The Volatility Risk Premium Embedded in Currency Options

The Volailiy Risk Premim Embedded in Crrency Opions Ben Sin Low and Shaojn Zhang JEL Classiicaion: G, G3 Keywords: volailiy risk premim, sochasic volailiy, crrency opion, erm srcre o risk premim Division o Banking and Finance, Nanyang Technological Universiy, Singapore 639798, abslow@n.ed.sg, phone: 65-67905753 Division o Banking and Finance, Nanyang Technological Universiy, Singapore 639798, asjzhang@n.ed.sg, phone: 65-6790440 We hank Phillipe Chen, Robin Grieves, William Lee, Pal Malaesa (he edior), Jn Pan, Yonggan Zhao, and he seminar paricipans a he Nanyang Bsiness School, he 6 h Annal Asralasian Finance and Banking Conerence, and he China Inernaional Conerence in Finance 004 or helpl commens and discssions. We especially hank an anonymos reeree or his consrcive commens and sggesions. Any remaining errors are or own responsibiliy. We also acknowledge inancial sppor rom he Cenre or Research in Financial Services a he Nanyang Bsiness School and he Cenre or Financial Engineering a he Nanyang Technological Universiy.

The Volailiy Risk Premim Embedded in Crrency Opions Absrac This sdy employs a non-parameric approach o invesigae he volailiy risk premim in he over-he-coner crrency opion marke. Using a large daabase o daily qoes on dela neral sraddle in or major crrencies he Briish Pond, he Ero, he Japanese Yen, and he Swiss Franc we ind ha volailiy risk is priced in all or crrencies across dieren opion mariies and he volailiy risk premim is negaive. The volailiy risk premim has a erm srcre where he premim decreases in mariy. We also ind evidence ha jmp risk may be priced in he crrency opion marke. I. Inrodcion I has been widely docmened in he lierare ha he price volailiy o many inancial asses ollows a sochasic process. This leads o he qesion o wheher volailiy risk is priced in inancial markes. Lamorex and Lasrapes (993), Coval and Shmway (00), and Bakshi and Kapadia (003), among ohers, presen sbsanial evidence ha volailiy risk is priced in he eqiy opion marke and ha he risk premim is negaive. However, alhogh oreign crrency rerns have sochasic volailiy (see, e.g., Taylor and X (997)), here is scan evidence on he marke price o volailiy risk in crrency opion markes. In his paper, we explore he volailiy risk premim in crrency opions.

Sarwar (00) sdies he hisorical prices o he Philadelphia Sock Exchange (PHLX) crrency opions on he U.S. Dollar/Briish Pond rom 993 o 995, and repors ha volailiy risk is no priced or he crrency opions in he sample. This inding conrass wih he overwhelming evidence o he exisence o volailiy risk premim in he eqiy opions markes. I is also inconsisen wih oher empirical indings in he crrency opion markes. For example, Melino and Trnbll (990, 995) repor ha sochasic volailiy opion models wih a non-zero price o volailiy risk have less pricing error and beer hedging perormance or crrency opions han do consan volailiy opion models. Frhermore, Black-Scholes implied volailiies or crrency opions have been shown o be biased orecass o acal volailiy (see Jorion (995), Covrig and Low (003), and Neely (003)). One possible explanaion or he bias is he presence o a volailiy risk premim. Invesors are presmably risk-averse and dislike volaile saes o he world. Wihin he ramework o inernaional asse pricing heory, Dmas and Solnik (995) and De Sanis and Gerard (998), among ohers, show ha oreign crrency secriies shold compensae invesors or bearing crrency risk in addiion o he radiional risk de o he covariance wih he marke porolio. As he volailiy o crrency price is also ncerain, i inrodces addiional risk ha invesors have o bear and shold be compensaed or. Asses ha lose vale when volailiy increases are more risky or invesors o hold han hose ha gain vale when volailiy increases, sch as crrency opions. Hence, nlike he case or oreign crrency secriies in he spo marke, where one may expec a posiive risk premim or bearing volailiy risk, his may no be he case or crrency opions. Coval and Shmway (00) ormalize his iniion in he

conex o mainsream asse pricing heory and show ha opions have greaer sysemaic risk han heir nderlying secriies. They provide evidence ha invesors are willing o pay a premim o hold opions in heir porolio as a hedge agains volaile saes o he world. Ths, his wold make he opion price higher han is price when volailiy risk is no priced. In oher words, he volailiy risk premim in crrency opions wold be negaive i i exiss. In his paper, we invesigae he volailiy risk premim in he over-he-coner (OTC) crrency opion marke. We exend a new mehodology proposed by Bakshi and Kapadia (003) rom eqiy index opions o crrency opions and apply i o a-hemoney dela neral sraddles raded in he OTC marke. An a-he-money dela neral sraddle is a combinaion o one Eropean call and one Eropean p wih he same mariy and srike price on he same crrency. A-he-money dela neral sraddles are he mos liqid opion conracs raded on he OTC marke. Becase heir prices are very sensiive o volailiy, hey are widely sed o hedge or speclae on changes in volailiy. Thereore, i volailiy risk is priced in he crrency opion marke, sraddles are he bes insrmens hrogh which o observe he risk premim. Or daabase incldes daily OTC average bid and ask implied volailiy qoes or Eropean a-he-money dela neral sraddles. The daabase covers he Briish Pond, Japanese Yen, and Swiss Franc (agains he U.S. Dollar) rom Jne 996 o December The OTC crrency opion marke is sbsanially more liqid han he exchange raded crrency opion marke. The annal rnover o crrency opions ha are raded on organized exchanges was abo US$.3 rillion in 995 and declined o US$0.36 rillion in 00 (see Bank or Inernaional Selemen (997, 003)), whereas he annal rnover on he over-he-coner marke was abo US$0.5 rillion in 995 and abo US$5 rillion in 00 (see Bank or Inernaional Selemen (996, 00)). 3

00, and he Ero rom Janary 999 o December 00. In OTC crrency opion marke, opion prices are qoed in erms o volailiy, expressed as a percenage per annm. For example, an opion ha is qoed a a 0% bid has he opion premim comped by sbsiing 0% as he volailiy igre ino he Garman-Kohlhagen (983) model, along wih he prevailing crren spo exchange rae and domesic and oreign ineres raes. One advanage o he daa is ha he OTC volailiy qoes apply o opion conracs o he same sandard mariy erm, regardless o which day he price is qoed. We sdy a-he-money sraddles or mariies o monh, 3 monhs, 6 monhs, and monhs. Or approach ensres ha or implied volailiy series are homogeneos wih respec o moneyness and mariy, and ha conclsions drawn rom analyzing his daabase are nlikely o be aeced by he mixre o moneyness and mariy. Or main indings inclde he ollowing. Firs, we ind ha volailiy risk is priced in or major crrencies he Briish Pond, Ero, Japanese Yen, and Swiss Franc across mariy erms beween monh and monhs. 3,4 Second, we provide direc evidence o he sign o he volailiy risk premim. The risk premim is negaive or all or major crrencies, which sggess ha In a sdy on he erm srcre o implied volailiies, Campa and Chang (995) se eqivalen OTC opion volailiy qoes rom December 989 o Ags 99. 3 The resls are conrary o he indings o Sarwar (00). The dierence may be becase Sarwar (00) ses daa or exchange-raded opions, which are mixed in mariy and moneyness, whereas he qoes o OTC crrency opions in or sdy have consan mariy and apply only or a-he-money opions. 4 Evidence o volailiy risk premim is also ond in or oher crrencies agains U.S. dollar and hree cross crrencies. The resls are repored in secion VI. 4

byers in he OTC crrency opion marke pay a premim o sellers as compensaion or bearing he volailiy risk. Third, and more imporan, we ind ha he volailiy risk premim has a erm srcre in which he risk premim decreases in mariy. This sdy is he irs o provide empirical evidence o he erm srcre o he volailiy risk premim. Previos sdies have docmened ha shor-erm volailiy has higher variabiliy han long-erm volailiy (e.g. X and Taylor (994) and Campa and Chang (995)), b none have invesigaed he implicaion on volailiy risk premims. Forh, we docmen ha jmp risk is also priced in he OTC markes. However, he observed volailiy risk premim is disinc rom and no sbsmed by he possible jmp risk premim. All o or indings are robs o varios sensiiviy analyses on risk-ree ineres raes, dieren sb-periods, and speciicaions o empirical models. The paper is organized as ollows. Secion II deails he mehodology sed in he sdy. Secion III explains he niqe eares o OTC crrency opion markes and he daa. Secion IV describes he empirical implemenaion o he mehodology. Secion V presens or main empirical indings and Secion VI repors some robsness sdies. Secion VII provides a smmary and conclsion. II. Mehodology Bakshi and Kapadia (003) propose a non-parameric mehod o invesigae volailiy risk premims in eqiy index opion markes. Under a general sochasic 5

opion-pricing ramework hey prove ha i volailiy risk is priced in he opion marke, hen he rern o a dynamically dela-hedged call opion on a sock index is mahemaically relaed o he volailiy risk premim. Hence, i is heoreically sond o iner he sign o he volailiy risk premim rom rerns on dynamically dela-hedged call opions. This approach allows he invesigaion o he volailiy risk premim wiho he imposiion o srong resricions on he pricing kernel or assming a parameric model o he volailiy process. Using his approach, Bakshi and Kapadia (003) show ha volailiy risk is priced in he S&P 500 index opion marke and ha he volailiy risk premim is negaive. We exend heir mehodology o he crrency opion marke. A. Theory Assme ha he spo price o a crrency a ime, x, ollows he process: dx () = md σ dz x () dσ = θ d δ dw where z and w are sandard Wiener processes, he random innovaions o which have insananeos correlaion ρ. Parameers m and σ are he insananeos dri and volailiy o he crrency spo price process; m can be a ncion o x and σ. The insananeos volailiy σ, ollows anoher dision process wih mean θ and sandard deviaion δ as speciied in eqaion (), where θ and δ may depend on σ b no on x. Le ƒ denoe he price o a Eropean sraddle on he crrency. By Io s lemma, ƒ ollows he sochasic process characerized by: 6

7 (3) d x x x x d dx x d = σ σ ρδ σ δ σ σ σ The price change, dƒ is properly inerpreed mahemaically as he ollowing sochasic inegral eqaion: (4) = τ τ τ τ σ σ ρδ σ δ σ σ σ d x x x x d dx x Using sandard arbirage argmens (see Cox, Ingersoll and Ross (985)), he sraddle price, ƒ, ms saisy he ollowing parial dierenial eqaion: (5) 0 ) ( ) ( = r x x q r x x x x σ λ θ σ σ ρδ σ δ σ where r and q denoe he domesic and oreign risk ree raes. The nspeciied erm, λ, represens he marke price o he risk associaed wih dw, which is commonly reerred o as he volailiy risk premim (see, e.g., Heson (993)). By rearranging eqaion (5) and sbsiing i ino he las inegral o eqaion (4), we obain: (6) ( ) ( ) = τ τ τ τ σ λ θ σ σ d x x q r r d dx x Sbsiing eqaion () ino (6), we can rewrie eqaion (6) and he sraddle price can be expressed as: (7) ( ) = τ τ τ τ τ σ δ σ λ dw d d x x q r r dx x We now consider a dynamically dela-hedged porolio ha consiss o a long sraddle posiion and a spo posiion in he nderlying crrency. The spo posiion is

adjsed over he lie o he sraddle ( o τ) o hedge all risks excep volailiy risk. The ocome o his dynamically dela-hedged porolio, hereaer reerred o as he delahedged sraddle proi (loss), is given by: x d x x τ τ (8) Π =, τ τ dx r ( r q) From eqaion (7) his can also be saed as: τ τ (9) Π, τ = λ d σ δ σ dw The second inegral in eqaion (9) is he Io sochasic inegral. Hence, he maringale propery o he Io inegral implies: (0) τ E ( Π =, τ ) E λ d σ The implicaion o eqaion (0) is ha i he volailiy risk is no priced (i.e., λ = 0), hen he dela-hedged sraddle proi (loss) on average shold be zero. I he volailiy risk is priced (i.e., λ 0), hen he expeced dela-hedged sraddle proi (loss) on average ms no be zero. Becase he vega o a long sraddle,, is posiive, he σ sign o he volailiy risk premim, λ, deermines wheher he average dela-hedged sraddle proi is posiive or negaive. B. Tesable Implicaions The heory sggess ha he rern on bying a sraddle and dynamically delahedging his posiion nil mariy is relaed o he volailiy risk premim. Alhogh he 8

heory reqires he assmpion o coninos hedging, in pracice, rebalancing akes place only a discree imes. 5 Sppose ha we rebalance he dela-hedged porolio a N eqally spaced imes over he lie o he sraddle beween ime and τ. Tha is, he hedging posiion is calclaed a ime n, n = 0,,,. N-, where 0 = and N = τ. We compe he dela-hedged sraddle proi a he mariy τ by: N N τ (), τ = Max ( x τ k, k x τ,0) - ( x x )- ( r ( ) ) r q n n n x n n n= 0 n= 0 N where, is he sraddle premim a ime, k is he srike price o he sraddle, x is he crrency price a he mariy τ, τ is he dela o he long sraddle, r and q are he n domesic and oreign ineres raes. The irs erm on he righ-hand side o he eqaion is he payo o he long sraddle a mariy τ, he second erm is he cos o bying he sraddle a ime, he hird erm is he rebalancing cos, and he las erm adjss or he ineres expenses on he second and hird erms. The dela-hedged sraddle proi (loss) allows s o es he ollowing wo hypoheses: Hypohesis : I, on average, Π, τ is non-zero, hen volailiy risk is priced in he crrency opion marke. Hypohesis : I, on average, Π, τ is negaive (posiive), hen he volailiy risk premim embedded in he crrency opion is negaive (posiive). 5 Bakshi and Kapadia (003) show ha he bias in he dela-hedged sraddle ocome cased by discree hedging is small relaive o he eec o a volailiy risk premim. Melino and Trnbll (995) provide simlaion evidence o show ha he discree dela hedging error wih daily re-balancing is very small. For example, heir average hedging errors, as a percenage o conrac size, or a -year a-he-money crrency opion are only 0.06% and 0.3% i volailiy is assmed o be consan or sochasic, respecively. 9

III. Descripion o he OTC Crrency Opion Marke and Daa The OTC crrency opion marke has some special eares and convenions. Firs, he opion prices on he OTC marke are qoed in erms o delas and implied volailiies, insead o srikes and money prices as in he organized opion exchanges. A he ime o selemen o a given deal, he implied volailiy qoes are ranslaed o money prices wih he se o he Garman-Kohlhagen ormla, which is he eqivalen o he Black-Scholes ormla or crrency opions. This arrangemen is convenien or opion dealers, in ha hey do no have o change heir qoes every ime he spo exchange rae moves. However, as poined o by Campa and Chang (998), i is imporan o noe ha his does no mean ha opion dealers necessarily believe ha he Black-Scholes assmpions are valid. They se he ormla only as a one-o-one nonlinear mapping beween he volailiy-dela space (where he qoes are made) and he srike-premim space (in which he inal speciicaion o he deal is expressed or he selemen). Second, compeing volailiy qoes o opion conracs are available on he marke everyday, b only or sandard mariy periods, sch as -week, -, 3-, 6-monh, and so on. For example, a 3-monh opion qoe on Monday will become an odd period (3 monhs less day) opion qoe on Tesday, and compeing qoes or his odd period opion are no available on Tesday. Third, mos ransacions on he marke involve opion combinaions. The poplar combinaions are sraddles, risk reversals, and srangles. Among hese, he mos liqid combinaion is he sandard dela-neral sraddle conrac, which is a combinaion o a call and a p opion wih he same srike. 0

The srike price is se, ogeher wih he qoed implied volailiy price, so ha he dela o he sraddle comped on he basis o he Garman-Kohlhagen ormla is zero. As he sandard sraddle by design is dela-neral on he deal dae, is price is no sensiive o he marke price o he nderlying oreign crrency. However, i is very sensiive o changes in volailiy. Becase o is sensiiviy o volailiy risk, dela-neral sraddles are widely sed by paricipans in he OTC marke o hedge and rade volailiy risk. I he volailiy risk is priced in he OTC marke, hen dela-neral sraddles are he bes insrmens hrogh which o observe he risk premim. For his reason, Coval and Shmway (00) ses dela-neral sraddles in heir empirical sdy o expeced rerns on eqiy index opions and ind ha volailiy risk premim is priced in he eqiy index opion marke. Or main opion daase consiss o daily average bid and ask implied volailiy qoes or he -, 3-, 6-, and -monh dela-neral sraddles on or major crrencies he Briish Pond, he Ero, he Japanese Yen, and he Swiss Franc 6 a heir U.S. Dollar prices. Or sample spans a period o approximaely 7.5 years, rom Jne 3, 996 o December 3, 003. The 003 daa are sed or calclaing he dela-hedged sraddle proi on he sraddle bogh in 00. The daa are obained rom Bloomberg, who colleced hem a 6 p.m. London ime rom large banks paricipaing in he OTC crrency opion marke. Oher daa colleced are synchronized daily average bid and ask spo U.S. Dollar prices o each crrency rom Bloomberg. Becase he Briish Pond, Ero, 6 We chose o sdy crrency opions on hese or crrencies becase hey are he mos liqid among all crrencies. According o he Bank or Inernaional Selemen (00), hey acconed or approximaely 67% o he oal size o he OTC crrency opions raded in April 00. As a robsness es we also sdy seven oher less liqid crrency opions and he resls are repored in secion VI.

Japanese Yen, Swiss Franc, and U.S. Dollar Treasry bill yields ha exacly mach each opion period are no available, he reprchase agreemen ineres raes (i.e., repo) or each crrency, which mach he opion mariy period, are sed. 7 Figre ses boxplos o show he disribion o he daily implied volailiy qoes on -, 3-, 6-, and -monh dela-neral sraddles or he or crrencies beween Jne 996 and December 00, excep he Ero. The solid box in he middle o each boxplo represens he middle 50% o he observaions ranging rom he irs qarile o he hird qarile, and he brigh line a he cener indicaes he median. Several paerns sand o in Figre. Firs, a all mariies, he Briish Pond has he lowes level o implied volailiy and he smalles variaion among he or crrencies, whereas he Japanese Yen has he highes level o implied volailiy and he greaes variabiliy. Take he one-monh mariy as an example. A one exreme, he implied volailiy o he Briish Pond has a median o 8.3% per annm and an iner-qarile range o %. A he oher exreme, he implied volailiy o he Japanese Yen has a median o.% and an iner-qarile range o 3.6%. In beween, he Swiss Franc and Ero have median implied volailiies o 0.6% and 0.8%, and iner-qarile ranges o.4% and 3.0%. Inser Figre here 7 The repo rae has a credi qaliy ha is closes o he yield o a Treasry bill. However, o examine he sensiiviy o or resls o he se o repo raes insead o he re risk-ree raes, we adjs or credi qaliy spread by sbracing % rom he repo raes and se he redced raes o rern he empirical ess. We obain qaliaively he same resls or he Briish Pond, Ero, and Swiss Franc. We cold no do his or he Japanese Yen becase is repo raes are very close o zero.

Second, here is a erm srcre in he variabiliy o he volailiy qoes or all or crrencies. Speciically, he variabiliy is a decreasing ncion o he ime-omariy as shor-daed opions have mch higher variabiliy han long-daed opions. Campa and Chang (995) observe a similar erm srcre in heir sample o OTC volailiy qoes or or major crrencies in a dieren ime period. X and Taylor (994) sdy he erm srcre o implied volailiy embedded in PHLX raded opions on or crrencies and repor ha long-erm implied volailiy has less variabiliy han shor-erm implied volailiy. The erm srcre in variabiliy o implied volailiy is consisen wih a mean-revering sochasic volailiy process (see Sein (989) and Heynen, Kemna, and Vors (994)). More imporanly, i has an implicaion or he volailiy risk premim. Becase he variabiliy o shor-erm volailiy is mch higher han ha o long-erm volailiy, i opion byers were o pay a volailiy risk premim, hen hey wold pay more in shor-erm opions. This means ha he volailiy risk premim shold have a erm srcre in which he risk premim is a decreasing ncion o mariy. We repor empirical evidence in relaion o his hypohesis in Secion V. Third, he boxplos show he skewness o he implied volailiy disribion. In each boxplo, he lower bracke conneced by whiskers o he boom o he middle box indicaes he larger vale o he minimm or he irs qarile less.5 imes he inerqarile range, while he pper bracke conneced by whiskers o he op o he middle box indicaes he lower vale o he maximm or he hird qarile pls.5 imes he iner-qarile range. For a righ-skewed disribion, he pper bracke is rher away rom he box han he lower bracke; he paern reverses or a le-skewed disribion. The lines beyond he lower or pper bracke represen oliers. The or crrencies dier 3

in skewness o implied volailiy. While he Briish Pond and he Swiss Franc have close-o symmeric disribions, he disribions o he Ero and he Japanese Yen are clearly righ skewed and he Yen has ar more oliers (i.e., a mch aer ail) han he oher hree crrencies. A close examinaion shows ha a majoriy o he oliers in he Yen occrred dring he Asian crrency crisis in 997 and 998. Figre shows he ime series plo o he implied volailiy or he 3-monh a-he-money sraddle or he Briish Pond, he Japanese Yen, and he Swiss Franc beween Jne 996 and December 00. The wo verical dash lines indicae he sar and he end o he Asian crrency crisis. The crisis dramaically changed he price process o he Japanese Yen dring ha period, b had lile eec on he Briish Pond and Swiss Franc. This sggess ha we shold condc a robsness analysis or he pos-crisis sb-period. Inser Figre here IV. Empirical Implemenaion Following he heory discssed in Secion II, we now consider porolios o bying dela-neral sraddles and dynamically dela-hedging or posiions nil mariy. We rebalance he dela-hedged porolio daily and measre he dela-hedged sraddle proi (loss) rom he conrac dae o he mariy dae τ, Π, τ, by he ormla N N τ ( x τ k, k x τ,0) - ( x x )- ( r ( r q ) x ) n n n n n N, τ = Max n= 0 n= 0 4

where k denoes he srike price o he sraddle, and on he Garman-Kohlhagen model, is he dela o he sraddle based n () = q e n ( n τ )[ N( d ) ] x n ln r q σ ( n τ ) n k d = where and N(d ) is he cmlaive sandard σ τ n n normal disribion evalaed a d. The Garman-Kohlhagen model, as an exension o he Black-Scholes model o crrency opions, is a consan volailiy model. Hence, he dela comped rom he Garman-Kohlhagen model may dier rom he dela comped rom a sochasic volailiy model. We miigae his problem by adoping a modiied Garman-Kohlhagen model, in which he volailiy ha is employed o compe he dela or daily rebalancing is pdaed based on he daily average bid and ask implied volailiy qoes. Chesney and Sco (989) conclde ha acal prices on oreign crrency opions conorm more closely o his modiied Garman-Kohlhagen model han o a sochasic volailiy model or o a consan volailiy Garman-Kohlhagen model. Bakshi and Kapadia (003) also provide a simlaion exercise o show ha sing he Black-Scholes dela hedge raio, insead o he sochasic volailiy conerpar, has only a negligible eec on dela-hedged resls. We repor a robsness sdy in secion VI ha examines he impac o poenial mis-measremen o he hedge raio More speciically, we ake he ollowing wo seps o mainain he dela-hedged porolio nil he mariy. Firs, we need o compe he money price or he sraddle. This is achieved by a one-o-one mapping beween he volailiy-dela space and he srike-premim space sing he Garman-Kohlhagen model. For an observed implied volailiy qoe on day, he srike price o he dela-neral sraddle is deermined by he 5

ormla ( r q 0.5σ )τ k = x e, where x denoes he synchronized spo oreign crrency price on day, r and q are he domesic and oreign ineres raes per annm, τ is he ime-o-mariy expressed in years, and σ is he average bid and ask implied volailiy qoe. 8 This srike price ormla is sed in he OTC marke by convenion. The sraddle premim is hen comped sing he Garman-Kohlhagen model, ogeher wih he comped srike price, he synchronized spo oreign crrency price, and ineres raes. Second, we orm and mainain he dela-hedged porolio nil he mariy. Sppose ha, on day, we bogh a sraddle conrac a he premim comped as above. As he sraddle is in isel dela neral, he dela-hedged porolio on day is composed o only he sraddle. However, on he ollowing day, he sraddle becomes an oddperiod conrac and may no longer be dela neral. Hence, o mainain a dela-hedged porolio, we need o sell he dela amon o he oreign crrency agains he U.S. Dollar a he day spo price. We hereore re-compe he dela sing Eqaion () wih he spo price, ineres raes, and he esimaed volailiy o an odd-period sraddle conrac a day. 9 We esimae he volailiy o an odd-period sraddle conrac sing he linear oal variance mehod (see Wilmo (998)) o inerpolae he volailiy o sandard mariies: (3) T T σ T σ ( T σ T σ ) T = T 3 T3 T T T 3 T 8 Eqaion () shows ha he dela o a sraddle is zero only when d = 0. The srike price is comped in sch a way ha d = 0. 9 We canno se he qoed volailiy on day becase volailiy qoes are valid only or sraddles o sandard mariies sch as monh or 3 monhs. Volailiy has o be esimaed or odd-period sraddles. 6

where T < T < T 3, σ T and σ T denoe he average bid and ask implied volailiy qoes 3 or he sandard mariies T and T 3 available in he marke, and σ T is he volailiy or he non-sandard mariy T ha we need o esimae by inerpolaion. We also pdae he ineres raes on a daily basis in rebalancing he dela-hedged porolio. The oreign crrency sold (bogh) is borrowed (invesed), and he corresponding long (shor) U.S Dollar cash (ne o he sraddle premim incrred a ime ) is invesed (borrowed) a heir respecively ineres raes. We conine o rebalance he dela-hedged porolio on a daily basis nil he sraddle mariy dae. The ne U.S. Dollar payo is he dela-hedged sraddle proi (loss). V. Empirical Resls A. Negaive Volailiy Risk Premim In his secion, we docmen or empirical indings. Table repors he saisical properies o he dela-hedged sraddle rerns on or crrencies a or mariies. The dela-hedged sraddle rerns are calclaed as he U.S. Dollar dela-hedged sraddle proi (loss) rom holding he dynamically dela-hedged porolio nil mariy, divided by he sraddle conrac size in U.S. Dollars. We annalize he rerns o make hem comparable across mariies. For each combinaion o crrency and mariy, delahedged sraddle rerns comprise a ime series o daily observaions. This arises becase, or each sandard mariy, we by an a-he-money dela neral sraddle each rading day and mainain a dela-hedged porolio o he sraddle and nderlying crrencies nil he sraddle mares. The irs colmn o Table liss he nmber o observaions in each 7

ime series. The Ero has ewer observaions han he oher hree crrencies becase i only came ino exisence in Janary 999. In he second colmn, we repor he percenage o daily sraddle observaions ha have negaive rerns when we by a sraddle and mainain he dela-hedged porolio nil mariy. The percenage is mch bigger han 50% in all cases, and is 85% or he Briish Pond a he -monh mariy. This indicaes ha or mos o he ime, OTC sraddle sellers earn posiive prois by selling sraddles and hedging heir exposres. According o he heory in Secion II, his sggess ha sch prois are a compensaion or bearing he risk o volailiy changes. The high percenage o negaive dela-hedged sraddle rerns indicaes ha or resls are no biased by oliers. Inser Table here We now invesigae wheher dela-hedged sraddle rerns are saisically signiican. The ncondiional means and sandard deviaions o dela-hedged sraddle rerns are lised in he hird and orh colmns o Table. The mean rern is negaive or all cases 0. The high sandard deviaions o he rerns make he means appear insigniicanly dieren rom zero. However, i is misleading o se he ncondiional sandard deviaion o es he mean, becase serial correlaion in he ime series o dela- 0 We have rn he empirical ess sing he average bid and ask sraddle prices o redce he impac o bid ask spread i here is any. We have also rn he empirical ess sing boh he bid prices and he ask prices. The resls are qaliaively similar o hose sing sraddle average bid and ask prices. 8

hedged sraddle rerns can case he sandard deviaion o be a biased measre o acal random error. The nex hree colmns o Table show ha he irs hree aocorrelaion coeiciens are qie large and decay slowly. This indicaes ha he ime series may ollow an aoregressive process. We calclae he parial aocorrelaion coeiciens in Table. The irs order parial aocorrelaion coeicien is large in all cases, while he second and hird order aocorrelaion coeiciens become mch smaller. The paern exhibied in boh aocorrelaion coeiciens and parial aocorrelaion coeiciens sggess iing an aoregressive process o order 3 (i.e., AR(3)) o he ime series o he dela-hedged sraddle rerns. An AR(3) process can be represened by he ollowing model: y = α β y β y β 3 y 3 ε, where ε is a whie noise process. Is ncondiional mean is given by he ormla E ( y α ) = β β β 3 which implies ha he nll hypohesis o a zero ncondiional mean is eqivalen o he nll hypohesis ha he inercep o he AR(3) process is eqal o zero. We esimae parameers o he AR(3) process and repor he esimaed inercep and is p-vale or he -saisic in he las wo colmns o Table. The inercep is signiicanly negaive in mos cases or he Briish Pond, he Ero, and he Swiss Franc, whereas i is negaive, alhogh insigniican, or he Japanese Yen. We sspec ha he non-signiicance o he Japanese Yen is de o he Asian crrency crisis. Hence, in In an nrepored analysis, we also i AR() and AR(5) processes o he daa and observe he same paerns. 9

Secion VI we repor a robsness analysis or he pos-crisis sb-period rom Jly 999 o December 00. Relying on he general eqilibrim model o Cox, Ingersoll and Ross (985), Heson (993) and Baes (000), among ohers, sgges ha he volailiy risk premim is posiively relaed o he level o volailiy. To conrol or he level o volailiy, we consider he ollowing model or he dela-hedged sraddle rerns: (4) y = α γσ βy βy β3y 3 ε where y is he dela-hedged sraddle rern, and σ is he implied volailiy. We inclde hree lagged variables y, y, and y 3, o conrol or serial correlaion. Table repors he resls o esimaing he above model or or crrencies a or mariies. The coeicien o implied volailiy, γ, is negaive or all crrencies a all mariies. To es is signiicance, we se he -es saisic based on Newey-Wes (987) heeroskedasiciy and aocorrelaion consisen sandard errors. The es shows ha γ is signiicanly negaive in all cases. This again provides evidence ha marke volailiy risk is priced in he OTC crrency opion markes. Inser Table here B. Eec o Overlapping Period We obain or daily dela-hedged sraddle rerns by prchasing a sraddle and mainaining a dela-neral porolio sing he spo crrency marke nil he sraddle mares. In calclaing he dela-hedged rern o he -monh sraddle bogh on a given 0

rading day, say day 0, we se he inormaion o day, day, p o day, assming ha here are rading days beore he sraddle mariy dae. Then, or he delahedged rern o he -monh sraddle bogh on day, we se inormaion o day, day 3, p o day 3. Conseqenly, he dela-hedged rerns o day 0 and day sraddles se inormaion rom an overlapping period beween day and day. There is a concern wheher or earlier evidence o a negaive risk premim is driven by he common inormaion in he overlapping periods. To address his isse, we adop he ollowing wo approaches: In he irs approach, we consrc a ime series o non-overlapping dela-hedged sraddle rern or each crrency. Speciically, or each crrency, we consrc a monhly series o dela-hedged rerns on he -monh sraddles bogh a he irs rading day o every monh in or sample period. As he dela-hedged rerns o he beginning-omonh sraddle only depend on he inormaion o he rading days in he same monh, hey are non-overlapping. Panel A o Table 3 repors he smmary saisics or he nonoverlapping rerns or he or major crrencies. Consisen wih he earlier resls in Table, he mean dela-hedged rerns are negaive across all or crrencies. To ascerain wheher he volailiy risk premim remains negaive or nonoverlapping series, we rn he ollowing regression (5) y = α γσ y ε where y is he dela-hedged sraddle rern, and σ is he volailiy qoe. I he volailiy risk is priced and has a negaive premim, hen we expec γ o be signiican We consider only he -monh mariy becase oo ew non-overlapping dela-hedged rerns are available or longer mariies or any meaningl analysis.

and negaive. Panel B o Table 3 repors he resls. The evidence sppors he conclsion ha volailiy risk is priced and has a negaive risk premim. Inser Table 3 here In he second approach, we remove he eec o he common inormaion rom he overlapping period by calclaing he dierence beween wo consecive delahedged sraddle rerns. In oher words, we sdy he dierence series, y = y y, where y is he dela-hedged rern on a dela-neral sraddle bogh on dae. Since he sraddle prices dier beween he wo days and he sraddle rern is proporional o he volailiy risk premim as eqaion (0) shows, we expec o observe he risk premim in he regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy. Thereore, we rn he ollowing regression or y : (6) y = γ σ β y β y β y D D D D D3 3 α ε where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he volailiy qoed or a dela-neral sraddle on dae, and σ is he dierence beween σ and σ. We inclde hree lags o he dependen variable o conrol or poenial serial aocorrelaion in risk premim exiss and is negaive, we expec he coeicien o and negaive. y. I he volailiy σ o be signiican Table 4 repors he esimaion resls o eqaion (6). The coeicien o implied volailiy, γ D, is negaive or all crrencies a all mariies. The -es saisic based on

Newey-Wes (987) heeroskedasiciy and aocorrelaion consisen sandard errors shows ha γ is signiicanly dieren rom zero in all cases. This evidence provides rher sppor ha marke volailiy risk is priced in crrency opion markes. Inser Table 4 here C. Term Srcre o Volailiy Risk Premim The empirical resls so ar have shown ha volailiy risk is priced in he OTC crrency opion markes and i has a negaive risk premim. However, hey reveal rher inormaion abo he erm srcre o he volailiy risk premims. Previos sdies sch as hose o Campa and Chang (995) and X and Taylor (994) repor ha volailiy isel is more volaile a shor-erm mariies han a long-erm mariies. The paern is also eviden in Figre. This means ha shor-erm opions carry higher volailiy risks han long-erm ones. Hence, i is reasonable o expec ha crrency opion byers pay a higher volailiy risk premim or shorer mariy opions as a compensaion o opion sellers or bearing higher volailiy risks. Consisen wih his expecaion, Table shows ha he average dela-hedged sraddle rern decreases when he opion mariy is exended. Table and able 4 show ha he coeiciens o implied volailiy, γ and γ D, also exhibi a erm srcre. Becase γ is he coeicien o volailiy, i can be regarded as he average dela-hedged sraddle rern per ni o volailiy. We call his he ni dela-hedged sraddle rern. We observe ha he magnide o he ni dela-hedged sraddle rern decreases wih mariy. Becase he vega o a-he-money 3

opions is an increasing ncion o mariy (see, e.g., Hll (003)), he downward sloping erm srcre in he magnide o he ni dela-hedged sraddle rern is nlikely o be de o vega. This evidence, combined wih eqaion (0), sggess ha he magnide o he volailiy risk premim decreases wih opion mariy, which is consisen wih he ac ha shor-erm volailiy is more volaile han long-erm volailiy. To es wheher he observed dierence beween he shor-erm volailiy risk premim and he long-erm volailiy risk premim is saisically signiican, we esimae he ollowing model on he combined ime series across he or mariies: (7) y = γ σ, I α I γ σ β y 3 α I 3, 3 I 3 3 β y α I γ σ α 6, I β y 3 3 6 6 6 I 6 γ σ ε, I where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ i, is he qoed volailiy or he i -monh mariy on dae, σ is he dierence beween σ and σ, and I i is he indicaor variable o an observaion or he i -monh mariy wih i being, 3, 6 or. We se α i o allow dieren inerceps a dieren mariies. The esimaion resls are repored in Table 5. The volailiy risk premim presens a erm srcre or all or crrencies. We se he Wald saisic o es wo nll hypoheses: one is γ = γ and he oher γ γ 3 = γ 6 γ. The es rejecs he wo nll hypoheses in all cases, which sggess ha he dierence beween shor-erm and longerm volailiy risk premims is signiican. This inding implies ha he opion byer is 4

paying a signiicanly higher volailiy risk premim o he opion seller or he shorer mariy opion. Inser Table 5 here VI. Robsness Analysis A. Oher Crrency Pairs To invesigae wheher or resls are a special eare o he or crrencies seleced or wheher hey apply more broadly, we examine seven oher crrency pairs. They are seleced based on he liqidiy o he opion conrac and he availabiliy o daa. For o he seven crrency pairs are he Asralian Dollar, Canadian Dollar, Norwegian Kroner, and New Zealand Dollar agains he U.S. Dollar. The oher hree are cross crrency pairs and hey are he Japanese Yen agains Briish Pond, he Japanese Yen agains Ero, and he Ero agains Swiss Franc. As hese seven crrency pairs are less liqid han he or seleced crrency pairs in he OTC opion marke, Bloomberg does no have complee daily sraddle qoes rom Jne 996 o December 003. We inclde only observaions beore December 003 ha have complee daa necessary or or empirical analysis. The sample size or each crrency pairs is repored in Table 6. We replicae he analysis based on Eqaions (6) and (7) or hese seven crrency pairs. Table 6 shows ha he coeicien o implied volailiy in eqaion (6) is 5

negaive and signiican. This is consisen wih he earlier observaions or he or major crrency pairs, and demonsraes ha volailiy risk is also priced in he OTC marke or hese seven crrency pairs. Inser Table 6 here Table 7 repors he esimaion resls o Eqaion (7) and provides evidence on he erm srcre in volailiy risk premim. Consisen wih he evidence in he Briish Pond, Ero, Japanese Yen and Swiss Franc agains U.S. dollar, he volailiy risk premim presens a erm srcre or all seven crrencies. The magnide o he shorerm risk premim is greaer han is long-erm conerpar. The wo Wald ess show ha he dierence is signiican. Inser Table 7 here B. Pos-Asian Crrency Crisis Period In his sbsecion, we repor a sb-period analysis ha serves wo prposes: o examine he emporal sabiliy o he volailiy risk premim, and o rle o he possibiliy ha or main indings may be aeced by he Asian crrency crisis. We replicae he analysis or he pos-crisis period beween Jly, 999 and December 3, 00. Table 8 repors he resls or he pos-crisis period. We observe he same properies o he dela-hedged sraddle rerns as in Table. The main dierence beween Table 8 and Table lies in he resls or he Japanese Yen. Dring he pos- 6

crisis period, he mean rern is signiicanly negaive or he Yen, whereas i is no or he whole period. A possible explanaion is ha he Asian crrency crisis aeced he Yen more han he oher hree crrencies, which cases he disribion o he Yen s implied volailiy o dier a lo dring and aer he crisis. Figre 3 shows ha he disribion is less righ-skewed or he Yen in he pos-crisis period han or he whole period. Inser Table 8 here Inser Figre 3 here Table 9 repors he esimaion resls o Eqaion (6) or he pos-asian crrency crisis period. I shows he same evidence as in Table 4 ha he volailiy risk premim is signiicanly negaive or he or crrencies and he or mariies nder or sdy. Inser Table 9 here C. Impac o Jmp Risk Recen sdies sgges ha opion prices accon or no only he sochasic volailiy in he rern disribion o nderlying asses, b also he poenially large ail evens. 3 Theoreical opion pricing models have been developed o incorporae boh sochasic volailiy and poenial jmps in he nderlying rern process (see Pan (00) 3 See, or example, Bakshi, Cao, and Chen (997), Baes (000), Die, Pan, and Singleon (000), Eraker, Johannes, and Polson (000), and Pan (00) or heoreical analysis and empirical evidence. 7

and he reerences herein). Hence, i is possible ha par o he risk premim observed in or empirical resls is de o jmp risk raher han volailiy risk. In his secion, we condc rher analysis o show ha he volailiy risk premim is disinc rom he jmp risk premim and is indeed a porion o he opion price. To isolae he poenial eec o jmp risk, we need o examine he dela-hedged sraddle rerns or a sample where jmp ears are mch less prononced. We do so by ideniying he days where jmp ears are high and exclde hem rom he sample. We arge ha large moves in crrency prices case marke paricipans o revise pwards heir expecaion o re large moves. This is consisen wih he ac ha GARCH ype models are adeqae or he rern process o inancial asses (see, e.g., Bollerslev, Cho, and Kroner (99)). Jackwerh and Rbinsein (996) repor ha aer he Ocober 987 marke crash, he risk-neral probabiliy o a large decline in he eqiy marke index is mch higher han beore he crash. Hence, jmp ears are likely o be high aer large moves in crrency prices. We ideniy he daes when crrency prices experienced large moves so ha he daily percenage change is wo sandard deviaions away rom he mean daily percenage change in or sample period. The irs colmn o Table 0 repors he nmber o days ha experienced a large move in crrency prices. We do no diereniae beween negaive and posiive jmps becase he sraddle price is eqally sensiive o moves in boh direcions. Take he Briish Pond as an example. Beween Jne 3, 996 and December 3, 00, 03 days (abo 6%) experienced large moves in he U.S. dollar price o he Briish Pond in eiher a posiive or a negaive direcion. The Ero has a lesser nmber o large move days becase o is shorer rading hisory. We ond ha he average dela-hedged sraddle rerns remain signiicanly negaive or all crrencies 8

aer we exclde rom he sample he dela-hedged sraddle rerns o hose days when large moves occrred and also he days immediaely aer. Frhermore, we compare he mean and he median o dela-hedged sraddle rerns in he day beore large moves wih hose in he day aer. The resls are repored in Table 0. The - and Wilcoxon ess show ha boh he mean and median o he delahedged sraddle rerns are signiicanly negaive in mos cases beore and aer large moves. 4 However, he mos salien eare is ha, or all crrencies and across all mariies, he dela-hedged sraddle rern is signiicanly more negaive in he day aer han he day beore. To conrol or he eec o variabiliy beween he days on he beore-verss-aer comparison, we calclae he rern dierence beween beore and aer or each large move day and compe he mean and median o he dierences. The resls are repored in he las wo colmns o Table 0. The wihin-day dierence clearly shows ha he magnides o dela-hedged sraddle rerns are signiicanly larger in he day aer han he day beore jmps. I is likely ha aer price jmps, he marke perceives a high risk o jmps and hs opion byers pay an addiional premim o opion sellers or bearing jmp risk on op o he volailiy risk. This inding provides evidence ha jmp risk is priced in he crrency opion marke. However, ha he delahedged sraddle rerns are negaive a mos imes, even on he days beore jmps when 4 A close look a he daes o large moves show ha he daes are se widely apar, which makes i sae o assme independence in he sample o dela-hedged sraddle rerns and se - and Wilcoxon ess. This also sggess ha alhogh large moves case marke paricipans o raise heir expecaions o re jmps, ew large moves o he same magnide happened consecively becase o he mean-revering nare o he rern process. 9

jmp risk is mch less prononced, indicaes ha he volailiy risk premim is disinc rom and no sbsmed by he jmp risk premim. Inser Table 0 here D. Eec o Mis-measremen in he Dela Hedge Raio Exan heory (e.g. Hll and Whie (987, 988), Heson (993) and ohers) sggess ha sing a dela hedge raio comped on he basis o a consan volailiy model sch as he Garman-Kohlhagen model may case bias in hedging perormance when he volailiy process is acally sochasic. The bias depends on he correlaion beween he volailiy process and he nderlying asse rern process. To miigae he poenial bias in he dela hedge raio, we se he modiied Garman-Kohlhagen model in comping he hedge raio, ha is, he volailiy is pdaed daily. Since his modiied Garman-Kohlhagen model may no have lly correced he mis-speciicaion, we esimae he ollowing model or he dela-hedged sraddle rerns: (8) y = α γσ ΩR, τ βy β y β3y 3 ε where he addiional variable, R,τ, is he rern in spo crrency prices over he sraddle mariy period rom o τ. We se R,τ o capre he poenial eec o a sysemaic hedging bias. We consrc wo ime series o a-he-money sraddles or each crrency or he -monh and he 3-monh mariy period: one series is or posiive R,τ, and he oher series is or negaive R,τ. The irs series is designed o capre a sample or 30

which he spo marke is pward rending. The second series represens a sample or which here is a downward rending spo marke 5. I he modiied Garman-Kohlhagen model sysemaically nder-hedges (overhedges), we expec Ω o be posiive (negaive) or an pward rending spo marke and Ω o be negaive (posiive) or a downward rending spo marke. 6 Inser Table here The regression resls are repored in Table. All he Ω coeiciens are posiive or pward rending markes and he majoriy is negaive or downward rending markes. However, no all o he Ω coeiciens are signiicanly dieren rom zero, pariclarly or downward rending markes. This sggess ha he modiied Garman-Kohlhagen model sed does no lly correc or he mis-speciicaion in all cases. I ends o nderhedge, so ha he dela-hedged sraddle rern is pward biased. However, he imporan hing is ha in all regressions, aer we explicily accon or he possible bias in he hedge raio, he γ coeicien o implied volailiy is signiicanly dieren rom zero and is negaive. The only excepion is he γ coeicien o he Ero crrency in he pward rending marke. This provides rher sppor or he negaive volailiy risk premim in he OTC crrency opion marke. 5 Time series or sraddles wih a 6-monh and -monh mariy period are no sed in his robsness es becase he se o he R,τ crieria canno lly capre he spo marke rend dring he opion lie. 6 Bakshi and Kapadia (003) employ a similar robsness es in heir sdy o he eqiy index opion marke. 3

We condc anoher es o assess he reasonableness o he esimaed delas hedge raio. In his es we compe wo daily dela neral sraddle rerns: () he irs day price changes o a new dela neral sraddle price and he esimaed second day price (R), and () he irs day price changes o sccessive new dela neral sraddle prices (R). We compe he wo price changes or -monh, 3-monh, 6-monh, and -year sraddles or GBP, CHF, JPY, and EUR. The daily rerns o a dela neral sraddle shold on average give an expeced rern less han he risk ree rae in he presence o a negaive volailiy risk premim. We also expec ha R on average is less han R. Since a long posiion in dela neral sraddle earns a large proi when here are jmps in he spo price, we exclde hose days when here are jmps in spo price. We deine a jmp as he daily percenage change in spo price ha is wo sandard deviaions away rom he mean daily percenage change in or sample period. This deiniion is consisen wih he deiniion we sed in secion VI, sbsecion C. Or empirical resls show ha he means o R are negaive and saisically signiican or all cases, excep -year GBP sraddles. Moreover, he means o R are higher han he means o R or all cases. The resls provide sppor on he reasonableness o or dela esimaes. 7 7 We hank an anonymos reeree or sggesing his robsness es. The empirical resls are available pon reqes. 3

E. Poenial Biases Dring Periods o Increasing or Decreasing Volailiy As a inal robsness check, we invesigae wheher he rend in he volailiy process aecs or conclsions. We ideniy wo rending periods or boh he Briish Pond and he Japanese Yen: one wih an increasing rend in observed volailiy qoes and he oher wih a decreasing rend. For he Briish Pond, he decreasing rend occrred beween Jne 6, 998 and Ags 4, 998, while he increasing rend occrred beween Jne 5, 999 and Sepember 7, 999. For he Japanese Yen, he decreasing and increasing rends exend rom Febrary 9, 999 o Jne 5, 999, and rom November 0, 000 o March 30, 00, respecively. We replicae he empirical analysis based on regressions (4) and (6) or he -monh and 3-monh dela-hedged sraddle rerns in hese or rending periods and obain similar evidence ha sppors he exisence o negaive volailiy risk premim. 8 VII. Conclsion Sbsanial evidence has been docmened ha volailiy in boh he eqiy marke and he crrency marke is sochasic. This exposes invesors o he risk o changing volailiy. Alhogh several sdies show ha volailiy risk is priced in he eqiy index opion marke and ha he volailiy risk premim is negaive, here are ew sdies abo he isse in he crrency opion marke. This paper conribes o he lierare in his direcion. 8 The empirical resls are no inclded, b available pon reqes. 33

Using a large daabase o daily ask volailiy qoes on a-he-money dela neral sraddles in he OTC crrency opion marke, we irs ind he volailiy risk is priced in or major crrencies he Briish Pond, Ero, Japanese Yen, and Swiss Franc - across a wide range o mariy erms beween monh and monhs. Second, we provide direc evidence o he sign o he volailiy risk premim. The risk premim is negaive or all or major crrencies, sggesing ha byers in he OTC crrency opion marke pay a premim o sellers as compensaion or bearing he volailiy risk. Third, we ind ha he volailiy risk premim has a erm srcre where he magnide o he volailiy risk premim decreases in mariy. This sdy is he irs o provide empirical evidence o he erm srcre o he volailiy risk premim. Alhogh previos sdies have docmened ha shor-erm volailiy has higher variabiliy han long-erm volailiy (e.g. Campa and Chang (995) and X and Taylor (994)), no sdy has invesigaed is implicaion on he volailiy risk premim. Forh, here is some evidence ha jmp risk is also priced in OTC marke. However, he observed volailiy risk premim is disinc rom and no sbsmed by he possible jmp risk premim. These indings are robs o varios sensiiviy analyses on risk-ree ineres rae, opion dela compaion, and speciicaion o empirical model. 34

REFERENCES Bakshi, G., C. Cao, and Z. Chen. Empirical Perormance O Alernaive Opion Pricing Models. Jornal o Finance, 5 (997), 003-049. Bakshi, G., and N. Kapadia. Dela-Hedged Gains and he Negaive Marke Volailiy Risk Premim. Review o Financial Sdies, 6 (003), 57-566. Bank or Inernaional Selemen. Cenral Bank Srvey o Foreign Exchange and Derivaives Marke Aciviy 995. Basle, (996). Bank or Inernaional Selemen. BIS Qarerly Review Inernaional Banking and Financial Marke Developmens. Basle, (997). Bank or Inernaional Selemen. Cenral Bank Srvey o Foreign Exchange and Derivaives Marke Aciviy 00. Basle, (00). Bank or Inernaional Selemen. BIS Qarerly Review Inernaional Banking and Financial Marke Developmens. Basle, (003). Baes, D. Pos- 87 Crash Fears in S&P 500 Fres Opions. Jornal o Economerics 94 (000), 8-38. Bollerslev, T., R.Y. Cho, and K.F. Kroner. ARCH Modeling in Finance. Jornal o Economerics, 5 (99), 5-59. Campa, J., and K.H. Chang. Tesing he Expecaions Hypohesis on he Term Srcre o Volailiies. Jornal o Finance 50 (995), 59-547. Campa, J., and K.H. Chang. "The Forecasing Abiliy O Correlaions Implied In Foreign Exchange Opions," Jornal o Inernaional Money and Finance, 7 (998), 855-880. 35

Chesney, M., and L. Sco. Pricing Eropean Crrency Opions: A Comparison o he Modiied Black-Scholes Model and a Random Variance Model, Jornal o Financial and Qaniaive Analysis, 4(989), 67-84. Coval, J., and T. Shmway. Expeced Opion Rerns. Jornal o Finance, 56 (00), 983-009. Covrig, V., and B.S., Low. The Qaliy o Volailiy Traded on he Over-he-Coner Crrency Marke: A Mliple Horizons Sdy. Jornal o Fres Markes, 3 (003), 6-85. Cox, J., J. Ingersoll, and S. Ross. An Ineremporal General Eqilibrim Model o Asse Prices. Economerica, 53 (985), 363-384. De Sanis, G., and B. Gerard. "How Big Is The Premim For Crrency Risk? " Jornal o Financial Economics, 49 (998), 375-4. Die, D., J. Pan, and K. Singleon. Transorm Analysis and Asse Pricing or Aine Jmp-disions. Economerica, 68 (000), 343-376. Dmas, B., and B. Solnik. "The World Price O Foreign Exchange Risk." Jornal o Finance, 50 (995), 445-479. Eraker, B., M.S. Johannes, and N.G., Polson. The Impac o Jmps in Rerns and Volailiy. Working paper, Universiy o Chicago, (000). Garman, B., and S. Kohlhagen. Foreign Crrency Opion Vales. Jornal o Inernaional Money and Finance, (983), 3-37. Heynen, R., A. Kemna, and T. Vors. Analysis o he Term Srcre o Implied Volailiies. Jornal o Financial and Qaniaive Analysis, 9 (994), 3-56. 36

Heson, S. A Closed Form Solion or Opions wih Sochasic Volailiy wih Applicaions o Bond and Crrency Opions. Review o Financial Sdies, 6 (993), 37-343. Hll, J., and A. Whie. "The Pricing O Opions On Asses Wih Sochasic Volailiies," Jornal o Finance, 4 (987), 8-300. Hll, J. and A. Whie. "An Analysis O The Bias In Opion Pricing Cased By Sochasic Volailiy," Advances in Fres and Opions Research, 3 (988), 9-6. Hll, J. Opions, Fres, and Oher Derivaives, 5 h ediion, Prenice Hall (003). Jackwerh, J., and M. Rbinsein. Recovering Probabiliy Disribions From Opion Prices. Jornal o Finance, 5 (996), 6-63. Jorion, P. Predicing Volailiy in he Foreign Exchange Marke. Jornal o Finance, 50 (995), 507-58. Lamorex, G., and W. Lasrapes. Forecasing Sock-Rern Variance: Toward An Undersanding o Sochasic Implied Volailiies. Review o Financial Sdies, 6 (993), 93-36. Melino, A., and S. Trnbll. Pricing Foreign Crrency Opions wih Sochasic Volailiy. Jornal o Economerics, 45 (990), 39-65. Melino, A., and S. Trnbll. Misspeciicaion And The Pricing And Hedging O Long- Term Foreign Crrency Opions. Jornal o Inernaional Money and Finance, 4 (995), 373-393. Neely, C. Forecasing Foreign Exchange Volailiy: Is Implied Volailiy he Bes We Can Do? working paper, Federal Reserve Bank o S. Lois (003). 37

Newey, W.K., and K.D. Wes. A Simple Posiive Semi-deinie, Heeroskedasiciy and Aocorrelaion Consisen Covariance Marix. Economerica, 55 (987), 70-708. Pan, J. The Jmp-risk Premia Implici in Opions: Evidence rom an Inegraed Time Series Sdy. Jornal o Financial Economics, 63 (00), 3-50. Sarwar, G. Is Volailiy Risk or Briish Pond Priced in U.S. Opions Markes? The Financial Review, 36 (00), 55-70. Sein, J. Overreacions in he Opions Marke. Jornal o Finance, 44 (989), 0-04. Taylor, S., and X. X. The Incremenal Volailiy Inormaion In One Million Foreign Exchange Qoaions. Jornal o Empirical Finance, 4 (997), 37-340. Wilmo, P. Derivaives, John Wiley & Sons (998). X, X., and S. Taylor. The Term Srcre O Volailiy Implied By Foreign Exchange Opions. Jornal o Financial and Qaniaive Analysis, 9 (994), 57-74. 38

Table : Smmary saisics o he dela-hedged sraddle rerns or he whole sample period The U.S. dollar dela-hedged sraddle proi (loss) rom holding he dynamically dela-hedged porolio nil mariy is comped by N N τ = Max x k, k x,0 - x x - r r q x ( τ τ ) ( ) ( ( ) ) n n n n n N, τ n= 0 n= 0 where ƒ is he sraddle premim a ime, x τ is he crrency price a he mariy τ, k is he srike price o he sraddle, n is he dela o he long sraddle, r and q are he domesic and oreign ineres raes. The dela-hedged sraddle rern is calclaed as Π, τ divided by he sraddle conrac size in U.S. Dollar. We annalize he rern o make i comparable across mariies. We repor smmary saisics on he rern or or crrencies a or sandard mariies. For each combinaion o crrency and mariy, he dela-hedged sraddle rern comprises a ime series o one observaion on each rading day. We i an AR(3) process o he ime series and es wheher he inercep is zero. This is eqivalen o a es o wheher he mean dela-hedged sraddle rern is eqal o zero. The sample period is rom Jne 996 o December 00 or he Briish Pond, he Japanese Yen, and he Swiss Franc, and Janary 999 o December 00 or he Ero. Mariy Obs. # % o negaive rern Sample mean Sample sandard Aocorrelaion coeicien Parial aocorrelaion coeicien AR(3) Inercep deviaion Lag Lag Lag 3 Lag Lag Lag 3 Esimae P-vale Panel A: Briish Pond (GBP) monh 705 7% -0.039 0.0504 0.863 0.77 0.700 0.863 0.07 0.05-0.008 0.000 3 monhs 705 77% -0.079 0.044 0.953 0.9 0.893 0.953 0.36 0.054-0.0007 0.0046 6 monhs 705 83% -0.05 0.067 0.957 0.9 0.904 0.957 0.067 0.4-0.0003 0.06 monhs 705 85% -0.030 0.0 0.948 0.90 0.95 0.948 0.07 0.78-0.000 0.09 Panel B: Ero (EUR) monh 07 58% -0.0 0.066 0.830 0.74 0.638 0.830 0.0 0.037-0.008 0.005 3 monhs 07 67% -0.05 0.059 0.95 0.96 0.888 0.95 0.0 0.074-0.0005 0.08 6 monhs 07 64% -0.0087 0.09 0.977 0.960 0.94 0.977 0.099-0.005-0.000 0.54 monhs 07 58% -0.007 0.054 0.989 0.979 0.970 0.989 0.04 0.046-0.000 0.966 39

Table (Con d) Mariy Obs. # % o negaive rern Sample mean Sample sandard Aocorrelaion coeicien Parial aocorrelaion coeicien AR(3) Inercep deviaion Lag Lag Lag 3 Lag Lag Lag 3 Esimae P-vale Panel C: Japanese Yen (JPY) monh 705 59% -0.005 0.0974 0.877 0.768 0.699 0.877-0.006 0.7-0.00 0.3507 3 monhs 705 60% -0.0085 0.0440 0.930 0.876 0.86 0.930 0.083 0.07-0.0005 0.50 6 monhs 705 68% -0.0055 0.0398 0.938 0.90 0.869 0.938 0.83 0.053-0.000 0.667 monhs 705 54% -0.0033 0.048 0.90 0.879 0.8 0.90 0.4-0.069-0.000 0.3659 Panel D: Swiss Franc (CHF) monh 705 58% -0.0047 0.0647 0.845 0.73 0.65 0.845 0.063 0.067-0.0005 0.5344 3 monhs 705 65% -0.0085 0.07 0.940 0.899 0.868 0.940 0.30 0.09-0.0004 0.0747 6 monhs 705 6% -0.0059 0.079 0.96 0.93 0.904 0.96 0.07 0.0-0.000 0.0696 monhs 705 66% -0.005 0.0 0.97 0.955 0.935 0.97 0. -0.09-0.000 0.038 40

Table : Regression o he dela-hedged sraddle rern on implied volailiy This able presens he esimaed resls or he regression model y = α γσ βy β y β3y 3 ε where y is he dela-hedged sraddle rern, and σ is he implied volailiy. We inclde hree lagged variables, y -, y -, and y -3, o conrol or serial correlaion in he ime series o dela-hedged sraddle rerns. We esimae one model or each combinaion o crrency and mariy. Or objecive is o es he nll hypohesis ha γ = 0. Independen variables monh 3 monhs 6 monhs monhs Panel A: Briish Pond (GBP) Inercep 0.08** 0.009** 0.0** 0.009** Implied vol. -0.5** -0.8** -0.36** -0.3** Sraddle rern lag 0.746** 0.694** 0.577** 0.634** Sraddle rern lag 0.064 0.49* 0.048 0.086 Sraddle rern lag 3 0.059-0.006 0.87 0.67 Adj. r-sqared 0.753 0.897 0.856 0.848 Obs. # 70 70 70 70 Drbin-Wason saisic.978.958.98.898 Panel B: Ero (EUR) Inercep 0.08** 0.004** 0.00 0.00 Implied vol. -0.78** -0.044** -0.05-0.009 Sraddle rern lag 0.859** 0.869** 0.97** 0.98** Sraddle rern lag 0.0-0.086 0.036 0.008 Sraddle rern lag 3-0.09 0.73 0.037 0.003 Adj. r-sqared 0.74 0.89 0.963 0.978 Obs. # 04 04 04 04 Drbin-Wason saisic.98.878.99.995 ** signiican a he % level * signiican a he 5% level 4

Table (Con d) Independen variables monh 3 monhs 6 monhs monhs Panel C: Japanese Yen (JPY) Inercep 0.7** 0.08** 0.050** 0.04* Implied vol. -.474** -0.94** -0.49** -0.0* Sraddle rern - lag 0.36** 0.7** 0.37** 0.498** Sraddle rern - lag 0.36** 0.4** 0.45** 0.5** Sraddle rern - lag 3 0.56** 0.09* 0.0** 0.06 Adj. r-sqared 0.6 0.686 0.74 0.798 Obs. # 70 70 70 70 Drbin-Wason saisic.7 0.858.48.58 Panel D: Swiss Franc (CHF) Inercep 0.09** 0.05** 0.0** 0.006** Implied vol. -0.68** -0.46** -0.0** -0.054** Sraddle rern - lag 0.789** 0.809** 0.886** 0.64** Sraddle rern - lag 0.04-0.05-0.05 0.35** Sraddle rern - lag 3 0.05 0.36** 0.079-0.005 Adj. r-sqared 0.777 0.879 0.908 0.9 Obs. # 70 70 70 70 Drbin-Wason saisic.974.973.98.85 ** signiican a he % level * signiican a he 5% level 4

Table 3: Non-overlapping sraddle rerns For each crrency, we consrc a non-overlapping monhly series o dela-hedged sraddle rerns on he - monh sraddles bogh a he irs rading day o every monh in he sample period. Panel A shows he smmary saisics or non-overlapping series. Panel B shows he resls o esimaing his regression: y α γσ ε = y where y is he dela-hedged sraddle rern, and σ is he implied volailiy qoe. I he volailiy risk is priced and has a negaive premim, we expec γ o be signiican and negaive. The sample period is rom Jne 996 o December 00 or he Briish Pond, he Japanese Yen, and he Swiss Franc, and Janary 999 o December 00 or he Ero. Briish Pond Ero Japan. Yen Swiss Franc Panel A. Smmary saisics Obs. # 77 48 77 77 Mean -0.08-0.00-0.00-0.0030 Sd. Dev. 0.054 0.0544 0.0835 0.073 % o negaive 66% 58% 63% 60% Aocorrelaion coeicien Lag -0.448-0.3 0.0746-0.0976 Lag -0.504-0.380 0.390-0.03 Lag 3 0.0300 0.0948-0.058 0.057 Panel B. regression agains implied vol Inercep 0.0359** 0.053 0.0360 0.0496* Implied vol. -0.45** -0.4988* -0.4073* -0.54* Lag 0.8969** 0.6944** 0.7660** 0.86** Adj. r-sqared 0.854 0.533 0.7548 0.8584 Drbin-Wason.808.030.485.74 ** signiican a he % level * signiican a he 5% level 43

Table 4: Regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy This able presens he esimaed resls or he regression model y = α γ σ β y β y β y ε D D D D D3 3 where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he volailiy qoed or a dela-neral sraddle on dae, and σ is he dierence beween σ and σ. We inclde hree lagged dependen variables o conrol or serial aocorrelaion. One regression model is esimaed or each combinaion o crrency and mariy. The objecive is o es he nll hypohesis ha γ = 0. D Independen variables monh 3 monhs 6 monhs monhs Panel A: Briish Pond (GBP) Inercep 0.000 0.000 0.000 0.000 Implied vol. -.94** -.56** -.089** -0.790** Sraddle rern lag -0.85** -0.68** -0.3466** -0.696** Sraddle rern lag -0.63** 0.035-0.859** -0.839 Sraddle rern lag 3-0.0796** 0.0336 0.073 0.0096 Obs. # 70 70 70 70 Adj. r-sqared 0.0 0.506 0.337 0.084 Drbin-Wason saisic.06.08.0589.994 Panel B: Ero (EUR) Inercep 0.0000-0.000-0.000-0.000 Implied vol. -.6459** -.670** -.664** -0.8543** Sraddle rern lag -0.0633-0.09** -0.0835** -0.0370 Sraddle rern lag -0.0360-0.930* -0.036 0.000 Sraddle rern lag 3-0.699* -0.0833-0.008-0.0457 Obs. # 03 03 03 03 Adj. r-sqared 0.47 0.768 0.34 0.337 Drbin-Wason saisic.080.89.098.054 ** signiican a he % level * signiican a he 5% level 44

Table 4 (Con d) Independen variables monh 3 monhs 6 monhs monhs Panel C: Japanese Yen (JPY) Inercep 0.0000 0.000 0.000 0.0000 Implied vol. -.4670** -.487** -0.87** -0.45** Sraddle rern - lag -0.0546* -0.05** -0.093-0.3* Sraddle rern - lag -0.057-0.0099-0.0799-0.035 Sraddle rern - lag 3-0.0355-0.006-0.0635-0.075 Obs. # 70 70 70 70 Adj. r-sqared 0.883 0.9545 0.844 0.599 Drbin-Wason saisic.64.3060.903.43 Panel D: Swiss Franc (CHF) Inercep 0.000 0.0000 0.0000 0.000 Implied vol. -.8374** -.6060** -.459** -0.765** Sraddle rern - lag -0.53** -0.49** -0.0039-0.46** Sraddle rern - lag -0.095** -0.59** -0.09-0.0077 Sraddle rern - lag 3-0.008-0.0883* -0.0476 0.085 Obs. # 70 70 70 70 Adj. r-sqared 0.477 0.950 0.3030 0.339 Drbin-Wason saisic.0398.5.9904.08 ** signiican a he % level * signiican a he 5% level 45

Table 5: Pooled regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy We pool he dela-hedged sraddle rerns or or mariies and esimae he ollowing model or each crrency y = α I α I α I α I γ σ, I γ σ β y 3 3, 3 I 3 3 β y γ σ 6, I β y 3 3 6 6 6 6 γ σ where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he qoed volailiy or he i -monh mariy on dae, σ i, is he dierence beween σ and σ, and I i is he indicaor variable o an observaion or he i -monh mariy wih i being, 3, 6 or. We se α i o allow dieren inerceps a dieren mariies. Or main ocs is on γ i s, which measres he ni dela-hedged sraddle rern a dieren mariies. The sample period is rom Jne 996 o December 00 or he Briish Pond, he Japanese Yen, and he Swiss Franc, and Janary 999 o December 00 or he Ero. ε, I Crrency Independen variables GBP EUR JPY CHF Dmmy - monh 0.000 0.0000 0.0000 0.000 Dmmy - 3 monhs 0.000-0.000 0.000 0.0000 Dmmy - 6 monhs 0.000-0.000 0.000 0.0000 Dmmy - monhs 0.0000-0.000 0.0000 0.000 Implied vol. - monh -.998** -.6396** -.4747** -.8370** Implied vol. - 3 monhs -.558** -.6** -.404** -.608** Implied vol. 6 monhs -.054** -.74** -0.896** -.5** Implied vol. monhs -0.7349** -0.8544** -0.470** -0.7804** Sraddle rern - lag -0.078** -0.0654-0.0495** -0.469** Sraddle rern - lag -0.98** -0.043-0.0439-0.0994** Sraddle rern - lag 3-0.056-0.645* -0.0334-0.053 Obs. # 6804 409 6804 6804 Adj. r-sqared 0.079 0.509 0.8933 0.64 Drbin-Wason saisic.0964.0786.04.045 Wald es or H 0 : γ = γ Saisic 73.8484 8.5774 703.0080 0.593 (p-vale) (0.00) (0.00) (0.00) (0.00) Wald es or H 0 : γ γ 3 = γ 6 γ Saisic 88.4576 96.8067 863.098 58.57 (p-vale) (0.00) (0.00) (0.00) (0.00) ** signiican a he % level * signiican a he 5% level 46

Table 6: Regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy or seven oher crrency pairs This able presens he esimaed resls o he ollowing regression model or seven oher crrency pairs y = α D γ D σ β D y β D y β D3 y 3 ε where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he volailiy qoed or a dela-neral sraddle on dae, and σ is he dierence beween σ and σ. We inclde hree lagged dependen variables o conrol or serial aocorrelaion. One regression model is esimaed or each combinaion o crrency and mariy. The objecive is o es he nll hypohesis, γ D = 0. Some crrency pairs in his able have smaller nmber o observaions. This is de o he ac ha Bloomberg does no have complee daily sraddle qoes rom Jne 996 o December 003. We inclde only observaions beore December 003 ha have complee daa necessary or he empirical analysis. Independen variables monh 3 monhs 6 monhs monhs Panel A: EUR/CHF Inercep 0.0000 0.000 0.0000 0.0000 Implied vol. -.6953** -.669** -.45** -0.834** Sraddle rern - lag -0.0958-0.05* -0.0747 0.098 Sraddle rern - lag -0.007-0.0930* -0.0894-0.0504 Sraddle rern - lag 3 0.0050-0.00* 0.07-0.0754 Obs. # 809 83 88 038 Adj. r-sqared 0.500 0.4065 0.359 0.550 Drbin-Wason saisic.0743.536.088.980 Panel B: YEN/GBP Inercep 0.0008 0.000 0.0000 0.0000 Implied vol. -.5397** -.6906** -.37** -0.675** Sraddle rern - lag -0.074-0.0665-0.6* 0.005 Sraddle rern - lag -0.064* -0.95** -0.089-0.078 Sraddle rern - lag 3-0.6** -0.6** -0.8-0.0639* Obs. # 3 3 3 3 Adj. r-sqared 0.960 0.3508 0.85 0.606 Drbin-Wason saisic.0067.9588.5.030 Panel C: YEN/EUR Inercep -0.0003 0.0000 0.0000 0.0000 Implied vol. -.6** -.5578** -.068** -0.7099** Sraddle rern - lag -0.0664-0.45* -0.069-0.0488* Sraddle rern - lag -0.0939* -0.76* -0.0365 0.003 Sraddle rern - lag 3-0.0045-0.0559-0.075 0.043 Obs. # 038 038 038 038 Adj. r-sqared 0.7 0.685 0.7806 0.5378 Drbin-Wason saisic.088.06.408.498 ** signiican a he % level * signiican a he 5% level 47

Table 6 (Con d) Independen variables monh 3 monhs 6 monhs monhs Panel D: Asralian (AUD) Inercep 0.0000 0.0000 0.0000 0.0000 Implied vol. -.5600** -.5400** -.000** -0.9300** Sraddle rern - lag -0.0974** -0.463** -0.87** -0.057* Sraddle rern - lag -0.0773* -0.049-0.0535-0.05 Sraddle rern - lag 3-0.069-0.0663* -0.00-0.088 Obs. # 679 678 680 73 Adj. r-sqared 0.0853 0.358 0.669 0.945 Drbin-Wason saisic.0574.065.094.37 Panel E: Canadian Dollar (CAD) Inercep -0.0009-0.0003-0.000-0.000 Implied vol. -.958** -.60** -0.8643** -0.7757** Sraddle rern lag -0.000** 0.0638-0.799** -0.3 Sraddle rern lag -0.590** -0.068-0.879** -0.699 Sraddle rern lag 3 0.0005 0.0305-0.075-0.786 Obs. # 98 68 37 60 Adj. r-sqared 0.663 0.5639 0.5337 0.378 Drbin-Wason saisic.84.678.047.083 Panel F: Norwegian Kroner (NOK) Inercep -0.0009 0.000 0.0000 0.0000 Implied vol. -.6090** -.340** -0.9449** -0.753** Sraddle rern lag -0.0** -0.03** -0.969** -0.989** Sraddle rern lag -0.640** -0.55-0.0545-0.0650 Sraddle rern lag 3-0.0903* -0.037 0.0367-0.0034 Obs. # 463 409 40 855 Adj. r-sqared 0.77 0.309 0.4749 0.703 Drbin-Wason saisic.096.59.970.395 Panel G: New Zealand Dollar (NZD) Inercep -0.003-0.000-0.000-0.000 Implied vol. -.6634** -.456** -0.856** -0.659** Sraddle rern - lag -0.80** -0.473* -0.47* -0.063 Sraddle rern - lag -0.0549-0.0690-0.0786 0.039 Sraddle rern - lag 3-0.038-0.49** -0.5* -0.0855 Obs. # 667 576 504 480 Adj. r-sqared 0.093 0.595 0.370 0.45 Drbin-Wason saisic.0985.0498.40.959 ** signiican a he % level * signiican a he 5% level 48

Table 7: Pooled regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy or seven oher crrency pairs We pool he dela-hedged sraddle rerns or or mariies and esimae he ollowing model or seven oher crrency pairs y = α I α I α I α I γ σ, I γ σ β y 3 3, 3 I 3 3 β y γ σ 6, I β y 3 3 6 6 6 6 γ σ where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he qoed volailiy or he i -monh mariy on dae, σ i, is he dierence beween σ and σ, and I i is he indicaor variable o an observaion or he i -monh mariy wih i being, 3, 6 or. We se α i o allow dieren inerceps a dieren mariies. Or main ocs is on γ i s, which measres he ni dela-hedged sraddle rern a dieren mariies. ε, I Crrency Independen variables EUR/CHF YEN/GBP YEN/EUR Dmmy - monh 0.0000 0.0008-0.0003 Dmmy - 3 monhs 0.000 0.000 0.0000 Dmmy - 6 monhs 0.0000 0.000 0.0000 Dmmy - monhs 0.0000 0.0000 0.0000 Implied vol. - monh -.6865** -.546** -.6094** Implied vol. - 3 monhs -.6384** -.6866** -.5646** Implied vol. 6 monhs -.47** -.307** -.0664** Implied vol. monhs -0.805** -0.6564** -0.770** Sraddle rern - lag -0.0956* -0.0746-0.0708* Sraddle rern - lag -0.039-0.956* -0.0930* Sraddle rern - lag 3-0.00-0.53-0.0097 Obs. # 355 548 45 Adj. r-sqared 0.899 0.09 0.073 Drbin-Wason saisic.0848.006.0905 Wald es or H 0 : γ = γ Saisic 4.4969 5.98 37.6506 (p-vale) (0.00) (0.00) (0.00) Wald es or H 0 : γ γ 3 = γ 6 γ Saisic 0.08 65.485 5.83 (p-vale) (0.00) (0.00) (0.00) ** signiican a he % level * signiican a he 5% level 49

Table 7 (Con d) Crrency Independen variables AUD CAD NOK NZD Dmmy - monh 0.0000-0.0009-0.0009-0.003 Dmmy - 3 monhs 0.0000-0.0003 0.000-0.000 Dmmy - 6 monhs 0.0000-0.000 0.0000-0.000 Dmmy - monhs 0.0000-0.000 0.0000-0.000 Implied vol. - monh -.5600** -.934** -.6085** -.6600** Implied vol. - 3 monhs -.5400** -.086** -.3094** -.476** Implied vol. 6 monhs -.0300** -0.8959** -0.9484** -0.8506** Implied vol. monhs -0.9000** -0.7963** -0.7393** -0.6393** Sraddle rern - lag -0.004** -0.8** -0.9** -0.76** Sraddle rern - lag -0.070* -0.504* -0.605** -0.0563 Sraddle rern - lag 3-0.0307-0.006-0.0875* -0.048 Obs. # 6750 663 37 7 Adj. r-sqared 0.45 0.904 0.985 0.80 Drbin-Wason saisic.06.378.0997.005 Saisic 35.609 5.7488 45.334 9.777 (p-vale) (0.00) (0.00) (0.00) (0.00) Wald es or H 0 : γ γ 3 = γ 6 γ Saisic 4.3990 3.340 46.889 38.6366 (p-vale) (0.00) (0.00) (0.00) (0.00) ** signiican a he % level * signiican a he 5% level 50

Table 8: Smmary saisics o he dela-hedged sraddle rern or he pos-asian crrency crisis sb-period The U.S. Dollar dela-hedged sraddle proi (loss) rom holding he dynamically dela-hedged porolio nil mariy is comped by N N τ = Max x k, k x,0 - x x - r r q x ( τ τ ) ( ) ( ( ) ) n n n n n N, τ n= 0 n= 0 where ƒ is he sraddle premim a ime, x τ is he crrency price a he mariy τ, k is he srike price o he sraddle, n is he dela o he long sraddle, r and q are he domesic and oreign ineres raes. The dela-hedged sraddle rern is calclaed as Π, τ divided by he sraddle conrac size in U.S. dollars. We annalize he rern o make i comparable across mariies. We repor smmary saisics on he rern or or crrencies a or sandard mariies. For each combinaion o crrency and mariy, he dela-hedged sraddle rern comprises a ime series o one observaion on each rading day. We i an AR(3) process o he ime series and es wheher he inercep is zero. This is eqivalen o a es o wheher he mean dela-hedged sraddle rern is eqal o zero. The pos- Asian crrency crisis sb-period is rom Jly 999 o December 00. Mariy Obs. # % o negaive rern Sample mean Sample sandard Aocorrelaion coeicien Parial aocorrelaion coeicien AR(3) Inercep deviaion Lag Lag Lag 3 Lag Lag Lag 3 Esimae P-vale Panel A: Briish Pond (GBP) monh 90 70% -0.03 0.0484 0.846 0.750 0.69 0.846 0.3 0.3-0.007 0.0065 3 monhs 90 78% -0.085 0.03 0.943 0.909 0.883 0.943 0.7 0.05-0.0009 0.004 6 monhs 90 8% -0.06 0.065 0.939 0.893 0.88 0.939 0.087 0.89-0.0003 0.0779 monhs 90 8% -0.033 0.00 0.96 0.885 0.883 0.96 0.93 0.34-0.0003 0.0666 Panel B: Ero (EUR) monh 90 59% -0.030 0.064 0.84 0.73 0.647 0.84 0.077 0.048-0.008 0.7 3 monhs 90 68% -0.03 0.064 0.959 0.97 0.900 0.959 0.09 0.047-0.0005 0.0674 6 monhs 90 67% -0.006 0.094 0.976 0.957 0.937 0.976 0.096-0.0-0.0003 0.0887 monhs 90 64% -0.0093 0.05 0.986 0.973 0.96 0.986 0.038 0.03-0.000 0.037 5

Table 8 (Con d) Mariy Obs. # % o negaive rern Sample mean Sample sandard Aocorrelaion coeicien Parial aocorrelaion coeicien AR(3) Inercep deviaion Lag Lag Lag 3 Lag Lag Lag 3 Esimae P-vale Panel C: Japanese Yen (JPY) monh 90 59% -0.038 0.073 0.878 0.78 0.705 0.878 0.044 0.049-0.007 0.653 3 monhs 90 66% -0.047 0.0363 0.886 0.86 0.745 0.886 0.44-0.007-0.004 0.0385 6 monhs 90 8% -0.055 0.078 0.803 0.7 0.68 0.803 0.86 0.038-0.000 0.084 monhs 90 70% -0.0096 0.0 0.80 0.7 0.579 0.80 0.5-0.45-0.007 0.0093 Panel D: Swiss Franc (CHF) monh 90 55% -0.000 0.0660 0.834 0.70 0.656 0.834 0.08 0.9-0.000 0.876 3 monhs 90 67% -0.0095 0.048 0.939 0.90 0.868 0.939 0.7 0.046-0.0006 0.0376 6 monhs 90 68% -0.0098 0.065 0.950 0.95 0.884 0.950 0.5 0.038-0.0004 0.0090 monhs 90 75% -0.0085 0.08 0.967 0.950 0.97 0.967 0.38-0.037-0.0003 0.0034 5

Table 9: Regression o he irs dierence o dela-hedged sraddle rern on he irs dierence o implied volailiy or he pos-asian crrency crisis sb-period This able presens he esimaed resls or he regression model y = α D γ D σ β D y β D y β D3 y 3 ε where y is he dela-hedged rern on a dela-neral sraddle bogh on dae, y is he dierence beween y and y, σ is he volailiy qoed or a dela-neral sraddle on dae, and σ is he dierence beween σ and σ. We inclde hree lagged dependen variables o conrol or serial aocorrelaion. One regression model is esimaed or each combinaion o crrency and mariy. The objecive is o es he nll hypohesis, γ D = 0. The pos-asian crrency crisis sb-period is rom Jly 999 o December 00. Independen variables monh 3 monhs 6 monhs monhs Panel A: Briish Pond (GBP) Inercep 0.0000 0.000 0.000 0.000 Implied vol. -.750** -.595** -0.870** -0.6980** Sraddle rern lag -0.6** -0.30** -0.4009** -0.796** Sraddle rern lag -0.60** 0.099-0.3309* -0.999 Sraddle rern lag 3-0.0977* 0.0690 0.064 0.043 Obs. # 898 898 898 898 Adj. r-sqared 0.99 0.488 0.444 0.878 Drbin-Wason saisic.080.3.033.9780 Panel B: Ero (EUR) Inercep 0.0000-0.000-0.000-0.000 Implied vol. -.5395** -.608** -.657** -0.8574** Sraddle rern lag -0.046-0.083-0.0807* -0.0389 Sraddle rern lag -0.0393-0.986-0.039-0.000 Sraddle rern lag 3-0.793* -0.099-0.004-0.0477 Obs. # 898 898 898 898 Adj. r-sqared 0.0958 0.8 0.3035 0.300 Drbin-Wason saisic.0775.889..057 ** signiican a he % level * signiican a he 5% level 53

Table 9 (Con d) Independen variables monh 3 monhs 6 monhs monhs Panel C: Japanese Yen (JPY) Inercep -0.000 0.0000 0.000 0.0000 Implied vol. -3.096** -.5436** -.3** -.063** Sraddle rern - lag -0.77** -0.973* -0.37** -0.0** Sraddle rern - lag -0.075* -0.0437-0.03* 0.0399 Sraddle rern - lag 3-0.0987* -0.9-0.037* -0.45 Obs. # 898 898 898 898 Adj. r-sqared 0.86 0.660 0.5 0.056 Drbin-Wason saisic.0697.036.0549.050 Panel D: Swiss Franc (CHF) Inercep -0.000-0.000 0.0000 0.000 Implied vol. -.590** -.6449** -.455** -0.738** Sraddle rern - lag -0.89** -0.75** 0.0063-0.368** Sraddle rern - lag -0.494** -0.06* -0.747-0.044 Sraddle rern - lag 3-0.030-0.700* -0.074 0.05 Obs. # 898 898 898 898 Adj. r-sqared 0.438 0.334 0.505 0.89 Drbin-Wason saisic.086.0904.9849.990 ** signiican a he % level * signiican a he 5% level 54

Table 0: Eec o jmps on he dela-hedged sraddle rern We ideniy he day on which he daily percenage change in he crrency price is wo sandard deviaions away rom he mean daily percenage change in or sample period. We examine he dela-hedged sraddle rerns in he day immediaely beore or aer hese price jmps. We do no diereniae beween negaive and posiive jmps becase he sraddle price is eqally sensiive o moves in boh direcions. The irs colmn repors he nmber o days when price jmped. Take he Briish Pond as an example. Beween Jne 3, 996 and December 3, 00, 03 days (abo 6%) experienced large moves in he U.S. Dollar price o he Briish Pond in eiher a posiive or a negaive direcion. Beore jmps Aer jmps Beore vs. aer # o jmps Mean Median Mean Median Mean Median (p-vale o (p-vale o (p-vale (p-vale o (p-vale (p-vale o Mariy -es) Wilcoxon es) o -es) Wilcoxon es) o -es) Wilcoxon es) Panel A: Briish Pond (GBP) monh 03-0.0009 0.003-0.0354-0.0357 0.0345 0.0307 (0.88) (0.89) (0.00) (0.00) (0.00) (0.00) 3 monhs 03-0.067-0.099-0.057-0.075 0.0090 0.0078 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 6 monhs 03-0.055-0.050-0.088-0.094 0.0033 0.005 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) monhs 03-0.049-0.077-0.066-0.088 0.007 0.000 (0.00) (0.00) (0.00) (0.00) (0.0) (0.00) Panel B: Ero (EUR) monh 53 0.04 0.065-0.05-0.06 0.0493 0.044 (0.03) (0.0) (0.0) (0.05) (0.00) (0.00) 3 monhs 53-0.00-0.007-0.006-0.00 0.004 0.0095 (0.00) (0.0) (0.00) (0.00) (0.00) (0.00) 6 monhs 53-0.0094-0.003-0.036-0.05 0.004 0.0053 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) monhs 53-0.07-0.057-0.05-0.047 0.0008 0.004 (0.00) (0.00) (0.00) (0.00) (0.30) (0.4) Panel C: Japanese Yen (JPY) monh 8 0.057 0.0367-0.0444-0.0474 0.070 0.0607 (0.0) (0.06) (0.0) (0.00) (0.00) (0.00) 3 monhs 8 0.003-0.005-0.037-0.063 0.059 0.034 (0.73) (0.84) (0.07) (0.0) (0.03) (0.00) 6 monhs 8-0.000-0.0059-0.0078-0.055 0.0068 0.007 (0.87) (0.6) (0.7) (0.0) (0.00) (0.00) monhs 8-0.0079-0.008-0.009-0.0 0.0030 0.0033 (0.0) (0.00) (0.00) (0.00) (0.0) (0.00) Panel D: Swiss Franc (CHF) monh 00 0.0353 0.035 0.0004-0.007 0.0349 0.0309 (0.00) (0.00) (0.95) (0.66) (0.00) (0.00) 3 monhs 00 0.0005 0.0003-0.0067-0.0 0.007 0.005 (0.87) (0.9) (0.0) (0.0) (0.00) (0.00) 6 monhs 00-0.0058-0.00-0.0087-0.030 0.0030 0.003 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) monhs 00-0.0066-0.0058-0.0076-0.007 0.0009 0.0006 (0.00) (0.00) (0.00) (0.00) (0.08) (0.07) 55

Table : Eec o mis-measremen in he dela hedge raio This able shows he esimaion resls or he regression model y = α γσ ΩR, τ βy β y β3 y 3 ε where y is he dela-hedged sraddle rern,σ is he implied volailiy, and R τ, is he rern in spo crrency prices over he mariy period rom o τ. We se R τ o capre he poenial sysemaic hedging bias. I he modiied Garman-Kohlhagen model sysemaically nder-hedges (over-hedges), we expec Ω o be posiive (negaive) or an pward rending spo marke and Ω o be negaive (posiive) or a downward rending spo marke. We inclde hree lagged variables, y -, y -, and y -3, o conrol or serial correlaion in he ime series o dela-hedged sraddle rerns. We esimae one model or each crrency or boh -monh and 3-monh mariies. The sample period is rom Jne 996 o December 00 or he Briish Pond, he Japanese Yen, and he Swiss Franc, and Janary 999 o December 00 or he Ero. Upward rending spo marke Downward rending spo marke Independen variables Briish Pond Ero Japanese Yen Swiss Franc Briish Pond Ero Japanese Yen Swiss Franc -monh mariy Inercep 0.009* 0.05* 0.0* 0.03** 0.09** 0.08 0.83** 0.08** Implied vol. -0.6** -0.3** -.366** -0.304** -0.368** -0.6* -.747** -0.6** Rern in spo marke 0.49** 0.34**.449** 0.337** -0.54** -0.87** -0.394-0.03 Sraddle rern - lag 0.678** 0.70** 0.30* 0.794** 0.73** 0.95** 0.09* 0.07** Sraddle rern - lag 0.073 0.069 0.09 0.064 0.068-0.06 0.4** 0.03 Sraddle rern - lag 3 0.06 0.053 0.3** 0.06 0.048-0.043 0.066* 0.749 Obs. # 849 438 74 708 836 575 947 98 Adj. r-sqared 0.76 0.687 0.64 0.806 0.773 0.770 0.7 0.7 Drbin-Wason sa..889.930.393.989.968.935 0.894.895 3-monh mariy Inercep 0.006** -0.00 0.03 0.03** 0.05** 0.007* 0.6** 0.04** Implied vol. -0.08** -0.07-0.78* -0.36** -0.3** -0.08** -.66** -0.9** Rern in spo marke 0.05 0.049** 0.5* 0.030* -0.7** -0.04* 0.04 0.00 Sraddle rern - lag 0.809** 0.88** 0.49* 0.799** 0.730** 0.790** 0.084* 0.099** Sraddle rern - lag 0.070-0.04 0.6** 0.05 0.099* 0.080 0.053* 0.053 Sraddle rern - lag 3 0.060 0.0* 0.054 0.07 0.073 0.087 0.05** 0.77* Obs. # 75 373 64 67 895 600 006 974 Adj. r-sqared 0.937 0.90 0.77 0.905 0.90 0.97 0.84 0.888 Drbin-Wason sa..99.70 0.963.937.833.766 0.455.833 ** signiican a he % level * signiican a he 5% level 56

Briish Pond Ero 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs Japanese Yen Swiss Franc 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs Figre : Disribion o implied volailiy qoes or he whole sample period This igre shows he boxplos o he daily implied volailiy qoes or -monh, 3-monh, 6-monh and -monh a-he-money sraddle beween Jne 996 and December 00 or Briish Pond, Japanese Yen, and Swiss Franc, and beween Janary 999 and December 00 or Ero. 57

Brisish Pond 3 monh Implied Volailiy 5 0 5 0 Mean 8.47% 5 0 3-Jn-96 3-Dec-96 3-Jn-97 3-Dec-97 3-Jn-98 3-Dec-98 3-Jn-99 3-Dec-99 3-Jn-00 3-Dec-00 3-Jn-0 3-Dec-0 3-Jn-0 3-Dec-0 Japanese Yen 3 monh Implied Volailiy 5 0 5 0 5 0 Mean.96% 3-Jn-96 3-Dec-96 3-Jn-97 3-Dec-97 3-Jn-98 3-Dec-98 3-Jn-99 3-Dec-99 3-Jn-00 3-Dec-00 3-Jn-0 3-Dec-0 3-Jn-0 3-Dec-0 Swiss Franc 3 monh Implied Volailiy 5 0 5 0 5 0 Mean 0.9% 3-Jn-96 3-Dec-96 3-Jn-97 3-Dec-97 3-Jn-98 3-Dec-98 3-Jn-99 3-Dec-99 3-Jn-00 3-Dec-00 3-Jn-0 3-Dec-0 3-Jn-0 3-Dec-0 Figre : Time series plo o implied volailiy qoes This igre shows he daily implied volailiy or he 3-monh a-he-money sraddle or he Briish Pond, he Japanese Yen and he Swiss Franc beween Jne 996 and December 00. The wo verical dash lines indicae he sar and he end o he Asian crrency crisis period. 58

Briish Pond Ero 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs Japanese Yen Swiss Franc 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs 0.0 0.0 0.0 0.30 monh 3 monhs 6 monhs monhs Figre 3: Disribion o implied volailiy qoes in he pos-asian crrency crisis sb-period This igre shows he boxplos o he daily implied volailiy qoes or -monh, 3-monh, 6-monh and -monh a-he-money sraddle beween Jly 999 and December 00 or all or crrencies. 59