Monopolistic Dealer versus Broker: Impact of Proprietary Trading with Transaction Fees

Transcription

1 Monopolistic Dealer versus Broker: Impact of Proprietary Trading with Transaction Fees Katsumasa Nishide (a) Yuan Tian (b) (a) Yokohama National University (b) Ryukoku University The latest version of this study is available at ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

2 Plan of Talk 1 Introduction 2 Model Setup 3 Equilibrium Solutions 4 Numerical Analysis 5 Conclusion 6 References ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

3 Introduction Introduction ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

4 Introduction Background Two types of trading system: Dealer market: Dealer (market maker) trades with other market participants with his/her own account (proprietary trading). Brokered market: Broker sets price just to clear orders from other market participants (no proprietary trading). Unclear which system is better for investors from the viewpoint of market activity, market liquidity, welfare of investors, etc. To answer the above question, the eect of proprietary trading needs to be examined. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

5 Introduction Literature review (1) Theoretical papers: Röell (1990), Fishman and Longsta (1992), Sarkar (1995), Viswanathan and Wang (2002), Bernhardt and Taub (2010). Agent setting the price is dierent from the one collecting transaction fees. Price setter is risk-neutral and perfectly competitive, implying that p = E[v F M ]. Empirical papers: Transaction fees are independent of the order amount. Pagano and Röell (1992, 1996), Huang and Stoll (1996), Heidle and Huang (2002). Result depends on the papers. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

6 Introduction Literature review (2) Sarkar (1995): Dual trading of investment banks, securities houses, etc., Market maker (p) (risk-neutral, competitive) Market maker (p) (risk-neutral, competitive) x 1 x 2 x 3 x 4 x 1 x 2 d x 3 x 4 Broker (c) Broker (c) c c c c c c c c Investor 1 (informed) Investor 2 (uninfomred) Investor 3 (uninformed) Investor 4 (informed) Investor 1 (informed) Investor 2 (uninfomred) Investor 3 (uninformed) Investor 4 (informed) In the dual trading, the broker is allowed to trade with his own account d. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

7 Introduction Literature review (3) Our study: Market system (dealer versus broker). Broker (c) (risk averse, monopolistic) Dealer (p, c) (risk averse, monopolistic) x 1 c x 1 x 2 c x 2 x 3 c x 3 x 4 c x 4 x 1 c x 1 x 2 c x 2 x 3 c x 3 x 4 c x 4 Market clearing (p) Investor 1 Investor 2 Investor 3 Investor 4 Investor 1 Investor 2 Investor 3 Investor 4 In the dealer market, the dealer can trade with his own account with the price set by the dealer himself. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

8 Introduction Aim of this study In this paper, we construct a one-shot CARA-Normal model with innitely many investors, a monopolistic and risk-averse dealer/broker who collects transaction fees. examine the eect of proprietary trading on equilibrium solutions. Main results: Proprietary trading always increases total welfare of investors. Economic interpretation: dealer sets a favorable price for investors to seek prots by proprietary trading. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

9 Model Setup Model Setup ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

10 Model Setup Financial Market There is only one risky asset. Risk-free interest rate is assume to be zero for simplicity. v: (random) payo of the risky asset. Two types of market participants: investors, a dealer or a broker. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

11 Model Setup Investors Let I denote the set of investors. {ω i } i I IIDN( ω,σ 2 ω): initial endowment of investors (Kim and Verrecchia, 1991). F i : information set of investor i. v N(µ i,σv 2 ). Fi Beliefs are heterogeneous with respect to the mean of v. Utility of investor i: U i = 1 [ (E a log e ay i where Y i is the nal wealth of investor i. ]) F i ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

12 Model Setup Dealer or broker Monopolistic. Collects transaction fees ($c per unit trade). Sets the price p for investors. F M : information set of dealer/broker. v N(µ M,σM 2 ). FM Utility function: U M = 1 ]) (E [e γ log γr(p,c) FM where R(p, c) is the nal wealth of the dealer (broker). ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

13 Model Setup Investor's utility maximization x i : trading amount of investor i. Y i is given by where sgn is the sign function. Y i = vω i + (v p)x i sgn[x i ]x i c, Since only v is random in Y i with respect to F i, x i can be solved as x i (p, c) = ω i }{{} + µ i p sgn[x i (p, c)]c. aσv 2 risk hedging }{{} prot seeking Let ζ i = µ i aσv 2 ω i. Then, we can rewrite x ζ i (p + c) ζ i (p c) i (p, c) = 1 {ζi >p+c} + 1 aσ {ζi <p c}. v 2 aσv 2 ζ i : investor i's subjective belief adjusted by inventory risk. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

14 Model Setup Broker's utility maximization Assumption 1 The broker sets (p b, c b ) to satisfy x i (p, c) = 0 i I and to maximize U M = 1 ]) (E [e γ log γr(p,c) FM where R(p, c) = c x i (p, c). i I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

15 Model Setup Dealer's utility maximization (2) Assumption 2 The dealer sets (p d, c d ) to maximize U M = 1 ]) (E [e γ log γr(p,c) FM where { } R(p, c) = (v p) ( x i (p, c)) + c x i (p, c). i I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

16 Model Setup Remark The utility of the dealer is higher than the one of the broker: Dealer has an additional control variable (the price p). The eect of proprietary trading by the dealer on investors is not so apparent: Dealer has a monopolistic power and may set an unfavorable price and transaction fees for investors. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

17 Equilibrium Solutions Equilibrium Solutions ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

18 Equilibrium Solutions Innitely many small investors To simplify the analysis, we assume a (continuously) innite number of investors (I = R). We also assume that µ i N(µ I,σ 2 I ) in I and independent of {ω i }. Can be justied by the central limit theorem if the belief biases of investors are IID (Hellwig, 1980). We dened ζ i = µ i aσ 2 v ω i : ζ i N(µ I aσ 2 v ω }{{} =µ ζ,σ 2 I + a 2 σv 4 σ 2 ω). }{{} =σ 2 ζ (risk-adjusted) belief ζ i solely represents heterogeneity in the model. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

19 Equilibrium Solutions Total order amount Let q I (ζ ) = 1 e (ζ µ ζ ) 2 2σ 2 ζ. 2πσ 2 ζ Under this setting, the total amount of orders is not random: = i I p c x i (p, c) ζ (p c) q I (ζ )dζ + aσv 2 p+c ζ (p + c) q I (ζ )dζ aσv 2 ] [ (φ(d + ) + d + Φ(d + )) (φ(d ) + d Φ(d )) = σ ζ aσv 2 where Φ and φ are the distribution and density functions of a standard normal, respectively, and d ± = ± µ ζ (p ± c). σ ζ ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

20 Equilibrium Solutions Equilibrium in brokered market Market clearing implies p b = µ ζ, and thus R(µ ζ, c) = 2 [ ) )] σ aσ ζ cφ ( cσζ c 2 Φ ( cσζ = U M. v 2 Proposition 1 The equilibrium price in the brokered market, p b, is given by p b = µ I aσ 2 v ω and the per-unit fee by c b = σ ζ z, where z < 0 is the solution of the equation z + 1 d logφ(z) = 0. 2 dz ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

21 Equilibrium Solutions Equilibrium in dealer market (1) The nal wealth of the dealer is given by R(p, c) = Utility of dealer: U M = i I { (v p) ( x i (p, c)) + c x i (p, c) = (v p c) σ ζ [φ(d + ) + d + Φ(d + )] aσv 2 +(v p + c) σ ζ [φ(d ) + d Φ(d )]. aσv 2 (µ M p) σ ζ aσv 2 σ 2 ζ +c aσv 2 γ σ 2 ζ σ M 2 2 a 2 σv 4 [( ) ( )] φ(d ) + d Φ(d ) φ(d + ) + d + Φ(d + ) [( d + φ(d + ) + d 2 +Φ(d + ) ) } ( )] + d φ(d ) + d Φ(d 2 ) [( ) ( 2. φ(d + ) + d + Φ(d + ) φ(d ) + d Φ(d ))] ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

22 Equilibrium Solutions Equilibrium in dealer market (2) Proposition 2 The equilibrium price and the per-unit fee in the dealer market, p d and c d, satisfy the simultaneous equation system (µ ζ µ M )Φ(ˆd ± ) σ ζ [φ(ˆd ± ) + 2ˆd ± Φ(ˆd ± )] γσ ζ σ [( ) ( )] M 2 Φ(ˆd ± ) φ(ˆd + ) + aσ ˆd + Φ(ˆd + ) φ(ˆd ) + ˆd Φ(ˆd ) v 2 = 0, where ˆd ± = ± µ I aσv 2 ω (p d ± c d ). σi 2 + a 2 σv 4 σ 2 ω ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

23 Equilibrium Solutions Relationship between the two systems Corollary 1 ( pd c d ) ( pb c b ) as γ. Proof. Note that R(p, c) = (p v) x i (p, c) + c x i (p, c) i I i I and V[R(p, c)] must be zero if γ. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

24 Numerical Analysis Numerical Analysis ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

25 Numerical Analysis Welfare analysis Basecase parameters: Note that a σ ω σ v µ I σ I µ M ω γ σ M µ ζ =.75 < µ M = 1. X : trading volume dened by X = x i = x i (ζ ) q I (ζ )dζ. i I I We dene the total welfare of investors by U I = U i (ζ )q I (ζ )dζ. I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

26 Numerical Analysis Eect of µ I (1) µ I : mean of E[v F i ], c: per-unit fee b d 0.65 c µ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

27 Numerical Analysis Eect of µ I (2) µ I : mean of E[v F i ], p: asset price b b(ask) b(bid) d d(ask) d(bid) p µ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

28 Numerical Analysis Eect of µ I (3) µ I : mean of E[v F i ], X : trading volume. 2 b d 1.8 X µ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

29 Numerical Analysis Eect of µ I (4) µ I : mean of E[v F i ], U I : total welfare of investors. 3 b d 2.5 UI µ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

30 Numerical Analysis Eect of µ I (5) µ I : mean of E[v F i ], U M : utility of dealer/broker. 1.5 b d 1.3 UM µ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

31 Numerical Analysis Eect of σ I (1) σ 2 I : variance of E[v F i], c: per-unit fee. 0.7 b d 0.55 c σ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

32 Numerical Analysis Eect of σ I (2) σ 2 I : variance of E[v F i], p: asset price b b(ask) b(bid) d d(ask) d(bid) p σ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

33 Numerical Analysis Eect of σ I (3) σ 2 I : variance of E[v F i], X : trading volume. 1.8 b d 1.4 X σ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

34 Numerical Analysis Eect of σ I (4) σ 2 I : variance of E[v F i], U I : total welfare of investors. 1.5 b d 1.3 UI σ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

35 Numerical Analysis Eect of σ I (5) σ 2 I : variance of E[v F i], U M : utility of dealer/broker b d 0.6 UM σ I ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

36 Numerical Analysis Comparison with Sarkar (1995) Sarkar (1995) our study Fee c b > c d c b < c d Trading volume X b > X d X b < X d Welfare U I b > U I d (if informed) U I b < U I d (on average) U I b < U I d (if uninformed) ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

37 Numerical Analysis Implication The nal wealth of dealer/broker: R(p, c) = (p v) x i (p, c) + c x i (p, c) i I i I }{{}}{{} random payo certain payo Prot by proprietary trading (random payo) can have a positive eect on the ex-ante utility of an investor, while fee revenue (certain payo) always has a negative eect. Proprietary trading is always benecial to investors in average. Dealer sets a favorable price for investors to seek prots by proprietary trading. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

38 Conclusion Conclusion ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

39 Conclusion Conclusion Research question: How does proprietary trading aects a nancial market with a monopolistic dealer/broker? Answer: It has a positive eect on both a monopolistic dealer and investors. Why? Prot seeking by dealer with proprietary trading induces a more favorable price for the average investor. ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

40 References References I Bernhardt, D. and B. Taub (2010), "How and When Is Dual Trading Irrelevant?", Journal of Financial Markets, 13(2), Heidle, H.G. and R.D. Huang (2002), "Informed-Based Trading in Dealer and Auction Markets: An Analysis of Exchange Listings", Journal of Financial and Quantitative Analysis, 37(3), Hellwig, M.F. (1980), "On the Aggregation of Information in Competitive Markets", Journal of Economic Theory, 22(3), Huang, R.D. and H.R. Stoll (1996) "Dealer versus Auction Markets: A Paired Comparison of Execution Costs on NASDAQ and the NYSE", Journal of Financial Economics, 41(3), Kim, O. and R.E. Verrecchia (1991), "Trading Volume and Price Reaction to Public Announcement", Journal of Accounting Research, 29(2), Nishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

41 References References II Pagano, M. and A. Röell (1992), "Auction and Dealership Markets. What Is the Dierence?", European Economic Review, 36(2-3), Pagano, M. and A. Röell (1996), "Transparency and Liquidity: A Comparison of Auction and Dealer Markets with Informed Trading", Journal of Finance, 51(2), Röell, A. (1990), "Dual-Capacity Trading and the Quality of the Market", Journal of Financial Intermediation, 1(2), Sarkar, A. (1995), "Dual Trading: Winners, Loser, and Market Impact", Journal of Financial Intermediation, 4(1), Viswanathan, S. and J.J.D. Wang (1995), "Market Architecture: Limit-Order Books versus Dealership Markets, Journal of Financial Markets, 5(2), Nishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42

42 Thank you for your attention ishide and Tian (YNU and Ryukoku U.) Monopolistic Dealer versus Broker March 21, / 42