UvA-DARE (Digital Academic Repository) Currency option pricing in a credible exchange rate target zone Veestraeten, D.J.M. Link to publication

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1 UvA-DARE (Digial Academic Reposiory) Currency opion pricing in a credible exchange rae arge zone Veesraeen, D.J.M. Link o publicaion Ciaion for published version (APA): Veesraeen, D. (202). Currency opion pricing in a credible exchange rae arge zone. Amserdam: Universiy of Amserdam, Deparmen of Economics. General righs I is no permied o download or o forward/disribue he ex or par of i wihou he consen of he auhor(s) and/or copyrigh holder(s), oher han for sricly personal, individual use, unless he work is under an open conen license (like Creaive Commons). Disclaimer/Complains regulaions If you believe ha digial publicaion of cerain maerial infringes any of your righs or (privacy) ineress, please le he Library know, saing your reasons. In case of a legiimae complain, he Library will make he maerial inaccessible and/or remove i from he websie. Please Ask he Library: hp://uba.uva.nl/en/conac, or a leer o: Library of he Universiy of Amserdam, Secrearia, Singel 425, 02 WP Amserdam, The Neherlands. You will be conaced as soon as possible. UvA-DARE is a service provided by he library of he Universiy of Amserdam (hp://dare.uva.nl) Download dae: 5 Aug 208

2 Currency opion pricing in a credible exchange rae arge zone Dirk Veesraeen Universiy of Amserdam Deparmen of Economics Roeerssraa 08 WB Amserdam The Neherlands July 7, 202 Absrac This aricle examines currency opion pricing wihin a credible arge zone arrangemen where inervenions a he boundaries push he exchange rae back ino is ucuaion band. Valuaion of such opions is complicaed by he requiremen ha he re ecion mechanism should preven he arbirage opporuniies ha would arise if he exchange rae were o spend nie ime on he boundaries. To preven he laer, we superimpose insananeously re ecing boundaries upon he familiar Geomeric Brownian Moion (GBM) framework. We derive closed-form expressions for European call and pu opion prices and show ha prices for he GBM model of Garman and Kohlhagen (983) arise as he limi case for in niely wide bands. We also illusrae ha aking accoun of boundaries is of considerable economic value as erroneously using he unbounded-domain model of Garman and Kohlhagen (983) easily overprices opions by more han 00%. Keywords: currency opion pricing, exchange rae arge zones, geomeric Brownian moion, re ecing boundaries, Brownian moion, risk-neural valuaion JEL Classi caion: F3; G3 Address correspondence o Dirk Veesraeen, Deparmen of Economics, Universiy of Amserdam, Roeerssraa, 08 WB Amserdam, The Neherlands; dirk.veesraeen@uva.nl. We are indebed o Hans Dewacher, Henk Jager, Franc Klaassen, Pie Sercu and Koen Vermeylen for simulaing discussions. The usual disclaimer applies.

3 I Inroducion In exchange rae arge zones, moneary auhoriies keep he price of heir currency beween lower and upper boundaries via foreign-exchange marke inervenions. The Breon Woods sysem and he Exchange Rae Mechanism (ERM) are well-known hisorical examples of such regimes. The ERM II arrangemen for he prospecive new members of he Euro zone coninues he presence of arge zones wihin he European Union. Swizerland insalled a one-sided arge zone vis-à-vis he Euro in Sepember 20 o sop he appreciaion of he Swiss Franc caused by he European sovereign deb crisis. Moreover, a number of emerging marke economies and ransiion counries implemened arge zones or are moving owards such regimes. For insance, he ucuaion range of he Chinese Renminbi (RMB) owards he US dollar widened from 0.3% around he cenral pariy in 2005 over 0.5% in 2007 o % since April 202 (People s Bank of China, 202). Chinese auhoriies inend o furher increase exibiliy for he RMB causing he foreign-exchange opions markes, ha a end-march 202 had 28 members (People s Bank of China, 202), o furher grow in relevance. Targe zone arrangemens have inensively been sudied following he publicaion of Krugman (99). Opion pricing echniques for such regimes, however, are sill no well-developed. The laer is primarily due o he requiremen ha inervenions a he boundaries should no yield arbirage opporuniies (see for insance Larsen and Sørensen (2007) for a recen discussion). More in paricular, he inervenion mechanism wihin he valuaion model mus ensure ha he exchange rae canno spend nie ime on he arge zone boundaries. Indeed, allowing he exchange rae o spend nie ime on he upper (lower) boundary implies ha subsequenly i can only decrease (increase) again and invesmen sraegies of no iniial oulay bu wih cerain gains would be enabled. This would hen violae he no-arbirage condiion of raional opion pricing. The boundaries of he arge zone hus should possess no probabiliy mass, i.e. he speed of re ecion away from hem should be in nie, which will be guaraneed in his aricle by choosing for insananeous and in niesimal re ecion in he 2

4 de niion of Skorokhod (96). This re ecion mechanism subsequenly will be superimposed upon he familiar sochasic se-up of Geomeric Brownian Moion (GBM) such ha we can discuss he resuling sochasic process as Re eced Geomeric Brownian Moion (RGBM). Employing RGBM for he valuaion of currency opions in credible arge zones, i.e. in zones of which susainabiliy is no quesioned by markes, has wo desirable characerisics. Firs, he familiar GBM-based currency opion model of Garman and Kohlhagen (983) emerges wihin our valuaion sraegy as is unbounded-domain limi. Second, he valuaion equaions are analyic closed-form formulas ha consis of in nie sums bu for which convergence is exremely fas such ha boh accuracy and workabiliy are guaraneed. Taking accoun of arge zone boundaries wihin he RGBM valuaion model also has srong economic implicaions as prices generally di er considerably from opion prices under GBM. Or, applying he unbounded-domain model of Garman and Kohlhagen (983) also o arge zone exchange raes ypically resuls in severe mispricing. In fac, in mos cases RGBM prices fall well below GBM prices as he upper boundary caps he upward ucuaion poenial of he exchange rae. Depending on he acual posiion of he exchange rae and he widh of he arge zone, GBM prices can hen easily surpass RGBM prices by more han 00%. The remainder of his aricle is organized as follows. Secion 2 develops our sochasic framework of RGBM in which due aenion will be given o he required absence of arbirage opporuniies. The ransiion probabiliy densiy funcion (pdf), i.e. he condiional densiy funcion, of RGBM will be obained in Secion 3. Secion 4 employs he laer densiy o derive European call and pu opion prices when wo-sided arge zones exis and also speci es he resuling hedge raios. Secion 5 hen specializes opion prices and hedge raios for one-sided arge zones, i.e. for se-ups where moneary auhoriies only defend an upper or lower boundary. Secion 6 concludes. 3

5 II Re eced Geomeric Brownian Moion (RGBM) Le (; F; P ) be a complee probabiliy space where is he oucome space conaining all evens!. F is a righ-coninuous increasing family F = (F ; > 0) of sub - elds of F and i is P -complee. P is a -addiive non-negaive measure on he measurable space (; F) represening a probabiliy on (; F) wih P () =. Finally, W = (W ; > 0) denoes a one-dimensional (F )-adaped Brownian moion. We assume ha he exchange rae, S, follows GBM wih drif and di usion coe ciens S and 2 S 2, respecively: ds S = d + dw. The arge zone arrangemen resrics he ucuaion poenial of he exchange rae by imposing wo boundaries upon he above sochasic process. A he lower boundary S and he upper boundary S, wih 0 < S < S, inervenions by moneary auhoriies will move he exchange rae back owards he cenre of he arge zone. Re ecion here is assumed o be insananeous and of in niesimal size, i.e. we adop he so-called re ecion funcions as de ned in Skorokhod (96). These re ecion funcions are he real righ-coninuous, non-negaive and non-decreasing funcions L and U ha specify he cumulaive amoun of upward and downward re ecion, respecively, and of which he poins of growh are locaed a he re ecing boundaries. The resuling sochasic process, ha will be referred o as RGBM, hen emerges as: ds S = d + dw + dl du. () Three properies of Equaion () are o be sressed here as we will exensively rely on hem laer when discussing opion valuaion. Firs, he incremens of L and U are of in niesimal magniude or he re ecion funcions are coninuous in S and and hus are of nie variaion. Also, he RGBM process is unique wih S, L and U being uniquely deermined by S and W, excep perhaps on a 4

6 se of measure zero (see Skorokhod, 96; Harrison and Reiman, 98; Ikeda and Waanabe, 98). Second, re ecion akes place insananeously, i.e. he speed of reurn from he boundaries is in nie. Formally, L and U have uncounably many poins of increase in nie ime on he boundaries, bu he se of all such poins has (Lebesgue) measure zero as discussed in more deail in, for insance, Harrison (985). The exchange rae hus spends no ime on eiher of he boundaries which will be crucial for arbirage pricing. Indeed, if he exchange rae were able o spend nie ime on he lower (upper) boundary, he price subsequenly would only be able o go up (down). Such perspecive would allow invesors o devise sraegies ha yield cerain gains wihou iniial invesmen. 2 Third, Iô s lemma for he RGBM process in Equaion () is a sraighforward exension of is formulaion under GBM. Indeed, L and U are (F )-measurable for all > 0 and hus are adaped o (F ) ha in urn is generaed by he Brownian moion process. Hence, S is composed of a local (F )- maringale wih coninuous sample pahs, namely he Brownian moion, and hree righ-coninuous (F )-adaped processes, namely he drif and re ecion componens. Thus, S is coninuous and boh he drif and re ecion componens have sample funcions of bounded variaion on any nie inerval (see, for insance, Harrison and Reiman, 98). As a resul, for any funcion f: R! R ha is dependen on S and and ha is wice coninuously di ereniable wih he moion of S as given in Equaion (), Iô s lemma yields: df(s ; ) ; ) ds ; ) d 2 f(s 2 (ds ) 2 ; df(s ; ) ; ) S (d + dw + dl du ) ; ) 2 f(s ; 2 2 S 2 d: (2) Under GBM, he erm (ds ) 2 equals 2 S 2 d as (dw ) 2 = d and dd = dw d = 0. This resul also carries over o RGBM due o he aforemenioned bounded-variaion naure of he re ecion For an inerpreaion of he re ecion funcions in erms of local ime, we can refer o Ikeda and Waanabe (98) and Harrison (985). 2 Alernaive re ecion mechanisms such as slow re ecion (Revuz and Yor, 994), delayed re ecion (Skorokhod, 96) and re ecion a so-called sicky barriers (Karlin and Taylor, 98) would hen clearly no be accepable for derivaive pricing since he speed of reurn from he boundaries in hese mechanisms is nie and hus posiive ime is spen on hem. 5

7 componens ha guaranees ha all addiional muliplicaive erms vanish, i.e. ddl = ddu = dw dl = dw du = dl dl = du du = dl du = 0. I is o be noed ha he incremens of he re ecion funcions remain presen in he rs righ-hand side erm in Equaion (2). 3 III The ransiion probabiliy densiy funcion of RGBM Applying Iô s lemma in Equaion (2) o he ransform s ln S yields: ds = 2 2 d + dw + dl du ; wih re ecion a s = ln S and s = ln S. The ransiion pdf for s is denoed by q (s; ; s 0 ; 0 ) and speci es he probabiliy of aaining s a ime given ha he process currenly, i.e. a 0, is a he source poin s 0. This densiy funcion, see for insance Risken (989), mus saisfy he Fokker-Planck equaion: 2 q (s; ; s 0 ; 0 (s; ; s0 ; 0 ) 2 2 (s; ; s 0; (3) for s < s 0 < s, s < s < s and > 0. Equaion (3) is o be solved subjec o an iniial condiion and wo boundary condiions. As noed earlier, insananeous re ecion ensures ha he wo arge zone boundaries have zero probabiliy or equivalenly ha all probabiliy mass is siuaed beween " # R s hem, i.e. q (s; ; s 0 ; 0 ) ds = Rs q (s; ; s 0 ; 0 ) ds = 0. Plugging Equaion (3) ino he laer s expression hen yields he following wo boundary condiions: lim s#s lim (s; ; s 0; 0 ) (s; ; s 0; 0 ) q (s; ; s 0 ; 0 ) = 0, (4a) q (s; ; s 0 ; 0 ) = 0. (4b) Finally, he iniial condiion for Equaion (3) is: lim [q (s; ; s 0 ; 0 )] = (s s 0 ) ( 0 ), (5) #0 3 We will come back o his propery when giving a formal jusi caion for he boundary behaviour of he opion price. 6

8 in which () denoes he Dirac dela funcion. This condiion guaranees ha all iniial probabiliy mass is locaed a he iniial value and he iniial poin of ime which by consrucion is he appropriae iniial condiion for processes based on Brownian moion. The ransiion pdf q (s; ; s 0 ; 0 ) is he soluion o he iniial-boundary value problem in Equaions (3)-(5) and an equivalen sysem has been solved in Veesraeen (2004). Adaping he soluion in Veesraeen (2004) and ransforming i in erms of RGBM for he exchange rae S wih he iniial poin of ime now se a and he end of he predicion inerval a T, i.e. Q (S T ; T ; S ; ), gives: where ( +X Q (S T ; T ; S ; ) = n= S T p 2 () exp n ln S ln S " ln S T + 2n(ln S ln S) ln S 2 exp 2 () #) () ( +X + n= S T p 2 () exp n ln S (n + ) ln S + ln S " 2n ln S 2 (n + ) ln S + ln S + ln S T 2 exp 2 () #) () +X exp n ln S (n + ) ln S + ln S T (6) S n=0 T " 2n ln S 2 (n + ) ln S + ln S + ln S T #!) () p +X + exp n ln S (n + ) ln S + ln S T S n=0 T " 2n ln S 2 (n + ) ln S + ln S + ln S T #) () p ; = ; [x] = Z x p 2 exp 2 y2 dy: I is fairly sraighforward, albei raher lenghy, o show ha he inegral of he ransiion pdf over is domain, i.e. SR Q (S T ; T ; S ; ) ds T, reurns uniy. Or, he boundaries under RGBM indeed possess S no probabiliy mass, which as argued before is essenial for arbirage-free opion valuaion. 7

9 IV European opion pricing under RGBM This secion values European call and pu opions via he risk-neural valuaion sraegy. 4 The riskneuralized homologue of he densiy in Equaion (6) is obained by replacing he drif facor by is risk-neural equivalen r r (see Garman and Kohlhagen, 983), where r and r denoe domesic and foreign risk-free ineres raes. The price a ime of a call opion wih ime o mauriy (T ), exercise price K and he arge zone boundaries S and S is given by: C S ; ; S; S = exp [ r ()] Z S S max [0; S T K] Q (S T ; T ; S ; )j =r r ds T : (7) Plugging he risk-neuralized ransiion pdf ino Equaion (7), assuming S T 6 K 6 S T, re-arranging and simplifying yields he following expression for he call opion price when domesic and foreign ineres raes di er: wih C S ; ; S; S; r 6= r = S exp [ r ()] K exp [ r ()] +X n= + d 3 S d 3 exp [ r ()] + exp [ r ()] + exp [ r ()] +X n= n S d 3 +X n=0 d = 2 r r ; d 2 = 2 r r ; 2 (r r ) d 3 = 2 ; +X n= n exp nd 2 ln S ln S h q ;n p i +X n= exp nd ln S ln S ( [q ;n ] [q 2;n ]) exp d (n + ) ln S n ln S ( [q 3;n ] [q 4;n ]) h q 2;n p io exp d2 n ln S (n + ) ln S (8) d3 d 2 KS h q 3;n + d 3 p T i h + d3 Kd 3 q 4;n + d 3 p T n exp d 2 n ln S (n + ) ln S n S d 3 KS d3 d 2 oo d3 Kd 3 ; 4 Complee derivaions of all resuls in his aricle can be obained from he auhor upon simple demand. ioo 8

10 and q ;n = ln S ln K 2n ln S ln S + r r () p ; q 2;n = ln S (2n + ) ln S + 2n ln S + r r () p ; q 3;n = ln S + (2n + ) ln S 2 (n + ) ln S r r () p ; q 4;n = ln S + ln K + 2n ln S 2 (n + ) ln S r r () p : The pricing formula in Equaion (8) requires a non-zero ineres rae di erenial in order o preven divisions by zero. In he case of idenical domesic and foreign ineres raes, l Hôpial s rule allows us o express he limi of Equaion (8) as: C S ; ; S; S; r = r = S exp [ r ()] K exp [ r ()] exp [ r ()] +X n= +X n= p exp [ r ()] + exp [ r ()] +X n=0 n exp n ln S +X n= ln S n exp (n + ) ln S n ln S n [q 3;n ] [q 4;n ] q 4;n p T +X n= oo exp n ln S ln S ( [q;n ] [q 2;n ]) h q ;n p i h q 2;n p io q 3;n p + KS exp (n + ) ln S n ln S ( [q3;n ] [q 4;n ]) n exp (n + ) ln S n ln S o KS + ln S ln K ; (9) wih [y] = p exp 2 2 y2 : The valuaion formulas in Equaions (8) and (9) are analyic closed-form expressions ha consis of in nie sums. Despie heir complex appearance, workabiliy is guaraneed as convergence o he limiing opion price occurs exremely fas. This is illusraed in Table where convergence for realisic 9

11 parameer se-ups requires a value of n of no more han 4 (in absolue value). 5 Table : around here As RGBM superimposes re ecing boundaries upon GBM, removing he boundaries gives he GBM, i.e. he unbounded-domain, price of Garman and Kohlhagen (983). Indeed, he limi for S! 0 and S! + yields: h C (S ; ) = S exp [ r ()] [q 5 ] K exp [ r ()] q 5 p i ; (0) wih q 5 = ln S ln K + r r () p : The familiar pu-call pariy also holds wihin exchange rae arge zones as re ecion keeps he currency opion conrac alive unil he mauriy dae such ha he invesmen sraegies ha underlie his pariy can also be developed when re ecing boundaries exis. 6 The value of he European pu opion, P S ; ; S; S, herefore emerges as: P S ; ; S; S = C S ; ; S; S S exp [ r ()] + K exp [ r ()] : () Targe zones have a srong impac on opion pricing as will be illusraed in Fig.. The doed lines depic he GBM opion prices of Equaion (0), whereas he solid nonlinear lines specify he RGBM opion prices of Equaion (8). The horizonal solid lines represen he maximum and minimum RGBM opion prices as he ucuaion limis for he exchange rae also creae a band for he opion price. Figure : around here 5 The prices for he unresriced-domain model of Garman and Kohlhagen (983) in Table will be used below o discuss he impac of applying he laer model also o arge zone exchange raes. 6 This consiues a crucial di erence o pricing of knockou opions where he pu-call pariy does no hold as argued in Kuniomo and Ikeda (992). In fac, knockou opions are nulli ed when he underlying asse reaches he boundaries and opion pricing hus urns ino a sopping-ime problem. Since he poin of ime a which he opion may be cancelled is unknown, he sraegies ha yield he pu-call pariy hen canno be applied. 0

12 Four properies of arge zone currency opion pricing are o be noed. Firs, he mos sriking feaure of Fig. is he S-shaped opion price funcion ha angenially nears he boundaries of is band. In fac, absence of arbirage opporuniies requires he opion o have zero movemen upon re ecion. Non-zero movemen would indeed allow for predicable gains if he opion were bough jus prior o inervenion. Or, he rs derivaive of he price funcion o he exchange rae is o equal zero a he boundaries. This requiremen can be illusraed wihin he following simple formal argumen. Iô s lemma in Equaion (2) applies boh jus before as well as a re ecion as argued in, for insance, Ikeda and Waanabe (98). Using Equaion (2) for he opion price hen speci es is insananeous change jus before and a re ecion a he lower boundary, respecively, as: dc S ; ; S; S S ; ; S; S S (d + dw 2 C S ; ; S; S 2 S 2 d; and 2 dc S ; ; S; S S ; ; S; S S (d + dw + dl 2 C S ; ; S; S 2 S 2 d; S ; ; S; S S ; ; S; S () (2) where we use he propery ha he re ecion funcion L only increases upon re ecion. Similarly, a he upper boundary he following expressions mus apply: dc S ; ; S; S S ; ; S; S S (d + dw 2 C S ; ; S; S 2 S 2 d; and 2 dc S ; ; S; S S ; ; S; S S (d + dw du 2 C S ; ; S; S 2 S 2 d. S ; ; S; S S ; ; S; S () (3) Precluding predicable pro s upon re ecion requires he insananeous change in he opion price jus prior o and a re ecion o be idenical. The expressions in Equaions (2) and (3) hen yield

13 he following wo condiions ha are o hold upon re ecion a he lower and upper boundaries, respecively: lim S #S lim S "S S ; ; S; S S S ; ; S; S S du # = 0; = 0: As he incremens in he re ecion funcions are sricly posiive upon re ecion and since he exchange rae is always larger han zero, we indeed obain he aforemenioned wo derivaive condiions ha are also prominenly presen in Fig.. Second, Fig. shows ha RGBM prices ypically fall below GBM prices as he upper boundary resrics he moneyness region and he likelihood of reaching i. However, and his may seem surprising a rs sigh, he panels also reveal ha RGBM prices can exceed GBM prices. In fac, he lower boundary can creae addiional value by prevening he exchange rae from moving far or farher below he exercise price and by pushing i upwards again. The resuling higher likelihood of gahering inrinsic value may hen even surpass he loss of value caused by he presence of he upper boundary. This e ec mus be larger for exercise prices ha are closer o he lower boundary as shown in panels (a) o (c). Third, widening he arge zone brings RGBM prices closer o GBM prices as illusraed across panels (d) o (f). Expanding he moneyness region by lifing he upper limi seps up RGBM prices, widens he band for he opion price and he price funcion sars more and more o resemble he unbounded-domain valuaion funcion. Fourh, using he Garman and Kohlhagen (983) model also for valuing opions on currencies ha acually evolve wihin a arge zone can resul in severe overpricing. 7 For insance, in he 25%-wide arge zone of panel (b) overpricing by GBM quickly surpasses 00%. Also he RGBM and GBM prices 7 As menioned earlier, underpricing is also possible. However, he required igh range of parameer values of exchange raes and exercise prices ha boh have o be (very) near o he lower arge zone limi makes his possibiliy of probably limied pracical relevance. 2

14 in Table con rm his subsanial degree of mispricing for he 25%-wide band. Given ha he widh of he laer zone can be seen as large by hisorical sandards, he documened degree of overpricing mus even be seen as raher conservaive since narrower bands furher consrain he moneyness region for RGBM opions. The RGBM model hus is of considerable economic value as erroneously applying he Garman and Kohlhagen (983) model also o arge zone exchange raes generally generaes (severe) overpricing. This has imporan implicaions for exchange rae risk managemen. Indeed, he higherhan-jusi ed cos of hedging could well depress demand for currency opions and as such creae larger exposure o exchange-rae risk. We proceed by deriving he opion dela or hedge raio, i.e. he rs derivaive of he opion price o he underlying exchange rae. This raio is of vial imporance o adequae managemen of exchange-rae exposure. The hedge raio for he call opion price in Equaion (8) is: S ; T ; S; S; r 6= = exp [ r ()] [q 2;n ] KS exp [ r ()] S d 3 exp [ r ()] +S exp [ r ()] +X n= +X n= [q 2;n ] +X n= exp nd ln S ln S [q ;n ] + [q ;n] p p (exp nd 2 ln S ln S q ;n p [q 4;n ] + d 3 +X n= S d 3 p T exp d (n + ) ln S n ln S [q 3;n ] d 3 [q 4;n ] p T exp d2 n ln S (n + ) ln S d3 d 2 KS q 3;n + d 3 p p q 2;n p!) p T [q 3;n ] p T (4) + d3 q Kd 3 4;n + d 3 p!) p : 8 The hedge raios for pu opions can be obained from heir call opion homologues via he derivaive of he pu-call pariy in Equaion (). 3

15 Idenical ineres raes a home and abroad call for careful evaluaion of he limi of Equaion (4) and his S ; T ; S; S; r = = exp [ r ()] [q 2;n ] KS exp [ r ()] S exp [ r ()] +X n= +X n= [q 2;n ] +X n= exp n ln S ln S [q ;n ] + [q ;n] p p (exp n ln S ln S q ;n p p T exp (n + ) ln S n ln S [q 3;n ] q 2;n p!) p [q 3;n ] p KS [q 4;n ] : As required, he limi of Equaion (4) for S and S going o 0 and +, respecively, yields he hedge raio of Garman and Kohlhagen (S ; = exp [ r ()] [q 5 ] : (5) Fig. 2 illusraes hedge raios for RGBM and GBM opions. The doed curves correspond o he hedge raio for he unbounded process ha increases from 0 for S! 0 in Equaion (5) o exp [ r ()] for S! +. The arge zone hedge raio on he conrary has a hump shape wih wo minima a zero as required by he abovemenioned no-arbirage condiion for he opion price funcion. Figure 2: around here Targe zone hedge raios never exceed values for he unbounded process and mosly fall well below hose levels. Panels (d) o (f) also illusrae ha he arge zone hedge raios for growing band widh coincide more and more wih he GBM hedge raio in Equaion (5) alhough he upper boundary keeps dragging he hedge raio in Equaion (4) o zero. In sum, hedge raios end o di er considerably beween he wo valuaion approaches such ha employing he Garman and Kohlhagen (983) model when acually a arge zone is presen can creae subsanial errors in hedging. 4

16 V One-sided arge zones This secion specializes he above resuls for se-ups in which moneary auhoriies defend a single lower or upper boundary. For insance, counries wih sizeable foreign-currency denominaed public and/or privae secor deb may wan o keep heir currency from depreciaing beyond some level in view of keeping he domesic-currency value of foreign-currency deb under conrol. On he oher hand, counries may for reasons of inernaional compeiiveness acively inervene in foreign-exchange markes o preven appreciaions of heir currency beyond a cerain level whils a he same ime no curbing depreciaions. The subsanial inervenions by Japanese moneary auhoriies in he yendollar marke in are ofen quoed in his respec (see Io, 2005; Hillebrand and Schnabl, 2008). Since 6 Sepember 20, he Swiss Naional Bank sands ready o buy foreign currency in unlimied amouns o keep he Swiss Franc (CHF) from falling below CHF.20 per Euro in view of limiing he de aionary impac ha he appreciaions of brough (Swiss Naional Bank, 202). We rs discuss call prices and hedge raios when only a lower boundary applies and subsequenly urn o he slighly more complex se-up of a sole upper boundary. We will also show ha he relaion beween RGBM and GBM prices now is unambiguous in he sense ha a sole lower (upper) boundary causes RGBM prices o be larger han or equal o (smaller han or equal o) GBM prices. 5

17 A sole lower boundary The call opion price when only a lower re ecing boundary a S is imposed will be denoed by C (S ; ; S) and arises as he limi of Equaion (8) for S! +: 9 h C (S ; ; S; r 6= r ) = S exp [ r ()] [q 5 ] K exp [ r ()] q 5 p i n h d3 Kd 3 S d 2 exp [ r ()] q 6 + d 3 p T + d 3 S d 3 S d exp [ r ()] f [q 6 ]g ; io (6) wih q 6 = ln S + ln K 2 ln S r r () p : For a zero ineres rae di erenial, he following expression applies: h C (S ; ; S; r = r ) = S exp [ r ()] [q 5 ] K exp [ r ()] q 5 p i p S exp [ r ()] (q 6 ( [q 6 ]) [q 6 ]) : (7) The rs and second erms in boh Equaions (6) and (7) specify he Garman and Kohlhagen (983) price and he remaining erms represen he non-negaive price e ec of he lower boundary. The exisence of he lower boundary indeed can generae addiional value when compared wih he GBM case as re ecion may increase he likelihood ha he opion ulimaely ends in he money. The RGBM opion price will exceed he GBM price provided ha he disance of he exchange rae versus he lower boundary and he exercise price is no oo large. Oherwise, RGBM and GBM prices will be indisinguishable. Formally, we know ha lim S#0 [C (S ; ; S)] = C (S ; ) and i is easy o see > 0. Or, C (S ; ; S) can increase in S as raising he lower boundary yields more scope for re ecion and hus may increase he poenial for he opion o end in he money, which 9 Veesraeen (2008) repors he call opion price when a lower barrier resrics he ucuaion range of he sock price. Again, pu opion prices and heir hedge raios can be obained via he pu-call pariy in Equaion (). 6

18 mus have a non-negaive e ec on is price. Hence, for decreasing S, C (S ; T ; S) mus approach C (S ; ) from above and hus C (S ; ; S) > C (S ; ) for all values of S. The hedge raio in he case of a sole lower boundary is speci ed as follows for a non-zero ineres rae di (S ; ; S; r 6= r ) = exp [ r ()] [q 5 ] + [q p KS exp [ r ()] q 5 p p + S d 3 S d exp [ r ()] +d3 S K d 3 S d 2 exp [ r ()] q 6 + d 3 p p ; [q 6 ] d 3 [q 6 ] p T and in he case of idenical domesic and foreign ineres raes i (S ; ; S; r = r ) = exp [ r ()] [q 5 ] + [q p KS exp [ r ()] q 5 p p + S S exp [ r ()] ( [q 6 ] ) : A sole upper boundary C S ; T ; S is he price of he call opion when only an upper arge zone boundary exiss and i emerges as he limi of he wo-boundary price in Equaion (8) for S! 0: C S ; ; S; r 6= r h = S exp [ r ()] f [q 5 ] [q 7 ]g K exp [ r ()] q 5 p i h + d3 S d 3 S d exp [ r ()] f [q 8 ] [q 9 ]g + d3 Kd 3 S d2 exp [ r ()] q 9 + d 3 p T h + d3 d 2S exp [ r ()] q 7 p i ; (8) i wih ln S ln S + r r + 2 q 7 = 2 () p ; ln S ln S r r + 2 q 8 = 2 () p ; q 9 = ln S + ln K 2 ln S r r () p : 7

19 For a zero ineres rae di erenial, he call opion price is: C S ; ; S; r = r h = S exp [ r ()] f [q 5 ] [q 7 ]g K exp [ r ()] q 5 p i S exp [ r ()] [q 8 ] q 8 p + p S exp [ r ()] ( [q 9 ] [q 8 ]) : [q 9 ] q 9 p T (9) Equaions (8) and (9) are slighly more complex han he pricing formulas when a sole lower boundary applies. This is due o he fac ha he upper boundary eners pricing no only hrough he condiional densiy funcion, bu now also emerges as he upper inegraion limi in Equaion (7). 0 The prices in Equaions (8) and (9) canno exceed he unbounded-domain price in Equaion (0), i.e. C S ; ; S 6 C (S ; ). This is due o he fac ha he upper boundary reduces he likelihood for he opion o end in he money and his e ec will be noiceable as long as he exchange rae is no oo far from he exercise price. Formally, we ;S) lim C S ; ; S = C (S ; ) S"+ > 0 as he RGBM price increases in he ceiling or a leas does no decrease in he laer. Indeed, a higher upper boundary raises he poenial for he call o end in he money and as a resul he RGBM price will near he unbounded-domain price from below. The hedge raio when he arge zone is characerized by he presence of a sole upper boundary is 0 The risk-neural valuaion Equaion (7) in he case of a sole lower boundary can be wrien as C (S ; ; S) = Z exp [ r ()] + K C S ; ; S = exp [ r ()] (S T K) Q (S T ; T ; S ; )j =r r ds T, whereas he price under a sole upper boundary is given by Z S K (S T K) Q (S T ; T ; S ; )j =r r ds T : 8

20 given S ; ; S; r 6= r = exp [ r ()] [q 5 ] + [q p [q 7 ] [q 7 ] p KS q 5 p exp [ r ()] p q 7 p! p S d 3 S d exp [ r ()] [q 8 ] d3 [q 8 ] [q 9 ] + d3 [q 9 ] +S S d2 exp [ r ()] S d 3 d p p 3 d 2 KS q 8 + d 3 p p + d q 3 Kd d 3 p! p : Finally, specializing his relaion for a zero ineres rae di erenial S ; ; S; r = r = exp [ r ()] [q 5 ] + [q p KS q 5 p q 7 p exp [ r ()] S S exp [ r ()] [q 8 ] p T [q 8 ] p T p T [q 9 ] : [q 7 ] + [q 8] p T [q 7 ] p T! VI Conclusions This aricle sudies currency opion pricing when he ucuaion range of he exchange rae is consrained by a arge zone arrangemen. Valuaion of such opions requires careful speci caion of he boundary behaviour. I mus be ascerained ha he exchange rae can spend no nie ime on he boundaries of he arge zone or insananeous re ecion upon inervenion is required. In fac, if he exchange rae were able o acually spend nie ime on eiher of he boundaries, i could subsequenly only move in one direcion. Invesmen sraegies wih cerain pro s would be enabled and his would rule ou arbirage pricing. We herefore superimpose insananeous and in niesimal re ecion upon he familiar sochasic framework of Geomeric Brownian Moion (GBM). This process is accordingly ermed Re eced Geomeric Brownian Moion (RGBM). Risk-neural valuaion subsequenly allows us o obain European 9

21 call and pu opion prices and heir hedge raios for wo-sided arge zones. As required, our pricing equaions reduce o he GBM prices of Garman and Kohlhagen (983) when evaluaing he limi for in niely wide arge zones. Despie he added complexiy in aking accoun of re ecion, he pricing relaions coninue o be analyic closed-form expressions. They conain in nie erms ha, however, converge exremely fas such ha accuracy and pracicabiliy are guaraneed. We subsequenly specialize resuls for se-ups in which moneary auhoriies mainain eiher a sole upper or a sole lower boundary. Such schemes arise when counries in order o, for insance, safeguard inernaional compeiiveness comba appreciaions beyond a cerain level wihou however limiing depreciaions as is he case in Swizerland since December 20. We illusrae ha he presence of exchange rae arge zones srongly a ecs opion prices and hedge raios such ha our resuls, nex o heir heoreical appeal, also have srong pracical and economic implicaions. In fac, neglecing arge zones in currency opion valuaion by erroneously applying he no-boundary GBM model of Garman and Kohlhagen (983) easily resuls in overpricing by more han 00%. Such overpricing could hen well depress demand for hedging and as such lead o excessive and poenially exremely cosly exposure o foreign-exchange risk. 20

22 References [] Garman, M. B. and Kohlhagen, S. W. (983) Foreign Currency Opion Values, Journal of Inernaional Money and Finance, 2, [2] Harrison, J. M. (985) Brownian moion and sochasic ow sysems, Wiley, New York. [3] Harrison, J. M. and Reiman, M. I. (98) Re eced Brownian moion on an orhan, Annals of Probabiliy, 9, [4] Hillebrand, E. and Schnabl, G. (2008) A srucural break in he e ecs of Japanese foreign exchange inervenion on yen/dollar exchange rae volailiy, Inernaional Economics and Economic Policy, 5, [5] Ikeda, N. and Waanabe, S. (98) Sochasic di erenial equaions and di usion processes, Norh- Holland, Amserdam. [6] Io, T. (2005) Inervenions and Japanese economic recovery, Inernaional Economics and Economic Policy, 2, [7] Karlin, S. and Taylor, H. M. (98) A second course in sochasic processes, Academic Press, New York. [8] Krugman, P. R. (99) Targe zones and exchange rae dynamics, Quarerly Journal of Economics, 06, [9] Kuniomo, N. and Ikeda, M. (992) Pricing opions wih curved boundaries, Mahemaical Finance, 2, [0] Larsen, K. S. and Sørensen, M. (2007) Di usion Processes for Exchange Raes in a Targe Zone, Mahemaical Finance, 7,

23 [] People s Bank of China (202) China Moneary Policy Repor: Quarer One 202, May. [2] Revuz, D. and Yor, M. (994) Coninuous maringales and Brownian moion, 2nd edn Springer- Verlag, Berlin. [3] Risken, H. (989) The Fokker-Planck Equaion. Mehods of Soluion and Applicaions, 2nd edn Springer-Verlag, Berlin. [4] Skorokhod, A. V. (96) Sochasic equaions for di usion processes in a bounded region, Theory of Probabiliy and is Applicaions, 6, [5] Swiss Naional Bank (202) 04h Annual Repor 20, March. [6] Veesraeen, D. (2004) The condiional probabiliy densiy funcion for a re eced Brownian moion, Compuaional Economics, 24, [7] Veesraeen, D. (2008) Valuing Sock Opions When Prices are Subjec o a Lower Boundary, Journal of Fuures Markes, 28,

24 Table : RGBM and Garman-Kohlhagen (GK) call opion prices.* n Time o mauriy (in years) 0:25 0:5 0:75 :5 = 0: :: :: :: :: : : : : : : 25 :: GK = 0: :: :: :: :: : : : : : : 25 :: GK = 0: :: :: :: :: : : : : : : 25 :: GK * Parameer values: S = 22; S = 20; S = 25; K = 22:5; r = 0:05 and r = 0:03: 23

25 Figure : Call opion prices under RGBM (solid lines) and GBM (doed lines) for r = 0.06, r * = 0.03, T = 0.5. σ = 0.5 and ( ) opion price opion price exchange rae exchange rae (a) S = 20, S = 25 and K = 20.5 (b) S = 20, S = 25 and K = opion price opion price exchange rae exchange rae (c) S = 20, S = 25 and K = 24 (d) S = 20, S = 25 and K = opion price opion price exchange rae exchange rae (e) S = 9, S = 26 and K = 20.5 (f) S = 5, S = 30 and K = 20.5

26 Figure 2: Hedge raios under RGBM (solid lines) and GBM (doed lines) for r = 0.06, r * = 0.03, σ = 0.5 and T = 0.5. ( ) hedge raio 0.5 hedge raio exchange rae exchange rae (a) S = 20, S = 25 and K = 20.5 (b) S = 20, S = 25 and K = hedge raio 0.25 hedge raio exchange rae exchange rae (c) S = 20, S = 25 and K = 24 (d) S = 9, S = 26 and K = 20.5 hedge raio 0.5 hedge raio exchange rae exchange rae (e) S = 7, S = 28 and K = 20.5 (f) S = 5, S = 30 and K = 20.5