Proceedings of the 48th European Study Group Mathematics with Industry 1

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1 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 1 ADR Opion Trading Jasper Anderluh and Hans van der Weide TU Delf, EWI (DIAM), Mekelweg 4, 2628 CD Delf jhmanderluh@ewiudelfnl, JAMvanderWeide@ewiudelfnl 1 Inroducion A company ha is seeking o raise capial o finance necessary invesmens, can issue socks, which are basically cerificaes of parial ownership in he company There are many rules for issuing socks, one of which is ha he company has is sea in he counry where he socks are issued If, neverheless, a non-us company, like Royal Duch NV, wans o raise capial in he US, i can issue ADRs ADR is an acronym for American Deposiory Receip, which is a cerificae issued by a US bank, represening a cerain amoun of sock of a non-us company on a non-us exchange Jus as US sock, ADRs can be raded, cleared and seled on American exchanges in accordance wih US marke regulaions ADRs are US securiies and are quoed and raded in US dollars This makes i easier for Americans o inves in non-us companies, due o he widespread availabiliy of dollar-denominaed price informaion, lower ransacion coss, and imely dividend disribuions The price of an ADR follows, accouning for he currency exchange rae, more or less he price in he home counry; if he US price ges oo far off from he price in he home counry, arbirageurs will sep in he marke and he arbirage opporuniy will soon cease o exis In order o provide he American invesor wih more invesmen possibiliies, opions are issued on hese ADRs These ADR opions are also lised on US markes, denominaed in US dollars and also he srike is specified in US dollars Non-US marke makers rading opions on a sock lised in heir domesic counry migh be ineresed in adding he corresponding ADR opions o heir porfolio The ineresing par of ADR opion rading, is he inegraion of he posiion in hese US lised opions wih he domesic opion posiion The advanage of his inegraion is ha we have - from a risk-managemen poin of view - a clear perspecive of he exposure he marke maker has wih respec o a single sock If we consider for example socks Royal Duch (RD), raded in Amserdam and heir corresponding US ADRs, we can - once we are able o manage his as one inegraed posiion - compue a single dela, gamma or vega for our Royal Duch posiion Furhermore we would like o exploi he

2 2 Jasper Anderluh and Hans van der Weide mis-pricing of US opions wih respec o heir Duch counerpars and so we need a pricing model o price he foreign US dollar denominaed ADR opions consisenly wih he domesic Euro denominaed opions and sock The marke model We sar building our marke model from he domesic sock price process {S } 0, ha we model as a Geomeric Brownian Moion, (1) ds S = µd + σdw S 0 = s 0 S = s 0 e (µ 05σ2 )+σw This is he classical approach o sock price modeling as is also used by [2 For he Euro/Dollar exchange rae process {F X } 0 we also assume ha i is given by a GBM, (2) df X F X = αd + Σ 1 dw + Σ 2 dz F X 0 = f 0 F X = f 0 e (r 05(Σ 1+Σ 2 ) 2 )+Σ 1 W +Σ 2 Z Here we used anoher sandard Brownian Moion Z independen of W o model a dependence srucure beween he domesic asse S and he exchange rae F X This is he same approach as in [3 and [5 We remark ha he direcion of he exchange rae is such ha he value F X is he number of Euros you have o pay for one US dollar a ime Denoe he ADR sock price process by {A } 0 We assume ha he marke is efficien, ie arbirageurs are acive o force he following relaion o hold, (3) A = S F X 0 This relaion is invesigaed in [1 and urned ou o be quie accurae looking a real markes where prices are formed concerning ransacion and conversion coss If we consider a European call opion wih srike K wrien on he ADR and herefore lised on he foreign marke, he pay-off in US dollars Φ C of his conrac can be wrien using he previous relaion by (4) Φ C (A T ) = Φ C ( ST F X T ) = ( ) + ST K F X T We remark ha also he srike K is denominaed in US dollars Now we need o find he equivalen maringale measure Q urning all he asses in our economy ino maringales in order o price his derivaive Firs we have o idenify he asses we can use building our porfolio As in he classical approach we use boh he domesic sock S and

3 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 3 he domesic bank-accoun process { } 0 as asses in our economy, where is given by, = e r d Here r d is he classical risk-free rae As an exra asse we inroduce he foreign or US bank-accoun process {B (f) } 0 We are considering our economy from he domesic perspecive (Euro-zone), so we should denominae all our asses in he same domesic currency and herefore we consider he US bank-accoun denominaed in Euros as a risky asse So he US bank accoun is no a bank accoun in he classical sense, ie from he domesic poin of view i is no offering he risk-free rae To illusrae his we ake a closer look a rading of his asse, which is convering one Euro a ime = 0 ino (F X 0 ) 1 US dollars and deposi his amoun on a US bank accoun A ime we earned he US riskfree rae on he deposied dollar amoun and in order o calculae is value in Euros we have o conver i again by he sochasic exchange rae F X We have for B (f), B (f) = F X F X 0 e r f Here r f is he foreign risk-free rae, ha we canno obain risk-free if we denominae he value in Euros Using he dynamics of he exchange rae F X we can compue he dynamics of B (f) by, db (f) = F X e r f r f d + e r f df X = B (f) [(r f + α)d + Σ 1 dw + Σ 2 dz From (3) we recognize ha he ADR price process is compleely deermined by he domesic sock price process S and he exchange rae F X and herefore we do no wan o inroduce A as an exra asse in our economy If we decide o choose as he numéraire, which is more or less a sandard choice, we can find he opion price by idenifying he equivalen maringale measure Q such ha he discouned asse price processes S [ 1 and B (f) [ 1 are maringales If we denoe he discouned sock price process by S, we are looking for a measure Q such ha boh he process W defined by, W = W + q 1 d W = dw + q 1 d is a sandard Brownian Moion and he discouned sock price process S is a maringale The exisence of such a Q is guaraneed by he Girsanov Theorem, see eg [4 Wriing he dynamics of S in erms of W we ge, d S = S [(µ r d )d + σdw = S [ (µ r d σq 1 ) d + σd W

4 4 Jasper Anderluh and Hans van der Weide For S o be a maringale, we se he drif erm equal o zero, so q 1 = µ r d σ Now we have solved q 1 we direcly obain he dynamics of S under Q by, ds = S [µd + σdw = S [µd + σd( W q 1 d) = S [r d d + σdw This is no a surprising resul, because i is equivalen o he classical risk-neural Black-Scholes dynamics of he sock price process, see [2 Now we proceed by changing he drif of he oher Brownian Moion Z such ha boh he process Z defined by, Z = Z + q 2 d Z = dz + q 2 d is a Q sandard Brownian Moion, independen of W and he process B is a maringale Here B denoes he discouned foreign bank accoun process B (f) [ 1 For he dynamics of B we obain, d B = 1 db (f) B d = B [(r f r d + α)d + Σ 1 dw + Σ 2 dz = B [(r f r d + α)d + Σ 1 (d W q 1 d) + Σ 2 (d Z q 2 d) [( ) = B µ r d r f r d + α Σ 1 Σ 2 q 2 d + Σ 1 d σ W + Σ 2 d Z Again we need he drif erm equal o zero for B a Q maringale, which is saisfied if we pu, q 2 = r µ r f r d + α Σ d 1 σ Σ 2 Now we find for he exchange rae process F X under he pricing measure Q: df X = F X [αd + Σ 1 dw + Σ 2 dz = F X [ (α Σ 1 q 1 Σ 2 q 2 )d + Σ 1 d W + Σ 2 d Z = F X [ (r f r d ) d + Σ 1 d W + Σ 2 d Z I is clear ha he exchange rae process F X does no ener ino our porfolio, because i is no possible o acually buy or sell he exchange rae We can however keep an amoun of foreign currency on a foreign bank accoun, ha is why we decided o ake he foreign bank accoun as a possible asse for our porfolio We used he fac ha all discouned

5 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 5 asses - where we used he domesic bank accoun as numéraire process - have o be maringales under Q o derive he dynamics of he exchange rae process F X under Q We can now use his measure and he corresponding asse-dynamics o price all aainable claims, expressed in he domesic currency Suppose {X } 0 is he vecor conaining all our asses in he economy, a claim Φ is aainable if we can find a self-financing sraegy φ such ha, (5) φ 0 X 0 + T 0 φ u dx u = φ T X T = Φ(X T ) Here he self-financing propery is used in he firs equaliy sign We used in our seup he domesic bank accoun as he numéraire and consruced he corresponding maringale measure If we consider he discouned asse price process X given by, X = X we have ha he self-financing propery of φ also holds for X Now we use ha he discouned asse prices are maringales and herefore - assuming he righ condiions on φ - we have ha he sochasic inegral represening he gains of he sraegy φ over ime is also a maringale We remark ha we have o impose some condiions on φ o guaranee he sochasic inegral o be a maringale insead of being a local maringale only From a no-arbirage argumen we have ha φ 0 X 0 mus be equal o he price of he conrac a ime 0 Now we can compue his price V Φ of he opion wih pay-off Φ using he maringale propery of he discouned asse prices by, (6) V Φ = 0 φ 0 X 0 = 0 EQ [ φ T X T = B(d) 0 T, EQ [Φ(X T ) = e r dt EQ [Φ(X T ) All he asses in our economy are in Euro currency and as we are using his asses o replicae our foreign opion (4), we also have o denoe he pay-off of ha opion in Euros Φ C by, Φ C (A T ) = F X T Φ C (A T ) = (S T F X T K) + (7) Now we use he pricing formula (6) o obain C for he price of he foreign opion in Euros: C for = e r dt EQ [ F X T (A T K) + = e r dt EQ [ (S T F X T K) + (8) As his opion is raded in he US marke we need an opion value in US dollars in order o compare our heoreical price o he marke price This is nohing else hen convering he Euro value C for ino a

6 6 Jasper Anderluh and Hans van der Weide value C for in US dollars by dividing by F X 0 In his secion we are considering European call opions, whereas in pracice he opions on ADRS are American A relaed maer of pracical concern is he exdividend dae of an ADR ha is ypically a few days earlier or laer han he ex-dividend dae of he non-us sock he ADR is based on Calibraion of he Model o he marke If we wan o use he model in pracice, we should come up wih a mehod o deermine he parameers σ, Σ 1 and Σ 2 as hey appear in (1) and (2) We will relae hese parameers o he volailiy of boh he sock price process and he exchange rae process, where he volailiy is defined as he sandard deviaion of he log-reurns, scaled by he square-roo of ime up o ime unis of 1 year Furhermore we relae he quaniies we have o esimae o he correlaion beween he log-reurns of he sock price process and he log-reurns of he exchange rae process The reason for doing so, is ha we can obain he volailiies and correlaion as described above direcly from an informaion sysem (eg Bloomberg) as hese sysems are used in a rading firm Suppose we have observaions of he price processes a ime poins { 0, 1,, M }, where we denoe he fixed ime sep i+1 i by Now we find for he log-reurn s i a ime i+1 of he sock price process S, (9) s i = ln S i+1 S i = (µ 1 2 σ2 ) + σ(w i+1 W i ) d = (µ 1 2 σ2 ) + σ U, where U N (0, 1) If we denoe he volailiy esimae we obain from our informaion sysem by ˆσ S, we immediaely can use i as an esimae for our σ As we choose our domesic sock model as he sandard GBM we of course expec ha he volailiy esimae from he informaion sysem is an esimaor for our volailiy parameer The more ineresing case is he deerminaion of he parameers Σ 1 and Σ 2 For he exchange rae process F X we have a similar expression for he log-reurns f i, f i = ln F X i+1 F X i = (α 1 2 (Σ 1 + Σ 2 ) 2 ) + Σ 1 (W i+1 W i ) + Σ 2 (Z i+1 Z i ) (10) d = (α 1 2 (Σ 1 + Σ 2 ) 2 ) + (Σ 1 U + Σ 2 V ) Here we have again V N (0, 1), where U and V are independen If we denoe he esimae for he volailiy of he exchange rae process by ˆσ F X, we can wrie, ˆ (11) ˆσ F X = Σ Σ 2 2

7 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 7 We need more informaion o deermine Σ 1 and Σ 2 separaely For he covariance beween he log reurns of he sock price process S and he exchange rae process F X a ime i+1 we have, COV(s i, f i ) = σ COV(U, Σ 1 U + Σ 2 V ) = σ Σ 1 For he correlaion ρ we have ρ = COV (s i, f i ) σ Σ 1 (12) = Σ Σ 2 2 Σ Σ 2 2 So using hese relaions we can, provided wih esimaes ˆ ˆ σ F X and σ S, ˆρ from he informaion sysem, come up wih esimaes for he parameers in our model, (13) σ = σ S, Σ 1 = ˆρ σ F X, Σ 2 = σ F X (1 ˆρ) 2 Conclusion In his paper we rea he opic of pricing opions on ADRs lised on US markes The imporance is in he fac ha he ADR price process is direcly relaed o he price process of he corresponding sock in he non-us marke This relaion exends o he possibiliy o replicae he foreign lised opion wih insrumens in he domesic marke We se up an economy consising of he domesic sock and bank accoun and moreover we inroduced he foreign bank accoun as an addiional risky asse Afer modeling hese insrumens, we derived an equaion for he price of a foreign call opion Finally we showed how we can relae sandard esimaes obained from a rading informaion sysem o he parameers of our model References [1 DP Miller, MR Morey The Inraday Pricing Behavior of Inernaional Dually Lised Securiies, Journal of Inernaional Financial Markes, Insiuions & Money, 6(4), 79-89, (1996) [2 F Black, M Scholes The pricing of opions and corporae liabiliies, J Poliical Econ 81, , (1973) [3 T Björk, Arbirage Theory in Coninuous Time, 1 Ediion, Oxford Universiy Press, Oxford, (1998) [4 I Karazas, S E Shreve, Brownian Moion and Sochasic Calculus, 2 Ediion, Springer-Verlag, Berlin, (1991) [5 M Musiela, M Rukowski, Maringale Mehods in Financial Modelling, 2 Prining, Springer-Verlag, Berlin, (1998)

8 8 Proceedings of he 48h European Sudy Group Mahemaics wih Indusry

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