Fx Alternativ And Strukturerade Biprodukter Pdf

FX Alternativ och Strukturerade Produkter 1 FX Alternativ och Strukturerade Produkter Uwe Wystup 7 april 2006 3 Innehåll 0 Förord ​​Omfattning av denna bok Läsarna Om författaren Erkännande Valutares alternativ A Resa genom alternativhistorik Tekniska problem för vanilj Alternativ Värde En anteckning på Framåtriktade grekiska identiteter homogenitetsbaserade relationer citat strejk i villkoren för delta volatilitet i villkoren för delta volatilitet och delta för en given strejker grekiska villkoren för delta volatilitet historisk volatilitet historisk korrelation flyktighet leende på-pengar volatilitet interpolation volatilitet leende konventioner at-the - Penningdefinition Interpolering av volatiliteten på mognadspilar Interpolation av volatiliteten Sprid mellan mognadspilar Volatilitet Källor Volatilitet Cones Stokastisk volatilitet 4 4 Wystup-övningar Grundläggande strategier innehållande vaniljalternativ Alternativ Call and Put Spread Risk Återgång Risk Reversal Flip Stradle Strangle Butterfly Seagull Övningar Första G Eneration Exotics Barrier Alternativ Digitala alternativ, Touch Options och Rabatter Compound and Installment Asiatiska Alternativ Alternativ Alternativ Framåt Start, Ratchet och Cliquet Alternativ Strömalternativ Quanto Alternativ Övningar Exotiska Korridorer Föder Exotiska Barriäralternativ Betalning Later Alternativ Steg upp och Steg ner Alternativ Sprid och Exchange Options Korgar Bästa och Sämsta Alternativ Alternativ och Framåt på Harmonic Average Variance and Volatility Swaps Övningar Strukturerade Produkter Framåt Produkter Direkt Framåt Deltagande Framåt Fade-In Framåt Utlåsning Framåt Shark Framåt Fader Shark Framåt 5 FX Alternativ och Strukturerade Produkter Butterfly Forward Range Framåt Räckvidd Fördröjning Framåt ackumulativ Framåt Boomerang Framåt Amortering Framåt Automatisk Förnyelse Framåt Dubbelshark Framåt Framåt Start Välja Framåt Fritt Stil Framåt Ökad SpotForward Tid Alternativ Övningar Serie Strategier Shark Framåt Serie Krage Extra Serie Övningar Inlåning och Lån Dual ValutaväxlingLöna Prestationslänkade Inlåning Tunnel InsättningLönade korridor InlåningLöna Turbo InlåningLöne Tower DeponeringLöna Övningar Räntesats och Kors Valutaväxling Kors Valutaväxling Hanseatisk Växling Turbo Kors Valutaväxling Buffert Kors Valutaväxling Växla Växla Korridor Växla Dubbellänk utan Växelkurva Återställ Växla Växla Korg Utbyte Växla Övningar Deltagande Noter Guld Deltagande Obs! Kurvlänk Obs! Utbyter Byt Flyttstreck Turbo Spot Obegränsad 6 6 Wystup 2.6 Hybrid FX Produkter Praktiska Materiel Handelarna Regelbundenhetskostnaden för Vanna och Volga Observationer Konsekvenskontroll Förkortningar för första generationens exotiska justeringsfaktorvolatilitet för riskåtergångar , Fjärilar och Teoretisk Värde Prissättning Barriär Alternativ Prissättning Dubbelbarriär Alternativ Prissättning Dubbelklicka Alternativ Prissättning Europeisk Stilalternativ Nollställbarhet Kostnaden för handel och dess påverkan på marknadspriset för Onetouch-alternativ Exempel Ytterligare tillämpningar Övningar Bjud Fråga Spr eads One Touch Spreads Vanilla Spreads Spreads för första generations Exotics Minimal Bud Fråga Spread Bid Ask Priser Övningar Uppgörelse Black-Scholes Modell för faktiska Spot Cash Settlement Leveransavräkning Alternativ med uppskjutna leveransövningar på bekostnad av försenade Fixing Announcements Valutan Fixing av Europeiska centralbankens modell - och utbetalningsanalysprocess Felbedömning Analys av EUR-USD Slutsats 7 FX Options och Strukturerade Produkter 7 4 Säkringsredovisning enligt IAS Inledning Finansiella instrument Överblick Allmän definition Finansiella tillgångar Finansiella skulder Avräkning av finansiella tillgångar och finansiella skulder Equity Instruments Compound Financial Instruments Derivat Embedded Derivatives Klassificering av Finansiella Instrument Utvärdering av Finansiella Instrument Initial Anmälning Initial Mätning Efterföljande Mätning Avkräkning Hedge Accounting Översikt Typer av Hedges Grundläggande Krav Stoppande Hedge Redovisning Metoder för Testi Ng Hedge Effectiveness Fair Value Hedge Cash Flow Hedge Testing for Effectiveness - En fallstudie av forward-plus-simulering av växelkursberäkning Beräkning av forward-plusvärdesberäkning av framåtriktad ränta Beräkning av prognostransaktion s Valutaväxlingsförskjutningsförhållande - prospektivt test för Effektivitetsvariabilitetsmått - Prospektivt test för effektivitetsregressionsanalys - Prospektivt test för effektivitetsresultat Retrospektivt test för effektivitet Konklusion Relevanta originalkällor för redovisningsstandarder Övningar 8 8 Wystup 5 valutamarknader En turnering genom marknadsförklaringen av GFI Group (Fenics), 25 Oktober Intervju med ICY Software, 14 oktober Intervju med Bloomberg, 12 oktober Intervju med Murex, 8 november Intervju med SuperDerivatives, 17 oktober Intervju med Lucht Probst Associates, 27 februari Programvara och systemkrav Fenics Position Håller prissättning direkt genom behandling av ansvarsfrister Handels - och försäljningsstopp rietary Trading Försäljningsdriven handel Interbankförsäljning Branchförsäljning Institutionell försäljning Företagsförsäljning Privatbankmarknad FX Options Trading Floor Joke 9 Kapitel 0 Förord ​​0.1 Räckvidd av denna bok Treasury management av internationella företag innebär att hantera kassaflöden i olika valutor. Därför består en investeringsbankens naturtjänst av en mängd olika penningmarknads - och valutaprodukter. Denna bok förklarar de mest populära produkterna och strategierna med fokus på allt bortom vaniljalternativ. Det förklarar alla FX-alternativen, gemensamma strukturer och skräddarsydda lösningar i exempel med ett särskilt fokus på applikationen med synpunkter från näringsidkare och försäljning såväl som från företagsklientperspektiv. Den innehåller faktiskt handlade avtal med motsvarande motivationer som förklarar varför strukturerna har handlats. På så vis får läsaren en känsla av hur man bygger nya strukturer för att passa kundernas behov. Övningarna är avsedda att träna materialet. Flera av dem är faktiskt svåra att lösa och kan tjäna som incitament för vidare forskning och testning. Lösningar till övningarna ingår inte i boken, men de kommer att publiceras på bokens hemsida, 0.2. Läsarnas förutsättning är grundläggande kunskap om valutamarknader som t ex från boken Foreign Exchange Primer av Shami Shamah, Wiley 2003, se 90. Målläsarna är forskarstuderande och fakulteten för finansiella ingenjörsprogram, som kan använda denna bok som en lärobok för en kurs med namnet strukturerade produkter eller exotiska valutaalternativ. 9 10 10 Wystup Traders, Trainee Structurers, Produktutvecklare, Försäljning och Quants med intresse för FX-produktlinjen. För dem kan det tjäna som en idékälla och en referensguide. Treasurers av företag intresserade av att hantera sina böcker. Med den här boken kan de själva strukturera sina lösningar. Läsarna som är mer intresserade av kvantitativa och modellerande aspekter rekommenderas att läsa utländsk valutarisk av J. Hakala och U. Wystup, Risk Publications, London, 2002, se 50. Denna bok förklarar flera exotiska valutakursalternativ med ett särskilt fokus på de underliggande Modeller och matematik, men innehåller inga strukturer eller företagskunder eller investerare. 0.3 Om författaren Figur 1: Uwe Wystup, professor i kvantitativt finans vid HfB Business School of Finance och Management i Frankfurt, Tyskland. Uwe Wystup är även VD för MathFinance AG, ett globalt nätverk av quants som specialiserat sig på kvantitativ finansiell, exotisk optionsrådgivning och Front Office Software Production. Tidigare var han finansiell ingenjör och strukturör i FX Options Trading Team på Commerzbank. Innan dess arbetade han för Deutsche Bank, Citibank, UBS och Sal. Oppenheim jr. Amp Cie. Han är grundare och chef för webbplatsen MathFinance. de och MathFinance Newsletter. Uwe har doktorsexamen i matematisk finansiering från Carnegie Mellon University. Han föreläser också matematisk finansiering för Goethe University Frankfurt, organiserar Frankfurt MathFinance Colloquium och är grundande chef för Frankfurt MathFinance Institute. Han har givit flera seminarier om exotiska alternativ, beräkningsfinansiering och volatilitetsmodellering. Hans specialiseringsområde är de kvantitativa aspekterna och utformningen av strukturerade produkter av utländska 11 FX Options and Structured Products 11 valutamarknader. Han publicerade en bok om valutarisk och artiklar i finans och stokastik och tidskriften av derivat. Uwe har gett många presentationer på både universitet och banker runt om i världen. Ytterligare information om hans curriculum vitae och en detaljerad publikationslista finns tillgänglig på 0.4. Tack till mina tidigare kollegor på handelsgolvet, framför allt Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Romerska Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala och alla mina kollegor och medförfattare, speciellt Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt och Robert Tompkins. Chris Swain, Rachael Wilkie och många andra av Wiley-publikationer förtjänar respekt eftersom de hade att göra med min ganska långsamma hastighet när de fyllde i boken. Nicole van de Locht och Choon Peng Toh förtjänar en medalj för seriös detaljerad korrekturläsning. 13 Kapitel 1 Valutakursalternativ FX Structured Products är skräddarsydda linjära kombinationer av FX-alternativ inklusive både vanilj och exotiska alternativ. Vi rekommenderar boken av Shamah 90 som en källa för att lära känna valutamarknader med fokus på marknadskonventioner, spot, forward och swap-kontrakt, vaniljalternativ. För prissättning och modellering av exotiska FX-alternativ föreslår vi Hakala och Wystup 50 eller Lipton 71 som användbara kamrater till den här boken. Marknaden för strukturerade produkter är begränsad till marknaden för nödvändiga ingredienser. Därför är det vanligtvis de mest strukturerade produkterna som handlas valutapar som kan bildas mellan USD, JPY, EUR, CHF, GBP, CAD och AUD. I det här kapitlet börjar vi med en kort historia av alternativ, följt av en teknisk sektion om vaniljalternativ och volatilitet, och hanterar vanliga linjära kombinationer av vaniljalternativ. Då ska vi illustrera de viktigaste ingredienserna för FX-strukturerade produkter: den första och andra generationen exotics. 1.1 En resa genom alternativets historia De allra första alternativen och terminerna handlades i det antika Grekland, när oliverna såldes innan de hade uppnått modenhet. Därefter utvecklades marknaden på följande sätt. 1600-talet Ända sedan 1500-talets tulpaner, som liknade deras exotiska utseende, odlades i Turkiet. Chefen för de kungliga medicinska trädgårdarna i Wien, Österrike, var den första som odlade de turkiska tulpaner framgångsrikt i Europa. När han flydde till Holland på grund av religiös förföljelse tog han glödlamporna med sig. Som ny chef för den botaniska trädgården i Leiden, Nederländerna, odlade han flera nya stammar. Det var från dessa trädgårdar som grymma handlare stal glödlamporna för att kommersialisera dem, för tulpaner var en stor statussymbol. 17th century De första futuresna på tulpaner handlades i Från och med 1634 kunde folk 13 14 14 Wystup köpa speciella tulpanstammar av vikten av sina glödlampor, för glödlamporna samma värde valdes som för guld. Tillsammans med den vanliga handeln gick spekulanterna in på marknaden och priserna höjdes. En pigg av stammen Semper Octavian var värd två vagnar av vete, fyra massor av råg, fyra feta oxor, åtta feta svin, tolv feta får, två vinklar vin, fyra fat öl, två fat smör, 1 000 pund ost , en äktenskapsäng med linne och en stor vagn. Folk lämnade sina familjer, sålde alla sina tillhörigheter, och till och med lånade pengar för att bli tulpanhandlare. När den här riskfria marknaden kraschade i 1637 gick handelsmän och privatpersoner i konkurs. Regeringen förbjöd spekulativ handel perioden blev känd som Tulipmania. 1700-talet År 1728 utfärdade den kungliga västindiska och guinea-bolaget monopolisten i handel med de karibiska öarna och den afrikanska kusten de första aktieoptionerna. Det var alternativ på inköp av den franska ön Ste. Croix, på vilken sockerplanteringar planerades. Projektet realiserades 1733 och papperslagren utfärdades. Vid sidan av beståndet köpte personer en relativ del av ön och värdesakerna samt företrädesrätten och bolagets rättigheter. 1800-talet År 1848 grundade 82 affärsmän Chicago Trade Board (CBOT). Idag är det den största och äldsta terminsmarknaden i hela världen. De flesta skriftliga dokumenten förlorades i den stora elden 1871, men det är allmänt trodde att de första standardiserade terminerna handlades, eftersom CBOT nu handlar flera terminer och framåt, inte bara T-obligationer och statsobligationer, utan även alternativ och guld. År 1870 grundades New York Cotton Exchange. År 1880 introducerades guldstandarden. 1900-talet År 1914 övergavs guldnivån på grund av kriget. 1919 byttes Chicago Produce Exchange, ansvarig för handel med jordbruksprodukter, till Chicago Mercantile Exchange. Idag är det den viktigaste terminsmarknaden för Eurodollar, utländsk valuta och boskap. År 1944 implementerades Bretton Woods System i ett försök att stabilisera valutasystemet. År 1970 övergavs Bretton Woods System av flera skäl. 1971 infördes Smithsonian-avtalet om fasta växelkurser. 1972 handlade den internationella monetära marknaden (IMM) futures på mynt, valutor och ädelmetall. 15 FX Alternativ och Strukturerade Produkter 15 21: a århundradet År 1973 lade CBOE (Chicago Board of Exchange) först handlade köpoptioner fyra år senare också alternativ. Smithsonianavtalet övergavs valutorna följde hanterade flytande. År 1975 sålde CBOT den första räntan framtid, den första framtiden utan någon verklig underliggande tillgång. År 1978 handlade den holländska aktiemarknaden de första standardiserade finansiella derivaten. År 1979 genomfördes det europeiska valutasystemet, och den europeiska valutaenheten (ecu) infördes. År 1991 undertecknades Maastrichtfördraget om en gemensam valuta och ekonomisk politik i Europa. År 1999 infördes euron, men länderna använde fortfarande pengar i sina gamla valutor medan valutakurserna hölls fasta. År 2002 infördes euron som nya pengar i form av kontanter. 1.2 Tekniska problem för vaniljalternativ Vi anser modellens geometriska Brownian motion ds t (rdrf) S t dt sigmas t dwt (1.1) för den underliggande växelkursen som anges i FOR-DOM (utländsk inhemsk), vilket innebär att en enhet av Den utländska valutan kostar FOR-DOM-enheter i den inhemska valutan. I händelse av EUR-USD med en plats på. Det innebär att priset på en euro är USD. Begreppet utländska och inhemska hänvisar inte till affärsenhetens placering, utan endast till denna noteringskonvention. Vi anger den (kontinuerliga) utländska räntan med r f och den (kontinuerliga) inhemska räntan med r d. I ett eget kapital scenario skulle r f representera en kontinuerlig utdelningshastighet. Volatiliteten betecknas av sigma, och Wt är en vanlig brunisk rörelse. Provbanorna visas i Figur 1.1. Vi betraktar denna standardmodell, inte för att den speglar växelkursens statistiska egenskaper (det gör det faktiskt inte), men eftersom det används allmänt i praktiken och frontkontorsystem och huvudsakligen fungerar som ett verktyg för att kommunicera priser i FX-alternativ . Dessa priser är vanligtvis citerade i fråga om volatilitet i den här modellen. Användning av Itocircs regel till ln S t ger följande lösning för processen S t S t S 0 exp sigma2) t sigmaw t, (1.2) vilket visar att S t är normalt distribuerad, mer exakt är ln S t normal Med medelvärdet ln S0 (rdrf12 sigma2) t och varians sigma 2 t. Ytterligare modellantaganden är 16 16 Wystup Figur 1.1: Simulerade vägar av en geometrisk brunisk rörelse. Fördelningen av platsen S T vid tiden T är logg normal. 1. Det finns ingen arbitrage 2. Handel är friktionslös, inga transaktionskostnader 3. Varje position kan tas när som helst, kort, lång, godtycklig fraktion, inga likviditetsbegränsningar. Utbetalningen för ett vaniljalternativ (europeisk uppsättning eller samtal) ges Av F phi (s TK), (1.3) där de kontraktsmässiga parametrarna är strejken K, utgångstiden T och typen phi, en binär variabel som tar värdet 1 vid ett samtal och 1 i fallet med en sätta. Symbolen x betecknar den positiva delen av x, dvs x max (0, x) 0 x Värde I Black-Scholes-modellen är värdet av utbetalningen F vid tidpunkten t om platsen är vid x betecknad med v (t, x ) Och kan beräknas antingen som lösningen av Black-Scholes partiell differential 17 FX Options och Structured Products 17 ekvation vtrdv (rdrf) xv x sigma2 x 2 v xx 0, (1.4) v (t, x) F. ) Eller likvärdigt (Feynman-Kac-teoremet) som det diskonterade förväntade värdet av utbetalningsfunktionen, v (x, K, T, t, sigma, rd, rf, phi) er dtau IEF. (1.6) Detta är anledningen till att grundläggande finansiell teknik huvudsakligen berörs av att lösa partiella differentialekvationer eller beräkningsförväntningar (numerisk integration). Resultatet är Black-Scholes formel Vi förkortar v (x, k, t, t, sigma, rd, rf, phi) phie r dtau fn (phid) KN (phid). (1,7) x: nuvarande pris för den underliggande tau T t: tid till mognad F IES T S t x xe (r d r f) tau. Framåtskridande pris för den underliggande theta plusmn rdrf sigma plusmn sigma 2 d plusmn ln x K sigmatheta plusmntau sigma tau In K plusmn sigma 2 2 tau sigma tau n (t) 1 2pi e 1 2 t2 n (t) N (x) xn (T) dt 1 N (x) Black-Scholes-formeln kan härledas med hjälp av den integrerade representationen av ekvation (1.6) ver dtau IEF e rttau IEphi (STK) (er dtau phi xe (rdrf 1 2 sigma2) tausigma tauy K) n (y) dy. (1.8) Nästa måste hantera den positiva delen och sedan slutföra torget för att få Black-Scholes formel. En avledning baserad på partiell differentialekvation kan göras med användning av resultat om den väl studerade värmeekvationen. 18 18 Wystup En notering på framåtriktningen Framåtkursen f är strejken som gör att nollvärdet av framåtkontraktet F S T f (1.9) är lika med noll. Det följer att f IES T xe (r d r f) T, dvs framåtkursen är det förväntade priset för den underliggande tiden vid tidpunkt T i en riskneutral inställning (driften av den geometriska bruniska rörelsen motsvarar kostnaden för bäring r d r f). Situationen r d gt r f kallas contango, och situationen r d lt r f kallas backwardation. Observera att i klassen Black-Scholes är klassen av framåtriktade kurvor ganska begränsad. Till exempel kan inga säsongseffekter inkluderas. Observera att värdet av terminskontraktet efter tidsnoll normalt skiljer sig från noll, och eftersom en av motparterna alltid är kort kan det finnas risk för att den korta parten är försummad. Ett terminsavtal hindrar den här farliga affären: det är i princip ett terminskontrakt, men motparterna har ett marginalkonto för att säkerställa att kontanta medel eller råvaror inte överstiger en angiven gräns. Greker Grekland är derivat av värdefunktionen med avseende på modell och kontraktparametrar. De är en viktig information för handlare och har blivit standardinformation som tillhandahålls av frontkontorsystem. Mer detaljer om grekerna och relationerna mellan grekerna presenteras i Hakala och Wystup 50 eller Reiss och Wystup 84. För vaniljalternativen listar vi några av dem nu. (Spot) Delta. V x phie r f tau N (phid) (1,10) Framåt Delta. Driftless Delta. Vf phie r dtau N (phid) (1.11) phin (phid) (1,12) Gamma. 2 v e rf tau n (d) x 2 xsigma tau (1,13) 19 FX Alternativ och strukturerade produkter 19 Hastighet. 3 v x 3 e r f tau n (d) x 2 sigma tau () d sigma tau 1 (1,14) Theta. V t e r f tau n (d) xsigma 2 tau phir f xe r f tau N (phid) r d Ke rta N (phid) (1.15) Charm. 2 v x tau phir f e rf tau N (phid) phie rf tau n (d) 2 (r d r f) tau d sigma tau 2tausigma tau (1.16) Färg. 3 v x 2 tau e r f tau n (d) 2xtausigma tau 2r f tau (r d rf) tau d sigma tau 2tausigma d tau (1.17) Vega. V sigma xe r f tau taun (d) (1.18) Volga. 2 v sigma 2 xe r f tau taun (d) d d sigma (1.19) Volga kallas också ibland vomma eller följa. Vanna. 2 v sigma x e r f tau n (d) d sigma (1,20) Rho. V r d phiktaue rttau N (phid) (1.21) v r fixtaue r f tau N (phid) (1.22) 20 20 Wystup Dual Delta. Dual Gamma. v K phie r dtau N (phid) (1.23) 2 v e r dtau n (d) K 2 Ksigma tau (1.24) Dual Theta. v T vt (1.25) Identiteter Samtalsparametern är förhållandet d plusmn d (1.26) sigma sigma d plusmn tau (1.27) rd sigma d plusmn tau (1.28) rf sigma xe rf tau n (d) Välja (d). (1.29) N (phid) IP phis T phik (1,30) N (phid) IP phis T phi f 2 (1.31) Kv (x, K, T, t, sigma, rd, rf, 1) K, T, t, sigma, rd, rf, 1) xe rf tau Ke r dtau, (1.32) vilket är bara ett mer komplicerat sätt att skriva den triviala ekvationen xx x. Delta-paritetsparametern är v (x, K, T, t, sigma, rd, rf, 1) xv (x, K, T, t, sigma, rd, rf, 1) x e r f tau. (1.33) I synnerhet lär vi oss att det absoluta värdet av ett putt delta och ett samtal delta inte exakt lägger till en, men endast till ett positivt tal e r f tau. De lägger till upp till en ungefär om antingen tiden till utgången tau är kort eller om den utländska räntan r f ligger nära noll. 21 FX Alternativ och Strukturerade Produkter 21 Valet K f ger identiska värden för samtal och sätta, vi söker den deltasymmetriska strejken som producerar helt identiska delar (spot, framåt eller driftfri). Detta villkor förutsätter d 0 och därmed fe sigma2 2 T, (1.34) i vilket fall det absoluta deltaet är erf tau 2. I synnerhet lär vi oss, det är alltid gt f, det kan inte vara en uppsättning och ett samtal med identiska värden och delta. Observera att strejken brukar väljas som mitten strejk när man handlar en strängle eller en fjäril. På samma sätt kan den dubbelt delta-symmetriska strejken circK fe sigma2 2 T härledas från förhållandet d Homogenitetsbaserade Relationer Vi kan önska att mäta värdet av det underliggande i en annan enhet. Detta kommer uppenbarligen att påverka alternativet prissättning formel som följer. av (x, K, T, t, sigma, rd, rf, phi) v (ax, ak, T, t, sigma, rd, rf, phi) för alla en gt 0. (1.35) Differentiering av båda sidor med respekt till a och sedan sätta en 1-utbyte v xv x Kv K. (1.36) Att jämföra koefficienterna för x och K i ekvationerna (1.7) och (1.36) leder till suggestiva resultat för delta vx och dual delta v K. Denna rymd - Homogenitet är orsaken till deltaformlernas enkelhet, vars tråkiga beräkning kan sparas på detta sätt. Vi kan utföra en liknande beräkning för de tidspåverkade parametrarna och få den uppenbara ekvationen v (x, k, t, t, sigma, rd, rf, phi) v (x, k, t a, ta, asigma, ar , ar f, phi) för alla en gt 0. (1.37) Differentiering av båda sidor med avseende på a och sedan inställning av 1 ger 0 tauv t sigmav sigma rdv rd rfv rf. (1.38) Det kan förstås också verifieras med direkt beräkning. Den övergripande användningen av sådana ekvationer är att generera dubbelkontrollmarkeringar vid beräkningen av grekerna. Dessa homogenitetsmetoder kan enkelt utvidgas till andra mer komplexa alternativ. Med samtalssymmetri förstår vi förhållandet (se 6, 7, 16 och 19) v (x, K, T, t, sigma, rd, rf, 1) Kfv (x, f2K, T, t, Sigma, rd, rf, 1). (1.39) 22 22 Wystup Strejken på puten och strejken i samtalet resulterar i ett geometriskt medelvärde som är lika med framåtriktningen f. Framåt kan tolkas som en geometrisk spegel som återspeglar ett samtal till ett visst antal satser. Observera att för köpoptionsalternativen (Kf) sammanfaller sysselsättningssymmetri med det speciella fallet av samtalspariteten där samtalet och satsen har samma värde. Direkt beräkning visar att siffra sjuktryck v v tauv (1.40) r d r f håller för vanilj alternativ. Detta förhållande gäller faktiskt alla europeiska alternativ och en stor klass av vägberoende alternativ som visas i 84. Man kan direkt verifiera förhållandet den utländska inhemska symmetrin 1 xv (x, K, T, t, sigma, rd , Rf, phi) Kv (1 x, 1 K, T, t, sigma, rf, rd, phi). (1.41) Denna jämlikhet kan betraktas som en av ansiktssymmetriska ansikten. Anledningen är att värdet av ett alternativ kan beräknas både i en inhemsk såväl som i ett främmande scenario. Vi betraktar exemplet på S t-modellering växelkursen på EURUSD. I New York kostar anropsalternativet (STK) v (x, K, T, t, sigma, r usd, r eur, 1) USD och därmed v (x, K, T, t, sigma, r usd, r eur, 1) x () 1. Euro. Alternativet EUR-call kan också ses som ett USD-put-alternativ med avbetalning K 1 KST Detta alternativ kostar Kv (1, 1, T, t, sigma, rx K eur, r usd, 1) EUR i Frankfurt, eftersom S T och 1 S t har samma volatilitet. Naturligtvis måste New York-värdet och Frankfurt-värdet komma överens, vilket leder till (1.41). Vi kommer också att lära senare att denna symmetri bara är ett möjligt resultat baserat på förändring av numeraire Citat Citat av den underliggande valutakurslikvationen (1.1) är en modell för växelkursen. Citatet är ett permanent förvirrande problem, så låt oss förtydliga detta här. Växelkursen betyder hur mycket av inhemsk valuta som behövs för att köpa en enhet i utländsk valuta. Om vi ​​till exempel tar EURUSD som växelkurs är standardnoteringen EUR-USD, där USD är inhemsk valuta och EUR är utländsk valuta. Termen inhemsk är inte på något sätt relaterad till näringsidkare eller något land. Det betyder bara numerairevaluta. Begreppen inhemska, numeraire eller basvaluta är synonymer som är utländska och underliggande. Under hela boken betecknar vi snedstrecket () valutaparet och med en streck (-) citatet. Slash () betyder inte en uppdelning. Exempelvis kan EURUSD citeras antingen i EUR-USD, vilket innebär att hur många dollar som behövs för att köpa en euro eller i USD-EUR, vilket innebär att hur många euro som behövs för att köpa en USD. Det finns vissa marknadsnoteringar som anges i tabell 1.1. 23 FX Alternativ och Strukturerade Produkter 23 Valutapar Standard Citat Exempel Citat GBPUSD GPB-USD GBPCHF GBP-CHF EURUSD EUR-USD EURGBP EUR-GBP EURJPY EUR-JPY EURCHF EUR-CHF USDJPY USD-JPY USDCHF USD-CHF Tabell 1.1: Standardmarknad Citat av stora valutapar med prissumma priser Trading Floor Language Vi kallar en miljon en buck, en miljard på varvet. Detta beror på att en miljard kallas milliarde på franska, tyska och andra språk. För den brittiska punden kallas en miljon ofta också en quid. Vissa valutapar har namn. Till exempel kallas GBPUSD kabel eftersom växelkursinformationen sänds via en kabel i Atlanten mellan Amerika och England. EURJPY kallas korset, eftersom det är korsfrekvensen för den mer likvida handeln USDJPY och EURUSD. Vissa valutor har också namn, t. ex. Nya Zeeland Dollar NZD kallas en kiwi, den australiensiska dollarn AUD heter Aussie, de skandinaviska valutorna DKR, NOK och SEK heter Scandies. Valutakurserna citeras i allmänhet upp till fem relevanta siffror, t. ex. I EUR-USD kunde vi observera ett citat av den sista siffran 5 kallas pipen, mellanniffen 3 kallas den stora siffran, eftersom växelkurserna ofta visas i handelsgolv och den stora siffran som visas i större storlek, är den mest relevanta informationen. Siffrorna kvar till den stora figuren är i alla fall kända, pipsna till höger om den stora figuren är ofta försumbar. För att klargöra kommer en ökning av USD-JPY med 20 pips att vara och en ökning med 2 stora siffror kommer att vara Notering av Alternativpriser Värden och priserna på Vanilla-alternativ kan citeras på de sex sätt som förklaras i Tabell 1.2. 24 24 Wystup namn symbolvärde i exempel exemplar inhemska kontanter d DOM 29 1448 USD utländska kontanter f FÖR 24 249 EUR inhemska d DOM per enhet DOM USD utländska f FÖR per enhet FOR EUR inhemska pips d pips DOM per enhet för USD pips per EUR utländska pips f pips FOR per enhet DOM EUR pips per USD Tabell 1.2: Standard marknadsnoteringstyper för optionsvärden. I exemplet tar vi FOREUR, DOMUSD, S 0. r d 3,0, rf 2,5, sigma 10, K. T 1 år, phi 1 (call), fiktiv 1, 000, 000 EUR 1, 250 000 USD. För piporna anges citatet USD pips per EUR också ibland som USD per 1 EUR. På samma sätt kan EUR-pips per USD även citeras som euro per 1 USD. Black-Scholes formel citerar d pips. De andra kan beräknas med hjälp av följande instruktioner. D pips 1 S 0 S 0 1 f K S d 0 S f pips 0 K d pips (1.42) Delta - och Premium-konventionen Spotpunktet för ett europeiskt alternativ utan premium är välkänt. Det kommer att kallas raw spot delta delta rå nu. Det kan citeras i någon av de två berörda valutorna. Relationen är delta omvänd raw delta raw S K. (1.43) Delta används för att köpa eller sälja plats i motsvarande belopp för att säkra alternativet upp till första order. För konsistens måste premien införlivas i deltahäcken, eftersom en premie i utländsk valuta redan kommer att säkra en del av optionens delta risk. För att göra det klart, låt oss överväga EUR-USD. I standard arbitrage-teorin anger v (x) värdet eller premien i USD av ett alternativ med 1 EUR notional, om platsen är x, och rå delta v x anger antalet EUR att köpa för deltahäcken. Därför är xv x det antal USD som ska säljas. Om premien betalas i euro snarare än i USD har vi redan VX EUR och antalet EUR att köpa måste minskas med detta belopp, dvs om EUR är premiumvaluta, måste vi köpa Vxvx EUR för Delta hedge eller motsvarande sälja xv xv USD. 25 FX Alternativ och Strukturerade Produkter 25 Hela FX-citat berättar i allmänhet en röra, för vi måste först avgöra vilken valuta som är inhemsk, vilken är utländsk, vad är alternativets nominella valuta och vad är premiumvalutan. Tyvärr är detta inte symmetriskt, eftersom motparten kan ha en annan uppfattning om inhemsk valuta för ett givet valutapar. På den professionella interbankmarknaden finns därmed en uppfattning om delta per valutapar. Normalt är det Fenics-skärmens vänstra sida om alternativet handlas i vänster sida premie, vilket normalt är standard - och höger sida delta om det handlas med höger sida premie, t. ex. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs, etc. Eftersom OTM-alternativ handlas mestadels är skillnaden inte stor och därigenom inte en stor punktrisk. Dessutom deltar deltaet per valutapar vänster sida i Fenics, eftersom de flesta fall används för att citera alternativ i volatilitet. Detta måste anges enligt valuta. Denna standard interbank-uppfattning måste anpassas till bankens reella delta-risk för ett automatiserat handelssystem. För valutor där bankens riskfria valuta är valutaens grundvaluta är det tydligt att deltaet är alternativets råa delta och för riskabelt premie måste detta bidrag inkluderas. In the opposite case the risky premium and the market value must be taken into account for the base currency premium, such that these offset each other. And for premium in underlying currency of the contract the market-value needs to be taken into account. In that way the delta hedge is invariant with respect to the risky currency notion of the bank, e. g. the delta is the same for a USD-based bank and a EUR-based bank. Example We consider two examples in Table 1.3 and 1.4 to compare the various versions of deltas that are used in practice. delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 (delta raw )SK Table 1.3: 1y EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 49.15EUR and the value is 4.427EUR. 26 26 Wystup delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 delta raw SK Table 1.4: 1y call EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 94.82EUR and the value is 21.88EUR Strike in Terms of Delta Since v x phie r f tau N (phid ) we can retrieve the strike as K x exp . (1.44) Volatility in Terms of Delta The mapping sigma phie r f tau N (phid ) is not one-to-one. The two solutions are given by sigma plusmn 1 tau(d d ). (1.45) tau Thus using just the delta to retrieve the volatility of an option is not advisable Volatility and Delta for a Given Strike The determination of the volatility and the delta for a given strike is an iterative process involving the determination of the delta for the option using at-the-money volatilities in a first step and then using the determined volatility to re determine the delta and to continuously iterate the delta and volatility until the volatility does not change more than 0.001 between iterations. More precisely, one can perform the following algorithm. Let the given strike be K. 1. Choose sigma 0 at-the-money volatility from the volatility matrix. 2. Calculate n1 (Call(K, sigma n )). 3. Take sigma n1 sigma( n1 ) from the volatility matrix, possibly via a suitable interpolation. 4. If sigma n1 sigma n lt , then quit, otherwise continue with step 2. 27 FX Options and Structured Products 27 In order to prove the convergence of this algorithm we need to establish convergence of the recursion n1 e r f tau N (d ( n )) (1.46) ( e r f ln(sk) tau (rd r f 1 ) 2 N sigma2 ( n ))tau sigma( n ) tau for sufficiently large sigma( n ) and a sufficiently smooth volatility smile surface. We must show that the sequence of these n converges to a fixed point 0, 1 with a fixed volatility sigma sigma( ). This proof has been carried out in 15 and works like this. We consider the derivative The term n1 e r f tau n(d ( n )) d ( n ) n sigma( n ) sigma( n ). (1.47) n e r f tau n(d ( n )) d ( n ) sigma( n ) converges rapidly to zero for very small and very large spots, being an argument of the standard normal density n. For sufficiently large sigma( n ) and a sufficiently smooth volatility surface in the sense that n sigma( n ) is sufficiently small, we obtain sigma( n ) n q lt 1. (1.48) Thus for any two values (1) n1, (2) n1, a continuously differentiable smile surface we obtain (1) n1 (2) n1 lt q (1) n (2) n, (1.49) due to the mean value theorem. Hence the sequence n is a contraction in the sense of the fixed point theorem of Banach. This implies that the sequence converges to a unique fixed point in 0, 1, which is given by sigma sigma( ) Greeks in Terms of Deltas In Foreign Exchange markets the moneyness of vanilla options is always expressed in terms of deltas and prices are quoted in terms of volatility. This makes a ten-delta call a financial object as such independent of spot and strike. This method and the quotation in volatility makes objects and prices transparent in a very intelligent and user-friendly way. At this point we list the Greeks in terms of deltas instead of spot and strike. Let us introduce the quantities phie r f tau N (phid ) spot delta, (1.50) phie r dtau N (phid ) dual delta, (1.51) 28 28 Wystup which we assume to be given. From these we can retrieve Interpretation of Dual Delta d phin 1 (phie r f tau ), (1.52) d phin 1 ( phie r dtau ). (1.53) The dual delta introduced in (1.23) as the sensitivity with respect to strike has another - more practical - interpretation in a foreign exchange setup. We have seen in Section that the domestic value v(x, K, tau, sigma, r d, r f, phi) (1.54) corresponds to a foreign value v( 1 x, 1 K, tau, sigma, r f, r d, phi) (1.55) up to an adjustment of the nominal amount by the factor xk. From a foreign viewpoint the delta is thus given by ( ) phie rdtau N phi ln( K ) (r x f r d sigma2 tau) sigma tau ( phie rdtau N phi ln( x ) (r K d r f 1 ) 2 sigma2 tau) sigma tau , (1.56) which means the dual delta is the delta from the foreign viewpoint. We will see below that foreign rho, vega and gamma do not require to know the dual delta. We will now state the Greeks in terms of x, . r d, r f, tau, phi. Värde. (Spot) Delta. v(x, . r d, r f, tau, phi) x x e r f tau n(d ) e r dtau n(d ) (1.57) Forward Delta. v f v x (1.58) e (r f r d )tau (1.59) 29 FX Options and Structured Products 29 Gamma. 2 v e r f tau n(d ) x 2 x(d d ) (1.60) Taking a trader s gamma (change of delta if spot moves by 1) additionally removes the spot dependence, because Gamma trader x 2 v e r f tau n(d ) 100 x 2 100(d d ) (1.61) Speed. 3 v e r f tau n(d ) x 3 x 2 (d d ) (2d 2 d ) (1.62) Theta. 1 v x t e r f tau n(d )(d d ) 2tau e r f tau n(d ) r f r d e r dtau n(d ) (1.63) Charm. Color. Vega. Volga. 2 v x tau 3 v x 2 tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d (d d ) 2tau(d d ) (1.64) e r f tau n(d ) 2r f tau (r d r f )tau d (d d ) d 2xtau(d d ) 2tau(d d ) (1.65) v sigma xe r f tau taun(d ) (1.66) 2 v sigma 2 xe r f tau taun(d ) d d d d (1.67) 30 30 Wystup Vanna. 2 v sigma x e r f tau taud n(d ) (1.68) d d Rho. Dual Delta. v e rf tau n(d ) xtau (1.69) r d e r dtau n(d ) v xtau (1.70) r f v K (1.71) Dual Gamma. K 2 2 v K 2 x 2 2 v x 2 (1.72) Dual Theta. v T v t (1.73) As an important example we consider vega. Vega in Terms of Delta The mapping v sigma xe r f tau taun(n 1 (e r f tau )) is important for trading vanilla options. Observe that this function does not depend on r d or sigma, just on r f. Quoting vega in foreign will additionally remove the spot dependence. This means that for a moderately stable foreign term structure curve, traders will be able to use a moderately stable vega matrix. For r f 3 the vega matrix is presented in Table Volatility Volatility is the annualized standard deviation of the log-returns. It is the crucial input parameter to determine the value of an option. Hence, the crucial question is where to derive the volatility from. If no active option market is present, the only source of information is estimating the historic volatility. This would give some clue about the past. In liquid currency 31 FX Options and Structured Products 31 Mat 50 45 40 35 30 25 20 15 10 5 1D W W M M M M M Y Y Y Table 1.5: Vega in terms of Delta for the standard maturity labels and various deltas. It shows that one can vega hedge a long 9M 35 delta call with 4 short 1M 20 delta puts. pairs volatility is often a traded quantity on its own, which is quoted by traders, brokers and real-time data pages. These quotes reflect views of market participants about the future. Since volatility normally does not stay constant, option traders are highly concerned with hedging their volatility exposure. Hedging vanilla options vega is comparatively easy, because vanilla options have convex payoffs, whence the vega is always positive, i. e. the higher the volatility, the higher the price. Let us take for example a EUR-USD market with spot. USD - and EUR rate at 2.5. A 3-month at-the-money call with 1 million EUR notional would cost 29,000 USD at at volatility of 12. If the volatility now drops to a value of 8, then the value of the call would be only 19,000 USD. This monotone dependence is not guaranteed for non-convex payoffs as we illustrate in Figure Historic Volatility We briefly describe how to compute the historic volatility of a time series S 0, S 1. S N (1.74) 32 32 Wystup Figure 1.2: Dependence of a vanilla call and a reverse knock-out call on volatility. The vanilla value is monotone in the volatility, whereas the barrier value is not. The reason is that as the spot gets closer to the upper knock-out barrier, an increasing volatility would increase the chance of knock-out and hence decrease the value. of daily data. First, we create the sequence of log-returns Then, we compute the average log-return r i ln S i S i 1, i 1. N. (1.75) r 1 N N r i, (1.76) i1 33 FX Options and Structured Products 33 their variance and their standard deviation circsigma 2 1 N 1 N (r i r) 2, (1.77) i1 circsigma 1 N (r i r) N 1 2. (1.78) The annualized standard deviation, which is the volatility, is then given by circsigma a B N (r i r) N 1 2, (1.79) where the annualization factor B is given by i1 i1 B N d, (1.80) k and k denotes the number of calendar days within the time series and d denotes the number of calendar days per year. The is done to press the trading days into the calendar days. Assuming normally distributed log-returns, we know that circsigma 2 is chi 2 - distributed. Therefore, given a confidence level of p and a corresponding error probability alpha 1 p, the p-confidence interval is given by N 1 N 1 circsigma a, circsigma chi 2 a, (1.81) N 11 chi 2 alpha N 1 alpha 2 2 where chi 2 np denotes the p-quantile of a chi 2 - distribution 1 with n degrees of freedom. As an example let us take the 256 ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 displayed in Figure 1.3. We get N 255 log-returns. Taking k d 365, we obtain r 1 N r i . N i1 circsigma a B N (r i r) N 1 2 10.85, i1 and a 95 confidence interval of 9.99, 11.89. 1 values and quantiles of the chi 2 - distribution and other distributions can be computed on the internet, e. g. at 34 34 Wystup EURUSD Fixings ECB Exchange Rate 403 4403 5403 6403 7403 8403 9403 Date 10403 11403 12403 1404 2404 Figure 1.3: ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 and the line of average growth Historic Correlation As in the preceding section we briefly describe how to compute the historic correlation of two time series x 0, x 1. x N, y 0, y 1. y N, of daily data. First, we create the sequences of log-returns Then, we compute the average log-returns X i ln x i x i 1, i 1. N, Y i ln y i y i 1, i 1. N. (1.82) X 1 N 1 N N X i, i1 N Y i, (1.83) i1 35 FX Options and Structured Products 35 their variances and covariance circsigma X 2 circsigma Y 2 circsigma XY and their standard deviations circsigma X circsigma Y 1 N 1 1 N 1 1 N 1 N (X i X) 2, (1.84) i1 N (Y i )2, (1.85) i1 N (X i X)(Y i ), (1.86) i1 1 N (X i N 1 X) 2, (1.87) i1 1 N (Y i N 1 )2. (1.88) i1 The estimate for the correlation of the log-returns is given by circrho circsigma XY circsigma X circsigma Y. (1.89) This correlation estimate is often not very stable, but on the other hand, often the only available information. More recent work by Jaumlkel 37 treats robust estimation of correlation. We will revisit FX correlation risk in Section Volatility Smile The Black-Scholes model assumes a constant volatility throughout. However, market prices of traded options imply different volatilities for different maturities and different deltas. We start with some technical issues how to imply the volatility from vanilla options. Retrieving the Volatility from Vanilla Options Given the value of an option. Recall the Black-Scholes formula in Equation (1.7). We now look at the function v(sigma), whose derivative (vega) is The function sigma v(sigma) is v (sigma) xe r f T T n(d ). (1.90) 36 36 Wystup 1. strictly increasing, 2. concave up for sigma 0, 2 ln F ln K T ), 3. concave down for sigma ( 2 ln F ln K T, ) and also satisfies v(0) phi(xe r f T Ke r dt ) , (1.91) v(, phi 1) xe r f T, (1.92) v(sigma , phi 1) Ke r dt, (1.93) v (0) xe r f T T 2piII , (1.94) In particular the mapping sigma v(sigma) is invertible. However, the starting guess for employing Newton s method should be chosen with care, because the mapping sigma v(sigma) has a saddle point at ( ) 2 T ln F K, phie r dt F N phi 2T ln FK KN phi 2T ln KF , (1.95) as illustrated in Figure 1.4. To ensure convergence of Newton s method, we are advised to use initial guesses for sigma on the same side of the saddle point as the desired implied volatility. The danger is that a large initial guess could lead to a negative successive guess for sigma. Therefore one should start with small initial guesses at or below the saddle point. For at-the-money options, the saddle point is degenerate for a zero volatility and small volatilities serve as good initial guesses. Visual Basic Source Code Function VanillaVolRetriever(spot As Double, rd As Double, rf As Double, strike As Double, T As Double, type As Integer, GivenValue As Double) As Double Dim func As Double Dim dfunc As Double Dim maxit As Integer maximum number of iterations Dim j As Integer Dim s As Double first check if a volatility exists, otherwise set result to zero If GivenValue lt Application. Max (0, type (spot Exp(-rf T) - strike Exp(-rd T))) Or (type 1 And GivenValue gt spot Exp(-rf T)) Or (type -1 And GivenValue gt strike Exp(-rd T)) Then 37 FX Options and Structured Products 37 Figure 1.4: Value of a European call in terms of volatility with parameters x 1, K 0.9, T 1, r d 6, r f 5. The saddle point is at sigma 48. VanillaVolRetriever 0 Else there exists a volatility yielding the given value, now use Newton s method: the mapping vol to value has a saddle point. First compute this saddle point: saddle Sqr(2 T Abs(Log(spot strike) (rd - rf) T))Your Search: 1 eBooks Search Engine We are pleased to introduce our wonderful site where collected the most remarkable books of the best authors. Only in one place together the best bestsellers for you dear friends. You can develop your knowledge and skills by downloading our books and guides. We are sure that you will enjoy our great project and it will make your life a little better. 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FX Options and Structured Products 1 FX Options and Structured Products Uwe Wystup 7 April 2006 3 Contents 0 Preface Scope of this Book The Readership About the Author Acknowledgments Foreign Exchange Options A Journey through the History Of Options Technical Issues for Vanilla Options Value A Note on the Forward Greeks Identities Homogeneity based Relationships Quotation Strike in Terms of Delta Volatility in Terms of Delta Volatility and Delta for a Given Strike Greeks in Terms of Deltas Volatility Historic Volatility Historic Correlation Volatility Smile At-The-Money Volatility Interpolation Volatility Smile Conventions At-The-Money Definition Interpolation of the Volatility on Maturity Pillars Interpolation of the Volatility Spread between Maturity Pillars Volatility Sources Volatility Cones Stochastic Volatility 4 4 Wystup Exercises Basic Strategies containing Vanilla Options Call and Put Spread Risk Reversal Risk Reversal Flip Straddle Strangle Butterfly Se agull Exercises First Generation Exotics Barrier Options Digital Options, Touch Options and Rebates Compound and Instalment Asian Options Lookback Options Forward Start, Ratchet and Cliquet Options Power Options Quanto Options Exercises Second Generation Exotics Corridors Faders Exotic Barrier Options Pay-Later Options Step up and Step down Options Spread and Exchange Options Baskets Best-of and Worst-of Options Options and Forwards on the Harmonic Average Variance and Volatility Swaps Exercises Structured Products Forward Products Outright Forward Participating Forward Fade-In Forward Knock-Out Forward Shark Forward Fader Shark Forward 5 FX Options and Structured Products Butterfly Forward Range Forward Range Accrual Forward Accumulative Forward Boomerang Forward Amortizing Forward Auto-Renewal Forward Double Shark Forward Forward Start Chooser Forward Free Style Forward Boosted SpotForward Time Option Exercises Series of Strategies Shark Forward Series Collar Extra Series Exercises D eposits and Loans Dual Currency DepositLoan Performance Linked Deposits Tunnel DepositLoan Corridor DepositLoan Turbo DepositLoan Tower DepositLoan Exercises Interest Rate and Cross Currency Swaps Cross Currency Swap Hanseatic Swap Turbo Cross Currency Swap Buffered Cross Currency Swap Flip Swap Corridor Swap Double-No-Touch linked Swap Range Reset Swap Basket Spread Swap Exercises Participation Notes Gold Participation Note Basket-linked Note Issuer Swap Moving Strike Turbo Spot Unlimited 6 6 Wystup 2.6 Hybrid FX Products Practical Matters The Traders Rule of Thumb Cost of Vanna and Volga Observations Consistency check Abbreviations for First Generation Exotics Adjustment Factor Volatility for Risk Reversals, Butterflies and Theoretical Value Pricing Barrier Options Pricing Double Barrier Options Pricing Double-No-Touch Options Pricing European Style Options No-Touch Probability The Cost of Trading and its Implication on the Market Price of Onetouch Options Example Further Application s Exercises Bid Ask Spreads One Touch Spreads Vanilla Spreads Spreads for First Generation Exotics Minimal Bid Ask Spread Bid Ask Prices Exercises Settlement The Black-Scholes Model for the Actual Spot Cash Settlement Delivery Settlement Options with Deferred Delivery Exercises On the Cost of Delayed Fixing Announcements The Currency Fixing of the European Central Bank Model and Payoff Analysis Procedure Error Estimation Analysis of EUR-USD Conclusion 7 FX Options and Structured Products 7 4 Hedge Accounting under IAS Introduction Financial Instruments Overview General Definition Financial Assets Financial Liabilities Offsetting of Financial Assets and Financial Liabilities Equity Instruments Compound Financial Instruments Derivatives Embedded Derivatives Classification of Financial Instruments Evaluation of Financial Instruments Initial Recognition Initial Measurement Subsequent Measurement Derecognition Hedge Accounting Overview Types of Hedges Basic Requirements Stopping Hedge Accou nting Methods for Testing Hedge Effectiveness Fair Value Hedge Cash Flow Hedge Testing for Effectiveness - A Case Study of the Forward Plus Simulation of Exchange Rates Calculation of the Forward Plus Value Calculation of the Forward Rates Calculation of the Forecast Transaction s Value Dollar-Offset Ratio - Prospective Test for Effectiveness Variance Reduction Measure - Prospective Test for Effectiveness Regression Analysis - Prospective Test for Effectiveness Result Retrospective Test for Effectiveness Conclusion Relevant Original Sources for Accounting Standards Exercises 8 8 Wystup 5 Foreign Exchange Markets A Tour through the Market Statement by GFI Group (Fenics), 25 October Interview with ICY Software, 14 October Interview with Bloomberg, 12 October Interview with Murex, 8 November Interview with SuperDerivatives, 17 October Interview with Lucht Probst Associates, 27 February Software and System Requirements Fenics Position Keeping Pricing Straight Through Processing Disclaimers Trading and Sales Proprietary Trading Sales-Driven Trading Inter Bank Sales Branch Sales Institutional Sales Corporate Sales Private Banking Listed FX Options Trading Floor Joke 9 Chapter 0 Preface 0.1 Scope of this Book Treasury management of international corporates involves dealing with cash flows in different currencies. Therefore the natural service of an investment bank consists of a variety of money market and foreign exchange products. This book explains the most popular products and strategies with a focus on everything beyond vanilla options. It explains all the FX options, common structures and tailor-made solutions in examples with a special focus on the application with views from traders and sales as well as from a corporate client perspective. It contains actually traded deals with corresponding motivations explaining why the structures have been traded. This way the reader gets a feeling how to build new structures to suit clients needs. The exercises are meant to practice the material. Several of them are actually difficult to solve and can serve as incentives to further research and testing. Solutions to the exercises are not part of this book, however they will be published on the web page of the book, 0.2 The Readership Prerequisite is some basic knowledge of FX markets as for example taken from the Book Foreign Exchange Primer by Shami Shamah, Wiley 2003, see 90. The target readers are Graduate students and Faculty of Financial Engineering Programs, who can use this book as a textbook for a course named structured products or exotic currency options. 9 10 10 Wystup Traders, Trainee Structurers, Product Developers, Sales and Quants with interest in the FX product line. For them it can serve as a source of ideas and as well as a reference guide. Treasurers of corporates interested in managing their books. With this book at hand they can structure their solutions themselves. The readers more interested in the quantitative and modeling aspects are recommended to read Foreign Exchange Risk by J. Hakala and U. Wystup, Risk Publications, London, 2002, see 50. This book explains several exotic FX options with a special focus on the underlying models and mathematics, but does not contain any structures or corporate clients or investors view. 0.3 About the Author Figure 1: Uwe Wystup, professor of Quantitative Finance at HfB Business School of Finance and Management in Frankfurt, Germany. Uwe Wystup is also CEO of MathFinance AG, a global network of quants specializing in Quantitative Finance, Exotic Options advisory and Front Office Software Production. Previously he was a Financial Engineer and Structurer in the FX Options Trading Team at Commerzbank. Before that he worked for Deutsche Bank, Citibank, UBS and Sal. Oppenheim jr. amp Cie. He is founder and manager of the web site MathFinance. de and the MathFinance Newsletter. Uwe holds a PhD in mathematical finance from Carnegie Mellon University. He also lectures on mathematical finance for Goethe University Frankfurt, organizes the Frankfurt MathFinance Colloquium and is founding director of the Frankfurt MathFinance Institute. He has given several seminars on exotic options, computational finance and volatility modeling. His area of specialization are the quantitative aspects and the design of structured products of foreign 11 FX Options and Structured Products 11 exchange markets. He published a book on Foreign Exchange Risk and articles in Finance and Stochastics and the Journal of Derivatives. Uwe has given many presentations at both universities and banks around the world. Further information on his curriculum vitae and a detailed publication list is available at 0.4 Acknowledgments I would like to thank my former colleagues on the trading floor, most of all Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Roman Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala, and all my colleagues and co-authors, specially Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt and Robert Tompkins. Chris Swain, Rachael Wilkie and many others of Wiley publications deserve respect as they were dealing with my rather slow speed in completing this book. Nicole van de Locht and Choon Peng Toh deserve a medal for serious detailed proof reading. 13 Chapter 1 Foreign Exchange Options FX Structured Products are tailor-made linear combinations of FX Options including both vanilla and exotic options. We recommend the book by Shamah 90 as a source to learn about FX Markets with a focus on market conventions, spot, forward and swap contracts, vanilla options. For pricing and modeling of exotic FX options we suggest Hakala and Wystup 50 or Lipton 71 as useful companions to this book. The market for structured products is restricted to the market of the necessary ingredients. Hence, typically there are mostly structured products traded the currency pairs that can be formed between USD, JPY, EUR, CHF, GBP, CAD and AUD. In this chapter we start with a brief history of options, followed by a technical section on vanilla options and volatility, and deal with commonly used linear combinations of vanilla options. Then we will illustrated the most important ingredients for FX structured products: the first and second generation exotics. 1.1 A Journey through the History Of Options The very first options and futures were traded in ancient Greece, when olives were sold before they had reached ripeness. Thereafter the market evolved in the following way. 16th century Ever since the 15th century tulips, which were liked for their exotic appearance, were grown in Turkey. The head of the royal medical gardens in Vienna, Austria, was the first to cultivate those Turkish tulips successfully in Europe. When he fled to Holland because of religious persecution, he took the bulbs along. As the new head of the botanical gardens of Leiden, Netherlands, he cultivated several new strains. It was from these gardens that avaricious traders stole the bulbs to commercialize them, because tulips were a great status symbol. 17th century The first futures on tulips were traded in As of 1634, people could 13 14 14 Wystup buy special tulip strains by the weight of their bulbs, for the bulbs the same value was chosen as for gold. Along with the regular trading, speculators entered the market and the prices skyrocketed. A bulb of the strain Semper Octavian was worth two wagonloads of wheat, four loads of rye, four fat oxen, eight fat swine, twelve fat sheep, two hogsheads of wine, four barrels of beer, two barrels of butter, 1,000 pounds of cheese, one marriage bed with linen and one sizable wagon. People left their families, sold all their belongings, and even borrowed money to become tulip traders. When in 1637, this supposedly risk-free market crashed, traders as well as private individuals went bankrupt. The government prohibited speculative trading the period became famous as Tulipmania. 18th century In 1728, the Royal West-Indian and Guinea Company, the monopolist in trading with the Caribbean Islands and the African coast issued the first stock options. Those were options on the purchase of the French Island of Ste. Croix, on which sugar plantings were planned. The project was realized in 1733 and paper stocks were issued in Along with the stock, people purchased a relative share of the island and the valuables, as well as the privileges and the rights of the company. 19th century In 1848, 82 businessmen founded the Chicago Board of Trade (CBOT). Today it is the biggest and oldest futures market in the entire world. Most written documents were lost in the great fire of 1871, however, it is commonly believed that the first standardized futures were traded as of CBOT now trades several futures and forwards, not only T-bonds and treasury bonds, but also options and gold. In 1870, the New York Cotton Exchange was founded. In 1880, the gold standard was introduced. 20th century In 1914, the gold standard was abandoned because of the war. In 1919, the Chicago Produce Exchange, in charge of trading agricultural products was renamed to Chicago Mercantile Exchange. Today it is the most important futures market for Eurodollar, foreign exchange, and livestock. In 1944, the Bretton Woods System was implemented in an attempt to stabilize the currency system. In 1970, the Bretton Woods System was abandoned for several reasons. In 1971, the Smithsonian Agreement on fixed exchange rates was introduced. In 1972, the International Monetary Market (IMM) traded futures on coins, currencies and precious metal. 15 FX Options and Structured Products 15 21th century In 1973, the CBOE (Chicago Board of Exchange) firstly traded call options four years later also put options. The Smithsonian Agreement was abandoned the currencies followed managed floating. In 1975, the CBOT sold the first interest rate future, the first future with no real underlying asset. In 1978, the Dutch stock market traded the first standardized financial derivatives. In 1979, the European Currency System was implemented, and the European Currency Unit (ECU) was introduced. In 1991, the Maastricht Treaty on a common currency and economic policy in Europe was signed. In 1999, the Euro was introduced, but the countries still used cash of their old currencies, while the exchange rates were kept fixed. In 2002, the Euro was introduced as new money in the form of cash. 1.2 Technical Issues for Vanilla Options We consider the model geometric Brownian motion ds t (r d r f )S t dt sigmas t dw t (1.1) for the underlying exchange rate quoted in FOR-DOM (foreign-domestic), which means that one unit of the foreign currency costs FOR-DOM units of the domestic currency. In case of EUR-USD with a spot of. this means that the price of one EUR is USD. The notion of foreign and domestic do not refer the location of the trading entity, but only to this quotation convention. We denote the (continuous) foreign interest rate by r f and the (continuous) domestic interest rate by r d. In an equity scenario, r f would represent a continuous dividend rate. The volatility is denoted by sigma, and W t is a standard Brownian motion. The sample paths are displayed in Figure 1.1. We consider this standard model, not because it reflects the statistical properties of the exchange rate (in fact, it doesn t), but because it is widely used in practice and front office systems and mainly serves as a tool to communicate prices in FX options. These prices are generally quoted in terms of volatility in the sense of this model. Applying Itocirc s rule to ln S t yields the following solution for the process S t S t S 0 exp sigma2 )t sigmaw t, (1.2) which shows that S t is log-normally distributed, more precisely, ln S t is normal with mean ln S 0 (r d r f 1 2 sigma2 )t and variance sigma 2 t. Further model assumptions are 16 16 Wystup Figure 1.1: Simulated paths of a geometric Brownian motion. The distribution of the spot S T at time T is log-normal. 1. There is no arbitrage 2. Trading is frictionless, no transaction costs 3. Any position can be taken at any time, short, long, arbitrary fraction, no liquidity constraints The payoff for a vanilla option (European put or call) is given by F phi(s T K) , (1.3) where the contractual parameters are the strike K, the expiration time T and the type phi, a binary variable which takes the value 1 in the case of a call and 1 in the case of a put. The symbol x denotes the positive part of x, i. e. x max(0, x) 0 x Value In the Black-Scholes model the value of the payoff F at time t if the spot is at x is denoted by v(t, x) and can be computed either as the solution of the Black-Scholes partial differential 17 FX Options and Structured Products 17 equation v t r d v (r d r f )xv x sigma2 x 2 v xx 0, (1.4) v(t, x) F. (1.5) or equivalently (Feynman-Kac-Theorem) as the discounted expected value of the payofffunction, v(x, K, T, t, sigma, r d, r f, phi) e r dtau IEF . (1.6) This is the reason why basic financial engineering is mostly concerned with solving partial differential equations or computing expectations (numerical integration). The result is the Black-Scholes formula We abbreviate v(x, K, T, t, sigma, r d, r f, phi) phie r dtau fn (phid ) KN (phid ). (1.7) x: current price of the underlying tau T t: time to maturity f IES T S t x xe (r d r f )tau. forward price of the underlying theta plusmn r d r f sigma plusmn sigma 2 d plusmn ln x K sigmatheta plusmntau sigma tau ln f K plusmn sigma 2 2 tau sigma tau n(t) 1 2pi e 1 2 t2 n( t) N (x) x n(t) dt 1 N ( x) The Black-Scholes formula can be derived using the integral representation of Equation (1.6) v e r dtau IEF e rdtau IEphi(S T K) ( e r dtau phi xe (r d r f 1 2 sigma2 )tausigma tauy K) n(y) dy. (1.8) Next one has to deal with the positive part and then complete the square to get the Black - Scholes formula. A derivation based on the partial differential equation can be done using results about the well-studied heat-equation. 18 18 Wystup A Note on the Forward The forward price f is the strike which makes the time zero value of the forward contract F S T f (1.9) equal to zero. It follows that f IES T xe (r d r f )T, i. e. the forward price is the expected price of the underlying at time T in a risk-neutral setup (drift of the geometric Brownian motion is equal to cost of carry r d r f ). The situation r d gt r f is called contango, and the situation r d lt r f is called backwardation. Note that in the Black-Scholes model the class of forward price curves is quite restricted. For example, no seasonal effects can be included. Note that the value of the forward contract after time zero is usually different from zero, and since one of the counterparties is always short, there may be risk of default of the short party. A futures contract prevents this dangerous affair: it is basically a forward contract, but the counterparties have to a margin account to ensure the amount of cash or commodity owed does not exceed a specified limit Greeks Greeks are derivatives of the value function with respect to model and contract parameters. They are an important information for traders and have become standard information provided by front-office systems. More details on Greeks and the relations among Greeks are presented in Hakala and Wystup 50 or Reiss and Wystup 84. For vanilla options we list some of them now. (Spot) Delta. v x phie r f tau N (phid ) (1.10) Forward Delta. Driftless Delta. v f phie r dtau N (phid ) (1.11) phin (phid ) (1.12) Gamma. 2 v e r f tau n(d ) x 2 xsigma tau (1.13) 19 FX Options and Structured Products 19 Speed. 3 v x 3 e r f tau n(d ) x 2 sigma tau ( ) d sigma tau 1 (1.14) Theta. v t e r f tau n(d )xsigma 2 tau phir f xe r f tau N (phid ) r d Ke rdtau N (phid ) (1.15) Charm. 2 v x tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d sigma tau 2tausigma tau (1.16) Color. 3 v x 2 tau e r f tau n(d ) 2xtausigma tau 2r f tau (r d r f )tau d sigma tau 2tausigma d tau (1.17) Vega. v sigma xe r f tau taun(d ) (1.18) Volga. 2 v sigma 2 xe r f tau taun(d ) d d sigma (1.19) Volga is also sometimes called vomma or volgamma. Vanna. 2 v sigma x e r f tau n(d ) d sigma (1.20) Rho. v r d phiktaue rdtau N (phid ) (1.21) v r f phixtaue r f tau N (phid ) (1.22) 20 20 Wystup Dual Delta. Dual Gamma. v K phie r dtau N (phid ) (1.23) 2 v e r dtau n(d ) K 2 Ksigma tau (1.24) Dual Theta. v T v t (1.25) Identities The put-call-parity is the relationship d plusmn d (1.26) sigma sigma d plusmn tau (1.27) r d sigma d plusmn tau (1.28) r f sigma xe r f tau n(d ) Ke rdtau n(d ). (1.29) N (phid ) IP phis T phik (1.30) N (phid ) IP phis T phi f 2 (1.31) K v(x, K, T, t, sigma, r d, r f, 1) v(x, K, T, t, sigma, r d, r f, 1) xe r f tau Ke r dtau, (1.32) which is just a more complicated way to write the trivial equation x x x. The put-call delta parity is v(x, K, T, t, sigma, r d, r f, 1) x v(x, K, T, t, sigma, r d, r f, 1) x e r f tau. (1.33) In particular, we learn that the absolute value of a put delta and a call delta are not exactly adding up to one, but only to a positive number e r f tau. They add up to one approximately if either the time to expiration tau is short or if the foreign interest rate r f is close to zero. 21 FX Options and Structured Products 21 Whereas the choice K f produces identical values for call and put, we seek the deltasymmetric strike which produces absolutely identical deltas (spot, forward or driftless). This condition implies d 0 and thus fe sigma2 2 T, (1.34) in which case the absolute delta is e r f tau 2. In particular, we learn, that always gt f, i. e. there can t be a put and a call with identical values and deltas. Note that the strike is usually chosen as the middle strike when trading a straddle or a butterfly. Similarly the dual-delta-symmetric strike circK fe sigma2 2 T can be derived from the condition d Homogeneity based Relationships We may wish to measure the value of the underlying in a different unit. This will obviously effect the option pricing formula as follows. av(x, K, T, t, sigma, r d, r f, phi) v(ax, ak, T, t, sigma, r d, r f, phi) for all a gt 0. (1.35) Differentiating both sides with respect to a and then setting a 1 yields v xv x Kv K. (1.36) Comparing the coefficients of x and K in Equations (1.7) and (1.36) leads to suggestive results for the delta v x and dual delta v K. This space-homogeneity is the reason behind the simplicity of the delta formulas, whose tedious computation can be saved this way. We can perform a similar computation for the time-affected parameters and obtain the obvious equation v(x, K, T, t, sigma, r d, r f, phi) v(x, K, T a, t a, asigma, ar d, ar f, phi) for all a gt 0. (1.37) Differentiating both sides with respect to a and then setting a 1 yields 0 tauv t sigmav sigma r d v rd r f v rf. (1.38) Of course, this can also be verified by direct computation. The overall use of such equations is to generate double checking benchmarks when computing Greeks. These homogeneity methods can easily be extended to other more complex options. By put-call symmetry we understand the relationship (see 6, 7,16 and 19) v(x, K, T, t, sigma, r d, r f, 1) K f v(x, f 2 K, T, t, sigma, r d, r f, 1). (1.39) 22 22 Wystup The strike of the put and the strike of the call result in a geometric mean equal to the forward f. The forward can be interpreted as a geometric mirror reflecting a call into a certain number of puts. Note that for at-the-money options (K f) the put-call symmetry coincides with the special case of the put-call parity where the call and the put have the same value. Direct computation shows that the rates symmetry v v tauv (1.40) r d r f holds for vanilla options. This relationship, in fact, holds for all European options and a wide class of path-dependent options as shown in 84. One can directly verify the relationship the foreign-domestic symmetry 1 x v(x, K, T, t, sigma, r d, r f, phi) Kv( 1 x, 1 K, T, t, sigma, r f, r d, phi). (1.41) This equality can be viewed as one of the faces of put-call symmetry. The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of S t modeling the exchange rate of EURUSD. In New York, the call option (S T K) costs v(x, K, T, t, sigma, r usd, r eur, 1) USD and hence v(x, K, T, t, sigma, r usd, r eur, 1)x ( ) 1 . Euro. This EUR-call option can also be viewed as a USD-put option with payoff K 1 K S T This option costs Kv( 1, 1, T, t, sigma, r x K eur, r usd, 1) EUR in Frankfurt, because S t and 1 S t have the same volatility. Of course, the New York value and the Frankfurt value must agree, which leads to (1.41). We will also learn later, that this symmetry is just one possible result based on change of numeraire Quotation Quotation of the Underlying Exchange Rate Equation (1.1) is a model for the exchange rate. The quotation is a permanently confusing issue, so let us clarify this here. The exchange rate means how much of the domestic currency are needed to buy one unit of foreign currency. For example, if we take EURUSD as an exchange rate, then the default quotation is EUR-USD, where USD is the domestic currency and EUR is the foreign currency. The term domestic is in no way related to the location of the trader or any country. It merely means the numeraire currency. The terms domestic, numeraire or base currency are synonyms as are foreign and underlying. Throughout this book we denote with the slash () the currency pair and with a dash (-) the quotation. The slash () does not mean a division. For instance, EURUSD can also be quoted in either EUR-USD, which then means how many USD are needed to buy one EUR, or in USD-EUR, which then means how many EUR are needed to buy one USD. There are certain market standard quotations listed in Table 1.1. 23 FX Options and Structured Products 23 currency pair default quotation sample quote GBPUSD GPB-USD GBPCHF GBP-CHF EURUSD EUR-USD EURGBP EUR-GBP EURJPY EUR-JPY EURCHF EUR-CHF USDJPY USD-JPY USDCHF USD-CHF Table 1.1: Standard market quotation of major currency pairs with sample spot prices Trading Floor Language We call one million a buck, one billion a yard. This is because a billion is called milliarde in French, German and other languages. For the British Pound one million is also often called a quid. Certain currency pairs have names. For instance, GBPUSD is called cable, because the exchange rate information used to be sent through a cable in the Atlantic ocean between America and England. EURJPY is called the cross, because it is the cross rate of the more liquidly traded USDJPY and EURUSD. Certain currencies also have names, e. g. the New Zealand Dollar NZD is called a kiwi, the Australian Dollar AUD is called Aussie, the Scandinavian currencies DKR, NOK and SEK are called Scandies. Exchange rates are generally quoted up to five relevant figures, e. g. in EUR-USD we could observe a quote of The last digit 5 is called the pip, the middle digit 3 is called the big figure, as exchange rates are often displayed in trading floors and the big figure, which is displayed in bigger size, is the most relevant information. The digits left to the big figure are known anyway, the pips right of the big figure are often negligible. To make it clear, a rise of USD-JPY by 20 pips will be and a rise by 2 big figures will be Quotation of Option Prices Values and prices of vanilla options may be quoted in the six ways explained in Table 1.2. 24 24 Wystup name symbol value in units of example domestic cash d DOM 29,148 USD foreign cash f FOR 24,290 EUR domestic d DOM per unit of DOM USD foreign f FOR per unit of FOR EUR domestic pips d pips DOM per unit of FOR USD pips per EUR foreign pips f pips FOR per unit of DOM EUR pips per USD Table 1.2: Standard market quotation types for option values. In the example we take FOREUR, DOMUSD, S 0 . r d 3.0, r f 2.5, sigma 10, K . T 1 year, phi 1 (call), notional 1, 000, 000 EUR 1, 250, 000 USD. For the pips, the quotation USD pips per EUR is also sometimes stated as USD per 1 EUR. Similarly, the EUR pips per USD can also be quoted as EUR per 1 USD. The Black-Scholes formula quotes d pips. The others can be computed using the following instruction. d pips 1 S 0 S 0 1 f K S d 0 S f pips 0 K d pips (1.42) Delta and Premium Convention The spot delta of a European option without premium is well known. It will be called raw spot delta delta raw now. It can be quoted in either of the two currencies involved. The relationship is delta reverse raw delta raw S K. (1.43) The delta is used to buy or sell spot in the corresponding amount in order to hedge the option up to first order. For consistency the premium needs to be incorporated into the delta hedge, since a premium in foreign currency will already hedge part of the option s delta risk. To make this clear, let us consider EUR-USD. In the standard arbitrage theory, v(x) denotes the value or premium in USD of an option with 1 EUR notional, if the spot is at x, and the raw delta v x denotes the number of EUR to buy for the delta hedge. Therefore, xv x is the number of USD to sell. If now the premium is paid in EUR rather than in USD, then we already have v x EUR, and the number of EUR to buy has to be reduced by this amount, i. e. if EUR is the premium currency, we need to buy v x v x EUR for the delta hedge or equivalently sell xv x v USD. 25 FX Options and Structured Products 25 The entire FX quotation story becomes generally a mess, because we need to first sort out which currency is domestic, which is foreign, what is the notional currency of the option, and what is the premium currency. Unfortunately this is not symmetric, since the counterpart might have another notion of domestic currency for a given currency pair. Hence in the professional inter bank market there is one notion of delta per currency pair. Normally it is the left hand side delta of the Fenics screen if the option is traded in left hand side premium, which is normally the standard and right hand side delta if it is traded with right hand side premium, e. g. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs, etc. Since OTM options are traded most of time the difference is not huge and hence does not create a huge spot risk. Additionally the standard delta per currency pair left hand side delta in Fenics for most cases is used to quote options in volatility. This has to be specified by currency. This standard inter bank notion must be adapted to the real delta-risk of the bank for an automated trading system. For currencies where the risk free currency of the bank is the base currency of the currency it is clear that the delta is the raw delta of the option and for risky premium this premium must be included. In the opposite case the risky premium and the market value must be taken into account for the base currency premium, such that these offset each other. And for premium in underlying currency of the contract the market-value needs to be taken into account. In that way the delta hedge is invariant with respect to the risky currency notion of the bank, e. g. the delta is the same for a USD-based bank and a EUR-based bank. Example We consider two examples in Table 1.3 and 1.4 to compare the various versions of deltas that are used in practice. delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 (delta raw )SK Table 1.3: 1y EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 49.15EUR and the value is 4.427EUR. 26 26 Wystup delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 delta raw SK Table 1.4: 1y call EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 94.82EUR and the value is 21.88EUR Strike in Terms of Delta Since v x phie r f tau N (phid ) we can retrieve the strike as K x exp . (1.44) Volatility in Terms of Delta The mapping sigma phie r f tau N (phid ) is not one-to-one. The two solutions are given by sigma plusmn 1 tau(d d ). (1.45) tau Thus using just the delta to retrieve the volatility of an option is not advisable Volatility and Delta for a Given Strike The determination of the volatility and the delta for a given strike is an iterative process involving the determination of the delta for the option using at-the-money volatilities in a first step and then using the determined volatility to re determine the delta and to continuously iterate the delta and volatility until the volatility does not change more than 0.001 between iterations. More precisely, one can perform the following algorithm. Let the given strike be K. 1. Choose sigma 0 at-the-money volatility from the volatility matrix. 2. Calculate n1 (Call(K, sigma n )). 3. Take sigma n1 sigma( n1 ) from the volatility matrix, possibly via a suitable interpolation. 4. If sigma n1 sigma n lt , then quit, otherwise continue with step 2. 27 FX Options and Structured Products 27 In order to prove the convergence of this algorithm we need to establish convergence of the recursion n1 e r f tau N (d ( n )) (1.46) ( e r f ln(sk) tau (rd r f 1 ) 2 N sigma2 ( n ))tau sigma( n ) tau for sufficiently large sigma( n ) and a sufficiently smooth volatility smile surface. We must show that the sequence of these n converges to a fixed point 0, 1 with a fixed volatility sigma sigma( ). This proof has been carried out in 15 and works like this. We consider the derivative The term n1 e r f tau n(d ( n )) d ( n ) n sigma( n ) sigma( n ). (1.47) n e r f tau n(d ( n )) d ( n ) sigma( n ) converges rapidly to zero for very small and very large spots, being an argument of the standard normal density n. For sufficiently large sigma( n ) and a sufficiently smooth volatility surface in the sense that n sigma( n ) is sufficiently small, we obtain sigma( n ) n q lt 1. (1.48) Thus for any two values (1) n1, (2) n1, a continuously differentiable smile surface we obtain (1) n1 (2) n1 lt q (1) n (2) n, (1.49) due to the mean value theorem. Hence the sequence n is a contraction in the sense of the fixed point theorem of Banach. This implies that the sequence converges to a unique fixed point in 0, 1, which is given by sigma sigma( ) Greeks in Terms of Deltas In Foreign Exchange markets the moneyness of vanilla options is always expressed in terms of deltas and prices are quoted in terms of volatility. This makes a ten-delta call a financial object as such independent of spot and strike. This method and the quotation in volatility makes objects and prices transparent in a very intelligent and user-friendly way. At this point we list the Greeks in terms of deltas instead of spot and strike. Let us introduce the quantities phie r f tau N (phid ) spot delta, (1.50) phie r dtau N (phid ) dual delta, (1.51) 28 28 Wystup which we assume to be given. From these we can retrieve Interpretation of Dual Delta d phin 1 (phie r f tau ), (1.52) d phin 1 ( phie r dtau ). (1.53) The dual delta introduced in (1.23) as the sensitivity with respect to strike has another - more practical - interpretation in a foreign exchange setup. We have seen in Section that the domestic value v(x, K, tau, sigma, r d, r f, phi) (1.54) corresponds to a foreign value v( 1 x, 1 K, tau, sigma, r f, r d, phi) (1.55) up to an adjustment of the nominal amount by the factor xk. From a foreign viewpoint the delta is thus given by ( ) phie rdtau N phi ln( K ) (r x f r d sigma2 tau) sigma tau ( phie rdtau N phi ln( x ) (r K d r f 1 ) 2 sigma2 tau) sigma tau , (1.56) which means the dual delta is the delta from the foreign viewpoint. We will see below that foreign rho, vega and gamma do not require to know the dual delta. We will now state the Greeks in terms of x, . r d, r f, tau, phi. Värde. (Spot) Delta. v(x, . r d, r f, tau, phi) x x e r f tau n(d ) e r dtau n(d ) (1.57) Forward Delta. v f v x (1.58) e (r f r d )tau (1.59) 29 FX Options and Structured Products 29 Gamma. 2 v e r f tau n(d ) x 2 x(d d ) (1.60) Taking a trader s gamma (change of delta if spot moves by 1) additionally removes the spot dependence, because Gamma trader x 2 v e r f tau n(d ) 100 x 2 100(d d ) (1.61) Speed. 3 v e r f tau n(d ) x 3 x 2 (d d ) (2d 2 d ) (1.62) Theta. 1 v x t e r f tau n(d )(d d ) 2tau e r f tau n(d ) r f r d e r dtau n(d ) (1.63) Charm. Color. Vega. Volga. 2 v x tau 3 v x 2 tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d (d d ) 2tau(d d ) (1.64) e r f tau n(d ) 2r f tau (r d r f )tau d (d d ) d 2xtau(d d ) 2tau(d d ) (1.65) v sigma xe r f tau taun(d ) (1.66) 2 v sigma 2 xe r f tau taun(d ) d d d d (1.67) 30 30 Wystup Vanna. 2 v sigma x e r f tau taud n(d ) (1.68) d d Rho. Dual Delta. v e rf tau n(d ) xtau (1.69) r d e r dtau n(d ) v xtau (1.70) r f v K (1.71) Dual Gamma. K 2 2 v K 2 x 2 2 v x 2 (1.72) Dual Theta. v T v t (1.73) As an important example we consider vega. Vega in Terms of Delta The mapping v sigma xe r f tau taun(n 1 (e r f tau )) is important for trading vanilla options. Observe that this function does not depend on r d or sigma, just on r f. Quoting vega in foreign will additionally remove the spot dependence. This means that for a moderately stable foreign term structure curve, traders will be able to use a moderately stable vega matrix. For r f 3 the vega matrix is presented in Table Volatility Volatility is the annualized standard deviation of the log-returns. It is the crucial input parameter to determine the value of an option. Hence, the crucial question is where to derive the volatility from. If no active option market is present, the only source of information is estimating the historic volatility. This would give some clue about the past. In liquid currency 31 FX Options and Structured Products 31 Mat 50 45 40 35 30 25 20 15 10 5 1D W W M M M M M Y Y Y Table 1.5: Vega in terms of Delta for the standard maturity labels and various deltas. It shows that one can vega hedge a long 9M 35 delta call with 4 short 1M 20 delta puts. pairs volatility is often a traded quantity on its own, which is quoted by traders, brokers and real-time data pages. These quotes reflect views of market participants about the future. Since volatility normally does not stay constant, option traders are highly concerned with hedging their volatility exposure. Hedging vanilla options vega is comparatively easy, because vanilla options have convex payoffs, whence the vega is always positive, i. e. the higher the volatility, the higher the price. Let us take for example a EUR-USD market with spot. USD - and EUR rate at 2.5. A 3-month at-the-money call with 1 million EUR notional would cost 29,000 USD at at volatility of 12. If the volatility now drops to a value of 8, then the value of the call would be only 19,000 USD. This monotone dependence is not guaranteed for non-convex payoffs as we illustrate in Figure Historic Volatility We briefly describe how to compute the historic volatility of a time series S 0, S 1. S N (1.74) 32 32 Wystup Figure 1.2: Dependence of a vanilla call and a reverse knock-out call on volatility. The vanilla value is monotone in the volatility, whereas the barrier value is not. The reason is that as the spot gets closer to the upper knock-out barrier, an increasing volatility would increase the chance of knock-out and hence decrease the value. of daily data. First, we create the sequence of log-returns Then, we compute the average log-return r i ln S i S i 1, i 1. N. (1.75) r 1 N N r i, (1.76) i1 33 FX Options and Structured Products 33 their variance and their standard deviation circsigma 2 1 N 1 N (r i r) 2, (1.77) i1 circsigma 1 N (r i r) N 1 2. (1.78) The annualized standard deviation, which is the volatility, is then given by circsigma a B N (r i r) N 1 2, (1.79) where the annualization factor B is given by i1 i1 B N d, (1.80) k and k denotes the number of calendar days within the time series and d denotes the number of calendar days per year. The is done to press the trading days into the calendar days. Assuming normally distributed log-returns, we know that circsigma 2 is chi 2 - distributed. Therefore, given a confidence level of p and a corresponding error probability alpha 1 p, the p-confidence interval is given by N 1 N 1 circsigma a, circsigma chi 2 a, (1.81) N 11 chi 2 alpha N 1 alpha 2 2 where chi 2 np denotes the p-quantile of a chi 2 - distribution 1 with n degrees of freedom. As an example let us take the 256 ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 displayed in Figure 1.3. We get N 255 log-returns. Taking k d 365, we obtain r 1 N r i . N i1 circsigma a B N (r i r) N 1 2 10.85, i1 and a 95 confidence interval of 9.99, 11.89. 1 values and quantiles of the chi 2 - distribution and other distributions can be computed on the internet, e. g. at 34 34 Wystup EURUSD Fixings ECB Exchange Rate 403 4403 5403 6403 7403 8403 9403 Date 10403 11403 12403 1404 2404 Figure 1.3: ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 and the line of average growth Historic Correlation As in the preceding section we briefly describe how to compute the historic correlation of two time series x 0, x 1. x N, y 0, y 1. y N, of daily data. First, we create the sequences of log-returns Then, we compute the average log-returns X i ln x i x i 1, i 1. N, Y i ln y i y i 1, i 1. N. (1.82) X 1 N 1 N N X i, i1 N Y i, (1.83) i1 35 FX Options and Structured Products 35 their variances and covariance circsigma X 2 circsigma Y 2 circsigma XY and their standard deviations circsigma X circsigma Y 1 N 1 1 N 1 1 N 1 N (X i X) 2, (1.84) i1 N (Y i )2, (1.85) i1 N (X i X)(Y i ), (1.86) i1 1 N (X i N 1 X) 2, (1.87) i1 1 N (Y i N 1 )2. (1.88) i1 The estimate for the correlation of the log-returns is given by circrho circsigma XY circsigma X circsigma Y. (1.89) This correlation estimate is often not very stable, but on the other hand, often the only available information. More recent work by Jaumlkel 37 treats robust estimation of correlation. We will revisit FX correlation risk in Section Volatility Smile The Black-Scholes model assumes a constant volatility throughout. However, market prices of traded options imply different volatilities for different maturities and different deltas. We start with some technical issues how to imply the volatility from vanilla options. Retrieving the Volatility from Vanilla Options Given the value of an option. Recall the Black-Scholes formula in Equation (1.7). We now look at the function v(sigma), whose derivative (vega) is The function sigma v(sigma) is v (sigma) xe r f T T n(d ). (1.90) 36 36 Wystup 1. strictly increasing, 2. concave up for sigma 0, 2 ln F ln K T ), 3. concave down for sigma ( 2 ln F ln K T, ) and also satisfies v(0) phi(xe r f T Ke r dt ) , (1.91) v(, phi 1) xe r f T, (1.92) v(sigma , phi 1) Ke r dt, (1.93) v (0) xe r f T T 2piII , (1.94) In particular the mapping sigma v(sigma) is invertible. However, the starting guess for employing Newton s method should be chosen with care, because the mapping sigma v(sigma) has a saddle point at ( ) 2 T ln F K, phie r dt F N phi 2T ln FK KN phi 2T ln KF , (1.95) as illustrated in Figure 1.4. To ensure convergence of Newton s method, we are advised to use initial guesses for sigma on the same side of the saddle point as the desired implied volatility. The danger is that a large initial guess could lead to a negative successive guess for sigma. Therefore one should start with small initial guesses at or below the saddle point. For at-the-money options, the saddle point is degenerate for a zero volatility and small volatilities serve as good initial guesses. Visual Basic Source Code Function VanillaVolRetriever(spot As Double, rd As Double, rf As Double, strike As Double, T As Double, type As Integer, GivenValue As Double) As Double Dim func As Double Dim dfunc As Double Dim maxit As Integer maximum number of iterations Dim j As Integer Dim s As Double first check if a volatility exists, otherwise set result to zero If GivenValue lt Application. Max (0, type (spot Exp(-rf T) - strike Exp(-rd T))) Or (type 1 And GivenValue gt spot Exp(-rf T)) Or (type -1 And GivenValue gt strike Exp(-rd T)) Then 37 FX Options and Structured Products 37 Figure 1.4: Value of a European call in terms of volatility with parameters x 1, K 0.9, T 1, r d 6, r f 5. The saddle point is at sigma 48. VanillaVolRetriever 0 Else there exists a volatility yielding the given value, now use Newton s method: the mapping vol to value has a saddle point. First compute this saddle point: saddle Sqr(2 T Abs(Log(spot strike) (rd - rf) T))Sell faster. Your next home is waiting. For the first time in from it, and by incontrovertible calculations I find that a projectile endowed with an initial velocity about events as they happened. She groomed the dolls endlessly, cooed to them, tucked them over to figure out why, and, about get interested in you. 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