Fx Optiot Ja Jäsennelty Tuotteet Pdf

Valuuttamääräiset optiot ja strukturoidut tuotteet 1 Valuuttamääräiset optiot ja strukturoidut tuotteet Uwe Wystup 7.4.2006 3 Sisältö 0 Esipuhe Tämän kirjan laajuus Lukijaluettelo Tietoja kirjoittajasta Kiitokset Valuuttaoptiot Vaihtoehtojen historia Vaihtoehtojen tekniset kysymykset Vanilja-optioiden arvot Arvo Huomautus Eteenpäin kreikkalaiset identiteetit Homogeenisyysperusteiset suhteet Lainausväki Delta-volatiliteettiluokituksen Delta-volatiliteetin ja delta-arvon perusteella Deltasin volatiliteetilta Delta-arvot Datalojen volatiliteetilta Historiallinen volatiliteetti Historiallinen korrelaatio Volatiliteetti Hymy hymyilemällä rahalla Volatiliteetti Interpolaatio Volatiliteetti Hymys Conventions At-The - Rahamääritelmä Volatiliteetin interpolointi maturiteetin pylväillä Volatiliteetin interpolaatio Eroa maturiteetin pilarien välillä Volatiliteetin lähteet Volatiliteetti Kynät Stokastinen volatiliteetti 4 4 Wystup-harjoitukset Perusstrategiat, jotka sisältävät vaniljavaihtoehtoja Kutsu ja laita Leviävä riski Käänteinen riski Peruutus Flip Straddle Stuttering Butterfly Seagull Harjoitukset Ensimmäinen G energation Exotics Barrier Options Digitaaliset lisävarusteet, kosketuspainikkeet ja hyvitykset Yhdistelmä - ja asennusvaihtoehdot Aasian vaihtoehdot Tarkasteluasetukset Eteenpäin Käynnistäminen, Ratchet ja Cliquet Options Virranhallintavaihtoehdot Quanto Options Harjoitukset Toinen sukupolvi Exotics-käytävät Faders Exotic Barrier - vaihtoehdot Maksulliset lisävarusteet Lisäykset ylös ja alas Vaihtoehdot Levitä ja Vaihto-oikeudet Vaihtoehdot Kortit Paras ja huonoin vaihtoehdoista Vaihtoehdot ja eteenpäin Harmonisen keskiarvon vaihtelu - ja volatiliteettisopimukset Harjoitukset Strukturoidut tuotteet eteenpäin Tuotteet eteenpäin eteenpäin Osallistuminen eteenpäin Häivytys eteenpäin Knock-out eteenpäin Shark Forward Fader Shark Forward 5 FX-vaihtoehdot ja strukturoidut tuotteet Butterfly Forward Range Forward Range Siirtyminen eteenpäin Kertynyt eteenpäin Boomerang Forward Jatkuva eteenpäin Automaattinen uudistaminen eteenpäin Double Shark Forward Forward Start Chooser Forward Vapaa Style Forward Boosted SpotForward Time Optio Harjoitukset Strategia sarja Shark Forward Series Collar Extra Sarjan harjoitukset Talletukset ja lainat Dual Valuutta TalletusLoan Suoritus Liitetyt Talletukset Tunnel DepositLoan Corridor TalletusLoan Turbo TalletaLoan Tower TalletusLoan Harjoitukset Korko ja rajat valuutan vaihtosopimukset Cross Currency Swap Hanseatic Swap Turbo Cross Valuutanvaihtopuskuri Puskuroitu Cross Valuutta Swap Flip Swap käytävä Vaihda Double-No-Touch-linkitetty Swap Range Reset Swap Basket Spread Swap Harjoitukset Osallistumismuistio Kulta Osallistuminen Huomautus Koripallotiedot Liikkeeseenlaskija Swap Moving Strike Turbo Spot Unlimited 6 6 Wystup 2,6 Hybrid FX - tuotteet Käytännölliset asiat Traders Rule of Thumb Tuen kustannukset Vannan ja Volgan havainnot Yhdenmukaisuuden tarkistus Ensimmäisen sukupolven lyhenteet Exotics Adjustment Factor Riskivahvistusten volatiliteetti , Perhoset ja teoreettinen hinnoittelu Barrier Options hinnoittelu Double Barrier Options hinnoittelu Double-No-Touch Options hinnoittelu Eurooppalainen tyyli vaihtoehtoja No-Touch todennäköisyys kauppakustannukset ja sen vaikutukset markkinoihin hintaan onetouch vaihtoehdoista Esimerkki lisäharjoituksista Harjoitukset tarjous kysy Spr eads One Touch Spreads Vanilla levittää leviämisen ensimmäisen sukupolven eksotikoille Minimaalinen tarjous Kysy spread-tarjouksia Kysy hinnoista Harjoitukset ratkaiseminen Black-Scholes - mallin todellinen Spot Cash Settlement toimitus Settlement Options kanssa viivästynyt toimitukset Harjoittavat viivästyneitä kiinnitys ilmoituksia Valuutta Fixing Euroopan keskuspankin mallin ja tuloksen analysointi Menettelytapa Virhearviointi EUR-USD-johtopäätöksen analyysi 7 Valuuttavaihtoehdot ja strukturoidut tuotteet 7 4 Suojauslaskenta IAS: n mukaan Johdanto Rahoitusinstrumentit Yleiskatsaus Yleiskatsaus Rahoitusvarat Rahoitusvelat Rahoitusvarojen ja rahoitusvelkojen eliminointi Oman pääoman ehtoiset instrumentit Yhdistetyt rahoitusinstrumentit Johdannaiset Sulautetut johdannaiset Rahoitusinstrumenttien luokittelu Rahoitusinstrumenttien arviointi Alkuperäinen kirjaaminen Alustavat arvot Myöhemmät arvot Tunnistamisen suojauslaskenta Yleiskatsaus Suojaustyypit Perusvaatimukset Testiin liittyvien suojauslaskentamenetelmien pysäyttäminen ng Suojausvaikutus Käyvän arvon suojaus Kassavirtasuojaus Suojauskoe tehokkuuden kannalta - Prosessin lisäpohjaisen simulointin tapaustutkimus ja valuuttakurssimuutosten laskeminen Forecaster Plus - arvon laskeminen Lähtökoron laskeminen Ennustamuunnoksen laskeminen n Arvo Dollar-Offset Ratio - Prospektiivinen testi Tehokkuuden varianssin vähentämistoimenpide - Tulevaisuuden tehokkuuden testi Regressioanalyysi - Tulevaisuuden tehokkuustesti Tulos Tehtävien tehokkuuskoe Päätelmät Merkittävät alkuperäiset lähteet laskentastandardeihin Harjoitukset 8 8 Wystup 5 Valuuttamarkkinat Markkinakatsaus GFI-konsernin (Fenics) lausunto, 25 Lokakuu Haastattelu ICY Softwaren kanssa, 14. lokakuuta Haastattelu Bloombergin kanssa, 12. lokakuuta Haastattelu Murexin kanssa 8. marraskuuta Haastattelu SuperDerivativen kanssa, 17. lokakuuta Haastattelu Lucht Probst Associatesin kanssa, 27. helmikuuta Ohjelmisto ja järjestelmävaatimukset Fenics Position Hinnoittelun pitäminen suorassa käsittelyssä Vastuuvapauslauseke Kaupankäynti ja myyntiehdotus rietary Kaupankäynti Liikevaihto Inter Bank Myyntiosasto Myynti Instituutio Myynti Yrityksen myynti Private Banking Listattu FX Options Kaupankäynti Lattia Joke 9 Luku 0 Esipuhe 0.1 Tämän kirjan soveltamisala Kansainvälisten yritysten rahoitusvarainhallinta edellyttää eri valuuttojen kassavirtojen käsittelyä. Siksi sijoituspankin luonnollinen palvelu koostuu erilaisista rahamarkkinavälineistä ja valuuttatuotteista. Tämä kirja kertoo suosituimmista tuotteista ja strategioista keskittyen kaikkiin vanilainvaihtoehtoihin. Se selittää kaikki FX-vaihtoehdot, yhteiset rakenteet ja räätälöidyt ratkaisut esimerkkeihin, joissa keskitytään erityisesti sovellukseen, jolla on näkemyksiä kauppiailta ja myynniltä sekä yritysasiakkaiden näkökulmasta. Se sisältää itse asiassa kauppaa käsitteleviä vastaavia motiiveja, jotka selittävät, miksi rakenteita on kaupattu. Näin lukija saa tunteen, kuinka rakentaa uusia rakenteita asiakkaiden tarpeiden mukaan. Harjoitukset on tarkoitettu käyttämään materiaalia. Useita niistä on todella vaikea ratkaista ja ne voivat toimia kannustimena lisätutkimuksille ja testauksille. Harjoitusten ratkaisut eivät ole osa tätä kirjaa, mutta ne julkaistaan ​​kirjan www-sivuilla. 0.2 Lukijalle asetettu edellytys on eräitä FX-markkinoiden perustietoja, kuten esimerkiksi Shami Shamah, Wiley 2003, ks. 90. Kohdeyleisön lukijat ovat jatko-opiskelijoita ja rahoitustekniikan ohjelmia käsittelevä tiedekunta, jotka voivat käyttää tätä kirjaa tekstikirjana kurssille, joka on nimeltään jäsennelty tuote tai eksoottinen valuuttavaihtoehto. 9 10 10 Wystup Traders, Trainee Structures, Product Developers, Myynti ja Quants kiinnostunut FX tuotevalikoimasta. Niille se voi toimia ideoiden lähteenä sekä referensseinä. Kirjojen hallinnointiin kiinnostuneiden yritysten rahastonhoitajat. Tämän kirjan avulla he voivat rakentaa itse ratkaisujaan. Kirjoittajat, jotka ovat kiinnostuneempia kvantitatiivisista ja mallinnusnäkökohdista, suosittelevat J. Hakala ja U. Wystup, Risk Publications, London, 2002, ks. 50. Tämä kirja kertoo useista eksoottisista FX-vaihtoehdoista, joissa keskitytään erityisesti taustalla oleviin malleja ja matematiikkaa, mutta ei sisällä rakenteita eikä yritysasiakkaita tai sijoittajia. 0.3 Tietoja tekijöistä Kuva 1: Uwe Wystup, kvantitatiivisen rahoituksen professori HfB: n talous - ja johtamiskoulussa Frankfurtissa, Saksassa. Uwe Wystup on myös MathFinance AG: n toimitusjohtaja, kvantitatiivisen rahoituksen kvantitatiivinen verkosto, Exotic Options advisory and Front Office Software Production. Aiemmin hän oli Commerzbankin FX Options Trading Team - yrityksen taloudellinen insinööri ja rakennuttaja. Ennen hän työskenteli Deutsche Bank, Citibank, UBS ja Sal. Oppenheim jr. amp Cie. Hän on MathFinance. de-sivuston ja MathFinance-uutiskirjeen perustaja ja ylläpitäjä. Uwe on PhD matematiikan rahoituksesta Carnegie Mellonin yliopistosta. Hän myös luennoi Goethe University Frankfurtin matemaattisesta rahoituksesta, järjestää Frankfurt MathFinance Colloquium - lehden ja on Frankfurt MathFinance - instituutin perustajajohtajana. Hän on antanut useita seminaareja eksoottisista vaihtoehdoista, laskennallisesta rahoituksesta ja volatiliteetin mallintamisesta. Hänen erikoistumisalueensa ovat ulkomaisten 11 FX Options ja Structured Products 11 - markkinoiden määrälliset näkökohdat ja strukturoitujen tuotteiden suunnittelu. Hän julkaisi kirjan valuuttakurssiriskistä ja artikkeleista Rahoitus ja Stohastics ja Journal of Derivatives. Uwe on antanut monia esityksiä molemmissa yliopistoissa ja pankkeissa ympäri maailmaa. Lisätietoja ansioluettelosta ja yksityiskohtaisesta julkaisuluettelosta on saatavilla osoitteessa 0,4. Kiitokset Haluan kiittää entisiä kollegojani kauppapuutarhasta, etenkin Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Roman Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala ja kaikki kollegani ja kirjailijat, erityisesti Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt ja Robert Tompkins. Chris Swain, Rachael Wilkie ja monet muut Wiley-julkaisut ansaitsevat kunnioitusta, koska he käsittelivät melko hidasta nopeutta tämän kirjan täyttämisessä. Nicole van de Locht ja Choon Peng Toh ansaitsevat mitalin vakavista yksityiskohtaisista todisteiden lukemisesta. 13 Luku 1 Valuutanvaihto Options FX Structured Products ovat räätälöityjä lineaarisia yhdistelmiä FX Options, mukaan lukien vanilja ja eksoottiset vaihtoehdot. Suosittelemme Shamah 90: n lähdekoodia oppimaan FX Marketsista, jossa keskitytään markkinoiden yleissopimuksiin, spot-, forward - ja swap-sopimuksiin, vanillavaihtoehtoihin. Eksoottisten FX-vaihtoehtojen hinnoittelusta ja mallinnuksesta suosittelemme Hakalan ja Wystup 50: n tai Lipton 71: n hyödyllisiä kumppaneita tähän kirjaan. Rakenteellisten tuotteiden markkinat rajoittuvat välttämättömien raaka-aineiden markkinoihin. Täten tyypillisesti on useimmiten strukturoituja tuotteita, joilla käydään kauppaa valuuttaparien välillä, jotka voidaan muodostaa USD: n, JPY: n, EUR: n, CHF: n, GBP: n, CAD: n ja AUD: n välillä. Tässä luvussa aloitamme lyhyen historian vaihtoehdoista, jota seuraa vanilavaihtoehtojen ja volatiilisuuden tekninen osa sekä käsittelemme yleisesti käytettyjä lineaarisia yhdistelmiä vaniljavaihtoehdoista. Sitten esitellään FX-rakenteisten tuotteiden tärkeimmät aineosat: ensimmäisen ja toisen sukupolven eksoottiset aineet. 1.1 Matka läpi optioiden historiaa Ensimmäisiä vaihtoehtoja ja futuureja vaihdettiin antiikin Kreikassa, kun oliiveja myydtiin ennen kuin ne olivat kypsyneet. Sen jälkeen markkinat kehittyivät seuraavalla tavalla. 1600-luvulta lähtien siitä lähtien, kun 1500-luvun tulivuoret, joita pidettiin eksoottisen ulkonäön vuoksi, kasvoivat Turkissa. Itävallan Wienin kuninkaallisen lääketieteellisen puutarhan päällikkö oli ensimmäinen turkkilaisten tulppaanien viljely onnistuneesti Euroopassa. Kun hän pakeni Hollantiin uskonnollisen vainon vuoksi, hän otti lamput pitkin. Alankomaiden Leidenin kasvitieteellisen puutarhan uudeksi johtajaksi hän viljelsi useita uusia kantoja. Näistä puutarhoista loistavat kauppiaat varastivat lamput markkinoimaan niitä, koska tulppaanit olivat loistava tilasymboli. 1700-luku Ensimmäiset tulppaanien futuurit myytiin vuonna 1634, 13-14 14 Wystup ostaa erityisiä tulppakantoja niiden sipulien painolla, sillä sipulit valitsivat saman arvon kuin kulta. Säännöllisen kaupankäynnin myötä keinottelijat tulivat markkinoille ja hinnat kiipesivät. Semper Octavian-kannan polttimon arvo oli kaksi vehnää, neljä kuormitusta ruista, neljä rasvatonta härkää, kahdeksan rasvaa sisältävää sikaa, kaksitoista rasvaa lampaita, kaksi viinirypäleitä, neljä tynnyriä olutta, kaksi tynnyriä voita, 1 000 kiloa juustoa , yksi aviopari, jossa on liinavaatteet ja suuri vaunu. Ihmiset jättivät perheensä, myivät kaiken omaisuutensa ja jopa lainattivat rahaa tulemaan tulipoli kauppiaille. Kun vuonna 1637 tämä väistämättä riskittömät markkinat kaatui, kauppiaat ja yksityishenkilöt joutuivat konkurssiin. Hallitus kieltäytyi spekulatiivisesta kaupankäynnistä ajanjakso tuli kuuluisa Tulipmania. 1700-luku 1728-luvulla Royal Caribbean Islands ja Afrikan rannikolla toimiva monopolisti Royal West-Indian ja Guinea Company antoivat ensimmäiset optio-oikeudet. Nämä olivat vaihtoehtoja Ranskan saaren Ste. Croix, johon oli suunniteltu sokerintuotantoa. Hanke toteutettiin vuonna 1733 ja paperiraaka-aineistoa myönnettiin Yhdessä varastossa, ihmiset ostivat suhteellisen osan saaresta ja arvoesineistä sekä yrityksen oikeudet ja oikeudet. 1800-luku Vuonna 1848 perustettiin 82 liikemiehiä Chicagon kauppakamari (CBOT). Tänään se on maailman suurimmat ja vanhin futuurimarkkinat. Useimmat kirjalliset asiakirjat menetettiin vuoden 1871 suuressa tulessa, mutta yleisesti uskottiin, että ensimmäiset standardoitavat futuurit käydään kauppaa CBOT: n kanssa, sillä nyt käydään kauppaa useilla futuureilla ja eteenpäin, ei vain T-joukkovelkakirjalainoilla ja valtiollisilla joukkovelkakirjalainoilla, vaan myös vaihtoehdoilla ja kulta. Vuonna 1870 New York Cotton Exchange perustettiin. Vuonna 1880 kultainen standardi otettiin käyttöön. 1900-luku Vuonna 1914 kulta-standardi hylättiin sodan takia. Vuonna 1919 Chicago Produce Exchange, joka vastaa maataloustuotteiden kaupasta, nimettiin Chicagon Mercantile Exchangeiksi. Tänään se on Eurodollarin, valuutanvaihto - ja karjanmerkin tärkeimpiä futuurimarkkinoita. Vuonna 1944 Bretton Woods - järjestelmä toteutettiin pyrkimyksenä vakauttaa valuuttajärjestelmä. Vuonna 1970 Bretton Woods System hylättiin useista syistä. Vuonna 1971 otettiin käyttöön Smithsonian sopimus kiinteistä valuuttakursseista. Vuonna 1972 Kansainvälinen valuuttarahasto (IMM) kävi kauppaa futuureja kolikoita, valuuttoja ja jalometallia vastaan. 15 FX Options ja Structured Products 15 21. vuosisata Vuonna 1973 CBOE (Chicago Board of Exchange) kävi ensimmäisen kerran kaupankäynnin kohteena vaihtoehtona neljä vuotta myöhemmin. Smithsonian sopimus hylättiin valuuttojen, joita seurasi hallittu kelluva. Vuonna 1975 CBOT myi ensimmäisen koron tulevaisuuden, ensimmäisenä tulevaisuudessa ilman todellista kohde-etuutta. Vuonna 1978 Hollannin osakemarkkinat vaihtoivat ensimmäiset vakioidut johdannaiset. Vuonna 1979 toteutettiin Euroopan valuuttajärjestelmä ja otettiin käyttöön Euroopan valuuttayksikkö (ECU). Vuonna 1991 allekirjoitettiin Maastrichtin sopimus yhteisestä rahasta ja talouspolitiikasta Euroopassa. Vuonna 1999 otettiin käyttöön euro, mutta maat käyttivät yhä käteistä vanhoja valuuttojaan ja valuuttakurssit pysyivät kiinteinä. Vuonna 2002 euro otettiin käyttöön rahana rahana. 1.2 Vanilla-optioihin liittyvät tekniset kysymykset Huomataan FOR-DOM: n (ulkomaiset) laskuttamassa taustalla olevassa valuuttakurssissa geometrinen Brownian motion ds t (rdrf) S t dt sigma t dw t (1.1), mikä tarkoittaa, että valuuttamääräiset kustannukset ovat kotimaisen valuutan FOR-DOM-yksiköitä. Jos kyseessä on EUR-USD, jossa on piste. tämä tarkoittaa, että yhden euron hinta on USD. Ulkomaisten ja kotimaisten käsite ei viittaa kaupankäynnin kohteena olevan yksikön sijaintiin, vaan ainoastaan ​​tähän lainausyleissopimukseen. Merkitsemme (jatkuva) ulkomaisen koron rf: llä ja (jatkuvalla) kotimaisella korolla r d: llä. Osakeselvityksessä rf edustaisi jatkuvaa osinkoa. Volatiliteettia merkitään sigmalla, ja Wt on vakio Brownian-liike. Näytteiden polut näkyvät kuvassa 1.1. Pidämme tätä standardimallia, ei siksi, että se heijastaa valuuttakurssin tilastollisia ominaisuuksia (itse asiassa, se ei t), vaan koska sitä käytetään laajasti käytännössä ja etupaneelissa ja toimii pääasiassa viestintävälineenä FX-vaihtoehdoissa . Nämä hinnat on yleensä mainittu volatiliteetin suhteen tämän mallin mukaisesti. Sovellettaessa Itocircin sääntöä ln S t: lle saadaan seuraava ratkaisu prosessille S t S t S 0 exp sigma2) t sigmaw t, (1.2), joka osoittaa, että S t on log-normaalisti ja tarkemmin sanottuna ln S t on normaali keskiarvolla ln S 0 (rdrf 1 2 sigma2) t ja varianssi sigma 2 t. Muut mallin oletukset ovat 16 16 Wystup Kuva 1.1: Simuloituja polkuja geometrisesta Brownian-liikkeestä. Paikan S T jakauma ajankohtana T on log-normaali. 1. Ei ole arbitraasia 2. Kaupankäynti on kitkatonta, ei transaktiokustannuksia 3. Jokainen asema voi olla milloin tahansa lyhyt, pitkä, mielivaltainen murto, ei likviditeettitarpeita Vanilin vaihtoehdon (eurooppalaisen putin tai puhelun) (1.3), jos sopimusparametrit ovat lakko K, vanhentumisaika T ja tyyppi phi, binaarimuuttuja, joka ottaa arvon 1 puhelun tapauksessa ja 1 jos kyseessä on laittaa. Symboli x merkitsee x: n positiivista osaa eli x max (0, x) 0 x Arvo Black-Scholes - mallissa maksun F arvo t hetkellä t jos piste on x: llä on merkitty v (t, x ) ja se voidaan laskea joko Black-Scholesin osittaisdifferentiaalisen 17 FX Options ja Structured Products 17: n ratkaisuna vtrdv (rdrf) xv x sigma2 x 2 v xx 0, (1,4) v (t, x) F. (1,5 ) tai vastaavasti (Feynman-Kac-Theorem) palkkatoiminnon diskontattuun odotettuun arvoon v (x, K, T, t, sigma, rd, rf, phi) er dtau IEF. (1.6) Tämä on syy siihen, miksi perusrahoitusjärjestelyt koskevat enimmäkseen osittaisdifferentiaaliyhtälöiden tai laskentatavan odotuksia (numeerinen integraatio). Tulos on Black-Scholes-kaava Lyhennä v (x, K, T, t, sigma, r d, rf, phi) phie r dtau fn (phid) KN (phid). (1.7) x: taustalla olevan tau T t: n nykyinen hinta T t: aika eräpäivään asti f IES T S tx xe (r d r f) tau. (t) 1 2pi e 1 2 t2 n (t) N (x) xn (t) N (x) xn (t) dt 1 N (x) Black-Scholes-kaava voidaan johtaa käyttämällä yhtälön (1.6) kiinteää esitystä käyttäen IEF: tä (STK) (er dtau phi xe (rdrf 1 2 sigma2) tausigma tauy K) n (y) dy. (1.8) Seuraavaksi on käsiteltävä positiivista osaa ja täytettävä neliö saadakseen Black-Scholes-kaavan. Osittaisdifferentiaaliyhtälöön perustuva derivaatio voidaan tehdä käyttämällä hyvin tutkittua lämpö-yhtälöä koskevia tuloksia. 18 18 Wystup Huomautus eteenpäin Terminen hinta f on lakko, joka tekee termiinin F S T f (1.9) nollan arvon nollaksi. Tästä seuraa, että f IES T xe (r d r f) T, eli termiinihinta on taustalla odotettu hinta hetkellä T riski-neutraalissa asennossa (geometrinen Brownian liike vastaa ajankohtaa r d r f). Tilannetta r d gt f f on nimeltään contango, ja tilannetta r d lt r f kutsutaan taaksepäin. Huomaa, että Black-Scholes - mallissa tuontihintojen käyrät ovat melko rajoitettuja. Esimerkiksi kausittaisia ​​vaikutuksia ei voi sisällyttää. Huomaa, että termiinikaasun arvo nollan jälkeen on yleensä erilainen kuin nolla ja koska yksi vastapuoli on aina lyhyt, voi olla lyhytpuolueen oletusriski. Futuurisopimus estää tämän vaarallisen tapauksen: se on pohjimmiltaan termiinisopimus, mutta vastapuolilla on oltava marginaalilaskuri sen varmistamiseksi, ettei käteisvarojen tai hyödykkeiden määrä ylitä määritettyä rajaa. Kreikkalaiset kreikkalaiset ovat arvon funktion johdannaisia ​​suhteessa malliin ja sopimusparametrit. Ne ovat tärkeä tieto elinkeinonharjoittajille, ja niistä on tullut etujärjestelmiin perustuvia vakiomuotoisia tietoja. Lisätietoa kreikkalaisista ja kreikkalaisten väliset suhteet esitetään Hakala ja Wystup 50 tai Reiss ja Wystup 84. Vanilla-vaihtoehdoissa luetellaan joitain niistä nyt. (Spot) Delta. v x phie rf tau N (phid) (1.10) Eteenpäin 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 Valuuttamääräiset optiot ja strukturoidut tuotteet 19 Nopeus. 3 v x 3 e r f tau n (d) x 2 sigma tau () d sigma tau 1 (1,14) Theta. (d) xsigma 2 tau phir fxe rf tau N (phid) r d Ke rdtau N (phid) (1.15) Kaulus. 2 v x tau phir f e r f tau N (phid) fie rf tau n (d) 2 (r d r f) tau d a sigma tau 2tausigma tau (1.16) Väri. 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 rf tau taun (d) (1,18) Volga. 2 v sigma 2 xe rf tau taun (d) d d sigma (1.19) Volgaa kutsutaan joskus myös nimellä vomma tai vulgamma. 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 rf 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) Identiteetit Put-call-pariteetti on suhde 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) Ke rdtau n (d). (X, K, T, t, sigma, rd, rf, 1) v (x, f, f) K, T, t, sigma, rd, rf, 1) xe rf tau Ke r dtau, (1.32), joka on vain monimutkaisempi tapa kirjoittaa triviaali yhtälö xx x. Put-call-delta-pariteetti on v (x, K, T, t, sigma, r d, rf, 1) x v (x, K, T, t, sigma, rd, rf, 1) x e rf tau. (1.33) Erityisesti opimme, että put delta ja call delta absoluuttinen arvo eivät täsmää täsmälleen yhteen, vaan vain positiiviseen numeroon e r f tau. Ne lisäävät yhteen suunnilleen, jos joko tau: n aika on lyhyt tai jos ulkomainen korko r f on lähellä nollaa. 21 Valinta K f tuottaa identtiset arvot puhelulle ja laittamiselle, etsimme deltasymmetristä lakkoa, joka tuottaa aivan identtisiä delta-arvoja (spot, forward tai driftless). Tämä edellytys merkitsee d0 ja siten fe sigma2 2 T, (1.34), jolloin absoluuttinen delta on erf tau 2. Erityisesti opimme, että aina gt f, t voi olla t ja putki, joilla on identtiset arvot ja deltat. Huomaa, että lakko valitaan tavallisesti keskimmäisenä iskeenä, kun kauppaa haaksirikkoutuu tai perhonen. Vastaavasti dual-delta-symmetrinen lakko circK fe sigma2 2 T voidaan johtaa ehdosta d Homogeenisyysperusteiset suhteet Haluamme haluta mitata taustalla olevan arvon eri yksiköissä. Tämä vaikuttaa tietenkin hinnoittelukehitykseen seuraavasti. (1.35) Molempien puolien erottaminen toisistaan ​​suhteessa (av, x, k, t, t, sigma, rd, rf, phi) v (ax, ak, T, t, sigma, rd, rf, phi) (1.7) ja (1.36) x ja K kertoimien vertailu johtaa delta vx: n ja dual delta v: n kantoihin. Tämä avaruus - homogeenisuus on syy delta-kaavojen yksinkertaisuuden kannalta, jonka tylsi laskenta voidaan säästää tällä tavoin. (X, K, t, t, sigma, rd, rf, phi) v (x, K, T a, ta, asigma, ar d , ar f, phi) kaikille nollakulmalle 0. (1.37) Molempien puolien erottaminen suhteessa 1: een ja asettamalla sitten 1: n saadaan 0 tauv t sigmav sigma rdv rd rfv rf. (1.38) Tietenkin tämä voidaan myös todentaa suoralla laskennalla. Tällaisten yhtälöiden yleinen käyttö on luoda kaksinkertaisen tarkkailun vertailuarvoja laskettaessa kreikkalaisia. Nämä homogeenisuusmenetelmät voidaan helposti laajentaa koskemaan muita monimutkaisempia vaihtoehtoja. Put-call-symmetrialla ymmärrämme suhde (ks. 6, 7,16 ja 19) v (x, K, T, t, sigma, rd, rf, sigma, rd, rf, 1). (1.39) 22 22 Wystup Laitteen lakko ja puhelun lakko tuottavat geometrisen keskiarvon, joka on yhtä suuri kuin eteenpäin f. Etenemistä voidaan tulkita geometriseksi peiliksi, joka heijastaa puhelun tiettyyn määrään laitetta. Huomaa, että rahanvaihto-optioilla (K f) put-puhelun symmetria on sama kuin puhelun pariteetin erityinen tapaus, jossa puhelulla ja matkalla on sama arvo. Suora laskenta osoittaa, että tasojen symmetria v v tauv (1.40) r d r f pätee vaniljavaihtoehtoihin. Tämä suhde pätee itse asiassa kaikkiin eurooppalaisiin vaihtoehdoihin ja laajaan luokkaan, jotka ovat riippuvaisia ​​polusta riippuvista vaihtoehdoista, kuten on esitetty kohdassa 84. Suhteesta voidaan suoraan todeta ulkomaisen kotimaisen symmetrian 1 xv (x, K, T, t, sigma, rd , rf, phi) Kv (1 x, 1 K, T, t, sigma, rf, rd, phi). (1.41) Tätä tasa-arvoa voidaan pitää yhdeksi put-call-symmetrian kasvoista. Syy on, että vaihtoehdon arvo voidaan laskea sekä kotimaisessa että ulkomaisessa skenaariossa. Pidämme esimerkkiä S euron kurssin mallintamisesta. New Yorkissa puheluvaihtoehto (STK) maksaa v (x, K, T, t, sigma, r usd, r eur, 1) USD ja siten v (x, k, t, t, sigma, r usd, r eur, 1) x () 1. Euroa. Tätä EUR-call-vaihtoehtoa voidaan pitää myös USD-put-optiossa, jonka voitto K 1 KST Tämä vaihtoehto maksaa Kv (1, 1, T, t, sigma, rx K eur, r usd, 1) t ja 1 S t ovat samaa haihtuvuutta. Tietenkin New Yorkin arvo ja Frankfurtin arvo on sovittava, mikä johtaa (1.41). Opimme myös myöhemmin, että tämä symmetria on vain yksi mahdollinen tulos numeerisen muutoksen perusteella Kohdeajankohdan kurssikehitys (1.1) on valuuttakurssin malli. Tarina on pysyvästi sekava asia, joten selvenkäämme tämän täällä. Valuuttakurssi kertoo, kuinka paljon kotimaista valuuttaa tarvitaan ostamaan yksi valuuttayksikkö. Jos esimerkiksi otetaan EURUSD valuuttakurssiksi, oletushinta on EUR-USD, jossa USD on kotimainen valuutta ja EUR on valuutta. Kotimainen termi ei millään tavoin liity kauppiaan tai maan sijaintiin. Se merkitsee vain numeerista valuuttaa. Termit kotimainen, numeerinen tai perusvaluutta ovat synonyymeja, jotka ovat ulkomaisia ​​ja taustalla olevia. Koko tämän kirjan yhteydessä merkitään raja () valuuttaparin ja viivan (-) avulla. Slash () ei tarkoita jakamista. Esimerkiksi EURUSD voi olla noteerattu joko EUR-USD: ssa, mikä tarkoittaa sitä, kuinka monta dollaria tarvitaan yhden euron tai USD-EUR: n ostamiseen, mikä tarkoittaa sitä, kuinka monta euroa tarvitaan yhden dollarin ostamiseen. Taulukossa 1.1 on lueteltu tiettyjä markkinatilanteen mukaisia ​​noteerauksia. 23 FX Options ja Structured Products 23 valuutan parin oletushinnasto näytesarja GBPUSD EUR-USD GBPCHF EUR-GBP EURJPY EUR-JPY EURCHF EUR-CHF USDJPY USD-JPY USDCHF USD-CHF Taulukko 1.1: Vakiomarkkinat suurimpien valuuttaparien noteeraus näytteen spot-hintoilla Kauppa Lattia Kieli Me kutsumme miljoona dollaria, miljardi pihaa. Tämä johtuu siitä, että miljardia kutsutaan milliardiksi ranskaksi, saksaksi ja muille kielille. Ison-Britannian punta pidetään usein miljoona. Tietyillä valuuttaparilla on nimiä. Esimerkiksi GBPUSD-nimitystä kutsutaan kaapeliksi, koska valuuttakurssitietoja käytetään lähettämään kaapelin kautta Atlantin valtamerellä Amerikan ja Englannin välillä. EURJPY: tä kutsutaan ristiksi, koska se on ristikkäisempää likviditeetteisempää USDJPY: tä ja EURUSD: tä. Tietyillä valuutoilla on myös nimiä, esim. Uuden-Seelannin dollaria NZD kutsutaan kiiviksi, Australian dollaria AUD on nimeltään Aussie, Skandinavian valuutat DKR, NOK ja SEK kutsutaan Scandiesiksi. Valuuttakurssit ovat yleensä viiteen viiteen numeroon, esim. EUR-USD: ssa voisimme tarkkailla viimeistä numeroa 5, jota kutsutaan pipiksi. Keskimmäistä numeroa 3 kutsutaan suureksi, koska valuuttakursseja näytetään usein kaupankäynnin lattioissa ja isossa kuvassa näkyvässä suuressa kuvassa, on tärkein tieto. Numerot, jotka on jätetty suurelle hahmolle, ovat joka tapauksessa tunnettuja, suuren kuvion pipsit ovat usein vähäpätöisiä. Jotta selvennettäisiin, USD-JPY nousu 20 pistettä ja nousu kahdella suurella luvulla on vaihtoehtopisteen hintatarjous. Vanillavaihtoehtojen arvot ja hinnat voidaan mainita taulukossa 1.2 esitetyllä kuudella tavalla. 24 24 Wystup-nimimerkin arvo esimerkin yksiköissä kotimainen käteisvaro DOM 29 188 USD ulkomaiset käteisvarat f 24 245 euroa kotimaan d DOM per yksikkö DOM USD ulkomaiset f FOR per yksikkö FOR EUR kotimainen pipsi d pipsi DOM per yksikkö FOR USD pipsiä per EUR ulkomaiset pipot f pipit FOR per DOM EUR pips per USD Taulukko 1.2: Optioiden arvojen standardimarkkinatyyppityypit. Esimerkissä otamme FOREUR, DOMUSD, S 0. r d 3,0, rf 2,5, sigma 10, K. T 1 vuosi, phi 1 (call), nimellinen 1 000, 000 EUR 1, 250, 000 USD. Pipsin osalta noteeraus USD pipsiä euroina ilmoitetaan joskus myös USD: na 1 euroksi. Samoin euromääräisiä dollareita voidaan noteeraa myös euroina 1 Yhdysvaltain dollareina. Black-Scholes-kaava lainaa d-pipoja. Muut voidaan laskea käyttäen seuraavia ohjeita. d pips 1 S 0 S 0 1 f K S d 0 S fips 0 K dips (1.42) Delta - ja Premium-yleissopimus Eurooppalaisen vaihtoehdon ilman palkkioita on tunnettua. Sitä kutsutaan raaka-aineeksi delta delta - raaka-aineeksi nyt. Se voidaan mainita jommassa kummassa kahdessa kyseessä olevassa valuutassa. Suhde on delta-käänteinen raaka-delta-raaka S K. (1.43) Deltaa käytetään ostamaan tai myymään pistettä vastaavassa summassa suojaamaan vaihtoehtoa ensimmäiseen tilaukseen saakka. Johdonmukaisuuden vuoksi palkkio on sisällytettävä delta-suojaukseen, koska ulkomaisessa valuutassa oleva palkkio suojaa jo osa optiokurssin riskeistä. Jotta tämä olisi selkeä, harkitse EUR-USDa. Tavallisessa arbitraasiteemissa v (x) tarkoittaa optio-ohjelman arvoa tai palkkioa, joka on 1 euron nimellisarvo, jos piste on x: llä, ja raaka delta vx tarkoittaa delta-suojauksen euromääräistä määrää. Siksi xv x on USD: n myyntiarvo. Jos nyt palkkio maksetaan euroina eikä dollareina, niin meillä on jo vx eu, ja euromääräistä euromääräistä summaa on vähennettävä tällä summalla eli jos euron hinta on valuuttamääräinen, meidän on ostettava vxvx EUR Delta-suojaus tai vastaava myydä xv xv USD. 25 FX-optiot ja strukturoidut tuotteet 25 Koko FX-lainauskerran tarina tulee yleisesti sotku, koska meidän on ensin selvitettävä, mikä valuutta on kotimainen, mikä on ulkomaista, mikä on opinnäytetyön valuuttamääräinen valuuttavaluutta ja mikä on palkkiovaluutta. Valitettavasti tämä ei ole symmetrinen, koska vastapuolella saattaa olla toinen käsitys kotivaluutasta tietyn valuuttaparin osalta. Tästä syystä ammattimaisilla pankkien välisillä markkinoilla on yksi käsitys delta per currency pair. Normaalisti se on Fenics-näytön vasemmanpuoleinen delta, jos vaihtoehtoa vaihdetaan vasemmanpuoleisella palkkiolla, joka on tavallisesti normaali ja oikeanpuoleinen delta, jos sitä käydään oikealla kädellä. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs jne. Koska OTM-vaihtoehdoista käydään kauppaa suurimman osan ajasta, ero ei ole valtava eikä siten luo valtavaa spot-riskiä. Lisäksi Fenicsin tavanomaisen delta per valuutan parin vasemmanpuoleisen delta Fenicsin tapauksessa useimmiten käytetään lainaamaan volatiliteetin vaihtoehtoja. Tämä on määritettävä valuuttana. Tämä tavanomainen pankkien välinen käsitys on mukautettava pankin todelliseen delta-riskiin automaattisen kaupankäyntijärjestelmän osalta. Valuutoissa, joissa pankin riskittömät valuutat ovat valuutan perusvaluutta, on ilmeistä, että delta on vaihtoehdon raaka delta ja riskialttiille palkkioille tämä palkkio on sisällytettävä. 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. Arvo. (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 . Euroa. 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. Arvo. (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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