Fx Optionen Und Strukturierte Produkte Pdf

FX Optionen und strukturierte Produkte 1 FX Optionen und strukturierte Produkte Uwe Wystup 7. April 2006 3 Inhalt 0 Vorwort Geltungsbereich dieses Buches Die Leserschaft Über den Autor Danksagungen Auslandsaustausch Optionen Eine Reise durch die Geschichte der Optionen Technische Probleme für Vanilla Optionen Wert A Hinweis auf die Vorwärts Griechen Identitäten Homogenität basierte Beziehungen Zitat Streik in Delta-Volatilität in Delta-Volatilität und Delta für einen geteilten Streik Griechen in Deltas-Volatilität Volatilität Historische Volatilität Historische Korrelation Volatilität Lächeln At-the-Money Volatilität Interpolation Volatilität Lächeln Konventionen At-The - Geld Definition Interpolation der Volatilität auf Fälligkeit Säulen Interpolation der Volatilität Spread zwischen Fälligkeit Säulen Volatilität Quellen Volatilität Kegel Stochastische Volatilität 4 4 Wystup Übungen Grundlegende Strategien, die Vanilla-Optionen enthalten Call and Put Spread Risk Reversal Risiko Reversal Flip Straddle Strangle Schmetterling Seemöwe Übungen Erste Generation Exotik Barriere Optionen Digitale Optionen, Touch-Optionen und Rabatte Compound und Raten Asiatische Optionen Lookback-Optionen Vorwärts Start-, Ratschen - und Cliquet-Optionen Power-Optionen Quanto-Optionen Übungen Zweite Generation Exotik Korridore Faders Exotische Barrier-Optionen Pay-Later Optionen Step-up und Step-down Optionen Spread und Exchange-Optionen Baskets Best-of-und Worst-of-Optionen Optionen und Forwards auf die harmonische durchschnittliche Abweichung und Volatilität Swaps Übungen Strukturierte Produkte Weiterleiten von Produkten Weiterleiten Vorwärts Teilnahme Vorwärts Einblendung Vorwärts Knock-Out Vorwärts Hai Vorwärts Fader Haifisch Vorwärts 5 FX Optionen und strukturierte Produkte Schmetterling Vorwärts Bereich Forward Range Accrual Forward Accumulative Forward Boomerang Forward Amortizing Forward Auto-Erneuerung Vorwärts Double Shark Vorwärts Vorwärts Start Chooser Forward Freier Style Forward Boosted SpotForward Time Option Übungen Serie von Strategien Shark Forward Series Kragen Extra Serie Übungen Einzahlungen und Darlehen Dual Währung DepositLoan Performance Linked Einlagen Tunnel DepositLoan Korridor DepositLoan Turbo DepositLoan Tower DepositLoan Übungen Zinssatz und Cross Currency Swaps Cross Currency Swap Hanseatic Swap Turbo Cross Währung Swap Gepuffert Cross Currency Swap Flip Swap Korridor Swap Double-No-Touch verknüpft Swap Range Reset Swap Basket Spread Swap Übungen Teilnahme Anmerkungen Gold Teilnahme Hinweis Warenkorb Verknüpfte Anmerkung Emittent Swap Moving Strike Turbo Spot Unlimited 6 6 Wystup 2.6 Hybrid FX Produkte Praktische Angelegenheiten Die Trader Daumennummer Kosten für Vanna und Volga Beobachtungen Konsistenzprüfung Abkürzungen für Erstausfuhr Exotik Anpassungsfaktor Volatilität für Risikoumkehrungen, Schmetterlinge und Theoretische Wertschätzung Barriere Optionen Pricing Double Barrier Optionen Preise Double-No-Touch Optionen Preise European Style Optionen No-Touch Wahrscheinlichkeit Die Kosten des Handels und seine Auswirkungen auf den Markt Preis von Onetouch Optionen Beispiel Weitere Anwendungen Übungen Bid Ask Spreads One Touch Spreads Vanilla Spreads Spreads für First Generation Exotik Minimal Bid Ask Spread Bid Ask Preise Übungen Abrechnung Das Black-Scholes-Modell für die Tatsächliche Spot Barausgleichsabwicklung Optionen mit aufgeschobenen Auslieferungsübungen auf die Kosten der verspäteten Fixing Ankündigungen Die Währung Fixierung der europäischen Zentralbank Modell und Auszahlung Analyse Prozedur Fehler Schätzung Analyse von EUR-USD Fazit 7 FX Optionen und strukturierte Produkte 7 4 Hedge Accounting unter IAS Einführung Finanzinstrumente Überblick Allgemeine Definition Finanzielle Vermögenswerte Finanzielle Verbindlichkeiten Verrechnung von finanziellen Vermögenswerten und finanziellen Verbindlichkeiten Eigenkapitalinstrumente zusammengesetzte Finanzinstrumente Derivate Eingebettete Derivate Klassifizierung von Finanzinstrumenten Bewertung von Finanzinstrumenten Instrumente Anfangserkennung Initiale Messung Nachfolgende Messung Ausbuchung Hedge Accounting Übersicht Arten von Hedges Grundlegende Anforderungen Stoppen Hedge Accounting Methoden zur Prüfung Hedge-Effektivität Fair Value Hedge Cash Flow Hedge Testing für Effektivität - Eine Fallstudie der Forward Plus Simulation von Wechselkursen Berechnung der Forward Plus Wertberechnung der Forward-Rate Berechnung der Prognose-Transaktion s Wert-Dollar-Offset-Verhältnis - Prospektiver Test für Effektivitätsabweichung Reduktionsmaßnahme - Prospektiver Test für Effektivitäts-Regressionsanalyse - Prospektiver Test für Effektivitätsergebnis Retrospektiver Test für Effektivitäts-Schlussfolgerung Relevante Originalquellen für Rechnungslegungsstandards Übungen 8 8 Wystup 5 Devisenmärkte Eine Tour durch die Markterklärung der GFI Group (Fenics), 25. Oktober Interview mit ICY Software, 14. Oktober Interview mit Bloomberg, 12. Oktober Interview mit Murex, 8. November Interview mit SuperDerivativen, 17. Oktober Interview mit Lucht Probst Associates, 27. Februar Software - und Systemanforderungen Fenics Position Halten Preisgestaltung Straight Through Processing Disclaimer Handel und Vertrieb Eigenhandel Trading Sales-Driven Trading Inter Bank Vertrieb Filiale Vertrieb Institutionelle Verkäufe Corporate Sales Private Banking Listing FX Optionen Trading Floor Witz 9 Kapitel 0 Vorwort 0.1 Geltungsbereich von Dieses Buch Treasury Management von internationalen Konzernen beinhaltet den Umgang mit Cash Flows in verschiedenen Währungen. Daher besteht der natürliche Service einer Investmentbank aus einer Vielzahl von Geldmarkt - und Devisenprodukten. Dieses Buch erklärt die beliebtesten Produkte und Strategien mit einem Fokus auf alles über Vanille-Optionen. Es erklärt alle FX-Optionen, gemeinsame Strukturen und maßgeschneiderte Lösungen in Beispielen mit besonderem Fokus auf die Applikation mit Blick auf Händler und Vertrieb sowie aus Kundensicht. Es enthält tatsächlich gehandelte Geschäfte mit entsprechenden Motivationen, die erklären, warum die Strukturen gehandelt wurden. Auf diese Weise bekommt der Leser das Gefühl, wie man neue Strukturen für die Bedürfnisse der Kunden baut. Die Übungen sollen das Material üben. Mehrere von ihnen sind eigentlich schwer zu lösen und können als Anreize für weitere Forschung und Prüfung dienen. Lösungen für die Übungen sind nicht Teil dieses Buches, aber sie werden auf der Webseite des Buches veröffentlicht werden, 0.2 Die Leser Voraussetzung ist einige grundlegende Kenntnisse der Devisenmärkte, wie zum Beispiel aus dem Buch Foreign Exchange Primer von Shami Shamah, Wiley genommen 2003, siehe 90. Die Zielleser sind Graduate Students und Fakultät für Financial Engineering Programme, die dieses Buch als Lehrbuch für einen Kurs namens strukturierte Produkte oder exotische Währungsoptionen verwenden können. 9 10 10 Wystup Trader, Trainee Structurers, Produktentwickler, Vertrieb und Quants mit Interesse an der FX Produktlinie. Für sie kann es als Ideenquelle und als Referenzführer dienen. Schatzmeister von Unternehmen, die sich für die Verwaltung ihrer Bücher interessieren. Mit diesem Buch können sie ihre Lösungen selbst strukturieren. Die Leser, die sich für die quantitativen und modellierenden Aspekte interessieren, werden empfohlen, um das Devisenrisiko von J. Hakala und U. Wystup, Risk Publications, London, 2002, siehe 50 zu lesen. Dieses Buch erklärt einige exotische FX-Optionen mit besonderem Schwerpunkt auf dem Basiswert Modelle und Mathematik, enthält aber keine Strukturen oder Firmenkunden oder Investoren. 0.3 Über den Autor Abbildung 1: Uwe Wystup, Professor für quantitative Finanzierung an der HfB Business School of Finance und Management in Frankfurt am Main. Uwe Wystup ist zudem CEO der MathFinance AG, einem globalen Netzwerk von Quants, spezialisiert auf Quantitative Finance, Exotische Optionsberatung und Front Office Software Production. Zuvor war er als Finanzingenieur und Structurer im FX Options Trading Team bei der Commerzbank tätig. Davor arbeitete er für die Deutsche Bank, die Citibank, UBS und Sal. Oppenheim jr. Amp Cie. Er ist Gründer und Manager der Website MathFinance. de und der MathFinance Newsletter. Uwe hält einen Doktortitel in der mathematischen Finanzierung von der Carnegie Mellon University. Er referiert auch über die mathematische Finanzierung der Goethe-Universität Frankfurt, organisiert das Frankfurter MathFinance-Kolloquium und ist Gründungsdirektor des Frankfurter MathFinance Instituts. Er hat mehrere Seminare zu exotischen Optionen, Computational Finance und Volatilitätsmodellierung gegeben. Sein Spezialisierungsgebiet sind die quantitativen Aspekte und die Gestaltung von strukturierten Produkten ausländischer 11 Devisenmärkte und strukturierte Produkte 11 Börsenmärkte. Er veröffentlichte ein Buch über Devisenrisiken und Artikel in Finanzen und Stochastik und das Journal of Derivatives. Uwe hat viele Vorträge an Universitäten und Banken rund um die Welt gegeben. Weitere Informationen zu seinem Lebenslauf und einer ausführlichen Publikationsliste finden Sie unter 0.4 Danksagungen Ich möchte mich bei meinen ehemaligen Kollegen auf dem Trading Floor bedanken, vor allem Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Roman Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala und alle meine Kollegen und Co-Autoren, besonders Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt und Robert Tompkins. Chris Swain, Rachael Wilkie und viele andere von Wiley Publikationen verdienen Respekt, wie sie mit meiner ziemlich langsamen Geschwindigkeit zu tun hatten, um dieses Buch zu vervollständigen. Nicole van de Locht und Choon Peng Toh verdienen eine Medaille für ernsthafte detaillierte Beweis lesen. 13 Kapitel 1 Devisenoptionen FX Structured Products sind maßgeschneiderte Linearkombinationen von FX-Optionen inklusive Vanille und exotischen Optionen. Wir empfehlen das Buch von Shamah 90 als Quelle, um über FX Markets mit einem Fokus auf Marktkonventionen, Spot-, Forward - und Swap-Verträge, Vanille-Optionen zu lernen. Für die Preisgestaltung und Modellierung von exotischen FX-Optionen empfehlen wir Hakala und Wystup 50 oder Lipton 71 als nützliche Begleiter zu diesem Buch. Der Markt für strukturierte Produkte ist auf den Markt der notwendigen Zutaten beschränkt. Daher gibt es in der Regel meist strukturierte Produkte, die die Währungspaare gehandelt werden, die zwischen USD, JPY, EUR, CHF, GBP, CAD und AUD gebildet werden können. In diesem Kapitel beginnen wir mit einer kurzen Geschichte der Optionen, gefolgt von einem technischen Abschnitt über Vanille-Optionen und Volatilität, und befassen sich mit gängigen linearen Kombinationen von Vanille-Optionen. Dann werden wir die wichtigsten Zutaten für FX strukturierte Produkte veranschaulichen: die Exotik der ersten und zweiten Generation. 1.1 Eine Reise durch die Geschichte der Optionen Die ersten Optionen und Futures wurden im alten Griechenland gehandelt, als Oliven verkauft wurden, bevor sie Reife erreicht hatten. Danach entwickelte sich der Markt auf folgende Weise. 16. Jahrhundert Seit dem 15. Jahrhundert wurden Tulpen, die für ihr exotisches Aussehen gemocht wurden, in der Türkei angebaut. Der Chef der königlichen medizinischen Gärten in Wien, Österreich, war der erste, der diese türkischen Tulpen erfolgreich in Europa kultivierte. Als er wegen der religiösen Verfolgung nach Holland floh, nahm er die Glühbirnen mit. Als neuer Leiter der botanischen Gärten von Leiden, Niederlande, kultivierte er mehrere neue Stämme. Es war von diesen Gärten, dass geizige Händler die Glühbirnen gestohlen haben, um sie zu vermarkten, weil Tulpen ein großartiges Statussymbol waren. 17. Jahrhundert Die ersten Futures auf Tulpen wurden im Jahre 1634 gehandelt, die Leute konnten mit dem Gewicht ihrer Glühbirnen spezielle Tulpenstämme kaufen, für die Glühbirnen wurde der gleiche Wert wie für Gold gewählt. Zusammen mit dem regulären Handel traten die Spekulanten in den Markt ein und die Preise stiegen. Eine Birne der Sämaschine Semper Octavian war zwei Wagenladungen von Weizen, vier Laugen Roggen, vier fette Ochsen, acht Fettschweine, zwölf fette Schafe, zwei Hogsheads Wein, vier Fässer Bier, zwei Fässer Butter, 1.000 Pfund Käse , Ein Ehebett mit Leinen und einem großen Wagen. Die Leute verließen ihre Familien, verkauften alle ihre Sachen und gaben sogar Geld, um Tulpenhändler zu werden. Als im Jahre 1637 dieser angeblich risikofreie Markt abgestürzt war, gingen Händler und Privatpersonen in Konkurs. Die Regierung verboten spekulativen Handel die Zeit wurde berühmt als Tulipmania. 18. Jahrhundert Im Jahre 1728 veröffentlichte die Royal West-Indian und Guinea Company, der Monopolist im Handel mit den Karibischen Inseln und der afrikanischen Küste die ersten Aktienoptionen. Das waren Optionen auf den Kauf der französischen Insel Ste. Croix, auf dem Zuckerpflanzungen geplant waren. Das Projekt wurde im Jahre 1733 realisiert und Papierbestände wurden ausgegeben. Zusammen mit der Aktie kauften die Menschen einen relativen Anteil an der Insel und den Wertsachen sowie die Privilegien und die Rechte des Unternehmens. 19. Jahrhundert Im Jahre 1848 gründeten 82 Geschäftsleute das Chicago Board of Trade (CBOT). Heute ist es der größte und älteste Futures-Markt der Welt. Die meisten schriftlichen Dokumente waren im großen Feuer von 1871 verloren, aber es wird allgemein geglaubt, dass die ersten standardisierten Futures von CBOT gehandelt wurden, handeln mehrere Futures und Forwards, nicht nur T-Bonds und Treasury-Anleihen, sondern auch Optionen und Gold. 1870 wurde die New York Cotton Exchange gegründet. Im Jahre 1880 wurde der Goldstandard eingeführt. 20. Jahrhundert 1914 wurde der Goldstandard wegen des Krieges aufgegeben. Im Jahr 1919 wurde die Chicago Produce Exchange, verantwortlich für den Handel mit landwirtschaftlichen Produkten umbenannt in Chicago Mercantile Exchange. Heute ist es der wichtigste Futures-Markt für Eurodollar, Devisen und Viehbestand. 1944 wurde das Bretton-Woods-System in einem Versuch eingesetzt, das Währungssystem zu stabilisieren. 1970 wurde das Bretton-Woods-System aus mehreren Gründen aufgegeben. 1971 wurde das Smithsonian-Abkommen über feste Wechselkurse eingeführt. 1972 hat der Internationale Währungsmarkt (IMM) Futures auf Münzen, Währungen und Edelmetall gehandelt. 15 FX Optionen und strukturierte Produkte 15 21. Jahrhundert 1973 hat die CBOE (Chicago Board of Exchange) erstmals vier Jahre später Call-Optionen gehandelt. Die Smithsonian-Vereinbarung wurde aufgegeben die Währungen folgte verwaltet schwimmenden. Im Jahr 1975 verkaufte die CBOT die erste Zinsfazilität, die erste Zukunft ohne realen Basiswert. 1978 hat die niederländische Börse die ersten standardisierten Finanzderivate gehandelt. 1979 wurde das Europäische Währungssystem eingeführt und die Europäische Währungseinheit (ECU) eingeführt. 1991 wurde der Maastrichter Vertrag über eine gemeinsame Währung und Wirtschaftspolitik in Europa unterzeichnet. Im Jahr 1999 wurde der Euro eingeführt, aber die Länder nutzten immer noch Bargeld ihrer alten Währungen, während die Wechselkurse festgesetzt wurden. Im Jahr 2002 wurde der Euro als neues Geld in Form von Bargeld eingeführt. 1.2 Technische Probleme für Vanilla-Optionen Wir betrachten das Modell geometrische Brownsche Bewegung ds t (rdrf) S t dt sigmas t dw t (1.1) für den zugrunde liegenden Wechselkurs, der im FOR-DOM (fremdländisch) angegeben ist, was bedeutet, dass eine Einheit von Die Fremdwährungskosten der inländischen Währung. Im Falle von EUR-USD mit einer Stelle von. Dies bedeutet, dass der Preis von einem EUR USD ist. Der Begriff der ausländischen und inländischen nicht verweisen die Lage der Handelseinheit, sondern nur auf diese Zitat-Konvention. Wir bezeichnen den (kontinuierlichen) Fremdzins um r f und den (kontinuierlichen) Inlandszins um r d. In einem Equity-Szenario würde rf eine kontinuierliche Dividendenrate darstellen. Die Volatilität ist mit Sigma bezeichnet, und W t ist eine Standard-Brownsche Bewegung. Die Beispielpfade sind in Abbildung 1.1 dargestellt. Wir betrachten dieses Standardmodell, nicht weil es die statistischen Eigenschaften des Wechselkurses widerspiegelt (in der Tat ist es nicht), sondern weil es in der Praxis und in den Frontbürosystemen weit verbreitet ist und hauptsächlich als Werkzeug dient, um die Preise in FX-Optionen zu vermitteln . Diese Preise sind in der Regel in Bezug auf die Volatilität im Sinne dieses Modells zitiert. Die Anwendung von Itocirc s-Regel auf ln S t ergibt die folgende Lösung für den Prozeß S t S t S 0 exp sigma2) t sigmaw t, (1.2), die zeigt, daß S t log-normal verteilt ist, genauer gesagt ln S t ist normal Mit Mittelwert ln S 0 (rdrf 1 2 sigma2) t und Varianz Sigma 2 t. Weitere Modellannahmen sind 16 16 Wystup Abbildung 1.1: Simulierte Wege einer geometrischen Brownschen Bewegung. Die Verteilung der Stelle S T zum Zeitpunkt T ist log-normal. 1. Es gibt keine Arbitrage 2. Der Handel ist reibungslos, keine Transaktionskosten 3. Jede Position kann jederzeit getroffen werden, kurzer, langer, willkürlicher Bruch, keine Liquiditätsbeschränkungen Die Auszahlung für eine Vanille-Option (European put or call) ist gegeben (1.3), wobei die Vertragsparameter der Streik K, die Ablaufzeit T und der Typ phi, eine binäre Variable, die im Falle eines Aufrufs den Wert 1 annimmt, und 1 bei a stellen. Das Symbol x bezeichnet den positiven Teil von x, dh x max (0, x) 0 x Wert Im Schwarz-Scholes-Modell wird der Wert der Auszahlung F zum Zeitpunkt t, wenn der Fleck bei x ist, mit v (t, x ) Und kann entweder als Lösung des Black-Scholes-Teildifferentials berechnet werden 17 FX-Optionen und Strukturierte Produkte 17 Gleichung vtrdv (rdrf) xv x sigma2 x 2 v xx 0, (1.4) v (t, x) F. (1.5 ) Oder äquivalent (Feynman-Kac-Theorem) als den ermäßigten Erwartungswert der Auszahlungsfunktion v (x, K, T, t, Sigma, rd, rf, phi) er dtau IEF. (1.6) Dies ist der Grund, warum die grundlegende Finanzplanung vor allem mit der Lösung partieller Differentialgleichungen oder Computing-Erwartungen (numerische Integration) beschäftigt ist. Das Ergebnis ist die Black-Scholes-Formel Wir verkürzen v (x, K, T, t, Sigma, r d, r f, phi) phie r dtau fn (Phid) KN (Phid). (1.7) x: aktueller Kurs des zugrunde liegenden Tau T t: Zeit bis zur Fälligkeit f IES T S t x xe (r d r f) tau. Vorwärtspreis des zugrunde liegenden Theta plusmn rdrf 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) xn (T) dt 1 N (x) Die Black-Scholes-Formel kann unter Verwendung der Integraldarstellung von Gleichung (1.6) ver dtau IEF e rdtau IEphi (STK) (er dtau phi xe (rdrf 1 2 sigma2) tausigma tauy K) N (y) dy (1.8) Als nächstes muss man mit dem positiven Teil umgehen und dann den Platz vervollständigen, um die Black-Scholes-Formel zu bekommen. Eine Ableitung auf der Grundlage der partiellen Differentialgleichung kann unter Verwendung von Ergebnissen über die gut untersuchte Wärmegleichung durchgeführt werden. 18 18 Wystup A Hinweis zum Vorwärts Der Vorwärtspreis f ist der Streik, der die Zeit Nullwert des Terminkontrakts F S T f (1,9) gleich Null macht. Daraus folgt, daß der Terminkurs der erwartete Preis des Basiswertes zum Zeitpunkt T in einem risikoneutralen Aufbau ist (Drift der geometrischen Brownschen Bewegung ist gleich den Kosten des Trages r d r f). Die Situation r d gt r f heißt contango, und die Situation r d lt r f heißt Rückwärts. Beachten Sie, dass im Black-Scholes-Modell die Klasse der Terminkurven sehr eingeschränkt ist. Zum Beispiel können keine saisonalen Effekte aufgenommen werden. Beachten Sie, dass der Wert des Terminkontrakts nach der Zeit Null in der Regel von Null verschieden ist und da einer der Kontrahenten immer kurz ist, kann es zu einem Ausfallrisiko der kurzen Partei kommen. Ein Futures-Vertrag verhindert diese gefährliche Angelegenheit: Es handelt sich grundsätzlich um einen Terminkontrakt, aber die Gegenparteien müssen eine Margin-Rechnung haben, um sicherzustellen, dass der Geldbetrag oder die gelieferte Ware eine bestimmte Grenze nicht überschreitet. Griechen Griechen sind Ableitungen der Wertfunktion in Bezug auf das Modell Und Vertragsparameter. Sie sind eine wichtige Information für Händler und sind Standardinformationen von Front-Office-Systemen. Weitere Details über Griechen und die Beziehungen zwischen den Griechen werden in Hakala und Wystup 50 oder Reiss und Wystup 84 präsentiert. Für Vanille-Optionen führen wir einige von ihnen jetzt auf. (Spot) Delta. V x phie r f tau N (Phid) (1.10) Vorwärts Delta. Driftloses 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 Optionen und strukturierte Produkte 19 Geschwindigkeit. 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) Farbe. 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) Wolga. 2 v sigma 2 xe r f tau taun (d) d d sigma (1.19) Volga wird auch manchmal Vomma oder Volgamma genannt. Vanna 2 v sigma x e r f tau n (d) d Sigma (1,20) Rho. Vr d phiktaue rdtau N (Phid) (1.21) v rf phixtaue rftau 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) Identitäten Die Put-Call-Parität ist die Beziehung 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) (1.2) 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, rd, rf, 1) v (x, K, T, t, Sigma, rd, rf, 1) xe rf tau Ke r dtau, (1.32), was nur eine kompliziertere Weise ist, die triviale Gleichung xx x zu schreiben. Die Put-Call-Delta-Parität ist 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) Insbesondere erfahren wir, dass der Absolutwert eines Put Delta und eines Call Delta nicht genau zu einem, sondern nur zu einer positiven Zahl e r f tau addiert wird. Sie addieren sich etwa, wenn entweder die Zeit bis zum Verfall tau kurz ist oder wenn der Fremdzins r f nahe Null ist. 21 FX Optionen und strukturierte Produkte 21 Während die Wahl K f identische Werte für Call und Put erzeugt, suchen wir den deltasymmetrischen Streik, der absolut identische Deltas erzeugt (Spot, Vorwärts oder Driftlos). Diese Bedingung impliziert d 0 und damit Fe sigma2 2 T, (1.34) wobei in diesem Fall das absolute Delta erf tau 2 ist. Insbesondere lernen wir, dass immer gt f, dh es kann t ein put und ein Anruf mit identischen Werten sein Und deltas Beachten Sie, dass der Streik in der Regel als der mittlere Streik beim Tragen einer Straddle oder ein Schmetterling gewählt wird. Ähnlich kann der Dual-Delta-symmetrische Streik-CircK fe sigma2 2 T aus der Bedingung abgeleitet werden d Homogenitätsbasierte Beziehungen Wir können den Wert des Basiswertes in einer anderen Einheit messen. Dies wird offensichtlich die Option Preisformel wie folgt ausführen. (X, K, T, t, Sigma, rd, rf, phi) v (ax, ak, T, t, sigma, rd, rf, phi) für alle gt 0. (1.35) Differenzierung beider Seiten mit Respekt (1.36) Ein Vergleich der Koeffizienten von x und K in den Gleichungen (1.7) und (1.36) führt zu suggestiven Ergebnissen für die Delta vx und Dual Delta v K. Diese Raum - Homogenität ist der Grund für die Einfachheit der Delta-Formeln, deren langwierige Berechnung auf diese Weise gespeichert werden kann. Wir können eine ähnliche Berechnung für die zeitbeeinflußten Parameter durchführen und die offensichtliche Gleichung v (x, K, T, t, sigma, rd, rf, phi) v (x, K, T a, ta, asigma, ar d (1, 37) Die Differenzierung beider Seiten in bezug auf a und dann das Setzen von 1 ergibt 0 tauv t sigmav sigma rdv rd rfv rf. Deutsch:. Englisch: v3.espacenet. com/textdoc? DB = EPODOC & ... PN = (1.38) Natürlich kann dies auch durch direkte Berechnung überprüft werden. Die Gesamtnutzung solcher Gleichungen besteht darin, bei der Berechnung von Griechen doppelte Prüfnoten zu erzeugen. Diese Homogenitätsmethoden lassen sich problemlos auf andere komplexere Optionen erweitern. Durch die Put-Call-Symmetrie verstehen wir die Beziehung (siehe 6, 7,16 und 19) v (x, K, T, t, Sigma, rd, rf, 1) K fv (x, f 2 K, T, t, Sigma, rd, rf, 1). (1.39) 22 22 Wystup Der Streik des Puts und der Streik des Anrufs führen zu einem geometrischen Mittel, das dem Vorwärts f entspricht. Der Vorwärts kann als geometrischer Spiegel interpretiert werden, der einen Aufruf in eine bestimmte Anzahl von Puts widerspiegelt. Beachten Sie, dass die Put-Call-Symmetrie für den Out-the-Money-Optionen (K f) mit dem Sonderfall der Put-Call-Parität zusammenfällt, wo der Call und der Put den gleichen Wert haben. Direkte Berechnungen zeigen, dass die Raten symmetrie v v tauv (1,40) r d r f gilt für Vanille-Optionen. Diese Beziehung gilt in der Tat für alle europäischen Optionen und eine breite Klasse von wegabhängigen Optionen, wie in 84 gezeigt. Man kann direkt die Beziehung der Fremd-Inlands-Symmetrie 1 xv (x, K, T, t, Sigma, rd , Rf, phi) Kv (1 x, 1 K, T, t, Sigma, rf, rd, phi). (1.41) Diese Gleichheit kann als eine der Gesichter der Put-Call-Symmetrie betrachtet werden. Der Grund dafür ist, dass der Wert einer Option sowohl in einem inländischen als auch in einem ausländischen Szenario berechnet werden kann. Wir betrachten das Beispiel der S t-Modellierung des Wechselkurses von EURUSD. In New York kostet die Call-Option (STK) v (x, K, T, t, Sigma, r usd, r eur, 1) USD und damit v (x, K, T, t, sigma, r usd, r Eur, 1) x () 1. EUR. Diese EUR-Call-Option kann auch als USD-Put-Option mit Auszahlung K 1 KST angesehen werden. Diese Option kostet Kv (1, 1, T, t, Sigma, rx K eur, r usd, 1) EUR in Frankfurt, da S T und 1 s t haben die gleiche Volatilität. Natürlich müssen der New Yorker Wert und der Frankfurter Wert zustimmen, was zu (1.41) führt. Wir werden auch später erfahren, dass diese Symmetrie nur ein mögliches Ergebnis ist, das auf der Änderung der Zahl basiert. Zitat Zitat der zugrunde liegenden Wechselkursgleichung (1.1) ist ein Modell für den Wechselkurs. Das Zitat ist ein dauerhaft verwirrendes Problem, also lasst uns das hier klären. Der Wechselkurs bedeutet, wie viel der inländischen Währung benötigt werden, um eine Einheit von Devisen zu kaufen. Zum Beispiel, wenn wir EURUSD als Wechselkurs nehmen, dann ist das Standard-Angebot EUR-USD, wobei USD die inländische Währung ist und EUR die Fremdwährung ist. Der Begriff Inland ist in keiner Weise mit dem Standort des Händlers oder eines Landes verbunden. Es bedeutet nur die numerische Währung. Die Begriffe Inland, Zahlen oder Basiswährung sind Synonyme wie ausländische und zugrunde liegende. Während dieses Buches bezeichnen wir mit dem Schrägstrich () dem Währungspaar und mit einem Bindestrich (-) das Zitat. Der Schrägstrich () bedeutet nicht eine Teilung. Zum Beispiel kann EURUSD auch in EUR-USD notiert werden, was bedeutet, wieviel USD benötigt wird, um einen EUR zu kaufen, oder in USD-EUR, was bedeutet, wie viele EUR benötigt werden, um einen USD zu kaufen. In Tabelle 1.1 sind bestimmte Marktstandards angegeben. 23 FX Optionen und strukturierte Produkte 23 Währungspaar Standardzitat Musterzitat GBPUSD GPB-USD GBPCHF GBP-CHF EURUSD EUR-USD EURGBP EUR-GBP EURJPY EUR-JPY EURCHF EUR-CHF USDJPJ USD-JPY USDCHF USD-CHF Tabelle 1.1: Standardmarkt Zitat der wichtigsten Währungspaare mit Beispiel Spot Preise Trading Floor Language Wir nennen eine Million ein Dollar, eine Milliarde ein Yard. Das ist, weil eine Milliarde Milliarde in Französisch, Deutsch und anderen Sprachen genannt wird. Für das britische Pfund wird eine Million auch oft als quid bezeichnet. Bestimmte Währungspaare haben Namen. Zum Beispiel wird GBPUSD als Kabel bezeichnet, weil die Wechselkursinformationen verwendet wurden, um durch ein Kabel im Atlantischen Ozean zwischen Amerika und England gesendet zu werden. EURJPY heißt das Kreuz, denn es ist der Cross-Rate der liquor gehandelten USDJPY und EURUSD. Bestimmte Währungen haben auch Namen, z. B. Der Neuseeland-Dollar NZD heißt Kiwi, der Australische Dollar AUD heißt Aussie, die skandinavischen Währungen DKR, NOK und SEK heißen Scandies. Wechselkurse werden in der Regel bis zu fünf relevante Zahlen zitiert, z. B. In EUR-USD konnten wir ein Zitat von Die letzte Ziffer 5 als Pip, die mittlere Ziffer 3 heißt die große Figur, da Wechselkurse werden oft in den Handelsböden und die große Figur, die in größerer Größe angezeigt wird, Ist die relevantesten Informationen. Die Ziffern, die der großen Figur überlassen wurden, sind sowieso bekannt, die Pips rechts der großen Figur sind oft vernachlässigbar. Um es klar zu machen, wird ein Anstieg von USD-JPY um 20 Pips und ein Anstieg um 2 große Zahlen werden Zitat der Option Preise Werte und Preise von Vanille-Optionen können in den sechs Möglichkeiten, die in Tabelle 1.2 erklärt werden. 24 24 Wystup-Namenssymbolwert in Einheiten des Beispiels inländischen Bargeld d DOM 29,148 USD ausländisches Geld f Für 24.290 EUR inländisches d DOM pro Einheit von DOM USD fremd f je pro Einheit von EUR EUR Pips d Pips DOM pro Einheit von USD USD Pips pro Stück EUR ausländische Pips f Pips pro Stück DOM EUR Pips pro USD Tabelle 1.2: Standard-Marktzitiertypen für Optionswerte. Im Beispiel nehmen wir FOREUR, DOMUSD, S 0. R d 3,0, r f 2,5, Sigma 10, K. T 1 Jahr, Phi 1 (Aufruf), Nominal 1, 000, 000 EUR 1, 250, 000 USD. Für die Pips wird das Zitat USD Pips pro EUR auch manchmal als USD pro 1 EUR angegeben. Ebenso können die EUR-Pips pro USD auch als EUR pro 1 USD angegeben werden. Die Black-Scholes-Formel zitiert d Pips. Die anderen können mit der folgenden Anweisung berechnet werden. D Pips 1 S 0 S 0 1 f K S d 0 S f Pips 0 K d Pips (1.42) Delta und Premium Convention Das Spot-Delta einer europäischen Option ohne Prämie ist bekannt. Es wird jetzt roher Punkt Delta Delta Roh genannt. Es kann in einer der beiden Währungen zitiert werden. Die Beziehung ist Delta Reverse Rohdelta Roh S K. (1.43) Das Delta wird verwendet, um Spot in der entsprechenden Menge zu kaufen oder zu verkaufen, um die Option bis zur ersten Bestellung abzusichern. Für die Konsistenz muss die Prämie in die Delta-Hedge einbezogen werden, da eine Prämie in Fremdwährung bereits einen Teil der Option s Delta-Risiko absichern wird. Um dies klar zu machen, betrachten wir EUR-USD. In der Standard-Arbitrage-Theorie bezeichnet v (x) den Wert oder die Prämie in USD einer Option mit 1 EUR fiktiv, wenn der Spot bei x ist und der Rohdelta v x die Anzahl der zu zahlenden EUR für die Delta-Hedge bezeichnet. Deshalb ist xv x die Anzahl von USD zu verkaufen. Wenn nun die Prämie in EUR anstatt in USD gezahlt wird, dann haben wir bereits vx EUR, und die Anzahl der zu erwerbenden EUR muss um diesen Betrag gesenkt werden, dh wenn EUR die Prämienwährung ist, müssen wir vxvx EUR kaufen Die Delta-Hecke oder gleichwertig verkaufen xv xv USD. 25 FX Optionen und strukturierte Produkte 25 Die gesamte FX-Zitat-Story wird in der Regel ein Chaos, denn wir müssen zuerst herausfinden, welche Währung inländisch ist, was fremd ist, was ist die fiktive Währung der Option und was ist die Prämienwährung. Leider ist das nicht symmetrisch, da das Gegenstück einen anderen Begriff der inländischen Währung für ein gegebenes Währungspaar haben könnte. Daher gibt es im professionellen Interbankmarkt einen Begriff von Delta pro Währungspaar. Normalerweise ist es das linke Delta des Fenics-Bildschirms, wenn die Option in der linken Seite Prämie gehandelt wird, die normalerweise das Standard - und rechtsseitige Delta ist, wenn es mit rechtsseitiger Prämie gehandelt wird, z. B. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs, etc. Da OTM Optionen die meiste Zeit gehandelt werden, ist der Unterschied nicht riesig und erzeugt daher kein großes Spotrisiko. Zusätzlich wird das Standard-Delta pro Währungspaar links Delta in Fenics für die meisten Fälle verwendet, um Optionen in der Volatilität zu zitieren. Dies muss nach Währungen angegeben werden. Diese Standard-Interbank-Idee muss an das echte Delta-Risiko der Bank für ein automatisiertes Handelssystem angepasst werden. Für Währungen, in denen die risikofreie Währung der Bank die Basiswährung der Währung ist, ist klar, dass das Delta das Rohdelta der Option ist und für eine riskante Prämie diese Prämie einbezogen werden muss. 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. Value. (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 Seagull 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 Deposits 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 Applications 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 Accounting 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 . EUR. 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. Value. (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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