Inter-Dealer Trading in Financial Markets*

Transcription

1 S. Viswanathan Dke University James J. D. Wang Hong Kong University of Science and Technology Inter-Dealer Trading in Financial Markets* I. Introdction Trading between dealers who act as market makers is a common featre of most major financial markets. With the exception of eqity markets that deal with relatively small order sizes, in most financial markets the cstomer order is filled by one dealer who then retrades with other dealers. Inter-dealer trading is an integral part of market design, particlarly for instittional markets that deal with large orders. While the exact strctre of inter-dealer trading is ndergoing significant change (as we discss below) two distinct kinds of inter-dealer trading, seqential trading and limit-order books are dominant in the marketplace. In eqity markets like the London Stock Exchange, inter-dealer trading constittes some 40% of the total volme and occrs mainly via an anonymos limit-order book, althogh some * We thank Kerry Back, Dan Bernhardt, Francesca Cornelli, Phil Dybvig, Joel Hasbrock, Hong Li, Rich Lyons, Ananth Madhavan, Ernst Mag, Tim McCormick, Venkatesh Panchapagesan, Ailsa Röell and seminar participants at University of Arizona, Dke University, Hong Kong University of Science and Technology, University of Michigan, University of Sothern California, Virginia Polytechnic University, Washington University in St. Lois, the 1999 Nasdaq-Notre Dame Microstrctre Conference, and the 1999 WFA Meetings in Santa Monica, California, for their comments. The sggestions of an anonymos referee considerably improved the paper. (Jornal of Bsiness, 2004, vol. 77, no. 4) B 2004 by The University of Chicago. All rights reserved /2004/ $10.00 We compare the following mlti-stage inter-dealer trading mechanisms: a one-shot niform-price action, a seqence of nit actions (seqential actions), and a limitorder book. With ninformative cstomer orders, seqential actions are revenepreferred becase winning dealers in earlier stages restrict qantity in sbseqent actions so as to raise the price. Since winning dealers make higher profits, dealers compete aggressively, ths yielding higher cstomer revene. With informative cstomer orders, winning dealers se their private information in sbseqent trading, redcing liqidity. Seqential trading breaks down when the cstomer order flow is too informative, while the limit-order book is robst and yields higher revenes. 1

2 2 Jornal of Bsiness inter-dealer trading occrs throgh direct negotiation between dealers. The evidence in Hansch, Naik, and Viswanathan (1998), Naik and Yadav (1997), and Reiss and Werner (1998) sggests that the layoff of large orders (risk sharing) is important reason why inter-dealer trading occrs in London. On the NASDAQ market, market makers can trade with each other on the SperSoes system, the SelectNet system and on electronic crossing networks (ECNs) like Instinet. The SperSoes systems allows market makers to place limit orders that are hit by other market makers, electronic crossing networks (ECNs) and day traders. Volme on SperSoes is 20% of NASDAQ volme, a significant portion of this volme is inter-dealer trading. Market makers can also place qotes on SelectNet system. If these qotes are hit by other market makers (who mst place orders that are a hndred shares more than the qoted size), the qoting dealer has the discretion to execte the order or withdraw the qote (this is called discretionary exection). Inter-dealer trading on SelectNet is arond 10% of total NASDAQ volme. Finally market makers can trade with each other on ECNs like Instinet (which incldes direct trades between instittions); Instinet has arond 15% of total NASDAQ volme. 1 In the market for U.S. Treasries, two-thirds of the transactions are handled by inter-dealer brokerage firms sch as Garban and Cantor- Fitzgerald, while the remaining one-third is done via direct interactions between the primary dealers. To improve market transparency in the U.S. Treasries market, in 1990 the primary dealers and for inter-dealer brokers fonded GovPX, which acts as a disseminator of transaction price and volme information. Hence, the trading between primary dealers can occr on a nmber of competing inter-dealer brokerage systems bt is reported via GovPX to financial instittions. 2 More recently, the Bond Market Association responded to SEC pressre for more transparency in the bond market by setting p a single reporting system like GovPX for investment grade bonds. 3 In the foreign exchange market, inter-dealer trading far exceeds pblic trades, acconting for abot 85% of the volme. 4 Traditionally, inter-dealer trading on the foreign exchange market has been either by direct negotiation or brokered. Mch of the inter-dealer trading via direct negotiation is seqential (an otside cstomer trades with dealer 1 who trades with dealer 2 who trades with dealer 3 and so on) and 1. See and for this information. 2. The GovPX web sit provides sefl information on the history of GovPX. 3. See the Bond Market Association web site or the related website In an appearance before a Hose sbcommittee on September 29, 1998, then SEC Chairman Arthr Levitt pshed for greater transparency in bond trading. 4. See Lyons (1995) for more details.

3 Inter-Dealer Trading in Financial Markets 3 involves very qick interactions; hence, it is often referred to as hot potato trading. Today, 90% of this direct inter-dealer trading is done via the Reters D system which allows for bilateral electronic conversations in which one dealer asks another for a qote, which the originating dealer accepts or rejects within seconds. 5 Brokered trading is also exected via one of two electronic limit-order book systems, the Reters Dealing 2002 system and the Electronic Brokerage System (EBS Spot Dealing System). The Reters Dealing 2002 system was lanched in 1992 and is discssed extensively by Goodhart, Ito, and Payne (1996). The EBS system was started to compete against the Reters system and claims average volmes in excess of $80 billion a day. 6 The preceding discssion makes clear that two distinct kinds of inter-dealer trading, seqential trading (as in the Reters D system) and limit-order books (as in the EBS system, the SperSoes system) are most prevalent. In or view, the literatre on market microstrctre does not explain how the strctre of inter-dealer trading shold differ according to the nderlying trading environment. 7 While the seminal work of Ho and Stoll (1983) sggests that risksharing is a strong motive behind inter-dealer trading, it is not entirely clear why risk-sharing needs cold not be met optimally by direct cstomer trading with several dealers. Even if inter-dealer trading is desirable, it is nclear which method of inter-dealer trading is more appropriate. Or view is echoed by Lyons (1996a) in his discssion of empirical reslts on the foreign exchange market where he states that a microstrctral nderstanding of this market reqires a mch richer mltiple-dealer theory than now exists (see e.g. Ho and Stoll 1983). In this paper, we provide models of inter-dealer trading that inclde both single-price procedres with seqential trading (reflecting the traditional voice-brokering methods of inter-dealer trading) and mltipleprice procedres like limit-order books (reflecting the recent move towards electronic books in the foreign exchange market). Also, we ask how the strategic behavior of dealers and the exection prices for cstomer orders differ with the exact strctre of the inter-dealer market (single-price action verss limit-order book) and with the motivation of cstomer trades (inventory verss information). We believe that 5. This information is provided in Evans and Lyons (2002) that ses the Reters-2001 dataset. 6. The EBS partnership is a consortim of banks that are the leading market makers in the foreign exchange market. One key advantage of the EBS spot dealing system is that it links atomatically with FXNET, a separate limited partnership of banks that provides for atomated netting and hence redces settlement risk. EBS s web site provides more details. The Reters web site provides details on the Reters Dealing 2002 system. 7. In the canonical models of Glosten and Milgrom (1985) and Kyle (1985), the otside order is taken completely by one dealer and no retrading occrs.

4 4 Jornal of Bsiness realistic modeling of inter-dealer trading is crcial to improve or nderstanding of dealership markets like the foreign exchange market, the U.S. Treasry market and the NASDAQ market. Initially, we compare cstomer welfare between (1) two-stage trading that involve inter-dealer trading after a cstomer-dealer trade and (2) one-shot trading traditionally analyzed in the literatre. We identify a key advantage of two-stage trading that is absent in one-shot trading environments. Since Wilson (1979), it is known that single-price divisible good actions are plaged with a demand redction problem (see also Back and Zender (1993), Wang and Zender (2002), and Asbel and Cramton (2002)). That is, niform-price actions have eqilibria in which prices deviate sbstantially from economic vales becase bidders act strategically and steepen their demand crves. Two-stage trading alleviates this problem becase cstomers do not split orders in the first stage, and in the second stage the dealer who wins in the first stage acts strategically. In particlar, the dealer who wins in the first stage restricts the spply of the good in the second stage, i.e., engages in spply redction. This raises the price in the second stage. Since the winner in the second stage has a higher tility in the first stage, there is an incentive to bid aggressively for the whole qantity in the first stage. This leads to higher revene in the two-stage procedre. The ability of two-stage trading to elicit greater bidder competition gives the seller a potentially powerfl weapon to cope with strategic bidding in single-price action. This idea is worth emphasizing and is sefl in nderstanding why mlti-stage trading occrs in the real world. To gain a deeper nderstanding of the natre of inter-dealer trading, we extend the analysis to the case when the dealers rely on a limitorder book at the second-stage. There are important differences between inter-dealer trading at one price (dealership) and inter-dealer trading at mltiple prices (a limit-order book). In contrast to the dealership market, the benefit to a limit-order book market decreases with cstomer order size. In a single-price action strategic bidding (i.e., a departre from bidding according to marginal valations) is greatest at large qantities. In a limit-order book it is the greatest at small qantities. 8 Therefore, for a limit-order book, the revene improvement of a two-stage verss a one-shot trading is mainly for small cstomer orders. We generalize or two-stage, single price procedre to a seqential action with mltiple ronds of nit-actions. In the seqential action, the winning dealer (in any rond) keeps some fraction of the object and 8. These differences in market makers trading strategies across market strctres are emphasized in Viswanathan and Wang (2002). The hybrid market strctre considered there was not two-stage trading, bt rather a concrrent setp which rotes orders to different marketplaces sing a size criterion.

5 Inter-Dealer Trading in Financial Markets 5 sells the remaining via a nit-action to one of the remaining dealers in that rond. The procedre is in the spirit of the voice-brokering market where a cstomer sells to dealer 1 who then sells to dealer 2 and so on. We show that in sch a seqential action market liqidity falls as the action progresses, and in each sccessive rond the winning dealer in that rond keeps a larger fraction of the order flow for himself. Frther, we show that the seller is better off with more ronds of actions. These reslts rationalize the se of seqential actions and explain the phenomenon of hot potato trading. We extend or analysis to environments where the cstomer order flow contains payoff-relevant information. Absent reporting of trades, the information asymmetry between the cstomer and the market makers imposes a cost on inter-dealer trading that adversely affects the attainment of efficient risk-sharing. More asymmetric information in order flow lowers market liqidity and yields niformly lower price competition between dealers. With large information asymmetry, a (linear strategy) eqilibrim does not exist in a seqential action. In contrast, inter-dealer trading with a limit-order book is less sensitive to private cstomer information and does not break down. Ths limitorder books are more robst to market breakdown than seqential actions. The paper proceeds as follows. Section II discsses the prior literatre and or relation to it. Section III presents a model of two-stage trading in a dealership setting. The basic intitions of the model are laid ot in this simple context by comparing cstomer welfare between two-stage trading and one-shot trading. Section IV analyzes a seqential action procedre that extends two-stage trading. The important case of limit-order book trading is taken p in Section V. Section VI deals with informative cstomer trades; we also compare the cstomer s expected revenes nder different trading mechanisms. Section VII concldes. II. Literatre Review Naik, Neberger, and Viswanathan (1999) and Lyons (1997) consider the vale of disclosing cstomer trades and inter-dealer trades in dealership markets. The focs of or paper is not on disclosre of cstomer trades bt rather on the comparison of differing inter-dealer systems. Vogler (1997) considers the special case of a dealership market when dealers have the same pretrading inventory and concldes that inter-dealer trading always dominates the one-shot dealership market. Potential costs to two-stage trading in or model are absent from Vogler s model becase his is not a model of private information and becase he assmes homogeneos inventories across the dealers. None of the above papers considers the limit-order book

6 6 Jornal of Bsiness as a possible mode of inter-dealer trading or considers seqential action procedres. 9 Or modeling of the limit-order book allows dealers to se limit-orders of arbitrary sizes in the inter-dealing stage and for optimal bidding for the cstomer trade in the first rond. This paper is also related to the recent literatre on limit-order books de to Biais, Martimort, and Rochet (2000), Viswanathan and Wang (2002) and Röell (1998). Viswanathan and Wang (2002) characterize the eqilibria of a dealership market (a single-price setp) and a limit-order book (a mlti-price setp) when the risk-averse dealers compete for the cstomer order via downward sloping demand crves. Röell (1998) provides related reslts when the order flow is drawn from the exponential distribtion. Bias, Martimort, and Rochet (2000) characterize the limit-order book when the marginal valation crve is downward sloping de to information in the order flow. Viswanathan and Wang (2002) discsses the relationships between these papers and their relation to the earlier work of Glosten on the monopoly specialist (1989) and on the competitive limit-order book (1994). While these papers characterize limit-order books, they do not allow for inter-dealer trading. Finally, or paper is related to the literatre on retrading in actions by Haile (2000) and Gpta and Lebrn (1999). In these models, bidders are risk-netral and re-trading occrs either becase the object was not allocated to the bidder with the highest valation (Gpta and Lebrn) or becase valations changed and the bidder with the highest valation in the second stage is not the highest bidder in the first stage (Haile). In contrast, retrading in or model occrs becase dealers wish to share risk and not hold all of the object themselves. III. Inter-Dealer Trading With a Single-Price Mechanism A. The Model There are N > 2 risk-averse dealers (market makers or liqidity providers) in the game. Each dealer can potentially fill a sell order from a risk netral otside cstomer 10 of size z, which is distribted over the nit interval [0, 1]. The dealers act to maximize a mean-variance derived tility of profit with the risk aversion parameter r. 11 A typical 9. Werner (1997) presents a doble action model of inter-dealer trading with initially identical dealers. Following the extensive literatre on doble actions with nit demands, all dealers in Werner (1997) sbmit limit orders to by or sell a fixed amont in the second stage. Inter-dealer trading is also taken as given in Lyons (1996b), where the focs is on the transparency of inter-dealer trades. 10. Cstomer bys are analyzed analogosly. 11. With single-price inter-dealer trading (modeled as a niform-price action), this qadratic objective fnction may be derived from the standard assmption of constant absolte risk-aversion tility fnctions and normally distribted payoffs. For the limit-order book (akin to a discriminatory action), these assmptions do not imply a qadratic objective fnction in the dynamic optimization problem.

7 Inter-Dealer Trading in Financial Markets 7 dealer, generically referred to as dealer k (k =1,2,..., N ), is endowed with an ex ante inventory of Ĩ k, which is drawn from some commonly known distribtion that has spport, [ I; Ī ]. We denote the average (per dealer) initial inventory as Q ð1=nþs N k¼1ĩk. The nderlying asset vale, ū, is normally distribted as N(ū; t 1 ), and ũ is independent of the ex ante inventories, Ĩ k. To isolate the difference between two-stage trading and one-shot trading, we initially assme that the ex ante dealer inventories are common knowledge among all dealers. The main conclsions are not qalitatively different when this assmption is relaxed (see Section III.D). Frthermore, we assme that the cstomer order does not contain information abot the fndamental vale of the traded asset. The conseqences of relaxing this assmption will be evalated in Section VI. As a benchmark, we consider a one-shot trading setp in which the dealers sbmit demand schedles to compete for fractions of the cstomer order which is split among the N dealers. The focs of this section, however, is two-stage trading in which the otside order is first filled in its entirety by a single dealer before it is divided among all dealers via inter-dealer trading. Order splitting dring cstomer-dealer trading is disallowed. 12 Partly becase most dealership markets are not anonymos, the cstomer expects to sell to one dealer instead of splitting the order among several dealers. As sch, competition among the market makers in the first rond is similar to bidding for an indivisible good. We refer to the dealer who wins the cstomer order as dealer W, and to any dealer who loses the cstomer order as dealer L (8L =1,2,..., N, L 6¼ W ). 13 After the cstomer-dealer trading, the cstomer order size and transaction price become known to dealer W, bt not to the other dealers. 14 Conseqently, dring inter-dealer trading, dealer W can condition his trading strategy on z, while dealer L cannot. Inter-dealer trading among all N dealers starts shortly after the first rond. 15 The 12. The all-or-nothing assmption is adopted in part to avoid the difficlty of choosing among the mltiple eqilibria that reslt from analyzing a share action. The main conclsions of the paper regarding the speriority of two-stage trading over one-shot trading are not driven by this assmption. What is important is the observation that the winning dealer in the initial cstomer-dealer trading engages in spply redction to raise the inter-dealer price. 13. Throghot the paper, the sbscript k denotes qantities for a typical dealer k, whereas the sperscript W (or L ) denotes qantities for any specific dealer identified as the winning (or losing) dealer in the first rond. 14. The isse of trade disclosre is important if the second stage is rn as a limit-order book, since a key aspect of a book is its inability to condition on qantity. In this paper, we do not prse isses related to disclosre bt focs on the comparison of varios modes of inter-dealer trading. See Naik, Neberger, and Viswanathan (1999) for a paper that focses on disclosre and cstomer welfare in a dealership market. 15. In a dynamic setting, dealer W wold face the tradeoff between waiting for the next cstomer by to arrive and initiating trading with other dealers right away. By focsing on inter-dealer trading that occrs soon after the cstomer order is filled by one particlar dealer, we are essentially stdying markets where the need to lay off the risk associated with an nbalanced portfolio is very significant.

8 8 Jornal of Bsiness efficacy of sch two-stage trading will be compared to one-shot trading in which the cstomer can directly trade with many market makers. It is convenient to visalize inter-dealer trading as involving three steps. First, dealer W hands over his entire holding of the asset, I W þ z, to an actioneer (the inter-dealer broker). Then, the actioneer solicits bids from all dealers (inclding dealer W )intheform of combinations of price and qantity. Frther, we restrict the analysis to demand schedles that are continosly differentiable and downward slopping. A typical dealer k s trading strategy, as a fnction of the eqilibrim price and possibly his own pretrading net position, is the qantity that is awarded to him by the actioneer, x k. 16 The eqilibrim price of the single-price inter-dealer trading, p 2,isdetermined by eqating demand and spply. after the actioneer collects payment from all winning bids (those with price levels at or above the eqilibrim price), the total proceeds are then retrned to dealer W and are his to keep. Note that, at the conclsion of the inter-dealer trading, dealer W s net holding is x W,whiledealerL s net position is I L þ x L. In other words, x W denotes dealer W s final allocation, while x L is dealer L s trade qantity. Reslts in the paper need to be interpreted with this convention in mind. In the two-stage game, the winning dealer sbmits a spply crve dring inter-dealer trading. An alternative is for the winning dealer to se the qantity choice as a strategy. However, with two stages of trading and no adverse selection problem, we can show that this alternative is revene inferior for the winning dealers. The case of winning dealers making seqential qantity choices is analyzed in Section IV. A distinct featre of the dealership market is that most transactions take place at a single price. Therefore, dealer k s profit at the conclsion of inter-dealer trading is: p W k ¼ ũ xw k þ p 2 I k þ z x W k ; if k previosly won the cstomer order; p L k ¼ ũ I k þ x k L p2 x k L ; if k did not win the cstomer order: Differences between inter-dealer trading via a dealership setp (which involves trading at a single-price) and inter-dealer trading via a limitorder book (which involves trading at mltiple prices) are discssed in Section V. 16. It is often convenient to express the bidding strategies as the inverse demand schedle, i.e., price as a fnction of qantity and inventory. To save notation, argments to the demand schedles are often sppressed.

9 Inter-Dealer Trading in Financial Markets 9 For ease of presentation, 17 we maintain the following restrictions throghot the paper: I k T ū ; 8k; 1 N T ū : With tractability in mind, we restrict or analysis to eqilibria of the model that are characterized by linear trading strategies. Given the symmetry of the problem, we will search for eqilibria in which the strategies of the nonwinning dealers (8L 6¼ W ) take the same fnctional form. B. Benchmark: A One-Shot Dealer Market A one-shot model where the cstomer can trade directly with N competing market makers serves as a benchmark for comparison with the two-stage trading model jst introdced. The following is a necessary condition for the eqilibrim strategies in a single-price dealership market: 18 X N j6¼i Bx j p; I j x i ðpþ ¼ Bp ū p ½I i þ x i ðp; I i ÞŠ : A niqe symmetric, linear soltion to the above eqation is the following: 19 x k ðp; I k Þ ¼ g d ū I k p ; 8k ¼ 1; 2;...; N; where The eqilibrim price and allocations are: N 2 g d ¼ : ð1þ ðn 1Þ p d ¼ ū ðn 1Þrt 1 Q N 2 x k ¼ z N þ N 2 N 1 ðq I kþ: z N ; ð2þ 17. We make these assmptions in order to restrict the discssion to jst one side of the market, i.e., a cstomer sells and the market makers by. Relaxing these parameter restrictions does not affect the conclsions of the paper in any sbstantive way. 18. See Viswanathan and Wang (2002) for a derivation. 19. This is similar to Kyle (1989).

10 10 Jornal of Bsiness The cstomer s total revene is R d ¼ z p d : At the end of trading, the net inventory positions are as follows: I k þ x k ¼ z N þ N 2 N 1 Q þ ¼ z N þ 2I k N þ N 2 N I k N P 1 j6¼k I j N 1 where the term in parenthesis in Eq. (3) is the average inventory of the other N 1dealers. In the ending positions of the dealers (Eq. (3)), the cstomer order is eqally shared. However, each individal dealers inventory is overweighted by one extra share and the average inventory of the other dealers is nderweighted correspondingly. 20 We will se this observation to provide intition for the revene speriority of two-stage trading. In sbmitting bids for the given qantity, each dealer nderstands the impact of his trade on the price. Hence a dealer who wishes to by finds it optimal to steepen his demand crve to redce the price. This is referred to as demand redction (see Asbel and Cramton (2002) for a general discssion of this phenomenon in niform-price actions) and reslts in a price lower than the dealer s marginal valation of the object (see Eq. (2)). Symmetrically, a dealer who wishes to sell has strategic incentives to engage in spply redction, i.e., to restrict the spply at every price. This strategic power leads to each dealer overweighting his own inventory by one extra share of inventory. 21 The two-stage trading model that we present exploits this strategic incentive. C. Eqilibrim Analysis of the Two-Stage Trading Model The two-stage trading model is solved sing backward indction. Under a dealership strctre, the second-stage (i.e., inter-dealer competition) can be modeled as a single-price divisible good action. The first rond bidding between the market makers for the cstomer order can be viewed as a nit demand action with private bidder valation. With pblicly known dealer inventories, the bidders valation is common knowledge and the otcome of the first stage bidding is standard. 20. Withot this overweight, the ending inventory wold be z N þ Ik N þ 1 P N j6¼k I j. The overweight of one s own inventory implies that inventory hedging is incomplete by the end of trading. 21. Note that dealers have eqal strategic power with respect to the cstomer order which is eqally shared. ; ð3þ

11 Inter-Dealer Trading in Financial Markets The Second Rond: Inter-Dealer Trading When the inter-dealer trading reslts in all trades clearing at a single price, the dealers eqilibrim trading strategies are provided below. Note that the soltion in Proposition 1 forms an ex post eqilibrim strategy in that it is independent of the distribtion of cstomer order size or the dealer inventories. Proposition 1. If inter-dealer trading occrs at a single price, it has a niqe linear strategy eqilibrim characterized by the following trading strategies: x W p; z; I W ¼ g2 ðū pþþ I W þ z N 1 ; x L ð4þ p; I L ¼ g2 ðū pþ N 2 N 1 I L ; 8L 6¼ W; ð5þ where the price elasticity of demand, g 2,isgivenby: g 2 ¼ N 2 : ð6þ ðn 1Þrt 1 Proof. See Appendix I. Using the above eqilibrim strategies, it is straightforward to show: p 2 ¼ ū Q þ z ; N x W ¼ N 2 Q þ z N 1 N þ I W þ z ; N 2 x L ¼ N 2 Q þ z N 1 N I L ; 8L 6¼ W: ð7þ Ths, the dealers net positions at the end of two ronds of trading will be: x W ¼ 2 z N þ N 2 N 1 Q þ I W N 1 ¼ 2 ð z þ I W Þ þ N 2 P! j6¼w Ĩj ; N N N 1 ð8þ I L þ x L ¼ N 2 z N 1 N þ N 2 N 1 Q þ I L N 1 ¼ 2I L N þ N 2 z þ P! j6¼l Ĩj : N N 1 These two expressions are similar to the last two terms of Eq. (3). Becase this second rond of trading involves retrading among the dealers, no additional external spply is available, which explains the absence of a ð9þ

12 12 Jornal of Bsiness term similar to the first term in Eq. (3). The original cstomer order, z, shows p as part of dealer W s pre-inter-dealer-trading inventory. Ths, it receives one share of overweight in W s ending inventory position, mch like the one-shot trading model where each dealer s own pre-trading inventory is overweighted in the final qantity allocation. This is de to the spply redction engaged in by dealer W so as to raise the price in the second stage. Comparing Eq. (7) with Eq. (2), we have p 2 > p d.hence, dealerw s spply redction raises the price, and conseqently dealer L has an ending inventory that nderweights the cstomer order. At the conclsion of inter-dealer trading, the cstomer order is split nevenly among the dealers, with dealer W retaining an above average fraction, 2/N, and every other dealer retaining a below average fraction, (N 2) [N(N 1)], of the original cstomer order z. In addition, the price is higher dring inter-dealer treading than in the one-shot trading model ( p 2 > p d ). This is an important difference between two-stage trading and one-shot trading. Next we write dealer k s ( 8k =1,2,..., N ) second-stage (certainty eqivalent) tility, depending on whether or not he gets to fill the cstomer order in the first rond: Ũk W ¼ ū x W k rt 1 W 2 x 2 k þ p2 Ik þ z x W k ¼ ū N 2 Q þ z N 1 N þ I k þ z rt 1 N 2 Q þ z N 2 2 N 1 N þ I k þ z 2 N 2 þ ū Q þ z I k þ z N 2 Q þ z N N 1 N þ I k þ z ; N 2 Ũk L ¼ ū I k þ x L k I k þ x L 2 k p2 x L k 2 ¼ ū I k þ N 2 Q þ z N I k rt 1 I k þ N 2 Q þ z 2 N I k N 1 N 1 ū Q þ z N 2 Q þ z N N 1 N I k : From the above, it is straightforward to compte the tility difference between the winning dealer and the losing dealers, which is sefl in analyzing the first stage of cstomer-dealer trading. Corollary 1. If inter-dealer trading occrs at a single price, the difference in dealer k s second-stage (certainty eqivalent) tility between winning and losing the cstomer order, z, is: ( Ũk W Ũk L rt 1 ¼ z ū ðn 1Þ 2 I k þ NðN 2Þ Q þ N 3 ) z ; 2 8k ¼ 1; 2;...; N: ð10þ 2

13 Inter-Dealer Trading in Financial Markets The First Rond: Cstomer-Dealer Trading For any given cstomer order z, Corollary 1 sggests that dealer k s incentives for filling the cstomer order are based on the following private vale fnction: 22 Ṽk ¼ Ũ W k Ũ L k ; ð11þ whichisstrictlydecreasinginhisowninventorylevel,i k.ths,withot loss of generality, we can index the dealers in ascending order of their inventory positions, i.e., I 1 < I 2 <...< I N. The first stage of trading is an nit action nder complete information. The dealer with the lowest ex ante inventory, dealer 1, wins the cstomer order, and he pays an amont eqal to the reservation price of the dealer with the second-lowest inventory, dealer 2. Proposition 2. Assme that dealer inventories are pblicly known. If inter-dealer trading occrs at a single price, the cstomer receives the following revene in the first rond of trading: ( R 1 ¼ z ū rt 1 ðn 1Þ 2 I 2 þ NðN 2ÞQ þ N 3 ) z z p 1 ; ð12þ 2 where I 2 is the second-smallest inventory among all the dealers. Proof. From Eq.s (10) and (11), it is clear that, for any given z, we have Ṽ 1 > Ṽ 2 >...> Ṽ N. Althogh the dealers do not know in advance the size of the incoming cstomer order, z, they compete by sbmitting a series of qantity-payment pairs, i.e., a demand schedle that states what the total payment to the cstomer, B k, will be at each potential cstomer order size. In particlar, the following is a set of eqilibrim bidding strategies (as a fnction of the cstomer order size, z): B 1 ðþ¼v z 2 ðþ; z B k ðþ¼v z k ðþ; z 8k 6¼ 1: The tie-breaking rle is that the lower-nmbered dealer wins when the same bid is sbmitted by more than one dealer. Regardless of the actal cstomer order sizes, the otcome is always that dealer 1 wins and pays the cstomer dealer 2 s reservation price of Ṽ De to the two-stage natre of the model it is not always tre that one cold directly work with the certainty eqivalent tility in compting the dealers optimal trading strategy in the first rond. See Lemma 1 in Appendix II for a proof of the validity of the certainty eqivalent approach in or model.

14 14 Jornal of Bsiness Cstomer revene from two-stage trading verss one-shot trading can be evalated by comparing Eq.s (12) and (2): p 1 p d ¼ rt 1 N 2 2 ðn 1Þ 2 2NðN 2Þ z þ ð Q I 2Þ : ð13þ rt 1 In Eq. (13), there are two effects that cold potentially work in opposite directions. The firs is a strategic effect that always works in favor of the two-stage trading game. This is captred by the term proportional to z. The second effect arises de to the srpls extracted by the winning dealer in the first rond of a two-rond trading model. This bidding effect is the term proportional to Q I 2. The intition for the strategic effect is as follows. Previosly we established that in eqilibrim all dealers overweight their own inventory. This reslts in the winning dealer retaining an extra share of the cstomer order: dealer W restricts qantity dring the inter-dealer trading stage. Restricting the qantity available ( spply redction ) to other bidders raises the price in the second rond and yields higher profits to the winning dealer. Since the winner of the cstomer order expects to receive higher profits in inter-dealer trading, all dealers compete more intensely for the cstomer order. This yields the strategic term, ð Þ z : 23 The strategic effect favors two-stage trading. Also, as the cstomer order size becomes larger, the winning dealer s potential profit in the second rond becomes greater, which implies more intense competition in the first rond. Therefore, the net benefit of this strategic effect to two-stage trading (as opposed to one-shot trading) increases as the cstomer order becomes larger. The bidding effect is related to the srpls extracted by the winning bidder in the first stage. Since the winning bidder has the lowest inventory, it mst be that Q I 1 > 0, i.e., the winning bidder wishes to by additional nits. This works in favor of the two-stage trading game as the cstomer order is allocated to a trader who desires it the most. However, the transaction price is not set by the dealer with the lowest inventory, bt rather by the dealer with the second lowest inventory. If Q I 2 > 0, the dealer with the second lowest inventory wants to by and hence the price favors the two-stage game. If Q I 2 > 0, the dealer with the second lowest inventory wants to sell. In this sitation, setting the transaction price based on a dealer who does not want to add to his inventory will have a negative impact on the first-stage price. N 2 2 ðn 1Þ 2 2N N Note that the cstomer receives p 1, not p 2. The strategic effect is less than p 2 p d ¼ rt 1 NðN 2Þ z becase the winning bidder has to be compensated for the risk of holding additional inventory of size z N, ths lowering the benefit of rnning the two-stage trading game for the cstomer.

15 Inter-Dealer Trading in Financial Markets 15 Notice that this bidding effect arises only becase of the srpls obtained by the winning bidder. One way to see this is to rewrite Eq. (13) as p 1 p d ¼ rt 1 N 2 2 ðn 1Þ 2 2NðN 2 Þ z þ ð Q I 1Þ rt 1 ðn 1Þ 2 ð I 2 I 1 Þ; ð14þ where the last term is the only negative term in the eqation and reflects the srpls extracted by the winning bidder. Note that the size of the bidding effect is affected by the distribtion of dealer inventories and not by the cstomer order size. Corollary 2. Assme that dealer inventories are pblicly known. Relative to the eqilibrim price in a one-shot game, p d, the eqilibrim price in the first rond of the two-stage game, p 1, has two properties: (i) p 1 as a fnction of the cstomer order size z is always flatter (i.e., more price-elastic) than p d ;(ii) p 1 has a higher intercept (i.e., small-qantity qote) than p d if and only if Q > I 2. The bidding effect does not change as cstomer order sizes change, while the strategic effect is proportional to the order size. Ths, the strategic effect dominates at large order sizes. Therefore, the two-stage dealership market generally provides better exection than its one-shot conterpart for large-sized order flows. See Figre 1 for an illstration. In empirical work, Naik and Yadav (1997) and Reiss and Werner (1998) find that the bid-ask spread on inter-dealer trades in smaller than the spread on cstomer-dealer trades. The model here prodces the following reslt, which is consistent with the empirical observation. Corollary 3. If Q I 2 < ðn 2Þ z 2N,thenthebid-askspreadoninterdealer trades is smaller than the spread on cstomer-dealer trades. From Eq.s (7) and (12), it is easy to see that p 2 p 1 ¼ rt 1 ðn 2Þ 2NðN 1Þ 2 z rt 1 ðn 1Þ 2 ðq I 2Þ: Ths, when the bidding effect (the second-term above) does not dominate (e.g., when the dealer inventories are relatively homogeneos), we have p 2 > p 1. In this case, the bid price in the second stage is higher than in the first stage of trading. An analogos analysis of by orders reveals that ask prices in the second stage are lower than the prices in the first stage of trading. Ths, the bid-ask spread is smaller on inter-dealer trades. D. Privately Known Dealer Inventories Now, any other dealer s higher private valation will manifest itself throgh a lower vale of Q in dealer k s private vale (notice the appearance of Q in Eq. (10)). Ths, when the dealers do not know

16 16 Jornal of Bsiness Fig. 1. Eqilibrim price vs cstomer order size in a dealership market. The solid line is for one-shot trading. The three dashed lines are for the initial stage of a two-stage trading model (the second lowest dealer inventory is 0.8,1.0,1.2 from top to bottom). The price for two-stage trading is generally higher than its one-shot conterpart at large cstomer order sizes, althogh at small cstomer order sizes the comparison is inflenced by dealer inventory. other dealers inventory positions, the dealers private vales are affiliated in the sense of Milgrom and Weber (1982). Now the celebrated revene eqivalence theorem does not hold and or choice of action form is relevant. For concreteness, we will assme the dealers have exponential preferences and model the cstomer-dealer trading as a second-price action. Proposition 3. Sppose all dealers have exponential preferences and their inventories are exponentially distribted with a pdf of f(h) = me mh and a cdf of F(h) =1 e mh, where the parameter m > If the cstomer-dealer trading is a second-price action, then the cstomer s expected revene is: ( ) R 1 z p 1 ¼ z ū rt 1 ð2ī þ zþ þ ðn 2Þ r ðn 1Þ 2 N 3 2 ðm kþī 2 ln m m k ; 24. For ease of comptation, we do not impose the inventory restrictions listed in Section III.A. With the exponential distribtion, inventories can be very large. Hence some dealers cold be sellers instead of byers at the single price that clears the second stage.

17 Inter-Dealer Trading in Financial Markets 17 where, Ī 2 is the expected vale of the second lowest inventory given by: and we reqire k < m: Ī 2 ¼ 2N 1 mnðn 1Þ ; k ðn 2Þr2 t 1 ðn 1Þ 2 Proof. See Appendix II. We find that the price the cstomer expects to receive is sally higher with two-stage trading than with one-shot trading (for one-shot trading we find p d by integrating Eq. (2) over Q): " # N 2 ð2n 3Þ p 1 p d ¼ z þ rt 1 2 Q 2NðN 2ÞðN 1Þ ðn 1Þ 2 Ī2 ðn 2Þ m þ ðm kþī 2 ln : ð15þ r z m k As in the analysis preceding Eq. (14) in the known inventory case, we can decompose the price difference into two components. The first is the strategic effect which arises becase of the spply redction dring inter-dealer trading. This effect is the first term in Eq. (15) which is strictly positive and proportional to the cstomer order size. The second line in Eq. (15) is the bidding effect. As in the known inventory case, it is typically positive (so long as the log term does not dominate). This effect consists of two effects, the srpls effect and the winner s crse (the winner s crse did not exist with known inventories). As before, the srpls effect arises becase the price is set by the second lowest inventory in Eq. (15). The winner s crse arises becase the winner does not know the inventories of the bidders below him and has to find their conditional expectation. This indces him to nderbid and is reflected in the last term in Eq. (15). Overall, the bidding effect is negative when k approaches m form below, that is, when the cstomer order is large and/or when the dealer inventories are drawn from a distribtion with high variance. For intition, we recognize that the variance of the exponential distribtion is 1/m 2.Sowhenm is small, the variance (and mean) of inventories is high, sggesting more srpls to the highest type (the dealer with the lowest inventory). Frther, the winner s crse is higher when the variance of inventories is higher. Under these circmstances, the bidding effect works against the two-stage action. z:

18 18 Jornal of Bsiness The models we have analyzed in Section III all lead to similar conclsions. Two-stage trading enjoys a pricing advantage (from the cstomer s perspective) over one-shot trading. In two-stage trading, the winning dealer strategically restricts the amont sold in the interdealer stage in order to raise the resale price. This strategic effect favors two-stage trading and is linear in the cstomer order size. In addition, there is a bidding effect that favors two-stage trading nless the cstomer order is large and inventories are drawn from a distribtion with a high variance. IV. Seqential Actions: Rationalizing Hot Potato Trading Mch of the inter-dealer trading in the foreign exchange markets is done via voice-brokering and has the following featre that was emphasized in the introdction: The cstomer trades with dealer 1 who trades with dealer 2 who trades with dealer 3, and so on. The qick seqence of bilateral inter-dealer trades following a cstomer trade is often referred to as hot potato trading. Given or finding in Section III that two-stage trading (an nit action followed by single-price trading) is generally favored over one-shot, single-price trading, a logical qestion to ask is whether cstomer welfare is improved by having more trading ronds. We constrct a seqential trading model of a dealership market where the cstomer first sells his qantity z to one of N dealers, who then resells a portion of the cstomer qantity to another dealer, and so on. This contines ntil there is a total of m 4 dealers left, at which point the dealer who has boght in the previos rond resells a fraction of his qantity to the other m 1 dealers sing a niform-price action. Hence the selling dealer in each rond chooses a qantity to trade rather than a spply crve. Ths, the model in this section differs slightly from that in Section III in that a dealer sbmits a qantity rather than a spply crve. As we will see, the nderlying intition of restricting spply to raise the price will still hold. In this section or focs is on the strategic bidding cased by seqential trading; ths we assme all dealers are symmetric in their inital inventory positions (set to zero). In other words, the bidding effect is absent here. Also, a dealer who has sold some qantity to another dealer cannot trade again. We will refer to the n-dealer stage of seqential trading, which eventally ends when it gets down to m dealers, as an (n, m) trading game, where n m. When necessary, we se the sperscript m and sbscript n to denote qantities in the (n, m) trading game. At the n-dealer stage, the dealer who prchased the qantity q n +1 from the previos rond resells a portion of it, q n, to the other n 1 dealers at the price of p n (q n ). We denote the selling dealer s expected tility U n and the other dealers expected tility V n.

19 Inter-Dealer Trading in Financial Markets 19 We conjectre that the inter-dealer trading price in an (n, m) game takes the following form: p n ðq n Þ¼ū lm n q n: ð16þ The parameter l m n is an inverse measre of the market liqidity: the lower is l m n, the more liqid is the inter-dealer trading. Proposition 4. In the seqential trading model above, the liqidity parameter is determined by the following iteration formla: l m l m nþ1 ¼ lm m 1 n 2ðm 1Þ 1 þ 2l m þ n ðm 1Þð1 þ 2l m m Þ2 ð1 þ 2l m mþ1 Þ2...ð1 þ 2l ; m n Þ2 8n m 4; ð17þ with l m =(m 2)=[(m 1)(m 3)]. Proof. See Appendix III. The next reslt characterizes the evoltion of market liqidity, the dealers trading volme and the eqilibrim price in the sccessive ronds of the seqential action game. Corollary 4. As inter-dealer trading progresses in the (n, m) trading game (i.e., as n becomes smaller), market liqidity decreases and the selling dealer retains a larger proportion of the qantity obtained in the previos rond. Frthermore, the eqilibrim price increases in later ronds. Proof. An inverse measre of market liqidity is l m n. We first prove by indction that l m n is decreasing in n. It is straightforward to verify that l m +1 < l m. Now by assming l m n < lm n 1,wehave: l m nþ1 ¼ l m l m m 1 n 2ðm 1Þ 1 þ 2l m þ n ðm 1Þð1 þ 2l m m Þ2 ð1 þ 2l m mþ1 Þ2...ð1 þ 2l m n Þ2 l m < lm m 1 n 1 2ðm 1Þ 1 þ 2l m þ n 1 ðm 1Þð1 þ 2l m m Þ2 ð1 þ 2l m mþ1 Þ2...ð1 þ 2l m n 1 Þ2 ¼ l m n : From his optimization problem (see Eq. (A-15) in Appendix III), the selling dealer retains the following qantity: q nþ1 q n ¼ 2lm n 1 þ 2l m q nþ1 : n ð18þ

20 20 Jornal of Bsiness Ths, the proportion he retains is increasing in l m n.asn becomes smaller, l m n becomes greater, and therefore, the selling dealer sells less and chooses to retain a greater share. As for the last statement, mltiplying (17) by q n +1 and sing Eq. (18), it is easy to show that l m n q n is increasing in n. Ths, according to Eq. (16), price increases as trading progresses (i.e., as n decreases). Corollary 4 demonstrates that the inter-dealer market becomes more illiqid as trading progresses. This deterioration of liqidity occrs for the following reasons. Mechanically, as seqential trading evolves, fewer prospective byers remain. Ths, the potential for risk-sharing with the remaining participants diminishes. This has a negative impact on the liqidity of the market. Frthermore, the winning dealers in sccessive ronds engage in more qantity restriction by selling less and withholding a greater share from the inter-dealer market. The fact that price increases over time is consistent with the twostage model in Section III (see Corollary 3). As trading nfolds the sellers get higher prices at the expense of worsening liqidity and declining trading volme. The effects of seqential inter-dealer trading on market liqidity, transactions volme, and dealer competition are illstrated in Figres 2 and 3. Corollary 4 describes the evoltion of liqidity, prices and volme along an action seqence where the total nmber of trading ronds is fixed. To draw a closer parallel to the previos comparison of one-shot trading and two-stage trading, we stdy the cstomer revene when the nmber of trading ronds is increased. Corollary 5. A risk-netral cstomer prefers a seqential dealership market with more stages of inter-dealer trading. For any given nmber of dealers, when there are more ronds of inter-dealer trading, the trading volme is higher. Proof. The cstomer-dealer trade can be viewed as the first rond of trading in the seqential trading game (N +1,m). Ths, the cstomer s expected revene is: E½ R m Nþ1 Š¼E½ zp Nþ1ð zþš ¼ Z 1 0 ðū lm Nþ1zÞzgðzÞdz: ð19þ Now consider a seqential action with one fewer rond of interdealer trading, i.e., seqential trading begins as an (N +1,m +1) game. In this case, the cstomer s expected revene is: E½ R mþ1 Nþ1 Š¼E½ zp Nþ1ð zþš ¼ Z 1 0 ðū lmþ1 Nþ1 zþzgðzþdz: To show that the risk-netral cstomer always prefers more ronds of inter-dealer trading, it is sfficient to show that E[ R m m N +1 ] > E[ R +1 N +1 ],

21 Inter-Dealer Trading in Financial Markets 21 Fig. 2. The inverse liqidity parameter, l, and inter-dealer trading volme (indicated by the thick bars) vs the ronds of trading in a seqential inter-dealer action market ( N = 12). As trading progresses, the inter-dealer transaction volme as a percentage of the cstomer order is decreasing, and market liqidity is also decreasing. m or eqivalently, l N +1 m < l +1 N +1. This is established by explicit calclation. Lastly, from Eq. (A-15) in Appendix III, it is easy to see that q n m > q n m +1,sinceq n m is inversely related to l n m. Corollary 5 states that, starting with a given nmber of dealers, the more ronds of inter-dealer trading, the higher the cstomer s expected revene. The intition is related to the strategic effect discssed in Section III. In the seqential trading game, dealers restrict qantities at each stage of inter-dealer trading. With more ronds of trading to go, the dealer who wins the cstomer order will have the incentive to withhold a smaller share. In eqilibrim, a higher trading qantity implies higher liqidity, which ltimately benefits the cstomer. Therefore, extends or earlier reslt regarding the speriority of two-stage trading over one-shot trading to the mlti-stage setting. 25 Corollary 5 offers an explanation as to why seqential actions may be beneficial as a trading instittion. It demonstrates that more ronds 25. The seqential action considered here mst end when m = 4, with one dealer trading with three other dealers who share the good eqally. An alternative wold be to psh the seqential action analogy frther and let the selling dealer sell via a nit action to the three remaining dealers. The dealer who wins wold then rn a nit action to sell to the two remaining dealers. In sch an end-game, one dealer gets no qantity allocation at all. We find that, for a risk-netral cstomer, this new seqential action ending with a nit action is preferred to a seqential action ending with a share action. This is consistent with or reslt that more ronds of inter-dealer trading improve cstomer welfare.

22 22 Jornal of Bsiness Fig. 3. Eqilibrim price vs cstomer order size in a seqential inter-dealer action market ( N = 10). The solid line is the benchmark price for which there is no trading srpls for the dealers. The dashed lines are the dealers demand crves with 1, 3, 5, and 7 ronds of trading remaining in the seqential game. With more ronds of inter-dealer trading remaining, dealer competition is more intense and the cstomer s expected revene is higher. of inter-dealer trading leads to higher expected revene for the cstomer. Conseqently, it provides a rationale for the hot potato phenomenon in the foreign exchange market. Nmerical calclation shows that the model can generate trading volmes that rival the sbstantial inter-dealer trading in the foreign exchange market. For example, with 12 dealers and 10 ronds of trading, the total interdealer trading volme is 4.59 times the initial cstomer volme: interdealer trading is 82% of total volme. This is in line with the nmber reported by Lyons (1995) who states that 85% of trading in the foreign exchange market is attribtable to inter-dealer trading. V. Inter-Dealer Trading with a Limit-Order Book While voice-brokering via seqential trading has traditionally been sed in the foreign exchange markets, mch volme has migrated to electronic limit-order book trading. In particlar, as discssed in the introdction, both the EBS partnership and Reters Dealing 2002 offer systems which have featres of a limit-order book. Here we analyze a two-stage model where the inter-dealer competition occrs within a limit-order book which is akin to a discriminatory action. In or analysis of the limit-order book, we assme that all dealer

23 Inter-Dealer Trading in Financial Markets 23 inventories are identical, I k = Q, k =1,2,..., N. Then, dealer k s trading profit is: p W k ¼ ũ x W k þ p0 2ðQ þ z x W k ÞþXN m6¼k Z p Z p p L k ¼ ũðq þ x L k Þ p0 2 x L k x L k ðyþdy; p 0 2 p 0 2 x L m ðyþdy; where p is the intercept of the demand schedle with the price axis, and p 0 2 is the market clearing price dring the inter-dealer trading stage. We emphasize that, to rn the inter-dealer market as an anonymos limitorder book, the cstomer order in the first stage cannot be disclosed to the dealers who do not receive the order in the first stage. In contrast to the eqilibrim in a single-price setting (Section III) which is independent of distribtional assmptions, in this section we sppose that the distribtion of cstomer order sizes has a linear hazard ratio. In particlar, the pdf and cdf for z 2 [0, 1] are the following: gðzþ ¼ 1 ð1 zþ1 1 ; GðzÞ ¼1 ð1 zþ 1 ; where is a positive parameter related to the first two moments of the distribtion as follows: E½ zš¼ 1 þ ; 2 Var½ zš¼ ð1 þ Þð1 þ 3 þ 2 2 Þ : Note that the case of = 1 corresponds to the niform distribtion. 26 A. Benchmark: A One-Shot Limit-Order Book For comparison prposes, a benchmark is presented in which the cstomer can trade directly with N competing market makers in a oneshot limit-order book. The following is a necessary condition for the eqilibrim strategies in a limit-order book market: 27 X N j6¼i Bx j ð pþ Bp ð1 zþ ¼ ū p ½x ið pþþqš : 26. The only other distribtion that yields a linear soltion is the exponential distribtion, which is a limiting case of the linear-hazard ratio class stdied here. This can be seen by taking the limit of GðzÞ ¼1 ð1 lzþ 1 as approaches infinity, which is G(z) =1 e l z. 27. See Viswanathan and Wang (2002) for a derivation.

24 24 Jornal of Bsiness The niqe symmetric, linear soltion to the above eqation is given by: x k ð pþ ¼g b ū Q Nð1 þ Þ 1 p ; 8k ¼ 1; 2;...; N: where g b ¼ The eqilibrim price and allocations are: Nð1 þ Þ 1 ðn 1Þ : ð20þ p b ¼ ū Q Nð1 þ Þ 1 ðn 1Þrt 1 Nð1 þ Þ 1 z N ; ð21þ x k ¼ z N : The cstomer s total revene in this case is: R b ¼ p b z þ XN Z p k¼1 p b x k ðyþdy: Note that dealer positions at the end of trading are as follows: I k þ x k ¼ z N þ Q: B. The Second Rond: Inter-Dealer Trading With identical ex ante inventories, each dealer has an eqal probability of winning the cstomer order in the first stage of trading. Withot loss of generality, we designate the dealer who wins the cstomer order in the first rond of trading as dealer W, and all other dealers are referred to as dealer L. It trns ot that dealer W s eqilibrim strategy is independent of the distribtional assmptions abot inventory or cstomer order size. Proposition 5. If inter-dealer trading is rn as a limit-order book, dealer W s eqilibrim strategy is: 8 < ū p x W ð p; zþ ¼ : Q þ z if z 2½s; 1Š; if z 2½0; sš: ð22þ

25 Inter-Dealer Trading in Financial Markets 25 That is, dealer W will sell a nonzero qantity in the inter-dealer market if and only if the cstomer order he fills in the first stage exceeds the following threshold size: 2 s ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð23þ ðn 1Þð1 þ Þþ2 þ ðn 1Þ 2 ð1 þ Þ 2 4 Proof. See Appendix IV. The analysis of the optimal strategy for a dealer who did not get to fill the cstomer order, dealer L, involves solving a dynamic optimization problem. Proposition 6. Assme that the dealer inventories all eqal Q. Ifinterdealer trading is rn as a limit-order book, the trading strategy for dealer L is: x L ð pþ ¼m 0 2 g 0 2 p; 8L 6¼ W ; ð24þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þð1 þ Þ 2 þ ðn 1Þ 2 ð1 þ Þ 2 4 g 0 2 ¼ 2ðN 2Þ ; ð25þ 2 3 m 0 2 ¼ g ū Q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: ðn 1Þð1 þ Þþ2 þ ðn 1Þ 2 ð1 þ Þ 2 4 ð26þ Dealer L gets a nonzero qantity allocation if and only if the cstomer order size is greater than s. Proof. See Appendix V. For z 2 [s, 1], we can solve for the eqilibrim price in the second stage as: p 0 2 ¼ ū Q rt 1 2ðN 1Þ p ðn 1Þð1 Þþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þ 2 ð1þþ þðn 1Þ g0 2 þ z : ð27þ Therefore, we can express dealer W s qantity allocation after the second stage trading, x W, and dealer L s acqired qantity from inter-dealer trading, x L, as follows: x W ¼ ū p0 2 X ; ð28þ x L ¼ m 0 2 g 0 2 p 0 2 I L Ỹ I L ; 8L 6¼ W: ð29þ

26 26 Jornal of Bsiness Ths, the net positions for the dealers at the conclsion of inter-dealer trading are: Since x W ¼ X ; I L þ x L ¼ Ỹ: X Ỹ ¼ 4NðN 2Þð1 þ Þð1 zþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðN 1Þ 2 ð1 þ Þ 2 þðn 1Þ ðn 1Þ 2 ð1 þ Þ 2 4Š 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0; ½ðN 1Þð1 Þþ ðn 1Þ 2 ð1 þ Þ 2 4Š ð30þ the winner of the cstomer order in the first rond retains a fraction of the cstomer order that is at least as large as the other dealers fraction at the end of inter-dealer trading. Therefore, regardless of the natre of pricing rles (limit-order book vs dealership), the dealer who fills the cstomer order in the first rond always retains a larger share and engages in qantity restriction. The winning dealer does this to raise the price and revene in the second stage. In eqilibrim, this leads to more dealer competition in two-stage trading than in one-shot trading. For a typical dealer k, his (certainty eqivalent) tility from interdealer trading is: Ũ W k ¼ ū X rt 1 2 X 2 þðq þ z X Þ p 0 2 þ 1 X N 2g 0 ðỹ QÞ 2 ; 2 j6¼k if he wins the cstomer order in the first stage. If dealer k did not win the cstomer order dring the first rond of trading, the corresponding tility will be: Ũ L k rt 1 ¼ ūỹ 2 Ỹ 2 ðỹ QÞ p ðỹ QÞ 2 : 2g 0 2 Therefore, Ũ W k Ũk L ¼ðū p0 2Þð X Ỹ Þ rt 1 2 ð X 2 Ỹ 2 Þ þ p 0 2 z þ 1 X N 2g 0 ðỹ QÞ 2 : 2 j¼1 ð31þ Corollary 6. When inter-dealer trading is rn as a limit-order

27 Inter-Dealer Trading in Financial Markets 27 book, the difference in dealer k s second-stage (certainty eqivalent) tility between winning and losing the cstomer order is given by: Ũ W k Ũk L 8 ¼ ðū p 0 2 Þð X ỸÞ rt 1 2 ð X 2 Ỹ 2 Þþ p 2 z 0 þ 1 X N >< 2g ðỹ 0 j¼1 QÞ2 if z 2½s; 1Š; 2 ūðq þ zþ rt 1 ðq þ zþ2 ũq rt 1 >: 2 2 Q2 if z 2½0; sš: ð32þ C. The First Rond: Cstomer-Dealer Trading For any given cstomer order z, Corollary 6 shows that all the dealers have the same reservation price, Ṽ Ũ W k Ũ L k,(8k =1,2,..., N ). Assme that the dealer inventories are all eqal to Q. Ifinter-dealer trading is rn as a limit-order book, the cstomer receives the following revene in the first rond of trading: R 0 1 ¼Ṽ 8 >< ¼ >: 2 ð X Ỹ Þ 2 þ p 0 2 z þ 1 X N 2g ðỹ 0 j¼1 QÞ2 if z 2½s; 1Š; 2 ū z rt 1 2 ð2q z þ z 2 Þ if z 2½0; s Š: ð33þ where the excess qantity retained by the winning dealer, X Ỹ, is given by Eq. (30), and the eqilibrim price dring inter-dealer trading is p 2, 0 by Eq. (27). Proof. It is straightforward to show: ū p 0 2 rt 1 2 ð X þ Ỹ Þ¼ rt 1 2 ð X Ỹ Þ0: Eq. (33) follows from rewriting Eq. (31) sing the last eqation and Eq. (30). Next we compare R 1 0 with the revene that a cstomer receives from a one-shot benchmark model in Section V.A: R b ¼ p b z þ 1 2g b X N j¼1 z 2 : ð34þ N 0 When z 2 [0, s], we can show that R 1 > R b. 28 Thiscanbeviewedas an extreme case of qantity restriction by the winning dealer: the winner 28. To see this, we first show that R 1 R b ¼ FðzÞ is a concave fnction of z. We then compte F(0) = 0, F(s) > 0, and F (0) > 0 ; ths FðzÞ > 0; 8z 2½0; sš.

28 28 Jornal of Bsiness gets the cstomer order and no inter-dealer trading takes place afterwards. In this sitation, the cstomer is strictly better off selling to a single dealer who does not retrade rather than selling to N dealers simltaneosly. For cstomer orders that are not too small ( z 2 [s, 1]), there will be inter-dealer trading following the initial cstomer-dealer trade. Comparing the first line of Eq. (33) with Eq. (34), we note that R 1 0 has an extra term proportional to ( X Ỹ ) 2. This is directly attribtable to the winning dealer keeping a larger share of the cstomer order. A second reason why R 1 0 tends to be higher than R b is that the eqilibrim price dring inter-dealer trading is higher than the corresponding price in one-shot limit-order book trading. 29 Finally, comparing the last terms in Eqs. (33) and (34), we find that there is a cost to two-stage trading related to the se of a flatter demand crve. The preceding discssion is smmarized in the following reslt (see Figre 4 for an illstration). Corollary 7. Assme that dealer inventories are all eqal to Q. Comparing the cstomer s revene from one-shot trading in a limit-order book, R 0 b, with the revene from two-stage trading with a limit-order book, R 0 1, we find that: (i) for small cstomer order sizes, z 2 [0, s ], it is always tre that R 1> 0 R b.for z 2 [ s, 1], however, there are the following tradeoffs: (ii) the benefits to two-stage trading come from qantity restriction by the winning dealer and the size of the benefits is small for large otside orders; (iii) there is a cost to two-stage trading becase a lesser amont of price discrimination srpls is extracted from the dealers. As in Section III, the benefits of two-stage trading with a limitorder book are also directly related to the winning dealer s tendency to redce trading qantity in order to raise price. There are two key differences, however. First, the cost of a two-stage trading with limit-order book trading manifests itself throgh a redced amont of price discrimination srpls that the cstomer can extract from the competing dealers. Second, the benefits of two-stage trading with a book become less important for larger cstomer order sizes. This is qite different from the sitation with a single-price inter-dealer trading, where the strategic effect is proportional to the cstomer order flow. This difference has its origin in the different ways that strategic bidding (i.e., departre from pricing according to marginal valation) operates in a single-price action verss a discriminatory action: The extent of bid redction decreases with qantity levels in a limit-order book, whereas it increases with qantity levels in a single-price clearing mechanism. 29. Comparison of Eq. (27) with Eq. (21) shows that: p 0 2 p b 0:

29 Inter-Dealer Trading in Financial Markets 29 Fig. 4. Average price received by the cstomer vs cstomer order size in a limitorder book market. The solid line is for one-shot trading. The dashed line is for the initial stage of a two-stage trading model. All dealer inventories are identical. The price for two-stage trading is generally higher than its one-shot conterpart at small cstomer order sizes, althogh at large cstomer order sizes the reverse is tre. VI. Informative Cstomer Trades In this section, we stdy the impact of private cstomer information on inter-dealer trading. Conditional on the cstomer order size, z, the dealer is assmed to face a simple inference problem: E½ũj z ¼ zš ¼ũ x z; ð35þ where x is a strictly positive parameter, i.e., a larger cstomer sell order implies a greater downward adjstment to the expected vale of the asset. 30 Since or focs here is on asymmetric information, we set all dealer inventories to zero, i.e., I k =0,8k =1, 2,..., N. We assme no disclosre of information abot previos trades, which is consistent with trading rles in foreign exchange and other instittional markets. The above linear pdating rle can be exploited to restate the two benchmark one-shot trading models in a convenient way. When the cstomer order is informative, the only change to the one-shot dealership model is to replace Eq. (1) with: g d ðxþ ¼ N 2 ðn 1ÞðNx þ 1Þ : ð36þ 30. For simplicity, we assme that other aspects of the asset vale distribtion, sch as variance, do not change with z.

30 30 Jornal of Bsiness Similarly, in the one-shot limit-order book model (assming the cstomer order is niformly distribted, i.e., = 1), we can se: g b ðxþ ¼ 2N 1 ; ð37þ ðn 1ÞðNx þ 1Þ in place of Eq. (20). These reslts are qite intitive becase, with a worsening adverse selection problem (a larger x vale), the dealers bid less aggressively by steepening their demand crves. From the preceding discssion, private cstomer information tends to make the one-shot trading less competitive. A. Seqential Actions Next we explore the effect of private cstomer information on the seqential action model. Since a linear pdating rle is assmed for the cstomer-dealer trading stage, we conjectre that in sbseqent inter-dealer trading the asset vale has a similar correlation strctre with the trading qantity. That is: E½ũjq n Š¼ū x nq n ; ð38þ in the (n, m) trading model. Note that, by definition, x N +1 x and q N +1 z. Sppose inter-dealer trading prices take the form: p n ðq n Þ¼ū l nq n ; we have the following reslt. 31 Proposition 8. In the seqential trading model with private information, the liqidity parameter and information parameter are determined by the following iteration formlas: l nþ1 ¼ 2l nð1 þ 2x nþ1 Þ x 2 l m x m nþ1 þ 2ð1 þ 2l n Þ 8N þ 1 n m 4; x x nþ1 ¼ n ; 1 þ 2l n x n 1 2ðm 1Þ m 1 x n þ 1 2 ; x m with l m =(m 2)[(m 1)x m +1] [(m 1)(m 3)] and x N +1 = x. 31. In contrast to Section IV, we omit the sperscript m in this section to redce the notational complexity. It is nderstood that the nmber of dealers in the last stage of the seqential game is m.

31 Inter-Dealer Trading in Financial Markets 31 Proof. See Appendix VI. Corollary 8. As inter-dealer trading progresses in a seqential trading game (i.e., as n becomes smaller), both the adverse selection problem and market liqidity worsen (i.e., x n +1 < x n and l n +1 < l n ). Proof. See Appendix VII. In contrast to the one-shot models where linear strategy eqilibria always exist, with seqential trading the existence of a linear strategy eqilibrim is not assred and the presence of informed cstomer trades may lead to a market breakdown. 32 Proposition 9. When the informativeness of the order flow is sfficiently high, market breakdown will always occr in seqential actions. Fixing the total nmber of dealers, the parameter region with market breakdown expands with the nmber of trading ronds. Proof. See Appendix VIII. The intition for a market breakdown is as follows. Becase only the winning dealer observes the cstomer order flow in the first rond, he possesses information abot the asset vale that other dealers do not have. Recognizing the incentives of the informed dealer to sell a greater share when he perceives a lower asset vale, the other dealers respond by steepening their demand crves. In a mlti-action trading environment, information asymmetry worsens along the action path. For the same level of qantity traded, the dealers in later trading stages infer a lower asset vale becase this qantity mst have reslted from a larger qantity sold by the cstomer (which is taken as a bad signal). As trading progesses, market liqidity worsens and the dealers demand crves become more inelastic. At high enogh vales of the initial adverse selection parameter x, the inference parameter in the last rond (x m ) becomes negative, indicating the nonexistence of a linear strategy eqilibrim. Not srprisingly, the region of breakdown becomes larger when there are more trading ronds and is the smallest when there is one rond of inter-dealer trading. The more ronds of trading, the worse is the market liqidity in the last rond. With enogh initial information asymmetry and enogh ronds of trading, we find that the final rond of trading collapses. If the initial information asymmetry is sfficiently high, more market breakdown occrs (i.e., market breakdown occrs not jst in the last rond, bt in the final two ronds, final three ronds, etc.) This is discssed in Proposition 9 and illstrated in Figre 5. Comparing the above reslt with Corollary 5, we see that tension exists between the strategic advantage implied by rnning more 32. This no-trade reslt is different from other examples of market breakdown in the literatre (see, e.g., Glosten (1989), Bhattacharya and Spiegel (1991)) in that it occrs with two-stage trading bt not with one-shot trading.

32 32 Jornal of Bsiness Fig. 5. The maximm feasible and optimal nmbers of trading ronds vs the information parameter, x, in a seqential action inter-dealer market ( N = 15). In this figre, the higher staircase represents the maximm feasible ronds of trading; the lower staircase represents the optimal (from the cstomer s perspective) ronds of trading. When there is little private cstomer information (when x is small), the nmber of ronds of inter-dealer trading is relatively high and the two staircases coincide. If the extent of private information is very large (e.g., when x is greater than 2), inter-dealer trading will be made impossible (i.e., the two staircases fall to zero level, which is not shown in the figre). At intermediate vales of x, the optimal nmber of ronds is smaller than the maximm feasible ronds of trading. actions and the information disadvantage associated with more interdealer trading. In other words, withot private information, more trading ronds benefit the cstomer becase dealers compete for the opportnity to win and gain higher trading profits in sbseqent trading. With private information, more trading ronds exacerbate the information asymmetry problem and have an offsetting effect on cstomer welfare. This implies that an interior nmber of ronds can be optimal (see Figre 5 for illstration). B. Two-Stage Limit-Order Book Trading Given the increasing se of limit-order books in the context of interdealer trading, it is important to nderstand how limit-order book trading is affected by the presence of private information. For this prpose, we modify the model in Section V by adding a linear inference problem (Eq. (35)) to the cstomer-dealer trading stage. Proposition 10. Assme that cstomer orders are niformly distribted over the interval [0,1]. If a limit-order book is sed dring

33 Inter-Dealer Trading in Financial Markets 33 inter-dealer trading, there exists a linear strategy eqilibrim in which the strategy for the winning dealer of the cstomer order is: 8 < ū x x W z p ð p; zþ ¼ if z 2½s; 1Š; ð39þ : z if z 2½0; sš: The inter-dealer trading strategy for dealer L is x L ( p) =m gp, where g ¼ m ¼ s ¼ s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðN 2Þ Nx þ 4NðN 2Þð1 þ xþþn 2 x 2 2ðN 2Þ ð1 þ NxÞ ; 1 þ s ū sð1 þ xþ ð1 þ xþþðn 1Þðrt 1 x sþ ; gū m ; ð1 þ xþg 1 ðn 2Þg þ 1 : ð40þ Proof. See Appendix IX. It is interesting to compare the way market makers in different interdealer trading systems respond to the problem of private information. In a dealership setting (e.g., a seqential action), dealers se increasingly inelastic demand crves when adverse selection worsens. This phenomenon eventally leads to a market breakdown. With a limit-order book, however, the winning dealer in the cstomer-dealer rond makes two adjstments in response to asymmetric information: he lowers the intercept of his demand crve, and he decreases the no-trade zone (i.e., increasing s ).It trns ot that, for cstomer orders above a certain threshold size, a linear strategy eqilibrim always exists in limit-order book trading. The above reslt demonstrates that, when inter-dealer trading takes place in a limit-order book, a linear strategy eqilibrim exists even when there is a severe adverse selection problem. This contrasts with seqential action s ssceptibility to private information. It sggests that, in environments where the concentration of informed traders is expected to be high, inter-dealer trading might well take the form of a limit-order book rather than a seqential action.

34 34 Jornal of Bsiness C. The Cstomer s Expected Revene To a large extent, the sccesses and failres of inter-dealer trading systems are measred by the cstomer welfare achievable nder sch systems. Ths, we are interested in comparing the expected cstomer revene nder both the seqential action and the limit-order book strctre when private information may be an isse. The revene comparisons in this section are established throgh nmerical comptation based on the following relations (for simplification, we set all dealer inventories to be zero). From Eq. (19), the expected cstomer revene in a seqential action is: E½ R S Š¼ Z 1 0 ½ū l Nþ1zŠzgðzÞdz: ð41þ Using (33), the cstomer s expected revene in limit-order book trading is: E½ R B Š¼ Z 1 s 2 ðx YÞ2 þpzþ N 2g Y 2 gðzþdz þ Z s 0 ū rt 1 2 z zgðzþdz; ð42þ where: p ¼ ū þðn 1Þrt 1 m ð1 þ xþrt 1 z 1 þðn 1Þ g ; X ¼ ū xrt 1 Y ¼ m gp: z p ; The following reslts smmarize the impact of inter-dealer trading system designs on cstomer welfare in the absence of asymmetric information. See Figres 6 and 7 for illstrations. Proposition 11. Sppose the cstomer trades do not carry private information and there is no pblic disclosre of trades. With two stages of trading, a risk-netral cstomer prefers inter-dealer trading in a limit-order book to a seqential action. With more than two stages of trading, a seqential action is revene preferred. Proposition 12. The cstomer s expected revenes in both the seqential action and the limit-order book decrease when the cstomer trades contain more information abot the asset s vale. When the information asymmetry is significant (i.e., at larger vales of x), a risknetral cstomer favors the two-stage limit-order book over the seqential action.

35 Inter-Dealer Trading in Financial Markets 35 Fig. 6. The cstomer s expected revene vs the total nmber of dealers in the absence of private cstomer information (x = 0). Except at very small nmber of dealers, the cstomer revene is higher nder an inter-dealer trading system with a seqence of actions than with a limit-order book. We assme that seqential trading takes the maximm ronds of actions (i.e., m = 4). Fig. 7. The cstomer s expected revene vs the information parameter, x. For the seqential inter-dealer market ( N = 15), we assme that seqential trading takes the optimal ronds of actions. At smll x vales, the cstomer revene in seqential trading is higher than the cstomer revene from an inter-dealer trading system based on a limit-order book. At higher vales of x, however, the cstomer prefers limit-order book trading.

36 36 Jornal of Bsiness From Figre 7, it is clear that both E[ R S ]ande[ R B ] are decreasing fnctions of x. That is, the presence of asymmetric information has a negative impact on cstomer welfare in the seqential action and in the limit-order book. However, the extent to which private information adversely affects the cstomer s expected revene differs depending on the strctre of inter-dealer trading. When private information is nonexistent or nimportant (zero or small x vales), the action market (two or more stages) tends to be favored by a risk netral cstomer over the two-stage limit-order book. When private information is pervasive (large vales of x), the limitorder book is revene sperior to the seqential action regardless of the nmber of action ronds inter-dealer trading may take. Proposition 12 reinforces the notion that a limit-order book is a better vene for interdealer trading when there is a severe adverse selection problem. VII. Conclsion In this paper, we stdy whether mlti-stage trading mechanisms that involve inter-dealer trading provide welfare improvement for cstomers over the one-shot settings traditionally analyzed in the market microstrctre literatre. Important determinants of sch a comparison inclde the pricing rles in the inter-dealer market, the size and distribtion of the cstomer orders, the information content of the cstomer orders, and, in the case of seqential actions, the nmber of ronds of trading. We identified a key advantage of mlti-stage trading that are absent in one-shot trading environments: the dealer who wins the cstomer order restricts the qantity he sells in sbseqent ronds so as to raise the price. As a reslt, the winning dealer gets an above average fraction of the cstomer order, ths providing strong incentives for all dealers to compete more aggressively for the cstomer order. This intensified competition leads to improved cstomer welfare relative to a one-shot trading. Ths, mlti-stage procedres are better able to deal with collsive bidding among market makers that occrs in a one-shot single price mechanism. The combination of an all or nothing first stage bidding and an active inter-dealer trading phase serves the dal prpose of facilitating competitive bidding and attaining risk-sharing among the dealers. We analyzed a seqential action model that approximates the traditional voice-brokering in foreign exchange trading. In the seqential action, there is repeated bilateral trading between dealers. A back-ofthe-envelope calclation shows that the model can generate inter-dealer volmes consistent with the trading volme reported in the literatre. Absent private information in the order flow, seqential actions yield higher revene than limit-order books.

37 Inter-Dealer Trading in Financial Markets 37 When informative order flows, in seqential actions the winning dealer in any given trading rond ses his private information in sbseqent ronds, redcing the liqidity of the inter-dealer market. Althogh in eqilibrim all information is revealed, the presence of private information distorts risk-sharing and works against the seqential action. As a reslt, the cstomer s expected revene in mltistage markets decreases with increases in the informativeness of the cstomer order. Of the two kinds of inter-dealer trading, the seqential action is more ssceptible to market breakdowns than is the limitorder book. A market breakdown always occrs in seqential actions when asymmetric information is high. This arges in favor of sing the limit-order book in inter-dealer trading. Or reslts provide strong spport for inter-dealer mechanisms that se a seqential action or a limit-order book. The first corresponds to the traditional voice-brokering services in the foreign exchange market (and its electronic sccessor, the Reters System) while the second is closer to the electronic limit-order books like the Reters Dealing 2002 System and the EBS Spot Dealing System. Appendix A Proof of Proposition 1 Given the information partition of the dealers, we conjectre the following linear eqilibrim strategies: x W ¼ m 0 g 0 p þ b 0 ði W þ zþ; x L ¼ m gp bi L ; 8L 6¼ W: Note that dealer W can condition his strategy on z, btdealerl cannot becase the actal cstomer order size is only known to dealer W in the first rond of trading. From the market clearing condition I W þ z ¼ x W þ XN L6¼W we can back ot dealer W s residal spply crve: " # p ¼ l x W þðn 1Þm I W z b XN I L ; x L ; L6¼W where l ¼ 1 ðn 1Þg : ða-1þ

38 38 Jornal of Bsiness Since dealer W already possesses I W + z nits of the asset to begin with, he will receive a payment on the I W + z x W nits that he resells to the other dealers dring inter-dealer trading. For any given set I L, L 6¼ W, dealerw s trading profit is therefore: p W ¼ ũx W þ pði W þ z x W Þ: ða-2þ Expected tility maximization leads to the following first-order condition for dealer W : 33 ths, 0 ¼ ū p þ lði W þ z x W Þ xw ; ða-3þ x W ¼ ū p þ lði W þ zþ : ða-4þ þ l Comparing Eq. (A-4) with the conjectred fnctional form, we have: m 0 ¼ ū þ l ; ða-5þ b 0 l ¼ þ l ; ða-6þ g 0 1 ¼ þ l : ða-7þ Note that the soltion obtained does not explicitly depend on I L ; ths, it is optimal for all realizations of I L, L 6¼ W. The analysis of dealer L s optimal strategy proceeds along similar lines; the only significant difference is that dealer L cannot condition his strategy on z. In the proof below we fix a specific realization of z and find the trading strategy for dealer L. Since this strategy is independent of z, it is an optimal strategy for all possible vales of z. From dealer L s perspective, the market clearing condition I W þ z ¼ x L þ x W þ XN k6¼l;w can be sed to back ot his residal spply crve: " # p ¼ s x L þðn 2Þm I W z b XN I k þ m 0 þ b 0 ði W þ zþ ; k6¼l;w x k ; 33. Since the objective fnction is concave in x W, the first-order condition is necessary and sfficient.

39 Inter-Dealer Trading in Financial Markets 39 where s ¼ 1 g 0 þðn 2Þg : ða-8þ If he acqires x L nits in inter-dealer trading to agment his initial inventory, dealer L will have the following trading profit: p L ¼ ũði L þ x L Þ px L : ða-9þ Expected tility maximization leads to the following first-order condition for L; ths; 0 ¼ ū p sx L ði L þ x L Þ; x L ¼ ū p rt 1 þ s I L : ða-10þ Comparing Eq. (A-10) with the conjectred fnctional form, we have: m ¼ ū þ s ; ða-11þ b ¼ rt 1 þ s ; ða-12þ 1 g ¼ þ s : ða-13þ The coefficients (m 0, b 0, g 0 ) and (m, b, g) can be solved from Eq.s (A-1), (A-5), (A-6), (A-7), (A-8), (A-11), (A-12), and (A-13). The soltions are smmarized in the proposition. Appendix B Proof of Proposition 3 First we prove the following lemma which establishes that it is appropriate to se the difference in certainty eqivalent tilities from the second stage (depending on whether the dealer wins or loses in the first rond) as the private vale for the nitdemand action in the first rond. Lemma 1. Optimal trading strategies in the first rond can be determined by assming that each market maker has the private vale, Ṽ k, which is eqal to the difference in certainty eqivalent tility levels when he wins the cstomer order and when he does not.

40 40 Jornal of Bsiness Proof. Becase a dealer ses a demand crve as his trading strategy, the dealer can condition the total payment he makes to the cstomer (pon winning) on the size of the cstomer order. Conseqently, in the following we will integrate over the tre vale of the asset, ũ, and other dealers inventories, Ĩ k, bt not over z. Let d be an indicator fnction which takes on vale 1 if dealer k wins the cstomer order and vale 0 otherwise. Denoting the eqilibrim bidding strategy in a second-price action as B k and the eqilibrim price in the second-rond as p 2, dealer k s objective fnction in the first rond is: Eũ; Ĩ k ½ e rfd½ũ xw k þ p2ðikþ z xw k Þ BkŠþð1 dþ½ũðikþ x L k Þ p2 x L k Šg Š ¼ Eũ; Ĩ k ½ e rũ½d xw k þð1 dþði kþ x L k ÞŠ rd½ p 2ðI k þ z x W k Þ B kš rð1 dþ p 2 x L k Š ¼ EĨ k ½ e r fũ½d xw k þð1 dþði kþ x L k rd½ p 2ðI k þ z x W k Þ BkŠ rð1 dþ p2 x L k ¼ EĨ k ½ e rdðũ W k BkÞ rð1 dþũ L k Š ¼ EĨ k ½ e rũ L k ŠEĨ k ½e rdðũ W k Ũ L k B kþ Š: rt 1 ÞŠ 2 ½d xw k þð1 dþði kþ x L k ÞŠ2 g Notice that Ũ k L is independent of Ĩ k and ths we are left with an objective fnction which is proportional to the objective fnction of a bidder in a standard single-nit action with private vale of Ũ W k Ũ L k. The above proof assmes that the inter-dealer trading occrs at a single price. It can be easily adapted to show that the certainty eqivalent approach is also valid when the cstomer order is informative abot the vale of the asset according to Eq. (35). Sppose dealer l holds the smallest inventory among all dealers other than k. Let m be any dealer who is neither k nor l. A reslt from Milgrom and Weber (1982) ( page 1114) states that the eqilibrim bidding strategy in a second-price action as a fnction of private vales, B k (x), is the niqe soltion to: h i E e r½ṽ k B k ðxþš ji k ¼ x; I l ¼ x; I m x; 8m 6¼ k; l ¼ e r0 ; ða-14þ where Ṽ k is bidder k s private vale of the asset, given by Eq. (11): ( Ṽ k ¼ z ũ rt 1 ðn 1Þ 2 I k þ NðN 2Þ Q þ N 3 ) z : 2

41 Inter-Dealer Trading in Financial Markets 41 For any given z, Eq. (A-14) can be inverted to give: B k ðxþ ¼ 1 r lne½e rṽ k ji k ¼ x; I l ¼ x; I m xš " n ¼ 1 r lne e r z ji k ¼ x; I l ¼ x; I m x ū rt 1 hĩ ðn 1Þ kþðn 2Þ 2 # P N Ĩ k þ Ĩ l þ m6¼k; l ( ¼ z ũ rt 1 ðn 1Þ 2 x þ 2ðN 2Þx þ N 3 ) z 2 N 2 r " Z # Ĩ ðn 2Þr 2 t 1 ln e ðn 1Þ 2 zy f ðyþdy : x Ĩ m þðn 3 2 Þ z io Therefore, dealer k s bidding strategy (i.e., total payment for a cstomer order of size z) expressed as a fnction of his inventory is: " B k ði k Þ¼ z ũ rt 1 ðn 1Þ 2 N 3 # ð2i k þ zþ 2 N 2 r " Z # Ĩ ðn 2Þr 2 t 1 ln e ðn 1Þ 2 zy f ðyþdy : I k The seller s expected revene takes the expectation of the above expression over Ĩ k = Ĩ 2 (the second-lowest inventory): " R 1 ¼ z ũ rt 1 ðn 1Þ 2 N 3 # ð2ī 2 þ zþ 2 Z Ī I h NðN 1ÞðN 2Þ r " Z # Ī ðn 2Þr 2 t 1 ln e ðn 1Þ 2 zy f ðyþdy f ðhþfðhþ½1 FðhÞŠ N 2 dh: where Ī 2 denotes the expected second-lowest inventory level N(N 1) R Ī I h f ðhþfðhþ½1 FðhÞŠN 2 dh. The above explicit soltion involves the distribtion characteristics of the dealer inventories. In general, the payment fnction B k is not qadratic in the cstomer

42 42 Jornal of Bsiness order size z, i.e., the nit (bid) price is nonlinear in z. 34 Take the example of exponentially distribted dealer inventories with the parameter m > 0. That is, f (h) = me mh and F(h) =1 e mh. For a given z, the cstomer s expected revene is: ( R 1 ¼ z ũ rt 1 ðn 1Þ 2 N 3 ) 2ð2N 1Þ 2 mnðn 1Þ þ z þ ðn 2Þ r where we reqire k < m. m k 2N 1 m NðN 1Þ ln m m k ; Appendix C Proof of Proposition 4 Analyzing the selling dealer s maximization problem: max q n U n ¼ ūðq nþ1 q n Þ rt 1 2 ðq nþ1 q n Þ 2 þ q n p n ðq n Þ; provides the following optimal qantities q n ¼ 1 þ 2l m q nþ1 ; n U n ¼ q nþ1 ū lm n 1 þ 2l m n q nþ1 ða-15þ : ða-16þ Notice that the above expected tility for the selling dealer is net of the price that he paid for the qantity q n +1 obtained in the previos rond of trading. For each dealer to be jst indifferent between receiving or not receiving q n +1, it mst be that: U n q nþ1 p nþ1 ðq nþ1 Þ¼V n : For dealers other than the seller, their expected tility is a constant at any stage of the game, i.e., V n = Constant, 8n. To determine this constant, we examine the last inter-dealer trading stage (i.e., when n = m). Using Eq. (2), the selling price is: p m ðq m Þ¼ū ðm 2Þrt 1 ðm 3Þ q m m 1 ū rt 1 lm m q m: 34. With known dealer inventories, the nit price is linear in z. See Eq. (12). 35. Since l n m > 0, the second-order condition is always satisfied.

43 Inter-Dealer Trading in Financial Markets 43 From the above formla, l m m ¼ m 2. The expected tility for a dealer ðm 1Þðm 3Þ who bys in the last rond is: V m ¼ ū qm m 1 rt 1 2 ¼ l m m 1 2ðm 1Þ m 1 q2 m : Ths, sing Eq. (A-15) recrsively, we have: qm 2 pm ðq m Þ m 1 qm m 1 V n ¼ V m ¼ lm m 1 2ðm 1Þ m 1 ¼...¼ lm m 1 2ðm 1Þ m 1 q2 m ¼ lm m 1 2ðm 1Þ qm þ 1 2 m 1 ð1 þ 2l m m Þ qn þ 1 2 ð1 þ 2l m m Þð1 þ 2lm mþ1 Þ...ð1 þ 2lm n Þ : Finally, the price at the (n +1,m) trading game can be determined as follows: p nþ1 ðq nþ1 Þ¼ 1 q nþ1 ðu n V n Þ ¼ ū rt 1 lm n 1 þ 2l m n q nþ1 lm m 1 2ðmþ1Þ m 1 q nþ1 ð1 þ 2l m m Þ2 ð1 þ 2l m mþ1 Þ2...ð1 þ 2l m n Þ2 ū lm nþ1 q nþ1: Ths, we have the following iteration formla for the liqidity parameter in an (n +1,m) game: lm n l m nþ1 ¼ 1 þ 2l m n þ l m m 1 2ðm 1Þ ðm 1Þð1 þ 2l m m Þ2 ð1 þ 2l m mþ1 Þ2...ð1 þ 2l m n Þ2 : Appendix D Proof of Proposition 5 Sppose the dealer Ls plays the strategy, x L ( p), which does not depend on the realization of z (only dealer W observes the tre cstomer order size going into the inter-dealer trading stage).

44 44 Jornal of Bsiness If dealer W intends to retain x W < Q + z nits after the second rond of trading, then his inter-dealer trading profit is: p W ¼ ũx W þ pðq þ z x W Þþ XN Z p j6¼w p x j ðyþdy; ða-17þ where p is the intercept of dealer j s demand schedle with the price axis. Maximization of dealer W s expected tility is eqivalent to maximizing the following objective fnction: ūx W þ pðq þ z x W Þ rt 1 2 ðxw Þ 2 þ XN which leads to the first-order condition for dealer W: j6¼w Z p p x j ðyþdy; 0 ¼ ū p þ Bp Bx W ðq þ z xw Þ xw Bp X N Bx W x j ð pþ j6¼w ¼ ū p xw : The market clearing condition Q + z = x W + P N j6¼w x j( p) is sed in the last step. Ths far, we have proved that the trading strategy: x W ð pþ ¼ū p ; is the eqilibrim strategy for dealer W. In the proof of Proposition 6, we will show that there exists a niqe ctoff cstomer order size, s, sch that dealer W is a net seller dring inter-dealer trading if and only if z 2 [s, 1]. Shold dealer W decide to se the strategy x W ( p) when he observes a z vale that is less than s (based on the incorrect assmption that all other dealers get a positive amont), it can be shown that dealer Ls qantity allocation wold be negative in that case. This cannot happen in eqilibrim, therefore, dealer W retains all of the cstomer order he fills if it is sfficiently small (i.e., when z < s). Appendix E Proof of Proposition 6 This proof consists of three parts. By conjectring that dealer W will trade in the interdealer market only if the cstomer order is greater than a threshold size 0 < s < 1, we first establish an ordinary differential eqation (ODE) as the necessary condition for dealer L s eqilibrim trading strategies. Then we explicitly solve for a linear soltion to the ODE. In the last step, we verify the existence of sch a threshold cstomer order size, s.

45 Inter-Dealer Trading in Financial Markets 45 In the following we characterize dealer i s optimal trading strategy, x i ( p) (we sometimes write dealer i s pward-sloping residal spply crve as h( p)) in response to the strategy sed by the other dealers, x j ( p)(8 j 6¼ i ). We se g(z) and G(z) to denote the pdf and cdf for the cstomer order size z when it falls within [ s, 1]. We can write dealer i s ncertain trading profit as follows: p i ¼ ũx i ð pþ TP i ;i 6¼ W; where his total payment is: and Z xið pþ TP i ¼ð1 aþpx i ð pþþ 0 " X N ¼ð1 aþpx i ð pþþ j¼1 ð1 aþpx i ð pþþ½a BŠ: pðqþdq Z xjð pþ 0 pðqþdq XN j6¼1 Z xjð pþ We note the following relations for se in sbseqent calclation: BA Bz ¼ XN j¼1 BB Bp ¼ XN j6¼i p Bx j Bz p Bx jð pþ Bp ¼ p BðQ þ zþ Bz where we make se of the market clearing condition: ¼ p; 0 # pðqþdq Bhð pþ ¼ p Bp ; ða-18þ Q þ z ¼ x w þ XN We assme that dealer i chooses his optimal trading strategy by maximizing the following derived mean-variance tility fnction: E z i6¼w x i : ūhð pþ rt 1 2 h2 ð pþ TP i : Defining A(z) as the state variable and p(z) the control variable, we can analyze the problem sing the following Lagrangian: L ¼ gðzþ ūhð pþ rt 1 2 h2 ð pþ AðzÞþBð pþ þ lp:

46 46 Jornal of Bsiness The optimality condition is: 0 ¼ BL Bp Bhð pþ ¼ gðzþ½ū p rt 1 hð pþš þ l: Bp ða-19þ The adjoint eqation is: Bl Bz ¼ BL BA ¼ gðzþ; ða-20þ with the transversality condition lðsþ ¼ 1: Using the transversality condition, the adjoint eqation can be integrated to obtain lðzþ ¼ ½1 GðzÞŠ: ða-21þ Combining Eqs. (A-19) and (A-21), we have: which is eqivalent to: ½ū p Bhð pþ hð pþš Bp ¼ 1 GðzÞ ; ða-22þ gðzþ X N j6¼i x 0 jð pþ ¼ ½1 GðzÞŠ=gðzÞ ū p x ið pþ : ða-23þ Motivated by dealer W s se of a linear bidding strategy, we now search for a linear strategy eqilibrim of the form: x k ¼ m gp; ða-24þ for dealers k. With this, the market clearing condition is: Q þ z ¼ x W þ x k þðn 2Þðm gpþ: The hazard ratio for z over the interval [ s, 1] is: 1 GðzÞ gðzþ ¼ ð1 zþ ¼½Q þ 1 x W x k ðn 2Þðm gpþš: ða-25þ

47 Inter-Dealer Trading in Financial Markets 47 Plgging Eqs. (22), (A-24), and (A-25) into Eq. (A-23), we obtain: 1 þðn 2Þg ¼ Q þ 1 ū p ū p ðn 1Þðm gpþ ðq þ x ; kþ which can also be written as: x k ¼ 1 ð1þ s ðū pþþsðn 1Þðm gpþ sðq þ 1Þ Q ; with 1 s ¼ : 1 þðn 2Þg Becase of the symmetry among the N 1 dealers other than dealer W, we mst have: g ¼ 1 m ¼ 1 1 þ s 1 þ s þ sðn 1Þg ; ū þ sðn 1Þm sðq þ 1Þ Q : The soltions to the above eqations are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þð1 þ Þ 2 þ ðn 1Þ 2 ð1 þ Þ 2 4 g ¼ 2ðN 2Þ ; m ¼ g ū Q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: ðn þ 1Þð1 þ Þþ2 þ ðn 1Þ 2 ð1 þ Þ 2 4 Given the above soltions, it can be checked that the following choice of threshold cstomer order size: 2 s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð0; 1Þ; ðn 1Þð1 þ Þþ2 þ ðn 1Þ 2 ð1 þ Þ 2 4 is a niqe nmber that satisfies the reqirements that Q + z x W and x k 0, 8k 6¼ W for z 2 [ s, 1]. This last statement completes the proofs of Propositions 5 and 6.

48 48 Jornal of Bsiness Appendix F Proof of Proposition 8 We can analyze the selling dealer s maximization problem: maxu n ¼ðū x nþ1q nþ1 Þðq nþ1 q n Þ rt 1 q n 2 ðq nþ1 q n Þ 2 þ q n p n ðq n Þ; which provides the following optimal qantities: 36 q n ¼ 1 þ x nþ1 1 þ 2l n q nþ1 d n q nþ1 ; U n ¼ q nþ1 ū 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ q nþ1 : ða-26þ Using Eq. (36), the eqilibrim strategy in the last stage of the game (where one dealer sells to m 1 other dealers) is: x k ðxþ ¼g d ðxþðū pþ; where g d ðxþ ¼ m 3 : ðm 2Þ½ðm 1Þx m þ 1Š Ths, the eqilibrim price there is: with p m ¼ ū l mq m ; l m ¼ ðm 2Þ½ðm 1Þx m þ 1Š ðm 1Þðm 3Þ ða-27þ In addition, the expected tility for those bidding dealers is: V m ¼ l m x m 1 2ðm 1Þ m 1 q2 m : Using Eq. (A-26), the above can be rewritten as: 2ðm 1Þ V n ¼ V m ¼ l m x m 1 ðm 1Þ ðq mþ1d m Þ 2 ¼...¼ l m x m 1 2ðm 1Þ ðm 1Þ ðq nþ1d m d mþ1...d n Þ 2 : 36. The second-order condition, 1 + 2l n > 0, is satisfied since l n > 0 in eqilibrim.

49 Inter-Dealer Trading in Financial Markets 49 Note that the expected tility for the nonwinning dealers in each stage mst be the same. Ths, the price at the (n + 1)-th stage can be determined as follows: p nþ1 ðq nþ1 Þ¼ 1 q nþ1 ðu n V n Þ ¼ ū 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ l m x m 1 2ðm 1Þ m 1 q nþ1 ðd m d mþ1...d n Þ 2 q nþ1: Ths, we derive the following iteration formla for the liqidity parameter: l nþ1 ¼ 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ 2ðm 1Þ þ l m x m 1 m 1 ðd m d mþ1...d n Þ 2 : ða-28þ Becase the optimal qantity q n is linear in q n +1 (the qantity traded in the previos rond; see Eq. (A-26)), q n mst be linear in z, the original cstomer order. Hence, the information pdating is linear: Therefore, we have: x n ¼ Covð; q nþ ¼ Covð; d nd nþ1...d N zþ Varðq n Þ Varðd n d nþ1...d N zþ 1 Covð; zþ ¼ ¼ rt 1 x Nþ1 : d n d nþ1...d N VarðzÞ d n d nþ1...d N x n ¼ x Nþ1 d n d nþ1...d N : ða-29þ Using the last relation, we have: d n ¼ x nþ1 x n : ða-30þ From Eq. (A-26), d n is defined as: d n ¼ 1 þ x nþ1 1 þ 2l n : ða-31þ Ths, x n +1 can be solved from the last two relations as: x nþ1 ¼ x n 1 þ 2l n x n ; ða-32þ

50 50 Jornal of Bsiness which provides the iteration formla for the information parameter of the model. Note that, sing Eq. (A-30), we can also rewrite Eq. (A-28) as: l nþ1 ¼ 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ þ l m x m 1 2ðm 1Þ x 2 nþ1 : ða-33þ m 1 x m Appendix G Proof of Corollary 8 We assme that x n > 0, 8 n. Using Eq. (A-32), we note that the reslt x n +1 < x n follows directly from l n > x n. Ths, we first prove that l n > x n by indction. From Eq. (A-27), it is clear that l m > c m. Now assme this is tre for stage n, i.e., l n > x n. Then, we also have l n > x n > x n+1. Using this and Eq. (A-33) (the last term of which is easily shown to be positive), we obtain the following: l nþ1 > 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ > 2l n þ 4l n x nþ1 ð2l n Þx nþ1 2ð1 þ 2l n Þ ¼ 2l n þ 4l n x nþ1 x 2 nþ1 2ð1 þ 2l n Þ ¼ l n 1 þ x nþ1 1 þ 2l n ¼ l n x nþ1 x n : From this, it follows that: l nþ1 x nþ1 > l n x n > 1; which proves that l n +1 > x n +1. To prove l n +1 < l n, we introdce the following variable: c n 2l n x n : Using Eqs. (A-31) and (A-33), we have: l nþ1 ¼ 2l nð1 þ 2x nþ1 Þ x 2 nþ1 2ð1 þ 2l n Þ þ B m d n...d m ¼ 2l nð1 þ x nþ1 Þþx nþ1 ½ð1 þ 2l n Þ ð1 þ x nþ1 ÞŠ 2ð1 þ 2l n Þ ¼ l n d n þ x nþ1 2 ð1 d nþþb m d n...d m ¼ d n 2 ð2l n x nþ1 Þþ x nþ1 2 þ B md n...d m ; where B m > 0isshortforl m x m 1 (2m 2)] (m 1). þ B m d n...d m

51 Inter-Dealer Trading in Financial Markets 51 Ths, 2l nþ1 x nþ1 ¼ d n ð2l n x nþ1 Þþ2B m d n...d m ; ¼ d n ð2l n x n Þþd n ðx n x nþ1 Þþ2B m d n...d m ; or: c nþ1 ¼ d n c n þ d n ðx n x nþ1 Þþ2B m d n...d m; From Eqs. (A-30) and (A-32), Ths, c nþ1 ¼ d n ¼ x nþ1 x n ¼ 1 1 þ c n < 1: c n 1 þ c n ð1 þ x nþ1 Þþ2B m d n...d m : To prove by indction, we can explicitly verify that c m+1 < c m. Frthermore, if we make the indction assmption c n < c n 1,the above eqation can be sed to show: c nþ1 < c n 1 1 þ c n 1 ð1 þ x n Þþ2B m d n 1...d m ¼ c n : Having proved c n+1 < c n, 8n, we can rewrite it as: or 2l nþ1 x nþ1 < 2l n x n ; 2ðl nþ1 l n Þ < x nþ1 x n < 0: This completes the proof that l n+1 < l n. Appendix H Proof of Proposition 9 Sbstitting Eq. (A-27) into Eq. (A-32), we have: x mþ1 ¼ 1 m 1 m 3 þ m2 2m 1 ðm 1Þðm 3Þx m < m 3 m 1 ; as long as x m is positive and m 4. Using Corollary 8, we then get x = x N +1 < x N <... < x m +1 < (m 3) (m 1).

52 52 Jornal of Bsiness Ths, for any given seqential action model, we can find an initial information parameter that violates the above relations (e.g., by choosing x (m 3) (m 1)). This shows that a market breakdown will occr when there is a severe adverse selection problem at the cstomer-dealer trading stage. As for the second statement, we observe that, with more trading ronds (smaller m vales), there are more x vales that satisfy the market breakdown relation x > (m 3) (m 1). Appendix I Proof of Proposition 10 The analysis of inter-dealer trading with a limit-order book nder conditions of asymmetric information parallels the previos analysis withot informed trades. First, assme that dealer L gets a positive qantity allocation from inter-dealer trading; then dealer W ses the following eqilibrim strategy (the derivation is very similar to that in Appendix IV): 8 < ū x x W z p ð p; zþ ¼ if z 2½s; 1Š; ða-34þ : z if z 2½0; sš; where s2½0; 1Š will be determined later. As in the analysis in Appendix V, dealer i s (i 6¼ W ) strategies are described by the following first-order condition: X N j6¼i x 0 ½1 GðzÞŠ=gðzÞ jð pþ ¼ ū x z p rt 1 x ið pþ 1 z ¼ ū x z p rt 1 x ið pþ : Conjectring a linear soltion x i ( p) =m gp, from the market clearing condition we can solve for: z ¼ ū p þ rt 1 ðn 1Þðm gpþ ð1 þ xþ : Sbstitting the above into dealer i s first-order condition: where 1 s ¼ 1 ū xrt 1 z p ū x ðn 1Þðm gpþ z p ; rt 1 1 s ¼ ðn 2Þg þ 1 : x i

53 Inter-Dealer Trading in Financial Markets 53 Collecting terms to match the original conjectre, we have: g ¼ 1 þ s x x þ 1 ðn 1Þsg 1 þ s m ¼ 1 þ s Therefore, the soltions are: ū ½1 þ rt 1ðN 1ÞgŠ x x þ 1 ; ða-35þ ½ū þ rt 1ðN 1ÞmŠ s þ sðn 1Þm: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðN 2Þ Nx þ 4NðN 2Þð1 þ xþþn 2 x 2 g ¼ 2ðN 2Þ ð1 þ NxÞ ;! 1 þ s ū sð1 þ xþ m ¼ ð1 þ xþþðn 1Þðrt 1 x sþ : The reqirements that z x W and x i 0, 8i 6¼ W are satisfied with the following choice of threshold cstomer order size: s ¼ gū m : ð1 þ xþg Finally, we can verify that l > 0. Ths, the second-order condition is always satisfied and the soltion constites an eqilibrim strategy. References Asbel, L., and Cramton, P Demand Redction and Inefficiency in Mlti-Unit Actions. University of Maryland Working Paper. Back, K., and Zender, J Actions of Divisible Goods: On the Rationale for the Treasry Experiment. Review of Financial Stdies 6: Bhattacharya, U., and Spiegel, M Insiders, Otsiders, and Market Breakdowns. Review of Financial Stdies 4: Biais, B., Martimort, D., and Rochet, J. C Competing Mechanisms in a Common Vale Environment. Econometrica 68: Evans, M. D. D., and Lyons, R. K Order Flow and Exchange Rate Dynamics. Jornal of Political Economy 110: Glosten, L Insider Trading, Liqidity, and the Role of Monopolist Specialist. Jornal of Bsiness 49: Glosten, L Is the Electronic Open Limit Order Book Inevitable? Jornal of Finance 49: Glosten, L., and Milgrom, P Bid, Ask, and Transaction Prices in a Specialist Market with Heterogeneosly Informed Traders. Jornal of Financial Economics 13:

54 54 Jornal of Bsiness Goodhart, C., Ito, T., and Payne, R A Stdy of the Reters D Dealing System in the Microstrctre of Foreign Exchange Markets, edited by J. Frankel, G. Galli, and A. Giovannini, University of Chicago Press: Gpta, M., and Lebrn, B First Price Actions With Resale. Economics Letters 64: Haile, P Actions with Private Opportnities and Resale Opportnities. Forth-coming in Jornal of Economic Theory. Hansch, O., Naik, N., and Viswanathan, S Do Inventories Matter in Dealership Markets? Evidence from the London Stock Exchange. Jornal of Finance 53: Ho, T., and Stoll, H The Dynamics of Dealer Markets nder Competition. Jornal of Finance 38: Kyle, A Continos Actions and Insider Trading. Econometrica 53: Kyle, A Informed Speclation with Imperfect Competition, Review of Economic Stdies 56: Lyons, R Tests of Microstrctral Hypotheses in the Foreign Exchange Market. Jornal of Financial Economics 39: Lyons, R. 1996a. Foreign Exchange Volme: Sond or Fry Signifying Nothing? In J. Frankel, G. Galli, and A. Giovannini (eds.) The Microstrctre of Foreign Exchange Markets. Chicago: University of Chicago Press. Lyons, R. 1996b. Optimal Transparency in a Dealer Market with an Application to Foreign Exchange. Jornal of Financial Intermediation 5: Lyons, R A Simltaneos Trade Model of the Foreign Exchange Hot Potato. Jornal of International Economics 42: Milgrom, P., and Weber, R A Theory of Actions and Competitive Bidding. Econometrica 50: Naik, N., Neberger, A., and Viswanathan, S Trade Disclosre Reglation in Markets With Negotiated Trades. Review of Financial Stdies 12: Naik, N., and Yadav, P Risk Sharing among Dealers: Evidence of Inter-Dealer Trading on the London Stock Exchange. London Bsiness School Working Paper. Reiss, P., and Werner, I Does Risk Sharing Motivate Inter-Dealer Trading? Jornal of Finance 53: Röell, A Liqidity in Limit Order Book Markets and Single Price Actions with Imperfect Competition. Princeton University Working Paper. Viswanathan, S., and Wang, J Market Architectre: Limit-Order Books verss Dealership Markets. Jornal of Financial Markets 5: Vogler, K Risk Allocation and Inter-Dealer Trading. Eropean Economic Review 41: Wang, J., and Zender, J Actioning Divisible Goods. Economic Theory 19: Werner, I A Doble Action Model of Inter-Dealer Trading. Stanford University Working Paper. Wilson, R Actions of Shares. Qarterly Jornal of Economics 93:

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