Option-Implied Currency Risk Premia

Transcription

1 Opon-Impled Currency Rsk Prema Jakub W. Jurek and Zhka Xu Absrac We use cross-seconal nformaon on he prces of G10 currency opons o calbrae a non-gaussan model of prcng kernel dynamcs and consruc esmaes of condonal currency rsk prema. We fnd ha he mean hsorcal reurns o shor dollar and carry facors (HML F X ) are sascally ndsngushable from her opon-mpled counerpars, whch are free from peso problems. Skewness and hgher-order momens of he prcng kernel nnovaons on average accoun for only 15% of he HML F X rsk premum n G10 currences. These resuls are conssen wh he observaon ha crash-hedged currency carry rades connue o delver posve excess reurns. FIRST DRAFT: MARCH 2013 THIS DRAFT: DECEMBER 2013 Jurek: Bendhem Cener for Fnance, Prnceon Unversy and NBER; jjurek@prnceon.edu. Xu: Deparmen of Operaons Research and Fnancal Engneerng, Prnceon Unversy; zhkaxu@prnceon.edu. We hank John Campbell, Peer Carr, Perre Colln- Dufresne, Valenn Haddad, Ian Marn, Peer Rchken, and semnar parcpans a Case Wesern Reserve Unversy, Darmouh, Georgeown Unversy, Harvard Busness School, Prnceon Unversy, Unversy of Rocheser, he 2 nd EDHEC-Prnceon Insuonal Asse Managemen Conference, and he Prnceon-Lausanne Quanave Fnance Workshop for provdng valuable commens.

2 The absence of arbrage lnks he exchange rae beween wo currences and he economes respecve sochasc dscoun facors. We explo hs lnk o exrac he dynamcs of sochasc dscoun facors from he cross-secon of currency opons and produce a me seres of opon-mpled currency rsk prema. The oponmpled currency rsk prema are free of peso problems, and allow us o elucdae he mporance of global jump rsks n he deermnaon of prema for common rsk facors n currency reurns. Esmaes of currency rsk prema are commonly derved by sudyng he me-seres and cross-seconal properes of hsorcal (realzed) reurns. Ths paper develops an alernave approach o hs queson, whch does no rely on hsorcal currency reurns, bu nsead uses cross-seconal daa on he prcng of exchange rae opons. We demonsrae ha, even hough no sngle currency opon s nformave abou he expeced reurn of he underlyng currency par, observng a suffcenly broad collecon of opons on currency cross-raes allows us o deduce he srucure of he sochasc dscoun facors, and herefore he dynamcs of currency rsk prema. 1 In order o operaonalze hs mehodology, we mpose a facor srucure on he prcng kernel dynamcs, whch assumes prcng kernels are drven by a combnaon of common (global) shocks and dosyncrac (counry-specfc) shocks, wh ndvdual counres dfferng n her exposure o global shocks. 2 Cross-seconal dfferences n global facor loadngs generae varaon n opon-mpled exchange rae dsrbuons, whch we explo o nfer he srucure of he prcng kernel nnovaons. We show ha currency rsk prema can be expressed usng he cumulan generang funcons of hese nnovaons, smlar o Backus, e al. (2011) and Marn (2013), generang a wo-facor prcng model for currency reurns. The global facor n our model corresponds o he HML F X rsk facor denfed by Lusg, e al. (2011) and s refleced n he reurns o currency carry rade sraeges whle he second facor represens compensaon demanded by nvesors for beng shor her own (local) currency. The model of prcng kernel dynamcs we calbrae o exchange rae opon daa can be nerpreed as a dscrezed verson of a connuous-me model wh me-changed Lévy ncremens (Carr and Wu (2004)). The global and counry-specfc prcng kernel nnovaons n our model are comprsed of a Gaussan componen and a non-gaussan componen, each of whch have me-varyng dsrbuons. The non-gaussan componen can nuvely be hough of as capurng he jon effecs of nra-perod sochasc volaly and jumps ( dsaser 1 Our ably o recover rsk prema from exchange rae opons can be raced back o he fac ha opon-mpled exchange rae dsrbuons reveal he condonal, counry-level prcng kernel dsrbuons. I s no an applcaon of he Recovery Theorem (Ross (2013)), and herefore does no rely on s underlyng assumpons (e.g. fne-sae Markov chans, me-homogenous dynamcs, saevarables followng bounded dffusons), dealed n Carr and Yu (2012). 2 The presence of asymmerc global facor loadngs can be used o raonalze volaons of uncovered neres pary (Backus, e al. (2001)), s conssen evdence from hsorcal currency reurns (Lusg, e al. (2011), Hassan and Mano (2013)), as well as, evdence from exchange rae opon markes (Baksh, e al. (2008)). Models wh mperfec rsk-sharng provde a mcrofoundaon for asymmerc global facor loadngs (Verdelhan (2010), Ready, Roussanov, and Ward (2013)). 1

3 rsks ). 3 These feaures combne o produce a model ha no only accommodaes non-gaussan nnovaons a each pon n me, bu also flexbly ncorporaes sochasc varaon n second and hgher momens, characersc of currency opon daa (Carr and Wu (2007), Baksh, e al. (2008)). 4 The avalably of a closed-form expresson for he generalzed Fourer ransform of he log currency reurn, enables us o effcenly recalbrae he model on each day n our sample (1999:1-2012:6; 3520 days) o mach he cross-secon of observed G10 exchange rae opon prces (up o 45 cross-rae pars x 5 srkes). 5 Fnally, usng he calbraed me-seres of model parameers, we compue esmaes of opon-mpled currency rsk prema for he currency pars n our panel. We confron he opon-mpled currency rsk prema wh he hsorcal reurns o emprcal facor mmckng porfolos denfed by Lusg, e al. (2011, 2013), and examne he model s ably o forecas currency par reurns n he cross-secon and me seres. Crucal o noe s ha due o lmaons on he avalably of exchange rae opons our nvesgaon focuses only on he G10 (developed marke) currences, unlke Lusg, e al. (2011, 2013) who examne boh developed and emergng marke currences. We fnd ha our model maches he hsorcal reurns on common facor mmckng porfolos, ndcang a subsanal degree of negraon n he prcng of currency rsks across he spo and opon markes. Moreover, he lack of a sascally sgnfcan wedge n he hsorcal and opon-mpled rsk prema suggess hsorcal esmaes of HML F X rsk prema n developed markes are no plagued by peso problems (Burnsde, e al. (2011)). Our model s never rejeced n cross-seconal (Fama-MacBeh) ess, and acheves adjused R 2 values up o 50%, whch are roughly double hose acheved from usng neres rae dfferenals alone. Fnally, we decompose he model-mpled currency rsk prema across nnovaons o he log prcng kernels, dfferenang beween: (a) he ype of shock (Gaussan vs. non-gaussan); and, (b) he ndvdual momens of he shock (varance, skewness, ec.). These resuls provde a srucural decomposon of rsk prema, whch complemens reduced-form sudes usng crash-hedged sraeges o examne he effec of al rsks on currency rsk prema (Burnsde, e al. (2011), and Farh, e al. (2013), Jurek (2013)). We show ha he skewness and hgher-order momens of he log prcng kernel nnovaons, on average, accoun for only 15% of he model-mpled HML F X rsk premum. Consequenly, he man 3 Frameworks wh (me-varyng) dsaser rsks have been used o mach he equy rsk premum n consumpon models (Marn (2013)), explan aggregae sock marke volaly (Wacher (2013)), and as a mechansm for generang volaons of uncovered neres rae pary (Farh and Gabax (2011). Gabax (2012) demonsraes he power of hs mehodology n conex of en classcal macro-fnance puzzles. 4 Our specfcaon of prcng kernel dynamcs shares he feaures of he connuous-me model of Baksh, e al. (2008), whch allows boh global and counry-specfc nnovaons o be non-gaussan. However, raher han mposng a fla erm srucure, our model addonally lnks he dynamcs of he neres raes o he underlyng sae varables. Ths creaes a lnk beween he level of neres raes and rsk prema, whch s necessary for a conssen, rsk-based explanaon of volaons of uncovered neres rae pary. The specfcaon we propose generalzes he condonally Gaussan affne model n Lusg, e al. (2011, 2013), and he jump-dffusve seup n Farh, e al. (2013), whch does no allow for counry-specfc jumps. 5 The avalably of he full cross-secon of G10 cross-raes (45 pars) plays an mporan role n model denfcaon n a dscree-me seng. The model can also be denfed by specfyng me-conssen dynamcs for he sae varables governng he me-change, and usng mulple enors n he calbraon, as n Baksh, e al. (2008). 2

4 channel hrough whch non-gaussan nnovaons ac o deermne rsk prema n our model boh global and counry-specfc s as conrbuors of prcng kernel varance. For comparson, he conrbuon of skewness and hgher-order momens o equy rsk prema s on average 35% n models calbraed o hsorcal consumpon dsasers (Barro (2006), Barro and Ursua (2008), Barro, e al. (2013)), bu only 2% n a model calbraed o mach he prcng of equy ndex opons (Backus, e al. (2011)). Our preferred calbraon focuses on prcng opons on a se of weny four currency pars, desgned o capure he mos lqudly raded exchange rae opons, as well as, opons on ypcal carry rade currences. Ths se ncludes he nne X/USD pars, and an addonal ffeen pars formed from currences whch had eher he hghes or lowes neres rae among G10 counres a some pon n me n our sample. The preferred model calbraon delvers a roo mean squared opon prcng error measured n volaly pons of 1.12, whch s n lne wh ypcal bd-ask spreads for exchange rae opons (Table I, Panel C). Usng he calbraed model, we compue par-level esmaes of condonal currency rsk prema. We hen aggregae hese opon-mpled rsk prema usng he porfolo weghs used n he consrucon of emprcal facor mmckng porfolos, o produce he correspondng model-mpled porfolo rsk prema. We examne he hsorcal reurns o condonal facor mmckng porfolos sored on he bass of conemporaneous neres raes and uncondonal facor mmckng porfolos sored on he bass of backward lookng averages of neres raes. We fnd ha opon-mpled rsk prema for he condonal and uncondonal HML F X facor mmckng porfolos are 3.55% (3.44%) per annum, respecvely (Table IV). Neher of hese quanes s sascally dsngushable from he correspondng mean realzed reurns of 4.96% and 3.32% per annum; he -sascs of he dfferences are equal o 0.43 and -0.16, respecvely. We hen decompose he model-mpled HML F X rsk no conrbuons semmng from Gaussan and non-gaussan prcng kernel shocks, as well as, across he momens of he prcng kernel nnovaons. The decomposon reveals ha (on average) 58% of he rsk premum for he HML F X facor mmckng porfolo s due o non-gaussan nnovaons n he global facor. To explore he mechansm hrough whch hese nnovaons conrbue o he HML F X rsk premum, we decompose he rskpremum across he momens of he global prcng shock. Ths reveals a srkng resul. Alhough non-gaussan shocks are he domnan drvers of he HML F X rsk premum, hey exer her nfluence prmarly by conrbung varance, raher han skewness (or hgher momens), o he global nnovaon. Specfcally, we fnd ha (on average) 85% of he oal rsk premum s arbued o he varance of he prcng kernel shocks, wh only 10% due o skewness of he shocks, and roughly 5% due o he hgher momens. Ths parallels he emprcal resuls repored by Jurek (2013) based on an analyss of reurns o crash-hedged G10 porfolos, bu conrass wh Farh, e al. (2013), who argue ha dsaser rsk accouns for more han a hrd of he currency rsk prema n devel- 3

5 oped economes. Ths dscrepancy reflecs wo sgnfcan dfferences n he emprcal denfcaon sraeges. Frs, Farh, e al. (2013) only observe daa for X/USD opons (N 1 opon-mpled dsrbuons), and are herefore unable o fully denfy he prcng kernel dynamcs (N kernels) exclusvely usng exchange rae opon daa unlke hs paper. Insead, hey measure he oal HML F X rsk premum from hsorcal reurns, and only use exchange rae opons o pn down he dsaser rsk componen. Second, her denfcaon of he dsaser rsk componen leans heavly on he assumpon of Gaussan counry-specfc nnovaons. Ths assumpon s a odds wh evdence n Baksh, e al. (2008) and he resuls of our calbraon, boh of whch pon o sgnfcanly non-gaussan counry-specfc nnovaons (Table A.I). For example, he average skewness values of he counryspecfc nnovaons range from (SEK) o (AUD) n he cross-secon; kuross values are rounely n excess of en. We conduc a smlar analyss for he rsk prema demanded by nvesors n each of he G10 counres for beng shor her own local currency (Table III). We refer o hs rsk premum as he shor reference rsk premum, and show ha whn our model s deermned exclusvely by he even momens of he counry-specfc prcng kernel nnovaons. Emprcally, we fnd ha he mean share of he rsk premum accouned for by he varance of he counry-specfc nnovaons s greaer han 98% n all en G10 currences. The shor reference rsk prema range from 0.54% (AUD) o 1.33% (SEK) per annum. We fnd ha he hsorcal reurns (3.32% per annum) o a sraegy whch s uncondonally shor he U.S. dollar agans an equal-weghed baske of he G10 currences, are sascally ndsngushable from he model-mpled premum of 1.59% per annum (dfference -sa: 0.63). However, he model s unable o explan he hgh hsorcal reurns o he shor dollar carry rade denfed n Lusg, e al. (2013), whch goes long (shor) he U.S. dollar agans a baske of foregn currences when foregn neres raes are low (hgh). To furher evaluae he ably of he model o explan realzed currency reurns we conduc regressons of currency par excess reurns ono he opon-mpled rsk prema (Table V). We fnd ha he sngle model-mpled varable acheves an adjused R 2 of 30% n cross-seconal (Fama-MacBeh) regressons, whch s comparable o ha achevable usng he neres rae dfferenal (26%). The explanaory power of he model-mpled quanes ncreases even furher when he rsk premum s dsaggregaed no s wo consuens (47%). The oponmpled rsk prema are never drven ou by neres rae dfferenals, hough hey also do no subsume her crossseconal forecasng power, suggesng ha currency opons and neres raes carry non-redundan nformaon abou currency rsk prema. Alhough he null hypohess of he model s never rejeced n hese regressons, he rejecons of he null of no predcably are relavely weak. The model fares poorly n pooled panel regressons, mmckng he emprcal dffcules of denfyng a sascally sgnfcan and posve, rsk-reurn radeoff n 4

6 equy markes (Campbell (1987), French, e al. (1987), Campbell and Henschel (1992)). Our preferred calbraon allows for me-varyng global facor loadngs, ξ, whch are parameerzed as a funcon of he prevalng neres rae dfferenals, ξ = ξ Ψ (r,+1 rus,+1 ). Ths parameerzaon capures he emprcal regulary ha porfolos sored on prevalng neres dfferenals (condonal facor mmckng porfolos) ouperform porfolos sored on hsorcal averages of neres rae dfferenals (uncondonal facor mmckng porfolos). Indeed, we fnd a negave whn-me-perod/across par assocaon beween he loadngs and neres rae dfferenals (Ψ > 0), and a negave across-me/across-par relaon beween he value of ξ and he uncondonal mean neres rae dfferenal (Table I, Panel A), conssen wh a rsk-based explanaon for currency excess reurns. The average esmae of he cross-seconal sensvy, Ψ, s 0.42 (-sa: 13.09). Alhough hs resul s conssen wh he preferred ( unresrced ) model specfcaon n Lusg, e al. (2011), he covaraon beween loadngs and exchange raes ndcaed by exchange rae opons s quanavely weak whn developed marke currences. As a resul, he mean opon-mpled rsk premum for he condonal HML F X replcang porfolo s only 11bps hgher han for he correspondng uncondonal facor mmckng porfolo n developed marke currences. Equvalenly, generang a large wedge beween he model-mpled condonal and uncondonal rsk prema by lnkng global facor loadngs o neres rae dfferenals, nduces excessve comovemen of opon-mpled exchange rae momens wh neres rae dfferenals, relave o ha observed n he daa. Fnally, we documen ha whle allowng for me-varyng loadngs leads o a modes mprovemen n he model s f o opon prces, also leads o an overall declne n model-mpled rsk prema. The nuon for hs resul s ha snce he models are consraned o mach opon prces (.e. exchange rae volales), an ncrease n he global facor loadng spread requres some combnaon of a declne n he global and/or counry-specfc sae varables governng he quany of rsk, such ha he level of he rsk premum may declne. The remander of he paper s organzed as follows. Secon 1 nroduces he model of counry-level prcng kernel dynamcs and derves key resuls for currency reurns, neres raes, rsk prema, and currency opon prces. Secon 2 descrbes he daa, model calbraon, and parameer esmaes. Secon 3 repors he me seres of opon-mpled currency rsk prema, conducs varous decomposons, and compares he model rsk prema o hsorcal reurns on emprcal facor mmckng porfolos. Secon 4 dscuses he effec of mevaryng loadngs on currency rsk prema whn our opon prcng calbraon, and examnes he mplcaons of lnkng me-varaon n loadngs o neres rae dfferenals for opon-mpled momens. Secon 5 concludes. Appendx A provdes auxlary resuls relaed o he cumulan generang funcon of he me-changed Lévy ncremens. 5

7 1 Prcng Kernels, Currency Rsk Prema, and FX Opon Prcng We develop a parsmonous model of exchange raes based on a specfcaon of prcng kernel dynamcs drven by a combnaon of global and counry-specfc wealh shocks, and derve s mplcaons for he meseres and cross-secon of currency rsk prema. Unlke prevous papers whch have suded hese quesons usng hsorcal (realzed) currency reurns, we exrac esmaes of nsananeous currency rsk prema from daa on foregn exchange opons. The calbraed model allows us o: (1) oban a me-seres of opon-mpled rsk prema for he currency carry facor (HML F X ; Lusg, e al. (2011)) and counry-specfc facors (e.g. such as premum for shor dollar exposure; Lusg, e al. (2013)); (2) decompose par- and porfolo-level rsk prema across momens of he prcng kernel nnovaons; and, (3) assess he mporance of non-gaussan (jump) rsks n he deermnaon of currency rsk prema, complemenng earler emprcal work (Burnsde, e al. (2011), Farh, e al. (2013), Jurek (2013)). Our model begns wh a specfcaon of counry-level log prcng kernel dynamcs of he form: m +1 m = α ξ L g Z L Y (1) where L g Z and L Y counres; he L Y are ndependen non-gaussan shocks. The L g Z shock s global, and s common o all shocks are counry-specfc, and are cross-seconally ndependen. Ths facor represenaon corresponds o a dscrezed verson of he connuous-me dynamcs n he me-changed Lévy model of Baksh, e al. (2008), and shares he spr of he affne models of Backus, e al. (2001), and Lusg, e al. (2011, 2013). The dsrbuon of he shocks s me-varyng, and depends on he underlyng model sae varables, Z and Y, whch correspond o he me-change parameers of a dynamc model (Carr and Wu (2004)). Inuvely, he values of he sae-varables can be hough of as deermnng he perodc varance of he prcng kernel shocks (.e. he prces of rsk). We leave he dynamcs of he sae varables unspecfed, and recover hem perod-byperod from cross-seconal daa on foregn exchange opons. 6 Cross-seconal dfferences n prcng kernel dynamcs are drven by a combnaon of dfferences n: (a) he level of α ; (b) counres loadngs on global shocks, ξ ; (c) he me-varyng level of he counry-specfc saevarables, Y ; and, (d) parameers deermnng he hgher-momens of he counry-specfc nnovaons, L. The Y drfs, α, have no effec on currency rsk prema, and can be subsued ou usng he yelds of bonds maurng a + 1 for he purpose of opon prcng. The mporance of allowng for asymmerc counry-level loadngs on 6 A supplemenary onlne Techncal Appendx derves he connuous-me analog of our model. The connuous-me model requres: (a) specfyng he dynamcs of he model sae varables, Z and Y ; and, (b) mposng he requremen ha global facor loadngs reman consan. 6

8 global nnovaons, ξ, for raonalzng volaons of uncovered neres pary was frs hghlghed by Backus, e al. (2001). 7 Snce hen, hs feaure has been ncorporaed n he conex of opon prcng (Baksh, Carr, and Wu (2008)), and n he denfcaon of a facor srucure n currency reurns (Lusg, e al. (2011, 2013)). Followng Lusg, e al. (2011), we allow he global facor loadngs o be me-varyng, and explore he mplcaons of hs feaure for he deermnaon of currency rsk prema. 1.1 Prcng kernel nnovaons The dynamcs of he prcng kernels and exchange raes n our model are drven by me-changed Lévy ncremens, such ha each shock can be hough of as a combnaon of a Gaussan (dffusve) shock and non- Gaussan (jump) shock. 8 Moreover, he dsrbuons of hese wo componens are me-varyng hrough her dependence on he sae-varables, Z, and, Y, conrollng he me-change. These feaures combne o produce a model ha no only accommodaes non-gaussan nnovaons a each pon n me, bu also flexbly ncorporaes sochasc varaon n second and hgher momens, characersc of currency opon daa (Carr and Wu (2007)). In our model, boh he global, L g Z, and counry-specfc ncremens, L, are allowed o be non-gaussan. Y Ths mmcs he feaures of he connuous-me model of Baksh, e al. (2008), bu s dfferen from Farh, e al. (2013), who allow for global jump shocks, bu fx he counry-specfc componen o be Gaussan. The laer assumpon s resrcve snce forces he model o mach any non-normales presen n exchange rae opon daa usng a combnaon of global facor loadngs and he parameers governng he global jump dsrbuon. Conssen wh Baksh, e al. (2008) we fnd ha allowng for non-gaussany n counry-specfc nnovaons plays an mporan role n machng he daa. To gan nuon abou he prcng kernel nnovaons, s useful o frs consder her non-me-changed equvalens (.e. a model wh..d nnovaons). We denoe he non-me-changed prcng kernel shocks correspondng o he me nerval beween and + 1, L g and L, o dfferenae hem from her me-changed counerpars, L g Z and L. Whou loss of generaly we normalze he non-me-changed nnovaons o have Y un varance and decompose each of hem (j {g, }) no he sum of wo componens: L j = W j (1 η j ) + Xj η j (2) 7 Backus, Telmer and Fores (2001) show ha n order o accoun for he anomaly n an affne model, one has o eher allow for sae varables o have asymmerc effecs on sae prces n dfferen currences or abandon he requremen ha neres raes be srcly posve. 8 Snce we are n a dscree-me seng he dsncon beween dffusve and jump shocks s somewha semanc. More generally, n he connuous-me verson of he model he dffusve shock can also be non-gaussan, f he nsananeous nnovaons o he saevarable governng he me change and he nnovaons o he prcng kernels are correlaed (Baksh, e al. (2008)). Consequenly, he non-gaussan componen n our dscree-me nnovaon s desgned o smulaneously capure he effecs of sochasc varaon n he sae-varable whn he modeled me nerval, and he effec of jumps. 7

9 ) where W (1 j (1 η) s a Gaussan nnovaon wh varance η j j and X j η j s a non-gaussan nnovaon wh varance η j, and ηj [0, 1]. Wh hs normalzaon, he parameer, ηj, s nerpreable as he me-varyng share of nnovaon varance due o jumps. The jump componen has a CGMY dsrbuon, nroduced by Carr, e al. (2002), whch s characerzed by a quare of poenally me-varyng parameers, {C, G, M, Y } j : µ j [dx] = C j egj x x Y j 1 dx x 0 C j e M j x x Y j 1 dx x > 0 (3) The C j parameer s a scalng facor, whch s se such ha he varance of he jump componen s ηj ; Gj and M j Y j deermne he exponenal dampenng of he dsrbuon for negave and posve jumps, respecvely. The parameer can be nerpreed as measurng he degree of smlary beween he jump process and a Brownan moon. The CGMY process ness compound Posson jumps ( 1 Y < 0), nfne-acvy jumps wh fne varaon (0 Y < 1), as well as, nfne-acvy jumps wh nfne varaon (1 Y < 2). Specfcally, we assume ha: (1) global jumps are one-sded, allowng only for posve shocks o margnal uly (M g = ); and, (2) counry-specfc jumps are wo-sded, capurng boh posve and negave dosyncrac shocks. Fnally, we use he (me-change) sae-varables, Z and Y, o se he perodc volaly of he Lévy ncremens. Appendx A dscusses our specfcaon n more deal. Smlar o Marn (2013), we rely on he cumulan generang funcons (CGF) of he prcng kernel nnovaons, k [u], o express quanes of neres such as yelds, currency rsk prema and and opon prces. 9 The cumulan generang funcon for he non-me-changed Lévy ncremens, L j, are repored n Appendx A. To derve he CGF for he correspondng me-changed ncremens we rely on Theorem 1 n Carr and Wu (2004). In our seup, he me-change s conrolled by pre-deermned sae-varables, Z and Y, whch allow he model o have non-dencally dsrbued nnovaons over me. Unlke n a more ypcal sochasc volaly model, he me-change varables affec no only volaly, bu also he hgher order momens of he prcng kernel, enablng he model o beer mach he emprcal feaures of foregn exchange opon daa. Theorem 1 of Carr and Wu (2004) saes ha for a generc me change, T, he cumulan generang funcon of he me-changed Lévy process, L T, s gven by k T [k L [u]], where k L [u] s he cumulan generang funcon of he non-me-changed 9 Recall ha he cumulan generang funcon of a random varable, ɛ +1, s defned as follows: k ɛ[u] = ln E [exp (u ɛ +1)] = κ j u j where κ j, are he cumulans of he random varable, whch can be compued by akng he j-h dervave of k ɛ[u] and evaluang he resulng expresson a zero. The cumulan generang funcon of he sum of wo ndependen random varables s equal o he sum of her cumulan generang funcons. j=1 j! 8

10 process and k T [u] s he cumulan generang funcon of he me-change. In our case, he me-change varables are fxed whn he measuremen nerval (.e. hey follow a degenerae sochasc process wh zero drf and volaly), such ha: [ ] k L g [u] = k Z k Z L g [u] k L Y [u] = k Y k L [u] [ ] = k L g [u] Z = k L [u] Y (4a) (4b) Unless specfcally noed wh superscrps, cumulan generang funcons are compued under he hsorcal (objecve) measure, P. Usng hese resuls, along wh he cumulan generang funcons of he Gaussan and non- Gaussan componens (Appendx A), he cumulan generang funcon of he me-changed Lévy ncremens, k L j S [u] for he emprcally relevan case when Y {0, 1} s gven by: k L j S [u] = k W j = (1 η j ) S [u] + k X j η j S ( ) 1 η j u2 2 S + η j [u] ((M u) Y M Y + (G + u) Y G Y ) Y (Y 1) (M Y 2 + G Y 2 ) S (5) To fx nuon abou he analycal resuls, we wll occasonally specalze o he case of a condonally Gaussan model, n whch case he cumulan generang funcon of he ncremens can be obaned by seng, η j = Term srucure of neres raes Snce he sochasc dscoun facors mus prce he rsk-free clams n her respecve economes, we have [ ] E M +1 = M exp ( y,+1 τ). Usng he CGFs of he Lévy ncremens we can express he yeld on a one-perod bond n counry as: y,+1 = α k L g Z [ ξ ] k L Y [ 1] = α k L g [ ξ ] Z k L [ 1] Y (6) In our model, yelds across counres share an exposure o he common global facor, Z, wh each counry havng s own poenally me-varyng loadng, ξ. If he loadngs are consan and he sae-varables addonally follow affne dffusons under he relevan measures, e.g. as n Baksh, e al. (2008), he erm-srucure of neres raes wll obey a wo-facor affne model. In he cross-secon, hs expresson lnks neres raes o underlyng rsk exposures, ξ, esablshng an mporan channel for a rsk-based explanaon of volaons of uncovered 9

11 neres pary. For example, f he global nnovaon s Gaussan (η g = 0), he cumulan generang funcon of he global ncremen, k L g [ ξ ] = 1 2 (ξ ) 2, such ha counres wh hgher (lower) global facor loadngs end o have lower (hgher) neres raes, all else equal. More generally, f he frs dervave of he cumulan generang funcon of he global ncremen evaluaed a ξ s negave, k L g [ ξ] < 0, and he second dervave s posve, k L g [ ξ] < 0, neres raes wll be nversely relaed o global facor loadngs. A suffcen condon for hs relaonshp o hold s ha G > ξ and Y < 1, whch we show s sasfed under our preferred model calbraon. In he me-seres, he expresson places a resrcon on he dynamcs of neres raes (yelds) as a funcon of he global, Z, and counry-specfc, Y, sae varables, and he parameers of he cumulan generang funcons. Fnally, he above expresson hghlghs ha erm-srucure daa can be used o exrac nformaon abou currency dynamcs, whch s an approach aken up by Brennan and Xa (2006), Ang and Chen (2010) and Sarno, e al. (2012). Whle erm srucure dynamcs reflec all momens of he underlyng prcng kernels, hs nformaon s acually more han suffcen for characerzng he dynamcs of rsk prema, snce hese depend only on momens wo and hgher. Our approach explos hs feaure o exrac currency rsk prema exclusvely from exchange rae opon daa. 1.3 Exchange raes and currency rsk prema In he absence of resrcons on he rade of fnancal asses, no arbrage requres ha he prcng kernel n counry I prce all asses denomnaed n currency I, as well as, he currency-i reurns of asses denomnaed n all foregn currences, J. Ths lnks he dynamcs of he exchange rae, S j +1 specfed as he currency I prce of currency J and he realzaons of he respecve prcng kernels, M +1 and M j +1, hrough: E P [ M +1 M E P ] Sj +1 S j x j +1 [ M ] +1 x +1 M = E P = E P [ M j +1 M j M j +1 M j x j +1 ] ( ) S j 1 +1 S j x +1 (7) (8) where x +1 and xj +1 are payoffs of he raded asses denomnaed n currences I and J, respecvely. If we could consruc Arrow-Debreu secures correspondng o each (N + 1)-uple of realzaons, {L g Z, {L k } N Y k k=1 }, he marke would be complee and each of he above no arbrage resrcons would have o hold sae-by-sae, such ha he exchange rae would be gven by he rao of he correspondng prcng kernels (Fama (1984), Dumas 10

12 (1992), Backus, e al. (2001), Brand, e al. (2006), Baksh, e al. (2008)): 10 S j +1 S j = M j +1 M j ( M ) 1 +1 (9) M The se of radeable clams n our model economy ncludes rsk-free bonds, exchange raes, and exchange rae opons, such ha markes are ncomplee. To see hs, noe ha alhough we observe exchange rae opons on all N (N 1) 2 pars, only N 1 are non-redundan, prevenng us from nferrng he realzaon of he N underlyng sochasc dscoun facors. When markes are ncomplee, he resrcon, (9), provdes a suffcen, bu no necessary, condon for he exchange rae o sasfy he par of no arbrage condons. For example, any exchange rae process formed by akng he rao of a par of canddae, srcly posve, sochasc dscoun facors s admssble. In hese crcumsances, exchange raes are non-redundan asses, such ha radng n exchange rae opons can be vewed as helpng complee markes. We nerpre he prcng kernel dynamcs, (1), as a represenaon of canddae sochasc dscoun facors n ncomplee markes. Correspondngly, we form a canddae exchange rae process usng he rao of he proposed prcng kernels, and recover s dynamcs by machng he prces of exchange rae opons. Gven our prcng kernel paramerzaon he log currency reurn can be wren as: s j +1 sj = ( ) m j +1 mj = α j + α ( m +1 m ) ( ξ j ξ ) L g Z L j + L Y j Y (10) Our prmary neres s he characerzaon of he model-mpled rsk premum for a zero-nvesmen poson whch s long currency J, and s fnanced n he nvesor s home currency I. Rsk prema on generc long-shor posons, whch do no nvolve he nvesor s home currency I, can hen be smply formed as he dfference beween he rsk prema on long posons n he wo foregn currences. Whou loss of generaly, f we ake he perspecve of an nvesor n counry I, he excess reurn from nvesng n a currency J s gven by: E P [ exp ( ) y j,+1 Sj +1 S j exp ( y,+1) ] ( = exp y j,+1 + ln EP ln E P [ ]) S j +1 S j exp ( y ),+1 [ ] S j +1 S j + y j,+1 y,+1 (11) 10 Gravelne and Burnsde (2013) crque economc nference abou real exchange rae deermnaon and rsk-sharng based on models n whch he exchange rae s represened as as a rao of prcng kernels ( asse-marke vew ). In parcular, hey emphasze he mporance of modelng rade frcons, he span of radable asses, and dfferences n preferences and consumpon bundles. By conras, our neres s n buldng an arbrage free model of exchange raes, raher han sudyng he economc model gvng rse o hem. 11

13 In our emprcal analyss, we sudy he relaon beween he measured currency excess reurns (lef hand sde), and he model-mpled rsk prema (rgh hand sde) obaned by calbrang he model o exchange rae opons. An aracve feaure of focusng on smple excess reurns s ha boh sdes of he equaon can be aggregaed usng porfolo weghs o oban model rsk prema for emprcal facor mmckng porfolos (e.g. for he HML F X and dollar carry facor mmckng porfolos suded by Lusg, e al. (2011, 2013)), whch s no he case for log excess reurns. 11 We verfy ex pos ha he use of he approxmaon resuls n quanavely neglgble errors. Followng Baksh, e al. (2008), we defne he currency rsk premum for currency par J/I as he dfference beween he log expeced reurn on he currency adjused for he neres rae dfferenal: λ j ln E P [ ] S j ( ) +1 S j + y j,+1 y,+1 (12) Subsung n he expressons for he rsk-free raes, and usng he cumulan generang funcon o re-wre he condonal expecaon we oban: λ j = = ( ) ( k L g [ξ Z ξ j ] + k L g [ ξ Z ] k L g [ ξ j Z ] + ( ) ( k L g [ξ ξ j ] + k L g[ ξ ] k L g [ ξj ] Z + k L Y [1] + k L Y k L [1] + k L [ 1] ) [ 1] ) Y (13) Ths expresson llusraes ha he expeced excess reurn on an ndvdual currency par s comprsed of wo componens. The frs componen conrolled by he global sae varable, Z represens compensaon for exposure o he global slope facor (HML F X ). The second componen conrolled by he counry-specfc sae varable, Y reference (local) currency: 12 represens he compensaon demanded by an nvesor n counry for beng shor hs own λ j HML, = λ reff X, = ( k L g [ξ ξ j ] + k L g[ ξ ] k L g ) ( k L [1] + k L [ 1] ) [ ξj ] Z (14) Y (15) 11 An alernave [ measure of he currency rsk premum used by Backus, e al. (2001) and Lusg, e al. (2013) s he mean log excess reurn, E P s j +1 ] ( sj + y j,+1,+1) y. Alhough he log rsk premum can be compued whou relance on an approxmaon, he dsadvanage of hs measure s ha canno be aggregaed lnearly usng porfolo weghs o produce model-mpled esmaes of porfolo rsk prema. The log rsk premum can be expressed as a seres expanson n erms of he cumulans of he me-changed Lévy ncremens as: E P [ ] ( ) s j +1 s j + y j,+1 y,+1 = ( 1) k k=2 ( (ξ ) k ( ) ξ j k ) κ k L g + κ k Z L Y k! κ k L j Y j 12 The onlne Techncal Appendx demonsraes ha n a model wh paramerc sae-varable dynamcs (e.g. as n Baksh, e al. (2008)), here wll generally be a full erm-srucure of opon-mpled currency rsk prema. Lusg, e al. (2013) sudy he correspondng erm-srucure of realzed rsk prema. 12

14 To faclae nerpreaon, consder he case where boh he global and counry-specfc nnovaons are Gaussan (η g = ηj = η ), such ha he cumulan generang funcon of he ncremens s gven by, k L [u] = u2 2. In hs case, he wo rsk prema are equal o: ( ) λ j HML, = ξ ξ ξ j Z λ reff X, = Y (16) Suppose ha as n a ypcal currency carry rade he foregn currency J has an neres rae hgher han he home counry I. Snce loadngs are nversely lnked o neres rae levels, (6), he frs rsk premum s posve and reflecs he compensaon demanded by an nvesor n counry I for exposure o a rsky asse (.e. he ( ) exchange rae), whose loadng on he global facors s gven by ξ ξ j. The magnude of he rsk premum s me-varyng and gven by he level of he global sae-varable, Z. Consequenly, he rsk premum for exposure o he global rsk facor, L g Z, s nerpreable as exposure o he HML F X ( slope ) facor denfed by Lusg, e al. (2011). The nvesor n counry I also requres compensaon for beng shor exposure o hs local (reference) currency shocks, L ; he magnude of hs rsk premum s conrolled by he local sae varable, Y Y. By combnng he rsk prema on currency pars J/I and K/I (e.g. AUD/USD and JPY/USD) we fnd ha he rsk premum demanded by an agen n counry I for a generc currency par, J/K, no nvolvng hs home currency ( ) are: λ jk HML, = ξ ξ k ξ j Z and λ reff X, = 0. Fnally, o gan more nuon abou he wo rsk premum componens, we explore wo decomposons. The frs decomposon expresses each rsk premum n erms of he momens (varance, skewness, ec.) of he correspondng prcng kernel shocks (L g Z and L ). These decomposon shares he flavor of he analyss n Y Backus, e al. (2011), who sudy dsaser rsk models by relang he prcng kernel enropy o he cumulans of log consumpon growh. 13 The second decomposon expresses each rsk premum n erms of he conrbuons arbuable o he Gaussan (W g Z and W Y nnovaons. ) and non-gaussan ( X g Z and X ) componens of he prcng kernel Y HML F X rsk premum In order o express he wo rsk prema as a funcon of he momens of he underlyng prcng kernel nnovaons, we ake advanage of he fac ha cumulan generang funcons are lnked o he cumulans of he correspondng random varable hrough an nfne seres expanson. Usng hs expanson, he conrbuon o he 13 The enropy of he prcng kernel s defned as, L(M +1) = lne [ M ] E [ ln M +1 ], and provdes an upper bound on he log excess reurn of any asse n economy I. For deals see Appendx B n Backus, e al. (2011). 13

15 excess reurn of currency par J/I from exposure o he global facor (HML F X ) s gven by: λ j HML, = n=2 ( ) n ξ ξ j ( ) ( ) + ξ n n ξ j n! κ n L g Z (17) where κ n L g denoes he n-h cumulan of he global ncremen, L g. If we approxmae hs expresson wh erms up o order hree capurng he premum for bearng varance and skewness we oban: λ j HML, ( ( ) ( ξ ξ ξ j V g 1 1 )) 2 ξj Sg V g Z + O(ξ 4 ) (18) where V g = κ2 L g and S g = κ3 L g ( ) 3 κ 2 2 L g, are he varance and skewness of he Lévy ncremens of he global facor, L g. In parcular, gven our ncremen normalzaon, we have Vg = 1. The compensaon for a par s HML F X rsk s hgh whenever: (a) he loadng dfferenal s large and posve (ξ ξ j 0); (b) he prce of rsk s hgh (.e. volaly of global prcng kernel shocks, Z, s hgh); and, (c) he skewness of he prcng kernel nnovaons s negave (S g < 0). Alhough he fourh order erms can be shown o be posve, we fnd ha hey are emprcally neglgble. Rsk-based explanaons of volaons of uncovered neres rae pary requre a lnk beween neres rae dfferenals and currency rsk prema. To llusrae hs feaure n our model consder a smlar, hrd-order expanson of he neres rae dfferenal: ( y j,+1 y,+1 = α j α + k L g ( α j α + ξ j ξ ) [ ξ ] k L g [ ξj ] Z k L j ( ) ( ξ 2 ξ j ) κ 1 L g + 2 ) 2 [ 1] Y j κ 2 L g + + k L [ 1] Y ( ) 3 ξ j ( ) ξ 3 6 κ 3 L g Z k L j[ 1] Y j + k L [ 1] Y + O(ξ 4 ) (19) Whenever he global jump nnovaons have negave mean (κ 1 L g < 0) and are negavely skewed (κ 3 L g < 0), he neres rae dfferenal wll ncrease wh he par s loadng dfferenal, ξ ξ j. These pars are also predced o have hgher currency excess reurns whn our model, conssen wh emprcal evdence on volaons of uncovered neres pary. An alernave mehodology for decomposng he HML F X componen of he currency par rsk premum s o compue he conrbuon from Gaussan and non-gaussan componens of he global prcng kernel nnovaon, L g Z. Ths decomposon formalzes he nuon underlyng he papers by Burnsde, e al. (2011), Farh, e al. (2013), and Jurek (2013), whch sudy he conrbuon of al rsk prema o he excess reurns of carry rade 14

16 porfolos by comparng he realzed reurns o unhedged and crash-hedged porfolos. Specfcally, snce he dffusve and jump componens of he Lévy ncremens are ndependen a each pon n me, he cumulan generang funcons appearng n he rsk premum formula, (14), can be expressed as sums of he cumulan generang funcons of he me-changed dffusve and jump shocks. Ths allows us o solae he componen of he par s HML F X rsk premum due o jump shocks, and represen as a share, φ j HML, of he oal rsk premum: φ j HML, = k X g k L g [ ] ξ ξ j [ ] ξ ξj [ ] + k X g ξ kx g + k L g [ ξ ] kl g [ ξ j [ ξ j ] ] η g (20) Shor reference rsk premum We perform he same momen and shock-ype decomposons on he premum demanded by an nvesor n counry for beng shor hs local, or reference, currency. Applyng he seres expanson o he cumulan generang funcon of he counry-specfc shock, L, governng he shor reference rsk premum, (14), we Y oban: λ reff X, = 1 + ( 1) k k=1 k! κ k L Y = n=1 1 (2 n)! κ2 n L Y (21) A sark feaure of he reference currency premum s ha does no depend on he odd momens of he counryspecfc shock. In oher words, economc raonalzaons of a reference currency premum, e.g. a shor dollar premum, canno be based on one-sded evens, such as crashes or sporadc flghs o qualy. Fnally, we compue he fracon, φ reff X,, of he shor reference rsk premum drven by he non-gaussan (jump) componen of he prcng kernel nnovaon. φ reff X, = k X [1] + k X [ 1] k L [1] + k L [ 1] η (22) Snce our model does no ascrbe a specal role o any parcular reference currency, we compue shor reference rsk prema for all currences n our sample. Emprcally, Lusg, e al. (2013) fnd evdence of large compensaon for shor dollar exposure. We revs hese resuls n he model calbraon and resuls secon (Secon 3). 15

17 1.4 FX opon prcng Gven he choce of our model paramerzaon, opon prcng can be effcenly accomplshed usng sandard Fourer ransform opon prcng mehods descrbed n Carr and Madan (1999). The cenral npu o hs prcng mehodology s he cumulan generang funcon (or characersc funcon) of he log exchange rae, s j = log S j, compued under he prcng measure. In general, f neres raes are me varyng over he lfe of he opon, s convenen o work under he rsk-forward measure, F τ, whose numerare s he τ-perod zero coupon bond n counry. In our dscrezed model, we assume he neres rae s fxed over he lfe of he opon a he level gven by (6), such ha he rsk-forward and rsk-neural, Q, measures concde. Alhough we wre he subsequen formulas under he rsk-forward measure, hese can be nerpreed as correspondng o rsk-neural quanes n he model. Fnally, s mporan o emphasze, ha he prcng measure depends on he home counry of he nvesor, snce nvesors n dfferen counres have dsnc prcng kernels. The rsk-forward measure for an nvesor from counry I, assocaed wh a zero-coupon bond maurng a me + 1, s defned as follows: df dp = M +1 M exp ( y,+1 ) (23) where P denoes he physcal (hsorcal) measure. The vrue of workng under he rsk-forward measure s ha we can prce clams denomnaed n uns of currency I (e.g. bonds or exchange rae opons on he currency pars J/I) as he produc of her expeced payoff, x(s j +1 ), under he F measure mulpled by he value of he one-perod zero-coupon bond n counry I. To see hs, recall ha any payoff sasfes he followng prcng equaon: V = E P [ M +1 M recognzng he measure change we oban: x(s j +1 ) ]. Dvdng boh sdes by he value of he zero-coupon maurng a + 1, and V = Z,+1 E P [ M +1 M exp ( ] y,+1 ) [ ] x(s j +1 ) = Z,+1 E F x(s j +1 ) (24) where Z,+1 = exp ( y,+1) s he value of he numerare, zero-coupon bond. In order o apply hs valuaon approach o exchange rae opons, we rely on Fourer prcng mehods and characerze he dsrbuon of he exchange rae a + 1 usng s cumulan generang funcon under F. To derve he cumulan generang funcons of he global and counry-specfc Lévy ncremens under he rsk-forward measure, F, we proceed drecly from he defnons of he cumulan generang funcon and he 16

18 measure change: k F L g [u] = ln E F Z ( = k F L [u] = Y k L g k F [u] = k L j L j [u] Y j Y j [ ( exp u L g )] [ ( Z = ln E P exp y,+1 + ( m +1 m ) + u L g [ ] u ξ kl g ( k L [u 1] k L [ 1] Z )] [ ξ ] ) Z (25a) ) Y (25b) (25c) Conrasng hese expressons wh he correspondng values under he objecve measure, P, we see ha he change of measure resuls n: (1) a change n he dsrbuon of he global facor dependen on counry s loadng on he global facor, ξ; (2) a change n he dsrbuon of he local (reference) shock, L ; and, (3) leaves he Y dsrbuon of foregn, counry-specfc shocks unchanged. To oban he cumulan generang funcon for he exchange rae a me + 1 under he rsk forward measure, F, we subsue (10), no he defnon of he CGF o oban: k F s j [u] = ( ) [( ) ] s j α j + α u + k F L g ξ ξ j u Z + k F [u] Y L + k F [ u] Y j L j (26) where each of he rsk-forward CGFs can be evaluaed usng he formulas above. In order o subsue ou he α and α j erms, we ake advanage of he fac ha: kf [1] = s j s j + y,+1 yj,+1.14 Fnally, he exponenal of he ( ) cumulan generang funcon, (26), evaluaed a u, exp k F [ u], corresponds o he generalzed Fourer ransform of he log currency reurn. From here, he Fourer ransform can be nvered numercally o produce prces for calls and pus, as n Carr and Madan (1999). s j Opon-mpled momens An aracve feaure of our model, whch we explo n he emprcal calbraon, s ha he cumulans of he dsrbuon are lnear funcons of he sae varables, {Z, Y, Y j }. For example, he second and hrd 14 To oban hs resul consder prcng an nvesmen n currency J from he perspecve of an nvesor n counry : [ M ( ) ] [ +1 1 = E Sj +1 exp y j M ( ) ( ) ] +1 M S j,+1 = E exp y M,+1 Sj +1 exp y j S j,+1 y,+1 = E F [ S j ( +1 exp y j S j,+1 y,+1) ] = exp (k F s j [1] + y j,+1 y,+1 ) 17

19 cumulans under he rsk-forward measure, F are gven by: κ 2,F s j κ 3,F s j = = ( ) 2 ( ξ ξ j ( ) 3 ( ξ ξ j k L g k L g ) [ ξ ] ( ) Z + k L [ 1] Y + ) [ ξ ] ( ) Z + k L [ 1] Y ( k L j ( k L j ) [0] Y j ) [0] Y j (27a) (27b) Gven a se of model parameers a me, he sae varables can be recovered usng a cross-seconal regresson of cumulans measured from conemporaneous exchange rae opon daa ono he model-mpled coeffcens. Alhough clams on cumulans are no raded, hey can be readly reconsruced from opon prces. Specfcally, usng he nsghs from Breeden and Lzenberger (1978) and Baksh, e al. (2003), we compue opon-mpled swap raes for varance ( ˆV s j ) and skewness (Ŝs j ). The correspondng emprcal esmaes of he rsk-forward cumulans can hen be recovered from he defnons lnkng momens and cumulans, ˆκ 2,F Ŝ s j ( ˆVs j ) 3 2 s j = ˆV s j, and ˆκ 3,F s j. Wh he prces of hese clams n hand, he values of he sae varables can be nferred drecly by means of a cross-seconal lnear regresson, sdeseppng more complcaed flerng procedures. Fnally, hese expressons hghlgh ha he momens of he opon-mpled exchange rae dsrbuon and, more generally, opon prces depend no only on he loadng dfferenals, bu also on he levels of he loadngs, ξ. The dependence on he level of he loadngs s crucal for model denfcaon, snce oherwse oponmpled momens would only allow us o pn down loadng dfferenals, whch are nsuffcen for recoverng rsk prema, e.g. (14). For example, f he global and counry-specfc nnovaons are Gaussan, he hgher order cumulans (j 3) are equal o zero, and he second cumulan s only a funcon of he loadng dfferenal, precludng denfcaon of currency rsk prema. Smlarly, he P-measure momens only depend on he loadng dfferenals. 15 = 2 Daa and Model Calbraon We calbrae he model of prcng kernel dynamcs descrbed n Secon 1 usng panel daa on G10 currency exchange rae opons, and he me-seres of G10 exchange raes and one-monh LIBOR raes. A novel feaure of our approach s ha provdes cross-seconal esmaes of nsananeous currency rsk prema, whou relyng on he me-seres of realzed (hsorcal) reurns. The opon-mpled esmaes of rsk prema are free from peso problems (Burnsde, e al. (2011)), and complemen he evdence on he facor srucure n currency reurns documened by Lusg, e al. (2011, 2013). We hen use he model o decompose he HML F X rsk premum 15 The cumulans under he hsorcal measure, P, are gven by he same expressons, bu wh he dervaves of he consecuve cumulan generang funcons evaluaed a zero, raher han { ξ, 1, 0}. 18

20 and he premum for shor exposure o varous G10 currences no: (a) dffusve and jump shocks; and (b) conrbuons from varous facor momens (e.g. varance, skewness, ec.). 2.1 Daa The key daase used n he analyss ncludes prce daa on foregn exchange opons coverng he full crosssecon of 45 G10 cross-pars, spannng he perod from January 1999 o June 2012 (T = 3520 days). 16 The daase provdes daly prce quoes n he form of mpled volales for European opons a consan maures and fve srkes, and was obaned va J.P. Morgan DaaQuery. FX opon prces are quoed n erms of her Garman-Kohlhagen (1983) mpled volales, whch correspond o Black-Scholes (1973) mpled volales adjused for he fac ha boh currences pay a connuous dvdend gven by her respecve neres raes. We focus aenon on consan-maury one-monh exchange rae opons. For each day and currency par, we have quoes for fve opons a fxed levels of opon dela (10δ pus, 25δ pus, 50δ opons, 25δ calls, and 10δ calls), whch correspond o srkes below and above he prevalng forward prce. In sandard FX opon nomenclaure an opon wh a dela of δ s ypcally referred o as a 100 δ opon; we adop hs convenon hroughou. The specfcs of foregn exchange opon convenons are furher descrbed n Wysup (2006), Carr and Wu (2007), and Jurek (2013). In general, an opon on par J/I gves s owner he rgh o buy (sell) currency J a opon expraon a an exchange rae correspondng o he srke prce, whch s expressed as he currency J prce of one un of currency I. The remanng daa we use are one-monh Eurocurrency (LIBOR) raes and daly exchange raes for he nne G10 currences versus he U.S. dollar obaned from Reuers va Daasream. 2.2 Calbraon: Inuon The fnal goal of our model calbraon procedure s o produce a me seres of nsananeous currency rsk prema. Snce hese rsk prema are expressed n erms of he cumulan generang funcon of he prcng kernel nnovaons, (13), he calbraon procedure can be undersood n erms of pnnng down he cumulans n he seres expanson of he respecve CGFs. Even more precsely, he denfcaon of currency rsk prema only requres knowledge of cumulans of order wo or hgher, as can be seen n (17) and (21). Our calbraon procedure recovers hese cumulans from nformaon n he opon-mpled exchange rae dsrbuon. To presen he nuon behnd our emprcal denfcaon sraegy s useful o begn by specalzng o a world n whch all prcng kernel nnovaons are Gaussan (η j = 0), such ha here are no CGMY jump 16 The G10 currency se s comprsed of he Ausralan dollar (AUD), Canadan dollar (CAD), Swss franc (CHF), Euro (EUR), U.K. pound (GBP), Japanese yen (JPY), Norwegan kronor (NOK), New Zealand dollar (NZD), Swedsh krone (SEK), and he U.S. dollar (USD). There are a oal of 45 possble cross-pars. 19

21 parameers o esmae. In hs case, exchange rae dsrbuons are Gaussan, and he only cumulan of order wo or hgher eher n he dsrbuon of he prcng kernel nnovaons, or n he exchange rae dsrbuon whch s non-zero, s he second cumulan. In parcular, he second cumulan (or, varance) of he opon-mpled dsrbuon s mmedaely revealed by he mpled volaly of any one of he fve, quoed opons. The quoes on he remanng four opons are redundan. The denfcaon of he model parameers proceeds on he bass of equaon (27a), whch lnks he varance of he opon-mpled dsrbuon under he rsk-forward measure wh he cumulans of he prcng kernel nnovaons. Gven we have daa on all exchange rae cross-pars, we have a oal of N (N 1) 2 observaon equaons for he opon-mpled varances. The model parameers o be pnned down on each day are: (1) he N counry loadngs, ξ ; (2) he varance of he global nnovaon, Z ; and, (3) he varances of he N counry-specfc nnovaons, Y. Snce we can normalze one of he loadngs o one, hs leaves us wh 2 N parameers. In parcular, he G10 currency se has en currences, yeldng weny parameers and fory fve observaon equaons. Consequenly, he parameers of he Gaussan model are fully-denfable usng our cross-seconal nformaon on each day ndvdually. Our ably o denfy he Gaussan model owes o he avalably of cross-rae opons. By conras, consder he denfcaon sraegy proposed by Farh, e al. (2013), whch only uses nformaon on X/USD opons (.e. opons on he exchange rae of foregn currences agans he U.S. dollar). Ths approach yelds a oal of nne N 1 observaon equaons for he weny model parameers, such ha her model s no fully denfed even n he Gaussan seng. Ths denfcaon problem does no go away when dsrbuons are non-gaussan, bu raher hghlghs ha he varance of he prcng kernels whch plays a frs-order role n deermnng he rsk premum conrbuons of he Gaussan and non-gaussan shocks remans undenfed n her seup. Imporanly, he non-denfcaon problem can be resolved by specfyng me-conssen dynamcs for he saevarables and smulaneously calbrang o X/USD opons wh mulple enors, e.g. as n Baksh, e al. (2008). How does he denfcaon nuon change f we go o he more general seng wh arbrary opon-mpled dsrbuons? To focus aenon on he concepual ssues of model denfcaon, raher han he emprcal mplemenaon, suppose ha one observes opons a a connuum of srke values. Consequenly, one could use he mehodology of Breeden and Lzenberger (1976) o exrac he opon-mpled dsrbuons for any exchange rae for whch one has opon daa. In hs case, he goal a each pon n me s o denfy he N + 1 facor dsrbuons (global + N counry-specfc nnovaons) on he bass of he exchange rae dsrbuons exraced from he opon daa. Ths s clearly no possble usng nformaon solely on he N 1 X/USD exchange rae opons. Agan, our ably o denfy he model reles on he avalably of cross-rae opons, whch yeld a oal of N (N 1) 2 opon-mpled dsrbuons o mach. From here, he denfcaon exercse reles on he specfc 20