) + Ne 1 k=c,m k=c,m N 0 j,t+1 = δ N1 t π α k G(ε k j,t ) + N0 t 1 π + π α k G(ε k j,t ) + Ne 0. k=c,m k=c,m Dealers observe past volume and (know N e q ) they know N q t when setting (Bk t, A k t ). 13 / 29

18 Monopolist Dealer s Prices Dealer with a captive customer chooses (A m t, Bt m ) to maximize [ N 0 ( t E j,ω 1 G(ε m Nt 0 + Nt 1 j,t ) ) ] (A m Nt 1 t v j ) + G(ε m Nt 0 + Nt 1 j,t )(v j Bt m ) 14 / 29

19 Monopolist Dealer s Prices Dealer with a captive customer chooses (A m t, Bt m ) to maximize [ N 0 ( t E j,ω 1 G(ε m Nt 0 + Nt 1 j,t ) ) ] (A m Nt 1 t v j ) + G(ε m Nt 0 + Nt 1 j,t )(v j Bt m ) Why? Recall that Both traders and dealers are small, so take future beliefs as given Dealers can hold unrestricted positions, have deep pockets Market-wide information dominates learning from an individual meeting Implications 1 No motive for experimentation, no benefit to waiting pricing decision is static 2 Sell (buy) choice unaffected by ask (bid) separates the bid/ask problems 3 Aggregate positions known irrelevant for pricing, only beliefs µ t matter 14 / 29

20 Monopolist Dealer s Prices (given beliefs µ t ) A m t = E j v j + 1 E [ ( )] ( ( ) ) j,ω G ε m [ ( j,t g ε m )] + Cov [ j,t ( )], v E j,ω g ε m j,t E j,ω g ε m j j,t }{{}}{{} market power asymmetric information [ ( )] ( ) ( )] = µ tv h + (1 µ t)v l + 1 E j,ω G ε m j,t E ω [g ε m h,t g ε m l,t ( )] + µ t(1 µ t)(v h v l ) ( )] E j,ω [g ε m j,t E j,ω [g ε m j,t }{{}}{{} market power asymmetric information Bt m = E j v j E [ ( )] j,ω G ε m j,t [ ( )] E j,ω g ε m j,t = µ tv h + (1 µ t)v l E [ ( )] j,ω G ε m j,t E j,ω [ g ( ε m j,t )] µ t(1 µ t)(v h v l ) Eω ( ( ) ) g ε m j,t Cov [ ( )], v E j,ω g ε m j j,t [ ( ) ( )] g ε m l,t g ε m h,t [ ( )] E j,ω g ε m j,t ) 15 / 29

21 Competitive Prices Bertrand competition zero profits (a la Glosten-Milgrom) 0 = A c t E [ ( j,ω vj 1 G(ε c j,t ) )] [( E j,ω 1 G(ε c j,t ) )] = Bc t E [ j,ω vj G(ε c j,t [ )] E j,ω G(ε c j,t ) ] Re-arranging, ( 1 G ( ε c j,t ) A c t = E tv j + Cov [ ( )], v E j,ω 1 G ε c j j,t }{{} asymmetric information ) Bt c = E tv j + ( ( ) ) G ε c j,t Cov [ ( )], v E j,ω G ε c j j,t }{{} asymmetric information 16 / 29

22 Evolution of Beliefs Information: Dealers see volume at end of period (buys and sells) As if they see thresholds (recall they know aggregate positions) ɛ k t = Bk t R t ω t or ɛ k t = A k t R t ω t where R t = R j,t if asset is of quality j As if they see a signal S t = R t + ω t signal extraction problem 17 / 29

23 Evolution of Beliefs Information: Dealers see volume at end of period (buys and sells) As if they see thresholds (recall they know aggregate positions) ɛ k t = Bk t R t ω t or ɛ k t = A k t R t ω t where R t = R j,t if asset is of quality j As if they see a signal S t = R t + ω t signal extraction problem Updating: what would ω t have to be in state k {l, h} to generate S t? ω k,t = S t R k,t µ t+1 = µ tf (ω h,t) µ tf (ω h,t ) + (1 µt)f (ω l,t ) 17 / 29

24 Evolution of Beliefs Information: Dealers see volume at end of period (buys and sells) As if they see thresholds (recall they know aggregate positions) ɛ k t = Bk t R t ω t or ɛ k t = A k t R t ω t where R t = R j,t if asset is of quality j As if they see a signal S t = R t + ω t signal extraction problem Updating: what would ω t have to be in state k {l, h} to generate S t? ω k,t = S t R k,t µ t+1 = µ tf (ω h,t) µ tf (ω h,t ) + (1 µt)f (ω l,t ) Volume depends on beliefs (through R j,t ) µ t+1 is fixed point of µ = µ t µ t + (1 µ t) f (ωt +R j,t+1(µ ) R l,t+1 (µ )) f (ω t +R j,t+1 (µ ) R h,t+1 (µ )) 17 / 29

25 Evolution of Beliefs Information: Dealers see volume at end of period (buys and sells) As if they see thresholds (recall they know aggregate positions) ɛ k t = Bk t R t ω t or ɛ k t = A k t R t ω t where R t = R j,t if asset is of quality j As if they see a signal S t = R t + ω t signal extraction problem Updating: what would ω t have to be in state k {l, h} to generate S t? ω k,t = S t R k,t µ t+1 = µ tf (ω h,t) µ tf (ω h,t ) + (1 µt)f (ω l,t ) Volume depends on beliefs (through R j,t ) µ t+1 is fixed point of µ = µ t µ t + (1 µ t) f (ωt +R j,t+1(µ ) R l,t+1 (µ )) f (ω t +R j,t+1 (µ ) R h,t+1 (µ )) Speed of learning depends on R h,t R l,t Trading more informative when the reservation values are very different 17 / 29

26 Equilibrium A recursive equilibrium is a collection of functions for 1 Reservation values: R j (µ) j {h, l} 2 Thresholds: ε k j (µ, ω), εk j (µ, ω) k {c, m} 3 Prices: A k (µ), B k (µ) 4 Beliefs: µ (µ, ω) such that 1 Taking beliefs (and prices) as given, thresholds are optimal for traders 2 Taking threshold functions as given, prices are optimal for dealers 3 Beliefs evolve according to Bayes rule 4 Reservation values are consistent with future beliefs and prices 18 / 29

27 a tractable case 19 / 29

28 The Uniform-Uniform Model Assumptions v j = c j. j = h, l ε i,t U( e, e) and ω t U( m, m) e, m are sufficiently large so that the thresholds are always interior Together, these assumptions simplify both the pricing and learning formulae Volume signal either uninformative or fully revealing: p(µ) Pr(full revelation) = R h(µ) R l (µ) 2m Closed form expressions for prices and spreads, e.g. under monopoly A m = E j v j + e B m = E j v j e 20 / 29

29 comparative Statics 21 / 29

30 Search Frictions and Spreads How does a higher π affect prices and learning? Key: effect on R h R l. R h R l = (1 δ) (c h c l ) + δ E[R h R l ] + δπ E(Ω h Ω l ) where Ω j = option value of selling option value of buying Proposition R h R l is decreasing in π. Ω h Ω l < 0: Option to sell (buy) is worth less (more) when quality is high Higher π increases the weight of the net option value, bringing R h and R l closer less adverse selection (given µ), but also slower learning spreads tighter initially (since initial beliefs are given), but eventually wider 22 / 29

31 Information and Spreads Introduce a one time signal (at time 0) about j a mean-preserving spread of µ t effect on spreads depends on second derivative of spreads w.r.t. µ Findings A c B c is concave, so the signal reduces average spreads under competition High e interior ε m j,t, εm j,t A m B m constant, so no effect of addl info Low e A m B m convex when adverse selection severe (v h v l suff high) 23 / 29

32 The Normal-Normal Model Assumptions ω t N(0, σω) 2 ε i,t N(0, σε) 2 v j = c j + ξ No analytical solution but easy to solve numerically (see paper for algorithm) Effect of π (true state is j = h) 1 Beliefs 5.4 Spreads Low π HIgh π Time Time Higher π Lower (R h R l ) Less learning Wider spreads eventually 24 / 29

33 The Normal-Normal model: Stationary Version Asset quality j changes over time (with probability ρ) Other elements exactly the same as before Non-trivial belief distribution in the long run (stochastic steady state) 2 Distribution of beliefs 2.59 Average Spreads Low π Med π High π V High π µ π Higher π Lower (R h R l ) Less learning Wider spreads 25 / 29

34 Conclusion A dynamic model with two canonical frictions asymmetric information and infrequent trading opportunities/market power Frictions interact in novel ways mitigating one could lead to wider spreads Next steps disentangling the two frictions? / 29

35 Dealers Indexed by i [0, 1] They come into each period with x i,t units of the asset Payoff: where (1 δ) s t [ d i,t P t + q i,t p t + δv j (x i,t + d i,t + q i,t )] s=t d i,t { 1, 0, 1} P t {A t, B t} x i,t+1 = x i,t + d i,t + q i,t p t: price in the interdealer market; competitive 27 / 29

36 Dealers Conjecture that future bid and ask only a function of aggregate information and independent of individual positions. Radner: REE in the inter-dealer market p t = E t [ v j {d i,t } i [0,1] ]. Dealers are small: E t[v j p t, d i,t ] = E t [ v j {d i,t } i [0,1] ] Act as if they are short-lived dealers and only care about E t[v j ] where expectation is common across all dealers Back 28 / 29

37 Equilibrium A recursive equilibrium is a set of functions: R j (µ) A(µ) and B(µ) s.t. R j = (1 δ) c j + δ E[R j (µ j)] + δ π Ω j (µ) A = B = Ev j g(a R j (µ j) ω) + 1 EG(A R j (µ j) ω) Eg(A R j (µ j ) ω) Ev j g(b R j (µ j) ω) EG(B R j (µ j) ω) Eg(B R j (µ j ) ω) where µ j = µ µ + (1 µ) L j (ω, R h (µ j ) R l(µ j )) Ω j (µ) = E [ max(b(µ) R j (µ j) ω ɛ, 0) max(r j (µ j) + ω + ɛ A(µ), 0) ] 29 / 29

38 Equilibrium Start with a guess for R j (µ) beliefs µ j = µ µ + (1 µ) L j (ω, R h (µ j ) R l(µ j )) 29 / 29

39 Equilibrium Compute optimal prices: A(µ) and B(µ) A = B = Ev j g(a R j (µ j) ω) + 1 EG(A R j (µ j) ω) Eg(A R j (µ j ) ω) Ev j g(b R j (µ j) ω) EG(B R j (µ j) ω) Eg(B R j (µ j ) ω) µ j = µ µ + (1 µ) L j (ω, R h (µ j ) R l(µ j )) 29 / 29

40 Equilibrium Update/verify the guess R j = (1 δ) c j + δ E[R j (µ j)] + δ π Ω j (µ) A = B = Ev j g(a R j (µ j) ω) + 1 EG(A R j (µ j) ω) Eg(A R j (µ j ) ω) Ev j g(b R j (µ j) ω) EG(B R j (µ j) ω) Eg(B R j (µ j ) ω) µ j = µ µ + (1 µ) L j (ω, R h (µ j ) R l(µ j )) Ω j (µ) = E [ max(b(µ) R j (µ j) ω ɛ, 0) max(r j (µ j) + ω + ɛ A(µ), 0) ] 29 / 29