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.