An Intraday Pricing Model of Foreign Exchange Markets

Transcription

1 WP/03/5 Revised: /3/06 An Inraday Pricing Model of Foreign Exchange Markes Rafael Romeu

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3 003 Inernaional Moneary Fund WP/03/5 Revised: /3/06 IMF Working Paper Inernaional Capial Markes Deparmen An Inraday Pricing Model of Foreign Exchange Markes Prepared by Rafael Romeu Auhorized for Disribuion by Donald J. Mahieson June 003 Absrac The views expressed in his Working Paper are hose of he auhor(s) and do no necessarily represen hose of he IMF or IMF policy. Working Papers describe research in progress by he auhor(s) and are published o elici commens and o furher debae. Dealers learn abou asse values as hey se prices and absorb porfolio flows. These flows causes invenory imbalances. Previous work argues ha dealers deviae from heir esimaes of asse values o induce flows ha unwind invenory imbalances. This sudy models dealer price-seing using muliple insrumens o smooh invenory imbalances and updae priors abou asse values. This approach shows ha canonical models in which price-seing is he only insrumen for invenory conrol, and incoming order flow is he only source of asymmeric informaion, are misspecified. Thus, esimaes of canonical models rejec prediced asymmeric informaion and invenory effecs because of omied and exraneous variables. These esimaions miss informaion from sources oher han incoming order flow, and hey overemphasize price shading in managing invenories. Esimaes of he model presened suppor hereofore elusive invenory and asymmeric informaion effecs. Price shading is found o have smaller role in invenory managemen and informaion effecs are shown o be sronger han previously esimaed. Addiionally, his approach yields direc measures of he srucural liquidiy cos parameers in he model akin o Kyle s Lambda. For example, esimaes presened sugges ha a sandard $0 million incoming purchase pushes price up by roughly one basis poin, and dealers expec o immediaely lay-off one-hird of every incoming order. JEL Classificaion Numbers: G5; F3 Keywords: Foreign Exchange, Microsrucure, Inernaional Macroeconomics Auhor s Address: rromeu@imf.org I am graeful o Roger Beancour, Michael Binder, Rober Flood, Carmen Reinhar, and John Shea for guidance; and also o Richard Lyons for guidance and daa. Thanks also o Richard Adams, Juan Blyde, Alain Chaboud, Giovanni Dell Ariccia, Andrew DePhillips, Larry Evans, Marin Evans, Sefan Hubrich, Andrei Kirilenko, Don Mahieson, José Pineda, Eswar Prasad, Mike Pries, Pedro Rodriguez, Francisco Vázquez, Ciibank FX, seminar paricipans a he Board of Governors, he IMF Research Deparmen, he Universiy of Maryland Economics Deparmen and R.H. Smih School of Business Finance Deparmen, and he Bank of Canada for commens.

4 - - Conens Pages I. Inroducion... 3 II. Inraday Price Discovery In Markes wih Muliple Dealers... 6 A. The Marke... 7 B. The Informaion Srucure... C. The Dealer s Opimizaion... D. A Comparison wih Exising Models... 4 III. Daa Consideraions... 5 IV. Esimaion... 7 V. Conclusions... APPENDIXES I. Model Soluion Invenory Carrying Cos Dealer s Beliefs The Dealer s Problem... 3 The Informed Trader s Problem REFERENCES Figures The Timing of he Model... 4 Canonical Models Informaion Effec: Incoming Order Flow and Price... 5 New Informaion Effec: Cumulaive Invenory Shocks and Price... 5 Tables Invenory Conrol: Firs Five Enries of Lyons (995) Daase... 6 Descripive Saisics... 6 Observed Incoming Order Flow and Ougoing Trades... 6 Informaion Effec: Daily Correlaion of Order Flow Variables wih Price... 7 Sysem of Esimable Equaions... 7 Price Formaion wih Muliple Insrumens... 8 Canonical Model Esimaes... 8 The Impac of a Federal Reserve Inervenion... 9

5 - 3 - I. INTRODUCTION The evidence supporing a igh relaionship beween a marke s absorpion of porfolio flows and is asses reurns is mouning. A he very leas, i implies ha asse reurns depend on how dealers inerac wih each oher and wih end users. The quesion now is how long marke rading affecs asse reurns. Assuming ha asse fundamenals follow a random walk, here could be permanen effecs if rading reveals new informaion. For example, dealers aggregaing porfolio flows may also aggregae informaion dispersed in he economy. Conversely, he marke s emporary indigesion from absorbing large porfolio shifs may imply ransiory effecs, as in microsrucure invenory models. A he level of he individual dealer, however, here is surprisingly lile (if any) evidence supporing heoreically prediced invenory effecs. This paper presens a new model of asse rading ha shows evidence of boh informaion and invenory effecs a he individual dealer level. The empirical resuls link porfolio flows o asse prices a he highes resoluion, and provide direc esimaes of he cos of liquidiy, asymmeric informaion, and invenory effecs. The resuls suggess ha previous models have underesimaed, if no missed or rejeced hese effecs in markes wih muliple dealers, such as bond or foreign exchange markes. An example illusraes why. Consider a foreign exchange (FX) dealer who is rading U.S. dollar-euro and waching he price of he currency flucuae hroughou he day. Assume ha he dealer is consrained wih a finie invenory (or, equivalenly, invenory coss). If random-walk asse values drive incoming rades, she mus respond wih an invenory-managemen sraegy or exhaus her supply. Pas models sugges ha his dealer diver her price away from he equilibrium full-informaion value o induce rades ha compensae for invenory imbalances. Bu changing prices o induce rades equaes o inenionally selling low or buying high. Wha if here is anoher way? In markes wih muliple dealers she can call oher dealers and unload her invenory imbalances on hem. This allows he dealer anoher insrumen for managing invenory and learning abou asse values. In his example, he dealer s insrumens are o change prices o induce incoming rades (i.e., incoming order flow), or o call ohers and use ougoing rades (i.e., ougoing order flow). Canonical singledealer models fail o consider how his affecs price formaion. Canonical modeling of dealer price-seing is grounded in he wo general microsrucure-pricing effecs. The firs is he invenory effec, in which he dealer mus manage a finie sock of he asse agains a demand ha responds o a random-walk fundamenal value. 3 In his siuaion, if he dealer passively fills orders, he probabiliy of a Examples in equiy markes include Froo, O Connell and Seasholes (00), and Froo and Ramadorai (00a). Examples in foreign exchange markes include Evans and Lyons (00), Froo and Ramadorai (00b), and Rime (00). Examples in bond markes include Massa and Simonov (00). 3 For example, Soll (978), Amihud and Mendelson (980), Ho and Soll (98, 983), O Hara and Oldfield (986) among ohers.

6 - 4 - sock ou is uniy. Hence, invenory models argue ha dealers changes prices away from he expeced asse value o induce rades ha unwind undesired posiions. The second effec is he asymmeric informaion effec, where, for example, he dealer faces a marke where some insiders have informaion abou he asse s liquidaion value. 4 Recognizing ha incoming order flow parially reflecs his informaion, he dealer changes her price accordingly. When muliple increasing-marginal-cos insrumens are available for managing invenory, as in he example, he dealer opimally spreads her invenory managemen across all of hem. Furhermore, communicaion wih oher dealers hrough ougoing calls is as informaive as communicaion hrough incoming rades. The dealer may use his informaion o updae her prior beliefs abou asse values and adjus invenory levels. Hence, par of observed invenory and price changes may be correlaed wih innovaions in informaion, bu be unrelaed o eiher invenory carrying coss or incoming order flow. This paper models his phenomenon in he conex of foreign exchange markes. In he model, he abiliy o make ougoing rades alers boh invenory driven price changes, and learning abou asse values. Ignoring ougoing orders leads o boh neglecing he role of informaion learned from hese orders and overemphasizing price changes in invenory managemen. Modeling price seing wihou considering hese effecs explicily leads o misspecified ess of informaion and invenory effecs. While empirical evidence of asymmeric informaion based on canonical dealer pricing models abounds, 5 ess for invenory effecs have failed. For example, Madhavan and Smid (99) and Hasbrouck and Sofianos (993) rejec expeced invenory effecs in equiy and fuures markes, respecively. Madhavan and Smid (993) only find evidence of unexpecedly long-lived effecs by modeling invenory mean reversion wih shifs in he desired invenory level. Manaser and Mann (996) acually find robus effecs opposie o heoreical predicions. Lyons (995) exends microsrucure models o foreign exchange markes and does find invenory effecs; however, Romeu (005) overurns he Lyons (995) resul supporing canonical models invenory specificaions specifically, invenory and informaion effecs are no simulaneously presen in subsamples. Oher sudies of foreign exchange markes also fail o find invenory effecs, and hence, he evidence supporing hese is a bes a mixed bag. 6 4 For example, Kyle (985), Glosen and Milgrom (985), Admai and Pfleiderer (988), Easley and O Hara (987, 99), among ohers. 5 For example, Hasbrouck (99 a, b), Hasbrouck (988), Madhavan and Smid (99, 993), Lyons (995), Evans and Lyons (00), Yao (998), Bjonnes and Rime (000), Ausubel and Romeu (005), among ohers. 6 In foreign exchange markes Yao (998) and Bjonnes and Rime (000) find no evidence of invenory effecs. The former suggess ha i is due o dealers aversion o revealing heir posiion (or privae informaion) hrough invenory-induced bid shading, whereas he laer sugges ha he inroducion of elecronic brokering is he cause. The model here suggess ha misspecificaion is he cause. More generally, see O Hara (995) on he empirical difficulies of prediced invenory effecs.

7 - 5 - The model presened here ness canonical dealer pricing models, and demonsraes why hey fail empirically. Previous models are misspecified insofar as hey neglec boh alernaives o conrolling invenory hrough price-induced flows, and alernaive sources of marke informaion. The model presened uses decenralized markes wih muliple dealers o underscore he impac of hese alernaives on price seing. A is hear is he idea ha dealers exploi every alernaive when rebalancing porfolios, raher han relying solely on price-induced order flow o change heir porfolio composiion. As dealers face increasing marginal losses for inducing flows hrough price shading, hey urn o oher mehods of unloading unwaned posiions. Compeiive dealer markes offer a clear opporuniy o observe his phenomenon. Previous work on price formaion in decenralized markes, boh a he dealer and a he marke-level, suppor he model presened here. For example, in discussing invenory conrol, O Hara (995) singles ou foreign exchange dealers abiliy o lay off orders on one anoher. A he dealer level, he Ho and Soll (983) framework permis inerdealer rading (alhough i does no arise in he model soluion) which is he basis of he approach presened here. Moreover, Romeu (005), Lyons (995) and Mello (996) all speculae ha nonlineariies in dealer pricing models relaed o iner-ransacion ime or muliple invenory conrol insrumens may be presen in canonical esimaions of dealer behavior boh of which are cenral o he model presened here. A he general-equilibrium level, he ho poao model of Lyons (997) favors dealer pricing wih muliple insrumens. In ha framework, high rading volume in he FX marke resuls from dealers passing on invenory imbalances. Marke makers in all ypes of markes have an incenive o minimize guaraneed losses from inducing rades via price changes, no jus in FX. While laying off invenory on ohers is an alernaive in muliple dealer seings such as FX or bond markes, here is evidence ha similar phenomenon exis in more cenralized markes as well. For example, Madhavan and Sofianos (997) find ha New York Sock Exchange (NYSE) specialiss engage in selecively rading o balance invenory. Hence, previous equiy marke sudies possibly overemphasize he role of prices in invenory managemen and miss oher invenory effecs. In addiion, if previous models accoun perfecly for invenory coss, hey sill overlook price changes resuling from new informaion ha alernaive insrumens yield. Accouning for boh hese effecs presens more complex behavior, where he marke maker is using muliple insrumens o boh manage invenory and updae priors. Empirical ess presened here suppor he model and offer several novel resuls. For example, asymmeric informaion effecs driving price changes are likely wice as large as previously esimaed no only is he price response o order flow effec larger, bu here are more insrumens. One can graphically compare prices wih he new informaion signals ha he dealer sees. Invenory pressure on prices is lower, perhaps as low as one-fourh previous esimaes. This makes sense since muliple insrumens will keep invenory managemen coss a he lower end of an increasing marginal cos curve. Afer conrolling for invenory

8 - 6 - and informaion effecs, he base bid-ask spread is wider han previously esimaed, and saisically indisinguishable from he marke spread convenion (3 pips). 7 When seing prices, he dealer plans o rade ou abou one-hird of he difference beween her curren and he opimal invenory posiions. A sandard ($0 million) incoming rade moves he dealer s price less han pips or $,000, and he expeced cos of execuing an ougoing rade is abou double ha amoun. Accordingly, he dealer is observed acceping incoming rades abou nine imes more ofen han ougoing rades, and five imes more volume is handled hrough incoming rades. A Federal Reserve inervenion of $300 million in he daa emporarily moves prices abou 6.7 pips per $00 million. 8 This increases he asymmeric informaion impac of rades on price changes by fifeen percen, which suggess ha order flow becomes more informaive as he marke learns of he inervenion. Tha is, he esimaes of how much our dealer shades her price in response o invenory imbalances is fairly robus o inervenion. This, aken wih he resul on asymmeric informaion, suggess ha he cenral bank inervenion was ransmiing informaion raher han inducing porfolio balance effecs. Finally, he base spread ighens by five percen when he inervenion is included in he esimaion. While boh ransiory and permanen effecs are presen in he daa, he resuls sugges a sronger permanen impac of porfolio flows on prices. Wih muliple insrumens, marke paricipans share inraday invenory more efficienly. Tha is, dealers exhaus he gains from sharing a large invenory posiion more quickly and wih less price impac in his model. As a resul, he ransiory effecs of invenory imbalances are presen, albei less imporan in deermining inraday price changes han esimaed previously. Furhermore, muliple insrumens faciliae a more efficien aggregaion of he dispersed informaion embedded in order flow, which can be inerpreed as favoring permanen price movemens. The paper is organized as follows. Secion II describes he heoreical framework and he model soluion, which is deailed in he Appendix I. Secion III shows empirical esimaes, ess of he model, and discusses inervenion effecs. Secion IV concludes. Esimaion deails are in Appendix II. II. INTRADAY PRICE DISCOVERY IN MARKETS WITH MULTIPLE DEALERS This secion generalizes he Madhavan and Smid (993) framework in which an uninformed marke maker wih invenory carrying coss ses prices in a marke wih informed agens. Opimally, he marke maker exracs informaion from arriving order flow, and ses prices o induce invenory-balancing rades. The Madhavan and Smid (993) framework is 7 A pip is he smalles price incremen in a currency. The value depends on he currency pair. The daa used here are dollar/deusche mark, so a pip is DM This amoun observed concords wih sudies of inervenion, e.g. Evans & Lyons (999) esimae 5 pips and Dominguez and Frankel (993) esimae 8 pips per $00 million.

9 - 7 - represenaive of he canonical microsrucure hypohesis of price formaion. In acualiy, however, his absracion may miss imporan alernaives available o dealers in compeiive dealer markes, such as bond and FX markes. For example, an FX dealer only ses prices when she passively receives an order (i.e., anoher dealer iniiaes he rade). 9 This priceseing is he focus of his sudy. Besides seing prices, however, she can iniiae inerdealer bilaeral dealer rades, iniiae brokered dealer rades, or iniiae IMM Fuures rades, as well as receive informaion from hese, or her sales and floor managers or fellow raders, among oher sources. A no ime does she se inerdealer prices under any of hese alernaives; however, hey may indirecly affec her price seing. I is inracable o model all hese alernaives explicily. 0 Furhermore, he daa available (invenory levels, incoming orders, and heir corresponding prices) would limi empirical ess of any such model. These limiaions wihsanding, he dealer modeled here has wo insrumens for balancing invenory: inducing order flow hrough price changes, and iniiaing ougoing rades wih ohers a heir prices. She also has wo insrumens for updaing priors: informaion refleced in incoming quaniies, and informaion refleced in unplanned (a he ime of price-seing) ougoing quaniies. The opimal price updaes priors from boh informaion sources and spreads invenory coss across boh insrumens, hence he misspecificaion in canonical models. The following secions formalize his modeling approach. Subsecion A describes he model seing: he marke, invenory, capial, and informaion variables. Subsecion B shows he opimal updaing using muliple informaive signals. Subsecion C shows he opimal invenory managemen, and he model soluion. Subsecion D shows he model nesing previous work, and heir misspecificaions. Proofs are in he appendix. A. The Marke Consider an economy where a dealer holds a porfolio of hree asses. She only makes markes in he firs, a risky asse wih a full informaion value denoed by v, which evolves as a random walk. Wrie his value as: v = v + θ, θ ~ N(0, σv). () 9 An exensive descripion of he Foreign Exchange (FX) marke s insiuional make-up can be found in Lyons (00). FX is raded bilaerally, over he couner, and privaely, via compuer ing sysems called Reuers Dealing. There are also elecronic brokers similar o bullein boards, provided by Reuers or EBS. Mos large rades are done via he Reuers Dealing sysem, and he spread is fixed by convenion. 0 Tha is, he reurn in economic insigh o modeling compeiive dealers is likely o be small relaive o he cos of overcoming he inracabiliy, paricularly in erms of he necessary assumpions. See O Hara (995) on precisely his inracabiliy.

10 - 8 - The second is an exogenously endowed risky asse ha is correlaed wih he firs, and generaes income y. The hird is capial, he risk-free zero-reurn numeraire, denoed by K. The disribuion of he wo risky asses is: v v σv σ vy N, y. () 0 σvy σ y The dealer s oal wealh is: W = vi + K + y, (3) Wih I being he dealer s invenory or risky asse posiion. The marke is open for =,,..., T periods. The erminal dae T is unknown, however, a he beginning every period = T wih probabiliy ( ρ). Hence, every period he probabiliy ha he marke closes is ( ρ), a which ime he dealer liquidaes her posiion and pays a invenory carrying cos. Wih probabiliy ρ, T, so he dealer engages in rading aciviies, pays he invenory carrying cos, and goes on o he nex period. Figure (page 4) depics he iming of he model. The oal change in he dealer s invenory from one even o he nex occurs in wo sages. In he firs sage, he dealer faces an incoming order (denoed by q j ) and knows her invenory (denoed by I ). Par of q j comes from informed dealers who know he full informaion value ( v ). The informed par of q j, denoed by Q, is driven by differences beween he dealer s price, denoed p, and he asse value v : Q = δ ( v p), δ > 0. (4) The res of he incoming order is an uninformed or liquidiy componen, denoed by X : X N( 0, σ X ). (5) One can hink of he uninformed as quaniies demanded by paries no monioring he markes or consrained o rade independen of price, for reasons no modeled here. The dealer only observes he aggregae order, (q j ), and ses he price. Hence, he incoming order flow is: q = Q + X = δ ( v p ) + X. (6) j When our dealer ses her price a (incoming) rade, she knows she can also call ou ohers and iniiae ougoing rades (denoed q ). These ougoing rades are depiced in he ou upper box of Figure. q indicaes our dealer s desired ougoing quaniy in expecaion, and condiional on informaion available a he ime of price seing. Because he dealer has Noe ha his is a one-period-ahead condiional disribuion, as he uncondiional disribuion would have a ime-varying variance. The invenory carrying cos, shown below, follows Madhavan and Smid (993). I is a cos proporional o he variance of he dealer s wealh.

11 - 9 - ou his ool of ougoing rades ( q ) available, she does no conrol invenory solely hrough ou price induced order flow. In his sense q capures he planned amoun he dealer prefers o lay off by iniiaing rades raher han by shading price o induce incoming rades. The role of ougoing rades ( q ou ) in price formaion is a deparure from canonical dealer models. In considering muliple dealer markes, i is an empirical realiy one ypically has daa on rade prices for only a subse of all dealer rades (his is paricularly because hey are relaively unregulaed wih much lower reporing requiremens). We wan o model he subse of available rades o he fulles exen possible, while a he same ime recognizing he role of rades no in ha subse. In his case, prices, invenories, and quaniies raded are available only for incoming rades. Invenory, however, summarizes all quaniies: incoming and ougoing. Tha is, we a leas have quaniy informaion for inegraing he rades wihou price daa ino he analysis of he available daa. Thus, we decompose he oal change in invenory from one rade o he nex ino hree componens: he observed incoming rade (q j ), he expeced ougoing rade ( q as shown in Figure. ou ), and unexpeced quaniy shocks o invenory, Denoe he unexpeced quaniy shocks o invenory as γ. While our dealer is rading ou ou q, hese exogenous quaniy shocks change her invenory beyond he ougoing rade ( q ) planned a he ime of seing prices. The source of hese shocks can be unplanned rading wih cliens of our dealer s bank (her employer), oher bank dealers, brokered rading, he ou rading floor manager, and so on. Accordingly, he oal quaniy ( q + γ ) will be he invenory change apar from he incoming rade (q j- ) from - o. Hence, las even s invenory (I - ), adjused for he las incoming rade (q j- ), as well as he oal realized ou ougoing quaniy ( q + γ ), yields nex even s invenory (I ). An example using acual dealer ransacions helps moivae he key assumpions ou regarding q and γ. Table (page 6) shows he firs five incoming rades received by a NY based foreign exchange dealer on a given rading day (hese daa are discussed in deail below). The firs column indexes he rades according o heir order of arrival; he second column shows he price se by he dealer a each incoming rades. The nex columns show incoming order flow, followed by he invenory a he beginning of he rade. The las column shows q ou + γ, which are observed joinly. Consider, for example, he hird incoming rade, which was a sale o he dealer of $8.5 million. A he ime of he rade, he dealer was long $ million, as refleced in her invenory. Canonical models of price formaion assume ha incoming orders are he only insrumen by which a dealer can adjus invenory levels and updae prior informaion. If one assumes ha his were he case, and since he dealer buys $8.5 million, her invenory a enry four should be $9.5 million long (he nex incoming rade). Insead, he dealer is shor $.5 million a enry four, which implies ha her invenory declined by $30.5 million beween he hird and he fourh rade. ou This decline is refleced in he las column, q + γ. I capures he invenory evoluion ha incoming order flow did no generae. This column is expressed as he sum of wo

12 - 0 - ou componens because q reflecs he opimal amoun ha he dealer should rade given he informaion available a he ime of he incoming rade. I is a firs order condiion. Any ou deviaion from q mus be a resul of new informaion, and is refleced in γ. Therefore, he par of invenory changes no generaed by incoming rades is he sum of planned and unplanned ougoing rades, q γ. ou + Hence, new informaion and evens occurring in he clock ime beween evens - and are assumed o be driving he quaniy shocks ( γ ) ; The shock γ is informaive ou because afer he dealer chooses her ougoing quaniy ( q ), she should rade his quaniy and nohing else unless new informaion moivaes a revision in he ougoing rade. Tha is, he choice made a - is opimal unil new informaion (a he nex incoming order, q j ) arrives. 3 Hence, he only reason our dealer would deviae from he opimal ougoing quaniy ( q ou ) beween - and is ha new informaion is revealed. For his reason, he evoluion of v can be inferred fromγ, and he oal ougoing quaniy will reflec he ou desired quaniy ( q ) plus he quaniy driven by new informaion ( γ ). γ capures ha informaion in he dealer s decision process beyond sricly wha is derived from incoming order flow, while keeping he analysis racable. 4 In summary, he ideniy ha describes he evoluion of invenory is: ou I I δ ( v p ) X + q + γ (7) In conras, a he ime of seing prices, he dealer s expecaion of nex period s invenory is: E[ I + Ω ] = I q + q. (8) j ou j Our dealer manages invenory because she pays a cos every period ha is proporional o he variance of her porfolio wealh, which includes he cash value of he invenory. One can moivaed his cos, for example, by risk aversion or marginally increasing borrowing coss. Assume ha he dealer incurs a capial charge due o he γ shocks. Tha is, any gains (losses) enering ino he dealer s wealh due o γ are subraced 3 An alernaive o his inerpreaion of γ is ha i could be an (uninformaive) sysemaic facor missing in he analysis of available price changes. This would no inroduce ineresing alernaive economics because dealers can anicipae his fully here is no news in i. 4 Alhough hey include muliple informaive signals, incoming order flow is he only source of privae informaion in Madhavan and Smid (99) or Lyons (995).

13 - - (added) from (o) he dealer s capial, K a a cos v. 5 Incorporaing his charge, a rade he dealer s wealh posiion is given by: W = v E[ I Φ ] + γ + E[ K Φ ] vγ + y. (9) ( ) ( ) This assumpion implies ha he dealer only pays he invenory carrying cos on he expeced wealh, and he invenory carrying cos due o quaniy shocks is canceled by he capial charge. The appendix shows ha he invenory cos is a funcion of he deviaions d from he opimal hedge raio of he risky asses, given by I. This hedge raio opimally smoohes he dealer s wealh, and eners he invenory cos as: d c ( ) = ω σ W = ω φ0 + φ I I. (0) B. The Informaion Srucure Wha is of ineres is how he dealer ses prices, which occurs only in he even of an incoming rade. The incoming rade is, in par, based on he equilibrium asse value, v. The dealer wishes o learn his value, and she will esimae he full informaion value of he asse based on her rading hisory and any publicly available informaion. The appendix shows he soluion o he dealer s learning problem modeled as a raional expecaions consisen Kalman filer. 6 This secion oulines he wo sources of informaion available for learning v and updaing prior beliefs in his model. Denoe he dealer s expecaion of he full informaion value of he risky asse as: Ev Φ = µ. () [ ] The dealer has wo ways of updaing µ and learning abou he full informaion value of he asse v. The firs is he incoming rade, q j. From his incoming quaniy he dealer exracs a signal of he asse value, v. Denoe his signal by s. The second source of 5 This assumpion simply eases he exposiion of he problem a hand, and keeps i in a discree ime framework. As discussed below, γ has a ime-varying variance. This complicaes calculaing he variance of he porfolio his would involve moving he enire model o a coninuous ime framework. Because of he discree-ime arrival process of incoming calls, his would make for a cumbersome soluion wih lile added payoff in relaion o he problem of how dealers se prices on incoming orders. I would no, however, change he model s conclusions regarding price seing wih muliple insrumens. 6 In he empirical esimaion, his sudy uses oal incoming orders (raher han he unexpeced componen) as signals, as in Lyons (995), Madhavan and Smid (99), Yao (998) and ohers. The s represens a funcion ha reflecs he informaion in incoming order flow, and κγ ( ) represens a funcion reflecing he informaion in invenory shocks, consisen wih he approach Hasbrouck (99a), Madhavan and Smid (993), and ohers. Generally, esimaions are robus o eiher approach, as is he case here.

14 - - informaion abou v is he informaion learned while execuing he ougoing rade, which is refleced in a funcion of he invenory shock, κ ( γ ). While boh κ ( γ ) and s are used o updae µ, assumed ha he variance of κ ( γ ) is increasing in he real ime (i.e., clock ime) elapsed beween incoming rades. Tha is, assume ha var( s ) = σ w and var( κγ ( )) = σ w τ, wih τ being he clock ime elapsed beween evens - and. As he appendix shows, his gives an updaing as a funcion of: µ µ = τ s + κ( γ ). () ( + τ) ( + τ) In equaion (), as elapsed iner-ransacion ime ges larger ( τ ) he dealer places he majoriy of he weigh on he incoming order s informaion, s. The longer he ime in beween rades, he less relevan is he informaion from ha ime in relaion o he incoming rade s informaion. Inuiively, () says ha he momen he dealer is seing p, s has jus arrived because i is based on he incoming order iself (q j ). The quaniy shock signal ( κγ ( ) ) also serves o signal he new innovaion, bu i arrives beween - and, and hence i is no assumed o have he same precision as s. Insead i is assumed ha κ ( γ ) s precision decreases (i.e., variance increases) as he clock-ime elapsed from even o increases. As more ime has passed in beween rades, κ ( γ ) has more noise. 7 Finally, he appendix shows ha he esimae of he full-informaion asse value, µ, generaes an unbiased esimae of he liquidiy rade, X. We denoe his saisic as EX [ Ω ] = x. C. The Dealer s Opimizaion Here he problem is se up as a sochasic dynamic programming problem; ~ denoe random variables, and he soluion is given in he appendix. The dealer solves: J( I, x, µ, K ) = E ρ v I + K + y c + ρ J( I, x, µ, K ), (3) max {( )[ ] } ou p, q subjec o he following evoluion consrains: Invenory: i ou E I + Φ = I δ ( µ p) x + q, (4) Noise Trading: i E x + Φ = 0, (5) 7 One migh argue ha as 0, τ he dealer has less ime o carry ou planned ransacions, bu she can always elec o no answer he incoming calls unil he par of planned ransacions she wans done are saisfied. Furhermore, he increasing frequency of incoming calls and shorening of iner-ransacion ime would iself be a source of new informaion for he dealer, as suggesed by Easley and O Hara (99). Indeed, Lyons (995) finds evidence supporing ha longer iner-ransacion clock imes increases he informaiveness of incoming order flow, as inerpreed in his sudy.

15 - 3 - Informaion: i E µ + Φ = µ, (6) Capial: i ou ou E K + Φ = K + pδ ( µ p) + px ( µ + q ) q c (7) Equaions (0), and (3) hrough (7) comprise he opimizaion problem. (4) consrains invenory evoluion. (5) consrains liquidiy rades o be zero in expecaion. (6) consrains he asse o a random walk. (7) consrains he capial evoluion, and specifies ha ou when he dealer rades q, she expecs o pay a price cenered on he full-informaion value, ou and wih a price impac ( µ + q ). capures he price impac of a marginal increase in her ougoing quaniy. Hence he dealer, while no a monopolis in he inerdealer marke, does face a downward sloping demand curve in her rades. Assuming ha he dealer faces when rading ou is similar o assuming ha here is marginal declining revenue from selling o an informed agen (recall ha revenue from he sale is pδ ( µ p) ). Modeling ouside prices explicily requires a general equilibrium framework ha normally mues dealer level pricing effecs. 8 The appendix shows he model soluion o be: d + δ ( β ) ( ) ( ) p = µ + β( ) I I + x; (8) A ( A ) ( + δ ) δ( + δ) ou d q = ( I I ) ( p) x + δ µ + ; (9) d ( + β ) I = I + β ( I I ) x, (0) ( ( ))( ou ψη j β + δ γ ) ψ ( η) γ δ( + δ) + δ ( β ) ( ) p = q + q + + q + + x () Equaion (8) shows he price of he dealer as a funcion of he esimaed asse value, d ( µ ), he deviaion from opimal invenory, ( I I ), and he liquidiy shocks (x ). In (9) he ougoing quaniy shows ha as he price impac of ougoing rades goes o zero, i.e., 0, ougoing rades fully adjuss invenories o he opimal level (in he appendix, A <0 is shown). In his case, he price will depend only on he esimae of v and he liquidiy demand. In equaion (), s is he informaion from incoming order flow (q j ) and he elapsed ime is measured by η = τ + τ. This equaion shows ha he incremen in dealer price conains informaion-driven componens from boh he curren incoming order ( η s ), and he previous invenory shock ( ( η ) γ ), boh weighed by he Bayesian updaing erm, ψ. The ou ( q + γ + q ) erm capures componen of he price change aribuable o invenory pressure i is he change in he invenory. Finally, he dealer changes her price due o he noise-rading componen ( x ). 8 For example, he Evans and Lyons (00) assumes ha dealers submi bids simulaneously and ransparenly, which in equilibrium implies ha prices be based on common informaion only. This paper avoids such resricions because he focus is on inerdealer price dynamics, bu his comes a he expense of he marke-wide price deerminaion of such models.

16 - 4 - Inuiively, he dealer would like o mainain invenory a he opimal level, bu as a marke maker she mus accep incoming orders ha consanly disurb her invenory posiion. ou As incoming orders arrive, she ries o resore balance o her invenory wih q and price changes. Adjusing back o he opimal level I d ou via q implies absorbing he coss from he ougoing order s price impac ( ). Adjusing invenories via price induced orders implies absorbing he cerain loss o he informed dealers, via δ ( µ p). The coefficiens in () reflec he balance beween hese compeing losses. Furhermore, he price is cenered on he bes guess of v, which is derived from wo informaion sources, s and κγ ( ). The respecive coefficiens reflec he informaion exracion, which involves weighing hese signals by he ime elapsed beween evens. D. A Comparison wih Exising Models This secion shows how he model presened ness he previous dealer-level frameworks. Resricing he model o no ougoing rades, and consequenly no invenory shocks, he soluion would be (). This is he Madhavan and Smid (993) pricing behavior for an equiy marke specialis; d ou p = s + ζ ( I I ) + ζ x γ q 0 T. () This model suggess, however, ha hese resricions may shu down oher avenues of invenory managemen available o specialiss. Tha is, as NYSE specialiss face increasing marginal coss o invenory managemen hrough price changes, hey opimally spread hese coss across differen avenues available. For example, Madhavan and Sofianos (997) find evidence supporing his. Hence, resricions ha yield () would lead o biased esimaes of invenory effecs since hey overemphasize he role of changing prices o manage invenory. Romeu (005), Bjonnes and Rime (000), Yao (998), Lyons (995) and Madhavan and Smid (99) posulae ha prices are se according o: Equaion (3) yields he price change as: d p = µ ( I I ) + γd (3) p = β + β q + β ( I q + q + γ ) + β I + β D + β D (4) ou 0 j j, Wih he daa used here, Romeu (005) shows ha esimaes of (4) are misspecified. Breaks presen in he daa coincide wih sysemaic differences in he lengh of inerransacion ime ( τ ). Previous sudies using canonical dealer pricing models have indeed noed ha iner-ransacion imes imply changes in he precision of incoming order flow, however, here are, in fac, changes in boh informaive variables ( q, γ ). The model presened here shows why iner-ransacion imes would cause breaks. Rewriing (4) consisen wih his paper s daa generaion process, noe he omied erm in brackes weighed by ( η ) below: j

17 - 5 - p = ϕ + ϕ q + ϕ ( q + q + γ ) + ( ϕ ϕ ) I + ϕ x + ( η )[ ϕ κ( γ ) ϕ q ] ou 0 j j j exraneous erm omied erm The daa generaing process under he hypohesis of muliple insrumens places zero weigh on lagged invenory (he exraneous erm), which would end o bias ( ϕ3 ϕ) oward zero. However, he esimaed coefficien ϕ capures no only he invenory effec, bu i parially reflecs informaion from γ which is conained in he invenory erm. Thus, he omied erm would normally ransmi informaion from γ o prices, bu is absence drives he invenory erm o parially reflec his informaion. Hence, he variaion in he informaiveness of γ will affec he invenory erm. When iner-ransacion imes are long τ ( τ and ( + τ ) η ), he omied erm should be irrelevan. A such imes, one should expec he incoming order flow coefficien ( ϕ ) o be significan, and var( κγ ( )), hence γ will be mosly noise, and uncorrelaed o price changes. This would in urn make ϕ less correlaed wih he informaion effec in p, since he invenory erm picks up he informaion in γ in lieu of he omied erm. Hence, one would expec o see he invenory effec dampened a hese imes. When iner-ransacion imes are shor ( τ 0, and η 0 ), one would see he order flow coefficien ( ϕ ) become less significan, whereas he coefficiens on he invenory erms would be more significan, and pick up he invenory effec more clearly. Hence, canonical models fail o find invenory effecs because hey are confounded wih informaion effecs, or hey include exraneous variables ha are assigned he invenory role. III. DATA CONSIDERATIONS This secion discusses he daa sources employed in esing he model, and hen presens he daa graphically o moivae boh he new invenory and he asymmeric informaion effecs prediced here, as well as hose prediced by canonical models. The daa se consiss of one week of a New York based foreign exchange dealer s prices, incoming order flow, invenory levels, and ransacion clock imes. Hence, p, q j, I, and τ (and η ) come direcly from he recordings of a Reuers Dealing rading sysem. Ou of he 843 ransacions, four overnigh price changes are discarded since he model a hand deals exclusively wih inraday pricing. A few measuremen errors are presen in ransacion clock imes, and hese are reaed wih a dummy variable in he esimaion. 9 Table (page 6) presens some descripive saisics. One observes ha he dealer keeps he average invenory a $. million, however, i deviaes as much as ±$50 million. Given a median 9 The daa are for he dollar/dm marke from Augus 3 7, 99. See Lyons (995) for an exensive exposiion of his daa se. The ransacion clock ime measuremen errors show up when he sequenial order of he rades is no consisen wih he clock-imes, e.g. rade canno have occurred earlier han rade.

18 - 6 - incoming order of roughly $3 million, reversing a one sandard deviaion swing in invenory necessiaes abou five sequenial incoming rades. This suggess alernaive measures of invenory managemen oher han inducing incoming rades are a work, which is also suggesed by oher FX sudies. 0 Table 3 (page 6) shows he observed incoming rades received by he dealer, as well as bilaeral rades ha our dealer iniiaes wih oher dealers in he FX marke. The able shows on average 0 ougoing rades per day iniiaed by our dealer. These, however, are ou concepually differen from q, which represens an ougoing quaniy planned a he ime of price-seing ha capures alernaives o shading he incoming ransacion price for ou invenory conrol. Thus q is unobservable in ha i represens he dealer s commimen o an ougoing rade a he momen of price seing only. A his momen she commis ou irreversibly o rading a a price who s opimaliy depends on being able o rade q ; one of ou he messages of his model is ha he price se by he dealer would be differen if q were no available for invenory conrol. Observed ougoing quaniies differ from he planned because he dealer reopimizes in response o unanicipaed informaion, fricions, or differences in he rading venues uilized o execue he ougoing rade. For example, a each incoming rade, because a price is se, here necessarily exiss an expeced ougoing rade. However, he dealer may no execue an ougoing rade before hen nex incoming rade is observed in he sample. Alhough hey are unobservable, he model soluion provides ou equaions which allow esimaion of q and γ. Table hree shows ha he spread on boh incoming and ougoing rades is ighly mainained a he marke s convenion of 3 pips. Diverging from his spread is frowned upon by ohers in he marke, as i is inerpreed as failing o provide predicable over he couner liquidiy. Hence, poin esimaes of he model ha imply widening or narrowing he spread should be inerpreed as heoreical consrucs ha in pracice manifes hemselves in oher ways, e.g. as shifs in he midpoin of he spread. The fundamenal quesion of ineres is how dealers se prices, i.e. equaion (). Is esimaion requires decomposing he invenory change so as o ge a he ougoing orders, ou q and invenory shocks. Because γ is driven by new informaion, he model soluion reflecs his informaion in our dealer s esimae of he liquidaion value of he asse. Tha is, price changes depend on updaing priors using wo sources of informaion: he incoming order flow, and he unexpeced ougoing order flow ( γ ). Canonical models ypically employ incoming order flow as a source of informaion; however, he use of γ as a source of informaion is new. To ge a feel for his variable, Figure (page 5) superimposes ou q 0 For example, Lyons (995) finds evidence ha observed ougoing bilaeral inerdealer rades and brokered dealer rading are used o conrol invenory in he conex of a canonical dealer pricing model. These do no include a small amoun of brokered rading (which occurs a 5 percen of he sample) which he dealer also engages in.

19 - 7 - cumulaive daily unexpeced order flow on he price, and Figure 3 does he same for cumulaive daily invenory shocks (i.e., cumulaive daily γ ). The verical lines represen he end of each day of he five-day sample (Monday hrough Friday). The correlaion of wo signals wih price seems o vary. For example, on Monday and Wednesday, incoming order flow appears o be a more precise signal of price han invenory shocks, whereas on Friday he opposie seems o be rue. In he model, elapsed clock-ime affecs he relaive precision beween hese signals. Table 4 (page 6) repors he daily correlaions and average iner-ransacion clock-ime. Alhough hese are cumulaive signals, Friday gives an example of shor iner-ransacion clock-ime, and higher correlaion in he (cumulaive) invenory shocks han (cumulaive) order flow shocks. Hence, hese signals seem o complimen each oher and are weighed by iner-ransacion ime in he model. IV. ESTIMATION The framework presened provides sufficien idenifying relaionships so as o permi an almos direc sysem esimaion of he model soluion. Only leveling consans, an auoregressive error on he invenory equaion, and bid-ask bounce dummies on he pricing equaion are added. Table 5 (page 7) lays ou he sysem of equaions given in he model soluion (he firs column), wih he empirical implemenaion of he soluion (he second column), and he parameers recovered from each equaion (hird column). The firs equaion in he sysem, he invenory evoluion, yields he opimal invenory level. The second equaion idenifies he opimal ougoing order q and γ. This is simplified as: ou ( ˆ ) ˆ c qˆ = c I + I + q, wih I = and ( q + γ ) ( I + q ) ou d d ou 3 j j ( c) (5) Solving for γ by adding and subracing c 3 I, yields: ( ˆd ) ( ˆd) ( I + q ) c I + I + q = ( c )( I + q ) + c I I (6) j 3 j 3 j 3 Hence, he ransformaion of (6) allows he esimaion of he proporion of incoming rade ha is expeced o be raded ou, c 3, as a moving average of he ne ougoing order flow ˆd. Moreover, in he pricing ( I qj ) +, and he deviaion from arge invenory ( I I ) equaion (he hird row of Table 5), removing expeced ougoing rade, as well as he incoming rade, from he invenory change idenifies he ougoing rade shock ˆ γ. However, since (6) is a funcion of erms such as I ha are already presen in he pricing equaion, i is necessary o ransform i so as o eliminae mulicollineariy. Thus, (6) is simplified for he pricing equaion o: ou ( qˆ ) ( ˆd) ( ˆd ) ( c )( I + q ) + c I I = ( c ) I + q + c I I q (7) 3 j 3 3 j 3 j

20 - 8 - one can express (7) in a more concepual way using q ˆou : ( ˆd ) ( c ) I + q + c I I q = ( c ) I + ( q qˆ ) (8) ou 3 j 3 j 3 j Equaion (8) idenifies ˆ γ as a weighed funcion of he invenory change which he dealer did no rade, less he par of he las incoming order ha he dealer did no rade ou. Grouping he erms on I in (8) wih he invenory effec permis esimaion of he sysem wihou mulicollineariy in he pricing equaion. In esimaing he incoming order flow s informaion conen canonical models use eiher order flow or is unexpeced componen. This sudy uses order flow direcly in he price equaion, so as o mainain comparabiliy o FX marke sudies, such as Lyons (995), however, esimaion is robus o eiher measure. In addiion, he model predics ha he only difference in he informaiveness of incoming and ougoing order flow is due o he clock ime beween rades,η. Thus, he soluion allows he idenificaion of he informaion effec from he differen componens of (8) since he iner-ransacion imes are observed. Hence, since he model soluion predics idenical coefficiens on hese erms, he componens of γ oulined above are accordingly consrained o have he same coefficien as incoming order flow afer accouning for η. Two direcion-of-rade dummy variables are included o capure he fixed coss such as order processing coss, and pick up he base spread for quaniies close o zero. These variables equal uniy if he incoming order is a purchase (i.e., he caller buys), and negaive one if he incoming order is a sale (i.e., he caller sells). The elapsed ime in beween ransacions is measured o he minue, and esimaes are robus o monoonic ransformaions of η. 3 Finally, scaling consans are included in all hree equaions, and he firs equaion is esimaed wih an AR() error o conrol for auocorrelaion. The sysem is esimaed simulaneously using Seemingly Unrelaed nonlinear leas squares. Table 6 (page 8) shows he esimaions of he model. Below, Table 7 presens canonical model esimaes of he same daa as a basis for comparison. For example, Hasbrouck (99) and Madhavan and Smid (993) use he unexpeced componen of incoming order flow, and esimae his measure as a residual of a vecor auoregression. In he case of he FX daa used here, hese auoregressions end o have lile explanaory power, making he residual almos idenical o he incoming order flow. Esimaing he model wih independen informaion coefficiens on incoming order flow and gamma is possible, and suppor he resricion imposed here. However, under such esimaions some invenory erms canno be grouped as presened here, and collineariy prevens saisfacory esimaions of he invenory effec, hence hese esimable forms are no used. 3 Some measuremen error in he ime samps leads o he inclusion of a dummy ineraced wih he absolue value of he clock ime (which urns ou o be insignifican).

21 - 9 - The esimaions in Table 6 indicae ha he model fis he daa fairly well. The main resuls are he very significan and properly signed coefficiens on he informaion and invenory effecs, c and c, as well as he prediced invenory evoluion and ougoing rade esimaes, c, c, and c 3. Canonical model esimaes are presened in Table 7 as a basis for comparison. Noe ha he canonical esimaes are no robus o subsample esimaion. Specifically, canonical model predicions of invenory effecs are rejeced in he firs half of he sample, and similarly, prediced informaion effecs are rejeced in he second half of he sample. 4 The model presened here is robus o subsample esimaion, nowihsanding he lower p-values of esimaed coefficiens in he firs sub-sample. Moreover, all hree equaions in he sysem are joinly significan as prediced, and he esimaes fail o rejec any of he esable resricions. Hence, his model rejecs canonical model poin esimaes of asymmeric informaion and invenory effecs. The model predics ha he dealer plans o rade ou roughly one-hird of each incoming rade ( ĉ 3 =0.34) each ime she quoes a price. Addiionally, he model esimaes he dealer s arge invenory a abou wo million ( I ˆd =.09). From Table he average invenory is.6, which is saisically indisinguishable from our dealer s observed average. 5 Asymmeric informaion The asymmeric informaion componen (c ) is significan and larger han canonical model esimaes given by β in Table 7 (0 5 muliply he pricing equaion coefficiens). One way o inerpre he esimaes is ha he dealer widens her spread by 3.5 pips per $0 million of incoming order flow or invenory shocks (wice c, since orders are quoed based on absolue size). These esimaes indicae a more inense asymmeric informaion effec han previously esimaed; no jus because of he higher esimaed effecs, bu because here are wo sources of privae informaion boh incoming and ougoing order flow boh pushing price changes. In erms of economic significance, he esimaes sugges ha he marginal $ million dollar order pushes he dealer s price by abou basis poins, given he average exchange rae in he sample of roughly DM.5 per US dollar, or percenage poins per excess US$ billion raded. This is higher han marke-wide esimaes of he price impac of US$ billion of excess order flow, which flucuae around half a percen. 6 However, hese laer hese esimaes are no comparable because of he inheren difficulies of linearly inerpolaing one dealer s behavior o he marke-wide equilibrium. These difficulies are paricularly acue since he dealer generaing hese daa predominanly provides inerdealer liquidiy, no end-user liquidiy. The ho poao hypohesis of Lyons (997) would sugges 4 Noe ha Romeu (005) documens evidence of model misspecificaion and srucural breaks presen in hese esimaes of he canonical dealer pricing model used here for comparison. 5 A Wald es fails o rejec equaliy of he mean o he arge wih a p-value of See Evans and Lyons (00) or Chaboud, e. Al. (006).