CURRENCY RANSLAED OPIONS Dr. Rober ompkins, Ph.D. Universiy Dozen, Vienna Universiy of echnology * Deparmen of Finance, Insiue for Advanced Sudies Mag. José Carlos Wong Deparmen of Finance, Insiue for Advanced Sudies ** ABSRAC his paper examines opions based upon a foreign asse bu wih a payou ha occurs in anoher currency. Differen ypes of currency ranslaed opions are covered: Flexos, Compos or Join Opions and Quanos. A special emphasis is pu on his final currency ranslaed opion, since i is he only produc ha compleely eliminaes he currency risk o he equiy invesor. JEL classificaions: C5, G3 his piece of research was parially suppored by he Ausrian Science Foundaion (FWF) under gran SFB (Adapive Informaion Sysems and Modelling in Economics and Managemen Science). Keywords: Currency ranslaed Opions, Dynamic Hedging. * Favoriensrasse, A-4 Wien, Ausria, Phone: +43--76-99, Fax: +43--79-6753 ** Sumpergasse 56, A-6 Wien, Ausria, Phone: +43--337576
, INRODUCION In his paper, we will examine opions based upon a foreign asse bu he payou of he opion occurs in anoher currency. he value of he coningen claim is affeced by he co-movemen of he underlying asse price and he movemen in some currency exchange rae. Even hough only currency-proeced opions on equiies are considered, he mehodology discussed applies o all such producs. hese producs are generally referred o as Guaraneed Exchange Rae Opions or Quanos. We will also examine relaed producs, which vary he degree of he currency guaranee, including Flexos and Compos or Join Opions.., CURRENCY RANSLAED OPIONS: A DESCRIPION As a general overview, hese producs differ from sandard opions in ha he value of he opion is expressed in anoher currency. In echnical jargon his is referred o as a change in he numeraire. his allows us o express he value of he coningen claim in anoher currency. o simplify our exposiion, consider an a-he-money call opion on a paricular equiy, for example he French sock Michelin wih a srike price of 5 FF. he price for he opion is quoed a he Paris Bourse a French Francs per share and he conrac represens he righ o buy shares of he sock. herefore, he oal cos of he opion would be French Francs for each opion conrac. Consider he value of his opion o a US invesor. Since his reference currency is he US Dollar, he would conver he currency value (numeraire) of he sock opion price by dividing he price of he opion by he number of French Francs per US Dollar. Assuming he exchange rae is 5 FF/US Dollar, he value of he opion is now $ for shares or $ in oal. he ulimae underlying he invesor is ineresed in remains he foreign shares in French Francs. he currency exchange rae simply allows he expression of he coningen claim in his domesic currency. As he exchange rae of US Dollars for French Francs is also a sochasic variable, hese producs address he secondorder effec his inroduces in pricing his currency adjused coningen claim. hese producs as a class have become known as Currency ranslaed Opions. he mos popular of hese producs are he Quanos. his is an abbreviaion for quaniy adjused opion. As wih all coningen claims he payoff of he coningen claim is dependen upon he underlying asse price. Wih his produc, he exchange rae a which he purchaser convers he currency is fixed a he sar of he opion. In our example, he payoff depends upon he value of he Michelin share price a expiraion and herefore can be examined as a simple equiy opion. he difference is due o he fac ha he impac of he change in he currency exchange rae (FF/US Dollar) depends upon he quaniy of he exposure which is also a funcion of he value of he Michelin share price a expiraion. If he opion finishes a expiraion ou-of-he-money (which is a funcion of he foreign price of he equiy and he srike price), he impac of he exchange rae is zero. If, on he oher hand, he opion finishes in-he-money, he impac of he exchange rae on
he payoff of he opion depends on he amoun of he final inrinsic value of he opion (in he foreign currency). his second-order impac requires he seller of he opion o coninually adjus a currency hedge o assure ha only he opion premium remains a a fixed exchange rae in he oher currency. Clearly, his will inroduce risk o he seller and will depend upon he correlaion beween he underlying asse movemen (which in urn defines he inrinsic value of he opion) and he exchange rae movemens. As wih mos exoic coningen claims, Currency ranslaed Opions are offered wih a variey of differen feaures. here are four basic versions of hese producs: () A foreign equiy call sruck in he foreign currency () A foreign equiy call sruck in he domesic currency (3) An equiy linked foreign exchange call, and (4) A fixed exchange rae foreign equiy call he simples Currency ranslaed Opion is he flexible exchange rae version, which is simply an opion on he foreign equiy wih no proecion agains he movemen in he currency. he flexible elemen refers o he fac ha he final payou of he opion depends upon he final value of he exchange rae. hese producs have been referred o as Flexos. his is he firs version of he Currency ranslaed Opion ha in realiy is simply a foreign equiy call sruck in he foreign currency. he second and hird versions of he produc are called Compos. hese producs offer more currency proecion han he Flexo due o he fac ha he srike price of he opion is in he invesor s home currency bu he value of he underlying asse remains in he foreign currency. he second version is for he equiy invesor rying o achieve some proecion agains exchange rae flucuaion. hese opions are also referred o as a Join Opion. his is an opion on an underlying, ofen a sock index, which is denominaed in a second currency. Unlike a guaraneed exchange rae opion (or Quano), in which exchange raes are fixed, he purchaser of a join call opion benefis from upside in he currency in which he asse is originally denominaed. he hird version is essenially he same produc bu expressed in erms of he currency marke. his invesor is a currency rader who is relaing he payoff of he opion o he performance of a foreign equiy marke. Ulimaely, his produc is a currency opion bu a minimum floor level has been esablished which is a funcion of he foreign equiy marke. In his produc, he invesor is exposed o he fall in he foreign equiy marke bu is proeced agains a fall in he currency marke. One may no immediaely realise why hese producs are almos idenical. he reason why hey are he same is he same raionale why a call opion on US Dollars for Deusche Marks is idenical o a pu opion on Deusche Marks for US Dollars. For he Compo opions, one version is he righ o exchange he currency for a foreign equiy and he oher version is he righ o exchange he foreign equiy for he currency. hey are an opposie reflecion of each oher like an image in a mirror. he rue Quano is he fixed exchange rae version ha assures ha he final payoff of he opion can be convered back o he invesor s home currency a 3
a fixed exchange rae. As he reader can appreciae, he Compo provides more proecion agains exchange rae changes han he Flexo bu less han he rue Quano.., CURRENCY RANSLAED OPIONS: APPLICAIONS Such opions have become increasingly popular as invesors desire exposure o foreign asses wihou assuming he foreign exchange rae risk. Mos of he demand is for bond and sock index opions. A currency ranslaed opion can in heory exis for any asse or liabiliy denominaed in a currency oher han ha in which i is usually raded. One example of such a srucure which is offered on an organised exchange is he Chicago Mercanile Exchange s Nikkei 5 sock index conrac, which uses he nominal price of he yen-denominaed index applied o as US dollar noional principal. In he Unied Saes and in Europe, much of ineres for hese producs has come from fund managers who are invesing inernaionally. One of he major problems when invesing in oher counries has been he currency risk. Cerainly, derivaive producs such as exchange raded opions and fuures have made he process easier by only ying up a relaively small amoun of capial ha is he iniial margin in he case of he fuures or premium in he case of he opion. However, when eiher he opion become in-he-money or fuures prices change (which has cash-flow impacs in he margin accoun), he invesor will face currency risks. he Currency ranslaed Opions have been developed o eliminae he currency risks associaed wih foreign invesmens., PRICING OF CURRENCY RANSLAED OPIONS Given ha currency ranslaed opions can be seen as sandard European call or pu opions on a share (if he payoff is expressed in he foreign currency), he firs sep is o evaluae he opion wihou he currency impac. Black and Scholes (973) have done his for European opions on non-dividend paying socks and Meron (973) exended his o coninuous dividend paying shares [discree dividend paying shares have also been considered see Roll (977)]. Once he sandard opion valuaion is compleed, adjusmens mus be made due o he currency effecs. he firs adjusmen ha mus be made o he opion price depends upon he correlaion beween he exchange rae and he underlying sock. he degree of impac depends upon he srucure of he currency ranslaed produc. For he Flexo, here is no impac and for Compos and Quanos, he effec is more pronounced. he second adjusmen o he price of a sandard opion depends on he difference in he ineres raes in he wo currencies. For he pricing of all coningen claims, he curren price in a risk neural framework is simply he expeced erminal payoff discouned back o presen value. Clearly, he risk-free ineres rae appropriae for discouning mus be in he same currency as he underlying asse. In he example of a sandard call opion on Michelin shares, he ineres rae used for discouning he payoff is in French Francs. If he opions payoff were expressed in US Dollars (he Quano, for example), hen i would be consisen o discoun he expeced value of his payou by he appropriae US 4
ineres rae for he period. However, i may be helpful a his poin o consider all impacs of ineres raes on opion valuaion. his firs effec of ineres raes is he deerminaion of he expeced sock price a he expiraion of he opion. In a risk-neural world, he expeced fuure u( ) sock price of Michelin shares (in he absence of dividends) is F = S e. Where u is he risk-free ineres rae in French Francs, is oday's dae and is he expiraion of he opion. Likewise, he French Franc ineres rae is relevan in he borrowing cos required o mainain he dynamic hedge. his can be seen as he u( ) righ-hand porion of he Black Scholes (973) formula: X e. A his sage, US Dollar ineres raes are irrelevan o he analysis, as he valuaion of he opion is simply he evaluaion of a European call payoff. However, once his expeced payoff has been deermined, i is hen re-expressed in US Dollar erms. Since he iniial premium paymen is in US Dollars, which is he presen value of he expeced payoff (also in US Dollars), he relevan ineres rae o link he presen and fuure value mus also be in US Dollars. From he sandpoin of he US Invesor, he cos of he Michelin call will depend upon he relaive levels of ineres raes in France and American. Consider he case of a Flexo. his opion is he presen value (discouned using French Franc ineres raes) of he expeced payoff of he Michelin call. his is simply convered a he curren spo exchange rae. In he case of a Quanoed Michelin call, he invesor pays he presen value (discouning using US Dollar ineres raes) of he expeced payoff of he same call opion. If he agreed exchange rae for he conversion is fixed equal o he curren spo rae, hen differences in he premium amoun paid by he invesor depend on he differences in he ineres raes beween he wo currencies. I has been popular o offer such opions where he payoff is in a higher ineres rae currency, as he premium expendiure will appear cheaper. Prior o he inroducion of he Euro, his was ofen done for Ialian invesors wishing o buy opions on Swiss socks (and fixed in Ialian lire) because of he wide ineres rae differenial beween he lira and he Swiss Franc. his simplified analysis ignores addiional impacs, which are inroduced when a correlaion exiss beween he levels of he underlying asse and he foreign exchange rae. he exac impac of he ineres rae differenial on he value of he currency-ranslaed opion will depend on he specific srucure of he produc. A his poin, we will examine hese effecs in a more rigorous manner by firs defining he four major caegories of currency ranslaed opions and hen providing formulas for he payoff of hese producs a expiraion. For his purpose, he following noaion will be used: Domesic Currency: US$ Foreign Currency: FF = iniial dae of opion = expiraion dae of opion = agreed fixed value hrough he life of he opion S = price of he sock in FF a he expiraion dae e = spo exchange rae in US$/FF e = agreed fixed exchange rae in US$/FF e = final exchange rae in US$/FF 5
X = srike price of he opion in FF k = srike price of he opion in US$, ransformed using: X.e (for Quano payoff) k = agreed fixed srike price of he opion in US$ (for Compo payoff) u = French ineres rae r = US ineres rae., PRICING FLEXO CURRENCY RANSLAED OPIONS As he Flexo is simply a foreign equiy call sruck in he foreign currency, pricing is easy. Le us reurn o our example of he US invesor who was ineresed in purchasing a hree monh 5 call on Michelin when he price of he sock was also 5. he price of he sandard call opion was FF and he conrac size was shares, so he oal expendiure would be FF per opion. However, since he invesor is US Dollar based, he mus conver enough US Dollars o yield FF and wih his amoun pay a French broker for he opion. A he end of he life of he opion, he proceeds will depend on he sock price level being higher han 5 and if ha is he case, he opion will yield back he inrinsic value in French Francs. he invesor would hen conver his amoun back o US Dollars a he hen prevailing exchange rae. his srucure allows an invesmen in he foreign equiy o be esablished wih proecion agains he price of he equiy falling bu no proecion agains foreign exchange risk. he payoff of his mos basic of currency ranslaed opion is: C = e MAX(,S X) () As far as he wrier of he Michelin call in Paris is concerned, he is indifferen as o wheher or no he produc is a currency ranslaed opion. I makes no difference o him wha he holder of he opion does wih his winnings afer hey have been handed over. All he seller is concerned wih is meeing his obligaion in French Francs. Of course, i will maer o he US invesor, as he Dollar proceeds of he opion depend upon he prevailing exchange rae a ha dae. Due o he fac ha he holder of he Flexo has a complicaed payoff which is a funcion of wo variables, he price of Michelin shares and he FF/$ exchange rae, Figure displays he payoff across hree dimensions. 6
f (S X) (7) f (S X) (7),4, $ PAYOFF, 8 6 4-4 4.3 4.6 4.9 FF/$ 5. 5.5 5.8 9 75 6 45 5 3 MICHELIN PRICE Figure. he Payoff of a Flexo a Expiraion as a Funcion of he Price of he Underlying Share and he Foreign Exchange Rae he reader can see ha in many ways his srucure resembles a sandard call bu he payoff curves even higher when he French Franc srenghens o 4. and he profis are diminished when he French Franc weakens o 6. FF/$. he value of his mos simple of currency ranslaed opions is he value of a sandard call opion (which could be esimaed using he Black and Scholes formula) muliplied by he curren or spo exchange rae. here will no be any impac from he correlaion or ineres rae facors since he holder of he opion bears all hese risks. hus, he heoreical pricing for Flexos is rivially simple. An American invesor simply buys a FF securiy for US $ a he presen exchange rae; in ime he sells he securiy for US $ a he hen prevailing exchange rae. Nohing myserious abou his; if he securiy is an opion and is he ime o payou, he US $ value of his opion is jus: e. f (S,X) () where f (S,X) is he value of he simple opion in FF and can be priced using he Black Scholes (973) formula.., PRICING COMPO CURRENCY RANSLAED OPIONS As was previously indicaed, he Compo comes in wo varieies: () A foreign equiy call sruck in he domesic currency and () An equiy linked foreign exchange call. We will discuss boh for he sake of compleeness, however, he only Compo ha has commonly appeared in he marke is he foreign equiy call sruck in he domesic currency. he inuiion behind his produc is also relaively sraighforward. his srucure allows an invesmen in he foreign equiy wih proecion agains he price of he equiy falling and some proecion agains foreign exchange risk. he payoff for he holder of his currency ranslaed opion is: C = MAX(,[S e ] k ) (3) 7
his Compo is exchanging k unis of he domesic currency for one uni of he foreign sock, S. In our example, he US invesor is buying a call opion on Michelin sock bu for a srike price of which is fixed (for him a) $5. While he srike is fixed, he payoff of he sock is muliplied by he exchange rae of $/FF. his is he same as dividing he price by he exchange rae of FF/$. In he case ha he sock price is 5 FF and he exchange rae is equal o 5 FF/$, he opion will be a-he-money for he US invesor. A firs glance, i appears ha he only sochasic variable is a Compounded variable, S e, wih a fixed srike price. Given ha he produc of lognormal variables will also be lognormally disribued, i would appear ha one could apply he Black Scholes (973) formula wih his Compound variable as he underlying. While his is rue, one should consider he payoff of he srucure from he sandpoin of he opion seller. Even hough he srike price is fixed for he foreign invesor, he value of he srike price (for he seller) depends upon he exchange rae. hus, his is an opion o exchange one risky asse for anoher. Margrabe (978) found he soluion o his problem as a generalisaion of he Black Scholes (973) formula. From he sandpoin of he seller of he Compo equiy call he promised payoff can be expressed as: C ' = MAX(,S [ k / e ] ) (4) he seller of his opion in Paris mus pay he difference beween he final share price of Michelin and he srike price of 5 FF divided by he exchange rae of FF/$. Since he srike price of 5 FF is fixed, he variaion of he US Dollar adjused srike price is solely deermined by he sochasic process driving he exchange raes. he reader is referred o Figure which displays he hree dimensional graph of he payoff of his produc as a funcion of Michelin share prices and he FF/$ exchange rae.,5, $ PAYOFF,5, 5-4 4.4 FF/$ 4.8 5. 5.6 6 9 75 5 3 45 6 MICHELIN PRICE Figure. he Payoff of an Equiy Compo a Expiraion as a Funcion of he Price of he Underlying Share and he Foreign Exchange Rae 8
f ( h k ) f ( h k ) (8) (8) As he reader can see, his payoff funcion differs fundamenally from eiher ha of a sandard European call or he Flexo. his payoff funcion is he produc of he wo asses and is defined by wo sochasic processes. hus, his produc is clearly defined as a Margrabe ype opion. Suppose he French sock were also quoed on he NY Sock Exchange as an American Deposiory Receip. his grealy simplifies he pricing since his ADR would represen he price of he French sock in US Dollars. Absence of arbirage would require he New York price in $ a ime o be h = e S. he US $ value of his opion is hen e. f ( h, k) (5) where h is he presen New York price of he French sock. he formula (or procedure) for f (h, k) depends of course, on he precise opion being considered. his formula will depend on σ h, he volailiy of he New York price of he French sock. his may be obained direcly, by observing hisorical New York price movemens. Alernaively, if he paricular sock is no quoed in New York, he effecive volailiy can be calculaed from he volailiies of he French sock price and he exchange rae as follows: σ = σ = σ + σ + h Se S e ρ σ σ Se S e (6) his volailiy appears similar o he volailiy inpu required for he Margrabe model bu in his insance he (wo imes) covariance erm is added o he wo individual variances (for he sock and he currency exchange rae). he alernaive Compo srucure, he equiy linked foreign exchange call, is simply he mirror image of his firs produc. he logic is simple: a call on US Dollars for Euros is he same as a pu on Euros for US Dollars. hus, if we price a call opion on Michelin shares for US Dollars, his is equivalen o a pu on he US Dollar for Michelin shares. he pricing follows direcly. he holder s payoff of his ype of Compo a expiraion is simply: C = S MAX(,e e ) where e represens he srike price of he foreign exchange. If we muliply (by he principle of homogeneiy) he sock price, S across he erminal opion payoff, he resul can be shown o be equivalen o equaion 3..3, PRICING QUANO CURRENCY RANSLAED OPIONS he Quano is he real produc of ineres, because his produc compleely eliminaes he currency risk o he equiy invesor. his produc fixes a he incepion he exchange rae a which he proceeds of a foreign equiy call will be paid. he pricing of his produc incorporaes boh he correlaions beween he sock price and foreign exchange and he ineres rae differenials of he wo counries. As hese effecs inroduce complicaions, neiher he sandard Black & Scholes (973) nor he generalisaion of Black and Scholes by Margrabe (978) applies. (7) 9
he payoff for he holder of he Quano opions can be expressed as: C = e MAX(,S X) = MAX(,[S e ] k) (8) While i may appear (from he righ hand expression of he equaion) ha we are exchanging a risky asse, S e, for anoher risky asse which is k, his is no he case. As wih he Compo opion, he firs risky asse is a Compounded variable, which would be in our example, he price of Michelin or 5 imes he fixed exchange rae of 5 or 5. However, he srike remains fixed equal o he French Franc srike price of 5 muliplied by he fixed exchange rae of 5. hen, as boh sides of he equaion are divided by he final exchange rae, any source of ' C = e / e MAX(,S X) he uncerainy for he srike price is removed and muliplied across he enire funcion. When his is expressed in his way, he seller views he Quano payoff as: o see why his produc is compleely hedged agains movemens in he exchange rae, he reader is referred o Figure 3. In his graph, he final payoff of he Quano call on Michelin shares is ploed versus boh he prices of he share and he exchange rae. In his figure, he US Dollar profi is he same regardless of he currency exchange rae and for all purposes has no currency risk. (9), 9 8 7 $ PAYOFF 6 5 4 3-4 4.3 4.6 4.9 FF/$ 5. 5.5 5.8 9 75 6 45 3 5 MICHELIN PRICE Figure 3. he Payoff of a Quano for he Holder a Expiraion as a Funcion of he Price of he Underlying Share and he Foreign Exchange Rae All currency risks are borne by he seller of he Quano. For his Michelin Quano call wih an exchange rae fixed a 5 FF/$ and a nominal srike price on he sock opion a 5, Figure 4 displays he sellers risks o boh he levels of he sock price and he erminal foreign exchange rae.
$ PAYOFF 8 6 4 4 4.3 4.6 FF/$ 4.9 5. 5.5 5.8 9 75 6 45 3 5 MICHELIN PRICE Figure 4. he Payoff of a Quano for he Seller a Expiraion as a Funcion of he Price of he Underlying Share and he Foreign Exchange Rae I is clear ha from his figure ha he risk of he payou is a funcion of he movemens in he equiy and he exchange rae. Essenially, he seller has he same risk as was previously defined as a Flexo. If he seller assumes he equiy and he exchange rae are uncorrelaed, hen one can ake expecaions of he producs separaely and add hem ogeher o esimae he value of he Quano; oherwise, adjusmens mus be made. o see he pure impac of he currency movemen relaive o he equiy price, we have subraced he risks of he seller of he Michelin Quano from he fixed payoff o he buyer. his can be seen in Figure 5. 5 5 5 $ PAYOFF -5 5 5 - -5 - MICHELIN PRICE 75 3 4 4.3 4.6 4.9 5. 5.5 FF/$ 5.8 Figure 5. he Pure Currency Risk of a Quano for he Seller a Expiraion as a Funcion of he Price of he Underlying Share and he Foreign Exchange Rae he perspecive of his graph has been shifed o beer assess he impac of he final exchange rae. Clearly, if he opion is ou of he money, here is no
S() () S ( σ W () + γ ) ( σ (W 3 () + v ) (3) (4) currency risk. In addiion, if he erminal foreign exchange rae is equal o he iniial fixed level of 5 FF/$, here is also no foreign exchange risk. On he oher hand, as he sock price rises, he seller becomes more exposed o changes in he exchange rae. he wors case scenario presened in his figure is ha he dollar srenghens o 4 FF/$ as he sock price rises o 3 FF. herefore, he holder of he Quano would pay more for he produc compared o a sandard call if his scenario were likely o occur. he oher elemen ha could conribue o he value of he Quano relaive o a sandard call opion would be he impacs of ineres raes. he seller mus deermine he expecaion of e / e in equaion 9. his is he difference beween he spo exchange and he final exchange rae a he expiraion of he opion. A he sar of he Quano s life, his is simply he raio of he spo exchange rae o he forward exchange rae for he expiry dae of he opion. his difference is relaed direcly o he ineres rae differenial beween he wo currencies, French Franc and US Dollar for he period. herefore, he valuaion of Quanos requires analysis o be compleed for a complicaed produc of wo sochasic processes. A his poin, i is convenien o review he following propery of wo price processes ha are driven by Geomeric Brownian moion. If W and W are wo independen Brownian moions and he correlaion coefficien beween hese processes, ρ, is a consan beween and, i can be shown ha he process W = ρ + ρ. 3 W W () is also a Brownian moion wih a correlaion ρ wih he Brownian moion W. In essence, we use wo correlaed bu independen Brownian moions o creae a new one. his propery will now be used for he firs sep: he value of he Quano depends direcly on wo random processes: e (), which is he process followed by he $/FF exchange rae; and S (), which is he process followed by he sock price in French francs. hese wo processes are driven by wo independen Brownian moions: W () and W (), respecively. herefore we can specify a wo facor model ha akes ino accoun wo sochasic sae variables: S() = S e e() = e e (σ W () + γ ) ( σ (W () + ) Addiionally, here is a dollar cash bond 3 v () () r B() = e and a FF cash bond u C() = e, wih r and u represening consan US Dollar and French Franc ineres raes, respecively. From hese equaions we noice ha he process S () is affeced by W while boh W and W affec he process e (). An inuiive way o undersand his is by analysing he payoff of he seller. he seller, knowing ha is assuming a currency risk, will be ineresed o price his insrumen using all possible informaion ha can have a poenial link o he behaviour of he exchange rae
and consequenly o he payoff of he opion. herefore he uses he correlaion beween S () and e (). Since S () and e () are no raded securiies in he US marke, we deermine he equivalen porfolio o replicae hese securiies and hereafer rea he resul as hough a raded asse exised: Y ()=e () C ()/B () and Z ()=e () S ()/B (). he nex sep is o use Io s calculus o find he sochasic differenial equaions for Y () and Z (). his is done by firs plugging equaions for he processes S() and e() ino he discouned equivalen securiies Y() and Z(), and hen applying Io s calculus o reach: dy = Y ρσdw () + ρ.σdw () + ( v + σ + u r)d (3) dz = Z (σ + ρσ )dw() + ρ.σdw () + ( γ + v + σ + ρσσ + σ r)d (4) hese formulae sugges i is possible o generae an equivalen payoff o a US Dollar denominaed raded securiy. Even so, his securiy is no acually a raded securiy. he pricing of derivaives on non-raded asses requires he inclusion of a marke price of risk. his can be solved by he applicaion of he Girsanov heorem o specify he marke price of risk. When his is done, hese equaions are modified such ha evaluaion is done under he risk neural measure (so ha he mean(s) or drif(s) cancels ou). hese adjusmens can be expressed as: + σ + ρσσ u λ = γ (5) σ v + λ = σ + u r ρλ σ ρ.σ (6) * W i () = W i () + λ i (s)ds ; i =, (7) Subsiuion of hese adjusmens ino Equaions 3 and 4 yield: e() = e e * ( σ W () ( ρσ σ + u σ )) S() = S e * * ρσ W () ρ.σ W () ( + + r u σ ) (8) (9) Finally, he new Brownian moions are replaced in he original model o obain he maringale SDEs for S () and e (). Consequenly, we will have finally specified he model under he risk-neural probabiliy measure. o faciliae he analysis rewrie (7) in he following way: 3
( σ Z ( ρσ σ ) σ) S() = S e u. e () Discouning he Quano payoff o presen value yields: e r r EQ ( S k) = e ( σ ) (.S ) ρσ u e e k () In order o avoid arbirage opporuniies, he presen value is se equal o. o deermine he value of a Quano forward conrac, we choose a delivery price such ha he equaliy holds: F Q u ( ρσσ ) = S e e or F = Fe Q ( ρσσ ) () where F represens he local currency forward price (French francs). Noice ha when ρ =, he value of he sandard forward is he same as he value of he Quanoed forward. Having he Quanoed forward price we evaluae he Quanoed call opion. o find E (Max (, S e k)) we can use he log-normal call formula (bu plugging in he Quanoed forward insead of he sandard forward). o ge he price of he produc a ime, we discoun he expecaion by he US ineres rae: V = e ( r ) F log Q F + σ log Q σ k k. FQφ kφ σ σ (3) o beer undersand he impacs of he correlaion relaionship and he ineres rae differenials beween he wo currencies involved in he value of he Quano, a hree monh call on Michelin shares was evaluaed using equaion 3. his opion had a curren spo price and srike price equal o 5 FF, 5% volailiy and assumed no dividends. he US ineres rae was fixed a 7% and he French Franc ineres rae was varied beween 5%, 7% and 9%. Finally, he correlaions beween he Michelin share price and he FF/$ exchange raes were assumed o be -%, % or +%. In able he reader will see he values of he Quanoed opions and he value of a sandard Michelin call opion wih all he prices expressed in US Dollars (convered a he curren spo rae of 5 FF/$). US Ineres Rae 7% 7% 7% FF Ineres Rae 5% 7% 9% Correlaion CALL CALL CALL -%.56.7.83 %.76.9 3.5 +%.98 3. 3.8 Sandard Call.78.9 3.4 able. he pricing of Quano Call Opions as a Funcion of he Ineres Rae Differenial and Varying Correlaion Levels. 4
he resuls are consisen wih wha was suggesed previously. When he share price is uncorrelaed o he foreign exchange rae, he only impac of Quanoing is due o he ineres rae differenial. In he case ha he US ineres rae is higher han he FF ineres rae (7% o 5%), he cos of he opion is slighly reduce in US Dollar erms. When he sock price is negaively correlaed o he exchange rae (-% in his example), his suggess ha increases in sock prices (and increases in call opions values) are associaed wih a weakening of he French Franc. his implies ha he gain o he US invesor is reduced o he noncurrency-proeced sandard European call (Flexo). Likewise, if he price of Michelin shares is % posiively correlaed o he exchange rae, he holder of he Quano is proeced from likely adverse currency movemens. hus, he would be willing o pay more han for he unproeced sandard European call opion. o visualise hese impacs beer, anoher series of hree dimensional graphs have been produced which plo he difference beween he price of he Quanoed call opion on Michelin shares and he price of a sandard call as a funcion of he ineres rae differenial beween French Franc and US Dollar ineres raes and of he level of he correlaion. his is porrayed in Figure 6...5..5..5 $ VALUE. -.5 -. -.5 -. -.5 -% -8% -6% -4% -% % CORRELAION % 4% 6% 8%.%.5%.%.5%.% -.5% RAE DIFF. -.% -.5% %-.% Figure 6. he Difference beween a Quanoed Call and a Sandard Call as a Funcion of he Ineres Rae Differenial and he Correlaion In his Figure, he reader can see ha here is no difference beween he values of a sandard call and a Quanoed call when boh he ineres differenial and he correlaion coefficien is zero. However, he impacs of Quanoing he opion can be seen o be a smooh plane shape, which is he resul of hese wo facors: he ineres rae differenial and he level of he correlaion coefficien. Of he wo facors, he ineres rae differenial (he US raes less French Franc raes) has a relaively small impac on he price of he Quanoed opion. In he insance where he ineres rae differenial is posiive, his will slighly reduce cos of he Quanoed opion. he correlaion beween he asses is of much greaer impac. Even so, as can be seen in he above able and Figure 6, he impac of Quanoing he call opion has a mos a % impac on he cos of he normal call opion. 5
3, HEDGING CURRENCY RANSLAED OPIONS he degree of difficuly in hedging hese producs depends upon he complexiy of he srucures. For he Flexo srucure, he produc is a sandard European call or pu opion. herefore, he seller should hedge Flexos in exacly he same manner as sandard European opions. For he Compo srucures, Margrabe (978) discussed he hedging of hese producs. Given ha he opion o exchange one risky asse for anoher is a generalisaion of he Black Scholes (973) formula, a similar dynamic hedging sraegy is proposed. he difference for he Margrabe case is ha here are wo Delas ha mus be esimaed and boh risky asses mus be dynamically hedged. For he firs ype of Compo call, he N (d) erm indicaes he quaniy of he equiy ha mus be held and N (d) indicaes he quaniy of foreign exchange o be sold. hese currency-ranslaed opions are commonly referred o as firs-order correlaion dependen producs as he wo underlying asses define he correlaion relaionship. For example, if he noional amoun of he ransacion were million US Dollars and he Compo were a he money, he seller would have o mainain posiions roughly equal o 5 million dollars in boh he equiy and he foreign exchange. ompkins (997) examined via simulaion he sabiliy of such a dynamic hedging sraegy when marke fricions are included. Such marke imperfecions included hedging discreely (once per day), ransacions coss and sochasic volailiy and correlaions. ompkins (997) found ha he hedge performance of hese opions is exremely sensiive o sochasic correlaions and poins ou ha his risk is no hedgeable wih exising securiies. While he Quano may appear o be more complicaed ha he Compo, he hedging is acually easier. he simple reason is ha Quanos are second order correlaion dependen securiies. For hese producs, he correlaion risks are beween he foreign exchange and he opion premium (no he underlying equiy holding). In mos insances he value of he opion premium will be a small percenage of he noional amoun of he equiy in he ransacion. hus, given a smaller quaniy is a risk when he correlaion varies, reduces he risk. herefore, he firs sep when hedging a Quano is o hedge he sandard coningen claim. In he example of he sock opion on Michelin paid ou a a fixed exchange rae in US Dollars, one would hedge he sock opion in he normal dynamic manner. his would enail selling he opion in French Francs and borrowing in French Francs o buy he Dela amoun of he underlying shares and rebalancing hrough ime. A simpler alernaive would be o simply borrow French Francs and purchase he call opion from a marke marker on he Paris Bourse. For he sake of simpliciy, we will assume ha he curren marke price of a six-monh Michelin call raded on he Paris Bourse is 5 FF. Given he number of shares represened in each opion conrac are, his would require 5 FF o be borrowed o fund he purchase of he opion. he US Dollar invesor will pay he iniial premium in US Dollars, which will be deermined by equaion 3. Assuming he agreed exchange rae (and spo rae) is 5 FF/US$, his would require $ per share or a oal expendiure of $. In he insance of he Flexo, his amoun is immediaely convered a he 6
prevailing spo exchange rae o fund he purchase of he Michelin call in French Francs. In he insance of he Quano, his amoun is no convered o French Francs bu placed in a US Dollar Deposi accoun. he resul of his hedging sraegy is ha he seller has a US Dollar deposi and a French Franc borrowing (assuming he funding of he Michelin call purchase by borrowing in French Francs and no dynamically hedging i). hus, his will be equivalen o a forward foreign exchange conrac on he curren value of he opion premium. o mainain he hedge of he sock opion, he seller mus evaluae everyday he value of he Michelin call in French Francs and assess he curren holding in he US Dollar deposi accoun. Consider on he day afer he Quano has been sold, he Michelin call price has risen on he Paris Bourse o 6 FF. he seller has an unrealised profi of FF. his amoun is he discouned presen value of he expeced addiional erminal payoff and can be logically hough of as a FF deposi of FF. A he agreed exchange rae of 5. FF/US$, he US Dollar accoun should hold US$ (imes shares). his will no be he case as he accoun only has $ plus one day s ineres. he seller mus borrow US$ o op up he accoun. his will amoun o a reversal of he original porfolio ha was equivalen o a foreign exchange forward conrac: now he seller will be holding a FF deposi and a US Dollar borrowing. As he price of he opion changes overime, he hedge simply mainains he value of his opion conrac in he US Dollar deposi accoun. If he opion finishes in he money a expiraion, he final holding in he US Dollar accoun is simply paid o he US invesor. By definiion, his means ha he Michelin call will also be in-he-money in Paris and he proceeds of he purchased call will be applied o covering he iniial borrowing required o purchase i. In pracice, he seller does no ener he US Dollar money markes o borrow or lend o mainain he value of he premium in Dollars. his is achieved by foreign exchange conracs o coninually convered back (from) ino he US Dollar o mainain he value of he payou in dollars. Variable asse and foreign exchange exposures will arise as changes in he foreign exchange rae and he underlying equiy occurs. his is where he correlaion facor comes in. If he correlaion beween he wo asses is +%, hen he movemen of he sock will be offse by he movemen in he currency and he hedger would find ha he dollar value of he produc may be unchanged. If he correlaion is zero beween he wo asses, he requiremens for currency hedging he US dollar value of he premium is a compleely random funcion of he price of he underlying equiy. hus, his siuaion is similar o he expeced reurn from a currency forward conrac where he divergence from he iniial price is also assumed o be oally random and hus he expeced presen value is zero. ha is why here is no appreciable difference beween he sandard call and he Quanoed call when he correlaion is zero and he ineres rae differenial is also zero. he zero ineres rae differenial means ha he expeced forward price of FF/$ is equal o he spo rae. When he ineres rae differenial is nonzero, hen he impac on he value of he Quano is simply he ineres rae differenial beween he wo currencies. ha is he expeced value of he difference beween 7
quano f = = e. f quano e (7) S S = B (S, ) (8) (6) he spo and forward prices. his would suggess ha he appropriae currency insrumen o use when hedging he risk of he US Dollar value of he premium would be he forward conrac of FF for $ on he expiraion dae of he Quanoed produc. For boh elemens of he Quano, he equiy risk and he foreign exchange risk mus be coninually dynamically hedged in a similar fashion o oher opion producs. We will now formally describe wha is required for he hedge. A Quano can be replicaed by a porfolio conaining sock (FF), US $ cash and FF borrowing as follows Where f quano = S+ f = s e = e. f B e quano e quano e ( S, ) (4) e (5) his is he amoun of US $ cash held in he hedge. Essenially, his is he curren value of he opion in US Dollars. So he premium is reained in US Dollars. Subsiuing his back in he las equaion gives S = B(S, S e ) (6) Pu anoher way, when hedging any Quano, we deposi he Quano premium in he $ marke. o hedge he equiy risk of he ransacion, we apply he sandard approach for hedging a sandard FF equiy opion. We simply borrow in French Francs he appropriae amoun and use his amoun o buy he dela amoun of he French equiy. In he insance of a Quanoed pu opion, we would be required o shor he FF sock and place he proceeds on deposi a he appropriae French Franc ineres rae. As was suggesed previously, a simpler sraegy would be o borrow sufficien French Francs and purchase he raded opion. ompkins (997) examined he effeciveness of his dynamic hedging approach when perfec marke condiions were relaxed. his research found ha he expeced cos of he suggesed dynamic hedging sraegy was insignificanly differen from he heoreical value deermined by equaion 3 (for relaively frequen daily rebalancing). As wih Compos, hedge performance suffered when correlaions were no consan, however, he order of magniude of he errors was 5% less. his is because he foreign exchange risk is limied o mainaining he value of he opion premium a an agreed exchange rae and no he value of he underlying equiy. he problem remains ha he correlaion canno be hedged. Mos praciioners choose o assume ha he correlaion beween he asse (commonly equiy) and he foreign exchange rae is zero. If he correlaion is generally weak and frequenly changing sign, i is reasonable ha his will average ou a zero over he life of he opion. While, his is somewha unsaisfying from a heoreical sandpoin, able indicaes he order of magniude of pricing errors given a range of correlaion beween -% o +%. his demonsraes ha such exreme correlaions have a relaively small impac on he value of he opion. 8
Given ha if significan correlaions are observed, hey end o be posiive, a seller assumes no correlaion, hedges accordingly and if he correlaion relaionship is posiive, less hedging was required. Essenially, he movemen in he foreign exchange has auomaically rebalanced he porfolio and he seller has less hedging coss han was iniially anicipaed. However, one pracical word of cauion wih Quanoed opions wih ρ =: marke praciioners ofen value American call opions using European models. his ends o work if dividend yields are fairly low and he opion is ou-of or ahe-money. Bu if he opion is Quanoed ino a high ineres currency, European and American prices can diverge sharply. he ransformaion for a Quano is o replace q (he dividend yield) by a higher value; bu a high q is wha leads o an American opion being exercised. 4, Conclusions In his paper, we have examined correlaion dependen coningen claims ha are linked o currency movemens. hese producs are based upon a foreign asse bu he payou of he opion occurs in anoher currency. Some of he producs examined can be considered sandard opions while ohers have somewha exoic feaures. A Flexo is simply a sandard opion repackaged by spo foreign exchange ransacions. he second ype of currency ranslaed opion, he Compo or join opion, is an exension of he Margrabe formula for exchanging one risky asse for anoher. he only currency-ranslaed opion ha compleely eliminaed he currency risk is he Quano. hese producs can parially be replicaed by he purchase of sandard opions; however, he guaranee of he fixed foreign exchange rae requires separae dynamic hedging. hese producs provide invesors wih he abiliy o inves in foreign asses wihou incurring foreign exchange risks. Given ample evidence exiss ha inernaional diversificaion provides a superior risk/reward profile, hese producs have proven o be imporan ools for invesors wih a global perspecive. 9
REFERENCES Baxer, M. & Rennie, A. (996) Financial Calculus Cambridge Universiy Press, pp. -7. Black F. and M. Scholes, (973) he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, Volume 8, pp. 637-659. Jamshidian, Farshid, (994) Corralling Quanos, Risk Magazine, Volume 7 (3), pp. 7-75. Margrabe, William (978) he Value of an Opion o Exchange One Asse for Anoher. Journal of Finance, Volume 33 (978), pp.77-86. Nefci, Sahli., (996) An Inroducion o he Mahemaics of Financial Derivaives Academic Press pp. 73-9. Reiner, Eric. (99) Quano Mechanics, Risk Magazine, Volume 5 (3), pp. 59-63. ompkins, Rober (997), Dynamic versus Saic Hedging of Exoic Opions: Evaluaion of Hedge Performance via Simulaion, Ne Exposure, Volume, Number (November) 997. he rademark for all hese producs is held by Goldman Sachs Besides he mahemaical explanaion given above, we can refer o economic crieria o jusify he specificaion of he second equaion. he idea is ha flows of foreign invesmen in a domesic sock marke (le s say from he US o he French sock marke) can exer pressure on exchange raes, eiher hrough higher demand of French francs or a higher supply of dollars. Of course, our model does no ry o measure any macroeconomic effec bu ries o ake ino accoun he link beween S and E hrough heir degree of correlaion and use i for pricing purposes.