Continuous Time Decision-Making in a Partially Decentralized Multiple Dealership Forex Market, and the Equilibrium Exchange Rate.

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1 Contnuou Tme Decon-Mkng n rtlly Decentrlze Multle Delerh Fore Mrket, n the Equlrum Echnge Rte Ale Dervz ABSTRACT The er ntrouce moel o /k rce ormton n n merectly centrlze ore elerh mrket n contnuou tme. The eler hve cotly cce to et quote whle nterretng gnl rom the jont eler -cutomer orer low n ecng uon ther own rce quote n ctvte n the nter-eler mrket. The lowet k/hghet rce o oregn currency ecome the ngle trncton rce n equlrum. Ech eler ue the oerve orer low to mrove the ujectve etmte o relevnt ggregte vrle, whch re the ource o uncertnty. Thee re: return on ometc n oregn et, the equlrum FX trncton rce n the ze o the ggregte eler trncton n the FX mrket. Thee uncertnte hve uon orm n re elt wth ccorng to rncle o ortolo otmzton n contnuou tm e. The rce normton content o the locl orer low oerve y the eler, h to o wth the etence o glol nvetor n the eler cutomer e. Glol nvetor hve cce to -k quote o more thn one eler, long wth ueror normton out unmentl. Thee two ctor hel them generte etter etmte o the uture nter-eler trncton thn oth the locl nvetor.e. thoe who only tre wth ther ngle locl eler n ny o the eler themelve. otn the ormul or the eler re roun the equlrum rce. lo how how normtonl cutomer-eler ymmetre generte non-ttonry cumultve orer low well evton o the nter-eler trncton rce rom uncovere rty o ntonl et return. JEL Clcton: F3, G, G9, D49, D8. ntroucton Clong the ertent g etween oregn echnge mrket mcrotructure moel n trtonl chter o nterntonl mcroeconomc n mtou theoretcl chllenge tht h lo ecome n ncrengly mortnt ue n the eye o olcy mker ne the centrl nk. The ore mcrotructure theory orgnlly eclre the ntenton to el wth the echnge rte ormton n term tht coul e recognze y nncl mrket rcttoner, ooe to tetook mcroeconomc wom tht eem more oten thn not, to l to e relecte ccurtely y FX-trer. The rolem eeclly urgent n the eye o monetry uthorty whoe very ron être the reume lty to mlement ere mcroeconomc ojectve through cton tken n the money n FX mrket. At reent, however, the et-known mcrotructure moel oerte wth concetul hortcut n normton-theoretc contructon tht mke ther mege even more, not le, tnt rom the lnguge oken n the eler room thn tnr rooton o clroom mcroeconomc. 3 A rolem n tel to erve rom ny o thee moel n emrclly menngul corollry tetle on vlle t. Monetry Det., the Czech Ntonl Bnk, N rkoe 8, CZ-5 3 rh, n nt. o normton Theory n Automton, o vorenkou vez 4, CZ-8 8 rh 8, Czech Reulc; e -ml: [email protected] or [email protected]. Comment n vlule vce y Rch Lyon rom U.C. Berkeley re grteully cknowlege. The uul clmer le. 3 The uthor eerence e on contct wth eole rom the oen mrket oerton ertment o numer o reerve nk h own,.e. the Czech Ntonl Bnk, the Bnk o Engln, n numer o other

2 The reent er nvetgte the lty o contnuou tme ortolo otmzton moel or FX eler n nvetor, to hghlght two grou o henomen n the FX mrket. The rt re the long run roerte o the echnge rte, uch generlze uncovere rty or comettve quotng y multle eler roun clerng rce. The econ grou, whch more hort run y nture, reer to normton etrcton roceure n Byen ele utng y rtcnt n the nter-eler mrket n the ce o chngng unmentl. n the coure o our nly o the hort-long run ctor nterly n the ecentrlze FX mrket, we hll ocu on the unmentl roerty clle the uncovere totl return rty UTR or the echnge rte. UTR octe the echnge rte eectton or gven ero wth the erence n totl return ntntneou ven over rce lu ctl gn on r o rereenttve ecurte. Thee totl return conce wt h yel to mturty n contnuou tme. Thu, return quote on econry mrket n not the money mrket rte whch re re-ene or the tme ntervl n queton, conttute the contnuouly ute meure o the eecte move n the echnge rte ee Dervz, 999, or etl. UTR woul e nturl reult n, rctclly, ny reonly ene ynmc moel o nterntonl et rcng n hll relce the much-comrome uncovere nteret rte rty o nïve no-rtrge moel. Emrcl vercton o UTR lo rove romng reult. Th rty theorem woul e vl ectly n ermnently n Wlrn uctoneer ettng whch gnore mcrotructure, wth rereenttve gent n mrket clerng t ech moment. However, when one tue the echnge rte ormton n elerh mrket, evton rom the unmentl UTR come out nturl conequence o gent heterogenety. Thee evton cn e ue to chrcterze the current tte o normton n nventory low etween eler n nvetor. The rooe moel etlhe lnk etween n nvully oerve orer low, Byen lterng o mrecely known unmentl y the eler, n eemngly non-ttonry ehvor o the country remum n the UTR ormul. The reult otne n th er tht evton rom UTR re mot ronounce t tme when ome eler re lernng new normton rom ther clent through the oerve orer low. Durng uch ero,. the echnge rte n the nter-eler orer low vrlty mut e hgher thn uner comlete ymmetrc normton. ynmc roceng o new gnl y the eler le to evton rom UTR tht ert untl the new normton ore; uner certn crcumtnce, Byen lernng y the eler cn le to ermnent revon o the country remum level n the uncovere rty relton. n th ene, the moel ecre tuton when the orer low eect re ertent. n orer to hghlght the rght rce covery roce trtng wth the en-uer o the oregn currency nvetor n gong through the eler who lern rom them, the etng mcrotructure lterture mke hr tncton etween clent n eler. Th hen oth n ngle eler equentl tre moel uch Gloten n Mlgrom, 985, or Eley rom emergng econome. The concluon, not urrngly, tht eler uully well vere n the uncovere nteret rte rty rgument n even know tht t oe not hol. He/he cn lo ely comment on the gnl mct o relee reh t out nlton or GD, on the echnge rte movement. t, however, totlly unreltc to eect eler to nlyze the ertng veru oolng nture o equlrum n trer-eclt gme o Eley n O Hr, 987, or Gloten, 989. Scholr my ue the ltter moel to nlyze the cton o rel elerh, ut not to communcte wth them.

3 n O Hr, 987, wth ngle rk-neutrl mrket mker, n n multle eler moel o the Evn n Lyon, 999, tye. Th tncton rely ole n rctce, n mot o vlle rce n tre ze t crry no tm o the uroe o ny gven FX-trncton. More mortntly, n rel le, the role o nvetor n eler n the ore mrket re oten comne n the erormnce o cooertng rt wthn one n the me rm. Mot tyclly, mjor commercl or nvetment nk h eler who ervce t cutomer n rtcte n the nter-eler mrket, ut lo oerte roretry ek tht eerce FX tre or the nee o t own ortolo mn gement. Acce to electronc cro-orer ecurty tre n normton ytem not rvlege o eler ether: the communty o Reuter n/or Bloomerg quote creen uer nclue conerle rt o nterntonlly ctve comne o very vertle role. Acce to thee clte ecome hghly ttrctve to ny comny whoe cro-orer oerton ttn certn ze. n vew o the gven crcumtnce, emrclly tetle moel o FX-tre cnnot, n houl not, rely on the trc t clent-eler erton. The roch tken here ume tht cce to the nter-eler mrket ole or non-emty uet o nvetor, who re le to oth erch or the et eler quote n eerce tre wth the eler they chooe. The only tncton etween uch glol nvetor n eler n the roer ene tht the ltter voluntrly tke on the uncton o mrket mker n therewth - the commtment to rove two-wy quote n tre wth nyoy rochng them. Art rom tht, eler hve the me ntrnc motve or holng oregn currency oton ny other nvetor, nce they re ctve n the me lne o nterntonl une the ltter. t th ynthetc eler-nvetor tht we re gong to moel. The mjorty o ore mcrotructure moel elng wth ymmetrc normton ene the gnl une etmte o ome true or unmentl vlue o the echnge rte. Th eture they hve nherte rom mcrotructure tue o other nncl mrket egment e.g. equte, where th true vlue octe wth revele ernng or ven t. However, n the ce o the echnge rte, no uch revelton o olute truth et. Thereore, creul choce o the content o the gnl, well jutcton o t relevnce or the rce ettng, neee. n the reent moel, th tk ullle y lettng the gnl rechng the eler contn normton out the other e o the mrket,.e. the ggregte ehvor o the recevng e o ther outgong tre. Th llow u to cloe the moel n nturl wy n erve ome ueul roerte o the rng equlr. Another eture o etng moel o nter-eler mrket tht comlcte emrc l lcton, the trct equencng o normton-len event n them. C. Evn n Lyon, 999: rt roun o eler tre wth cutomer gnlng unmentl normton, then n nter-eler roun to hre nventory rk evenly etween them, then nother roun o cutomer tre to unwn oen oton. On the contrry, t on FX tre o gven nncl nttuton relect rnom equence o event ertnng to own nvetment nee, the nter-eler n the cutomer-eler oerton. n other wor, the three nme ctegore o trncton generte three multneouly evolvng rocee. At ech moment, the eler mut eerce ele ute e on wht he h oerve o r. He hrly le to y whch trng roun norm hm out wht n rtculr. Uner uch conton, contnuou tme tochtc moel o the eler ehvor er to e the mot convenent. A non-neglgle rgument n vor o contnuou tme eler moel wth uon uncertnty t etter nlytcl trctlty comre to crete tme moel wth rtrry tttc o rk ctor. There lrey et lne o lterture tht evelo the nng o 3

4 the Kyle, 985, rk-neutrl mrket mker moel. For emle, Bck, 99, Bck n eeren, 998, n Bck, Co n Wllr,, work wth rce n cumultve orer n ecurty mrket lcle to ore n emmrtngle orm, wth uon comonent orgntng n the cton o noe trer. When one goe over to otmzng moel or rkentve gent, the tochtc mmum rncle ecome n even more owerul tool o nlyzng equlrum rce n orer low ynmc n contnuou tme, once the eler otmzton rolem roerly ene n th ettn g. The crucl chllenge here to enty n nterret the normton ymmetry n Byen ele utng henomen n ther role n the otne oluton. 4 Th tk n contnuou tme ortolo moel o the Co-ngeroll-Ro tye wth heterogeneou ele w rt ree y Detemle, 986. Ztero, 998, ue the me roch to erve reult on et rce voltlty uner ymmetrc normton n contnuou tme. Snce ech eler enjoy locl cutomer e tht tre wth hm ecluvely, there enough ce or non-trvl -k re n h quote. The well-known reon or etence o re, ente n mot moel wth normtonl heterogenety o trer, nmely the comenton o loe rom tre wth the norme, y rot me n tre wth the unnorme, mlctly reent n th ettng well. Ovouly, the role o the norme trer lye y the glol, whle tht o the unnorme y the locl trer. The ltter, when they tre t gven re, re uject to the eerce o monooly ower y ther eler. Th unertnng cloe to the e o Coeln n Gl, 983, n errun n Vtle, 996. However, the reent moel le to come u wth n elnton o non-trvl n vrle -k re even when there more thn one eler n the mrket. The etence o glol nvetor eten the cometton rom ure nter-eler ntercton to the here o eler-cutomer relton. n th reect, they eem to e key ctor n the uncton o ecentrlze nter-eler mrket, whch tycl or ore tre. The etence o normtonl heterogenety etween nvetor grou o erent reence votl n genertng non-trvl orer low n tonl ource o echnge rte voltlty n our moel. Tht why t mortnt to note tht uch heterogenety h lrey oun relecton n nnce lterture. For ntnce, Brennn n Co, 997, rgue tht ymmetry etween ometc n oregn nvetor n the knowlege o ometc et return h n mct on the recton n volume o cro-orer equty tre. More ecclly, recent workng er y Sehole,, rgue tht lrge oregn nvetor n emergng mrket hve n normtonl vntge n ch n hgher return on ometc equty, over the mjorty o ometc nvetor. Thereore, cro-orer normtonl ymmetry oe not lwy men n vntge or the reent. Our moel hre the Brennn n Co the tht reency-e normtonl heterogenety mtter, ut lo llow or erence n normtonl enowment etween locl n glol nvetor, n ccornce wth the Sehole conjecture. The er tructure ollow. Secton ecre the moel vrle, the tructure o uncertnty n the eler otmzton rolem Suecton., ter whch the otml eler trtege or ve t own quote n ctve t other eler quote FX tre re erve Suecton.. lo how how non-zero -k re n comettve m- 4 The revouly mentone contnuou tme moel o FX tre y Bck n collegue re e on Kyle uctoneer n o not nclue elct eler. Bee, thee moel trct rom the uul Brownn et return uncertnte ce y n nvetor n concentrte on the mrket nterretton o nrrowly ene eogenou gnl out the oregn currency vlue. 4

5 quote re reult o eler otmzton. Secton 3 nlyze the long-run FX trncton rce ynmc reultng rom the uncovere totl return rty. Secton 4 cue orer-low nterretton n utng o ele y eler, leng to evton rom uncovere rty o return. Secton 5 conclue n ncte olte o uture eveloment o the moel.. The nvetor-eler econ rolem. The moel The worl economy lt nto the home country wth legl tener M n the oregn country wth legl tener. By the echnge rte we unertn the rce o n M-term. nvetor reent n the home country ccount ther oerton n M, thoe reent n the oregn country ue the ccountng unt. 5 Among the nvetor o oth reence there uet o thoe who oer ore eler ervce to the other. Tht, they gree to rove n k quote whch the rce t whch they gree to ell n echnge or M, n rce t whch they gree to uy n echnge or M. We wll cll them nvetor -eler or mly eler. Ech eler h locl cutomer e o thoe nvetor who only eerce ore tre through hm t gven quote they re ree to vry the tre quntty rom zero to nnty eenng on the quote they ee. Bee them, there re other glol nvetor who re le to erch or the et quote mong ll eler, t cot. Every eler hmel h the lty o cotly covery o the et quote mong other eler. One o the tncton etween nvetor -eler n nvetor wthout the elng ccty tht the ormer ctegory ce lower cot o quote erch. Another one the rorty o the non-eler nvetor n lernng the new out unmentl, to e cue n Secton 4. n the ce o mller FX mrket e.g. Euroen currency not elongng to the euro zone or n emergng economy currency one cn looely octe glol nvetor wth multntonl n nterntonl nncl cororton reent n the locl economy. Altogether, the orer low tht eler oerve cont o three rt. The rt the uly o or emn or rom h locl cutomer, eenent only on h own quote vlue n n the economc unmentl nluencng the cutomer e. Th low unoerve y other rte. The remnng two orer low comonent come rom the nter-eler mrket. Thee re the orer o thoe glol nvetor n other eler, whoe quote erch h reulte n the choce o th eler or the ere tre n the gven ero. For every rty nvolve, rtcton n the nter-eler mrket the ource o n normton ute out oth the glol unmentl vrle n the uture echnge rte move, re o the receve tre. The nter-eler mrket rtlly ecentrlze n the ene tht ech rtcnt only oerve h/her own tre, where the knowlege o tnng et quote the reult o cotly erch or ech outgong tre. Thee erch cot wll e ncororte nto the overll trncton cot to e ene n erte rgrh elow. One cn thnk o n gent wth cce to electronc n voce roker who ervce rt o the mrket, ut who unure whether etter rce cnnot e oun y rochng certn eler nvully. Whle not moelng thee treo etween roker n nvul erch elctly, we ot the unertnng tht the et rce cn lwy e oun, ut the reource ecte to the erch grow ter thn the ere trncton mount. 5 The moel to e ene wll ccommote the Jenen nequlty-relte roerte o the eecte echnge rte chnge the reency-e ymmetry o the ltter n nve no-rtrge ettng known Segel ro. Thu, erent reence wll not cue orml contrcton. See Dervz, 999, or the etl o Segel ro reoluton n the correonng cl o moel. 5

6 The nterntonl nvetor eercng the uncton o n FX-eler wll e moele n gent chrcterze y our tte vrle. Thee re: ometc ch holng, holng o the comote ometc et, oregn ch holng, holng o the comote oregn et. The ometc et y out rnom rte o return r n M-unt. lo ume tht mntnng the nvetment ortolo requre contnuou nut o un mngement cot wthout whch the vlue eterorte t tochtc rte α. Th wll e convenent nce one cn then coner equlr wth contnt verge level o et holng. Let one unt o cot X unt o M. Anlogou vlue or the oregn nvetment ortolo re enote y r, α n X, reectvely the ltter rce, nturlly, n -unt. t convenent to ume tht the ch holng n ern ther own rte o return r n r, reectvely. n the mlet vrnt, t cn e the overnght money mrket rte or t roortonl rt the elementry ero ntr-y. More reltclly, one houl nclue n the ch vrle the nter-nk lon/eot oton n FX w oton. n tht ce, r r the ntntneou rte n the ometc oregn money mrket on the correonng orton o,.e. the genertor hort rte o the ometc oregn term tructure. For ny trctly otve to roce z, ymol z wll e horthn or z/z. The rt n uon coecent o the et return n rce rocee ntrouce ove wll e enote ollow: r = t Z, r = t Z, r = n t N Z, r = n t N Z, t α = A Z, α = t A Z, X = π t ρ Z, X = π t ρ Z. Snce t not the uroe o th er to tuy GARCH or other non-contnt vrnce eect n et rce, ll uon coecent, N, A n ρ re ume contnt. to rocee r, r, r, r, α n α elong to the eogenou ource o uncertnty n the moel. Jontly, thee rocee, together wth two more to e ntrouce n Suecton n Secton 4, reectvely, generte the normton lter F=F t t to whch ll tochtc rocee erng n the moel wll e te. F t enote the rtton o the event ce correonng to the ull normton out the oerton o the mrket, on whch the ojectve rolty meure vl t tme t ene. Let the uon te to F e nne y vector Z o mutully neenent tnr Brownn moton. The current FX trncton rce A mentone ove, t ume tht oth the glol nvetor n the nvetor -eler re llowe to erch n the nter-eler mrket or ccetle quote. Accorngly, let u ntrouce the noton o the et ttnle n k rce or gven eler, n. Thee re goo or one unt o. Tre ze -eenent non-lnerte wll e ccommote y men o conve trncton cot ee elow. Aumton For ech glol nvetor n ech eler, the et nvully ttnle hghet rce equl to the et ttnle lowet k rce: = =. The erence etween the et/tnng k n ometme clle ne re, or touch. Aumton et th ne re to zero. The rtonle o th umton le n the n turl normton-emnton lty o the nter-eler mrket n comettvene o t rtcnt. nee, one oerve >, then there woul e cler rtrge oortunty. n the oote ce < there woul e no nter-eler tre t ll untl the 6

7 too hgh k went own n/or the too low went u, to meet ech other n rove gn rom tre to thoe who eze the oortunty rt. Tht, we eleve tht there wll lwy e n gent who wll ee the elocke mrket n unercut/over the tnng quote n h vor. A oon ll the rtrge n mrket hre rorton olte hve een ehute, there remn grou o nter-eler mrket eller n nother grou o uyer. They trnct t the mutully ccetle rce. The uyer k rce n the eller rce quote re non-comettve: ove n the rt grou n elow n the econ. Thereore, the uyer o not hve to ell n the eller o not hve to uy. To ck u the legtmcy o Aumton, rt note tht the ne re mnhng wth the numer o cometng eler one o the outcome o the moel y Ho n Stoll, 983. Ther moel lo rect tht re o nvul eler woul not ll to zero even when ther numer ecome lrge. Th n lne wth the reult to e otne n the reent er. On the emrcl e, Goohrt et l., 996, oerve tht ntereler FX re on mjor currency r re uully tny mll or / o the ont or tnr mount. However, thee re the numer reorte y the Reuter D- ytem o electronc rokerge,.e., gn, the ne re, gvng erect mtch wth Aumton. The eler oulton lt nto eller n uyer t every gven moment. The mot trvl reon why omeone ecome eller n omeone ele uyer the heterogenety o et enowment. But, even n the ence o the ltter, there olty o erence n rvtely ccounte net return on et etween eler. For ntnce, oregn reent my hve hnc wth regr to ttrton n ther ometc et holng, n vce ver. Dometc eler my enjoy rvlege cce to the ometc money mrket, wth reultng hgher return on ometc ch lnce, n ymmetrc vntge my e enjoye y oregn reent wth reect to oregn ch. Other tye o heterogenety cn e mgne well, nclung thoe tht vry over tme. Th er wll ocu on heterogenety o ele. The lt-oerve nter-eler trncton rce the one ue n the eler ccountng. Ech gent vew t trctly otve to roce wth coecent ene y = π t nvetment erormnce meure For the lter uroe o enng reerence over the th o nvetor-eler cton, we re ntroucng the notonl ven rte δ to e ccounte or n every nnteml ero t. Our moel hll e le to work wth very hort tme te uch thoe o the rel-tme t o electronc FX-tre ytem. n th ce, the ene ven rte hll e unertoo n mgnry nnteml contruton o the eler ertment to the ven un o the rm, whoe ntegrl over longer ero, n relty, eng wthrwn rom the ch lnce t crete ntervl. Hvng th nnteml ven rte n the moel convenent wy to mke the eler ccountle or the ultmte erormnce o the FOREX ctvty o the rm, n, t the me tme, rng the reerence tructure o the gent cloe to the tnr ortolo otmzton etu. rce erch cot n orer to generte non-trvl uly n emn cheule or ntnce, Gloten, 989, ue non-lner rcng cheule n h nvetor -eler gme n oth monooltc n comettve eler ettng cng the other e o the mrket, t uul to ene trncton cot tht grow n conve wy n the trncton volume. t well known tht oervle trncton cot ncurre y g rel-le nvetor, let lone eler, re ρ Z. 7

8 uully neglgle or mll/tnr mount, ut grow rly when the tre ze ecee the tnr. Accorngly, non-lner rcng cheule re relty. Bee tht, ome orm o generlze trncton cot eem to le ehn the eler ehvor, even they re not rely meure. n th er, we tke the vew o thee cot rng rom the quote erch n the nter-eler mrket. Nmely, t logcl to ume tht n orer to tre volume v t o t rce whch the lt oerve hghet v < n the lt oerve lowet k v >, one mut y k v t unt o M. Th um ecome recet n enter the lnce wth negtve gn, ol,.e. v <. Here, k trctly ncreng n trctly conve uncton wth k. The mot nturl emle woul e qurtc uncton o the orm = k v=vk v /. For reon o contency, one houl ume the me erch cot mechnm to e reent n the two other mrket egment well. Tht, mount v n v o the ometc n the oregn et re urche er ero ol negtve, the M- n -lnce re reuce y mount X k v t n X k v t, reectvely. The eler orer low A w mentone n the ntroucton, t etremely cult to otn erte t on the cutomer n the nter-eler comonent o eler orer ook. nte, we tnguh etween the locl cutomer o the gven eler o not hve cce to or eerce FX tre wth other n the rtcnt o the nter-eler mrket. Roughly, the ltter et nclue ll thoe who cce the me creen. Both the nvetor who erch mong vlle eler quote glol nvetor, n eler themelve, elong there. Accorngly, the r o k n -rce,, nnounce y the eler, hll ervce everyone t the moment o nnouncement. However, we ume tht the ele r le to tnguh etween loclly rkle non-uon emn D t, loclly rkle uly S t, n rnom low Λ/ = ltz/, whch cn go oth wy, n h orer low. Altogether, the ccounte chnge n the eler M-lnce, nuce y tre t h quote, equl to D S t lt Z = D S t Λ, whle the correonng chnge n the -lnce [ S D ] t lt Z = [ S D ] t Λ. Oerve, n rtculr, the lt uon term n n : Z n Z. They relect the eler uncertnty out the orer low rom oth the glol nvetor n other eler. nee, one cn generte the eler uon-tye orer low lmt o two mll-ntervl crete low o very mll urche t n very mll le t wth non-zero egree o rnomne n the recton o the tre n t ect mgntue. Then the rt rt o the lmt low wll correon to the emn n uly tht re certn urng the nnteml ntervl t, nmely l l D t n S t, where l l =l. 8

9 The uon rt relect the remnng uncertnty. Conerng the eler n M-reent, we ccount or the rnom orer low n M-term y ereng t tochtc growth rte o the M-ch ccount. t cn e hown tht the -oton n tht ce evolve ectly hown y the lt term n equton 6. The reence o -roortonl rt l n the rt comonent o the orer low elne y the olty o the rolty revon y the eler n ccornce wth the Grnov theorem. The conequence o uch rolty meure chnge n the ce o Byen ute o ele wll e cue n Secton 4. the eler were oregn reent, the uncertn rt o the orer low tht he ce woul e -roortonl, n the certn FX uly - n emn-entre n equton,, comng rom the locl cutomer, houl e moe ccorngly. Summng u the enton gven ove, we come to the ollowng ytem o tte-trnton equton or the tte vector =[,,, ] T o n nterntonl nvetor wth the eler uncton wth ometc reence: = r r δ t X k v t k v t D S t lt Z, 3 = α v t, 3 [ S D ] t lt Z = r r X k v t v t, 3c = α v t. 3 reerence The moel n contnuou tme tht eng ene n th er, ntene to e lcle to hgh requency, e.g. ntr-y, t. Wth uch hort tme te, t not ovou whether tme reerence/count ctor o the tnr otmzton moel lcle. We hve lrey gree tht the ven rte δ n equton 3 to e unertoo n ntntneou contruton to the ven un, to e ntegrte over nte tme ero. Let there et uncton u tyng the conventonl growth n concvty conton, uch tht uδ meure the ero utlty o the hreholer, erve rom recevng contruton δ to the ven un. Smlrly, nnteml rte r n α o et return n eterorton n the moel cn e unertoo um o the er-ero contruton to the ntegrl eecte return over nte ero the rt rt n uneecte nnovton n thee ntegrl return the uon rt. Accorngly, n ntntneou tme reerence rte cn e ene n nnteml contruton to n ntegrl count ctor vl or ntely tnt ero n the uture. Another olty, to e elote here, to unertn th rte, enote y β, the rmeter o oon eth roce. Snce the reent moel nether requre eler to cloe the FX oton n reetermne nte tme, nor ccet the etence o n eogenou nl/unerlyng vlue o the currency, the ollowng contructon h the ojectve o relcng the rtect o the true lquton vlue uner covery, oten utlze n mcrotructure moel. 6 The trnton to the lmt o crete rocee generlzton o the well-known roceure o genertng geometrc Brownn moton lmt o rnom wlk wth rt, when the te ze goe to zero, e e.g. Dt n nyck, 994, Secton 3.. 9

10 For ech tme moment t, the eler wll hve to cloe own h ore tre une wthn the net tme ntervl t, wth rolty -e β t, n lqute the outtnng enggement n currency t current rce. The reult o lquton,.e. the lnce, evlute y men o trctly ncreng n concve et or equet utlty uncton G. A reult, t ny tme t the ometclly reent eler mmze E t e β t wth reect to control th δ, v, v, v { u δ βg } F t 4, t, uject to 3, gven the current vlue t o et holng. Note the ernce o rmeter β n ront o G n the ntegrn n 4. Snce, n ech ero, the rm lqute wth rolty -e -β, the eecte utlty erve rom the lqute oton equl to β β e e G = G βg. the eler ree ro, her lquton lnce enterng G hll e ccounte or n - unt n e ene. Otml olce o the eler. Followng the reult o the Aen, we cn chrcterze the oluton o the rolem 3, 4 y men o the how rce o the our hel et, whch re the jont rocee o the rolem erng n the mmum rncle. Gven the Hmltonn o the rolem, clculte n the Aen, the otml cton o the eler re chrcterze y the ollowng rt orer conton: δ = g, 5 k v =, 5 k v =, 5c X k v =, 5 X, = ε ε = where g enote the nvere uncton to u, locl -emn n y the eler. S S, 5e ε = the rce eltcty o the D D ε = the rce eltcty o the locl -uly, oth oerve

11 Further, ccorng to the jont equton tte n the mmum rncle o the Ae n, the how et rce ce y the eler ty the ytem o equton t G t lt Z t r lt Z t = β β, 6 = N r A N t βt α A t t = t r t G t β β r A N t βt α A, 6, 6c = t. 6 Nturlly, eler ree n the oregn country, then he ce the how rce whoe lw o moton mut e ymmetrclly jute veron o 6, n ccornce wth her unt o ccount. n other wor, n equton 6-6, one woul hve to wtch the ne r, n,. Alo, the term nlogou to Z n other term contnng woul er n equton 5c, 5 nte o 5, 5. Only equton 5e, chrcterzng the quote ettng, woul remn t. Sre An mmete conequence o the.o.c. 5e the ollowng ereon or the eler -k re: = ε. 7 ε eltcte ε n ε re hgh, the log o the rght hn e o 7 romtely equl to. The conjecture out hgh rce eltcte o emn n uly on the e o the ε ε eler cutomer e eem jute, nce the cutomer re not cng unque elermonoolt. t reonle to ume tht oon the vntge o trng wth monoolt or monoont ecome too event, cutomer cn lwy try to look u nother one,.e. turn glol ete the octe cot. n the equel, wll ume the etence o common uer oun or nvere eltcte e. ε n ε or ll cutomer Equton 7 woul come u n tnr wy n ny monoolt two-wy mrket mker rolem, regrle o the reence o uncertnty. However, the moel ene here contn two le tnr element. Frt, t le to generte re tht re vrle n tme n tochtc. Secon, y conerng eler who rtcte n comettve nter-eler mrket, the moel retrct the equlrum th o eler quote to thoe tht relect normton out unmentl, to the etent the ltter emnte y other rte cton ee Secton 4 elow. To ccount or the oerve vrle FX re, t necery to coner only uch uly n emn cheule ce y the eler, tht generte non-contnt non-etermntc rce

12 eltcte. Tycl unctonl orm or uncton S n D, nee, ty th requrement. To ee th, oerve tht ny nvetor cn e moele y men o equton nlogou to 3 n 4 ove, one elmnte the eler orer low rt rom trn ton equton 3 n 3c. Then the FX uly/emn cheule wll e gven y n equton ormlly entcl to 5. Deenng on the current vlue o how rce reltve to the oere mrket -rce, ome nvetor wll e eller n other uyer. n oth ce, the ggregte S n D tht the eler ee wll e lner uncton o the nvere rce, wth tochtc coecent. Such uncton gve re to vrle tochtc rce eltcte. The lw o moton 6 o the how et rce chrcterze them n term o the unmentl vrle o the ene economy. Thee unmentl n the reent moel re comre o vrou comonent o the totl et return ven, rce movement, erecton rte, lu vrle whch chrcterze the ggregte low o -un to/rom the ometc eler to e ene n Secton 4. The unmentl normton ccele to erent grou o mrket rtcnt cn hve erent qulty. olte to moel normton emnton rocee wth regr to unmentl wthn the reent roch wll e cue n Secton 4. But rt, here, to ee tht eler quote mut relect the rocee normton on unmentl comettvely therewth reucng monooltc welre lo eect, t necery to euce numer o roerte o ole mrket equlr. Th one n the net ecton. 3. Show rce rty, uncovere rty o totl return, n the ymtotc ynmc o the echnge rte The rt orer conton o otmlty 5 cn e ue to tte rly generl roerty o the echnge rte ynmc, whch cn e clle the generlze uncovere totl return rty conton. The unerlyng roerty o equlrum n the et mrket the me the one tht le to the conumton-e CAM. A mlr reult woul otn n mot nterntonl ortolo otmzton moel. n contrt to the very much crete uncovere nteret rte rty o the tetook, the totl return rty enjoy enough emrcl uort ee Dervz, 999, or moel n emrcl vercton; nother roch wth more t on leng currence cn e oun n Nl-De Smone n Rzzk, 999. To ormulte the reult, t necery to ene the totl ntntneou rte o return o the ometc n the oregn et: R = r X X α = y t υ Z r, R = X α = y t υ Z X. Net, ene the ulry vrle Y Y X =. X Vrle Y cn e clle the how rce rty ne. The reon the ct tht, ccorng to to lemm le to equton 6 n 6 or the how rce, t tochtc erentl te the ollowng roerty: Y = R R At. 8

13 Here, A the rty term the um o numer o covrnce term they come out conequence o to lemm. They re uully mll n, ccorng to our umton on the contncy o uon coecent o eogenou vrle, A contnt. The rt three term on the rght hn e o 8 ene the uncovere rty o totl return UTR or the echnge rte. Nmely, Y were contnt n the rty term A cloe to zero, then the reltve eecte chnge n etween tme t n tt woul e equl to the totl return erentl. Thu, the uncovere return rty cn e ormulte the equlty o the echnge rte erentl to the ntntneou return rte erentl etween rereenttve ometc n oregn ecurte lu covrnce term: = R R At. 9 A ror, there no reon to ee the how rce rty ne contnt. nte, y ntroucng the hortenng notton k = k v, k = k v, k = k v, we cn erve rom 5-5e the ollowng ereon or Y: k k Y =. k the mrket were chrcterze y the etence o rereenttve gent wth no elercutomer tncton n the clerng rce were et y Wlrn uctoneer, then ll ecurty urche rte woul hve to e et to contnt n equlrum. ole equlr woul then nclue one wth contnt et rce tren, contnt Y n the echnge rte tht te the uncovere return rty 9 ectly. To nlyze equlr n the r eence o eler, t ueul to mke mlyng umton out the ehvor o the et mrket egment other thn the ore. Secclly, ume tht the et holng n o every eler n nvetor oe long run verge lmt level,. Ech gent mly mntn the long run verge level o oth et holng n the ortolo y comentng, n the rt rt, or ther contnuou ttrton ecre y 3 n 3. Then the otm l urche rte v n v re otve contnt: = v, v v = v = =, mkng the et holng rocee revert to, n the men: = t A Z, t A Z =. Regrng the rce rocee X n X, we hll ume tht they re ure uon π =π =. The other three umton re me t lmtng the long run rce trjectore n the ore to the cl o oune one. Tht,. only equlr wth the echnge rte growth rte π oune rom oth e, wll e conere; 3

14 . the ttenton retrcte to econome wth entcl reerence n roceng cot uncton o the eler-ometc reent, eler-oregn reent, glol nvetor n locl cutomer e o ech eler entty n ech grou; c. ze o the clent currency uly n emn e, generte y non-eler nvetor re oune. The enet o the ove retrcton the etence o common equlrum uer n lower oun or nvul nter-eler orer v n, conequently, nvul rmeter k well. Now the otml quote equton 5e oer reult out the long run ehvor o the eler quote. rooton Uner the umton.-c. me ove, oth n k quote re ymtotclly equvlent to ± Y. ε, n the ove ormul, the how rce rty ne Y ymtotclly contnt. The necety or Y to e lmot contnt cler rom the etence o uer n lower oun or k. The ereon tel reult rom uttutng nto 5e the ollowng ereon or the rto /: k = Y k, whch ollow mmetely rom 5-5. The mege o the ove rooton the etence o trong lnk etween the eler ehvor towr h/her clent n the contrnt comng rom the nter -eler mrket. Although t otml or the eler to mntn otve re etween n k quote, the level on whch the m -ont etween n k et, nne own y the rto o the eler ch how rce. The ltter re te to the how rce o ometc n oregn et or, more ecclly, to the rouct o the lt oerve FX-clerng rce n the how rce rty ne. n other wor, the how rce rty ne govern the quote hng n the moel. Th ne unmentl- cro-orer et return erentl- rven n, thereore, comettvely etermne y ll eler n nvetor. Accorngly, no eler h n olute monooly ower over the clent. 4. Funmentl normton etrcton rom the orer low A uul n clent-eler moel o FX tre, eler n the reent etu re uoe to lern rom ulcly oerve event n rvte gnl contne n the orer low. n the reent moel, Byen chrcter o ele utng mle n the eler otmzton roceure tel, ollow rom the Grnov theorem le to the rolty chnge n otml control o uon ee Ellott, 98, or etl. However, t mole to rect unmguouly the recton o the echnge rte revon erceve rom the oute, reult o rtculr new rrvl n the correonng ele ute. Only the ect eccton o the lterng roceure y the eler cn gve tctory nwer. n th 4

15 ecton, we otn more ecc nwer out the equlrum trncton rce y mkng more ecc umton out the nture o ele n gnl. A reult, one wll e le to ormulte nrrower reult out the mct o ele chnge regrng the et return n the ggregte -un low, on the eler-nvetor cton. n generl, ny Byen ele ute n the reent moel mut e chrcterze y chnge n the rolty meure r tht eler ue n the otmzton rolem 3, 4. Uner the new rolty r*, roce Z nnng the eogenou uncertnte o the economy no longer Brownn. There nother vector, Z*, o tnr mutully neenent Brownn moton uner r*, relte to Z n ccornce wth the Grnov theorem. n t nve orm, t re Z=htZ*, 3 where h n F-te contnuou roce tyng numer o regulrty conton. The equlr o the moel cn e octe wth nvul trjectore o h tht generte the nter-eler mrket orer low trjectore v, recte y urchng eler towr ellng one ee the eler econ rolem n Suecton.. Every h lo generte the trjectore o the eler n k quote n the uyng n the ellng gr ou, n, therey, the clent orer low. The comlete ttement o ormton mechnm o h requre eccton o the ect lterng technque tht the gent ue when roceng the oervton. The mot oulr woul e, nturlly, the Klmn-Bucy lter. hll trt y ttng the c roerte tht requre mnml mount o lger ut llutrte the eler wy o workng wth the orer low. t wll hve to o wth the utng o ele on the eogenou vrle rt term ter new normton out the current tte o the economy h een regtere n relecte n the cutomer orer low. Frt, one hll ot the rule o clent gnl nterretton y the eler. To o th, ntrouce n tonl ource o eogenou uncertnty n the moel, the rnom roce M o the cumultve net outgong urche o oregn currency n the nter-eler mrket y eler ometc reent. n other wor, M meure o the cumultve nter-eler orer low rom M-reent to -reent. The rt n uon coecent o M re ene y M=mtµZ. Gven the homogenety o the ntonl eler oulton Aumton o Secton 3, we conclue tht m orme y ummton o entcl ctve nter-eler tre v cro the M- reent eler. Let u ene the oervton roce or the economy the vector wth comonent nvul ource o ujectvely erceve uncertnty n the moel. Tht, ene the vector uon roce Q y Q=[r, X, α, r, r, X, α, r, M] T. Symolclly, the evoluton o Q wll e wrtten Q=qtΘZ. Th the ynmc uner the ojectve rolty r, utlze y thoe who re le to enty the rt rt qt recely. To e le to enty Θ, t ucent to know only the vlue o Q cro tme, nce, wth th knowlege, the qurtc vrton o Q cn e comute. The ollowng umton mke ue o the ct. 5

16 Aumton Every eler oerve the current tte o the ytem, eree y Q, correctly, n ll eler gree uon the vlue o uon coecent Θ. However, eler hve mrece normton out the vlue o the rt term q. Conequently, n nvul eler normton lter F Q generte y Q n comlete to ty the uul conton cruer thn the comlete normton lter F ntrouce n Secton. Alo, the eler ujectve rolty meure r * er rom r. A regr the rt eght comonent o Q hll enote them y Q, the umton o ther oervlty y ny eler lule enough n oe not requre etene jutcton. On the other hn, t mght eem unnturl or ngle ometc/oregn eler to know ectly how much o -currency h/her comtrot oulton h ccumulte n ggregte. Aumton, n ct, weker. t only requr e tht noy gnl comrng the ull lt o uon nnng the uon rt o M, n n mrece meure o t rt, receve t every moment. Th me ole y oervton o the orer low, elne elow. Let the comonent o vector Z o Brownn moton genertng the rk o the economy, e lt nto the rt Z Q whch n the oervton roce Q, n the neenent rt Z w nnng the unoerve tte tht re reonle or the erence etween F Q n F. The menon o Z Q mut e equl to tht o Q. Moreover, the lw o moton o Q hll only nvolve Z Q, o tht t cn e wrtten Q=qtΘZ Q, wth non-ngulr uon mtr Θ tyng regulrty conton neee or the Grnov theorem to e lcle. Bee, ume tht the lt comonent o Q,.e. M, oerve through the rnom comonent Λ o the orer low receve y the eler ene n Suecton.. Th men, mong other thng, tht Λ nne y the totlty o comonent o Z Q n non-trvl wy.e. the rvte orer low relect the ull rnge o the oerve ource o uncertnty n the economy. wll requre even more content rom th orer low, y mkng the ollowng umton out the wy eler nterret Λ. Aumton 3 The rnom comonent Λ o the orer low comng rom the clent n other eler h the orm where coecent g otve. m Λ = g' Q' g g M, 4 Equton 4 hll e unertoo ollow. The eler relze tht the rte o reent eler urche o lnke to the cutomer urche rom M-reent eler y the lnce o yment entty. n vew o Aumton c me n Secton 3 zero verge cutomer orer low growth, t cler tht the cutomer orer low comonent o Λ cnnot contn ertent non-zero tren. The eler lnk the lnce o yment to Q n /, n, thereore, le to etlh one-to-one correonence etween Λ n M. Further, n uwr uture move n the echnge rte wll nuce the other to urche more ell le t h/her own k quote rther thn long tme y contctng other eler. Th eln the otve gn o g. Now we cn ormulte the lterng n Byen lernng roce or the eler. 6

17 The vector q o rt coecent o the oervton roce Q wll e vewe n to roce w wth the lw o moton w [ w] t Z w = 5 uner the ojectve rolty. All coecent re oune F Q -te rocee, whch, n ton, my e non-trvl uncton o tme. Th the tte roce unoervle rectly y the eler. nte, ech eler h ele out w, enote y w, whch contonl Q w = E w F. Th n F Q - eectton o w gven the eler current normton: [ ] te roce, wth ntl vlue w n the ntl vrnce-covrnce mtr ω ume to e gven. T ut B = Θ Θ. Accorng to the roerte o the Klmn-Bucy lter ee Lter n Shryev, 977, Ch., or etl, roce w te the..e. t t t w t = wt t ω t B [ Qt wt t] w B ω w w t Θ T { Q } Z = ω 6 uner the ojectve rolty r. The vrnce-covrnce roce ω te the etermntc Rcct erentl equton wth ntl conton ω. ω = ω ω T ω B ω t 7 By utrctng 5 rom 6, one otn the..e. or the eler etmton error: T Q w w w = ω B w w t ω Θ Z Z Th error mrtngle uner r n, mtr ωb - - tle, then wth tme, the ujectve etmte ecome cloer to the true vlue o w n the men. Recll tht we ume entc l eler wthn ech nton, n let the ze o the M-reent eler oulton e normlze to unty. Then, n equlrum, the otml outgong ntereler tre y the ometc eler, v, mut ty the trvl mrket clerng conton v t = mt or ll t, where the eler ujectve etmte o the ggregte men rte o ctve - urche rom non-reent, m, the lt comonent o the contonl eectton w o the unoerve tte roce w uner the gven normton tructure, the eler olve the otmzton rolem rom Secton uner rolty r* nte o r, n ll the rocee he work wth re F Q -rojecton o the ojectve F-te rocee. n the notton o Secton, the otml olcy o the eler mle. 7

18 X k k k = = θ = θ X k k k k v k. 8 k m Accorng to the umton o Secton 3, k n k re contnt. Further, the term θ whch relte to the how rce rty ne Y n to roce whoe erentl equl to the totl return erentl R -R lu covrnce term c t eenent only on the comonent o oervton covrnce mtr Θ. By umng the lner-qurtc erch cot uncton k gven n Suecton., we re le to wrte the ntntneou echnge rte chnge m k m k k m = R R c t υ υ t t km km km, 9 where m the lt row o the uon mtr T ω Θ rom 6. Equton 9 ecre the generlze uncovere totl return rty n n equlrum wth ymmetrc eler normton n Byen lernng rom the orer low gnl. Here, the rt three term on the rght hn e rereent the tnr UTR, whle the lt three term evton rom UTR, whch we cll rty lo known the country remum n the echnge rte lterture. Equton 9 llow u to mke three generl concluon out the roerte o th ymmetrc normton equlrum:. At tme when new normton out the ggregte orer low M eng rocee y the eler, the rty term non-trvl eemngly non-ttonry roce. the rrvl o new normton one -tme event, the rt comonent o the rty eventully er.. The voltlte o oth the echnge rte tel n the rty term n the generlze UTR equton houl e normlly hgher uner ymmetrc normton, thn n ore mrket wth ully norme eler. 3. A chnge n the erceve covrnce tructure o the rt o unmentl vrle mtr ω y the eler cn hve ermnent or, t let, very ertent eect on the erceve ynmc o the ggregte orer low rt roce m n, therey, on the rty contnt/country remum level. Thereore, the emrclly oerve revon o the country remum re elne n the moel conequence o reve nterretton o the orer low tttc montore y the eler. Among the ymmetrc normton equlr ecre n th ecton there re thoe wth unot roerte, well. The reon the el -ulllng nture o ele out roce m. The ct cn e llutrte y n emle where eler hve erect normton out the tttc o ll comonent o Q ecet M. Emle: mrece knowlege o the ggregte orer low only To tret th ce, t convenent to coner comonent Q eogenou rmeter o the moel, o tht reene rocee Q=M n w=m hve menon. Covrnce rmeter,,, ω, µ=θ re now clr. Let the true verge tren o the nter-eler orer low e mrtngle gven y 8

19 m m = m Z, wth >,.e. the true m revert to zero on verge. n uch n economy, ully norme eler woul hve generte ore mrket wth UTR te ectly u to n tonl rnom error term eenent on. ncomletely norme eler mut lern the true vlue o m rom the orer low oervton. The nnovton o ther ele out m ecre y the clr..e. ω ω M m = m m m t Z, µ µ n the evoluton o the recon o th ujectve etmte y the clr etermntc erentl equton ω ω& = ω. µ Th covrnce converge to the tle tey tte ω = µ, whch µ never vnhe unle the uon rmeter n the..e. or m zero. Tht, the ynmc o m never ully converge to tht o m. The coecent m ω n 9 equl to. µ Suoe tht the true ntl vlue o m zero, ut the ujectve ntl ele o the rereenttve ometc eler hen to e m >. Then, rom we conclue tht m ten to e negtve the echnge rte movement re more ownwr long thn wht UTR recrng. Equton 8 n 9 n th ce tell u tht t uch tme, the eler ujectve how vlue o the echnge rte more oten hgher thn the true vlue, nucng hm to ntte -urche n the nter-eler mrket n, therey, vlte h ele. ut erently, the ele tht everyone ele urche, mke hm urche well the herng eect. The ynmc o n 9 tht o the ownwr jutment o the echnge rte ter the ntl overhootng move mmetely ter the ormton o the ror ele m. Bee, on verge, m > m,.e. the rtlly norme eler > overrect on the new out the current movement n the FX mrket, genertng hgher voltlty regtere y oute oerver. The ove emle howe n etreme ce o el -ulllng ele n the nter-eler mrket, whle, oeng the mmum ole knowlege on other vrle, the eler h no ncentve to correct ther e etmte o the ggregte nter-eler orer low ucently. the etmte ute nvolve other mcro tte vrle well, one cn eect tht the unot eect n the m-vrle wll e mtgte. 9

20 5. Concluon The er h emontrte tht the long term mcro ctor nluencng the echnge rte n the hort term normton emnton out thee ctor mong FX eler n nvetor cn e hnle wthn common contnuou tme ortolo otmzton moel. roerte o equlr n th moel ccount or. unmentl roerte uch the uncovere et return rty o the echnge rte n t generlzton. the etence o vrle tochtc re n comettve mult-eler envronment 3. concentrton o nvul eler k n quote roun commonly oerve clerng rce 4. eler lernng out chngng unmentl rom rvte n nter-eler tre, n the coure o whch, new normton roceng cn cue ev ton rom the uncovere rty o totl return or the echnge rte 5. ermnent chnge n the rty contnt country remum reult o wtchng etween el -ulllng ele out the tttc o the nter-eler orer low. A w rgue n Secton 4, the orgn o chnge n el -ulllng ele out the ntereler orer low cn e the rrvl o new normton out the covrnce mtr o the unoerve tte roce. Th normton mot lkely to mrove the currently vlle one, n only ecetonlly, rouce totlly new ttern o co-movement etween the unmentl chrctertc o the economy, uch rouctvty, et return or the term tructure o nteret rte. Thu, one eclue etreme overhul o the long-term tructurl eenence n the economy, the orer low eect ecre n the er re unlkely to hve ermnent mct on the echnge rte. On the other hn, th mct my e very long-ltng or the reon o nte ee o le rnng y the gent, eree y the Klmn-Bucy lter equton eture n the tet. On the emrcl e, the ertence o orer low-cue eect conrme y ucently long eoe o evton o the oerve echnge rte rom the uncovere totl return rty. n te o the ee nght rove y the nter-temorl eler otmzton moel uner ull rtonlty, the lmt o lclty o th tye o moel eem to e ttne ue to hgh comlety o the oluton. Tht, wth the eceton o ew ecl ce, the gent unle to egn otml trng olce n rctcl term, even he rere to ollow the economt recommenton. Thereore, one o the chllenge or uture reerch the tton o the reently etng moel wth ull nvetor-eler rtonlty, to numerclly trctle rmework o ynmc ytem wth ounely rtonl gent. Aen: The Mmum rncle oluton o the eler rolem n contnuou tme The mterl o th en e uon Dervz, 999. The generl orm o the eler' otmzton rolem cue n Secton cn e ymolclly wrtten own m l E T e t [ ϑ] [ ϑ] U X ; t e t l t T B T, X T, A

21 wth reect to control l, uject to the tte -trnton equton X µ X, l t σ X, l Z =, A the vlue X o the tte roce t tme t= gven. The tte roce X vector wth n comonent n Z vector o mutully neenent tnr Brownn moton. B the o-clle nl equet uncton. t reent vlue mut oe lmt the tme horzon o the otmzton rolem nnte T =. Fnlly, [ ϑ] = t ϑτ τ the count ctor etween ero t n. The rolem A, A cn e olve y ormng the current vlue Hmltonn t, X, l,, Ξ = U X, l µ X, l tr Ξ σ X, l H, whch to e mmze wth reect to l. Here, n Ξ re the o clle rt- n econorer jont rocee o the me menon n X n Ξ n n-mtr, wth t Ξ = D σ long the otml th. When tte X tn or et holng, the jont roce cn e clle the how rce vector o the correonng grou o et. Let [,g] ene the rectle co-vrton o uon rocee n g, n ut, g =, g wth the tnr horthn or,. Then the rt-orer jont [ ] t roce te the tochtc erentl equton ϑ t A A t D Ut = n, A3 X wth the nn-mtr vlue roce A ene y A=DXµtDXσZ, A=n. The nl conton T=DXBT,XT, or n rorte trnverlty conton T=, mut e e to A3. The jont roce cn e lo ecre the X-grent o the vlue uncton o the rolem A, A, rove the ltter erentle. n the nvetor -eler rolem o Secton, the tte-trnton equton A lner n the tte vrle. Thereore, the coecent mtr A n t qurtc vrton A or th trnton equton re ely een to e equl to t r Λ A = Λ r α r r α,

22 = A A N N A A N A. The revou two mtr ereon, uttute nto equton A3, rener the jont equton ytem 6 o Secton. To erve the ereon or the Hmltonn o the eler rolem, one nee to clculte the term contnng the rt n the econ orer jont roce. Frt o ll, oerve tht = v S D v k X v n l v S D v k v k X n l δ µ, n the rt o the Hmltonn contnng the rt orer jont roce otne y clr multlcton o the ove column vector y the row vector. Further, = A N A N D σ, = A N A N σ. Thereore, the econ-orer jont roce rt o the Hmltonn equl to [ ],,, A N N A A N,.e. t oe not contn the control vrle. n hort, mmzng the Hmltonn wth reect to the control o the eler rolem equvlent to mmzng the ereon { } S D v k v k X { } S D v k X v v v δ.

23 Th mmzton ully ecre y the rt orer conton = u δ =, v k, v X k =, D D =, [ S S ] = S [ ] D X k v =, tht we ue n Suecton. o the mn tet. Reerence. Bck, K. 99 ner trng n contnuou tme. Revew o Fnncl Stue 5, No.3, Bck, K., H. Co, n G. Wllr merect cometton mong norme trer. Journl o Fnnce 55, No.5, Bck, K., n H. eeren 998 Long-lve normton n ntry ttern. Journl o Fnncl Mrket, Brennn, M., n H. Co 997 nterntonl ortolo nvetment low. Journl o Fnnce 5, No.5, Coeln, T., n D. Gl 983 normton eect n the -k re. Journl o Fnnce 38, Dervz, A. 999 Ajont rocee o the ortolo Otmzton rolem n Equlrum Aet rce. nttute o normton Theory n Automton. Reerch Reort No.954, June. 7. Dervz, A. 999 Generlze Aet Return rty n the Echnge Rte n Fnnclly Oen Economy. The Czech Ntonl Bnk, Monetry Det., W. No.. 8. Detemle, J. 986 Aet rcng n roucton economy wth ncomlete normton. Journl o Fnnce 4, No., Dt, A., n R. nyck 994 nvetment uner Uncertnty. rnceton, N.J.: rnceton Unv. re.. Eley, D., n M. O Hr 987 rce, tre ze, n normton n ecurty mrket. Journl o Fnncl Economc 9, Ellott, R.J. 98 Stochtc Clculu n Alcton. Srnger-Verlg.. Evn, M., n R. Lyon 999 Orer Flow n Echnge Rte Dynmc. NBER Workng er No. 737, Augut. 3. Gloten, L. 989 ner trng, lquty, n the role o the monoolt eclt. Journl o Bune 6, No., Gloten, L., n. Mlgrom 985 B, k n trncton rce n eclt mrket wth heterogeneouly norme trer. Journl o Fnncl Economc 4, Goohrt, C., T. to, n R. yne 996 One y n June 993: A tuy o the workng o Reuter - electronc oregn echnge trng ytem. n: J. Frnkel, G. Gll n A. Govnnn e. The Mcrotructure o Foregn Echnge Mrket. Chcgo: Unv. o Chcgo re; Ho, T., n H. Stoll 983 The ynmc o eler mrket uner cometton. Journl o Fnnce 38, Kyle, A. 985 Contnuou ucton n ner trng. Econometrc 53, Lter, R.S., n A.N. Shryev 977 Stttc o Rnom rocee, Vol. n. Srnger-Verlg. 3

24 9. Nl-De Smone, F., n W.A. Rzzk 999 Nomnl Echnge Rte n Nomnl nteret Rte Derentl, MF W/99/4.. errun, W., n. Vtle 996 ntereler tre n normton low n ecentrlze oregn echnge mrket. n: J. Frnkel, G. Gll n A. Govnnn e. The Mcrotructure o Foregn Echnge Mrket. Chcgo: Unv. o Chcgo re.. Sehole, M. Smrt Foregn Trer n Emergng Mrket, W, H School o Bune, Unverty o Clorn, Berkeley Jnury.. Ztero, F. 998 Eect o nncl nnovton on mrket voltlty when ele re heterogeneou. Journl o Economc Dynmc n Control,