Returns. BREAKING eccesso Returns. Determining GIU Eccesso i rendimenti in eccesso richiede la sottrazione del tasso privo di rischio, o tasso di riferimento, dal tasso effettivo raggiunto, ad esempio, se l'attuale tasso privo di rischio è di 1 2 e il portafoglio in esame ha ricevuto un ritorno di 8 , il rendimento in eccesso sarebbe il 6 8 differenza rendimenti in eccesso può essere positivo o negativo a seconda del risultato dell'equazione rendimenti in eccesso positivi dimostrano l'investimento hanno sovraperformato il tasso privo di rischio o di riferimento, mentre rendimenti negativi si verificano quando un investimento un rendimento inferiore rispetto a il tasso privo di rischio o benchmark. Widely usato come misura del valore aggiunto dal portafoglio di investimento o manager o il manager s capacità di battere il mercato, rendimenti in eccesso possono anche essere indicati come la alfa dopo essere regolato dal rischio valutato, noto come l'alfa e beta beta. Alpha e Beta. The sono entrambi metriche relative al livello di rischio o la volatilità sperimentato in un particolare titolo Mentre l'alfa fornisce una misura per quanto riguarda le prestazioni beni, la beta specifica il livello di rischio presente quando rispetto al capital asset pricing model CAPM Calcolato utilizzando una forma di analisi di regressione, la beta è una misura del bene s capacità di rispondere al mercato ad esempio fluctuations. For, si consideri un large-cap fondo comune degli Stati Uniti che ha lo stesso livello di rischio cioè beta 1 come l'indice SP 500 Se il fondo genera un ritorno di 12 in un anno in cui la SP 500 è avanzato solo 7, la differenza di 5 sarebbe considerato come l'eccesso di ritorno, o l'alfa generato dai rendimenti dei fondi manager. Excess e lungo termine Results. Critics di fondi comuni e altri portafogli gestiti attivamente sostengono che è quasi impossibile di generare rendimenti superiori su una base costante nel lungo periodo, a seguito della quale, la maggior parte dei gestori di fondi sovraperformare il benchmark nel corso del tempo Inoltre, i fondi attivi vengono spesso con tasse più alte che possono negare una parte dei guadagni con esperienza dai investor. This ha portato alla popolarità enorme di fondi indicizzati e fondi negoziati in borsa e ha portato ad alcune società di gestione di fondi, come ad esempio Legg Mason , offrendo prodotti ibridi supplementari Le nuove offerte sono progettate per attirare gli investitori che erano inclini a tirare i loro fondi di fondi gestiti e investire quei fondi in varie indice funds. You possono acquistare opzioni FX per coprire o sfruttare un exposure. There sono 2 tipi di options. A Mettere opzione dà all'acquirente il diritto, ma non l'obbligo di vendere la valuta sottostante ad una predeterminata opzione price. A chiamata dà al compratore il diritto ma non l'obbligo di acquistare la valuta sottostante ad un predeterminato price. In entrambi i casi i venditore dell'opzione viene pagato un premio da parte del compratore dell'opzione opzioni non sono adatti a tutti gli investitori, in quanto portano rischi significativi Solo consigliato per investors. Some esperto del Benefits. Potential per migliorare returns. By vendita di opzioni put in attesa di comprare una valuta ad un più basso di vendita di opzioni call price. By in attesa di vendere una valuta ad un più alto price. Some del Risks. Selling un'opzione generalmente comporta un rischio maggiore rispetto all'acquisto di opzioni e un venditore di opzioni può sostenere una perdita bene in eccedenti l'importo del premio received. For più information. 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FX ed eccesso restituisce un Multi Moment Termine struttura del modello di tasso di cambio Dynamics.1 FX opions e eccesso Reurns Un Muli Momen Term srucure modello di scambio Rae Dynamics Yu-chin Chen e Ranganai Gwai Ocober 2013 Absrac Il presente documento propone con dei cambi opions FX wih prezzi Srike differen e mauriies ha ERM srucure di volailiy sorride o capure boh expecaions FX e rischi uso opions quotidiane daa per sei principali coppie di valute, ci mostrano ettari che attraversano secion e ERM srucure di opions-implicite sandard deviaion, asimmetria e kurosis consisenly senza spiegare solo lui condiional dire bu anche lui enire disribuion condiional di valuta subsequen eccesso reurns per orizzonti che vanno da una settimana o welve monhs Questa robus paern empirica è consisen wih un represenaive expeced uiliy massimizzare invesor che, in addiion o preoccuparsi abou lui media e la varianza, si preoccupa anche Abou ha asimmetria e kurosis di lui piacerebbe ritornare disribuion troviamo anche cambio ha movemens Rae, che sono il modello o nooriously diffi empiricamente egli scambiare rae di puzzle disconnesso, sono in fac ben spiegato-by ha ERM srucures di premi in avanti e opions-implicita momens superiori nostri resuls sugge ettari ha perenni problemi affrontati da lui scambio empirica Rae lieraure sono mesi probabilmente a causa o assumpions eccessivamente resricive inheren a prevalente esing mehods, che non riescono o correttamente Accoun perché lungimirante nella proprietà di Raes cambio e asimmetria poenial o eccesso kurosis in lui condiional disribuion di Raes Parole cambio FX movemens reurns eccesso opions pricing volailiy sorriso ERM rischio srucure di volailiy implicita quanile regressione JEL codice E58 , E52, C53, F31, G15 Firs Draf Ocober ci sono indebed o lui LAE Keh-Hsiao Lin e Joe Huang per noi poining o ha relevan fonti daa Vorremmo anche o matassa Samir Bandaogo, Nick Basch, Yanqin Fan, David Grad, ji-Hyung Lee, Charles Nelson, Bruce Preson, Abe Robison, e Eric Zivo per erede utile commens e assisance Tutti gli errori rimanenti sono nostri Chen s e la ricerca Gwai s sono annuncio supportate da lui Gary Waerman borsa di studio e si Buechel Fellowship un lui Universiy di Washingon Chen Gwai Deparmen di Economia, Universiy di Washingon, Box Seale, WA 98.195,2 1 Inroducion Il lieraure scambio Rae economia ha più di lui anni affrontato molti puzzle empirici, o anomalie ettari sono difficili o spiegare in poi base di eiher suono heory economica o Caminate pracical Sarno 2005 come esempio, Alhough predics heory Raes cambio nominale hA dovrebbe dipendere Curren e expeced fuure fundamenals macroeconomiche, ha il consenso in lui lieraure è Raes cambio hA sono essenially empiricamente disconneced da lui macroeconomici variabili ettari si suppone o orlo deermine Questo disconnesso empirica viene a lui sotto forma di bassi correlaions beween Raes di cambio nominali e erede presunto macro-based deerminans e anche a lui sotto forma di scarse prestazioni dei modelli di scambio Rae macro-based in ou-di-campione forecasing vedere Engel 2013 per una recensione a relaed empirica anomalia ha ha ricevuto una notevole aenion in lui lieraure è scoprì ineres pariy UIP puzzle o lui di puzzle in avanti premio Il puzzle UIP è lui empirica irregulariy mostrando ettari egli divise Rae è un predicor parziale di Raes cambio fuure SPO uno manifesaion della sua empirica dell'aria regolare ir è ettari counries wih ineres superiori Raes finiscono o vedere valute erede successivamente lo appreciae e un sraegy carry-rade exploiing suo paern, in media, trasporta la valuta in eccesso reurns 1 Questa violaion di lui UIP condiion è comunemente aribued o ime-variante premi per il rischio e pregiudizi a Marke misurata expecaions Tuttavia, i proxy empirici basati su forecass intervistate o misure sandard di rischio - per insance, quelle Buil da Consumo di growh, reurns calzino marke, o lui Fama e French 1993 elementi caratteristici - non hanno avuto successo nello spiegare che Puzzle 2 come tale, pur riconoscendo che presenza di rischio, macroeconomico a base di approcci Raes cambio di modellazione o Ofen ignorare il rischio in esing empirica vedere di insance, Engel e Wes 2005 Mark 1995 in poi finanziare 1 a portare rade sraegy è o prendere in prestito valute a basso ineres e dare in high-ineres valute oppure o vendere valute a termine ettari sono aa premium e comprare in avanti valute wih un discoun avanti 2 Sede, Engel 1996 per un sondaggio di lui lieraure avanti premium, così come sudies recentemente ri quali Burnside e al 2011 e Bacchea e van Wincoop 2009 1.3 lato, effors volte o idenify porfolio elementi caratteristici di rischio piacerebbe ritornare a base di offrire qualche successo empirico per spiegare che cross-secional disribuion di reurns FX in eccesso, bu avere Lile o dire abou cambio bilaeral dinamiche Rae si veda ad esempio, Lusig e al 2011 Menkhoff e al 2012 Verdelhan 2012 3 Il puzzle UIP è Aken seriamente a lui scambiare Rae lieraure perché UIP condiion è un propery di mos modelli macroeconomici open-economy Questo abeti carta link che persisen puzzle empirici affrontati da lui scambiano rae economia lieraure o preferenza eccessivamente resricive e assumpions disribuional in mehods esing convenional Noi sosteniamo hA assumpions ausiliari ueste ofen inadequaely accoun per eiher lui propery di Raes di cambio nominali o asimmetria poenial eo FA affligge in lui disribuion di reurns FX Abbiamo gallina proporre con lui lungimirante ERM srucure di volailiy sorrisi o expecaions capure di condiions macroeconomici fuure così come marke percepito volailiy, crash e il rischio ail di cambio fuure rae realizaions Concepually, dal momento che i profitti di conracs opion dipendono da lui uncerain fuure realizaion di lui prezzo di lui sottostante asse, prezzi opion Mus riflessi marke senimens e credenze abou egli probabiliy di payoff fuure utilizzando i prezzi di un secion croce di opion conracs un-lui-denaro, inversioni di rischio e buerflies Vega-pesava 10 e 25 Delas che erogano vincite sotto realizaions fuure differenial di lui spo scambio Rae, scopriamo ex ane deviaions sandard, asimmetria, e kurosis di lui disribuion di scambio fuure expeced Rae movemens Wih opions daa al giorno per sei principali coppie di valute e sette enors, ci mostrano hA ueste misure ex-ane marke basati su FX volailiy, crash e il rischio ail grado di spiegare egli condiional mezzi di reurns valuta in eccesso o deviaions ex pos da UIP per orizzonti che vanno da una settimana o welve monhs Noi gallina utilizzano l'analisi di regressione quanile o demonsrae ha 3 Lusig e al 2011 e Verdelhan 2012 per esempio, idenify un bagaglio FACOR sulla base di croce secions di ineres reurns valuta Rae-sponsorizzata e una FACOR dollaro sulla base di secions trasversali di valuta bea-sponsorizzata reurns 2.4 ha opions a base di misure di rischio FX non solo spiegare che condiional dire bu anche lui enire disribuion condiional di deviaions subsequen da UIP Addiionally, abbiamo trovare i proxy hA per rischi globali FX opions-implied mostrare il potere explanaory RILIEVO per quarerly reurns eccesso Noi portiamo ou un baery di controlli robusness ha includono robus analisi LEA quadrati di regressione, analisi di regressione utilizzando non sovrapposti daa e momens opion-implicita exraced da 10- misure opions dela INSEAD di lui opions 25-dela utilizzati in lui principali regressioni così come sottocampione analisi nostri principali risultati empirici sopravvivere hese ess robusness, suggesing ha egli srong relaionship empirica beween reurns in eccesso e opions basati su FX maggiori rischi Momen è nessun essere guidato da questioni come il nostro uso di daa wih sovrapposizione observaions o egli presenza di ouliers nel nostro campione abbiamo gallina andare oltre l'analisi mached-frequenza e exend egli impostazione inaugurata da Hansen e Hodrick il 1980 e Laer utilizzato in Clarida e Taylor 1997 e Chen e Tsang il 2013, ha utilizzato lui avanti Raes o ineres differenials oltre ime e tutto coppie di valute o modello eccesso reurns Il componen ERM srucure aggiunge potenza explanaory RILIEVO, dimostrato da sE enormi aumenti in lui adjused R 2 s rispetto o resuls da analisi di frequenza mached 4 Abbiamo Führer spettacolo cambio ha movemens Rae, che si sono rivelati nooriously difficile o modello empiricamente nel corso egli anni, sono in fac ben spiegato da lui ERM srucures di abeti momens opion-implicita e ordine superiore momens Sandard UIP analisi scambio regresso rae movemens in poi abeti momen di lui percepiti disribuion di scambio rae movemens per lui stesso Enor, e lui explanaory potere di tali sono di solito molto bassi per il cambio quarerly Rae movemens, troviamo hA augmening tali regressioni da includendo anche informaion da lui ERM srucure di abeti momens pure mentre ERM srucure di opion-implicito secondo o momens fourh cede R 2 s che vanno da 70 o 85 e fi 4 Tha è, confrontando le colonne A e B di grado 7 3.5 molto bene Il bene fi dei nostri Muli-Momen ERM srucure Specificaions, mostrato nelle figure 4a - 4e, può essere conrased o ha scarsa fi di lui sandard regressioni UIP mostrati nelle figure 5a - 5e da un lato, ecco un enorme lieraure collega ha ERM srucure di ineres Rae Raes o curva dei rendimenti o dinamiche fuure expeced di fundamenals macroeconomici quali la politica moneary, inflaion e Oupu per esempio, Ang e Piazzesi 2003 Diebold e al 2006 e al 2006 Ang e Chen e Tsang 2013 exend sua srand di lieraure o ha aperto economia conex da noing ha egli ERM srucure di ineres rae differenials curva dei rendimenti relaive conain informaion Abou ha expeced dinamiche fuure di differenze di fundamenals macroeconomici in poi la mano Oher, ci sostengono in secion 2 ha ha ERM srucure di abeti opion-implicita momens capures egli stesso informaion mentre ERM srucure di ineres RAE differenials Pertanto, o egli EXEN ha egli curva dei rendimenti relaive conains informaion abou expeced PAH fuure di condiions macroeconomiche domesic e stranieri, i nostri risultati ha egli ERM srucure di abeti momens aiutano a spiegare scambio Rae movemens sugge Raes cambio di hA non sono disconneced da fundamenals macroeconomiche Le Robus risultati empirici a la sua carta sugge expecaions ha boh e il rischio deve essere attentamente accouned per la modellazione srucural ed empirica di Raes cambio in addiion, i nostri resuls sugge ha oltre-ha-couner opions FX marke capures boh conceps in pracice Questi resuls demonsrae inoltre ha ha perenni problemi affrontati da lui scambiare rae economia lieraure sono mesi probabilmente a causa o assumpions ausiliari eccessivamente resricive in lui esing empirica di lui modelli raher Han o limiaions di lui heoreical modelli hemselves derivaives semplici come lui avanti e fuures sono stati utilizzati exensively a spiegare reurns valuta in eccesso o scambio rae movemens veda, per esempio, Hansen e Hodrick il 1980 e il Clarida e Taylor 1997 tra molti ohers Payoffs da conracs avanti, 4.6 tuttavia, sono lineari a lui piacerebbe ritornare in poi valuta sottostante e come tale non fare conain come utile a se di informaion come lui non conracs - Lineare esaminiamo infatti, opions FX sono stati utilizzati o varianza proxy o rischio ail in varie conexs specifici come esing ha Porfolio modello di bilancio di scambio rae deerminaion Lione del 1998, misurando announcemens effecs Grad 2010 evaluaing evens rari heory Farhi e al 2009 e conducendo densiy forecass Chrisoffersen e Mazzoa un lato, sudies hA concentrarsi su ERM dinamiche srucure fine o solo concenrae su Raes di cambio a termine o ineres rae differenials, che sono abeti momens di lui disribuions di Raes fuure SPO per esempio Hansen e Hodrick 1980 Clarida e Taylor 1997 e Chen e Tsang 2013 in poi la mano Oher, sudies hA concentrano sul momen maggiori rischi fine o conduc mached frequenza analizza ad esempio, Malz 1997 e Lyons 1998 a SE il bes della nostra conoscenza, però, qui è stata alcuna sysemaic e completa esing di wheher lui ex-ane informaion conained in lui Erm srucure di volailiy sorride davvero predics ex pos moneta in eccesso reurns Il nostro lavoro si propone o colmare il suo gap 5 Il nostro uso di daa opions prezzo e relaed mehodologies empirica ha un certo numero di moivaing elementi caratteristici di Firs, opions sono lungimiranti da consrucion Campa e al 1998a poin ou ha il suo propery lungimirante significa prezzi opion dovrebbero incorporae informaion come forhcoming swiches regime o che la presenza di un problema di peso 6 in secondo luogo, i prezzi opion sono profondamente rooed nel comportamento paricipan Marke perché hey sono basati su paricipans marke wha fare INSEAD di wha hey diciamo 7 Furhermore, croce 5 Questo articolo conribues anche, più in generale o colmare lo scarto beween ha lieraure sulla moneta prezzi derivaive e ha in poi l'economia di Raes cambio Chen 1998 commens hA Mos Südens di economia finanziaria si concentrano in poi mahemaical ools di SE modelli di pricing opion wih enfasi Lile in poi l'economia di scambio rae deerminaion, mentre radiional macroeconomia-inernaional finiscono o rifuggire da lui echnicaliy di lui valuta derivaive, despie è evidente imporance in pracice Questa lacuna la ricerca accademica e pioggia ha creaed un problema per praciioners 6 Il problema peso si riferisce o si effecs su deduzioni causate da basso probabiliy Evens ha non fare si verificano in lui campione, che può portare o posiive eccesso piacerebbe ritornare 7 Come discusso in precedenza, conracs termine sono avanti - looking da consrucion, bu per una data coppia di valute e Enor, qui è solo un prezzo a termine wih dipendenza lineare su fuure spo realizaion I prezzi opion muliple per opions wih srikes differen offrono una informaion molto più ricca sare Lasly, consrucions sandard di expecaions marke e rischi percepiti sulla base di fundamenals macro o elementi caratteristici di finanza fanno 5,7 secions dei prezzi opion implicano un probabiliy disribuion subjecive di Raes cambio fuure SPO, che capures boh paricipans marke credenze e preferenze 8 in terzo luogo, echniques moderni come lui Vanna-Volga mehod vedono Casagna e Mercurio 2005 e ha mehodology di Bakshi e al 2003 faciliae compuaion elegan di opions-implicita più elevati momens ordine di scambio fuure rae cambia Questi mehods sono PARTICOLARMENTE aracive perché opions-implicato momens sono exraced wihou imporre assumpions sulle preferenze invesor e meccanismi expecaion-formaion, o specificando che sochasic processo di guida ha alla base lo scambio spo Rae Lasly, Chang e al 2013 litigare ha dato momens opion-implicita possono essere exraced aa frequenza più alta, ha opions approccio ci dà esimaes veramente condiional ed evita un problema rade-off encounered quando esimaing superiore momens da reurns Hisorical daa Quando si utilizza reurns Hisorical daa, finestre più lunghi sono necessari o aumentare la precisione, mentre le finestre shorer sono tenuti o obain condiional raher han esimaes uncondiional la res di lui carta è organizzato come segue secion 2 moivaes la nostra attenzione su alti rischi Momen e dinamiche ERM srucure a spiegare reurns in eccesso e lo scambio Rae movemens Secion 3 spiega come i prezzi FX opion possono capure boh lui expecaions previsionali e rischi FX percepiti dà deails della nostra sraegy per exracing ha Secion informaion 4 conains lui resuls empirici e discussione Secion 5 conains Führer inerpreaions e la discussione su He resuls empirici e secion 6 conclude che la carta non lavoro ben 8 Questo disribuion è comunemente indicato o come egli rischio-neurale disribuion, Hough I non implica ettari ha disribuion deriva in condizioni di rischio-neuraliy in poi conrary, i incorporaes boh ha expeced disribuion probabiliy fisica di scambio fuure rae realizaion così come lui rischiare premio, o compensaion richiesto o l'orso ha uncerainy 6.8 2 Perché Higher Order momens e Term Srucure 2 1 avanti Premium Puzzle e eccesso di valuta Reurns Il condiion Marke efficienze per lui straniera markes cambio, sotto expecaions raional, equaes croce ineres confine differenials colpa del II ha expeced Rae di depreciaion valuta nazionale, adjused perché rischi di tariffazione associaed wih posizioni valutarie, 9 II, E s 2 1 Questo condiion è expeced o attesa per tutti i invesmen orizzonti, wih ineres Raes ha sono mauriies mached Ignorando che rischia premio ehm, numerosi articoli sono ESED sua equaion dal Fama 1984 e trovare violaions sysemaical della sua UIP condiion SSII, e 0 H 0 1 2 2 wih un esimaed 0 e R 2 s ettari sono di solito vicino o pari a zero è colui cosiddetti ineres scoperte rae rompicapo pariy o lui di puzzle in avanti premio vedere Engel 1996 per un sondaggio di lui lieraure per vedere che Connecion wih Raes avanti, Noé ha ha ricoperto ineres condiion pariy, un empiricamente valido no-arbirage condiion, equaes lui in avanti fs premium, wih ineres differenials Il rischio-neurale UIP condiion sopra hus implica ha lui in avanti Rae dovrebbe essere un predicor imparziale per fuure spo rae e SF o SFU, dove e u 0 Dovremmo nex definire reurns FX in eccesso mentre rae del piacerebbe ritornare attraverso le frontiere a NE di movemen valuta, e si può vedere ha lui UIP o premio di puzzle in avanti può essere 9 nel suo documento, si definiscono gli scambiare Rae mentre domesic prezzo di valuta estera un aumento che cambio rae indicaes un depreciaion di lui a casa moneta Tuttavia, a casa non fa hanno un significato geografico bu seguono lui FX convenions marke Vedi grado 1A 7.9 espresso come non-zero in media eccesso piacerebbe ritornare sopra ime xr FSII, su 2 3 I è naural gallina o noe ha ha mancato empirica di lui rischio-neurale UIP condiion può essere aribuable o eiher egli presenza di un premio di rischio ime-varianti, o l'errore ha expecaion, u, potrebbe non essere significa IID zero nell'arco ime Se disribuion di eiher di hese è senza dire nulla su lui ime serie, esimaes empiriche di lui pendenza coefficien in regressione equaion 2 2 sarebbe probabilmente soffrono omied pregiudizi variabile o Oher complicaions Alcuni problemi wih ess convenional di UIP Il hypohesis non distorsione avanti è rue per una data disribuion di sa ogni poin in ime Se condiional disribuion di s è, comunque, non oltre IID ime - come suggesed da lui exraced momens opion-implicita e disribuions esempio nella figura 1 qui sotto - gallina esing ha hypohesis e sf usando serie ime daa migh non essere appropriae La regressione OLS basata su esing quadro equaion 2 2 fa ha ausiliari assumpion hA shock OS sono iid normale corso ime Tuttavia, reurns FX sono ben documened O hanno Faer affligge han normale, e in alcuni casi distorto 10 INSERIRE FIGURA 1 QUI 2 2 Perché superiori momens ordine 11 nel suo subsecion mostriamo ha in addiion o rischio neuraliy e expecaions raional assumpions, ha cerniere UIP condiion anche lui raher resricive assumpions ausiliari 10 Cincibuch e Vavra 2004 wrie I è reurns finanziari comune conoscenza hA sono non normale, ah ehi solito hanno pesante affligge e ha hey migh essere falsati Pertanto mi sembra efficienza o dispari ES, che coinvolge lui noion hA raional marke giocatori uilize tutto Informaion disponibili, e resric ha expecaion errore o essere normale 11 Maerial nel suo subsecion è da Mark 2001 reurns 8,10 ettari FX sono IID normale sopra ime e invesors hA avere Consan avversione al rischio absolue CARA uiliy Questi wo assumpions addiional riducono ha represenaive invesor s opimal asse allocaion problema Oa dire varianza problema opimizaion Ci sar wih ha problema di un invesor che, in ogni periodo, allocaes suo porfolio Tra asini rischiosi wih egli obiettivo di massimizzare gli expeced uiliy del periodo nex wealh In ogni periodo, ha invesor ha n asini rischiose o scegliere dalla vecor di reurns lordi è dato da R 1 R 1, 1 rn, 1 Se supponiamo W è arbirarily sE o 1, Q uando 1 r 1, in cui è n da 1 vecor di porfolio pesa Il problema invesors è o scegliere o ingrandimento lui espressione EUW 1 EU R 1 UW 1 FR 1 dr 1, 1 dr 2, 1 dr n , 1 2 4 subjec o ha condiion ha ni 1 I, 1, dove fr 1 è lui unirsi disribuion probabiliy di r CARA e Normaliy ridurre i problemi o dire varianza opimizaion Le ci Fuhrer assumere ettari ha invesor ha CARA uiliy e reurns hA sono condiionally normalmente disribued Il CARA uiliy assumpion significa che uiliy è dato da UW 1 ew 1, dove 0 è lui coefficien di avversione al rischio absolue Il disribuional assumpion r 1 N 1, 1 implica ha W 1 N p, 1, 2 p, 1, dove p, 1 1 e 2 p, 1 1 Wih lui sopra wo assumpions, espressione 2 4 riduce o 12 EUW 1 e ew 1 p, 2 2 p, 1 2 5 Equaion eq mvopim demonsraes ha sotto gli assumpions del CARA uiliy 12 La seconda equaliy da lui fac ha ew 1 LN p, 1, 2 2 p, 1, quindi e ew 1 p, 1 2 2 p, 1 9.11 funcion e normaliy condiional di reurns, egli generale porfolio allocaion problema 2 4 riduce o si intende - variance problema opimizaion 13 Se noi assumiamo Fuhrer ha il nostro invesor ha un porfolio 2-asse costituito da un legame domesic nominalmente sicuro e un legame estere, e ha lei allocaes un fracion della sua wealh o lui domesic legame, periodo NEX gallina wealh espresso in unis valuta locale è data da W 1 1 i 1 1 i S 1 W 2 6 S nel suo esempio 2-asse e CARA uiliy e reurns condiionally normali ha espressioni per lui condiional media e la varianza periodo nex wealh sono dato da p, 1 1 i 1 1 i ES 1 SW, 2 p, 1 1 2 1 i 2 V ar S 1 W 2 S 2 2 7 Collegamento che espressioni in equaion 2 7 ino funcion objecive 2 5, Ripresa egli abeti ordinare condiion wih rispet o e riorganizzare lui abeti ordinare i rendimenti condiion ha seguito equaion che deermines implicily ha opimal 1 i 1 i ES 1 w 1 1 i 2 V ar S 1 2 8 SS 2 Equaion 2 8 riduce o si UIP condiion se assumiamo ha tutte invesors sono rischi - neural 0 i 1 i ES 1 S 2 9 La regressione Fama in equaion 2 2 ess una versione logarihmic di equaion 2 9 I SEPS chiave nel derivare ha esable resricions in equaion 2 9 sono gli unirsi assumpions di 13 La quadraic uiliy funcion implica media varianza opimizaion per arbirary disribuion piacerebbe ritornare Tuttavia, ha quadraic uiliy implica l'aumento absolue avversione al rischio e saiaion Jondeau e al 2010, pagina 352 14 UIP terrà anche se 1, indipendentemente invesors grado di avversione al rischio 10.12 CARA uiliy e normaliy condiional di nex periodo wealh , che riducono lo invesor s opimizaion o dire varianza Le illusraes discussione di cui sopra ha derivanti lui UIP equaion ESED ttraverso espressione 2 2 dipende assumpions oher oltre expecaions raional e il rischio-neuraliy Se normaliy assumpion è sceso, per esempio, l'espressione gallina 2 9 sarà MOS probabilmente includono momens di ordine superiore in fac, Jondeau e al 2010 NOE ha sotto CARA uiliy, se lasciamo cadere lui normaliy assumpions, gallina ha invesor preferirebbero asimmetria posiive e bassa kurosis, come ha egli invesor s funcion objecive in equaion 2 5 sarà includono anche lui Hird e momens fourh di lui FX piacerebbe ritornare disribuion Sco e Horvah 1980 Vedi ha un individuo avverso al rischio sricly che preferisce sempre più o meno U 1 0 e ama asimmetria posiive a tutti i livelli wealh necessariamente antipatia alta kurosis 2 3 Perché ERM srucure Il srucure ERM dei prezzi opion ci permette o exrac informaion abou expeced fuure condiions macroeconomici Tornando o UIP equaion 2 1, riordinando e ieraing avanti, si può mostrare ettari ha di cambio nominale rae dipende Curren e ineres fuure expeced Rae differenials così come sul rischio expeced I ineres Raes sono variabili politiche moneary, e Hus dipendono fundamenals macroeconomici s e ijijj 0 Expeced ineres fuure differenials e jj 0 Expeced Fuure FX rischio 2 10 Wriing lui scambiare rae a lui formare in equaion 2 10 demonsraes ha imporance di capuring expecaions e rischi in esing modelli rae cambio sandard approcci empirici, tuttavia, imporre assumpions disribuional hA ridurre egli somma di fundamenals fuure expeced o uguali curren fundamenals e anche ignorano rischi vedere Engel e Wes 2005 Mark 1995 11.13 Chen e Tsang 2013 propongono utilizzando informaion conained in lui Erm srucure di ineres rae differenials o assumpions disribuional ueste laterali-Set Essi allo sfrutta informaion in lui Erm srucure di ineres relaive Rae differenials o proxy per modifiche expeced in fuure fundamenals macro e mostrare ettari Nelson e Siegel 1987 elementi caratteristici di exraced da curve dei rendimenti relaive Predic reurns FX bilaeral e spiegare la valuta in eccesso reurns uno MONH o wo anni a venire Clarida e Taylor 1997 e Clarida e al 2003 mostrano ettari anche se in avanti Rae è un predicor parziale di fuure spo rae ha in avanti i puzzle di alta qualità, ha ERM srucure dei premi avanti davanzale conains informaion utile per predicing scambio subsequen rae cambia Questa Recen successo dei modelli empirici ERM srucure ha demonsraes Fuhrer imporance di expecaions capuring proponiamo o uso informaion conained in lui Erm srucure dei prezzi opion o capure ha abeti nominale di 2 10 e utilizziamo lui secion croce dei prezzi opion o capure ha expeced componen rischio in subsecion 3 1, a nostro avviso ha lo ERM srucure di sorriso volailiy conains informaion abou expeced fuure condiions macro e rischi FX al di là di ettari conained in eiher ha relaive curva dei rendimenti o ERM srucure dei premi avanti 3 informaion Conen di valuta opions prezzi opion fornire un LEA hree disinc pezzi di informaion Abou paricipans marke expecaions e preferenze opions wih egli stesso sottostante coppia di valute e Enor bu differen prezzi Srike sorriso volailiy, opions wih egli stesso prezzo Srike e la stessa coppia di valute sottostante bu enors differen ERM srucure di volailiy implicita e lasly, i prezzi dei opions wih egli stesso Enor prezzo Srike bu differen coppie di valute sottostanti secion 3 ci spiegano che informaion heoreically conained in lui volailiy sorriso, ehm srucure dei prezzi opion e correlaions incrociata dei prezzi opion wih differen sottostante coppie di valute Noi gallina descrivono una mehodology per exracing suo informaion 12.14 3 1 Volailiy Sorriso, Termine Srucure e Cross Correlaions Breeden e Lizenberger 1978 spettacolo ha in Markes complee, ha chiamato opion prezzi funcion C e lui prezzo di esercizio K sono relaed come segue 2 CK 2 e RD QS, 3 1 dove RD e RF sono lui domesic e stranieri privi di rischio ineres Raes e QS è lui il rischio-neurale probabiliy densiy funcion pdf di fuure Raes SPO Equaion 3 1 implica ha, in linea di principio, siamo in grado di esimae lui tutta pdf di ime cambio S spo rae da ime volailiy sorriso una volta che disribuion è disponibile, mi diventa possibile o ge esimaes empirici di lui sandard deviaion, asimmetria, kurosis e ancora più elevati momens ordine di lui marke percepito densiy probabiliy di S dato informaion disponibile un ime In addiion o lui Breeden e Lizenberger 1978 Resul in equaion 3 1, abbiamo Noe ha Alhough marke paricipans possono essere reaed come se hey sono rischio-neurale per lui scopo di opion-pricing, informaion prezzi opion heoreically conain abou boh invesor convinzioni e preferenze di rischio, come illustrato da lui seguente formula per lui prezzo di un europeo-syle chiamata opion C, K, TKM, ds TSTKPST T e RD Preferenze credenze KSTKQST ds T 3 2 Boh In equaion 3 2, kernel M, è che i prezzi, che capures lui invesor s grado di avversione al rischio e PS è lui fisica funcion probabiliy densiy di Raes cambio fuure SPO 15 un CONRAC in avanti può in fac essere visto come una chiamata opion europea-syle wih un prezzo Srike zero per vedere i suoi, ricordiamo ettari, su da un lato, egli heoreical avanti scambiare Rae 15 in lui seconda espressione, ha prezzi kernel sta eseguendo boh lui rischi adjusmen e discouning funcions, mentre a lui Hird espressione funcions ueste sono divisi beween Q ed e rd 13,15 è data da lui formula F, T e rd in poi la mano Oher, evaluaing equaion 3 2 K 0 rendimenti 0 STQST ds T e terzo EQST 3 3 C, 0, T e Rd 0 STQST ds TF, T 3 4 Il relaionship beween opions e indietro nel equaion 3 4 suggess ha egli secion incrociata dei prezzi opion dovrebbe, aa minimo, conain tanto abou informaion invesor credenze e preferenze, come ha conained dei prezzi a termine Passando o ha ERM srucure dei prezzi opion, in un modo o moivae ha heoreical informaion Conen di lui ehm srucure dei prezzi opion è o sar da equaion 2 10 s E ijijj 0 Expeced ineres fuure differenials E jj 0 Expeced Fuure FX rischio 3 5 Ora, sotto ha empiricamente valida condiion CIP, ineres Rae differenial è uguale o lui in avanti premio per tutti enors j 16 ijijfjsrd standard di qualità ambientale j ln S Firs Momen di Q, Enor j 3 6 Jensen s inequaliy ERM Equaion 3 6 hus dice ettari, ignorando che Jensen s inequaliy ERM e lui Consan ehm rd, ha ineres rae differenial è uguale a lui abeti momen di lui opion implicita disribuion rischio-neurale della ln S j S per ogni j Enor I ineres Raes sono moneary 16 La seconda equaliy da dividendo 3 3 da S e aking logarihms 14.16 variabili di politica e herefore dipenderà fundamenals macroeconomici quali unemploymen e inflaion Quando combinati, equaions 3 6 e 3 5 demonsrae ius ha Ti piace lui curva dei rendimenti, ha ERM srucure di lui abeti momens di disribuions implicite capures anche informaion abou Curren e expeced fuure fundamenals macroeconomici Un secondo moivaion perché informaion Conen di lui Erm srucure dei prezzi opion proviene da he expecaion hypohesis for implied volailiy If he expecaions hypohesis holds in he FX marke, hen he implied volailiy for long daed opions should be consisen wih he implied volailiy of shor daed opions quoed oday and in he fuure For example, if he curren six monh implied volailiy is 10 and he curren hree monh implied volailiy is 5 , hen, under he expecaion hypohesis, hen he hree monh implied volailiy hree monhs from now should be 13 2 because 0 5 0 1 2 0 25 0 05 0 132 2 The expecaion hypohesis herefore suggess ha he erm srucure of opion-implied volailiy conain informaion abou he marke s percepion abou he fuure dynamics of shor erm implied volailiy Saring from he Hull and Whie 1987 sochasic volailiy model, Campa e al 1998b es he expecaion hypohesis for FX implied volailiy and fail o rejec he hypohesis A hird source of informaion from currency opions is by using correlaions of opions on differen currency pairs o consruc global measures of FX risk Opion-implied correlaions arise from hree way arbirage argumens For example if he exchange raes a ime are given by S AB S AC, and S BC, and assuming hey follow saionary processes, we have ha ln s AB, ln s AC, ln s BC, s AC, s BC, 3 7 15.17 Equaion 3 7 above implies ha V ar s AB V ar s AC V ar s BC 2Corr s AC, s BC V ar s AC 1 2 V ar s BC 1 2, 3 8 which can be rearranged o give Corr s AC, s BC V ar s AC V ar s BC V ar s AB 2V ar 1 2 s AC V ar s BC 1 2 3 9 If we use opion-implied variance o esimae he righ hand side of equaion 3 9 , hen he resuling esimae of s AC, s BC is opion - implied correlaion Siegel 1997 poins ou ha his opion-implied correlaion reveals marke senimen regarding how closely he currencies are expeced o move in he fuure The average opion-implied correlaion can be inerpreed as capuring global FX correlaion risk Recenly, effors aiming o idenify porfolio reurn - based global risk facors offer some empirical success in explaining he cross-secional disribuion of excess FX reurns The majoriy of exising research in his line of lieraure, however, use proxies of global risk consruced from hisorical reurns and focus on mached-frequency analysis Given he advanages of using opion price daa oulined in secion 1 , a naural quesion o ask is wheher opions-based measures of FX global risk add furher insighs o he srand of lieraure using global FX risk o explain FX excess reurns and FX reurns 17 Verdelhan 2012 and Lusig e al 2011 , for example, idenify a carry facor based on cross secion of ineres rae-sored currency reurns, and a dollar facor based on cross-secion of bea-sored currency reurns Raffery 2011 consrucs a global skewness risk facor using hisorical reurns from carry rade porfolios and shows ha higher average excess reurns co-vary more posiively wih global skewness 18 Menkhoff e al 2012 invesigae he role of global volailiy risk in explaining cross-secions of carry rade reurns, and conclude ha carry rade reurns are compensaion for exposure o global volailiy risk Mueller e al 2012 invesigae he role of global correlaion risk as a driver of currency reurns Cenedes e al 2012 show ha higher average is significanly relaed o large fuure carry rade losses, while lower average correlaion is significanly relaed o large gains 16.18 3 2 Exracing Opion-Implied Momens 19 We use he mehodology of Bakshi e al 2003 henceforh BKM o exrac model-free opion-implied sandard deviaion, skewness and kurosis from he volailiy smile Grad 2010 and Jurek 2009 also use he BKM mehodology o exrac FX opions-implied higher order momens 20 The exraced momens using he BKM mehodology are model-free because we make no assumpions regarding he ime series process governing he underlying spo exchange rae The model-free naure of he mehodology is aracive because i means he mehodology is equally applicable o all exchange rae regimes Campa e al 1998a argue ha having a mehodology ha does no presuppose a sochasic process followed by he underlying spo exchange rae is especially useful in siuaions where he FX regime is unknown or changing, or when he degree of governmen inervenion is unclear The BKM mehodology ress on he resuls of Carr and Madan 2001 , which show ha if we have an arbirary claim wih a pay-off funcion HS wih finie expecaions, hen HS can be replicaed if we have a coninuum of opion prices They also show ha if HS is wice-differeniable, hen i can be spanned algebraically by he following expression HSHSS SH SS where HSHSSH SS KSKS 0 H SS KKS dk, 3 10 and H SS 2 HS 2 Assuming no arbirage opporuniies, he price of a claim wih pay-off HS is given by he expression p HS SH SS e rd HSS Se rd SSH SS KC , , KH SS KP , , K dK 0 3 11 where K is he srike price, C , , K and P , , K are, respecively, he prices of a 19 Exracion of momens done in he R saisical language R Core Team 2013 20 In his secion we closely follow he exposiion and noaion in Grad 2010 17.19 European-syle call and pu opions S is some arbirary consan, usually chosen o equal curren spo price Equaion 3 11 indicaes ha any pay-off funcion HS can be replicaed by a posiion of HS SH SS in he domesic risk-free bond, a posiion of HS in he sock, and combinaions of ou-of-he-money calls and pus, wih weighs H SS K Suppose we have conracs wih he following pay-off funcions 21 RS 2, Volailiy Conrac HSRS 3, Cubic Conrac RS 4, Quaric Conrac, 3 12 where RS ln SS BKM show ha he variance, skewness and kurosis of he disribuion of R can be calculaed using he following formulas Sdev , e rd V , , 2 3 13a Skew , erd W , 3V , , e rd 2 , 3 e rd V , , 2 3 2 3 13b Kur , erd X , 4e rd , w , 6e rd , 2 V , 3 , 4 e rd V , , 2 2, 3 13c where he expressions for V ,,w , and X , and , are given in appendix A Derivaions of equaions in 3 13 and expressions for , , V , , W , and X , can be found in Bakshi e al 2003 and Grad 2010 The BKM mehodology described above requires a coninuum of exercise prices However, in he OTC FX opions marke implied volailiies are observed for only a discree number of exercise prices We herefore need a way o esimae he enire volailiy smile from a few 21 One can use he framework o price conracs wih higher order payoffs and herefore exrac momens of order higher han 4 The poin ha we wan o emphasize, ha higher order momens maer, is demonsraed even if we only sop a 4 h order 18.20 K pairs by inerpolaion and exrapolaion To his end, we use he Vanna Volga VV mehod described in Casagna and Mercurio 2007 The procedure allows us o build he enire volailiy smile using only hree poins Casagna and Mercurio 2007 noe ha if we have hree opions wih implied volailiy 1, 2, 3 and corresponding exercise prices K 1,K 2 and K 3 such ha K 1 K 2 K 3, hen he implied volailiy of an opion wih arbirary exercise price K can be accuraely approximaed by he following expression k 2 2 d 1 K d 2 K 2 2 D 1 KD 2 K , 3 14 d 1 K d 2 K where D 1 K ln K 2 K ln K3 ln K 1 K ln 2 KK 1 ln 3 K 1 ln KK 1 K 2 K 1 ln ln K 3 KK 3 K 2 2 ln ln KK 1 ln K 3 K 1 ln KK 2 K 3 K 2 3 2, D 2 K ln K 2 K ln K3 ln KK d 1 K 1 d 2 K 1 1 2 2 K 1 ln KK 2 K ln 2 KK 1 ln 3 ln K 3 KK 1 ln K 3 K 2 d 1 K 3 d 2 K 3 3 2 2 1 and d 1 x log S x rd rf 2 2 2 , d 2 xd 1 x 2 , x K, K1, K 2, K 3 Expression 3 14 allows us o find he implied volailiy of an opion wih an arbirary srike price We use K 1 K 25 p, K 2 K AT M and K 3 K 25 c The VV mehodology has a number of aracive feaures, which are explained in Casagna and Mercurio 2007 Firs, i is parsimonious because i uses only hree opion combinaions o build an enire volailiy smile This is he minimum number ha can be used if one wans o capure he hree mos prominen movemens in he volailiy smile change in level, change in slope, and change in curvaure 22 The VV mehod also has a solid financial moivaion Casagna and Mercurio 22 The ATM sraddle, VWB and he Risk Reversal capure hese movemens See discussions in Casagna 19.21 2007 show ha i is based on a replicaion argumen in which an invesor consrucs a porfolio ha, in addiion o hedging agains movemens in he price of he underlying asse CC , also hedges agains movemens in volailiy of he underlying asse V ega S In siuaions where volailiy is sochasic, i migh be useful o consruc porfolios ha, in addiion o hedging agains changes in he price of he underlying asse, he invesor also hedges agains for he Vega C , he Vanna 2 C and he Volga 2 C as migh be necessary 2 S in siuaions when volailiy is sochasic 3 3 Daa Descripion In he o--c marke, he exchange rae is quoed as he domesic price of foreign currency, which means a fall in he repored exchange rae represens an appreciaion of domesic currency Domesic and foreign, however, do no have any geographic significance, bu are in accordance o some marke quoing convenions Table 1A conains deails of he marke quoing convenions for he six currency pairs ha we use in his paper Compared o exchange-raded opions, here are several advanages ha come wih using o--c daa in our empirical analysis Firs, mos of he FX opions rading is concenraed in he o--c marke This means o--c currency opions prices are more compeiive and herefore more likely o be represenaive of aggregae marke beliefs compared o prices in he less liquid exchange marke Table 1C , obained from he 2010 BIS Triennial Survey, shows ha alhough he o--c opions marke is small relaive o he overall FX marke, i is very liquid and rapidly growing when we look ai in absolue erms INSERT TABLE 1C HERE A second advanage of using o--c opion price daa is ha fresh opions for sandard enors are quoed each day, making i possible o obain a ime series of FX opion prices wih 2010 and Malz 1998 20.22 consan mauriies This can be conrased wih exchange raded opions, whose prices are quoed for a specific expiry dae, such ha as we approach he expiry dae, he opion prices also incorporae he fac ha he enor is changing O--c opions make i possible o disenangle erm srucure, cross-secional and ime series aspecs embedded in opion prices Our hird and final moivaion for o--c opion daa is ha European-syle opions are raded in his marke, whereas exchange raded FX opions are usually American-syle When analyzing opion prices of a given enor, American-syle opions have o be adjused for he possibiliy of early exercise We nex explain some imporan OTC currency marke quoing convenions Firs, opion prices are given in erms of implied volailiy insead of currency unis while moneyness is measured in erms of he dela of an opion The dela of an opion is a measure of he responsiveness of he opion s price wih respec oa change in he price of he underlying asse If he prices of call and pu opions are given by C and P, hen opion price and implied volailiy are linked using he formula from Garman and Kohlhagen 1983 s exension of he Black-Scholes model o FX C e rd F d 1 K d 2 P e rd K d 2 F d 1 where d 1 log SK rd rf 2 2 2 , d 2 d 1 2 , There is a one - o-one relaionship beween opion price and implied volailiy when using he Black and Scholes 1973 formula 23 The expressions for call and pu delas are given by he expressions 23 Use of he Black-Scholes formula does no, however, mean raders agree wih he assumpions underlying he Black-Scholes model 21.23 ce rf d 1 pe rf d 1 , 3 15a 3 15b where is he sandard normal cumulaive densiy funcion cdf The absolue values of c and p are herefore beween 0 and 1, while pu-call pariy implies ha pc 1 The marke convenion is o quoe a dela of magniude x as a 100 x dela For example, a pu opion wih a dela of is referred o as a 25 pu Lasly, in he FX o--c opion marke, prices are quoed in combinaions raher han simple call and pu opions The mos common opion combinaions are a-he-money ATM 24 sraddle, risk reversals RR , and Vega-weighed buerflies VWB An ATM sraddle is he sum of a base currency call and a base currency pu , boh sruck a he curren forward rae This is he mos liquid srucure in he o--c FX opions marke A RR is se up when one buys a base currency call and sells a base currency pu wih a symmeric dela The mos liquid RR is he 25 , in which boh he call and pu have a dela of 25 percen Finally, a VWB is buil by buying a symmeric dela srangle and selling an ATM sraddle 25 The 25 combinaion is he mos raded opions VWB The ATM sraddle, risk reversal and srangle are usually inerpreed as shor cu indicaors of volailiy, skewness and kurosis of he perceived condiional disribuion of exchange rae movemens The profi diagrams in figure 6 demonsrae why i he sraddle becomes profiable if here is a movemen in he underlying asse s price ii he risk - reversal makes profi if here is a movemen in a paricular direcion iii he srangle becomes profiable if here is a big movemen in any direcion in he 24 ATM here means he dela of he opion combinaion is zero Tha is, he opion combinaion is dela - neural 25 In a srangle, you buy an ou of he money call and an equally ou of he money pu 22.24 underlying asse s price INSERT FIGURE 6 HERE The definiions of he hree opion combinaions are as follows 26 AT M, 0 c, 50 c 50 p 25 RR, 25 c, 25 p, 25 vwb, 25 c, 25 p, Srangle AT M, 3 16a 3 16b 3 16c Equaions 3 16 can be rearranged o ge he implied volailiy for 0 call, 25 call and 25 pu Expressions for backing ou implied volailiy of hese plain-vanilla opions from he prices of raded opion combinaions are given below 0 c, AT M 50 c, 50 p, 3 17a 25 c, AT M 25 vwb, 25 RR, 3 17b 25 p , AT M 25 vwb, 1 2 25 RR, 3 17c Finally, K 25 p, K AT M, K 25 c, he exercise prices corresponding o AT M, , 25 c, and 25 p, can be backed ou by using he expression for opion delas given in equaion 3 15 For example, o ge K AT M we use he fac ha he ATM sraddle has a dela of zero e rf ln SK AT M rdrf 2 AT M ln AT MSK AT M rdrf 2 AT M AT M 26 Table 1B conains sample opion price quoes for sandard combinaions and sandard mauriies 0 3 18 23.25 Since is a monoone funcion, we can solve equaion 3 18 for K AT M o ge K AT MS e rd rf 2 AT MF e 1 2 2 AT M 3 19 Using similar argumens, one can show ha he expressions for K 25 c and K 25 p K 25 c S e 1 1 4 erd 25 c, rdrf 2 25 c K 25 p S e 1 1 4 erd 25 p, rdrf 2 25 p , 3 20a 3 20b wih K 25 p K AT MK 25 c Casagna and Mercurio 2007 Our opions daa consiss of over he couner o--c opion prices for he six currency pairs lised in able 1A and covering he period 1 January 2007 o April The spo raes, forward raes and risk-free ineres raes are obained from Daasream 4 Empirical Sraegy and Main Resuls 4 1 Empirical properies of exraced opion-implied momens Summary saisics of he exraced momens are in able 2 27 The summary saisics show ha all he exraced momens are generally very persisen, wih AR 1 coefficiens as high as Zivo and Andrews 1992 uni roo ess, however, sugges ha almos all he implied momens are saionary wih srucural breaks in he means on daes around lae 2008 and early 2009 For he res of he analysis, we rea all variables as saionary Looking a he maximum and median for each series, as well as he ime series plos, we see ha here are a number of ouliers, especially for he 9m and 12m momens The ime 27 Summary saisics for he opion-implied momens of he oher five currency pairs are similar, and can be found in he online appendix Time series plos for 1W K, 2M, 3M, 6M, 9M and 12M are also in he online appendix 24.26 series plos in figure 2 show ha hese ouliers are found mosly beween lae 2008 and early INSERT TABLE 2 AND FIGURE 2 HERE 4 2 Can opion-implied momens forecas FX excess reurns Mached Frequency Analysis Predicive abiliy of he volailiy smile We sar by invesigaing wheher - period opion-implied momens can explain he condiional mean of subsequen excess reurns Thus, for each currency pair i and enor , we esimae he following regression model by OLS fi, E si 0, 1, sdev i, 2, skew i, 3, kur i, ui, 4 1 Noe ha he LHS variable is ex-ane excess currency reurns forward bias Under raional expressions, fi, E si is also equal o he risk premium Gereben 2002 and Malz 1997 also esimae regression specificaion 4 1 and inerpre he resuls in ligh of he ime-varying risk premia explanaion of he UIP puzzle Gereben 2002 argues ha if he forward bias is due o ime-varying risk premia, hen variables ha capure he naure of FX risk should be able o explain he dynamics of he forward bias The opion-implied momens on he RHS in regression equaion 4 1 , which capure perceived FX volailiy, ail and crash risk should herefore be able o explain he forward bias Malz 1997 also argues ha saisical significance of he coefficien on skew can be inerpreed as providing suppor for he peso problem explanaion of he UIP puzzle Going back o expression 4 1 , we noe ha E s is no observable If we assume ha marke paricipans have raional expecaions, hen E s and s will only differ by a forecas error 1 ha is uncorrelaed wih all variables ha use informaion a ime, such 25.27 ha s E s 1 4 2 Plugging equaion 4 2 ino equaion 4 1 and rearranging gives us he following esimable regression equaion xr 0, 1, sdev 2, skew 3, kur 4 3 where he error erm u and xr is ex-pos excess reurns defined in expression 2 3 To provide inuiion regarding expeced coefficien signs in he regression equaion 4 3 , we ake he poin view of a domesic invesor who invess in domesic bonds using money borrowed from abroad As shown in equaion 2 3 , such an invesor benefis from higher domesic ineres raes as well as appreciaion of domesic currency Le s also assume ha he home currency is riskier, such ha our invesor would demand higher excess reurns for higher sdev and kurosis in he exchange rae If invesors are averse o high variance and kurosis, hey would require higher excess reurns for holding bonds denominaed in unis of he riskier domesic and we would expec he coefficiens on sdev and kurosis o be boh posiive We expec he skew coefficien o be posiive for invesor s wih preference for posiive skewness Such an invesor will require higher compensaion for an increase in skew, which represens a higher perceived likelihood of domesic currency depreciaion Given he discussion in subsecion 5 2 , however, we noe ha pinning down he coefficien signs a priori is impossible wihou making furher assumpions abou he invesor s uiliy funcion or orhogonaliy of he momens In our regression analysis, we herefore focus mainly on join significance of he explanaory variables and model fi raher han on significance and signs of individual coefficiens Sub - sample analyses sugges he presence of srucural breaks in he mached-frequency 26.28 regression relaionships for he majoriy of currency pairs and enors We use he Bai and Perron 2003 srucural break es o idenify he dae for he mos prominen break 28 and esimae a modificaion of regression equaion 4 3 ha includes ineracions wih srucural break indicaor variable xr i 0, 00, D1 i, D1 i, 1, sdev i, 3, kur i, 4, sdev i, D1 i, 2, skew i, 5, skew i, 6, kur i, D1 i, i 4 4 where D1 i, is an indicaor variable ha is zero before he break dae and equal o one oherwise The mached-frequency resuls, shown in ables 3 a -3 f , demonsrae a consisen abiliy of opions-based measures of FX sandard deviaion, skewness and kurosis-proxying o explain excess currency reurns The coefficiens on he six non-inercep erms are always joinly significan, as shown by he low p-values for he Wald ess across currency pairs and across enors The adjused R 2 s are also generally high across currency pairs and enors, for example, ranging from 11 o 28 for 1M enors We carry ou a baery of robusness checks on he mached frequency resuls presened in able 3 Firs, we noe ha since we are using overlapping daa, he R 2 s will be inflaed 29 To ge an idea of he degree of R 2 inflaion and see if our resuls sill change when we use non-overlapping daa, we re-run he regression 4 4 for 1M enor using non-overlapping observaions We sill use he same break dae found from he regressions wih overlapping daa regressions, which are presened in able 3 The resuls of regressions wih non-overlapping daa regressions, shown in able 4a , sugges ha he mached frequency resuls presened in able 3 are no being enirely driven by our use of overlapping daa Resuls from sub-sample analysis and regressions using 10 opions insead of 25 are 28 We only focus on he major breaks, and herefore do no choose he number of breaks according o informaion crieria such as AIC 29 For ha reason, we do no inerpre he higher R 2 s for 12m regressions as represening beer fi a longer horizons 27.29 presened in ables 4 b and 4 c respecively Again, when we look a he adjused R 2 s and ess of join significance of coefficiens on he momens, we find ha here are no major differences wih he resuls presened in able 3 Our final robusness check addresses he issue of ouliers The summary saisics of he exraced momens show some huge ouliers In he presence of ouliers, ordinary leas squares migh give misleading resuls For 3M enor, we re-esimae he regressions specificaion 4 4 using robus leas squares Our esimaion mehod addresses he presence of ouliers in boh he dependen variable and independen variables Again, he main findings sill hold, as can be seen in able able 4d INSERT TABLE 4 HERE We digress from he bilaeral analysis we have done so far in his secion o invesigae wheher opions-based measures of global volailiy, skewness and kurosis can explain he dynamics of bilaeral excess reurns For he mached frequency global risk regressions, resuls of which are presened in 5 a , we exrac he firs hree componens from each of 3M sandard deviaion, skewness and kurosis across all currency pairs involving he USD The coefficiens on he pricincipal componens are joinly significan , wih adjused R 2 s ranging from 14 o 26 We hen exend he global risk regression o incorporae erm srucure informaion by using principal componens exraced from all currency pairs and from all enors as regressors The resuls from he erm-srucure of global risk regression, presened in able 5 b , show ha informaion from he erm srucure of global risk adds furher explanaory power, wih adjused R 2 s ranging from 16 o 40 INSERT TABLE 5 HERE We nex go beyond OLS regression, which models he condiional mean of he he dependen variable given he explanaory variables, by using quanile regression analysis QR o invesigae he predicive abiliy of opions-based FX risk measures for he enire 28.30 disribuion of ex-pos excess currency reurns By modeling he enire disribuion of he dependen variable, QR allows us o ge a more complee picure of he predicive abiliy of he opion-implied momens QR also has a furher advanage over OLS in ha i is robus o ouliers in he dependen variable and does no impose resricive disribuional assumpions on he error erms We esimae he following linear quanile regression model, modified o include one break Q xr i 0, 1, ST DEV i, 2, SKEW i, 3, KURT i, i, , 4 5 where Q xr i is he h quanile of excess reurns given informaion available a ime 30 Mached-frequency quanile regression resuls for 3M enor are shown in ables 6a - 6f We find ha he coefficiens on non-inercep erms are always joinly significan across quaniles for all currency pairs Adjused R 2 s are also consisenly high, ranging from 16 o 44 for AUDUSD and 10 o 26 for USDJPY for example Anoher consisen paern across currency pairs and enors is ha opion-implied momens have more predicive abiliy for lower and upper quaniles of excess reurns han he middle quaniles INSERT TABLES 6a - 6f HERE Can he erm srucure of implied momens explain FX excess reurns The mached-frequency resuls presened in subsecion 4 2 1 sugges ha opions-based measures of FX higher momen risks consisenly explain subsequen bilaeral excess reurns We now urn o sudying he predicive abiliy of he erm srucure of opions-implied momens for currency excess reurns We firs exend regression equaion 4 3 by regressing 3M bilaeral excess reurns on 1M, 3M and 12M opion-implied momens Tha is, for each currency pair i, we esimae 30 We esimae he quanile regression model using he same break daes obained in he OLS analysis 29.31 he following OLS regression xr i 3M 0,3M j 1, j sdev j, ij 2, j skew j, ij 3, j kur j, ii 3M, 4 6 where j Similar o he mached-frequency analysis in subsecion 4 2 1 , our final erm srucure regression model is a modificaion of 4 6 in which we include ineracions wih a srucural break indicaor variable D1 Regression resuls from specificaion 4 6 wih break are shown in column B of able 7 Compared o he mached frequency resuls presened in column A, we see a huge increase in he adjused R 2 s wih adjused R 2 s now ranging from 58 o 74 for he resuls from equaion 4 6 In column C of able 7 , we presen condensed resuls of regressions ha incorporae informaion from all enors no jus 1M,3M and 12M by using principal componens exraced from all enors Column C herefore conains resuls from he following regression 3 3 xr 3m i 0, 2,j PC j sdevt erm i 3,j PC j skewt erm ij 1 j 1 3 4,j PC j kurt erm ii 3M j 1 4 7 In equaion 4 7 , PC j xxxxt erm i refers o he jh principal componen exrac from he currency i erm srucure of opion - implied momen xxxx Resuls from esimaion regression equaion 4 7 are in column C of able 7 Lasly, we exend he specificaion in 4 7 by adding informaion from he erm srucure of firs momens as addiional regressors 30.32 3 3 xr 3m i 0, 1, j PC j meant erm i 2,j PC j sdevt erm ij 1 3 3,j PC j skewt erm ij 1 j 1 j 1 3 4,j PC j kurt erm ii 3M 4 8 As we argued earlier, he erm srucure of firs momens capures expecaions of he dynamics of fuure macroeconomic fundamenals We use he erm srucure of ineres rae differenials o exrac he principal componens of he erm srucure of firs momens of log SS As we noed in 3 1 , under CIP, he forward premium fs, which is he heoreical mean of he risk-neural probabiliy densiy of log SS is equal o he ineres differenial ii, Using yield curve daa o exrac he erm srucure of firs momens has he advanage of allowing us o also use ineres rae differenials for enors no covered by our opion price daa As wih our previous regressions, we esimae a version of regression model 4 8 ha includes ineracions wih a srucural break indicaor variable The condensed resuls from esimaing equaion 4 8 wih breaks are presened in column D of able 7 Acual vs fied plos from his regression are shown in figures 3 a -3 e INSERT FIGURE 3 AND TABLE 7 HERE The main finding from comparing columns C and D is ha informaion from he erm srucure of firs momens is no redundan The adjused R 2 s all show sizable increases, and Wald ess for he null hypohesis ha all coefficiens on he firs momen principal componens are zero sugges he firs momens are conribuing addiional explanaory power The main conclusion from analysis of he resuls presened in able 7 is ha FX risks, capured by he higher order momens, and expecaions, capured by he erm srucure of implied momens, have subsanial explanaory power for ex-pos excess currency reurns 31.33 4 3 Can opion-implied momens forecas currency reurns In subsecion 4 3 , we invesigae he abiliy of opions - based measures of higher momen risks and heir erm srucures o explain currency reurns s Can he volailiy smile predic currency reurns For each currency pair i, we sar by esimaing he sandard UIP regression sisifisii 4 9 We focus on model fi and join significance raher han esing wheher he coefficien is equal o 1 Fied vs Acual plos of esimaed regression 4 9 wih breaks are shown in figures 4 a - 4 e , while condensed resuls can be found in column A of able 8 We hen consider he predicive abiliy of - period opion - implied higher momens by esimaing he following augmened UIP regression sisi 1 fi, s 2 sdev i, 3 skew i, 4 kur i, i, 4 10 Equaion 4 10 herefore augmens he sandard UIP equaion 4 9 by sudying he predicive abiliy of he 1 s 4 h momens of he disribuion of log SS The condensed regressions resuls are shown in column B able 8 The adjused R 2 s for he mached-frequency augmened UIP regressions are consisenly high and he higher order momens are always joinly significan INSERT TABLE 8 AND FIGURE 5 HERE Can he erm srucure of implied momens predic currency reurns We move on o sudying wheher he erm srucure of opions-implied momens have predicive abiliy for subsequen FX reurns We sar by esimaing a erm srucure 32.34 modificaion of he sandard UIP equaion 4 9 ha uses informaion conained in he erm srucure of forward premia 3 sisi 0, 1,j PC j meant erm i 4 11 j 1 Condensed resuls from regression specificaion 4 11 are presened in column C of able 8 Comparing columns A and C in able 8 , we see ha adding he whole erm srucure of forward premia significanly improves he UIP regression fi Lasly, we regress exchange rae movemens on he erm srucure of 1 s 4 h momens 3 3 si 3M si 0, 1,j PC j meant erm i 2,j PC j sdevt erm ij 1 3 3,j PC j skewt erm ij 1 j 1 j 1 3 4,j PC j kurt erm ii 3M 4 12 Plos of acual versus fied values from regression 4 12 are shown in figures 4 and he condensed regression resuls are in column D of able 8 INSERT TABLE 8 AND FIGURE 4 HERE Comparing columns C and D in able 8 , we find ha he erm srucure of 1 s -4 h momens adds a significan amoun of explanaory power for exchange rae movemens The main conclusion from he resuls presened in able 8 is ha higher order momens and expecaions capured hrough erm srucure dynamics combine o explain subsequen exchange rae movemens 33.35 5 Furher Inerpreaion and Discussion 5 1 Higher Momens Maer Asse Pricing Derivaion of UIP Condiion 31 The fundamenal asse pricing equaion is given by EMR 1, 5 1 where M is he pricing kernel and RSS is he gross reurn on an asse Suppose ha asses can be denoed in domesic or foreign currency unis Under complee markes, he following relaionship holds MMSS 5 2 where M is he foreign pricing kernel By aking logs and condiional expecaions, expression 5 2 can be wrien as E ss E logm E logm , 5 3 where s log s 32 Applying pricing equaion 5 1 o price a forward conrac yields EMFS 0 5 4 31 The maerial in his subsecion is from Backus e al 2001 32 Wriing he reurns in he form 5 3 also makes i clear why macroeconomic fundamenals such as consumpion growh are expeced o explain currency excess reurns As poined ou by Backus e al 2011 , in macroeconomics, he pricing kernel is ied o macroeconomic quaniies such as consumpion growh Expression 5 7 herefore suggess ha he dynamics of FX reurns should be explained by domesic and foreign macroeconomic fundamenals 34.36 Dividing equaion 5 4 by S and using he resul in expression in equaion 5 2 gives he expression for he forward premium fs log e M log e M 5 5 Applying he asse pricing equaion 5 1 o price one-period domesic and foreign risk free bonds, we ge expressions for he shor raes i log e M and i log e M Equaion 5 5 and he expressions for shor raes give us he CIP condiion ii log e M log e M fs 5 6 Finally, he expression for ex-ane currency excess reurns or deviaion from UIP condiion is herefore given by f E sii E ss loge ME logm loge ME logm 5 7 Under risk-neuraliy he RHS of expression 5 7 is zero, and we ge he forward unbiasedness condiion f E s Risk aversion is capured in he pricing kernel Expression 5 7 is herefore someimes referred o as FX risk premium Equaion 5 7 makes i clear why he failure of he UIP condiion is usually aribued o ime-varying risk and expecaional errors Excess reurns should heoreically depend on ime-varying cross-counry differences in risk, capured hrough he pricing kernels This risk could include liquidiy risk, business-cycle relaed risks, poliical risk, and liquidiy risk In macroeconomics, pricing kernel is linked o macroeconomic fundamenals such as consumpion growh Thus, expression 5 7 also suggess ha currency excess reurns should depend on differences in expeced macroeconomic condiions As we menioned earlier, he difficuly faced by he lieraure is ha sandard proposed measures of risk do no appear o have srong correlaion wih excess reurns 35.37 Equaion 5 7 is also insighful in showing how excess reurns can poenially be explained by higher order momens This can be seen clearly by expressing loge M in erms of he cumulans of he condiional disribuion of logm loge M jj 1, where jj is he jh cumulan of he condiional disribuion of log m Cumulans are closely relaed o momens, and he expressions for he firs four cumulans are 1 1, 2 2, 3 3 and 4 4 3 2 2, where 1 is he condiional mean and j denoes he jh cenral momen of he disribuion of loge M Equaion 5 7 can herefore also be wrien in he form f E sjjj 2 j 5 8 where j and j are he jh order cumulan of log m and log m respecively Equaion 5 8 illusraes ha currency excess reurns will in general depend on he higher order momens of he disribuion of he pricing kernel Noe ha if we assume ha pricing kernels are log-normally disribued, hen expression 5 8 reduces o E s 2 2 2 f The preceding discussion ye again illusraes how disribuional assumpions can poenially lead oa disregard for higher order momens which migh crucial in empirical daa 5 2 Higher momens maer asse allocaion under higher order momens 33 We showed in subsecion 2 2 ha he assumpions of CARA uiliy and normaliy of reurns reduce he invesor s problem o mean-variance opimizaion However, if he disribuion of porfolio reurns is asymmeric, or he invesor s uiliy funcion is of a higher order 33 Maerial in his subsecion is from Jondeau e al 2010 36.38 han he quadraic, or he mean and variance do no compleely deermine he disribuion of asse reurns, hen higher order momens and heir signs mus be aken ino accoun in he porfolio asse allocaion problem In his subsecion we presen a framework for incorporaing higher order momens ino he asse allocaion problem The objecive in 2 4 can be inracable and i is usual o focus on approximaion of 2 4 based on higher order momens Jondeau e al 2010 consider a Taylor s series expansion of he uiliy funcion around expeced uiliy up o he fourh order UW 1 UEW 1 U 1 W 1 W 1 EW 1 1 2 U 2 W 1 W 1 EW 1 U 3 W 3 1 W 1 EW 1 U 4 W 4 1 W 1 EW 1 4, 5 9 where U n denoes he nh derivaive of he uiliy funcion wih respec o nex period wealh Taking he condiional expecaion of expression 5 9 yields EUW 1 UEW 1 U 1 W 1 W 1 EW 1 1 2 U 2 W 1 W 1 EW 1 U 3 W 3 1 W 1 EW 1 U 4 W 4 1 W 1 EW 1 4 5 10 Under he assumpion ha he invesor s uiliy funcion is CARA, expression 5 10 reduces o EUW 1 ep 1 2 2 2 p 3 6 s3 p 4 24 k4 p 5 11 In equaion 5 11 , s 3 p and kp 4 are he skewness and kurosis of porfolio reurn I is clear from equaion 5 11 ha under CARA uiliy, invesors prefer posiive skewness and dislike high variance and high kurosis Opimal porfolio weighs can hen be obained by maximizing expression 5 10 insead of he exac objecive funcion shown in expression 2 4 For CARA uiliy, he weigh he invesor pus on he higher order momens depends on he degree of risk aversion parameer In more general seings, however, he weigh on he nh momen depends on he nh derivaive of he uiliy funcion, and he signs of 37.39 sensiiviies of uiliy funcion o changes in higher momens canno be easily pinned down If he momens are no orhogonal o each oher, hen he effec of uiliy of increasing one momen migh no be sraigh forward Sco and Horvah 1980 esablish some general condiions for invesor preference for skewness and kurosis 5 3 Higher Momens Maer Higher order ICAPM 34 We sar wih he fundamenal pricing equaion, EMR i 1 5 12 for each asse i where M is he pricing kernel and R are - period gross reurns In consumpion-based asse pricing models, M is also equal o he invesor s marginal rae of subsiuion beween curren and fuure consumpion The sandard wo momen CAPM follows from assuming a linear relaionship beween he pricing kernel and, R , w he reurns o he world porfolio Harvey 1991 show ha if markes are globally inegraed, cross - counry porfolio reurns should be driven by condiional covariances of counry porfolio reurns and reurns o he world porfolio ER i R f R w V ar R w Cov R i , R w , 5 13 wih R i and R f denominaed in he same currency Equaion 5 13 is he wo-momen inernaional CAPM ICAPM To incorporae fourh order momens, we assume ha he pricing kernel can be approximaed by a hird order Taylor expansion of he marginal uiliy of world reurns M 1 WU 2 WU 1 W Rw WU 3 W 2 U 2 W Rw 2 WU 4 W 3 U 1 W Rw 3 5 14 34 From Guidolin and Timmerman 2008 38.40 Combining equaions 5 12 and 5 14 gives us he four-momen CAPM, ER i R f 1 Cov R , i R w 2 Cov R , i R w 2 3 Cov R , i R w 3 covariance coskewness cokurosis 5 15 Equaion 5 15 says ha he excess reurns on asse i will depend on he covariance, co-skewness and co-kurosis on he reurns o ha asse and he reurns on he world porfolio 6 Conclusion This paper has documened a robus abiliy of opions-implied measures of FX higher momen risks o explain subsequen excess currency reurns and FX reurns We also find ha he erm srucure of such risks, capuring forward-looking propery of he exchange rae, add furher explanaory power Our findings sugges ha expecaion and risk should be given more careful consideraion in he 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STANDARD DEVIATION AND KURTOSIS b AUDUSD 1M SKEWNESS c EURUSD 1M STANDARD DEVIATION AND KURTOSIS d EURUSD 1M SKEWNESS 47.49 Figure 2 Time Series Evoluion Of 1M Opion Implied Momens e GBPUSD 1M STANDARD DEVIATION AND KURTOSIS f GBPUSD 1M SKEWNESS g USDCAD 1M STANDARD DEVIATION AND KURTOSIS Noe Momens exraced using he mehodology developed in Bakshi e al 2003 48.50 Figure 2 Time Series Evoluion Of 1M Opion Implied Momens h USDCAD 1M SKEWNESS i USDJPY 1M STANDARD DEVIATION AND KURTOSIS j USDJPY 1M SKEWNESS Noe Momens exraced using he mehodology developed in Bakshi e al 2003 49.51 Figure 3 Quarerly FX Excess Reurns on Term Srucure of 1 so 4 h Momens Break a AUDUSD 3M b EURUSD 3M c GBPUSD 3M Noe Fied vs Acual plos from he regression of 3M excess reurn, as defined in expression 2 3 , on he firs hree principal componens from he erm srucure of exraced momens of Q ln STS Regression specificaion in expression 4 8 Condensed regression resuls are in column D of able 7 50.52 Figure 3 Quarerly FX Excess Reurns on Term Srucure of 1 so 4 h Momens Break d USDCAD 3M e USDJPY 3M Noe Fied vs Acual plos from he regression of 3M excess reurn, as defined in expression 2 3 , on he firs hree principal componens from he erm srucure of exraced momens of Q ln STS Regression specificaion in expression 4 8 Condensed regression resuls are in column D of able 7 51.53 Figure 4 Quarerly Exchange Rae Movemens on Term Srucure of 1 so 4 h Momens Break a AUDUSD 3M RET b EURUSD 3M RET c GBPUSD 3M RET Fied vs Acual plos from he regression of 3M log ST on he firs hree principal componens from he erm srucure of exraced momens of Q ln STS Regression specificaion in expression 4 12 Condensed regression resuls are in column D 8 S 52.54 Figure 4 Quarerly Exchange Rae Movemens on Term Srucure of 1 so 4 h Momens Break d USDCAD 3M RET e USDJPY 3M RET Fied vs Acual plos from he regression of 3M log ST on he firs hree principal componens from he erm srucure of exraced momens of Q ln STS Regression specificaion in expression 4 12 Condensed regression resuls are in column D of able 8 S 53.55 Figure 5 Quarerly Exchange Rae Movemens on Mached Frequency 1 s Momen Break a AUDUSD b EURUSD 3M UIP c GBPUSD 3M UIP Fied vs Acual plos from he regression of 3M log ST S on mached frequency forward premium sandard forward premium regression Regression specificaion in expression 4 9 Condensed regression resuls for all currency pairs are in column A of able 8 54.56 Figure 5 QuarerlyExchange Rae Movemens on Mached Frequency 1 s Momen Break d USDCAD 3M UIP e USDJPY 3M UIP Fied vs Acual plos from he regression of 3M log ST S on mached frequency forward premium sandard forward premium regression Regression specificaion in expression 4 9 Condensed regression resuls for all currency pairs are in column A able 8 55.
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