An Intraday Pricing Model of Foreign Exchange Markets

Transcription

1 An Inraday Pricing Model of Foreign Exchange Markes Absrac Marke makers learn abou asse values as hey se inraday prices and absorb porfolio flows. Absorbing hese flows causes invenory imbalances. Previous work has argued ha marke makers change prices o manage incoming flows and offse invenory imbalances. This sudy argues ha hey have muliple insrumens, or ways o manage invenory imbalances and learn abou evolving asse values. Hence, hey smooh invenory levels and updae prior informaion abou asses using muliple insrumens. In ignoring oher insrumens, previous sudies have ignored he informaion ha hese provide and overemphasize he role of price changes in invenory managemen. The model presened here provides new esimaes of asymmeric informaion and invenory effecs, he price impac of each insrumen, he cos of liquidiy, and he impac of an inervenion on hese coss.

2 - - I. INTRODUCTION The evidence supporing a igh relaionship beween a marke s absorpion of porfolio flows and is asses reurns is mouning. I implies ha asse reurns depend on how raders 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, raders 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 rader, 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 rader level. The empirical resuls link porfolio flows o asse prices a he mos micro level, and provide direc esimaes of he cos of liquidiy, asymmeric informaion, and invenory effecs. The model suggess ha previous sudies have underesimaed, if no missed or even rejeced hese effecs. 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? Specifically, in FX, 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). If muliple increasing-marginal-cos insrumens are available for managing invenory, he dealer opimally spreads her invenory managemen across all of hem. Furhermore, jus as incoming order flow provides informaion abou asse values, so do oher insrumens used for managing invenory, such as calling ou o oher dealers. 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 how dealers conrol invenory 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).

3 - 3 - hrough price seing and how hey learn 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. This implies ha previous microsrucure models misspecify and bias esimaes of informaion and invenory effecs. Microsrucure models sudy rading coss and payoffs refleced in asse reurns ha dividend discouning ignores. They poin o wo general microsrucure-pricing effecs. The firs is he invenory effec, in which he specialis mus manage a finie sock of he asse agains a demand ha responds o a random-walk fundamenal value. In his siuaion, if he specialis passively fills orders, he probabiliy of a sock ou is uniy. Hence, invenory models argue ha he specialis changes her price away from he expeced asse value o induce rades ha unwind undesired posiions. The second effec is he asymmeric informaion effec, 3 where he specialis faces a marke where some agens have more informaion abou he asse s equilibrium value. Here she will recognize ha he incoming order flow reflecs his informaion and will change her price accordingly. Empirical sudies have found evidence of asymmeric informaion in equiy markes; 4 however ess for invenory effecs are unsuccessful. For example, Madhavan and Smid (99) and Hasbrouck and Sofianos (993) do no find invenory effecs. 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 (00) revisis he Lyons (995) daa and finds evidence of model misspecificaion, and specifically ha invenory and informaion effecs are no simulaneously presen in subsamples. Oher sudies of foreign exchange markes also fail o find invenory effecs. 5 Clearly, he evidence supporing invenory effecs is a bes a mixed bag. To invesigae invenory effecs, he model in his sudy uses he over-he-couner insiuional srucure of FX markes. A is hear is he idea ha dealers exploi every alernaive when rebalancing porfolios, raher han relying on only price-induced order flow o change heir porfolio composiion. As dealers face increasing marginal losses for inducing flows, hey urn o oher mehods of unloading unwaned posiions. The decenralized naure For example, Soll (978), Amihud and Mendelson (980), Ho and Soll (98, 983), O Hara and Oldfield (986) among ohers. 3 For example, Kyle (985), Glosen and Milgrom (985), Admai and Pfleiderer (988), Easley and O Hara (987, 99), among ohers. 4 For example, Hasbrouck (99 a, b), Hasbrouck (988), Madhavan and Smid (99, 993), among ohers. 5 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.

4 - 4 - of he foreign exchange marke offers a clear opporuniy o observe his phenomenon. The conclusions sugges ha previous models are misspecified because hey neglec alernaives o conrolling invenory hrough price-induced flows. The model presened here shows why previous models are misspecified and fail empirically. Previous sudies address he decenralized naure of FX markes boh a he dealer level, and a he general equilibrium level. A he dealer level, he approach is o exend Madhavan and Smid (99) for example, see Lyons (995). The model presened here suggess ha he basic dealer-pricing behavior posulaed in hose sudies is no opimal. Mello (995) also conjecures ha a dealer may have muliple insrumens and non-linear pricing behavior, as presened here. A he general-equilibrium level, models such as Lyons (997) favor modeling dealer pricing wih muliple insrumens. Tha sudy argues ha high FX volume migh be due o dealers passing on invenory imbalances. Finally, he principle behind his paper is no limied o foreign exchange rading. All marke makers have an incenive o minimize guaraneed losses from inducing rades via price changes. Tha alernaives exis in FX is clear; however, in oher markes alernaives exis as well. For example, Madhavan and Sofianos (997) find ha New York Sock Exchange (NYSE) specialiss engage in selecively rading o balance invenory. 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 suppor he model and offer several novel resuls. For example, asymmeric informaion effecs driving price changes are wice as large as previously esimaed; one can graphically compare prices wih he new informaion signals ha he dealer sees. Invenory pressure is less presen in price changes han was previously esimaed. Afer conrolling for invenory and informaion effecs, he base bid-ask spread is wider han previously esimaed. When seing prices, he dealer plans o rade ou less han one-quarer of he difference beween her curren and he opimal invenory posiions. The cos of inducing a sandard ($0 million) incoming rade is abou.5 pips (a pip is he smalles price incremen in a currency 6 ) or $,000. The cos of execuing an ougoing rade of he same size is 9 o 0 imes higher. The dealer s observed incoming rades are, however, 9 o 0 imes greaer han ougoing rades, boh in number and in daily volume. Hence, ougoing orders rade a a premium, which is consisen wih oher sudies of dealer behavior. A Fed inervenion increases he aymmeric informaion impac of incoming rades on price changes bu decreases ha of ougoing rades. An inervenion lowers he cos by 50 percen of inducing an incoming rade and by 5 percen of execuing an ougoing rade. I also lowers invenory coss and ighens he spread. The inervenion emporarily moves 6 The value depends on he currency pair. The daa used here are dollar/deusche mark, so a pip is DM

5 - 5 - prices abou 6.7 pips per $00 million (Fed purchases of $300 million move prices abou 0 pips), which suppors oher sudies. 7 By comparison, a 0-pip move in our dealer s price induces a purchase of $ million. Finally, he model addresses he broader relaion beween porfolio flows and asse prices. I suggess ha 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. This informaion ranslaes ino permanen price movemens. Hence, while boh ransiory and permanen effecs are presen in he daa, he evidence favors a permanen impac of porfolio flows on prices. 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. AN INTRADAY FOREIGN EXCHANGE PRICING MODEL This secion exends 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. A real FX dealer only ses prices when she passively receives an order (i.e., anoher dealer iniiaes he rade). 8 These prices are he focus of his sudy. Besides seing prices, however, she can iniiae inerdealer bilaeral, brokered, or IMM Fuures rades, as well as receive informaion from hese, he financial press, news and advisory services, he sales and floor managers, and 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, and 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. 7 Evans & Lyons (999) esimae 5 pips and Dominguez and Frankel (993) esimae 8 pips per $00 million. 8 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.

6 - 6 - 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). () 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: 9 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. 0 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 9) 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 raders 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. (5) ( X ) 9 Noe ha his is a one-period-ahead condiional disribuion, as he uncondiional disribuion would have a ime-varying variance. 0 The invenory carrying cos, shown below, follows Madhavan and Smid (993). I is a cos proporional o he variance of he dealer s wealh.

7 - 7 - 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 A he ime of seing prices, he dealer knows ha laer she can call ohers and rade ou ou an ougoing quaniy (denoed q ). This is depiced in upper box of Figure. q indicaes he dealer s desired ougoing quaniy in expecaion, and condiional on informaion ou available a he ime of price seing. Because he dealer has q available, she does no ou conrol invenory solely hrough price induced order flow. In his sense, q capures he planned amoun he dealer prefers o rade ou raher han hold (and pay he invenory carrying cos), or dispense via price-induced incoming rades. While our dealer is rading ou q, however, here will be exogenous quaniy shocks o her invenory, as shown in Figure. The source of hese shocks can be unplanned rading wih cliens of our dealer s bank (her employer), oher bank dealers, brokered rading, he rading floor manager, and so on. These shocks perurb he dealer s invenory posiion ou beyond he planned quaniy, q. Denoe hese disurbances as γ. Accordingly, he oal ou 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 ou as he oal realized ougoing quaniy ( q + γ ), yields nex even s invenory (I ). An example illusraes he rading process. Suppose our dealer begins he day wih invenory a zero. The inerdealer compuer communicaes ha a dealer wans o sell her uni. Here, =, I =0, q j =-. Suppose our dealer opimally ses he price o 5 (p =5), and ou plans o buy unis from oher dealers when rade is hrough ( q =). In he daa we do no see our dealer s rading unil someone iniiaes anoher rade wih her over he inerdealer compuer. A he nex incoming rade (=), suppose ha I =6, and q j =-. Considering he las rade s invenory, incoming and planned ougoing quaniies, I should be hree. I is six, which implies ha γ was 3. Tha is, he dealer planned o aggressively buy afer she passively buys from he incoming rade. Her invenory should be hree, bu i is six, which implies ha he unplanned invenory shock was 3. Assume furher ha new informaion and evens occurring in he clock ime beween evens - and are driving he quaniy shocks ( γ ) ; hence, assume a linear funcion (denoed κ () ) such ha: The general form of (7) is chosen for ease of exposiion; he conclusions are robus o differen funcional forms.

8 - 8 - κγ α + αγ = θ+ ε ε σ. (7) ( ) 0, ~ N(0, ε ) In equaion (7), he unexpeced invenory shock (noisily) signals innovaions in he asse value ha occur while he dealer execues ougoing rades. γ is informaive because ou afer he dealer chooses her ougoing quaniy ( q ), she should rade his quaniy and nohing else; ha is, he choice made a - is opimal unil new informaion (a he nex incoming order, q j ) arrives. The only reason our dealer would deviae from he opimal ou ougoing quaniy ( q ) beween - and is ha new informaion is revealed. Hence, he evoluion of v can be inferred fromγ. If his is he case, hen he oal ougoing quaniy ou will reflec he 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. In summary, he ideniy ha describes he evoluion of invenory is: ou I I δ ( v p ) X + q + γ (8) In conras, a he ime of seing prices, he dealer s expecaion of nex period s invenory is: E[ I j ] ou + Ω = I qj + q. (9) 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 (added) from (o) he dealer s capial, K a a cos v. 3 In he previous example, he dealer ou would pay invenory coss on q, bu no on γ a he end of he rade. Incorporaing his charge, a rade he dealer s wealh posiion is given by: W = v E[ I Φ ] + γ + E[ K Φ ] vγ + y. (0) ( ) ( ) 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 Alhough hey include muliple informaive signals, incoming order flow is he only source of privae informaion in Madhavan and Smid (99) or Lyons (995). 3 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 very 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.

9 - 9 - 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 smoohs he dealer s wealh, and eners he invenory cos as: d c ( ) = ω σ W = ω φ0 + φ I I. () 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. 4 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 informaion abou v is he informaion learned while execuing he ougoing rade, which is refleced in a funcion of he invenory shock, kg ( - ). While boh kg ( - ) and s are used o updae µ, assumed ha he variance of kg ( - ) is increasing in he real ime (i.e., clock ime) elapsed beween incoming rades. Tha is, assume ha var( s ) = σ w and var( kg ( - )) = s wd, wih τ being he clock ime elapsed beween evens - and. As he appendix shows, his gives an updaing as a funcion of: m - m ( D - = ) s + ( ) k( g ) +D +D -. (3) In equaion (3), 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, (3) 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 ( kg ( - ) ) 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 kg ( - ) s 4 Alernaively, one could use a Bayesian updaing model, such as Lyons (995), Madhavan and Smid (99), Yao (998) and ohers. These use oal incoming orders (raher han he unexpeced componen) as signals.

10 - 0 - precision decreases (i.e., variance increases) as he clock-ime elapsed from even o increases. As more ime has passed in beween rades, kg ( - ) has more noise. 5 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 E[ X 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 in he appendix. The dealer solves: J( I, x, µ, K ) = E ρ v% I + K + y c + ρ J( I%, x%, % µ, K% ), (4) max {( )[ ] } ou p, q subjec o he following evoluion consrains: Invenory: i ou E I% + Φ = I δ ( µ p) x + q, (5) Noise Trading: i E x % 0 + Φ =, (6) Informaion: i E % µ + Φ = µ, (7) Capial: i ( ) ou ou E K + Φ = K + pδ µ p + px µ + αq ) q c (8) Equaions (), and (4) hrough (8) comprise he opimizaion problem. (5) consrains invenory evoluion. (6) consrains liquidiy rades o be zero in expecaion. (7) consrains he asse o a random walk. (8) 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 + αq ). α 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 α 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. 6 The appendix shows he model soluion o be: δα d + δ ( β+ δ ) p = µ + β( ( δα ) ) + I I + ( δ ( + δα ) ) x; (9) 5 One migh argue ha as τ 0, 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). 6 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 general equilibrium insighs.

11 A ( A ) - - ou d q = ( I I ) ( p) x α + δ µ + ; (0) d ( + β ) I ' = I + β ( I I ) + x, () ( ) α δa ( + β) + α p = ψηs + A( + β)( q + γ q ) + ψ ( η) κ( γ ) + x αδ ou j, Equaion (9) 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 (0) he ougoing quaniy shows ha as he price impac of ougoing rades goes o zero, i.e., α 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 ou previous invenory shock ( ( η) κγ ( ) ). The ( q + γ qj, ) erm capures componen of he price change aribuable o invenory pressure. Finally, he dealer changes her price due o he noise-rading componen ( x ). 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 orders arrive ( qj,, qj, ) 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 (α ). Adjusing invenories via price induced orders implies absorbing he cerain loss o he informed raders, 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 (3). 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. (3) 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 (3) would lead o biased esimaes of ()

12 - - invenory effecs since hey overemphasize he role of changing prices o manage invenory. Romeu (00), Bjonnes and Rime (000), Yao (998), Lyons (995) and Madhavan and Smid (99) posulae ha prices are se according o: Equaion (4) yields he price change as: p = µ + ( I I d ) + D. (4) p = β + β q + β ( I q + q + γ ) + β I + β D + β D (5) ou 0 j j, Wih he daa used here, Romeu (00) shows ha esimaes of (5) are misspecified. Breaks presen in he daa coincide wih sysemaic differences in he lengh of inerransacion ime ( τ ). The changes in iner-ransacion imes are exogenous and imply changes in he precision of he informaive variables ( q, γ ). To see why hese would cause breaks, rewrie (5) consisen wih his paper s daa generaion process, and noe he omied erm in brackes weighed by ( η ) below: j 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 3 = 0, so ha he rue process would place zero weigh on lagged τ invenory. When iner-ransacion imes are long ( τ and ( + τ ) η ), his omied erm should be irrelevan. A such imes, one should expec he incoming order flow coefficien ( ϕ ) o be significan. A such imes, var( kg ( - )) Æ, hence γ will be mosly noise, and uncorrelaed o price changes. This would in urn make ϕ less correlaed wih p since here is no erm picking up he informaion role for γ oher han he invenory erm. In addiion, since here is no role for I - in he equaion, i is uncorrelaed wih he price change, and as 0 ϕ ϕ end o zero. Hence, one would no expec Wih ( ϕ ϕ ) ϕ, his would make ( ) 3 o see invenory effecs a hese imes. When iner-ransacion imes are shor ( τ 0, and η 0 ), one would see he coefficien on q j become insignifican, whereas he coefficiens on he invenory erms would be significan.

13 - 3 - III. DATA AND EMPIRICAL TESTS This secion empirically ess he model. The firs subsecion discusses he daa; he second subsecion esimaes and ess he model (deails are in he appendix). The hird subsecion shows he impac of an inervenion on he esimaes, and on δ (he cos of inducing order flow) and α (he cos of placing ougoing orders). A. Daa Consideraions The daa se used o es he model consiss of one week of a New York based foreign exchange dealer s prices, incoming order flow, invenory levels, and ransacion clock imes. 7 Hence, p, q j, I, and τ (and η ) come direcly from he recordings of a Reuers Dealing 000- compuer rading sysem. Ou of he 843 ransacions, he four overnigh price changes are discarded, since he model a hand deals exclusively wih inraday pricing. The fundamenal quesion ha his model seeks o resolve is how dealers se prices, and hence, of ineres is he price change equaion (). Is esimaion requires knowledge of ou he unobservable ougoing orders, q, and invenory shocks, γ, γ. These are ou unobservable because q represens he dealer s commimen o make an ougoing rade a he momen of price seing only. I is a his momen ha she commis irreversibly o a price ou who s opimaliy depends on being able o rade q ; one of he messages of his model is ou ha he price would be differen if q were no available. Acual ougoing quaniies may be ou differen from he planned q due o unanicipaed informaion, fricions, or differences in he rading venues uilized o execue he ougoing rade. Alhough hey canno be observed, he model soluion provides equaions which allow esimaion of q and γ, γ ou. The model soluion also suggess ha price changes depend on updaing priors using wo sources of informaion: he unexpeced par of he incoming order flow ( s ), and he unexpeced ougoing order flow ( γ ). Oher 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 0) superimposes 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 7 The daa are for he dollar/dm marke from Augus 3 7, 99. See Lyons (995) for an exensive exposiion of his daa se.

14 - 4 - 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 (page ) repors he daily correlaions and average inerransacion clock-ime. Alhough hese are cumulaive signals, Friday gives an example of shor inerransacion clock-ime, and higher correlaion in he (cumulaive) invenory shocks han (cumulaive) order flow shocks. B. Esimaion Table (page ) lays ou he hree opimaliy condiions of he model and heir esimable forms. The firs is he opimal invenory evoluion, which is direcly esimaed and yields he opimal invenory level. The second equaion pariions he non-incoming ou invenory change ino he opimal ougoing order, q, and he invenory shock, γ. The final equaion hen esimaes he change in price as he sum of he hree componens in he model. The firs reflecs informaion effecs from incoming order flow. The second reflecs invenory accumulaion pressure on he price. The hird reflecs informaion from he invenory shock. 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 one 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. The esimae of he unobservable invenory shock proved o be problemaic. Is inclusion opens he door o mulicollineariy and is possibly correlaed wih he residual. As a resul, i is insrumened ou using is own lags. The unexpeced incoming order flow ( s ) is calculaed as he residual of a VAR. 8 ; ˆ j = j k + ε = k = q q s ε. (6) The final price change regression is esimaed using non-linear leas squares. The resuls are robus o changing iniial parameer values. Table 3 (page ) shows he esimaions of he model, as well as esimaes of he daa using he Lyons (995) specificaion. 9 The esimaions indicae ha he model fis he daa fairly well. The asymmeric informaion componens (c and c 3 ) are significan and much larger han previous esimaes given by β (0 5 muliply all coefficiens). The esimaes indicae ha he dealer widens her spread by beween roughly 3.5 and 9 pips per $0 million of unexpeced incoming order flow or invenory shocks (double c and c 3 ). This indicaes a more inense asymmeric informaion 8 As in Hasbrouck (99) and Madhavan and Smid (993). See appendix for esimaion deails on he generaed regressors and insrumenaion of he invenory shocks. 9 However, Romeu (00) finds evidence of model misspecificaion presen in he esimaes from Lyons (995) ha used here as a basis for comparison.

15 - 5 - effec; no only are hey higher han previous models sugges, bu here are wo sources of informaion pushing price changes. 0 In comparing informaion from incoming rades versus invenory shocks, price is more sensiive o he laer. This may resul from ougoing rades providing a more heerogeneous and richer informaion componen due o he variey of sources (end users, IMM Fuures, ec.). Anoher reason may be ha in pracice incoming rades are larger or more frequen, so ha he same uni increase will push prices less han a uni increase in invenory shocks. I may also resul from his dealer s focus on providing liquidiy for incoming rades, raher han aggressively rading ou. In addiion, he coefficien on invenory managemen b is significan and ends o be similar in size o he previous models β ; however, since previous models have wo invenory erms, he oal invenory effec is: ou ou β I + β I β ( I q + q + γ ) + β I β ( q + q + γ ) + ( β + β ) I. (7) 3 j 3 j 3 < 0 < 0; β > β3 Hence, he invenory impac on price of β is augmened by ( β + β3) I. Our dealer alers her price less in response o invenory accumulaion han he oal invenory driven change in previous models. These differences in he esimaes are due o he marke maker s use of muliple increasing marginal cos insrumens o manage invenory and learn new informaion. Consider he invenory overweighing of previous models. Esimaions ha ignore muliple insrumens will overweigh he invenory componen because price changes have such an imporan role in invenory managemen. The model presened here suggess ha price is bu one of muliple insrumens used o conrol invenory coss. As a resul, invenory accumulaion is no as imporan in explaining price changes. Consider now why previous work underesimaes he informaion componen. Even if pure invenory pressures were perfecly explained by previous models, here is a componen of invenory change driven by new informaion. Invenory heory canno explain his informaion-driven invenory componen. This componen is one of muliple signals ha, according o he model, vary in precision depending on elapsed clock-ime. This suggess ha incoming order flow can be relaively less informaive a differen imes, and should be weighed accordingly. Previous esimaions assign all informaion-driven price changes o he (a imes, noisy) incoming order flow, and mue is rue informaive impac. Finally, c 4 measures he effecive spread for q j close o zero. I suggess ha afer having conrolled for informaion and invenory effecs, he baseline spread is roughly.5-.8 pips (wice c 4 imes 0-5 ). Noe ha hese esimaes are remarkably close o he median inerdealer spread observed in he FX marke of 3 pips. 0 A cavea here is ha he model measures unexpeced incoming order flow as an informaion signal, which is no direcly comparable o previous esimaes ha use he enire order. In addiion, he esimaes are subjec o measuremen errors, however, he lower bound esimae is higher han he Lyons (995) esimae. Lyons (996) describes his dealer as a liquidiy machine. As noed in secion III. C. below, ougoing orders are observed abou one-ninh as ofen as incoming orders are received.

16 - 6 - The lef panel of Table 4 (page ) shows he esimae for he opimal ougoing ou quaniy, q. The esimae suggess ha a every even he dealer plans o rade ou less han one fourh of he difference beween he curren and opimal invenory, (I -I d ). The righ panel shows esimaes of he invenory evoluion, and he opimal invenory level, I d, which is abou wo million dollars long. The resuls wihsand various robusness ess. Firs, no evidence of he presence or locaion of (possibly muliple) srucural breaks is found. These ess are robus o possible heerogeneiy and auocorrelaion in he residuals. In addiion, he esimaed linear regressions use Huber/Whie robus sandard errors. Esimaion wih various monoonic ransformaions of η produces no significan change in he resuls. The adjused R indicaes a good fi for he daa. Finally, he appendix deails he esimaion of several linear approximaions o he non-linear equaion and various ess performed as furher evidence of robusness. C. Fed Inervenion and he Cos of Liquidiy The las five percen of recorded rades occurred while he Fed inervened o suppor he dollar. In Figure (page 0), he sharp appreciaion on he las day reflecs he marke reacion o he inervenion. I perhaps succeeded in slowing he slide of he dollar, bu was unsuccessful in susaining a reversal. The marke closed down on he day, and down from is high afer he sar of inervenion. I involved muliple dollar purchases oaling $300 million afer he close of European markes. The exac sar ime is unknown; however, from he financial press hree suspeced imes are shown in he firs column of Table 7 (page 4). 3 There are oo few observaions o meaningfully esimae he inervenion in isolaion. Insead, his secion compares esimaes of he model and he liquidiy coss wih and wihou he inervenion period (i.e., 95 percen of he sample, versus 00 percen). Non-linear Wald ess fail o rejec equaliy beween he wo ses of esimaes, which is foreseeable given nearly idenical underlying samples. Table 3 (page ) shows he impac of he inervenion on he esimaed parameers. The inervenion increases he asymmeric informaion effec of incoming order flow (c ) by over 50 percen. Surprisingly, he esimae for invenory shocks (c 3 ) declines, alhough by jus 5 percen. This asymmery may be idiosyncraic o his dealer; she is a major liquidiy provider in he marke, and consummaes many more incoming rades. Turning o invenory effecs (c ), hese become less negaive as he Fed dollar buys dollars and injecs liquidiy. Sup{F} ess based on Andrews (993)/Bai Perron (998). 3 Quoing he Wall Sree Journal, Augus 0, 99: The Federal Reserve Bank of New York moved o suppor he U.S. currency... as he dollar raded a.470. This is he mos precise documenaion available of he inervenion sar, and ha price corresponds o :3 pm. Oher imes are seleced because of repors of a mid-day sar (hence, :0 pm), and a :6 pm he price jumps 36 pips, suggesing a possible inervenion sar a ha poin.

17 - 7 - This is also consisen wih he ighening of he spread (c 4 ). From Table 4, he desired invenory, I d, declines by 4 percen, indicaing perhaps ha he dealer (prescienly) bes agains an appreciaion due o he inervenion. Finally, he dealer increases he planned ougoing rade by percen a every even. The model esimaes allow one o recover he respecive coss of liquidiy for rading ou or inducing incoming rades (show in he appendix). Tha is, esimaes of δ (from ou ou δ ( µ p) ) and α (from q ( µ + αq ) ) are recovered from he esimaed parameers in Table 5 (page 3). The cos of inducing a $0 million order (based on δ ) is presened and convered o pips, dollars, and as a percen of $0 million. I coss our dealer.63 pips (abou $,00 or jus over one basis poin of he order size) before he inervenion, and.36 pips (abou $900 or less han one basis poin) wih he inervenion. The cos of a $ million ougoing rade (α ) is significanly higher. I coss.48 pips wihou he inervenion and.4 pips including i. So, wihou he inervenion, i is abou nine imes more expensive o rade $ million ou han o induce $ million. During he inervenion, i is over en imes more expensive! This cos dispariy is sunning, bu Table 6 lends suppor o his resul and insigh as o why. Table 6 (page ) shows he observed number of ougoing and incoming rades by he dealer. Incoming rades ounumber ougoing rades 8.5 o. The able also gives he average and median sizes of incoming rades ( q ) and ougoing rades ( q ). Incoming rades are j slighly larger han ougoing rades, boh in mean and median. These, and he fac ha incoming rades are far more in number, imply incoming rades handle abou nine o en imes he daily volume of dollars ha ougoing rades do. This accords well wih he esimaed coss of he respecive insrumens. I suggess ha he dealer primarily handles incoming rades, bu occasionally uses ougoing rades, perhaps because of a need o rade wih end users, some oher segmen of he marke, or some special convenience ougoing rades provide. Table 7 (page 4) compares he price impac of he Fed s $300 million inervenion wih he price impac faced by our dealer. From he sar of he inervenion (:3 pm) o he highes price, he dollar appreciaed 0 pips, bu i hen closed 9 pips below he sar. Columns () and (3) show ha his implies -3 and 6.67 pips per $00 million of inervenion from he sar o he closing and high price, respecively. Columns (4) and (5) show ha if our dealer moves her price -9 (3 imes -3) or 0 (3 imes 6.67) pips, she could sell $55 million or buy $ million, respecively. For a given price impac, he Fed s purchases more dollars han he dealer does, since hese purchases are realized agains a falling marke. 4 These esimaes are similar o hose in previous work. ou 4 Esimaes are presened for oher suspeced inervenion sar imes. Because hese imes imply a much larger price movemen, hey also imply a much more effecive inervenion. This, however, is no consisen wih previous esimaes of inervenion price impac (pips per $00 million), he financial press repors of he inervenion s ineffeciveness, or he fac ha in subsequen days, furher inervenions were execued in suppor of he dollar. These wihsanding, hey are presened for comparison purposes.

18 - 8 - IV. CONCLUSIONS The model presened incorporaes he realisic opions available o marke makers for absorbing porfolio flows. Pas models say ha making markes enails moving prices away from he full informaion value o induce rades ha compensae invenory imbalances. Bu his equaes o inenionally selling low or buying high o avoid paying invenory coss. This paper suggess ha here is a beer way. One clear example is ha in FX, he dealer has he abiliy o call ohers in he marke and unload her unwaned invenory on hem. Of course his is no o sugges ha ougoing orders are a panacea for invenory problems, so hese are modeled wih price impac (i.e., increasing marginal coss). However, a he margin, she will equae he loss of rading unwaned invenory o incoming calls wih he marginal price impac (i.e., he loss of rading unwaned invenory in ougoing calls) and wih he marginal loss of he invenory imbalance (i.e., he marginal invenory carrying cos). In addiion, hese ougoing calls do no occur in a vacuum. As long as evens ranspire during he ougoing call period, he dealer will learn hrough rading a hose imes and updae her beliefs. These updaes bring abou price changes ha neiher invenory coss nor incoming order flow can explain. And FX dealers are jus one example of marke makers who smooh coss over muliple insrumens. This paper argues ha one should consider where dealers or specialiss migh be subsiuing away from convenional invenory coss when modeling price seing. Price-induced order flow is one of a mulipliciy of informaive insrumens available o marke makers. The esimaions suppor he proposed model and provide several novel empirical resuls. Generally, hese indicae ha previous sudies overemphasize he role of price changes in invenory managemen, since no oher insrumens are considered. This omission biases downward he role of informaion in price changes, can make invenory effecs appear insignifican, and ighens he bid-ask spread. A he ime of price seing, planned ougoing rades are less han one-quarer of he difference beween curren and opimal invenory posiions. The cos of inducing a $0 million rade is abou.5 pips, or $,000. Ougoing rades are observed roughly one-enh less frequenly han incoming rades, and are esimaed o be en imes more expensive. A Fed inervenion increases he informaiveness of order flow, and lowers he cos of liquidiy for he dealer. I also lowers invenory coss and ighens he spread. Finally, he model addresses he broader relaion beween porfolio flows and asse prices. The presence of invenory effecs suggess ha par of observed price changes is ransiory. However, wih muliple insrumens, dealers exhaus he gains from sharing a large invenory posiion wih less price impac. As a resul, he ransiory componen of price changes is less imporan han he informaion componens from he muliple insrumens. Hence, while boh ransiory and permanen effecs are presen in he daa, he model favors a permanen impac of porfolio flows on prices.

19 - 9 - Figure. The Timing of he Model Beween Evens & +; ou ou Trade: q a price ( µ + αq ) Shock: γ Elapsed clock-ime: τ Even ime Even T; Incoming order: q j Known: I γ ou Choose: P, q,, τ Even + T; Incoming order: q j+ Known: I,, + γ τ ou P, q Choose: + + The figure above describes he iming of he model. A every even:. if T, he dealer knows her curren invenory (denoed I ), and a new incoming rade (one source of informaion for updaing priors) occurs. The incoming quaniy is q j. ou. The dealer decides her price (denoed by P ) and plans her ougoing rade (denoed by q ). These are he alernae mehods available for offseing invenory disurbances caused by he incoming rade. ou 3. Beween evens, he dealer execues he planned ougoing rade ( q ), and faces a quaniy shock, (denoed by γ ). This is anoher source of informaion for updaing priors. 4. In addiion, he dealer observes ime elapsed beween rades (denoed by τ ). 5. A he nex even (+), he dealer uses he new incoming rade q j+ as well as he quaniy shock beween rades and he ime elapsed beween rades o updae priors on he evoluion of he asse value, and se prices.

20 - 0 - Figure. Daily Cumulaive Incoming Unexpeced Order Flow versus Price Dollar Price of FX Millions of Dollars Price Unexpeced Order Flow Figure superimposes price on cumulaive incoming unexpeced order flow, Augus 3-7, 99. Figure 3. Daily Cumulaive Invenory Shocks versus Price Dollar Price of FX Millions of Dollars Price Invenory Shocks Figure 3 superimposes price on cumulaive invenory shocks, Augus 3-7, 99.

21 - - Table. Daily Correlaion of Order Flow Variables wih Price Order Flow Unexpeced Invenory Mean Elapsed Order Flow Shocks Time* Monday Tuesday Wednesday Thursday Friday Table shows he daily correlaion beween price and he order flow variable used o updae priors. The firs column shows incoming unexpeced order flow and he second invenory shocks correlaions for each day, Augus 3-7, 99. The las column shows daily mean elapsed inerransacion ime. * Reporing errors imply mean absolue value ransacion ime. Table. Model Soluions and Esimable Equaions () Model Soluion Empirical Implemenaion Parameers Recovered β d ( + β ) ( ) I α a a I + ε ˆd aˆ 0 I = aˆ I ' = I + ( I I ) + x 0, () ou A d q = ( A ) ( I I ) ( p) x α + δ µ + ou ( ) ( ˆd q + γ = b I I ) + q + ε ( j), (3) ou p = ψηs + A( + β)( q + γ q j δa ( + β) + α ( η) κ( γ ) αδ x + ψ + wih ( q ) I I q ou + γ + + j ou p = c0( η) + ηcs + c( q + γ qj ) + c3( η) γ + c4d + c5d + ε3 ou wih q + γ q j I I and κγ ( ) = α + αγ 0 aˆ = + β ( ˆd ) ou qˆ = bˆ ( I I ) + q ˆ γ = ˆ ε bˆ j, = A A α cˆ, cˆ ˆ ˆ, c3, c4 cˆ = A( + β ) Liquidiy Cos Parameers δ ( aˆ bˆ ) = cˆ ˆ ( b ) ˆ = ab ˆ ˆ cˆ ( b ) α Table compares he algebraic soluion o he model (in he firs column) wih he esimable equaions hese imply (in he second column). The final column shows he parameers recovered for economic inerpreaion. Row () shows invenory evoluion, row () shows ougoing quaniy, and row (3) shows price changes. The boom shows how he srucural parameers measuring he cos of liquidiy are recovered from he esimaes.

22 - - Table 3. Price Change Equaion p = c + cηs + c ( q + γ q ) + c ( η) ˆ γ + c D + c D + ε 0 c c c 3 c 4 5 Adj. R ou 0 j, , c c Full No Fed Previous Esimaes = p β0 βqj βi β3i β4d β5d ma() β 0 β β β 3 β 4 β 5 Adj. R Esimae Table 3 compares he non-linear regression esimaes of he model presened here wih esimaes of invenory and asymmeric informaion effecs in hese daa using previous models. The upper panel shows wo ses of esimaed parameers (wih p-values in ialics). The row labeled Full includes a $300 million Fed inervenion in he las 5 percen of he sample, Augus 7, 99. The row labeled No Fed excludes his inervenion. The boom panel reproduces he previous esimaion of Lyons (995). All esimaes muliplied by 0 5. ( ˆd ) ε, Table 4. Model Esimaes ou ( q + γ ) = b ( I I ) + q + I = a + 0 ai + + ε, j b Adj. R a 0 a Adj. R I Full No Fed Table 4 shows he linear regression esimaes from he auxiliary regressions in rows () and () of Table, Augus 3-7, 99. P-values obained using Huber/Whie robus sandard errors are in ialics below poin esimaes. In he lef panel, he esimae ˆb represens he proporion of he undesired invenory ha he dealer rades ou. In he righ panel, I * indicaes he desired invenory posiion.

23 - 3 - Table 5. Cos of Liquidiy ou ou Incoming Liquidiy: q = δ ( v p ) + X ; Ougoing Liquidiy: q ( µ + αq ); j δ Incoming Cos of $0 M Sample Pips Dollars Percen α Cos Raio Full $ % No Fed $, % Table 5 shows he srucural parameer values recovered from he coefficien esimaes. δ measures he change in he incoming order from changing he price (in millions of dollars). The Incoming Cos of $0 M columns measure he cos of aracing a $0 million order (sandard size) in: pips (DM0.000), US dollars, and dollar cos as a percen of he $0 million order. α measures he price impac of augmening he ougoing order by $ million in pips. The final column measures he raio of coss of dealing $million hrough ougoing order flow (numeraor) versus incoming order flow (denominaor). Table 6. Incoming versus Ougoing Order Flow Daily Avg. Size (Abs. Val.) Daily Volume Raio Trades No. Median Average Median Average Incoming Ougoing Table 6 shows he observed rades ha he dealer made, Augus 3 7, 99. Incoming refers o rades ha he dealer made when conaced by ohers. Ougoing refers o rades ha iniiaed by conacing ohers. Size is he absolue value of he order size; he median and average are given. Daily volume raio gives he raio of he average incoming daily volume (average or median size imes average number of rades) o he average ougoing volume. These do no include brokered rade, for which he iniiaor of he rade is unknown.

24 - 4 - Table 7. Cos of Inervenion () () (3) (4) (5) Fed moves his many pips per $00 M Orders induced for an inervenionequivalen ($300 M) price change To Closing To High To Closing To High (Inervenion Sar) * :3 PM $55.3 $.5 -(9.0) (0.0) (Oher Sudies) Evans & Lyons (999) 5 Dominguez & Frankel (993) 8 (Oher possible sar imes) :0 PM $355.7 $53.90 (58.0) (87.0) :6 PM $8.76 $ (46.0) (75.0) Table 7 shows cos comparisons for he $300 million Fed inervenion on Augus 7, 99. The exac sar ime and sequence of he inervenion is unknown. Column () liss hree possible imes. Columns () and (3) show he price change in pips (DM 0.000) ha a $00 million Fed purchase induces from he sar of he inervenion o he closing price (), or high price (3). Columns (4) & (5) show how many millions of dollars he dealer could induce by changing her price he number of pips as he $300 million inervenion price change. * Wall Sree Journal, Augus 0, 99: The Federal Reserve Bank of New York moved o suppor he U.S. currency... as he dollar raded a.470. This is he mos precise documenaion available of he inervenion sar, and ha price corresponds o :3 pm. Oher imes seleced because of repors of a mid-day sar, and because beween :6 and :3 pm, he price jumped 36 pips, suggesing a possible inervenion sar here.

25 - 5 - Appendix I APPENDIXES I. MODEL SOLUTION Invenory Carrying Cos From equaions () and (3) he variance of he dealer s porfolio is σw = σ VI + σ y + Iσvy. (8) σ vy Add and subrac ino () o ge: σ v σvy σvy σvy σvy c = ω σ y σ + VI + Iσvy + ω σ = y σ + v I (9) σv σv σ v σ v Which is he righ-hand-side of () wih coefficiens: d σvy σvy I = φ = σv φ0 = σ y. (30) σv σv Dealer s Beliefs Given marke demand q j, he dealer creaes a saisic based on he inercep of he demand curve, which is independen of her price. Denoe his saisic as D. D = qj + δ p = δ( v p) + X + δ p = δv + X. (3) From he signal of marke demand D he dealer forms wo saisics. The firs is an innovaion in he full informaion value of he risky asse, which shall be denoed as s. The second is a signal of he liquidiy demand, which is denoed as (lower case) x, and will depend on he esimae of full informaion value, µ. X w = δ D = v + δ ; E[ w] = v (3) x = D δµ, E[ x] = X. (33) Consisen wih raional expecaions, assume ha he dealer s previous esimae, µ is he seady sae disribuion over he rue asse value v, and ha he variance of µ is proporional o he variance of w. Hence, one can wrie σ µ =Ω σ w. Given he variance of w, form a signal o noise raio given by: σ v, ϒ= wih σ w = δ σ x. (34) σ w The dealer uses he recursive updaing of a Kalman filer o form he expecaions over v. This implies ha she updaes he prior belief µ using he curren order flow w. The resuling poserior, µ, converges o a seady-sae disribuion whose ime varying mean is an unbiased esimae of he rue value of v. The recursive equaions o generae his esimae are given by:

26 - 6 - Appendix I ϒ+ ϒ + 4ϒ Ω=, (35) Hence, if he dealer had only informaion based on he incoming order, she would use he Z following esimae, which is denoed as µ, as he esimae of v : Z µ =Ω w + ( Ω ) µ. (36) Noe, however, ha he dealer also receives informaion for updaing µ hrough a linear funcion of he invenory shock which is denoed by κ ( γ ). Given κ ( γ ), an unbiased esimae of v is given by: γ µ =Ω [ µ + κ( γ )] + ( Ω ) µ = µ +Ω κ( γ ), (37) where he same Kalman filer algorihm as defined above is used. Hence here are wo signals γ of v a he ime of seing he price. Given he assumpion, he variance of µ is a linear Z funcion of he variance of µ. Tha is, Z γ var( µ ) = σ Z, var( µ ) = σz * τ, (38) where τ is he elapsed clock ime beween incoming order (-) and. The opimal signal for he dealer is hen: Z γ µ = ηµ + ( η ) µ = η [ Ω w + ( Ω ) µ ] + ( η )[ µ +Ω κ( γ )]. (39) wih η ( τ + τ ) =. Now grouping and rearranging: µ µ = η Ω( w µ ) + ( η) Ω κ( γ ) = ηω( δ D µ ) + ( η) Ω κ( γ ) (40) X Since w = δ D = v +, δ µ µ = ηω ( δ ( qj + δ p) µ ) + ( η) Ω κ( γ ) (4) Add and subrac δµ o ge: µ µ = ηωδ [ qj δ ( µ p) + δ ( µ µ )] + ( η) Ω κ( γ ) (4) Solving for ( µ µ ) yields, ( µ µ )[ Ω η] = ηωδ [ qj δ ( µ p)] + ( η) Ω κ( γ ) (43) Which gives he final relaionship for he updaing: ( µ µ ) = ξs + ξκ( γ ), (44) Where s = qj δ ( µ p) is he unexpeced order flow, and ηω ξ ( η) Ω ξ ξ = & > 0; ξ = & < 0. (45) δ( Ωη) η δ( Ωη) η Hence, ξ and ξ are inversely relaed wih respec o η, and as inerransacion ime is longer, more weigh is placed on he unexpeced incoming order flow signal s. Here, κγ ( ) is assumed o be some simple linear funcion: κ( γ ) = ω0 + ωγ, where ω0 may be assumed zero if desired.

27 - 7 - Appendix I The Dealer s Problem The dealer s problem is reproduced here: J( I, x, µ, K ) = E ρ v% I + K c + ρ J( I%, x%, % µ, K% ), (46) max {( )[ ] } ou p, q subjec o he following evoluion consrains: i ou E I% + Φ = I δ ( µ p) x + q, (47) i E x % 0 + Φ =, (48) i E % µ + Φ = µ, (49) i ( ) ( ou ou E K % + Φ = K + pδ µ p + px µ + αq ) q c, (50) For exposiional simpliciy, in wha follows he expecaion operaors on he evoluion equaions and he ime subscrips are dropped, and a forward lag is denoed by a superscrip. The firs order condiions are given by: p δ E[ JI I x µ K ] δµ δ p x E[ JK I x µ K ] ou ou q E[ JI I x µ K ] µ α q E[ JK I x µ K ] Subsiuing (5) ino (5), and assuming for now ha E[ J I x K ] : ( ', ', ', ') + ( + ) ( ', ', ', ') = 0, (5) : ( ', ', ', ') ( + ) ( ', ', ', ') = 0. (5) K ( ', ', µ ', ') 0 (I confirm his laer), price is: x ou p = µ + + αq. δ (53) Denoe from here on he value funcion wihou is argumens for noaional simpliciy, mainaining he convenion ha J () is he forward lag of J (). Furhermore, in wha follows a subscrip denoes he derivaive of he funcion wih respec o ha argumen. The envelope condiions for his problem are: d JI() = ( ρµ ) + ρe[ JI(')] ωφ( I I )[( ρ) + ρe[ JK(')] ] ; (54) Jx() = ρ ( E[ JI(')] pe[ JK(')] ) ; (55) ou Jµ () = ( ρ) I δρej [ I(')] + ρej [ µ (')] + ρ( δ p q ) EJ [ K(')] ; (56) JK() = ( ρ) + ρe[ JK(')] ; (57) Based on he envelope condiions, i is conjecured ha he value funcion akes on he funcional form: (,,, ) 0 ( d J I x µ K = A + µ I + K + A I I ) + Ax( I I ) + A3x (58) Using he conjecure, and he evoluion equaions, aking he derivaives wih respec o I and K updaing: d d E[ J% I (')] = E[ µ ' + A( I' I ) + Ax'] = µ + A( I' I ). (59) EJ [% K (')] = E[] =. (60) Plugging (59) and (60) ino (5) yields he opimal ougoing quaniy: A ( I d ou ' I ) = q. α (6) ou Subsiuing (6) ino (53) for q yields he pricing equaion:

28 - 8 - Appendix I ( ' d ) x p = µ + A I I +. (6) δ Taking he evoluion equaion for invenory, (47), one can subsiue (6) in for p and solve for I ' o ge: d ( + β ) I ' = I + β ( I I ) + ( ) x α, (63) wih A ( + δα) βα β = A = α A ( + δα) ( + β )( + δα). (64) Given he invenory evoluion of (63), one can solve for he opimal pricing policy funcion: d δa ( + β) + α p = µ + A ( + β)( I I ) + x. (65) αδ Taking firs differences of (65), and subsiuing in: ou δa ( + β) + α p = µ A( + β) Z + A( + β)( q + γ ) + x. (66) αδ Subsiuing he relaionship for he updaing of he µ given by (44) yields: ou δa ( + β) + α p = ψη s + A( + β)( q + γ qj, ) + ψ( η) γ + x (67) αδ Nex he conjecured funcional form of (58) is confirmed. Begin by aking he envelope condiion for x, (55), and solve for coefficiens A and A 3 of he conjecured funcional form s derivaive, which is: J% x = Ax + Aε 3 (68) Subsiuing he opimal policy funcions ino (55), as well as he updaed derivaives of he conjecured funcional form which are given by (59) and (60) yields: ρ( A ( + β) δ+ α) A = ρa( + β), A3 = 4 δα. Coninuing, he envelope condiion on I in (58) can be solved wih he conjecured funcional form s derivaive, which is given in (59). This ( ωφ ) yields A = ρ ( + β ). An economically sensible soluion requires A <0, hence, using he definiion for A, i is required ha: ωφ( + β)( + δα) β + = 0. (69) ρ( + β) This implies β (,0). As β, he righ-hand-side of (69) goes o negaive one. As β 0, he righ-hand-side of (69) is posiive. Hence, since (69) is a coninuous funcion, by β, 0 (69) holds. he Mean Value Theorem ( )

29 - 9 - Appendix II II. FURTHER ESTIMATION DETAILS Derivaion of Liquidiy Cos Parameers aˆ = + β; bˆ = ; cˆ = A( + β); β = α : A A( + δα ) A α α A( + δα) cˆ ( ˆ )( ˆ ) A b a ( a ) ( ˆ)( a ) = ( α), cˆ cˆ ˆ cˆ ( b ) ˆ b ˆ ab ˆ ˆ δ : = = ˆ. α α α A ( + δα) aˆ = + β = ( ) ˆ =, α A + δα a α ˆ cˆ ( b ) α A( + δα) ( ˆ )( ) ( )( ) ˆ ˆ = ˆ α = α A( + δα), a ab α cˆ ˆ ( b ) ˆ ˆ cˆ( b ) cˆ( b ) ( aˆ ˆ ) ( aˆ )( ) ( )( ) ( ) ˆ = ˆ ˆ ˆ ˆ a = α α A + δα b ab a b cˆ ˆ ( b )( aˆ ) ( ˆ ˆ ) = A ( + δα ), ab ˆ = A a ( cˆ ), cˆ( bˆ )( aˆ ) aˆ ( bˆ )( aˆ ) ( ab ˆ ˆ )( cˆ ) ( ˆ ) ab ˆ ( bˆ )( aˆ ˆ ) ab ˆ ( ˆ ˆ ) = δα ab α = ˆ cˆ ( b ) ab ˆ ˆ ( bˆ )( aˆ ) aˆ ˆ ˆ b b aˆ ( ˆ ( ˆ ) )( ˆ ˆ ) ( ˆ ( ˆ ) ) = = ( + δα ),,, = = δ ; c b ab c b Linear Approximaions o he Model ( ), The following linear approximaions o he model were also esimaed. The parameer esimaes obained were used o iniialize he non-linear leas squares, as were zeros. Sup-F ess for muliple srucural breaks were also run on hese equaions and were generally no found. The esimaes were consisen wih hose found here and are available from he auhor upon reques. Below, Z represens he observed incoming order, q j. ou p = β + β ( η) s + β ( qˆ Z ) + β ( η) ˆ γ + β D + β D + ε p = β + β ( η) Z + β ( qˆ Z ) + β ( η) ˆ γ + β D + β D + ε ou p = β + β ( η) s + β Z + β ( η) γ + β D + β D + ε p = β + β ( η) Z + β Z + β ( η) γ + β D + β D + ε ou sage : p = β + β ( η) s + β ( qˆ Z ) + βηγ ˆ + β D + β D + ε ou sage : p = β + β ( η) s β ( qˆ Z ) + β ˆ β + ˆ βη ) ˆ γ + β D + β D + ε (

30 Appendix II Insrumenal Variable Selecion I is difficul o find a good insrumen for γ since i is a financial marke shock. The insrumens chosen were a lead and a lag. Using only a lag, or only a lead, lowers he correlaion of he insrumen wih he variable. γ = L() + L( ) γ. ( )

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