How To Price An Opion

Transcription

1 HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May his Version Oc ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models when hey are used o price and hedge exoic opions. hese models are esed in he same way as marke praciioners use hem: models are fied o all he marke raded liquid opion prices and are recalibraed whenever models are used o mark-o-marke he opion under consideraion or o se up hedging porfolios. he es is based on heir effeciveness of hedging exoic opions. ince exoic opions are raded in he over-hecouner marke hisorical daa is no available and he radiional model esing approach of comparing marke prices wih model prices can no longer be applied. We propose a new mehodology o overcome his difficuly: model performance is based on he accuracy of a synheically creaed exoic opion. Using hisorical &P 500 fuures opion prices we show ha he frequenly recalibraed Black-choles model performs beer han he oher alernaive models for hedging ou-of-he-money barrier opions bu performs poorer for hedging compound opions. Our resuls also indicae ha he model performance depends on he degree of pah dependence of he opion under consideraion as noiced by Hull and uo 00.

2 Inroducion In he las decades he financial markes have winessed a remarkable growh in boh volume and complexiy of he conracs ha are raded in he over-he-couner marke. Banks and oher financial insiuions rely heavily on mahemaical models for pricing and hedging hose conracs. Alhough he Black-choles 973 model is sill widely used amongs praciioners for opion pricing as well as hedging a variey of empirical sudies have shown ha he model does no adequaely describe he underlying asse price process. A key assumpion of he Black-choles model is ha he underlying asse price follows a geomeric Brownian moion wih consan volailiy. However he implied volailiies from he marke prices of he opions end o vary across boh srike prices and mauriies. his phenomenon is usually referred o as volailiy smile and volailiy erm srucure see e.g. Rubinsein 994. As a resul inadequae use of he Black-choles model can lead o significan pricing and hedging errors. his is ermed as model risk arising from he use of inadequae models Green and Figlewski 999. o reduce he model risk researchers have proposed various alernaive models ha relax he unrealisic assumpions in he Black-choles model. hese exended models can be caegorized ino wo groups: One-facor models including he consan elasiciy of variance model ox 996 and he deerminisic volailiy funcion model Dupire 994 Derman and Kani 994 and Rubinsein 994; Muli-facor models including he jump diffusion model Meron 976 Baes 99 ec. and he sochasic volailiy model Hull and Whie 987 Heson 993 co 987 ein and ein 99 and Wiggins 987 among ohers. While hese models are more realisic he marke paricipans are sill exposed o he model risk. ince each model relaxes some assumpions of he Black-choles model model risk arises because exoic opions will

3 depend on he model chosen. I is herefore a very imporan empirical issue in finance o es wheher his ype of risk can be reduced by using a more complicaed alernaive model. A number of empirical ess on he performances of opion pricing models have been conduced in recen years including Bakshi ao and hen 997 Baes 996 and Dumas Fleming and Whaley 998 among ohers. 3 No surprisingly all hese empirical ess show evidence ha hose alernaive models perform beer han he Black-choles formula alhough relaive performances of hose models are differen. Mos of he works so far have been focusing on he model s ou-of-sample performance in he following way: Parameers of he model under consideraion are esimaed such ha he model prices for some European opions mach hose prices ha are observed in he marke e.g. from marke ransacions or broker quoes a a specific ime. he resuling model is hen used o price some oher European or American opions a a laer ime. hese model prices are hen compared wih he prices observed from he marke a his ime. However for valuing vanilla opions model specificaion is less imporan because hey are acively raded in he marke place and a grea deal of informaion on he way he insrumens are priced a any given ime is readily available from brokers and oher sources Hull and uo 00. Marke paricipans end o calibrae heir models in a way ha he models can fi he marke prices as close as possible and re-calibrae hem whenever hey mark he opion o markes or rebalance heir hedging porfolios. Moreover praciioners primarily use mahemaical models o compue he prices and 3 Bakshi ao and hen 997 conduc a very comprehensive empirical sudy on he pricing and hedging performance of various alernaive models for &P 500 index opions. he models hey es include he Black-choles model he sochasic volailiy model he sochasic volailiy and jump model and he sochasic volailiy and sochasic ineres rae model. Baes 996 has esed he performance of he Black-choles model he deerminisic volailiy funcion model and he sochasic volailiy and jump model using currency opions. Dumas Fleming and Whaley 995 on he oher hand focus on he performance of a few deerminisic quadraic volailiy models and he implied volailiy model.

4 hedging parameers for exoic opions ha are raded in he over-he-couner marke and he lack of hisorical daa on exoic opion prices makes he esing approach of comparing model prices and marke prices impossible. In oher words he approach models are esed in he curren lieraure is no exacly in he same way as models are used in pracice. he primary objecive of his paper is o empirically es he performance of various models ha are currenly used in pracice for valuing and hedging exoic opions such as barrier opions compound opions and lookback opions ec. 4 Exoic opion prices are much more sensiive o model misspecificaion han European opion prices. his is because marke prices for exoic opions are no available and marke paricipans canno calibrae heir model in he same way as when he model is used for valuing vanilla opions. Furhermore our empirical analysis ries o fi he model under consideraion o he cross-secional prices of all observed liquid opions and hen es he performance of he model on conemporaneously hedging exoic opions which is differen from he ou-of-sample esing approach in exising lieraure. he performance of a model along he ime-series dimension is no necessarily he same as ha along he cross-secional dimension. Mos of all we es models in he same way as he praciioners use models e.g. we recalibrae models frequenly o he marke daa. In his way he model risk is correcly esimaed. herefore our research will be of ineres o academics in finance as well as o praciioners and regulaors in invesmens and risk managemen. o overcome he difficuly of he lack of hisorical daa on exoic opion prices we propose a new esing mehodology: he model parameers are esimaed a ime and a replicaing porfolio is synheically creaed from he marke daa including he liquid 4 We mainly focus on barrier opions and compound opions in his paper. 3

5 opion prices and he underlying asse price. A he nex sep he model price is compared wih he value of he model s replicaing porfolio hen he model is recalibraed and he replicaing porfolio is rebalanced. his procedure coninues unil he mauriies of he exoic opions. If he model is specified correcly or if he model works well one uni of he exoic opion can be hedged by an offseing posiion in he replicaing porfolio and he expecaion and variance of he hedging errors should be very small. For his reason he average of he hedging errors can be used as an indicaor of he performance of he model under consideraion when i is used o hedge exoic opions. Our esing mehod is similar o ha of Melino and urnbull 995 in some ways. Melino and urnbull 995 examine he effecs of he sochasic volailiy upon he pricing and hedging of long-erm foreign currency opions. ince long-erm foreign currency opions are no acively raded and here is no daa available heir es analysis is also based on dynamic hedging errors. However our mehod is differen in ha he model is recalibraed frequenly and he performance is based on is effeciveness on hedging exoic opions. Green and Figlewski 999 invesigae he performance of he Black-choles model when i is recalibraed daily o hisorical daa. We recalibrae he model o curren marke daa raher han hisorical daa. Hull and uo 00 adop a similar approach o ours; however insead of using he marke daa hey assume here is a rue model ha generaes he rue vanilla and exoic opions daa. herefore he model performance es can sill be based on he comparison beween he candidae model prices and he rue observed prices. Using &P 500 fuures opions we consider he performances of he radiional Black-choles model and hree oher major alernaive models: he consan elasiciy of 4

6 variance EV model he jump diffusion model and he sochasic volailiy model. imilar o Bakshi e al 997 we employ wo differen ypes of hedging sraegies o gauge he relaive performance of differen models: he minimum variance hedging sraegy and he dela-vega neural hedging sraegy. he mehod can be easily adoped o es oher models ha are currenly used in pracice. Our finding indicaes ha model recalibraion does have some effecs on model relaive performances. he Black-choles model ouperforms alernaive models on hedging ou-of-he-money barrier opions in erms of dollar hedging errors. For hedging compound opions however he jump diffusion model or he sochasic volailiy model performs beer han he Black-choles model overall. In addiion our resuls show ha he model performance also depends on he degree of pah dependence of he opions as noed by Hull and uo 00. For hedging long-erm barrier opions he performances of all models are poor. Alhough he hedging performances are no necessarily he same as he pricing performance our resuls indicae ha he raders common pracice of recalibraing he Black-choles model for pricing and hedging less liquid and exoic opions may work well for some ypes of opions however i may no work well for ohers. he remainder of his paper is organized as follows. ecion briefly reviews various opion pricing models being esed in his paper and hen discusses he poenial model risk for he praciioner s models. ecion 3 discusses he esimaion and esing mehodologies. ecion 4 describes he &P 500 fuures and fuures opion daa. ecion 5 presens he models esimaed resuls and discuss heir in-sample fi. In ecion 6 we repor he empirical resuls. ecion 7 concludes his paper. 5

7 Opion Pricing Models. heoreical Opion Pricing Models: A Brief Review In addiion o he Black-choles model we consider hree alernaive compeing models in his paper: he consan elasiciy of variance EV model he jump diffusion JUMP model and he sochasic volailiy V model. For convenience he risk free ineres rae and he dividend rae of he underlying asse are denoed by r q respecively and are assumed o be consan over ime in his paper.. Black-choles Model he Black-choles 973 model assumes ha he underlying asse price follows a geomeric Brownian moion under he risk neural probabiliy measure: d = r q d dw where is he volailiy of he underlying asse and is assumed o be consan is he price of underlying asse a ime w is a sandard Brownian moion. In he Black-choles world he marke is complee and he derivaives wrien on he asse can be perfecly hedged by he underlying asse and a risk free invesmen. For any derivaive wrien on he asse and paying g a mauriy is price f a ime saisfies he following parial differenial equaion: f f f r q rf = 0 wih boundary condiion f = g. In paricular for a European call opion wih srike X he payoff a mauriy 6

8 is X and is price X a ime can be calculaed by solving equaion. I is given by: Where q r X = e N d Xe N d d ln r q X = d = d. 3 Equaion 3 implies ha here is a one-o-one correspondence beween he opion price and he volailiy. As a resul for each opion he implied volailiy can be compued by solving for he volailiy ha equaes he model price wih he observed marke price. Under he assumpions of he Black-choles model volailiies should be he same for opions on he same asse wih differen srikes. However empirical findings show ha he implied Black-choles volailiies vary sysemaically wih srikes a phenomenon usually referred o as he volailiy smile. In he equiy marke he implied volailiies for opions wih he same mauriy usually decrease as he srikes increase. In oher words he Black-choles model under-prices deep ou-of-he-money pu opions and over-prices deep ou-of-he-money call opions. his volailiy paern is paricularly noiceable since he 987 marke crash ee Rubinsein 994. he volailiy smile implies ha he implied asse reurn disribuion is negaively skewed wih higher kurosis han allowable in he lognormal disribuion assumed by Black-choles. o capure hese sylized facs observed in he empirical sudies wo major exensions are made o he Black-choles model in he lieraure: he firs exension relaxes he assumpion on he volailiy. he second exension allows for jumps in he dynamic process of he underlying asse price. he alernaive models can be eiher one of he exensions or combinaion of he exensions. In his paper we es wheher hese 7

9 exensions can improve he performance over he Black-choles model on hedging exoic opions such as barrier opions and compound opions.. onsan Elasiciy of Variance Model he consan elasiciy of variance model hereafer he EV model developed by ox 975 simply assumes ha he local volailiy of he underlying asse price depends on he price level. pecifically under he risk neural probabiliy measure he sochasic process of he underlying asse price is assumed as follows: d α = r q d dw 4 where w is a sandard Brownian moion; and α are consan parameers and α known as he elasiciy facor is resriced o he inerval [0. In he limiing case α = he EV model reduces o he Black-choles model. he general EV process also ness he square roo process α = and he absolue diffusion process α = 0 as special cases. equal o Under he EV process he insananeous volailiy of underlying asse reurns is α and hence is an inverse funcion of he underlying asse price. Boh empirical observaions and economic raionale suppor he inverse relaionship beween he underlying asse price and he volailiy. onsequenly by incorporaing he negaive correlaion beween he underlying asse price changes and he volailiy changes he EV model could beer describe he acual sock price behavior han he Black-choles model and his is confirmed by he empirical sudies of MacBeh and Merville 980 and Emanuel and Macbeh 98 among ohers. I can be easily shown ha he marke is sill complee under he assumpions in he EV model hus opions can be perfecly hedged by coninuously rebalancing a replicaing porfolio ha consiss of he underlying asse and a risk free asse. Using he 8

10 9 same argumen as in he Black-choles seing one can show ha a parial differenial equaion similar o equaion sill holds in his case: Le f be he price of an arbirary derivaive a ime i saisfies he following parial differenial equaion: 0. = rf f f q r f α 5 ox 996 derives a closed form soluion for he price of a European call opion wih srike X α X = Γ = 0 n n x q n kx n G x e e X α α α = Γ 0 n n x r n kx n G x e Xe α α α 6 where ; = q r e q r k α α ; q r k e x = α α. ] [ du u e m v m G m v u Γ = chröder 989 shows ha equaion 6 can be expressed in erms of he noncenral chi-square disribuions: ; ; x y Q Xe x y Q e X r q α α α = 7 Where α = kx y and ; k v z Q is he complemenary non-cenral chi-square disribuion funcion evaluaed a z wih v degrees of freedom and non-cenral parameer k.

11 o evaluae Q z v k in equaion 7 we use he simple and efficien algorihm suggesed by chröder 989 and when z or k is large we use he approximaion o he non-cenral chi-square disribuion derived by ankaran Pure Jump Diffusion Model Meron 976 develops a pure jump process o model he movemen of he sock price subjec o occasional disconinuous breaks. Under he risk neural probabiliy measure he model assumes ha he process of he underlying asse price is as follows: d = r q λ k d dw JdQ. 8 In equaion 8 is he volailiy of he underlying asse reurns condiional on no jump occurring and is assumed o be consan. λ is he annual frequency of jumps. k is he average jump size measured as a proporional increase in he asse price. J is he random percenage jump condiional on a jump occurring and ln J ~ Nln k δ Q is a Poisson couner wih inensiy λ i.e. Pr ob dq = = λd δ is he sandard deviaion of ln J and dw is a sandard Brownian moion and is assumed o be independen of dq. given by: δ Under hese assumpions he insananeous mean of he jump diffusion process is d E = µ λk. d he insananeous variance of he oal reurn of he process is given by:. d var = d δ λ k k e. 0

12 In his model he insananeous mean of he underlying asse reurns consiss of wo pars: he firs par is due o he normal underlying asse price changes and he second par is due o he abnormal underlying asse price changes. Accordingly he variance of he oal reurn of he underlying asse has wo componens as well: he componen of he normal ime variance and he componen of jump variance. If here is no jump i.e. = 0 λ hen his model reduces o he Black-choles model. ompared o he Black-choles model he jump diffusion model aribues he skewness and excess kurosis observed in he implied disribuion of he underlying asse reurns o he random jumps in he underlying asse reurns: he skewness arises from he average jump size and he excess kurosis arises from he magniude and variabiliy of he jump componen. herefore he jump diffusion model could be more capable of capuring he empirical feaures of underlying equiy reurn han he Black-choles model. If f is he price of an arbirary derivaive a ime hen using he riskneural argumen we can ge he following parial differenial equaion: 0. ] [ = rf f J f E f f k q r f λ λ 9 For a European call opion is price δ λ k X wrien analyically as Meron 976:! 0 = = n n n r n r d XN d N e n e e k X n λ δ λ λ 0 where k n k q r r n = ln λ. ln δ δ δ n d d n n r X d n n n n = =

13 Unlike he Black-choles model and he EV model he jump diffusion model has wo sources of uncerainy and herefore is a mulifacor model. 4. ochasic Volailiy Model V: he sochasic volailiy model inroduced by Hull and Whie 987 Heson 993 co 987 ein and ein 99 and Wiggins 987 among ohers assumes he volailiy of he underlying asse price follows a paricular sochasic process. As an example we consider he case where he volailiy follows a mean-revering Ornsein- Uhlenbeck OU hereafer process i.e. under he risk neural probabiliy measure he underlying asse reurn and volailiy processes are as follows: d dv = r q d v dw = k θ v ] d dz [ where v is he volailiy of he underlying asse reurns; k θ are he speed of adjusmen long-run mean and volailiy of volailiy parameers respecively; z and w are sandard Brownian moions wih a correlaion coefficien ρ. he sochasic volailiy model provides some addiional flexibiliy over he Black-choles model o capure he empirical feaures found in he disribuion of he underlying asse reurn. I aribues he skew effec o eiher he correlaion beween he underlying asse reurn and he volailiy or he volailiy of volailiy and aribues kurosis effec o he volailiy of volailiy. However he effecs on opion pricing may be small when he mauriy of he opion is shor. his is because he volailiy follows a coninuous diffusion process and he abiliy ha he volailiy process generaes enough shor-run skewness or excess kurosis is limied. Adding jumps in he process of underlying asse reurns offers anoher flexibiliy o capure empirical feaures in he shor run.

14 In his model he price of a derivaive f v a ime ha pays g a he mauriy saisfies he following parial differenial equaion: f f r q f k θ v v v f f v f ρv rf v = 0 wih he boundary condiion f v = g. pecifically for a European call opion wrien on he asse wih srike price X and mauriy he price v X saisfies he differenial equaion subjec o v X = max[ X 0]. A closed form soluion of v X can be expressed as see chöbe and Zhu 999: q r v X e P Xe P =. 3 Where P j = π 0 e Re[ iφ ln[ X ] f j v ; φ ] dφ iφ for j =. f j are he characerisic funcions of P j respecively and are given in Appendix A. ome numerical mehods are needed o calculae he inegrals in P in equaion 3. In his paper Gaussian quardraure procedures in NAG are used and hese inegrals can be evaluaed efficienly and accuraely for a broad range of reasonable parameers.. he Praciioner s Model. he Praciioner s Model Alhough researchers have made remarkable advances in developing more realisic opion pricing models he mos widely used valuaion procedure among praciioners is however he simples Black-choles model wih ad hoc adjusmens and j 3

15 recalibraions. his so-called praciioner s Black-choles PB hereafer approach can be described as follows. he Black-choles implied volailiies of all raded opions are calculaed a ime and he implied volailiies for oher vanilla opions are calculaed by inerpolaion across srike prices and mauriies o ge a volailiy surface a ime. Wih his volailiy surface oher opion prices a ime can hen be calculaed from he Black- choles formula using he volailiy obained from he corresponding poin on he surface. he above procedure is repeaed whenever he model is used. One of he key poins of he PB is o calibrae and recalibrae he Black-choles model o fi he cross-secional European opion prices exacly. In oher words in pracice marke paricipans end o calibrae heir models o fi he cross-secional observed marke prices exacly a a ime poin and recalibrae hem whenever hey mark he opion o markes or rebalance heir hedging porfolios. econd marke paricipans mainly use models o price or hedge less liquid or exoic opions. In fac he praciioner s model uses all observed liquid opion prices as he inpus insead of he oupus.. Poenial Pricing and Hedging Errors in he Praciioner s Model Alhough he praciioner s model fis he observed liquid opion prices almos exacly a any poin i does no compleely eliminae he model risk. Firs he dynamics of he underlying asse price obained by fiing he model o a cross-secion of observed opion prices migh be incompaible wih he no-arbirage evoluion of he underlying asse price. econd updaing he model from ime o ime implicily assumes he fied parameers can change over ime. his implies ha he model is inernally inconsisen Dybvig 989 and may permi arbirage opporuniies in derivaives Backus Foresi and Zin

16 he model risk becomes very imporan when he model is used o price or hedge exoic opions. his is because exacly pricing all European opions means he uncondiional probabiliy disribuion of he underlying asse price a all fuure ime is always correc. However differen models will give differen join disribuion of he underlying asse price a differen imes. onsequenly even hough he praciioner s model can correcly price a derivaive whose payoff is coningen on he asse price a any one paricular ime here is no guaranee ha i can correcly price a derivaive whose payoff is coningen on he underlying asse price a more han one ime Hull and uo 00. Exoic opions such as barrier opions and compound opions are examples of pah-dependen opions whose payoffs depend on he underlying asse price a differen imes and hey may be mis-priced by he praciioner s model. For his reason we look a exoic opions when we es he model risk. In addiion mos exoic opions are raded in he over-he-couner marke and he marke prices are no available. As a resul exoic opion prices are more sensiive o he model mis-specificaion han vanilla opion prices. Frequen recalibraion of a model migh also generae hedging errors. o illusrae his poin we ake he Black-choles model as an example. Assume ha he opion being hedged is an exoic opion whose prices are no available. onsequenly one has o rely on he model o price and hedge such opions. In he Black-choles model he marke is complee so he exoic opion can be perfecly hedged by aking he underlying asse and he risk free invesmen as hedging insrumens. ince he replicaing porfolio can only be rebalanced discreely in pracice opion hedging errors may arise eiher from discree adjusmens o he hedging porfolio or from model mis-specificaion. If he model is implemened inconsisenly i.e. he model is frequenly recalibraed i can also generae hedging errors. 5

17 6 o show his le us assume ha he rue volailiy of he underlying asse price is and i is mis-specified as in he Black-choles model a ime. Denoe he rue price and he model price of an opion by and respecively. he replicae porfolio based on he mis-specified model consiss of = unis of and B = unis in he risk free invesmen a ime. he value of he porfolio ha he wrier holds a ime in his case can be wrien as. = π 4 he opion price is he model price and because he marke price for his opions is no available. Neverheless such a hedging sraegy is sill useful in idenifying models ha can se up more accurae hedges for he arge opion. he hedging errors ignoring high order erms from ime o ime d are given by d d d rd d d d = π d d d rd d d d d = ε d d d d = ε 5 where ε is a random variable drawn from he sandard normal disribuion and. d d = he firs erm in equaion 5 arises from boh he model mis-specificaion and he discree adjusmens o he hedge. he expecaion and variance of his erm are zero if he model is correcly specified and he hedging porfolio is rebalanced coninuously.

18 Furhermore he size of his erm is proporional o he model s gamma hedge parameer. As a resul exoic opions and vanilla opions may have differen sensiiviies o he model mis-specificaion. Inuiively his is because he payoff of an exoic produc depends no only on he underlying asse price a mauriy bu also on is price hroughou he life of he opion. he second erm in equaion 5 arises from he deviaion of model prices a ime d due o he model recalibraion. If he model is correcly specified and he hedging porfolio is rebalanced frequenly Galai 980 and Lean 985 show ha he discree adjusmen errors are small relaive o he mis-specificaion hedging errors. Adoping a similar approach used in Galai 983 we can also rewrie equaion 5 in he following form dπ = d d df F rd d d d 6 where F = F d = d d and df = F d F. he firs line of equaion 6 is he hedging errors when he rue opion prices is used in he hedging porfolio while he second line conains he errors arising from he deviaion beween he model price and he rue price in addiion o he deviaion of he model prices due o model recalibraions. rd 3 Research Mehodologies In his secion we describe he model esimaion mehods and he model esing procedure. 3. Model Esimaion Mehods here are wo differen esimaion mehods for a given model: he firs one is o apply he economeric esimaion mehod such as he maximum likelihood mehod or he 7

19 generalized mehod of momens mehod o obain he required esimaes using hisorical daa of he underlying securiy price. One of he poenial problems of his approach as noed by Bakshi e al. 997 is is sringen requiremen on hisorical daa. In addiion for some models i is no possible o obain he parameer esimaes for he risk-adjused processes ha are necessary for he valuaion purpose i.e. he sochasic volailiy model. he oher esimaion mehod is o imply he model parameers from he observed opion prices. Parameers implied from he opion prices seem o be beer esimaors which is confirmed by a number of empirical sudies. 5 his is because opion prices reflec he marke paricipans view on he fuure movemens of he sock price. For he second esimaion mehod wo differen approaches usually apply: one is o esimae he parameers a each ime by using he cross-secional opion prices and hen average hem o ge he esimaors. he oher is o esimae he parameers by using he cross-secional ime series opion prices. In his sudy we use he implied parameer esimaion mehod. ince our esing approach is o recalibrae he model frequenly we esimae he model on each rading day using he cross-secional opion price daa. As a resul a ime series of esimaes for any parameer in he model are obained. Deails of he mehod are described as follows. For a model ha depends on a se of parameers Θ = a a a n le us wrie he price of a vanilla opion call or pu wih srike X and mauriy ime as Θ X where and represen he curren ime and sock price respecively. A each ime here are many vanilla opions wih differen srikes and mauriies raded in he marke place. ~ If we wrie he corresponding marke prices as X hen he parameer vecor Θ a ime is chosen o minimize he sum of he squared errors E i.e. 5 ee for example Melino and urnbull 990 Baes 996 among ohers. 8

20 ~ E = Min [ X Θ X ]. 7 Θ i j i Differen objecive funcions in model esimaion migh yield differen esimaion and performance resuls. he error funcion defined in equaion 7 can cause some problems e.g. i assigns more weigh o relaively expensive opions in-he-money opions and long ime-o-mauriy opions and less weigh o opions wih low values. One can use oher alernaive objecive funcions such as he percenage E or he implied volailiy E in esimaion. We choose he above funcion o esimae he models because i is widely used in he curren lieraure. 3. Hedging ess his esing approach for opion pricing models is from he perspecive of he raders who frequenly samples marke prices rebalances hedges and recalibraes heir models. More specifically firs he model is recalibraed on every rading day. By doing his he model parameers are allowed o change over ime and his may add flexibiliy o each model o capure changes in he disribuion of he underlying asse reurn ha consan parameer models fail o capure. econd he model parameers are obained by fiing he model o he cross-secional observed liquid opion prices a day as close as possible. he performance of he esimaed model on simulaneously valuing exoic opions such as barrier opions and compound opions is hen invesigaed. his is more a cross-secional approach o es he model performance a a poin in ime as opposed o a ime series approach o es over longer-horizons. his approach differeniaes our sudy from hose in he curren lieraure. he empirical es would have been similar o hose in he curren lieraure if we could observe he prices of he exoic opions under consideraion. However his is no he case for exoic opions because hey are raded in he over-he-couner marke and here is no hisorical daa available. o assess he model performances we sudy he j i j 9

21 errors beween he model predic price and he value of he replicaing porfolio insead of pricing errors. his approach allows us o es he model performance on valuing opions wihou hisorical daa. pecifically we adop he following approach for esing he effeciveness of a model for valuing or hedging an exoic opion: We firs divide he ime o mauriy ino m seps and assume ha hedging porfolios are rebalanced only a hese imes. he model parameers under consideraion are esimaed using he cross-secional vanilla opion prices observed from he marke a ime = j. he resuling model is hen used o price he exoic opion under consideraion and se up he replicaing porfolio. A he nex ime j he model parameers are re-esimaed. he model price of he exoic opion is compared wih he value of he replicaing porfolio wih he difference denoed by π j and hen he replicaing porfolio is rebalanced based on he new parameers esimaed for he model. 3 Repea he above seps unil he mauriy of he arge opion record all he hedging errors for j = m. π j 4 onsider a se of he same exoic opions indexed by i wih i = n repea seps from o 3 we will record mn hedging errors ha are denoed by. π ij Finally he oal average dollar hedging errors and absolue hedging errors are calculaed as: n m E π = π ij mn i= j= n m E π = π ij. mn i= j= 0

22 Ignoring he errors from ime discreizaion he hedging errors should be zero if he model is correcly specified. he average dollar hedging error measures he average losses or profis of he hedging porfolios over he rebalancing inerval and he average absolue hedging errors measures he average deviaions of he errors from zero over he rebalancing inerval. onsequenly he model performance on pricing and hedging exoic opions can be judged by he average dollar and absolue hedging errors. he exoic opions we consider in his paper include barrier opions and compound opions. hese are among he mos widely used opions by boh marke praciioners and academics. However wih he excepion for he Black-choles model here are no analyical formulae for hose opions. As a resul numerical mehods have o be used o value hese opions and o calculae he hedging raios. 6 In pracice some difficulies arise when hedging exoic opions. onsider for example an up-and-ou barrier call opion. he payoff of he opion is given a mauriy ime by max[ X 0]I {max0 < H }. he dela and gamma of he opion become large in absolue values near expiraion when he asse price is close o he barrier. A rader who adops he dela hedging sraegy would ake large shor or long posiions in he underlying asse and make large adjusmens o he hedging porfolio. o avoid such difficulies he hedging posiions are only rebalanced up o 7 days before he mauriy of he opion. 6 We adop he Mone arlo simulaion approach in his paper. o reduce he variance of he esimaors we use he aniheic variable echnique while carrying he simulaion.

23 4 Dae Descripion he daase used for our empirical analysis conains he daily closing prices of &P 500 index fuures and fuures opions raded on he hicago Mercanile Exchange ME from January 993 o December 993. his daase is chosen for he following reasons: Firs indices are beer represenaions of he economy han any arbirary choice of individual socks and hey are acively raded in he marke. econd index opions have been he focus of many previous sudies in he lieraure e.g. Bakshi e al 997 Baes 99 and Dumas e al 995. here are four conrac monhs for fuures conracs: March June epember and December in each year. he las rading daes of all he fuures conracs are on he hursday prior o he hird Friday of conrac monhs. he conrac monh of a fuures opion can be any monh in he year. A fuures opion is in American syle i.e. he holder can exercise i on any business day before i expires. For he opion ha expires in he March quarerly he underlying asse is he fuures conrac for he monh in which he opion expires and he las rading dae is he same as he underlying fuures conracs. For he opion ha expires in monhs oher han hose in he March quarerly cycle he underlying fuures conrac is he nex fuures conrac in he March quarerly cycle ha is neares he o expiraion of he opion and he las rading dae is he hird Friday of he conrac monh. ince he daa we use are he daily closing prices for boh fuures and heir opions synchronizaion problems may occur. However since he &P fuures and opions are acively raded we may ignore his issue. U -bill raes as a proxy for he risk free ineres rae. he daily -bill middle raes for he four mauriies one- hree- six- and -monh are obained from

24 Daaream. he discoun raes for periods oher han he four mauriies are calculaed hrough linear inerpolaion. he moneyness of an opion is defined as m = F / X where F is he underlying fuures price and X is he srike price of he opion. An opion is said o be deep ou-of-he-money if m < ou-of-he-money if 0.90 m a-he-money if 0.97 < m <. 03 in-he-money if.03 m. 0 deep in-he-money if m > he ime-o-mauriy of an opion is measured by he number of calendar days beween he valuaion and expiraion daes. An opion is classified as shor-erm if is days-o-expiraion is less han 60 days medium-erm if i is beween 60 and 80 days and long-erm if i is more han 80 days. here are 4000 opions in he raw daa of which 7699 are call opions. Mauriies of all of he opions are less han one year. o save compuing ime only prices of call fuures opions in he daase are used. Moreover some filers are applied o he daa: Firs opions wih less han 7 days o expiraion are excluded because hese opions have relaively small ime premiums and heir implied volailiies are exremely sensiive o liquidiy-relaed biases. econd opions wih m greaer han en percen are excluded because deep in- and ou-of-he-money opions have small ime premiums and conain lile informaion abou he volailiy. Furhermore hese opions are infrequenly raded and heir quoes are no updaed frequenly. hird opions ha violae he arbirage resricions F max[ 0 F X ] 7 his definiion is consisen wih Bakshi e al

25 are excluded. here are 3687 call opions in he filered daase wih an average of 54.7 ransacions per rading day. able repors he average prices and he number of observaions in each caegory of he filered daa. Noe ha 4.4 percen of he 3687 observaions are a-hemoney opions 35. percen are ou-of-he-money opions and.4 percen are in-hemoney opions. Opions wih mauriy days less han 80 days ake up 78.7 percen of he oal observaions. he average call opion prices range from 0. for shor-erm ou-ofhe- money opions o 4.69 for long-erm in-he-money opions. he fuures daase consiss of 0 fuures conracs in oal and here are 8 differen mauriies every rading day. he longes fuures are -year conracs. Figure shows he Black-choles implied volailiy paerns wih respec o moneyness and mauriies. he implied volailiies are obained by averaging he Black- choles implied volailiies wihin each moneyness-mauriy caegory and across he days in he sample. Obviously he fuures opions exhibi volailiy smile effec across moneyness and mauriies. 5 Model Parameer Esimaion Model parameers are implied from he observed opion prices every day as described in ecion 3.. ince fuures opions raded on ME are in American syle we need o ake ino accoun heir early exercise premiums. Alhough American opions can be valued hrough numerical mehods hese mehods are very compuing expensive for he purpose of our esimaion. For his reason we use he quadraic approximaion approach o value he American opions see Appendix B. Parameers for he given model are esimaed using Equaion 9 on each rading day. o see he sabiliy of he esimaes he summary saisics of he implied parameers for various models are repored in able. 4

26 he mean of he esimaed volailiy parameer for he Black-choles model is 0. over he sample period. However he esimaes vary from day o day wih he minimum value of 0.09 and he maximum value of 0.5. he sabiliy of he esimaed parameers can also be inferred from heir coefficiens of variaion which equals he raio beween he sandard deviaion and he mean. Alhough he esimaes of he volailiy for he Black-choles model indicae ha here is some variaion across ime he parameer is quie sable because he coefficien of variaion is only he sigma parameer esimaes in he EV model range from o.69 wih he mean of 6.33 and he sandard deviaion of hey vary significanly over ime. he high sandard deviaion for he sigma parameer esimaes is generally expeced since he variaion of he elasiciy facor α has an exponenial effec on he esimaes of he sigma parameer. onsequenly a small deviaion from he rue value of he elasiciy facor could lead o a large deviaion from he rue value of he sigma parameer in he EV model. he elasiciy facor is relaively sable compared wih he sigma parameer as he coefficien of variaion is only he mean of he elasiciy facor is which indicaes ha he fuures price changes and volailiy changes are negaively correlaed. he average of esimaed volailiy condiional on no jumps in he jump diffusion model is wih he sandard deviaion of 0.0. he jump diffusion model aribues he negaive skew and excess kurosis o he jump risk where jumps occur wih a mean annual frequency of.59 imes and a mean negaive jump size of 0.9. he sandard deviaion of jump sizes condiional on a jump is Excep for he volailiy parameer he coefficiens of variaion for oher parameers are greaer han one which shows ha hese esimaes are quie unsable. 5

27 he mean of he esimaed spo volailiy is 0. for he sochasic volailiy model. he mean of he speed of adjusmen of he volailiy is 4.77 he mean of he long run volailiy is and he mean of volailiy of volailiy is for he spo volailiy process. hese esimaes have relaive high coefficiens of variaion indicaing ha hey are observed wih significan errors. he correlaion coefficien beween he underlying asse reurns and is volailiy changes is negaive wih a mean of he esimaion resuls for he hree alernaive models indicae ha he disribuion of he underlying asse reurns is asymmeric which is a feaure ha he Black-choles model fails o capure. he improvemen of he alernaive models over he Black- choles model is furher evidenced by he Es of he models: he average of he Es of he Black-choles is 8.48 which is he highes one among all of he models considered. he averages of Es of he EV he jump diffusion and he sochasic volailiy models are and 4. respecively. In oher words hese alernaive models give a much beer in-sample fi han he Black-choles model which is expeced because hey have more parameers and herefore allow for more degrees of freedom. However if some of he parameers in he model are redundan and merely cause over fiings of he daa he model will be penalized when ou-of-sample pricing and hedging performance is used as a crieria. Overall he implied esimaes of he models considered over he sample reveal evidence of he parameric insabiliy. Models wih more parameers improve he insample fiing performance han models wih fewer parameers bu yield less sable esimaes. Noe ha heoreical models are derived under he assumpion of consan parameers. he divergence from heory indicaes ha hese models fail o capure some feaures of he process of he underlying asse price. 6

28 6 Hedging Performance on Exoic Opions In his secion we describe he hedging procedure and analyze he empirical resuls. o assess he hedging performance we consider wo differen hedging sraegies: he minimum variance hedging sraegy and he dela-vega hedging sraegy. 6. Minimum Variance Hedging raegy he minimum variance hedging sraegy involves only he underlying fuures conrac as he hedging insrumen. We consider he minimum variance hedging sraegy because he marke is no complee in he jump diffusion model and in he sochasic volailiy model and he hedging raio should reflec he jump risk and he volailiy risk. A minimum variance hedging porfolio consiss of one uni of he hedged opion and unis of he underlying fuures where he hedging raio he variance of he hedging porfolio value. X s X s is deermined by minimizing o be more specific suppose ha an opion rader wries one uni of opion. If he wrier relies on he minimum variance hedging sraegy o hedge his opion hen he value of he hedging porfolio a ime is: H = X s B where s B = X is he amoun of risk free invesmen. he hedging porfolio is selffinancing and he change of H from o he oal variance of dh is given by dcan be wrien as dh = d X s d Brd. Var dh Var d X svar d X sov d d =. By minimizing Var dh he hedging raio can be solved as: ov d d X s =. 8 Var d 7

29 In he Black-choles model and he EV model he marke is complee and an opion can be perfecly hedged by aking posiions in he underlying asse and risk free invesmen. For hese wo models he minimum variance hedging sraegy is he same as he dela neural hedging sraegy and he hedging raio is he dela of he hedged opion. However in he jump diffusion or he sochasic volailiy model he minimum variance hedging is no perfec in he sense ha one canno perfecly replicae he payoff of an opion by only aking posiions in he underlying asse and risk free invesmen. In he case of he jump diffusion model he minimum variance hedging raio is given by ee Appendix for deails X = V j λ V j E[ J J ] k 9 δ wherev j = λ k λ e k. Equaion 9 shows ha if here is no jump risk i.e. λ = 0 he minimum variance hedging is he same as he dela neural hedging. However if here is jump risk is impac will be refleced in he second erm of Equaion 9. In he sochasic volailiy model he minimum variance hedging raio is given by see Appendix X = ρ. 0 v v Equaion 0 shows ha if he volailiy is deerminisic or sock reurns are uncorrelaed wih volailiy changes hen he minimum variance hedging raio is he same as he dela raio. As we have menioned in ecion 5 ρ is usually negaive in he equiy markes and as a resul minimum variance hedging raio is usually less han he dela raio o reflec he impac of volailiy changes. 8

30 In his sudy he opions o be hedged are barrier opions and compound opions on &P 500 fuures. o see he effec of he mauriy of an opion on he hedging performance we consider he barrier or compound call opions wih hree differen erms o expiraion: shor-erm medium-erm and long-erm. he hedging procedure is described as follows: A ime model parameers are esimaed by fiing he model o he prices of he raded fuures opions. he price of he exoic opion under consideraion can hen be calculaed from he model. o hedge his exoic opion a hedging porfolio is consruced wih X s unis of fuures F and unis in he risk free invesmens. ince he fuures conracs require zero iniial cash oulay he oal cos of such a porfolio is zero: H X 0 = 0. = s A ime he hedging porfolio is rebalanced. Using model parameers esimaed a ime he value of he hedging porfolio is given by H = X F F r s. H is referred o as he hedging error over he rebalancing inerval. hese seps are repeaed up o 7 days before he opion s mauriy dae. his will give he average dollar hedging errors average absolue hedging errors for his hedging sraegy as described. his procedure racks he hedging errors for one realizaion of he opion being hedged. In order o perform empirical analysis he procedure is repeaed for every 7 days in he sample period and each repea represens a realizaion of he sample pah. Average dollar and absolue errors are calculaed for each model hrough hese hedging errors and he resuls are repored in ables 3 and 4. able 3 repors he hedging errors when he arge opions are up-and-ou call opions. he barrier level of each barrier opion is se equal o. imes of he underlying 9

31 fuures price while srikes are se equal o and.06 imes of underlying fuures price respecively. he hedging porfolios are rebalanced daily up o 7 days before he mauriy dae of he hedged opions. everal observaions can be made from he average dollar hedging errors in able 3. Firs he Black-choles model ouperforms he oher hree alernaive opion-pricing models for hedging ou-of-he-money barrier opions for all mauriies. he jump diffusion model on he oher hand performs he bes for hedging shor-erm in-he-money barrier opions and he sochasic volailiy model performs he bes for hedging longerm in-he-money barrier opions. econd he average hedging errors increase as mauriies of barrier opions increase for any given model. Based on he average absolue hedging errors his mauriy effec becomes much more significan. Noe ha he prices of barrier opions may decrease as he mauriies increase; his resul indicaes ha he hedging performance relaive o he opion value is quie poor for long-erm barrier opions for any given model. his confirms he resul noed by Hull and uo 00 ha model performance depends on he degree of pah dependence of he exoic opion. For a barrier opion he probabiliy of hiing he barrier becomes relaively large when he ime o mauriy increases and herefore he knockou feaure becomes more imporan. o reconfirm his conclusion we repea he above hedging procedure for barrier opions whose barriers are closer o he underlying fuures price e.g. he barrier is se equal o.05 imes of he fuures price. 8 he hedging errors as we expeced are larger in his case han heir counerpars in able 3. Based on he sizes of he absolue hedging errors however he Black-choles model yields he smalles sizes of hedging errors for almos all of he opions considered indicaing ha he hedging performance of he Black-choles model is quie sable. 8 he resuls are no repored here. 30

32 able 4 repors he hedging errors for compound opions. he compound opion considered is he call on call opion. he underlying call opion is a fuures opion wih 60 days o expiraion. he srike of he underlying call opion is se as he underlying fuures price. rikes of he call-on-call opions are se equal o and 4.0 respecively. Based on he dollar errors and he absolue hedging errors in able 4 he sochasic volailiy model and he jump diffusion model generally perform beer han he Black-choles model and he EV model for hedging mos of he compound opions. hese findings are in line wih hose in he curren lieraure. Anoher observaion is ha he average dollar and absolue hedging errors relaive o he values of he exoic opions being hedged for a given model do no change much when he mauriies of he arge opions increase. Inuiively increasing he mauriy of a compound opion does no affec he imporance of is exoic feaure or he degree of pah dependence. As a resul he hedging performance of a compound opion changes very mildly when is mauriy increases. For any given model we can also see ha he relaive performance for hedging he shor-erm barrier opions is beer han ha for hedging he shor-erm compound opions while he performance for hedging he long-erm barrier opions is poorer han ha for hedging he long-erm compound opions. For he Black-choles model for example he average dollar hedging errors relaive o he arge opion values are from 0.07% o.% for shor-erm barrier opions while hey are from 0.3% o.% for shor-erm compound opions. he relaive hedging errors are from 0.5% o.5% for long-erm barrier opions while hey are from 0.3%o 0.56% for long-erm compound opions. his evidence becomes much pronounced in erms of relaive absolue hedging errors. 3

33 6. Dela and Vega Neural Hedging raegy As we menioned earlier when here are more han one sae variable in he model he exoic opions can no longer be perfecly hedged by rading only he underlying asse and he risk free asse. For example opions wrien on he same asse are needed o hedge he addiional volailiy risk in he sochasic volailiy model. A hedging porfolio is called dela-vega neural if he porfolio value is insensiive o he changes in he underlying asse price and is volailiy. uppose one needs o hedge one uni of shor posiion in an exoic opion. E a ime. he replicaing porfolio consiss of a unis of an European opion b unis of he underlying asse and porfolio a he ime is hus B unis of he risk free asse. he value of he E π = Θ a b B. his porfolio is self-financing and dela and vega neural if π = 0 π = a E b = 0 π v = v a v E = 0. herefore he hedge parameers according o he specified model can be solved as a = v E v b v E = E v B E = a b 3 3