Protecting Target Zone Currency Markets from Speculative Investors

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Abner Jacobs
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1 Poecing Tage Zone Cuency Makes fom Speculaive Invesos Eyal Neuman Alexande Schied Januay 22, 217 axiv: v1 q-fin.mf] 23 Jan 218 Absac We conside a sochasic game beween a ade and he cenal bank on age zone makes. In his ype of makes he pice pocess is modeled as a diffusion which is efleced a one o moe baies. Such models aise when a cuency exchange ae is kep above a ceain heshold due o cenal bank inevenion. We conside a ade who wishes o liquidae a lage amoun of cuency, whee fo whom pices ae opimal a he baie. The cenal bank, who wishes o keep he cuency exchange ae above his baie, heefoe needs o buy is own cuency. The pemanen pice impac, which is ceaed by he ansacions of boh sides, uns he opimal ading poblems of he ade and he cenal bank ino coupled singula conol poblems, whee he common singulaiy aises fom a local ime along a andom cuve. We fis solve he cenal bank s conol poblem by means of he Skookhod map and hen deive he ade s opimal saegy by solving a sequence of appoximaed conol poblems, hus esablishing a Sackelbeg equilibium in ou model. Mahemaics Subjec Classificaion 21: 93E2, 91G8, 6J55 Key wods: make impac, age zone models, singula sochasic conol, local ime, Sackelbeg equilibium 1 Inoducion In Sepembe 16 of 1992, a day which is known as he black Wednesday, he Biish govenmen was foced o sep ou of he Euopean Exchange Rae Mechanism ERM. Sholy befoe his cucial dae, he Biish pound exchange ae was close o is lowe limi, and cuency make speculaos wee ying o ake advanage of his oppouniy o dop he exchange ae by making immense sho odes. Among he speculaos sood ou Geoge Soos who shoed Depamen of Mahemaics, Impeial College London, SW7 2AZ London, Unied Kingdom Depamen of Saisics and Acuaial Science, Univesiy of Waeloo, and Depamen of Mahemaics, Univesiy of Mannheim. alex.schied@gmail.com 1

2 1 billion pounds wihin a single day. The Biish easuy kep spending is foeign cuency duing ha day, in ode o buy Biish pounds, which wee becoming less valuable, as he exchange ae was apidly deceasing. By he end of he day, he Biish govenmen could no keep up wih he puchases and announced an exi fom he ERM. As a esul, Soos and he speculaos won. In one day, Geoge Soos made ove 1 billion Seling. On he ohe hand, he coss fo he Biish axpayes was esimaed a aound 3.3 billion Seling. The goal of his wok is o descibe such siuaions mahemaically and o analyze he wos-case scenaio a cenal bank may be facing when keeping up an exchange ae peg. To his end, we model he sysem of a speculaive inveso vesus he cenal bank o govenmen as Sackelbeg equilibium of a sochasic diffeenial game. One of he main ingediens is o model he cuency exchange ae as a age zone model. In age zone models he exchange ae is allowed o move inside a specific egime wih one o moe baies which ae enfoced by moneay auhoiies see Kugman 1] and 15, 4, 9, 2], among ohes. The pice pocess is ofen modeled by a diffusion which is efleced a he baies. Fo insance, he EUR/CHF exchange was kep above he 1.2 baie fom Sepembe 6, 211, o Januay 15, 215, by he Swiss Naional Bank see Figue 1 in 11]. Moe examples ae he Biish pound vs. Deusche Mak exchange ae which was kep above 2.8 befoe black Wednesday see in Figue 1 and Euo vs Czech Kouna Figue 2. One of he main effecs which needs o be aken ino accoun in he inveso-cenal bank model, is he make impac which is ceaed by he buy and sell odes of boh sides. Make impac efes o he empiical fac ha he execuion of lage odes affecs he pice of he undelying asse. Usually, his affec causes an unfavoable addiional execuion coss fo he side who pefoms he ansacion. One of he bes sudied make impac models is he Almgen Chiss model, which we will use hee in is coninuous-ime vesion inoduced in 1] see also he suvey 8] and he efeences heein. In 11] we consideed ading saegies fo an inveso fo whom pices ae opimal a he baie of he age zone and who ceaes only empoay pice impac. In he seing of 11], ading is consained fo he imes a which he pice pocess is locaed a he baie. Fo insance, fo a Swiss inveso wishing o puchase euos duing he peiod of a lowe bound on he EUR/CHF exchange ae, i would be naual o buy euos only a he lowes possible pice, which a his ime was equal o he baie of 1.2 EUR/CHF. Such saegies need o be absoluely coninuous wih espec o he local ime of he pice pocess a he baie and he ading speed should be defined as he coesponding Radon Nikodym deivaive. The minimizaion of a cos-isk funcional ha is associaed wih such saegies, was fomulaed in 11] as a singula sochasic conol poblem. This conol poblem was solved by means of a scaling limi of ciical banching paicle sysems, which is known as a caalyic supepocess. One of he main challenges in his pape is o incopoae he effecs of he pemanen make impac on age zone makes. Adding pemanen make impac o he seing of 11], uns he age zone pice pocess fom a efleced Bownian moion o a Skookhod SDE, as we descibe in geae deail in Secion 2. As a esul, we develop new ools in ode o solve he singula conol poblems which aise in his seing. 2

Figue 1: Plo of he GBP/DEM exchange ae fom Januay 1, 1992 o Decembe 31, 1992.

3 Figue 1: Plo of he GBP/DEM exchange ae fom Januay 1, 1992 o Decembe 31, On Sepembe 16, 1992, he Biish govenmen announced on exi fom he Euopean Exchange Rae Mechanism ERM, which was followed by a apid dop of he exchange ae. Figue 2: Plo of he EUR/CZK exchange ae fom Augus 1, 216 o Augus 1, 217. On Apil 6, 217, he Czech Naional Bank emoved he EUR/CZK floo, his was followed by a apid dop of he exchange ae. 3

4 2 Model seup and main esuls Le Ω, F, F, P be a fileed pobabiliy space saisfying he usual condiions and suppoing a sandad Bownian moion W. We suppose ha in he absence of a lage-inveso inevenion he exchange ae S beween wo cuencies is given by a Bachelie model, S = S + σw,, whee S and σ ae wo posiive consans. This unaffeced exchange ae pocess will be impaced by he pice impac componens of wo ypes of big playes. One of hese big playes models he cenal bank of one of he wo cuencies, wheeas he second one sands fo a saegic speculaive inveso o fo he accumulaed pice impac of a goup of such invesos. I is he goal of he cenal bank o mainain a one-sided age zone in which he acual exchange ae mus say above a specified level c. Such age zones ae fequenly obseved on financial makes; seveal well-known examples ae pesened in Secion 1. The cenal bank keeps up he age zone hough he pemanen pice impac of ades ha ae execued as soon as he exchange ae heaens o fall below he level c, heeby ceaing an eve inceasing invenoy. This accumulaion of invenoy is ofen poblemaic fo he cenal bank and fequenly leads o he abandonmen of he age zone egime. A nooious example is he beaking of he Bank of England by he inveso Geoge Soos on Sepembe 16, The saegy of he saegic inveso ove he ime hoizon, T ] is descibed hough he ading speed ξ in a linea Almgen Chiss-ype model, so ha X = ξ s ds is he accumulaed invenoy afe ading ove he ime ineval, ]. Hee, we use a Makovian conol ha may depend on he cuen ime and he cuen exchange ae S. I is impoan o noe hee ha S denoes he acual exchange ae afe cenal bank inevenion and no he unaffeced exchange ae pocess S, which will ypically no be obsevable o any make paicipan. Thus, we assume ha ξ is of he fom ξ = v, S,, T ], 2.1 whee v, x is a coninuous funcion on he domain D :=, T ] c, \ {, c} saisfying he following wo popeies: Fo evey compac subse K of D hee exiss L K L K x y fo all, x,, y K. such ha v, x v, y Thee exiss a consan C such ha v, x C1 + x fo all, x D. By V we denoe he class of all such funcions v. In he linea Almgen Chiss model, he pemanen pice impac geneaed a ime by he saegy ξ in 2.1 is of he fom γ ξ s ds = γ v, S d, whee γ > is he pemanen impac paamee 1]. Thus, he acual exchange ae pocess is of he fom S = S + γ v, S d + R, 2.2 4

5 whee R is he pemanen pice impac geneaed by he esponse saegy of he cenal bank. This saegy mus be such ha he sochasic inegal equaion 2.2 admis a soluion S saisfying S c fo all, T ] P -a.s. Moeove, he esponse R mus be adaped o he naual filaion of S. As fo he saegic inveso s saegy, we could insis ha R is absoluely coninuous in, bu since cenal banks ypically face less esicions on ansacion coss han egula invesos, we will only assume ha R ω is of bounded vaiaion fo P -a.e. ω Ω. Le us denoe by Rv he class of all pocesses R saisfying he peceding condiions fo a given saegy v V. The cenal bank has wo main goals. Fis, he age zone mus be mainained by guaaneeing ha S c fo all. Second, he invenoy accumulaed by keeping up he age zone mus be conolled. Le us denoe he invenoy accumulaed up o ime by Y. As fo he saegic inveso, Y is elaed o R by R = γy. Thus, we le Y v = { γr R Rv }. The following heoem idenifies ha he esponse of he cenal bank, fo any given v V, is opimal in he sense ha i minimizes he invenoy of he cenal bank in a pahwise sense. Theoem 2.1. Assume ha S > c. Fo any v V hee exiss a unique elemen in Y Y v ha is minimal in he sense ha Y Y fo all fo all, T ] P -a.s., fo all Y Y v. Moeove, fo we have whee L c 1 S = lim ε ε S = S + γ v, S d + 1 γ Y, 2.3 Y = 1 γ Lc S,, T ], σ 2 1 c,c+ε] S d S, S = lim ε ε is he local ime of he semimaingale S a c. 1 c,c+ε] S d, P -a.s., Fom now on, fo any v V, we will always conside he opimal esponse 1 Y = 1 γ γ Lc S fom he peceding heoem. Then he dynamics of he exchange ae pocess 2.3 depends only on v. Thus, o make his dependence explici, we will hencefoh wie S v fo he pocess S given by 2.3. Nex we conside he opimal saegy of a speculaive ade who ies o maximize he cenal bank s invenoy wih he goal of pushing i o is isk limis and so o foce i o abandon he age zone. The accumulaion of excessive isk is indeed one of he mos common easons why a cenal bank would abandon a age zone. Thus, he speculaive inveso aims o maximize he expeced cenal bank invenoy a a given fuue ime T. Tha is, he goal is o maximize EYT ] o, equivalenly, ELc T Sv ]. Accoding o he linea Almgen Chiss 5

6 model, he inveso s ading saegy ξ = v, S v ceaes ansacion coss, someimes also called slippage, popoional o T ξ 2 d = T v, S v 2 d. These coss aise, e.g., fom sho-em pice impac effecs and fom he need o incease he popoion of make vs. limi odes in a saegy wih high ading speed. We heefoe assume ha he goal of he inveso is o T ] maximize E L c T S v κ v, S v 2 d ove v V. 2.4 Ou main esul povides a closed-fom soluion o he peceding poblem of sochasic opimal conol and hus esablishes a Sackelbeg equilibium in ou sochasic diffeenial game beween ade and cenal bank. As a esul, we have singled ou he wos-case scenaio a cenal bank may be facing when keeping up a one-sided age zone. Theoem 2.2. Suppose ha S > c and le β = γ 2 /2κσ 2 and U, z = 1 ] E β log exp βσl z c/σ W, z c, 2.5 whee L x W is he local ime of he Bownian moion W a level x R. Then we have U, S = sup E L c S v κ v V ] v, S v 2 d. Moeove, U, z belongs o C 1,2, T ] c,, and hee exiss a unique saegy v V fo which he supemum is aained. I is given by v, z = γ 2κ zu T, z. 2.6 Remak 2.3. Fom Fomula on p. 161 in 5] we ge a closed-fom expession fo U, U, z = 1 z c ef β log σ 2 + e βz c+β2 σ 2 /2 z c 1 ef σ 2 βσ ], whee efx = 2 π x e y2 dy is he Gaussian eo funcion. See Figue 3 fo a plo of U, z and v, z. 6

Figue 3: The value funcion U, z lef and he opimal saegy v, z igh fo σ = γ = κ = 1 and c =. 3 An appoximae conol poblem In his secion, we conside a egulaized vesion of he conol poblem 2.4.

To his end, we define and he egulaized local ime G ε x := 1 2πε e x c2 /2ε. L c,ε S v := G ε S v d. Then we conside he following egulaized vesion of he conol poblem 2.

7 Figue 3: The value funcion U, z lef and he opimal saegy v, z igh fo σ = γ = κ = 1 and c =. 3 An appoximae conol poblem In his secion, we conside a egulaized vesion of he conol poblem 2.4. This conol poblem povides he basis fo he infomed guess of he value funcion and opimal saegy in Theoem 2.2. I is also ineesing in is own igh. To his end, we define and he egulaized local ime G ε x := 1 2πε e x c2 /2ε. L c,ε S v := G ε S v d. Then we conside he following egulaized vesion of he conol poblem 2.4: maximize E L c,ε S v κ ] vu 2 du ove v V. 3.1 I will be convenien o make he dependence of conolled eflecing diffusion S v on is iniial value z := S c explici by wiing S v,z. Wih his noaion, we define he value funcion of he poblem 3.1 by V ε, z := sup E L c,ε S v,z κ v V The geneao of S v,z is fomally given by ] vu 2 du. 3.2 G = γv, z z + σ2 2 zz, 3.3 7

8 wih Neumann bounday condiion a c. Hence, sandad heuisic agumens sugges ha he funcion V ε should solve he following Hamilon Jacobi Bellman equaion, wih iniial condiion U = σ2 2 zzu + G ε + supγv z U κv 2, in, T ] c,, 3.4 and Neumann bounday condiion v R U, z = fo all z c, 3.5 z U, c = fo all T 3.6 The maximum ove v R on he igh-hand side of 3.4 is aained in and so 3.4 becomes v = γ zu 2κ, 3.7 Le h, z be such ha U, z = 2κσ2 γ 2 U = σ2 2 zzu + G ε + γ2 4κ zu 2, in, T ] c,. 3.8 log h, z. Then, h mus solve h = σ2 2 zzh + γ2 2κσ 2 hg ε, in, T ] c,, 3.9 wih iniial condiion h, z = 1 and bounday condiion z h, c+ =. Poposiion 3.1. Le β = γ 2 /2κσ 2. Then he funcion U ε, z = 1 β log E e β Gεz+σW d ], belongs o C 1,2, R and is esicion o, T ] c, is a classical soluion o he iniial value poblem Equaion 3.7 suggess ha he opimal saegy v ε fo he poblem 3.1 is given by vε, x = γ zu ε T, x κ To make his saemen moe pecise, noe fis ha E z U ε G εz + σw d e β ] Gεz+σW d, z = E e β ] Gεz+σW d This funcion is clealy saisfies a unifom Lipschiz condiion in z. 8

9 Theoem 3.2. The funcion U ε is equal o he value funcion V ε in 3.2 and he saegy v ε is he P z -a.s. unique opimal saegy in V. Now we show ha he appoximae value funcions appoximae ou oiginal value funcion 2.5, which can also be epesened as U, z = 1 ] E β log exp βl c z σw As in Poposiion 3.1, we le β = γ 2 /2κσ 2. Poposiion 3.3. We have U ε, z U, z unifomly in, z, T ] R as ε. 4 Poofs 4.1 Poof of Theoem 2.1 Befoe we pove Theoem 2.1 we inoduce he following lemma. Lemma 4.1. Le z >, v V, and y C, T ] wih y =. Define H as he class of funcions k C, T ] which saisfy he following wo condiions. a k = and k is nondeceasing. b Thee exiss a nonnegaive soluion x o he inegal equaion x = z + vs, c + xs ds + y + k,, T ]. Then, if hee exiss k H such ha T 1 {x>} d k =, hen k is a minimal elemen in H in he sense ha k k fo all, T ] and k H. Poof. The esul is ivial if k is he unique elemen in H. Ohewise, we ake c = fo simpliciy and assume ha hee is an addiional k H and le x = z + x = z + vs, xs ds + y + k, vs, xs ds + y + k. We will show below ha x x fo all, T ]. Once his has been esablished, we assume by way of conadicion ha hee is some, T such ha k > k. Then we have x x = v, xs v, xs ds + k k v, xs v, xs ds. 9

10 Since < z = x = x, he ajecoies {, x, T ]} and {, x, T ]} ae boh conained in a compac subse K of D =, T ], \ {, }. Hence, hee exiss a consan L such ha x x = x x L xs xs ds. Thus, Gonwall s inequaliy yields xs = xs fo all s, ] and in un he conadicion k = k. Now we pove ou claim ha x x fo all, T ]. To his end, we assume by way of conadicion ha hee exiss some τ, T and ε, T τ such ha xτ = xτ and x < x fo τ, τ + ε. Then, since x > x, he funcion x has no zeos in τ, τ + ε, and we ge k = kτ fo all τ, τ + ε]. Moeove, we have = xτ xτ = τ v, xs v, xs ds + kτ kτ. Using ou assumpion x < x, he fac ha k is nondeceasing, and he Lipschiz condiion of he funcion v, we ge ha fo τ, τ + ε], x x = x x = τ τ v, xs v, xs ds + kτ k v, xs v, xs ds L xs xs ds. Hence, Gonwall s inequaliy implies ha ha x = x fo all τ, τ + ε], which is he desied conadicion. Poof of Theoem 2.1 Conside he following SDE wih eflecion: ds = σ dw + γv, c + S d + dr, S, S = z c, R is coninuous and nondeceasing, T 1 {S >} dr =. This is a sandad Skookhod equaion, fo which exisence and uniqueness was fis poved by Skookhod 14]. Moeove, Theoem in Secion 1.3 of 12] idenifies R as he local ime L S of he soluion S a. When leing S := c + S, we ge a soluion o ds = σ dw + γv, S d + dr, whee S c and R = L S = L c S. Fo P -a.e. ω Ω, he funcions x = S ω, y = σw ω, and k := L c Sω saisfy he condiions of Lemma 4.1. Theefoe, L c S is he desied minimal elemen of Y v. τ 4.1 1

11 4.2 Poofs of he esuls fom Secion 3 Poof of Poposiion 3.1. Le h, z := E e β Gεz+σW d ],, z R. Since G ε is bounded and smooh, we may apply Theoem 3.6 in Chape 4 of 7] o conclude ha h belongs o C 1,2, R and saisfies h = σ 2 2 zz h + γ2 2κσ 2 hg ε in, R wih iniial condiion h, z = 1. Since G ε is symmeic aound c, he same is ue fo h, and i follows ha z h, c =. Hence, he esicion of h o, T ] c, saisfies 3.9 ogehe wih he iniial and bounday condiions h, z = 1 and z h, c+ =. Reacing he seps ha led o 3.9 now complees he poof of he asseion. Poof of Theoem 3.2. Iô s fomula yields ha fo all v V P z -a.s., U ε, S v,z = U ε, z + σ + U ε, z + σ z U ε, S v,z dw + U ε, S v + γv, S v,z z U ε, S v,z dw z U ε, S v,z dl c S v,z 4.2 z U ε, S v,z Gε S v,z + σ2 2 zzu ε, S v,z d κv, S v,z 2 d, whee we have used he HJB equaion 3.4 as well as 3.6 ogehe wih he fac ha dl c S v,z is suppoed on { S v,z = c}. Since G ε and G ε ae bounded, i follows fom 3.11 ha z U ε is bounded, and so zu ε, S v,z dw is a ue maingale. Using he iniial condiion U ε, = and aking expecaions in 4.2 gives U ε, z E Gε S v,z κv 2 ] d. 4.3 Taking he supemum ove v V yields U ε, z V ε, z fo all and z. Nex, by 3.7, we will have equaliy in 4.2, and hence in 4.3, if and only if v, S v = γ 2κ zu ε, S v fo a.e., ], which gives v = v ε. Recall ha z U ε, z is bounded and coninuously diffeeniable in boh vaiables, hence v ε V. The following lemma will be needed fo he poof of Poposiion

12 Lemma 4.2. a Le p 1. Fo evey T, hee exiss C > such sup E z R b Fo evey λ > we have lim sup ε G ε z + σw u du L c z + σw u p ] Cε p/4, fo all ε, 1, sup E z R e λ T Gεz+σW d ] E exp λ sup x R ] L x T σw <. Poof. The poof uses ideas fom Lemma 2.2 in 3]. Fo simpliciy, in his poof we will wie L x fo L x σw. a Le p, and ε as in he hypohesis. Taking c = in Execise 1.33 in Chape VI.1 of 13], we obain he exisence of a consan C depending only on p and T such ha E sup L x L y p] C x y p/2, fo all x, y R. 4.4 T Fom he occupaion ime fomula we have P -a.s. G ε z + σw u du = G ε z + xl x dx = 1 2π Using 4.4 and Jensen s inequaliy we heefoe have E sup T G ε z + σw u du L c z R p ] b Fom 4.5 we ge ha fo any ε, 1, E e λ ] T Gεz+σW d = E e λ 2π 1 2π 1 2π Cε p/4. R R R e y2 /2 L c+ εy z T and he igh-hand side is finie accoding o Lemma 1 in 6]. R e x2 /2 L c z+ εx dx. 4.5 e y2 /2 E sup L c z+ εy T e y2 /2 C ε p/2 y p 1/2 dy ] dy E exp λ sup x R L c z p] dy L x T ], Poof of Poposiion 3.3. We wie again L c z he funcion log x fo x 1, we ge U ε, z U, z 1 β E sup e βls c z e β s Gεz+σW d ] s T 1 β E e βlc z T T fo L c z σw. Using he Lipschiz coninuiy of Gεz+σW d sup s T L c z s s ] G ε z + σw d. 12

13 The Cauchy Schwaz inequaliy hus yields U ε, z U, z 1 E e 2βLc z T + ] T 1/2E Gεz+σW d β Lemma 4.2 b shows ha e 2βLc z T + T lim sup ε sup E z R GεS+z d ] lim sup ε <. sup L c z s s T sup E z R s 2 ] 1/2 G ε z + σw d. 4.6 ] 1/2 e 4βLc z T E e 4β ] T 1/2 GεS+z d Using his bound along wih Lemma 4.2 a in 4.6 hus gives ha U ε, z U, z, unifomly in, z, T ] R, as ε. 4.3 Poof of Theoem 2.2 Poposiion 4.3. The funcion U saisfies he following paial diffeenial equaion, U, z = σ2 2 zzu, z + γ2 z U, z 2, 4κ in, T ] c,, 4.7 wih bounday condiion z U, c = 1,, T ]. 4.8 Poof. Le x ψ, x := ef σ 2 Fo >, we have ψ, x = βσ e x2 /2σ 2 + βσ 2π x ψ, x = βe βx+β2 σ 2 /2 xx ψ, x = 2 σ 2 ψ, x. + e βx+β2 σ 2 /2 x 1 ef 1 ef 2 ex βσ2 2 /2σ2 σ 2 βσ ]. 2 x βσ 2 ] 1 ef σ, 2 x βσ 2 ], σ 2 Fom 2.7 we have U, z = 1 log ψ, z c. I follows ha β In paicula, U, z = ψ, z c βψ, z c, zu, z = xψ, z c βψ, z c. 4.9 z U, c = xψ, βψ, = 1. Nex, he second z-deivaive of U coesponds o zz U, z = 2 xxψ, z c βψ, z c β x ψ, z c. βψ, z c Plugging eveyhing ogehe yields he asseion. 13

14 Now we ae eady o pove Theoem 2.2. Poof of Theoem 2.2 Recall ha we wie S v,z fo he eflecing diffusion 2.3 saing fom z = S > c wih given v V. Iô s fomula gives U, S v,z = U, z + σ + = U, z + σ + U, z + σ z U, S v,z dw + U, S v,z γv, S v,z + γv, S v,z z U, S v,z dw L c S v,z z U, S v,z γ2 4κ z U, c dl c S v,z z U, S v,z + σ2 2 zzu, S v,z z U, S v,z z U, S v,z dw L c S v,z + 2 d κv, S v,z 2 d, d whee we have used 4.7 and 4.8 in he second sep. I follows fom 4.9 ha z U is bounded, and so σ zu, S v,z dw is a ue P z -maingale. Using he iniial condiion U, = and aking expecaions gives U, z E L c S v,z κ ] v, S v,z 2 d. 4.1 Taking he supemum ove v V shows he inequaliy in Theoem 2.2. Noe ha we will have an equaliy in 4.1 if and only if v, x = γ 2κ zu, x fo a.e., ]. Finally, he fomulas deived in he poof of Poposiion 4.3 easily yield ha v V. Acknowledgemen. Eyal Neuman would like o hank he CFM - Impeial Insiue of Quaniaive Finance who suppoed his eseach. Refeences 1] R. Almgen. Opimal execuion wih nonlinea impac funcions and ading-enhanced isk. Applied Mahemaical Finance, 1:1 18, 23. 2] C. Ball and A. Roma. Deecing mean evesion wihin eflecing baies: applicaion o he Euopean Exchange Rae Mechanism. Applied Mahemaical Finance, 51:1 15, ] B. Béad Begey and P. Vallois. Appoximaion via egulaizaion of he local ime of semimaingales and Bownian moion. Sochasic Pocess. Appl., 11811:258 27,

15 4] G. Beola and R. J. Caballeo. Tage zones and ealignmens. The Ameican Economic Review, 823:pp , ] A. N. Boodin and P. Salminen. Handbook of Bownian moion facs and fomulae. Pobabiliy and is Applicaions. Bikhäuse Velag, Basel, second ediion, 22. 6] E. Csáki. An inegal es fo he supemum of Wiene local ime. Pobab. Theoy Relaed Fields, 831-2:27 217, ] R. Due. Sochasic calculus. Pobabiliy and Sochasics Seies. CRC Pess, Boca Raon, FL, A pacical inoducion. 8] J. Gaheal and A. Schied. Opimal ade execuion unde geomeic Bownian moion in he Almgen and Chiss famewok. Inenaional Jounal of Theoeical and Applied Finance, 14: , ] F. D. Jong. A univaiae analysis of ems exchange aes using a age zone model. Jounal of Applied Economeics, 91:pp , ] P. R. Kugman. Tage zones and exchange ae dynamics. The Quaely Jounal of Economics, 163: , ] E. Neuman and A. Schied. Opimal pofolio liquidaion in age zone models and caalyic supepocesses. Finance and Sochasics, 2:495 59, ] A. Pilipenko. An Inoducion o Sochasic Diffeenial Equaions wih Reflecion. Lecues in pue and applied mahemaics 1. Univesiäsvelag Posdam, ] D. Revuz and M. Yo. Coninuous Maingales and Bownian Moion. Spinge Velag, hid ediion ediion, ] A. V. Skookhod. Sochasic equaions fo diffusion pocesses in a bounded egion. Theoy of Pobabiliy & Is Applicaions, 63: , ] L. E. O. Svensson. The em sucue of inees ae diffeenials in a age zone: Theoy and Swedish daa. Jounal of Moneay Economics, 281:87 116,

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as