Understanding Exchange Rate Dynamics: What Does The Term Structure of FX Options Tell Us?

Transcription

1 Undersanding Exchange Rae Dynamics: Wha Does The Term Srucure of FX Opions Tell Us? Yu-chin Chen and Ranganai Gwai December 2014 Absrac This paper uses foreign exchange (FX) opions wih differen srike prices and mauriies ( he erm srucure of volailiy smiles ) o capure boh marke expecaions and perceived risks. Using daily opions daa for six major currency pairs, we show ha he cross secion and erm srucure of opions-implied sandard deviaion, skewness and kurosis consisenly explain no only he condiional mean bu also he enire condiional disribuion of subsequen currency excess reurns for horizons ranging from one week o welve monhs. We also find ha exchange rae movemens, which are nooriously difficul o model empirically, are in fac well-explained by he erm srucures of forward premia and opions-implied higher momens. The robus empirical paern is consisen wih a represenaive expeced uiliy maximizing invesor who, in addiion o he mean and variance, also cares abou higher momens in he reurn disribuion. The erm srucure linkage in urn provides suppor for an Epsein-Zin ype preference. Our resuls sugges ha he perennial problems faced by he empirical exchange rae lieraure are likely due o overly resricive assumpions inheren in he prevailing esing mehods, which fail o properly accoun for he forward-looking propery of exchange raes and poenial skewness or excess kurosis in he condiional disribuion of FX movemens. Keywords: exchange raes; excess reurns; opions pricing; volailiy smile; risk; erm srucure of implied volailiy; quanile regression JEL Code: E58, E52, C53,F31,G15 Firs Draf:Ocober We are indebed o he lae Keh-Hsiao Lin and Joe Huang for poining us o he relevan daa sources. We would also like o hank Samir Bandaogo, Nick Basch, Yanqin Fan, David Grad, Ji-Hyung Lee, Charles Nelson, Bruce Preson, Abe Robison, and Eric Zivo for heir helpful commens and assisance. All remaining errors are our own. Chen s and Gwai s research are suppored by he Gary Waerman Scholarship and he Buechel Fellowship a he Universiy of Washingon. Chen: yuchin@uw.edu; Gwai: gwair@uw.edu; Deparmen of Economics, Universiy of Washingon, Box , Seale, WA 98195

2 1 Inroducion The exchange rae economics lieraure has over he years faced many empirical puzzles. As an example, alhough heory predics ha nominal exchange raes should depend on curren and expeced fuure macroeconomic fundamenals, he consensus in he lieraure is ha exchange raes are essenially empirically disconneced from he macroeconomic variables ha are supposed o deermine hem. This empirical disconnec comes in he form of low correlaions beween nominal exchange raes and heir supposed macro-based deerminans and also in he form of poor performance of macro-based exchange rae models in ou-of-sample forecasing. 1 A relaed empirical anomaly ha has received considerable aenion in he lieraure is he uncovered ineres pariy (UIP) puzzle or he forward premium puzzle. The UIP puzzle is he empirical irregulariy showing ha he forward exchange rae is a biased predicor of fuure spo exchange raes. The UIP puzzle is aken seriously in he exchange rae lieraure because he UIP condiion is a propery of mos open-economy macroeconomic models. One manifesaion of his empirical (ir)regulariy is ha counries wih higher ineres raes end o see heir currencies subsequenly appreciae and a carry-rade sraegy exploiing his paern, on average, delivers excess currency reurns. 2 This violaion of he UIP condiion is commonly aribued o ime-varying risk premia and biases in (measured) marke expecaions. However, empirical proxies based on surveyed forecass or sandard measures of risk - for insance, ones buil from consumpion growh, sock marke reurns, or he Fama and French (1993) facors -have been unsuccessful in explaining he puzzle. 3 As such, while recognizing he presence of risk, macroeconomic-based approaches o modeling 1 See Engel (2013) for a review. 2 A carry rade sraegy is o borrow low-ineres currencies and lend in high-ineres currencies, or o sell forward currencies ha are a a premium and buy forward currencies wih a forward discoun. 3 See, Engel (1996) for a survey of he forward premium lieraure, as well as recen sudies such as Burnside e al. (2011) and Bacchea and van Wincoop (2009). 1

3 exchange raes ofen ignore risk in empirical esing. 4 5 This paper argues ha he persisen empirical puzzles faced by he exchange rae economics lieraure are mos likely due overly resricive preference and disribuional assumpions in convenional esing mehods. For example, researchers ypically assume ha exchange rae reurns are normally disribued, or ha invesors uiliy funcions depend only on he mean and variance. We argue ha hese auxiliary assumpions ofen inadequaely accoun for eiher he forward-looking propery of nominal exchange raes or poenial skewness and/or fa ails in he disribuion of FX reurns. We empirically demonsrae ha FX risks as capured by higher order momens of perceived FX reurns disribuions of perceived FX reurn disribuions, as well as expecaions, capured by he erm srucure of opion prices, do really maer in explaining exchange rae movemens. We highligh he usefulness of capuring risks and expecaions in sages. Firs, we show ha opions-implied sandard deviaions, skewness and kurosis of fuure exchange rae movemens are able o explain no only he condiional mean, bu he enire condiional disribuion of excess currency reurns. Second, we show ha informaion exraced from he erm srucure opions-implied risk measures add subsanial explanaory power for excess currency reurns. Finally, we show ha quarerly exchange rae movemens are well explained by he erm srucure of 1 s -4 h momens of opions-implied reurns disribuions, wih adjused R 2 s ranging from 58% for USDJPY o 84% for GBPUSD. Simple derivaives such as he forwards and fuures have been used exensively in explaining excess currency reurns or exchange rae movemens. 6 Payoffs from forward conracs, however, are linear in he reurn on he underlying currency and as such do no conain as 4 ( See for insance, Engel and Wes (2005); Mark (1995)) 5 On he finance side, effors aiming o idenify porfolio reurn-based risk facors offer some empirical success in explaining he cross-secional disribuion of excess FX reurns, bu have lile o say abou bilaeral exchange rae dynamics (see for example, Lusig e al. (2011); Verdelhan (2012)). Lusig e al. (2011) and Verdelhan (2012) for example, idenify a carry facor based on cross secions of ineres rae-sored currency reurns and a dollar facor based on cross secions of bea-sored currency reurns. 6 See for example, Hansen and Hodrick (1980) and Clarida and Taylor (1997) among many ohers. 2

4 useful a se of informaion as he non-linear conracs we examine. Concepually, since payoffs of opion conracs depend on he uncerain fuure realizaion of he price of he underlying asse, opion prices mus reflec marke senimens and beliefs abou he probabiliy of fuure payoffs. Our use of opions price daa and relaed empirical mehodologies has a number of moivaing facors. Firs, opions are forward-looking by consrucion, which means opion prices should herefore be able o incorporae informaion such as forhcoming regime swiches or he presence of a peso problem. 7 Second, opion prices are deeply rooed in marke paricipan behavior because hey are based on wha marke paricipans do insead of wha hey say.furhermore, cross secions of opion prices imply a subjecive probabiliy disribuion of fuure spo exchange raes, which capures boh marke paricipans beliefs and preferences. 8 Third, modern echniques such as he Vanna-Volga mehod 9 and he mehodology of Bakshi e al. (2003) faciliae elegan and model-free compuaion of opions-implied higher order momens of fuure exchange rae changes. Our empirical findings in his paper have hree implicaions. Firs, he exchange rae model based on UIP is no ha bad, and we can coninue using i in open economy macroeconomic models. However, we need o undersand ha if we pu lognormal shocks, hey will no fi will. Quick improvemens can be made by conrolling for he erm srucure of higher order momens, which can be obained from opion price daa. Second, on he financial side, conceps of risk which depend on only he mean and variance such as he Sharpe raio for porfolio performance evaluaion, perhaps ough o be modified o accoun for he imporance of higher momen risks such as skewness and kurosis. 7 The peso problem refers o he effecs on inferences caused by low-probabiliy evens ha do no occur in he sample, which can lead o posiive excess reurn. 8 This disribuion is commonly referred o as he risk-neural disribuion, hough i does NOT imply ha he disribuion is derived under risk-neuraliy. On he conrary, i incorporaes boh he expeced physical probabiliy disribuion of fuure exchange rae realizaion as well as he risk premium, or compensaion required o bear he uncerainy. 9 (See Casagna and Mercurio (2005)) 3

5 2 Why Higher Order Momens and Term Srucure? 2.1 Forward Premium Puzzle and Excess Currency Reurns The efficien marke condiion for he foreign exchange markes, under raional expecaions, equaes cross border ineres differenials i i wih he expeced rae of home currency depreciaion, adjused for he risk premium associaed wih currency holdings, ρ : 10 i τ i τ, = E s +τ + ρ +τ. (2.1) This condiion is expeced o hold for all invesmen horizons τ, wih ineres raes ha are a mached mauriies. Ignoring he risk premium erm, numerous papers have esed his equaion since Fama (1984), and find sysemaical violaions of his UIP condiion: s +τ s = α + β(i τ i,τ ) + ɛ +τ ; E [ɛ +τ ] = 0,, H 0 : β = 1 (2.2) wih an esimaed β < 0 and R 2 s ha are usually close o zero. This is he so-called uncovered ineres rae pariy puzzle or he forward premium puzzle (see Engel (1996), for a survey of he lieraure). To see he connecion wih forward raes, we noe ha he covered ineres pariy condiion, an empirically valid no-arbirage condiion, equaes he forward premium f +τ s, wih ineres differenials. The risk-neural UIP condiion above hus implies ha he forward rae should be an unbiased predicor for fuure spo rae: E s +τ = f +τ or s +τ = f +τ + u +τ, where E [u +τ ] = 0. We should nex define FX excess reurns as he rae of reurn across borders ne of currency movemen, and one can see ha he UIP or forward premium puzzle can be 10 In his paper, we define he exchange rae as he domesic price of foreign currency. A rise in he exchange rae indicaes a depreciaion of he home currency. However, home does no have a geographical significance bu follow he FX marke convenions. See able (1A) 4

6 expressed as a non-zero averaged excess reurn over ime: xr +τ = f +τ s +τ = (i τ i τ, ) s +τ = ρ +τ + u +τ (2.3) I is naural hen o noe ha he empirical failure of he risk-neural UIP condiion can be aribuable o eiher he presence of a ime-varying risk premium, ρ +τ, or ha expecaion error, u, may no be i.i.d. mean zero over ime. If he disribuion of eiher of hese is no mean zero over he ime series, empirical esimaes of he slope coefficien in regression equaion (2.2) would likely suffer omied variable bias or oher complicaions. 2.2 Why higher order momens? 11 We show ha in addiion o risk neuraliy and raional expecaions assumpions, he UIP condiion also hinges on he raher resricive auxiliary assumpions ha FX reurns are i.i.d. normal over ime and ha invesors have consan absolue risk aversion (CARA) uiliy. The wo addiional assumpions have he effec of reducing he represenaive invesor s opimal asse allocaion problem o a mean-variance opimizaion problem. We sar wih he problem of an invesor who, in each period, allocaes her porfolio among risky asses wih he goal of maximizing he expeced uiliy of nex period wealh. In each period, he invesor has n risky asses o choose from. The vecor of gross reurns is given by r +1 = (r 1,+1,..., r n,+1 ). If we suppose W is arbirarily se o 1, hen W +1 = α r +1, where α is an n by 1 vecor of porfolio weighs. The invesors problem is o choose α o maximize he expression E [U(W +1 )] = E [U(α r +1 )] =... U(W +1 )f(r +1 )dr 1,+1 dr 2,+1...dr n,+1 (2.4) subjec o he condiion ha n i=1 α i, = 1, where f(r +1 ) is he join probabiliy disribuion 11 Maerial in his subsecion is from Mark (2001) 5

7 of r CARA and Normaliy reduce problem o mean-variance opimizaion Le us furher assume ha he invesor has CARA uiliy and ha reurns are condiionally normally disribued. The CARA uiliy assumpion means he uiliy is given by U(W +1 ) = e γw +1, where γ 0 is he coefficien of absolue risk aversion. The disribuional assumpion r +1 N(µ +1, Σ +1 ) implies ha W +1 N(µ p,+1, σ 2 p,+1), where µ p,+1 = α µ +1 and σ 2 p,+1 = α Σ +1 α Wih he above wo assumpions, expression (2.4) reduces o 12 E [U(W +1 )] = E [e γw +1 ] = γµ p, γ2 σ 2 p,+1 (2.5) Equaion (2.5) demonsraes ha under he assumpions of CARA uiliy funcion and condiional normaliy of reurns, he general porfolio allocaion problem (2.4) reduces o he mean-variance opimizaion problem. 13 If we furher assume ha our invesor has a 2-asse porfolio made up of a nominally safe domesic bond and a foreign bond, and ha she allocaes a fracion α of her wealh o he domesic bond, hen nex period wealh expressed in local currency unis is given by W +1 = [ α(1 + i ) + (1 α)(1 + i ) S ] +1 W (2.6) S. In his 2-asse example and CARA uiliy and condiionally normal reurns he expressions 12 The second equaliy follows from he fac ha e γw+1 LN( γµ p,+1, γ 2 σ 2 p,+1), so E [e γw+1 ] = γµ p,+1 + γ 2 σ 2 p,+1 13 The quadraic uiliy funcion imply mean variance opimizaion for arbirary reurn disribuion. However, he quadraic uiliy implies increasing absolue risk aversion and saiaion (Jondeau e al. (2010), page 352). 6

8 for he condiional mean and variance of nex period wealh are given by: µ p,+1 = [ α(1 + i ) + (1 α)(1 + i ) ES +1 S ] W, σ 2 p,+1 = (1 α)2 (1+i )2 V ar (S +1 )W 2 S 2 (2.7) Plugging he expressions in equaion (2.7) ino objecive funcion (2.5), aking he firs order condiion wih respec o α and rearranging he firs order condiion yields he following equaion which implicily deermines he opimal α: (1 + i ) (1 + i ) E S +1 = γw (1 α)(1 + i ) 2 V ar (S +1 ). (2.8) S S 2 Equaion (2.8) reduces o he UIP condiion if we assume ha all invesors are risk-neural (γ = 0): i 1 + i = E S +1 S. (2.9) The Fama regression in equaion (2.2) ess a logarihmic version of equaion (2.9). The key seps in deriving he esable resricions in equaion (2.9) are he join assumpions of CARA uiliy and condiional normaliy of nex period wealh, which reduce he invesor s opimizaion o mean-variance. The above discussion illusraes ha deriving he UIP equaion esed hrough expression (2.2) depends on oher assumpions beyond raional expecaions and risk-neuraliy. If he normaliy assumpion is dropped, for example, hen expression (2.9) will mos likely include higher order momens. In fac, Jondeau e al. (2010) noe ha under CARA uiliy, if we drop he normaliy assumpions, hen he invesor would prefer posiive skewness and low kurosis, such ha he invesor s objecive funcion in equaion (2.5) will also include he hird and fourh momens of he FX reurn disribuion. Sco and Horvah (1980) show ha a sricly risk-averse individual who always prefers more o less (U (1) > 0) and likes posiive skewness a all wealh levels will necessarily dislike high 14 UIP will also hold if α = 1, regardless of invesors degree of risk Aversion. 7

9 kurosis Asse allocaion under higher order momens 15 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 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: U(W +1 ) = U(E W +1 ) + U (1) (W +1 )(W +1 E W +1 ) + 1 2! U (2) (W +1 )(W +1 EW +1 ) U (3) (W 3! +1 )(W +1 E W +1 ) U (4) (W 4! +1 )(W +1 E W +1 ) 4, (2.10) where U n (.) denoes he n h derivaive of he uiliy funcion wih respec o nex period wealh. Taking he condiional expecaion of expression ( 2.10) yields E [U(W +1 )] U(E W +1 ) + U (1) (W +1 )(W +1 E W +1 ) + 1 2! U (2) (W +1 )(W +1 E W +1 ) U (3) (W 3! +1 )(W +1 E W +1 ) U (4) (W 4! +1 )(W +1 E W +1 ) 4. (2.11) Under he assumpion ha he invesor s uiliy funcion is CARA, expression (2.11) reduces o E [U(W +1 )] e γµ p [ ] 1 + γ2 2 σ2 p γ3 6 s3 p + γ4 24 k4 p. (2.12) 15 Maerial in his subsecion is from Jondeau e al. (2010) 8

10 In equaion (2.12), s 3 p and kp 4 are he skewness and kurosis of porfolio reurn. I is clear from equaion (2.12) 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 (2.11) 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 γ. In more general seings, however, he weigh on he n h momen depends on he n h derivaive of he uiliy funcion, and he signs of 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. 2.3 Why erm srucure of opion-implied momens? Rearranging he UIP relaionship in equaion (2.1) and ieraing forward, we can show ha he nominal exchange rae depends on curren and expeced fuure ineres rae differenials as well as on expeced fuure risk: s = E (i +j i +j) j=0 } {{ } Expeced fuure ineres differenials E ρ +j j=0 } {{ } Expeced Fuure FX risk (2.13) Expression (2.13) highlighs he link beween he exchange rae and macroeconomic fundamenals. There is a huge lieraure linking he erm srucure of ineres rae raes(yield curve) o expeced fuure dynamics of macroeconomic fundamenals such as moneary policy, inflaion and oupu by observing ha shor erm ineres raes are moneary policy variables ha depend on macroeconomic variables such as inflaion and oupu while longer erm yields are risk-adjused averages of expeced fuure shor raes. 16 Chen and Tsang (2013) exend 16 see Diebold e al. (2005) for a survey. 9

11 his srand of lieraure o he open economy conex by noing ha he erm srucure of ineres rae differenials (relaive yield curve) conain informaion abou he expeced fuure dynamics of differences in macroeconomic fundamenals. We noe ha he relaive yield curve capures he same informaion abou expeced macroeconomic fundamenals as he erm srucure of opion-implied firs momens of fuure exchange rae movemens. We exend he lieraure on yield curve-exchange rae linkage by invesigaing he abiliy of enire opion-implied disribuions o explain exchange rae dynamics. Wriing he exchange rae in he form in equaion (2.13) also demonsraes he imporance of capuring expecaions and risk in he empirical modeling of exchange rae. Sandard empirical approaches usually impose disribuional assumpions ha reduce he sum of expeced fuure fundamenals o equal curren fundamenals and also ignore risk. 17 There is also a srand of lieraure ha documen he empirical success of empirical exchange rae models ha capure informaion in he erm srucure of forward premia.clarida and Taylor (1997) and Clarida e al. (2003) show ha even if he forward rae is a biased predicor of fuure spo rae (he forward premium puzzles),he erm srucure of forward premia sill conains informaion useful for predicing subsequen exchange rae changes. This line of lieraure is linked o Chen and Tsang (2013) by observing ha hrough he empirically valid covered ineres pariy (CIP) condiion, he forward premium equals he ineres rae differenial a all mauriies. 3 Informaion Conen of Currency Opions 3.1 Volailiy Smile and Term Srucure of Opion Prices Breeden and Lizenberger (1978) show ha in complee markes, he call opion pricing 17 see Engel and Wes (2005), Mark (1995). 10

12 funcion (C) and he exercise price K are relaed as follows: 2 C K 2 = e rdτ π Q (S +τ ), (3.1) where r d and r f are he domesic and foreign risk-free ineres raes and π Q (S +τ ) is he risk-neural probabiliy densiy funcion (pdf) of fuure spo raes. Equaion (3.1) implies ha, in principle, we can esimae he whole pdf of ime S +τ spo exchange rae from ime volailiy smile. Once he disribuion is available, i becomes possible o ge empirical esimaes of he sandard deviaion, skewness, kurosis and even higher order momens of he marke perceived probabiliy densiy of S +τ given informaion available a ime. In addiion o he Breeden and Lizenberger (1978) resul in equaion (3.1), we noe ha alhough marke paricipans can be reaed as if hey are risk-neural for he purpose of opion-pricing, opion prices heoreically conain informaion abou boh invesor beliefs and risk preferences, as shown from he following formula for he price of a European-syle call opion: C(, K, T ) = K M,T (S T K) π P (S } {{ } T ) ds } {{ } T = e rd τ Preferences Beliefs K (S T K) π Q (S T ) ds } {{ } T. (3.2) Boh In equaion (3.2), M,+τ is he pricing kernel, which capures he invesor s degree of risk aversion and π P (S +τ ) is he physical probabiliy densiy funcion of fuure spo exchange raes 18. A forward conrac can in fac be viewed as a European-syle call opion wih a srike price of zero. To see his, we recall ha, on one hand, he heoreical forward exchange rae is given by he formula: 18 In he second expression, he pricing kernel is performing boh he risk-adjusmen and discouning funcions, while in he hird expression hese funcions are divided beween π Q and e rdτ. 11

13 F,T = e rd τ On he oher hand, evaluaing equaion (3.2) a K=0 yields: 0 S T π Q (S T )ds T = e rdτ E Q (S T ). (3.3) C(, 0, T ) = e rd τ 0 S T π Q (S T )ds T = F,T. (3.4) The relaionship beween opions and forwards in equaion (3.4) suggess ha he cross secion of opion prices should, a a minimum, conain as much informaion abou invesor beliefs and preferences as ha conained in forward prices. Moving on o he erm srucure of opion prices, one way o moivae he heoreical informaion conen of he erm srucure of opion prices is o sar from equaion (2.13): s = E (i +j i +j) j=0 } {{ } Expeced fuure ineres differenials E ρ +j j=0 } {{ } Expeced Fuure FX risk. (3.5) Now, under he empirically valid CIP condiion, ineres rae differenial is equal o he forward premium for all enors j: 19 i +j i +j = f +j [ ( )] s = r d τ + E Q S+j ln S } {{ } Firs momen of π Q + ω }{{}, enor j. (3.6) Jensen s inequaliy erm Equaion (3.6) hus says ha, ignoring he Jensen s inequaliy erm ω and he consan erm r d τ, he ineres rae differenial equals he firs momen of he opion-implied risk-neural disribuion of ln ( ) S+j S for any given enor j. The ineres raes are moneary policy variables and herefore depend on macroeconomic fundamenals such as unemploymen 19 The second equaliy follows from dividing (3.3) by S and aking logarihms 12

14 and inflaion. When combined, equaions (3.6) and (3.5) demonsrae ha jus like he yield curve, he erm srucure of he firs momens of implied disribuions also capures informaion abou curren and expeced fuure macroeconomic fundamenals. A second moivaion for he informaion conen of he erm srucure of opion prices comes from he expecaion hypohesis for implied volailiy,ha he erm srucure of opion-implied volailiy conain informaion abou he marke s percepion abou he fuure dynamics of shor erm 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 Exracing Opion-Implied Momens 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. 21 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 H[S] wih finie expecaions, hen H[S] can be replicaed if we have a coninuum of opion prices. They also show ha if H[S] is wice-differeniable, hen i can be spanned algebraically by he following expression H[S] = (H[ S] + (S SH S [ S]) + S H SS [K](S K) + + S 0 H SS [K](K S) + dk, (3.7) 20 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) In his secion we closely follow he exposiion and noaion in Grad (2010). 13

15 where H S = H S and H SS = 2 H S 2. Assuming no arbirage opporuniies, he price of a claim wih pay-off H[S] is given by he expression p = (H[ S] SH S [ S])e rdτ +H S [ S]Se rdτ + S S H SS [K]C(, τ, K)+ H SS [K]P (, τ, K)dK 0 (3.8) where K is he srike price, C(, τ, K) and P (, τ, K) are, respecively, he prices of a European-syle call and pu opions. S is some arbirary consan, usually chosen o equal curren spo price. Equaion (3.8) indicaes ha any pay-off funcion H[S] can be replicaed by a posiion of (H[ S] SH S [ S]) in he domesic risk-free bond, a posiion of H[ S] 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: 22 [R (S +τ )] 2, Volailiy Conrac H[S] = [R (S +τ )] 3, Cubic Conrac [R (S +τ )] 4, Quaric Conrac, (3.9) ) where R (S +τ ) = ln( S +τ S. BKM show ha he variance, skewness and kurosis of he disribuion of R +τ can be calculaed using he following formulas: Sdev(, τ) = e rd τ V (, τ) µ(, τ) 2 (3.10a) Skew(, τ) = erdτ W (,τ) 3V (,τ)µ(,τ)e rdτ +2µ(,τ) 3 [e rdτ V (,τ) µ(,τ) 2 ] 3 2 (3.10b) Kur(, τ) = erdτ X(,τ) 4e rdτ µ(,τ)w (,τ)+6e rdτ µ(,τ) 2 V (,τ) 3µ(,τ) 4 [e rdτ V (,τ) µ(,τ) 2 ] 2, (3.10c) 22 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. 14

16 where he expressions for V (, τ),w (, τ) and X(, τ) and µ(, τ) are given in appendix (A). 23 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 (K σ) 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 σ 1,σ 2, σ 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: σ(k) = σ 2 + σ 2 + σ d 1 (K)d 2 (K)(2σ 2 D 1 (K) + D 2 (K)), (3.11) d 1 (K)d 2 (K) where D 1 (K) = ln [ [ ] [ K 2 K ln K3 ] ln [ ] [ K ]σ 1 + K ln 2 K K 1 ln 3 K 1 ln ] K K 1 [ ] K 2 K 1 ln ln [ ] K 3 K [ K 3 K 2 ]σ 2 + [ ln ln ] K K 1 [ ln [ ] K 3 K 1 ln ] K K 2 [ K 3 K 2 ]σ 3 σ 2, D 2 (K) = ln [ K 2 ] [ K ln K3 ] ln[ K [ ] [ K ]d 1 (K 1 )d 2 (K 1 )(σ 1 σ 2 ) 2 K + 1 ]ln[ K K 2 ] K ln 2 K K 1 ln 3 ln[ K 3 K K 1 ]ln[ K 3 K 2 ] d 1(K 3 )d 2 (K 3 )(σ 3 σ 2 ) 2 1 and d 1 (x) = log[ S x ] + (rd r f σ2 2)τ σ 2 τ, d 2 (x) = d 1 (x) σ 2 τ, x K, K1, K 2, K 3. Expression (3.11) allows us o find he implied volailiy of an opion wih an arbirary srike 23 Derivaions of equaions in (3.10) and expressions for µ(, τ), V (, τ), W (, τ) and X(, τ) can be found in Bakshi e al. (2003) and Grad (2010). 15

17 price. We use K 1 = K 25δp, K 2 = K AT M and K 3 = K 25δc. The VV mehodology is preferable because i is parsimonious as i uses only hree opion combinaions o build an enire volailiy smile. 24 Furhermore,he VV mehod also has a solid 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 (δ = C ), also hedges agains movemens in S volailiy of he underlying asse (V ega = C σ ). 3.3 Daa Descripion In he o--c marke, he exchange rae is quoed as he domesic price of foreign currency, so a fall in he exchange rae represens an appreciaion of domesic currency. 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. 25 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 consan mauriies. Lasly,unlike American-syle opions raded in he exchange marke, European-syle opions ha are raded in he o--c marke do no need 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 24 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.the ATM sraddle, VWB and he Risk Reversal capure hese movemens. See discussions in Casagna (2010) and Malz (1998) 25 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 a i in absolue erms. 16

18 responsiveness of he opion s price wih respec o a 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 Black-Scholes formula applied o FX: C = e rd τ [ F +τ Φ(d 1 ) KΦ(d 2 ) ] P = e rd τ [ KΦ( d 2 ) F +τ Φ( d 1 ) ] where d 1 = log[ S K ] + (rd r f σ2 2)τ σ 2 τ, d 2 = d 1 σ 2 τ. The expressions for call and pu delas are given by he expressions: δ c = e rf Φ(d 1 ) δ p = e rf Φ( d 1 ), (3.12a) (3.12b) where Φ(.) is he sandard normal cumulaive densiy funcion (cdf). The absolue values of δ c and δ p are herefore beween 0 and 1, while pu-call pariy implies ha δ p = δ c 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) 27 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. An RR is se up when one buys a base currency call and sells a base currency pu wih a symmeric dela. Finally, a VWB is buil by buying a symmeric dela 26 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 25δ pu. 27 ATM here means he dela of he opion combinaion is zero. Tha is, he opion combinaion is dela-neural 17

19 srangle and selling an ATM sraddle. 28 The 25δ combinaion is he mos raded opions VWB. 29 The definiions of he hree opion combinaions are as follows: 30 σ AT M,τ = σ 0δc,τ = σ 50δc + σ 50δp σ 25δRR,τ = σ 25δc,τ σ 25δp,τ σ 25δvwb,τ = σ 25δc,τ + σ 25δp,τ } {{ 2 } Srangle σ AT M,τ (3.13a) (3.13b) (3.13c) Equaions (3.13) can be rearranged o ge he implied volailiy for 0δ call, 25δ call and 25δ pu. Expressions for backing ou implied volailiy of hese plain-vanilla opions from he prices of raded opion combinaions are given below: σ 0δc,τ = σ AT M = σ 50δc,τ + σ 50δp,τ (3.14a) σ 25δc,τ = σ AT M + σ 25δvwb,τ σ 25δRR,τ (3.14b) σ 25δp,τ = σ AT M + σ 25δvwb,τ 1 2 σ 25δRR,τ. (3.14c) Finally, K 25δp, K AT M, K 25δc, he exercise prices corresponding o σ AT M,τ, σ 25δc,τ and σ 25δp,τ can be backed ou by using he expression for opion delas given in equaion (3.12 ). For 28 In a srangle, you buy an ou of he money call and an equally ou of he money pu 29 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 (1) 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 underlying asse s price. 30 Table (1B) conains sample opion price quoes for sandard combinaions and sandard mauriies. 18

20 example, o ge K AT M we use he fac ha he ATM sraddle has a dela of zero: e rf τ [ Φ ( ln[ S K AT M ] + (r d r f σ2 AT M )τ ) ( Φ ln[ σ AT M τ S K AT M ] + (r d r f σ2 AT M )τ σ AT M τ Since Φ(.) is a monoone funcion, we can solve equaion (3.15) for K AT M o ge: )] = 0. (3.15) K AT M = S e (rd r f σ2 AT M )τ = F +τ e 1 2 σ2 AT M. (3.16) Using similar argumens, one can show ha he expressions for K 25δc and K 25δp K 25δc = S e [ Φ 1 ( 1 4 erdτ )σ 25δc,τ τ+(r d r f σ2 25δc )τ] K 25δp = S e [Φ 1 ( 1 4 erdτ )σ 25δp,τ τ+(r d r f σ2 25δp )τ], (3.17a) (3.17b) wih K 25δp < K AT M < K 25δc (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 Time series plos of he exraced risk neural momens of log S T S are shown in figure (2). The exraced momens are 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 There are also some ouliers in some of he skewness and kurosis series, especially for 9m 19

21 and 12m enor. 31 INSERT FIGURE (2) HERE 4.2 Can he erm srucure of implied momens predic currency reurns? For each currency pair i, we sar by esimaing he sandard UIP regression s i +τ s i = α + β(f +τ i s i ) + ɛ i +τ. (4.1) We focus on model fi and join significance raher han esing wheher he β coefficien is equal o 1. Fied vs Acual plos of esimaed regression (4.1) (wih breaks) are shown in figures 4(a)-4(e), while condensed resuls can be found in column A of able (3). For all currency pairs, he forward premia coefficiens are saisically significan a he 1% level, and adjused R 2 of a leas 10% for each currency pair. We hen consider he predicive abiliy of τ-period opion-implied higher momens by esimaing he following augmened UIP regression: s i +τ s i = α + β 1 (f i,+τ s ) + β 2 sdev i,+τ + β 3 skew i,+τ + β 4 kur i,+τ + ɛ i,+τ (4.2) Equaion (4.2) herefore augmens he sandard UIP equaion (4.1) by sudying he predicive abiliy of he 1 s 4 h momens of he disribuion of log ( S+τ S ). The condensed regressions resuls are shown in column B able (3). The adjused R 2 s for he mached-frequency augmened UIP regressions are consisenly higher han hose from he sandard UIP specificaion in column A, ranging from 26% o 67%. Coefficiens on he higher order momens are always joinly significan a he 1% level. 31 Summary saisics of he exraced momens can be found in he online appendix o his paper. 20

22 We nex esimae a erm srucure modificaion of he sandard UIP equaion (4.1) ha uses informaion conained in he erm srucure of forward premia o predic exchange rae movemens: 3 s i +τ s i = γ 0,τ + γ 1,j P C j meant erm + ɛ i +τ. (4.3) j=1 Condensed resuls from regression specificaion (4.3) are presened in column C of able (3). Comparing columns A and C in able (3), we see ha adding he whole erm srucure of forward premia significanly improves he UIP regression fi. For example, he adjused R 2 jumps from 27% o 57% for AUDUSD, 15% o 40% for EURUSD and from 15% o 54% for USDCAD. Lasly, we regress exchange rae movemens on he erm srucure of 1 s 4 h momens: 3 3 s i +3M s i = γ 0,τ + γ 1,j P C j meant erm i + γ 2,j P C j sdevt erm i + j=1 3 γ 3,j P C j skewt erm i + j=1 j=1 j=1 3 γ 4,j P C j kurt erm i + ɛ i +3M. (4.4) Plos of acual versus fied values from regressions (4.4) and (4.1) are shown in figures (4). These plos show ha considered wih he sandard UIP regression, accouning for higher order momen risks and expecaions subsanially improves ha model fi. The condensed regression resuls for he higher momen erm srucure specificaion, shown in column D of able (3) show ha compared o he UIP specificaion in column A, accouning of for higher momens and expecaions,for example, increases adjused R 2 from 27% o 67% for AUDUSD, 15% TO 53% for EURUSD and from 10% o 57% for USDJPY. INSERT TABLE (3) AND FIGURE (4) HERE 21

23 4.3 Can opion-implied momens forecas FX excess reurns? Mached Frequency Analysis: Predicive abiliy of he volailiy smile For each currency pair i, we sar by invesigaing he predicive abiliy of τ period opion-implied measures of sandard deviaion, skewness and kurosis for subsequen excess currency reurns 32 : f i,+τ E (s i +τ) = γ 0,τ + γ 1,τ sdev i,+τ + γ 2,τ skew i,+τ + γ 3,τ kur i,+τ + u i,+τ. (4.5) Under raional expressions, f i,+τ E (s i +τ) is also equal o he risk premium. Gereben (2002) and Malz (1997) also esimae regression specificaion (4.5) 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 4.5), 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 +τ can be inerpreed as providing suppor for he peso problem explanaion of he UIP puzzle. Going back o expression 4.5), we noe ha E (s +τ ) is no observable. If we assume ha marke paricipans have raional expecaions, hen E (s +τ ) and s +τ will only differ by a forecas error ν +1 ha is uncorrelaed wih all variables ha use informaion a ime, such ha s +τ = E (s +τ ) + ν +1. (4.6) Plugging equaion (4.6) ino equaion (4.5) and rearranging gives us he following esimable 32 Excess reurns are a componen of he expeced exchange rae movemens since E (s i +τ ) s i can be decomposed ino excess reurns (f i,+τ E (s i +τ )) and he ineres differenial, which equals f s under CIP. 22

24 regression equaion: xr +τ = γ 0,τ + γ 1,τ sdev +τ + γ 2,τ skew +τ + γ 3,τ kur +τ + ɛ +τ (4.7) where he error erm ɛ +τ = u +τ + ν +τ and xr +τ is ex-pos excess reurns defined in expression (2.3). To provide inuiion regarding expeced coefficien signs in he regression equaion (4.7), 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 (2.2.2), 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 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 33 and 33 We only focus on he major breaks, and herefore do no choose he number of breaks according o informaion crieria such as AIC. 23

25 esimae a modificaion of regression equaion (4.7) ha includes ineracions wih srucural break indicaor variable: xr i +τ = γ 0,τ + γ 00,τ D1 i,τ + D1 i,τ γ 1,τ sdev i,+τ γ 3,τ kur i,+τ + γ 4,τ sdev i,+τ + D1 i,τ γ 2,τ skew i,+τ + γ 5,τ skew i,+τ + γ 6,τ kur i,+τ + D1 i,τ + ɛ i +τ. (4.8) where D1 i,τ is an indicaor variable ha is zero before he break dae and equal o one oherwise. The mached-frequency resuls, shown in ables 5(a)-5(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 a he 1% level. The adjused R 2 s for example, range from 13% (USDJPY) o 28% for 1 monh enor and from 20%(USDJPY) o 42% (EURJPY) for he 3M enor. 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 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 (θ.) = γ 0,τ + γ 1,τ ST DEV i,+τ + γ 2,τ SKEW i,+τ + γ 3,τ KURT i,+τ + ɛ i,+τ, (4.9) where Q xr i (θ.) is he θ h quanile of excess reurns given informaion available a ime. 34 Mached-frequency quanile regression resuls for 3M enor are shown in ables (7a)- 34 We esimae he quanile regression model using he same break daes obained in he OLS analysis 24

26 (7f). We find ha he coefficiens on non-inercep erms are always joinly significan across quaniles for all currency pairs. Adjused R 2 s range from 13% o 44% for AUDUSD, and 12% o 30% 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 (7a)- (7f)HERE Can he erm srucure of implied momens predic excess currency reurns? We firs exend regression equaion (4.7) by regressing 3M bilaeral excess reurns on 1M, 3M and 12M opion-implied momens. Tha is, for each currency pair i, we esimae he following OLS regression: xr i +3M = γ 0,3M + j γ 1,τ j sdev +τ j,i + j γ 2,τ j skew +τ j,i + j γ 3,τ j kur +τ j,i +ɛ i +3M, (4.10) where j {1M, 3M, 12M}. Similar o he mached-frequency analysis in subsecion (4.3.1), our final erm srucure regression model is a modificaion of (4.10) in which we include ineracions wih a srucural break indicaor variable D1. Regression resuls from specificaion (4.10) (wih break ) are shown in column B of able (2). Compared o he mached frequency resuls presened in column A, we see a huge increase in he adjused R 2 s. For example, adjused R 2 increases from 33% o 62% for AUDUSD, 34% o 49% for EURUSD, and from 20% o 36% for USDJPY. In column C of able (2), 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: 25

27 3 3 xr+3m i = γ 0,τ + γ 2,j P C j sdevt erm i + γ 3,j P C j skewt erm i + j=1 j=1 3 γ 4,j P C j kurt erm i + ɛ i +3M. j=1 (4.11) In equaion (4.11), P C j xxxxt erm i refers o he j h principal componen exrac from he currency i erm srucure of opion-implied momen xxxx. Resuls from esimaion regression equaion (4.11) are in column C of able (2). Lasly, we exend he specificaion in (4.11) by adding informaion from he erm srucure of firs momens as addiional regressors: 3 3 xr+3m i = γ 0,τ + γ 1,j P C j meant erm i + γ 2,j P C j sdevt erm i + j=1 3 γ 3,j P C j skewt erm i + j=1 j=1 j=1 3 γ 4,j P C j kurt erm i + ɛ i +3M. (4.12) The 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( S+τ S ). 35 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 (4.12) ha includes ineracions wih a srucural break indicaor variable. The condensed resuls from esimaing equaion (4.12) wih breaks are presened in column (D) of able (2). Acual vs fied plos from his regression are shown in figures 35 As noed earlier,he ) forward premium, which is he heoreical mean of he risk-neural probabiliy densiy of log( S+τ S is equal o he ineres differenial i τ i,τ. 26

28 (3(a)-3(e)). INSERT FIGURE (3) AND TABLE (2) HERE Compared o he mached frequency regressions column A of able (2), inclusinf he erm srucure of 1 s 4 h momens increases he adjused R 2 from 33% o 66% for AUDUSD, 34% TO 51% for EURUSD, 48% o 83% for GBPUSD, 48% o 62% for USDCAD and from 20% o 57% for USDJPY. These dramaic improvemens in he model fi furher highligh he imporance of properly accouning for expecaions and higher momen risks. 5 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 srucural modeling and empirical esing of exchange rae models. This paper can be exended in several direcions ha are useful o academics,moneary policy officials and invesmen professionals. Firs, how useful is he opion-based informaion for ou-of-sample forecasing of exchange rae. The abiliy o accuraely forecas exchange raes movemens for many purposes, including deermining he fuure value of foreign denominaed deb paymens and for hedging for invesmen managers exploiing inernaional invesmen opporuniies. Second, an empirical analysis of he macroeconomic variables and evens ha drive he opion-implied momens would furher shed ligh on he link beween exchange raes and fundamenals. Lasly, i migh be worhwhile o 27

29 References Bacchea, P. and E. van Wincoop (2009), Predicabiliy in financial markes:wha do survey daa ell us? Journal of Inernaional Money and Finance, 28, Bai, J. and P. Perron (2003), Compuaion and analysis of muliple srucural change models. Journal of Applied Economerics, 18, Bakshi, G., N. Kapadia, and D. Madan (2003), Sock reurn characerisics,skew laws, and he differenial pricing of individual equiy opions. Review of Finacial Sudies, 6, Breeden, D. and R. Lizenberger (1978), Prices of sae coningen claims implici in opions. Journal of Business, 51, Burnside, C., M. Eichenbaum, I. Kleschchelski, and S. Rebelo (2011), Do peso problems explain he reurns o he carry rade? Review of Financial Sudies, 24, Carr, P. and D. Madan (2001), Opimal posiioning in dervaive securiies. Quaniaive Finance, 1, Casagna, A. (2010), FX Opions and Smile Risk. WILEY, Unied Kingdom. Casagna, Anonio and Fabio Mercurio (2005), Consisen pricing of fx opions. SSRN Working Paper No Casagna, M. and F. Mercurio (2007), The vanna-volga mehod for implied volailiies. Risk, 20, Chen, Y-c. and K-P. Tsang (2013), Wha does he yield curve ell us abou exchange rae predicabiliy. Review of Economics and Saisics, 95,

30 Clarida, R.H., L. Sarno, M.P Taylor, and G. Valene (2003), The ou-of-sample success of erm srucure models as exchange rae predicors:a sep beyond. Journal of Inernaional Economics, 60, Clarida, R.H. and M.P. Taylor (1997), The erm srucure of forward exchange rae premiums and he forecasabiliy of spo exchange raes:correcing he errors. Review of Economics and Saisics, 79, Diebold, F.X., Piazzesi M., and Rudebusch G.D. (2005), Modeling bond yields in finance and macroeconomics. American Economic Review, 95, Engel, C. (1996), The forward discoun anomaly and he risk premium: A survey of recen evidence. Journal of Empirical Finance, 3, Engel, C. and K. D. Wes (2005), Exchange raes and fundamenals. Journal of Poliical Economy, 113, Engel, Charles (2013), Exchange raes and ineres pariy. In Handbook of Inernaional Economics,vol. 4,Forhcoming (Gia Gopinah, Elhanan Helpman, and Kenneh Rogoff, eds.), Elsevier. Eviews (2013), Eviews 8 guide ii. Fama, E. and K. R. French (1993), Common risk facors in he reurns on socks and bonds. Journal of Financial Economics, 33, Fama, Eugene F. (1984), Forward and spo exchange raes. Journal of Moneary Economics, 14, Gereben, A. (2002), Exracing marke expecaions from opion prices:an applicaion o over-he couner new zealand dollar opions. New Zealand Cenral Bank Working Paper. 29