No. 14. FX Basket Options. Jürgen Hakala, Uwe Wystup. September Frankfurt School of Finance & Management Standard Chartered Bank

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1 Cnr for Praccal Quanav Fnanc No. 4 FX Bask Opons Jürgn Hakala, Uw Wysup pmbr 008 Auhor: Prof. Dr. Uw Wysup Dr. Jürgn Hakala Profssor of Fnanc Quanav Dvlopmn - Global Marks Frankfur chool of Fnanc & Managmn andard Charrd Bank Frankfur/Man London u.wysup@frankfur-school.d jurgn.hakala@uk.sandardcharrd.com Publshr: Frankfur chool of Fnanc & Managmn Phon: Fax: onnmannsr. 9- D-6034 Frankfur/M. Grmany

2 Absrac W xplan h valuaon and corrlaon hdgng of Forgn Exchang Bask Opons n a mul-dmnsonal Black-chols modl ha allows ncludng h sml. h chnqu prsnd s a fas analyc approxmaon o an accura soluon of h valuaon problm. Ky words: Forgn Exchang Opos, Bask Opons, Corrlaon Rsk, Volaly ml Modllng, Io-aylor Expanson JEL classfcaon: C63, F3, G,G3, IN: hs arcl s an xnson of [] cf. Rfrncs and has bn commssond as a conrbuon o Wly s Encyclopda of Quanav Fnanc.

3 Conn. Inroducon Bask Opons Prcng Bask Opons Exac Mhod A mpl Approxmaon A Mor Acura and Equally Fas Approxmaon Corrlaon Rsk Praccal Exampl Uppr Bound by Vanlla Opons ml Adjusmn Conclusons... 0 Rfrncs... 3

4 . Inroducon In many cass corpora and nsuonal currncy managrs ar facd wh an xposur n mor han on currncy. Gnrally hs xposurs would b hdgd usng ndvdual srags for ach currncy. hs srags ar composd of spo ransacons, forwards, and n many cass opons on a sngl currncy. Nvrhlss, hr ar nsrumns ha nclud svral currncs, and hs can b usd o buld a mul-currncy sragy ha s almos always chapr han h porfolo of h ndvdual srags. As a promnn xampl w xplan bask opons n dal.. Bask Opons Bask opons ar drvavs basd on a common bas currncy, say, and svral ohr rsky currncs. h opon s acually wrn on h bask of rsky currncs. Bask opons ar Europan opons payng h dffrnc bwn h bask valu and h srk, f posv, for a bask call, or h dffrnc bwn srk and bask valu, f posv, for a bask pu rspcvly a maury. h rsky currncs hav dffrn wghs n h bask o rflc h dals of h xposur. For xampl, a bask call on wo currncs U-$ and JP- pays off max a + a ; K a maury, whr dnos h xchang ra of /$ and dnos h xchang ra of / a m, a h corrspondng wghs and K h srk. A bask opon procs agans a drop n boh currncs a h sam m. Indvdual opons on ach currncy covr som cass, whch ar no procd by a bask opon shadd rangular aras and ha s why hy cos mor han a bask. 4

5 h llpsods connc h pons ha ar rachd wh h sam probably assumng ha h forward prcs ar a h cnr. 3. Prcng Bask Opons Bask opons should b prcd n a conssn way wh plan vanlla opons. Hnc h basc modl assumpon s a log-normal procss for h ndvdual corrlad bask componns. A dcomposon no uncorrlad componns of h xchang ra procsss d µ d + N Ω j j dw j s h bass for prcng. Hr nrs ra of h -h currncy par, ncrmns. h covaranc marx s gvn by µ dnos h dffrnc bwn h forgn and h domsc dwj h j -h componn of ndpndn Brownan Cj ΩΩ j ρjσσ j. Hr volaly of h -h currncy par and ρ j h corrlaon coffcns. σ dnos h 3.. Exac Mhod arng wh h uncorrlad componns h prcng problm s rducd o h N - dmnsonal ngraon of h payoff. hs mhod s accura bu rahr slow for mor han wo or hr bask componns. 3.. A mpl Approxmaon A smpl approxmaon mhod assums ha h bask spo slf s a log-normal procss wh drf µ and volaly σ drvn by a Wnr Procss W wh soluon [ µ d + σdw ] d W σ + µ σ gvn w know h spo a m. I s a fac ha h sum of log-normal procsss s no log-normal slf, bu as a crud approxmaon s cranly a quck mhod ha s asy o mplmn. In ordr o prc h bask call h drf and h volaly of h bask spo nd o b drmnd. hs s don by machng h frs and scond momn of h bask spo 5

6 6 wh h frs and scond momn of h log-normal modl for h bask spo. h momns of log-normal spo ar: E µ+σ µ Ε W solv hs quaons for h drf and volaly σ µ ln ln E E E In hs formula w now us h momns for h bask spo:, j N j j N N k jk k j Ω Ω + +µ µ µ α α Ε α Ε h prcng formula s h wll-known Black-chols-Mron formula for plan vanlla call opons: σ ± σ υ ± µ + K F d F d K d F r d ln 0 0 N N Hr N dnos h cumulav normal dsrbuon funcon and d r h domsc nrs ra A Mor Acura and Equally Fas Approxmaon h prvous approach can b akn on sp furhr by nroducng on mor rm n h Iô- aylor xpanson of h bask spo, whch rsuls n d d K F d F d K d F r d λ σ + λ λ λ σ λ λ + + λ σ σ λ υ λ λσ + λ µ ln 0 0 N N h nw paramr λ s drmnd by machng h hrd momn of h bask spo and h modl spo. For dals s [].

7 Mos rmarkably hs major mprovmn n h accuracy only rqurs a margnal addonal compuaon ffor. 4. Corrlaon Rsk Corrlaon coffcns bwn mark nsrumns ar usually no oband asly. Ehr hsorcal daa-analyss or mpld calbraons nd o b don. Howvr, n h forgn xchang mark h cross nsrumn s radd as wll,.g., for h xampl abov $/ spo and opons ar radd, and h corrlaon can b drmnd from hs conrac. In fac, dnong h volals as n h rahdron, w oban formula for h corrlaon coffcns n rms of known mark mpld volals: ρ ρ 34 σ σ 3 σ σ σ σ 6 + σ σ 3 σ σ 4 σ 5 hs mhod also allows hdgng corrlaon rsk by radng FX mpld volaly. For dals s []. 5. Praccal Exampl W wan o fnd ou how much on can sav usng a bask opon. W ak as a bas currncy and consdr a bask of hr currncs $, and. For h volals w ak 7

8 h rsulng corrlaon coffcns ar h amoun of opon prmum on can sav usng a bask call rahr han hr ndvdual call opons s llusrad n h followng abl. h amoun of prmum savd ssnally dpnds on h corrlaon of h currncy pars. In h nx fgur w ak h paramrs of h prvous scnaro, bu rsrc ourslvs o h currncs $ and. 8

9 6. Uppr Bound by Vanlla Opons I s acually clar ha h prc of h wo vanlla opons n h prvous xampl s an uppr bound of h bask opon prc. I sms nuvly clar ha for a corrlaon of 00% h prc s h sam. urprsngly ha s jus h cas f a spcfc rlaon bwn h srk of h ndvdual opons and hr volals s sasfd. h bask srk has o sasfy whch lads o h naural choc K K a + a 0 K 0, K 0 K a + a. h ndvdual componns consrans can b sasfd by choosng K 0 µ + σ +χσ for som arbrary, bu common χ for all bask componns. 7. ml Adjusmn For h dscrbd prcng mhod hr s no sml consdrd. Gvn h volaly sml for vanlla opons σ K, wh h sam maury as h bask opon, h mpld dnsy P for ach currncy par n h bask can b drvd from vanlla prcs V. 9

10 r PK, V K, σ K, KK A mappng ϕ w can b drvd whch maps h log-normal random numbrs o smladjusd random numbrs for ach currncy par. h mplc consrucon solvs h problm for h probably of h mappd Brownan o b h sam as h sml mpld probably. Usng Mon Carlo smulaon o prc vanlla opons usng h mappng, can b shown ha n h lm h drvd prcs ar prfcly n ln wh h sml. h formula for h Mon Carlo smulaon for a ralzaon of a Brownan w s gvn by 0, w 0 µ + σ +ϕ w σ. o prc h bask opon usng h sml n Mon Carlo a squnc of ndpndn random numbrs s usd. hs random numbrs ar corrlad usng h squar-roo marx Ω as abov and hs ar fd no h ndvdual mappngs, hnc gnrang h smulad spo a h bask maury. Evaluang h pay off and avragng wll gnra a sml adjusd prc s abl blow. Black-chols prcs and sml adjusd prcs ar shown nx o ach ohr for a drc comparson. Bas currncy EUR nomnal n EUR 39, currncs UD JPY GBP nomnals 9% 30% 4% RR 5d -0.5% -4.30%.0% Fly 5d 0.30% 0.7% 0.5% B-valus n EUR ml valus n EUR Bask sml valu 554 um of ndvduals sml 680 Bask valu 563 um of ndvduals Conclusons Many corpora clns ar xposd o mul-currncy rsk. On way o urn hs fac no an advanag s o us mul-currncy hdg nsrumns. W hav shown ha bask opons ar convnn nsrumns procng agans xchang ras of mos of h bask componns changng n h sam drcon. A rahr unlkly mark mov of half of h currncs xchang ras n oppos drcons s no procd by bask opons, bu whn akng hs rsdual rsk no accoun h hdgng cos s rducd subsanally. h sml mpac on h bask can b calculad rahr asly whou rfrrng o a spcfc modl, bcaus h produc s pah ndpndn. 0

11 Rfrncs [] Hakala, J. and Wysup, U. 00. Forgn Exchang Rsk. Rsk Publcaons, London. [] Hakala, J. and Wysup, U. 00. Makng h mos ou of Mulpl Currncy Exposur: Procon wh Bask Opons. h Euromony Forgn Exchang and rasury Managmn Handbook 00. Adran Hornbrook.

12 FRANKFUR CHOOL / HFB WORKING PAPER ERIE No. Auhor/l Yar 00. chalas, Chrsoph / chanz, Kay-Mchal Hrsg. chafflr KG/Connnal AG m Lch dr CX Corp.-Enschdung ds U Dsrc Cour for h ouhrn Dsrc of Nw York Hölschr, Lus / Haug, Mchal / chwnbrgr, Andras Analys von uramnsdan 98. Hmr, homas / Arnd, basan h Gnss of h Black-chols Opon Prcng Formula 97. Hmr, homas / Hölschr, Lus / Wrnr, Mahas Ralf Accss o Fnanc and Vnur Capal for Indusral MEs Bögr, Marc / Guhoff, Anja / Hdorn, homas Loss Gvn Dfaul Modll zur chäzung von Rcovry Ras Almr, homas / Hdorn, homas / chmalz, Chrsan h Dynamcs of hor- and Long-rm CD-sprads of Banks Barhl, Erch / Wollrshm, Jua Kulurunrschd b Mrgrs & Acqusons: Enwcklung ns Konzps zur Durchführung nr Culural Du Dlgnc 93. Hdorn, homas / Kunz, Wolfgang / chmalz, Chrsan Lqudäsmodllrung von Krdzusagn rm Facls and Rvolvr Burgr, Andras Produkvä und Effznz n Bankn rmnolog, Mhodn und aus quo Löchl, Hors / Pchr, Floran h ragc Valu of Invsmns n Chns Banks by Forgn Fnancal Insuons chalas, Chrsoph / Morgnschws, Brnd / prngr, Hans Oo / Ockns, Klaas / achulz, Ranr / afran, Robr Dr dusch NPL Mark 007: Akull Enwcklungn, Vrkauf und Bwrung Brch und Rfra ds NPL Forums chalas, Chrsoph / ralkowsk, Ingo 0 Jahr dusch Buyous Bannr, Chrsna / Hrsch, Chrsan h Economcs of Rang Wachlss: Evdnc from Rang Changs Dmdova-Mnzl, Nadshda / Hdorn, homas Gold n h Invsmn Porfolo Hölschr, Lus / Rosnhal, Johanns Lsungsmssung dr Inrnn Rvson Bannr, Chrsna / Hänsl, Dnns Drmnans of banks' ngagmn n loan scurzaon Bannr, Chrsna moohng vrsus mlnss - Wann snd sabl Rangs opmal und wlch Anfordrungn snd an opmal Brchsrgln zu slln? 83. Bannr, Chrsna Hrognous Mulpl Bank Fnancng: Dos Rduc In±cn Crd-Rngoaon Incdncs? Crmrs, Hnz / Löhr, Andras Dskrpon und Bwrung srukurrr Produk unr bsondrr Brückschgung vrschdnr Marksznarn Dmdova-Mnzl, Nadshda / Hdorn, homas Commods n Ass Managmn Crmrs, Hnz / Walznr, Jns Rskosurung m Krddrvan unr bsondrr Brückschgung von Crd Dfaul waps Crmrs, Hnz / raughbr, Parck Handlungsalrnavn nr Gnossnschafsbank m Invsmnprozss unr Brückschgung dr Rskoragfähgk Grdsmr, Dr / Roffa, Barbara Monary Analyss: A VAR Prspcv Hdorn, homas / Kasr, Dr G. / Muschol, Andra Porfoloopmrung m Hdgfonds unr Brückschgung höhrr Momn dr Vrlung Job, Clmns J. / Ockns, Klaas / afran, Robr / chalas, Chrsoph Work-Ou und rvcng von noldndn Krdn Brch und Rfra ds HfB-NPL rvcng Forums

13 75. Abrar, Kamyar Fusonskonroll n dynamschn Nzskorn am Bspl ds Brbandkablskors chalas, Chrsoph / chanz, Ka-Mchal Wrpaprprospk: Marknführungspublzä nach EU-Prospkvrordnung und Wrpaprprospkgsz Dcklr, Robr A. / chalas, Chrsoph Dsrssd Db n Grmany: Wha s Nx? Possbl Innovav Ex rags Blk, Ansgar / Poll, horsn How h ECB and h U Fd s nrs ras Hdorn, homas / Hopp, Chrsan / Kasr, Dr G. Hrognä von Hdgfondsndzs Baumann, fan / Löchl, Hors h Endogny Approach of h hory of Opmum Currncy Aras - Wha dos man for AEAN + 3? Hdorn, homas / raumann, Alxandra Ndrschlagsdrva Hdorn, homas / Hopp, Chrsan / Kasr, Dr G. Möglchkn dr rukurrung von Hdgfondsporfolos Blk, Ansgar / Poll, horsn How Do ock Mark Rurns Rac o Monary Polcy? An ARDL Congraon Analyss for Grmany Dayns, Chrsan / chalas, Chrsoph Akull Rchsfragn ds Bank- und Kapalmarksrchs II: Dsrssd Db - Invsng n Duschland Grdsmr, Dr / Poll, horsn Masurs of xcss lqudy Bckr, Grno M. / Hardng, Prham / Hölschr, Lus Fnancng h Embddd Valu of Lf Insuranc Porfolos chalas, Chrsoph Modrnsrung dr Wassrwrschaf m pannungsfld von Umwlschuz und Wbwrb Brauch Duschland n Rchsgrundlag für d Vrgab von Wassrvrsorgungskonzssonn? 6. Bayr, Marcus / Crmrs, Hnz / Kluß, Norbr Wrschrungssragn für das Ass Managmn Löchl, Hors / Poll, horsn A cas for mony n h ECB monary polcy sragy Rchard, Jörg / chalas, Chrsoph / chanz, Kay-Mchal Unrnhmn m Prm andard - ayng Publc odr Gong Prva? - Nuznanalys dr Börsnnoz Hun, Mchal / chlnk, orsn Early Warnng ysms of Fnancal Crss - Implmnaon of a currncy crss modl for Uganda Hmr, homas / Köhlr, homas Auswrkungn ds Basl II Akkords auf ösrrchsch KMU Hdorn, homas / Myr, Brnd / Prowak, Alxandr Prformancffk nach Drcors Dalngs n Duschland, Ialn und dn Ndrlandn Grdsmr, Dr / Roffa, Barbara h Rlvanc of ral-m daa n smang racon funcons for h uro ara Barhl, Erch / Grg, Rauno / Kühn, Ilmhar-Wolfram Unrschdlch Ansäz zur Mssung ds Humankapals Andrs, Dmar / Bndr, Andras / Hsdahl, Ralf / chalas, Chrsoph / hön, homas Akull Rchsfragn ds Bank- und Kapalmarkrchs I : Non-Prformng-Loans / Faul Krd - Handl, Work-Ou, Ousourcng und cursaon Poll, horsn h lowdown n Grman Bank Lndng Rvsd Hdorn, homas / ragusano, ndaro D Anwndbark dr Bhavoral Fnanc m Dvsnmark chüz, Danl / chalas, Chrsoph Hrsg. Wdr d Vrschludrung von Unrnhmn durch Pfandvrsgrung Grhold, Mrko / Hdorn, homas Invsonn und Emssonn von Convrbl Bonds Wandlanlhn Chvalr, Prr / Hdorn, homas / Krgr, Chrsan mpraurdrva zur sragschn Abschrung von Bschaffungs- und Absazrskn Bckr, Grno M. / gr, Norbr Inrnaonal Cash Flow-Rchnungn aus Egnr- und Gläubgrsch

14 47. Bonkos, Wolfram / chmd, Wolfgang M. Nos on convxy and quano adjusmns for nrs ras and rlad opons Hss, Dr Drmnans of h rlav prc mpac of unancpad Informaon n U.. macroconomc rlass 45. Crmrs, Hnz / Kluß, Norbr / Köng, Markus Incnv Fs. Erfolgsabhängg Vrgüungsmodll duschr Publkumsfonds Hdorn, homas / Köng, Lars Invsonn n Collaralzd Db Oblgaons Kahlr, Holgr / gr, Norbr Blanzrung von Unrnhmnszusammnschlüssn nach U-GAAP Bräg von udrndn ds udngangs BBA 0 unr Bglung von Prof. Dr. Norbr gr Rchnungslgung m Umbruch - HGB-Blanzrung m Wbwrb m dn nrnaonaln andards nach IA und U-GAAP 4. Ovrbck, Ludgr / chmd, Wolfgang Modlng Dfaul Dpndnc wh hrshold Modls Balhasar, Danl / Crmrs, Hnz / chmd, Mchal Porfoloopmrung m Hdg Fonds unr bsondrr Brückschgung dr Rskokomponn Hdorn, homas / Kanwll, Jns En mprsch Analys dr pradunrschd von Fssazanlhn zu Floarn m Euroraum und drn Zusammnhang zum Prs ns Crd Dfaul waps Böchr, Hnnr / gr, Norbr Blanzrung von Fnanzdrvan nach HGB, EsG, IA und U-GAAP Moormann, Jürgn rmnolog und Glossar dr Banknformak Hdorn, homas Bwrung von Krdprodukn und Crd Dfaul waps Hdorn, homas / Wr, vn Enführung n d fundamnal Aknanalys gr, Norbr Inrnaonal Accounng andards IA Moormann, Jürgn / hlng, Frank ragc Posonng of E-Commrc Busnss Modls n h Porfolo of Corpora Bankng okolovsky, Zbynk / rohhckr, Jürgn F für dn Euro, mulaonsbasr Euro-Maßnahmnplanung für Drsdnr-Bank-Gschäfsslln Roßbach, Pr Bhavoral Fnanc - En Alrnav zur vorhrrschndn Kapalmarkhor? Hdorn, homas / Jasr, Olvr / Wllnr, Ulrch Evn Rsk Covnans Bswas, Ra / Löchl, Hors Rcn rnds n U.. and Grman Bankng: Convrgnc or Dvrgnc? Ebrl, Günr Gorg / Löchl, Hors D Auswrkungn ds Übrgangs zum Kapaldckungsvrfahrn n dr Rnnvrschrung auf d Kapalmärk Hdorn, homas / Kln, Hans-Dr / brch, Frank Economc Valu Addd zur Prognos dr Prformanc uropäschr Akn Crmrs, Hnz Konvrgnz dr bnomaln Oponsprsmodll ggn das Modll von Black/chols/Mron Löchl, Hors D ökonomschn Dmnsonn dr Nw Economy Frank, Axl / Moormann, Jürgn Grnzn ds Ousourcng: En Exploraon am Bspl von Drkbankn Hdorn, homas / chmd, Pr / lr, fan Nu Möglchkn durch d Namnsak 000. Bögr, Andras / Hdorn, homas / Graf Waldsn, Phlpp Hybrds Krnkapal für Krdnsu 000. Hdorn, homas Enschdungsornr Mndsmargnkalkulaon Wolf, Brg D Egnmlkonzpon ds 0 KWG Crmrs, Hnz / Robé, oph / hl, Drk Ba als Rskomaß - En Unrsuchung am uropäschn Aknmark

15 8. Crmrs, Hnz Oponsprsbsmmung Crmrs, Hnz Valu a Rsk-Konzp für Markrskn Chvalr, Prr / Hdorn, homas / Rüz, Mrl Gründung nr duschn rombörs für Elkrzäsdrva Dsr, Danl / Ehrlchr, vn / Hdorn, homas CaBonds Jochum, Eduard Hoshn Kanr / Managmn by Polcy MbP Hdorn, homas Krddrva 999. Hdorn, homas Krdrsko CrdMrcs 999. Moormann, Jürgn rmnolog und Glossar dr Banknformak Löchl, Hors h EMU and h hory of Opmum Currncy Aras Löchl, Hors D Gldpolk m Währungsraum ds Euro Hdorn, homas / Hund, Jürgn D Umsllung auf d ückak für dusch Akngsllschafn Moormann, Jürgn and und Prspkvn dr Informaonsvrarbung n Bankn Hdorn, homas / chmd, Wolfgang LIBOR n Arrars Jahrsbrch Eckr, homas / Moormann, Jürgn D Bank als Brbrn nr lkronschn hoppng-mall Jahrsbrch Crmrs, Hnz / chwarz, Wll Inrpolaon of Dscoun Facors Moormann, Jürgn Lan Rporng und Führungsnformaonssysm b duschn Fnanzdnslsrn 995 FRANKFUR CHOOL / HFB WORKING PAPER ERIE CENRE FOR PRACICAL QUANIAIVE FINANCE No. Auhor/l Yar 3. Wbr, Andras / Wysup, Uw Vrglch von Anlagsragn b Rsrrnn ohn Brückschgung von Gbührn. En mulaonssud zur Vrlung dr Rndn. Wbr, Andras / Wysup, Uw Rsrrn m Vrglch. En mulaonssud zur Vrlung dr Rndn 008. Wysup, Uw Vanna-Volga Prcng Wysup, Uw Forgn Exchang Quano Opons Wysup, Uw Forgn Exchang ymmrs Bckr, Chrsoph / Wysup, Uw Was kos n Garan? En sasschr Vrglch dr Rnd von langfrsgn Anlagn chmd, Wolfgang Dfaul waps and Hdgng Crd Basks Kln, Fodor Acclrang h Calbraon of ochasc Volaly Modls Grbsch, usann/ Kühn, Chrsoph / Wysup, Uw Insalmn Opons: A Closd-Form oluon and h Lmng Cas

16 04. Bonkos, Wolfram / chmd, Wolfgang M. Inrs Ra Convxy and h Volaly ml Bckr, Chrsoph/ Wysup, Uw On h Cos of Dlayd Currncy Fxng Bonkos, Wolfram / chmd, Wolfgang M. Cross currncy swap valuaon Wallnr, Chrsan / Wysup, Uw Effcn Compuaon of Opon Prc nsvs for Opons of Amrcan yl 004 HFB ONDERARBEIBERICHE DER HFB - BUINE CHOOL OF FINANCE & MANAGEMEN No. Auhor/l Yar 0. Ncol Kahmr / Jürgn Moormann ud zur Ausrchung von Bankn an Kundnprozssn am Bspl ds Inrn Prs: 0, Prnd don: shppng Download: Workng Papr: hp:// CPQF: hp:// Ordr addrss / conac Frankfur chool of Fnanc & Managmn onnmannsr. 9 D 6034 Frankfur/M. Grmany Phon: Fax: Mal: m.bmr@frankfur-school.d Furhr nformaon abou Frankfur chool of Fnanc & Managmn may b oband a: hp:// 6