Loss Functions in Option Valuation: A Framework for Selection. Christian C.P. Wolff, Dennis Bams, Thorsten Lehnert

Transcription

1 LSF Research Working Paper Series N Dae: Ocober 008 Tile: Auhor(s)*: Absrac: Loss Funcions in Opion Valuaion: A Framework for Selecion Chrisian C.P. Wolff, Dennis Bams, Thorsen Lehner In his paper we invesigae he imporance of differen loss funcions when esimaing and evaluaing opion pricing models. Our analysis shows ha i is imporan o ake ino accoun parameer uncerainy, since his leads o uncerainy in he prediced opion price. We illusrae he effec on he ou-of-sample pricing errors in an applicaion of he ad-hoc Black-Scholes model o DAX index opions. We confirm he empirical resuls of Chrisoffersen and Jacobs (004) and find srong evidence for heir conjecure ha he squared pricing error crierion may serve as a general purpose loss funcion in opion valuaion applicaions. A he same ime, we provide a firs yardsick o evaluae he adequacy of he loss funcion. This is accomplished hrough a daa-driven mehod o deliver no jus a poin esimae of he Roo Mean Squared Pricing Error (RMSE), bu a disribuion. Keywords: Opion pricing, loss funcions, esimaion risk, GARCH, implied volailiy JEL Classificaion: *Corresponding Auhor s Address: G1 Tel. : ; Fax : address: Chrisian.wolff@uni.lu The opinions and resuls menioned in his paper do no reflec he posiion of he Insiuion. The LSF Research Working Paper Series is available online: hp:// Papers/008 For ediorial correspondence, please conac: caroline.herfroy@uni.lu Universiy of Luxembourg Faculy of Law, Economics and Finance Luxembourg School of Finance 4 Rue Alber Borschee L-1511 Luxembourg

2 Loss Funcions in Opion Valuaion: A Framework for Selecion Dennis Bams, Thorsen Lehner Deparmen of Finance, Maasrich Universiy, P.O. Box 616, 600 MD Maasrich, The Neherlands and Chrisian C.P. Wolff Luxembourg School of Finance, Universiy of Luxembourg, 4, rue Alber Borschee, L-146 Luxembourg This version: Ocober 008 Absrac In his paper we invesigae he imporance of differen loss funcions when esimaing and evaluaing opion pricing models. Our analysis shows ha i is imporan o ake ino accoun parameer uncerainy, since his leads o uncerainy in he prediced opion price. We illusrae he effec on he ou-of-sample pricing errors in an applicaion of he ad-hoc Black-Scholes model o DAX index opions. We confirm he empirical resuls of Chrisoffersen and Jacobs (004) and find srong evidence for heir conjecure ha he squared pricing error crierion may serve as a general purpose loss funcion in opion valuaion applicaions. A he same ime, we provide a firs yardsick o evaluae he adequacy of he loss funcion. This is accomplished hrough a daa-driven mehod o deliver no jus a poin esimae of he Roo Mean Squared Pricing Error (RMSE), bu a disribuion. Keywords: JEL code: opion pricing, loss funcions, esimaion risk, GARCH, implied volailiy G1 *Corresponding auhor. Tel.: ; fax address: chrisian.wolff@uni.lu.

3 1. Inroducion The adequacy of an opion-pricing model is ypically evaluaed in an ou-of-sample pricing exercise. We naurally prefer he mehod ha minimizes he price differences o he observed marke prices. However, he choice of he paricular loss funcion for he in-sample esimaion and he ou-of-sample evaluaion influences he resul of ha model selecion process. Chrisoffersen and Jacobs (004) show ha he evaluaion loss can be minimized by aking he same loss funcion for in-sample esimaion and ou-ofsample evaluaion. In conras, empirical researchers are ofen inconsisen in heir choice of he loss funcions. They do no align he esimaion and evaluaion loss funcions and herefore he resuls of hese sudies may be misguiding. In he saisics lieraure i was already argued ha he choice of he loss funcion is par of he specificaion of he saisical model under consideraion (e.g. Engle (1993)). Therefore, i may happen ha a misspecified model ouperforms a correcly specified model, if differen loss funcions in esimaion and evaluaion are used. The majoriy of empirical opion valuaion sudies use differen loss funcions a he esimaion and evaluaion sages; examples are Huchinson e al. (1994), Bakshi e al. (1997), Chernov and Ghysels (000), Heson and Nandi (000) and Pan (00). The resuls of hese sudies regarding model selecion are herefore quesionable. In conras, Dumas e al. (1998) (DFW) and Lehner (003) esed he ou-ofsample performance of heir model using idenical loss funcions in he esimaion and evaluaion sages. While Chrisoffersen and Jacobs (004) show he imporance of he loss funcion in opion valuaion, hey do no recommend one paricular loss funcion. However, he paricular loss funcion used in heir empirical analysis characerizes he model specificaion under consideraion. Therefore, i is sill possible ha even if he loss funcions are aligned, a misspecified model ouperforms a correcly specified model when he inappropriae loss funcion is used. They correcly sugges ha he alignmen is more a ruleof-humb han a general heorem and ha he usefulness has o be evaluaed in empirical work. The general problem wih loss funcions is ha he choice of a paricular one is heavily subjecive and deermined by he user of he opion valuaion model. Depending on he paricular purpose of he model, like hedging, speculaing, or marke making, one or he oher loss funcion is preferred. Using differen

4 loss funcions, he user pus more or less weigh on he correc pricing of opions wih differen moneyness. In opion valuaion, no only he pricing model plays an imporan role, bu also he parameer values of hese pricing models. Parameers are usually esimaed based upon hisorical daa. When a paricular phenomenon is no presen in he hisorical daa, he parameers of he disribuion funcion ha are inended o accoun for he phenomenon are esimaed wih considerable uncerainy. Uncerainy in he parameer esimaes leads o uncerainy in he forecased fuure price process and, hence, uncerainy in he ou-of-sample RMSE. We will show ha i is imporan o ake ino accoun esimaion risk. Esimaion risk refers o he fac ha poin esimaes of parameers, resuling from an esimaion procedure, do no necessarily correspond o he underlying rue parameers. There is sill uncerainy abou hese rue values. The rade-off beween he RMSE associaed wih he parameer esimaes and he uncerainy embedded in repored pricing errors plays an imporan role here. The aim of his paper is o provide an empirical selecion approach o arrive a he mos suiable loss funcion for a given daa se and given he purpose of he model. A relaed mehod was proposed by Bams e al. (005) in order o evaluae Value-a-Risk models and in his paper we apply similar logic o he problem of loss funcion selecion in an opion valuaion conex. In our view, such an approach should deal wih uncerainy in he repored ou-of-sample pricing errors ha sems from parameer uncerainy. In he nex secion we se up he economeric framework. We explain our esing procedure using a sandard opion pricing model, he so-called ad-hoc Black-Scholes model. In secion 3 we describe he daa, secion 4 provides a descripion of our opion pricing procedure, and he empirical resuls for he sandard model are presened in secion 5. In secion 6 we demonsrae ha he resuls are insensiive o he choice of he underlying opion pricing model and replicae he analysis for a more sophisicaed GARCH opion pricing model. Finally, secion 7 provides conclusions and suggesions for fuure research. 3

5 . Economeric framework.1 Opion Model For he empirical analysis, we firs use an alernaive o he prominen ad-hoc Black-Scholes model of Dumas, Fleming and Whaley (1998) provided by Derman (1999). 1 We allow each opion o have is own Black-Scholes implied volailiy depending on he exercise price K and ime o mauriy T. The sample of alernaive values for ime-o-mauriy and he exercise price (and hence also moneyness) was spli in NT ime-o-mauriy values and N M alernaive moneyness values. The following funcional form for he opions implied volailiy will be applied in he remainder of he paper: IV = ω ω 0 + ω1m i + ω M i + ω3t j + ω 4T j + 5M it j i = 1, K, N M j = 1, K, N T, (1) where IV denoes he implied volailiy for a call opion wih moneyness M i and ime-o-mauriy For every exercise price and mauriy we can compue he implied volailiy and derive opion prices using he Black-Scholes model. Wihou loss of generaliy we focus on he price of a call opion, as well as on he goal o arrive a a model ha bes forecass ou-of-sample call prices. For a call opion wih moneyness M i and ime-o-mauriy T j he price of a call opion is defined as: ( IV, M i, T j ; p) i = 1, K, N M j = 1, K NT c = BS, where BS( IV M, T p), denoes he heoreical call price according o he Black-Scholes formula, c i j; is he observed call price. Wih = ( ω, K ω ) esimaed. 0, 5 T j. p we denoe he vecor of unknown parameers o be (). Loss Funcions Empirical esimaion of he model in equaions (1) and () requires he inclusion of an error erm, or saed oherwise he formulaion of a loss funcion. We propose and compare hree alernaive loss 1 In a laer secion, we generalize he analysis using a srucural model. 4

6 funcions which are based on RMSE o esimae he parameers of he ad-hoc Black-Scholes. The alernaive loss funcions read: N M N 1 T 1. he implied volailiy error loss funcion: L = I ( i, j ) ( IVˆ IV ) 1 N in i= 1 j= 1, wih 1 associaed error erm equal o: η = I Vˆ IV, i = 1, K, N j = 1, K, N M T N M N 1 T. he squared pricing error loss funcion: L = I( i, j ) ( cˆ c ) in N i= 1 j= 1, wih associaed error erm equal o: η = c ˆ c, i = 1, K, N j = 1, K, N M T N M N T 3. he squared relaive pricing error loss funcion L = I ( i, j ) 3 1 N in i= 1 j= 1 cˆ c c, wih 3 associaed error erm: η cˆ c = c, i = 1, K, N j = 1, K, N M T The loss funcion include ( i j) moneyness M i and ime-o-mauriy daabase includes = M T in N I( i j) I, which is he indicaor funcion, ha is equal o one if he combinaion of N N i= 1 j= 1 T j is available in he daabase, and zero oherwise. The in-sample, observaions. A paricular loss funcion, denoed wih L( η in ; p), is minimized in order o arrive a parameer esimaes for he vecor p. Le pˆ denoe he vecor of parameer esimaes and le in ηˆ denoe he vecor of associaed in sample residuals. In our case, hence, hree alernaive loss funcions are aken ino consideraion o arrive a parameer esimaes..3 Ou-of-Sample Applicaion The model may subsequenly be applied for purposes such as predicing opion prices a a paricular poin in ime. Bu also oher ou-of-sample objecives may apply. In our applicaion we evaluae he ou-of- 5

7 sample performance of he model agains he same hree loss funcions ha were applied for he in-sample model esimaion. The applicable observed moneyness and ime-o-mauriy ogeher wih he parameer esimaes pˆ resul in prediced opion prices. The ou-of-sample predicive power of he model can be assessed by he size of he loss funcion applied o he ou-of-sample daa evaluaed a he in-sample parameer esimaes, pˆ. Denoe he vecor of ou-of-sample residuals by ou εˆ. As such a hree by hree grid emerges in which for hree possible in-sample loss funcion crieria, hree ou-of-sample loss funcions are evaluaed. The oucome of a loss funcion evaluaion is a RMSE. We argue ha in-sample selecion of a loss funcion o arrive a parameer esimaes should be based on he ou-of-sample performance of he loss funcion disribuion under he alernaive in-sample selecions. Below we propose a measure ha simplifies he comparison of he alernaive loss funcion disribuions..4 Evaluaion Crierion For predicion purposes, he usual line of reasoning would be o prefer he in-sample loss funcion selecion and subsequen opimizaion rouine ha resuls in he lowes ou-of-sample loss funcion. We argue ha his line of reasoning is incomplee, in ha i does no ake accoun of parameer uncerainy. We propose a simulaion approach in order o derive a whole disribuion of ou-of-sample pricing errors by incorporaing he uncerainy in he parameer esimaes from he calibraion. In his paper we apply sandard boosrapping echniques o arrive a a disribuion for he ou-of-sample loss funcion. Consider Q boosraps from he in-sample residuals, resuling in Q boosrapped values for he parameer esimaes, denoed by ( ) ( Q ) ˆ1 K ˆ. The associaed ou-of sample vecors of residuals are denoed by p,, p ( ) ( Q ) ˆ1 K ˆ, and he resuling values for he associaed ou-of-sample loss funcion are given by ε L,, ε, K, Lˆ ( ) ( Q ). The resul of boosrapping from an in-sample loss funcion hence resuls in an RMSE ˆ1 disribuion funcion associaed wih a paricular ou-of-sample loss funcion. For a deailed descripion of he boosrapping procedure, we refer o he Appendix o his paper. 6

8 By comparison of he differen ou-of-sample RMSE disribuion funcions, he in-sample loss funcion crierion ha yields he bes resul can be deermined. In order o make RMSE disribuions comparable, we define a crierion, he ASC (Asymmeric Selecion Crierion) ha summarizes he ou-of-sample loss disribuion ino a single saisic: ( L) 1 ( L) F,5% ( L) ( L) F ( L) ASC 1 avg = (3) avg avg 1 97,5% where L denoes he vecor of boosrapped RMSEs, denoed wih Lˆ1 ( ), K, Lˆ ( Q ). The mahemaical 1 1 operaors avg ( ), F ( ) and ( ),5% F deermine he average, he,5% lower bound and he,5% 97,5% upper bound of a vecor. The ASC is essenially a ool o evaluae and compare disribuions of ou-ofsample RMSEs. The ASC has a number of, we hink, useful and inuiive feaures. I uses a lo of informaion abou he disribuion of opion pricing errors. We hink ha he ASC-crierion is an aracive decision-making ool ha should appeal o many individuals, because of he following characerisics: 1. A higher average RMSE resuls in worse performance, as measured by he ASC. This is refleced in he firs erm of equaion (3);. An RMSE below he average RMSE value implies ha he crierion works relaively well for hose cases, which resuls in beer performance as measured by he ASC. The numeraor of he second erm in equaion (3) accouns for his effec; 3. An RMSE above he average RMSE value means ha he crierion works relaively poorly for hose cases, which resuls in greaer uncerainy as measured by he ASC. The denominaor of he second erm in equaion (3) accouns for his effec; 1, which 4. If here is no uncerainy in he parameer esimaes, he measure converges o avg( L) implies ha he degree of performance ha is measured by he ASC is equal o he level associaed wih he average RMSE. 7

9 .5 Experimenal Design The core idea is o deermine he ASC from equaion (3) for a single ou-of-sample loss disribuion a he ime, based upon alernaive in-sample loss funcions. We compare he ASCs for he hree alernaives and he in-sample loss funcion wih he highes ASC is he preferred in-sample loss funcion. I allows us o arrive a bes performance in applying he opion model in an ou-of-sample conex. Given ha a single comparison does no provide srong suppor, we propose a repeiive algorihm in which he daa sample is spli up in a developmen sample ha is used o esimae he model and a holdou sample ha is used o evaluae he performance of he model. A moving window is applied o creae muliple developmen and hold-ou samples. Per sample a comparison of he hree ASCs akes place. The oucome of he enire moving window procedure is summarized as a percenage. The percenage indicaes how frequenly an in-sample loss funcion leads o he highes ou-of-sample ASC (based on coun). 3. Daa We use daily closing DAX 30 index opions and fuures prices for a period from January 000 unil December 000. The raw daa se is direcly obained from he EUREX, European Fuures and Opions Exchange. The marke for DAX index opions and fuures is he mos acive index opions and fuures marke in Europe. Therefore i is an ineresing marke for esing opion pricing models. For index opions he expiraion monhs are he hree neares calendar monhs, he hree following monhs wihin he cycle March, June, Sepember and December, as well as he wo following monhs of he cycle June, December. For index fuures he expiraion monhs are he hree neares calendar monhs wihin he cycle March, June, Sepember and December. The las rading day is he hird Friday of he expiraion monh, if ha is an exchange-rading day; oherwise, he opion expires on he exchangerading day immediaely prior o ha Friday. We exclude opions wih less han one week and more han 5 weeks unil mauriy and opions wih a price of less han Euros o avoid liquidiy-relaed biases and because of less useful informaion on 8

10 volailiies. Insead of using a saic rule and exclude opions wih absolue moneyness of more han 10% (see DFW), we exclude opions wih a daily urnover of less han 10,000 Euros. This rule was applied afer carefully analyzing he paricular daa se (see Lehner (003)). Among ohers DFW argue ha opions wih absolue moneyness of more han 10% are no acively raded and herefore conain no informaion on volailiies. Therefore an obvious soluion is o filer he available opion prices and include all opions ha are acively raded, inside or ouside he 10% absolue moneyness inerval. In paricular, in volaile periods deep ou-of-he-money opions are highly informaive if hey are acively raded. As a resul, each day we use a minimum of 3 and a maximum of 4 differen mauriies for he calibraion. The DAX index calculaion is based on he assumpion ha he cash dividend paymens are reinvesed. Therefore, when we calculae opion prices, heoreically we do no have o adjus he index level for he fac ha he sock price drops on he ex-dividend dae. Bu he cash dividend paymens are axed and he reinvesmen does no fully compensae for he decrease in he sock price. Therefore, in he conversion from e.g. fuures prices o he implied spo rae, we observe empirically a differen implied underlying index level for each mauriy 3. For his reason, we always work wih he underlying index level implied from fuures or opion prices. 4. Opion pricing procedure We use he following procedure for one paricular day o price opions on he following rading day: In our noaion, absolue moneyness is defined as K/F-1, where K is he exercise price and F is he forward price. 3 Since he socks underlying he index porfolio pay dividends, he presen value of expeced fuure dividends is differen for differen lifeimes of he fuures or opions conracs. 9

11 Firs, we compue he implied ineres raes and implied dividend adjused index raes from he observed pu and call opion prices. We are using a modified pu-call pariy regression proposed by Shimko (1993). Pu-call pariy for European opions reads: c j rf j T j ie p = [ X PV ( D )] K (4) where X is he underlying index level a ime, c and p are he observed call and pu closing prices, respecively, wih exercise prices K i and mauriy T j, PV(D j ) denoes he presen value of dividends o be paid from ime of opion valuaion unil he mauriy of he opion conrac and rf j is he coninuously compounded ineres rae ha maches he mauriy of he opion conrac. Therefore we can infer a value for he implied dividend adjused index level for differen mauriies, X -PV(D j ), and he coninuously compounded ineres rae for differen mauriies, rf j. In order o ensure ha he implied dividend adjused index value is a non-increasing funcion of he mauriy of he opion, we occasionally adjus he sandard pu-call pariy regression. Therefore we check o ensure ha he value for X -PV(D j ) is decreasing wih ime o mauriy, T j. Since we use closing prices for he esimaion, one alernaive is o use implied index levels from DAX index fuures prices assuming ha boh markes are closely inegraed. Second, we esimae he parameers of he ad-hoc Black-Scholes model by minimizing he paricular loss funcion (e.g. he difference beween he marke implied volailiies - from daily closing prices - and he implied volailiies of heoreical opion prices for calls and pus prediced by he model). Given reasonable saring values, we price European calls and pus wih exercise price K i and ime o mauriy T j. We repea his procedure wih he usual opimizaion mehod (Newon-Raphson mehod) and obain he parameer esimaes ha minimize he paricular loss funcion. The goodness of fi measure for he opimizaion is he roo mean squared error crierion. Third, having esimaed he parameers in-sample, we urn o ou-of-sample valuaion performance and evaluae how well each day s esimaed models value he raded opions a he end of he following day. We filer he available opion prices according o our crieria for he in-sample calibraion. The fuures marke is he mos liquid marke, and he opions and fuures markes are closely inegraed. Therefore i 10

12 can also be assumed ha he fuures price is more informaive for opion pricing han jus using he value of he index. For every observed fuures closing price, we can derive he implied underlying index level and evaluae he opion. Given a fuures price F j wih ime o mauriy T j, spo-fuures pariy is used o deermine X -PV(D j ) from X PV ( D ) = F (5) j e -rf j T j j where PV(D j ) denoes he presen value of dividends o be paid over he lifeime of he fuures conrac, T j, and rf j is he coninuously compounded ineres rae (he inerpolaed EURIBOR rae) ha maches he mauriy of he fuures conrac (or ime o expiraion of he opion). If a given opion price observaion corresponds o an opion ha expires a he ime of delivery of a fuures conrac, hen he price of he fuures conrac can be used o deermine he quaniy X -PV(D j ) direcly. The mauriies of DAX index opions do no always correspond o he delivery daes of he fuures conracs. In paricular, for index opions he wo following monhs are always expiraion monhs, bu no necessarily delivery monhs for he fuures conrac. When an opion expires on a dae oher han he delivery dae of he fuures conrac, he quaniy X -PV(D j ) is compued from various fuures conracs. Le F 1 be he fuures price for a conrac wih he shores mauriy, T 1 and F and F 3 are he fuures prices for conracs wih he second and hird closes delivery monhs, T and T 3, respecively. Then he expeced fuure rae of dividend paymen d can be compued via spo-fuures pariy by: r3t3 rt log (F3 / F ) d = (6) (T T ) 3 Hence, he quaniy X PV D) X e dt ( = associaed wih he opion ha expires a ime T in he fuure can be compued by 4 X e dt ( ( rf1 d) T1 dt ) = F1 e. (7) 4 See e.g. he appendix in Poeshman (001) for deails. 11

13 This mehod allows for a perfec mach beween he observed opion price and he underlying dividend adjused spo rae. Given he parameer esimaes and he implied dividend-adjused underlying we can calculae opion prices and compare hem o he observed opion prices of raded index opions. 5. Empirical resuls For each rading day of he year 000, we esimaed he model in equaions (1) and () using closing prices of raded opions ha fulfil our crieria. On average 84 opion prices are used for he calibraion and evaluaion of he models, wih a minimum of 6 and a maximum of 155. The model is esimaed hree imes and each ime wih a differen loss funcion. Therefore, we esimae he model by minimizing one loss funcion and deermine he roo mean squared error (RMSE) according o he loss funcion used. A he same ime, we also deermine he RMSE according o he oher wo loss funcions. Table 1, Panel A, repors he average RMSEs over he whole period (January 1 s, 000-December 9 h, 000). The diagonal elemens of he able correspond o he RMSE from using he same loss funcion a he esimaion and evaluaion sages. The off-diagonal enries repor he RMSE from using differen loss funcions in- and ou-of-sample. As expeced, calibraing he model using a paricular loss funcion also resuls in a minimum RMSE for ha paricular loss funcion. [Table 1] In a nex sep, we use he model calibraed on one rading day o price all raded opions 1, 5 and 0 days ou-of-sample. Again, we esimae he model using hree alernaive loss funcions, and also evaluae he model using he same hree loss funcions. This resuls in a hree-by-hree able, ha allows for a comparison of he effec of in-sample loss funcion selecion on he ou-of-sample performance of a paricular loss funcion. Table 1, Panel B repors he average RMSE for he ou-of-sample loss funcions. Sill, as expeced, using he same loss funcion a he esimaion and evaluaion sages minimizes he pricing error a he evaluaion sage. The resuls of he moving window are summarized in parenheses. 1

14 Ineresingly, while he diagonal sill displays he highes coun on preferred in-sample/ou-of-sample combinaion, he numbers sugges ha his preference is miigaed as he ou-of-sample period becomes longer. We confirm he resuls of Chrisoffersen and Jacobs (004)). Based upon average RMSE he preferred in-sample loss funcion crierion is he same as he corresponding ou-of-sample loss funcion. The resuls in able 1 do no ake ino accoun he effec of parameer uncerainy. In able hese are included. Because of he low number of observaions per (daily) cross-secion, we decided o pool he residuals from he las 1 rading days for boosrapping purposes. Furhermore, he cross-secional dependence of moneyness and mauriy moivaed a block boosrap. In he laer we define a wodimensional grid which is based upon moneyness inervals ( M 0.9, 0.9 < M 0.96, 0.96 < M 1.00,1.00 < M 1.04, M 1. 04) and ime-o-mauriy inervals ( T 4, 4 < T 84, T > 84 ). The residuals arising from he pas 1 rading days are divided over he 15 resuling wo-dimensional grid. Boosrapping occurs by random sampling from he residuals in he relevan cell of he wo-dimensional grid. Table, panels A and B, repors he average values of he ASC measure (see equaion (3)). In addiion, he able repors he resuls from he moving window analysis. Based on coun, we have deermined how ofen a paricular loss funcion is preferable over he ohers. [Table ] The resuls show ha our crierion can serve as a useful ool o evaluae differen loss funcions for he purpose of opion valuaion. From he analysis, we see ha he conjecure of Chrisoffersen and Jacobs (004) can be confirmed: from he in-sample analysis, as well as from he ou-of-sample analysis. However, as he ou-of-sample period increases he squared error loss funcion becomes a beer alernaive in all cases. This is refleced by he number of imes he ASC in he moving window exercise shows he highes ASC. A shorer horizons, he squared error loss funcion already is preferable over he 13

15 IV RMSE loss funcion. A longer horizons, he squared error loss funcion is preferable in virually all cases. Only in case of he relaive RMSE loss funcion ou-of-sample crierion, he performance of he squared error loss funcion is similar o he relaive RMSE loss funcion. 6. Comparison wih a GARCH Opion Pricing Model So far he empirical analysis has focused on esing he impac of differen loss funcions using an ad-hoc Black-Scholes model as he underlying opion pricing model. We now consider he popular GARCH opion pricing models and invesigae wheher he previous resuls can be generalized for more srucural models. To documen his, we replicae he analysis for a parsimonious GARCH specificaion, which only conains volailiy clusering and a leverage effec. In a Gaussian discree-ime economy he value of he index a ime, X, can be assumed o be governed by he following dynamics (see e.g. Duan (1995)): e r X = ln X 1 + d = µ + σ e Ω 1 ~ N(0,1) under probabiliy measure P, (8) ln ( σ ) ω + α ln( σ ) + β( e γ e ) = where he condiional mean is defined as µ = +, d is he dividend yield of he index porfolio, rf λσ rf is he risk-free rae, λ is he price of risk and Ω -1 is he informaion se in period -1 and he combinaion of β, γ, b and δ capures he leverage effec. Duan (1995) shows ha under he Local Risk Neural Valuaion Relaionship (LRNVR) he condiional variance remains unchanged, bu under he pricing measure Q he condiional expecaion of r is equal o he risk free rae rf : 14

16 [ r ) ] exp( rf ) Q E exp( Ω 1 =, (9) Therefore, he LRNVR ransforms he physical reurn process o a risk-neural dynamic. The risk-neural Gaussian GARCH process reads 5 : r = rf 1 σ +σ ε ε Ω 1 ~ N(0,1) under risk-neuralized probabiliy measure Q, (10) ln ( σ ) ω + α ln( σ ) + β ε λ γ ( ε λ) ( ) = where he erm gives addiional conrol for he condiional mean. In Equaion (1), ε is no 1 σ necessarily normal, bu o include he Black-Scholes model as a special case we ypically assume ha is a Gaussian random variable. The uncondiional volailiy level is equal ε o ω + β E exp [( ε λ γ ( ε λ) ) ] 1 α and can be evaluaed numerically. The parameer α measures he persisence of he variance process. The locally risk-neural valuaion relaionship ensures ha under he risk neural measure Q, he volailiy process saisfies Var Q P [ r Ω ] = Var [ r Ω ] = σ 1 1. (11) A European call opion wih exercise price K i and mauriy T j has a ime price equal o: c Q ( rf T ) E max( X K,0) [ Ω ] = exp j i 1 (1) For his kind of derivaive valuaion models wih a high degree of pah dependence, compuaionally demanding Mone Carlo simulaions are commonly used for valuing derivaive securiies. We use he proposed simulaion adjusmen mehod, he empirical maringale simulaion (EMS) of Duan and Simonao (1998), which has been shown o subsanially accelerae he convergence of Mone Carlo price 5 This ype of GARCH specificaion is sufficien for he purpose a hand. Neverheless, exensions of he model can improve he pricing performance (see e.g. Lehner (003)). 15

17 esimaes and o reduce he so called simulaion error. As saring values for he calibraion, we make use of he ime-series esimaes from he equivalen ime-series GARCH model using approximaely hree years (75 rading days) of hisorical reurns. In addiion, we use wo ime-series parameer esimaes for he opion calibraion: he long run volailiy σ equal o he relaively sable 3-year hisorical sandard deviaion and he risk premium parameer λ. Using he ime-series esimaes for he price of risk is common pracice, bu our variance argeing approach is differen o he one used in oher sudies (e.g. Heson and Nandi (000)). They perform a consrained calibraion in which he parameers λ and he local volailiy are resriced o he ime-series GARCH-esimaes. In conras, we esimae he local volailiy ogeher wih he oher parameers. Fixing he saionary volailiy level sabilizes he esimaion process dramaically wihou influencing he pricing performance of he model. In paricular, in recen years here is some suppor for he hypohesis ha he informaion provided by implied volailiies from daily opion prices is more relevan in forecasing volailiy han he volailiy informaion provided by hisorical reurns (e.g. Blair e al. (001)). Therefore, an esimae of he local volailiy from opion prices direcly migh be more informaive han he ime-series esimae. I is also ineresing o noe ha he saionary volailiy level is known o be unsable over ime when esimaed from opion prices; a fac ha is ypically no discussed in empirical opion pricing sudies. In he following, we precisely replicae he empirical analysis described in he previous secions using he GARCH model as he underlying opion pricing models. The in- and ou-of-sample RMSEs of he model are presened in Table 3, Panels A and B. In general, he resuls confirm he findings of he previous secion. Therefore, using he same loss funcions a he esimaion and evaluaion sages minimizes he in- -sample pricing error a he evaluaion sage (in bold). The ou-of-sample evaluaion shows he same findings as for he Ad Hoc Black Scholes model. Alignmen of he in-sample and ou-ofsample loss crierion yields he bes resuls in erms of average RMSE. The moving window analysis suggess ha a longer horizons he performance of he squared error loss funcion increases relaive o he alernaives. 16

18 [Tables 3 and 4] Table 4 presens he in- and ou-of-sample resuls for he relaive uncerainy in he prediced pricing error. Again, he resuls sugges ha i is imporan o consider esimaion risk, defined as he uncerainy ha poin esimaes of parameers, resuling from an esimaion procedure, do no necessarily correspond o he underlying rue parameers. Therefore, when looking a he ASC of he pricing error disribuions, again he findings of he previous secion can be confirmed. Using he squared error loss funcion a he esimaion sage minimizes he uncerainy in he loss funcion a he evaluaion sage regardless which loss funcion is used o evaluae he model. This holds in paricular a longer ou-of-sample horizons. Therefore, he resuls from he previous secion are robus and seem o be independen of he underlying opion pricing model under consideraion. 7. Conclusions This paper invesigaes he imporan empirical issue concerning model selecion in an opion valuaion conex. So far, he empirical lieraure has mainly focused on he relaive performance of various opion valuaion models. The role and he imporance of he loss funcions a he esimaion and evaluaion sages have been overlooked frequenly. We propose a daa-driven mehod ha allows us o evaluae he relaive performance of differen loss funcions. The resuls in his paper are far-reaching for he opion valuaion lieraure, because researchers are ypically inconsisen in heir choice of he loss funcions and resuls are herefore incomparable. Our approach allows us o promoe a paricular loss funcion. Using he squared pricing error crierion a he esimaion sage minimizes he uncerainy in he in- and ou-of-sample pricing errors, regardless which loss funcion is used a he evaluaion sage. We confirm he empirical resuls of Chrisoffersen and Jacobs (004) and find srong evidence for heir conjecure ha he squared pricing error crierion may serve as a general purpose loss funcion in opion valuaion applicaions. Of course, he choice of he loss 17

19 funcion is subjecive, bu he framework proposed in his paper offers a firs plausible yardsick o selec he mos appropriae loss funcion. Noe ha we do no view he ASC as he key innovaion of he paper, bu he whole ou-of-sample framework. In his conex he ASC is one plausible way of looking a he disribuions of RMSEs. We envisage fuure applicaions of our framework no only in he area of pricing, bu also in he conex of hedging decisions and risk managemen. 18

20 Appendix: Non-parameric block boosrapping mehodology Assume ha we have a se of observed marke prices of call opions, denoed wih M i and ime-o-mauriy ĉ wih moneyness T j ( i = 1, K, N, j = 1, K, N ) and le c denoe he heoreical call price M according o a paricular opion pricing model. In he following we describe he block boosrapping mehodology for he example of he squared pricing error loss funcion. The procedures for he oher loss funcions are, of course, similar. T The loss funcion is minimized in order o arrive a parameer esimaes for he vecor p, and reads: L = 1 N in N M N T ( cˆ c ) i= 1 j= 1 (A.1) Le pˆ denoe he vecor of parameer esimaes and le residuals. The error erm is equal o: in ηˆ denoe he vecor of associaed in sample η in = c ˆ c, i = 1, K, N j = 1, K, N M T (A.) Associaed wih he vecor of parameer esimaes is a confidence inerval. We sugges ha considering he full parameer disribuion and applying i in an ou-of-sample applicaion provides useful informaion. Boosrapping is in essence he generaion of independen variables from he parameer esimaes pˆ and a random sample of he in-sample residuals in ηˆ. For each sample ha is generaed he associaed parameer esimaes are deermined. Repeaing his exercise many imes resuls in a disribuion or confidence inerval associaed wih he original parameer esimaes. 19

21 The measuremen of parameer uncerainy urns ou o be nonrivial, for wo reasons. Firs, he small sample - around 100 observaions for one day - does no allow he use of parameric mehods. Second, he cross secional dependence across moneyness and mauriies does no allow he use sandard sampling echniques. To cope wih hese issues we rely on non-parameric block boosrapping echniques. This includes he following: 1. To block-boosrap for a paricular observaion dae, he boosrapping sample includes all residuals of daily esimaions of he las rading monh (abou 1 rading days). Thus, he boosrapping sample consiss no jus of he residuals for a paricular day, bu is enlarged by inclusion of all residuals of he las rading monh.. Residuals are grouped in a wo-dimensional grid which is based upon moneyness inervals and ime-o-mauriy inervals: M 0.9, 0.9 < M 0.96, 0.96 < M 1.00,1.00 < M 1.04, M 1.04 and T 4, 4 < T 84, T > 84. The residuals arising from he pas 1 rading days are divided over he 15 sub-samples of he resuling wo-dimensional grid. Boosrapping occurs by random sampling from he residuals in he relevan sub-samples of he wo-dimensional grid. Consider Q boosraps from he in-sample residuals, resuling in Q boosrapped values for he parameer esimaes, denoed by ( ) ( Q ) ˆ1 K ˆ. The associaed vecors of ou-of-sample residuals are denoed by p,, p ( ) ( Q) ˆ1 K ˆ, and he resuling values for he associaed ou-of-sample loss funcion are given by ε L,, ε, K, Lˆ ( ) ( Q ). The resul of boosrapping for an in-sample loss funcion hus resuls in an RMSE ˆ1 disribuion funcion associaed wih a paricular ou-of-sample loss funcion. 0

22 References Bakshi, C., C. Cao, Z. Chen Empirical performance of alernaive opion pricing models. Journal of Finance Bams, D., T. Lehner, C.C.P. Wolff 005. An Evaluaion Framework for Alernaive VaR Models. Journal of Inernaional Money and Finance Black, F., M. Scholes The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy Blair, B.J., S.-H. Poon, S.J. Taylor 001. Forecasing S&P100 volailiy, The incremenal informaion conen of implied volailiies and high frequency index reurns. Journal of Economerics Chernov, M., E. Ghysels 000. A sudy owards a unified approach o he join esimaion of objecive and risk-neural measures for he purpose of opion valuaion. Journal of Financial Economics Chrisoffersen, P., K. Jacobs 004. The Imporance of he Loss Funcion in Opion Valuaion. Journal of Financial Economics Derman, E Regimes of Volailiy. RISK Duan, J.-C The GARCH Opion Pricing Model. Mahemaical Finance Duan, J.-C Cracking he Smile. RISK Duan, J.-C., J.-G. Simonao Empirical Maringale Simulaion for Asse Prices. Managemen Science Dumas, B., J. Fleming, R.E. Whaley Implied Volailiy Funcions: Empirical Tess. The Journal of Finance Engle, R.F A commen on Hendry and Clemens on he limiaions of comparing mean squared forecas errors. Journal of Forecasing Heson, S.L., S. Nandi 000. A Closed-Form GARCH Opion Valuaion Model. The Review of Financial Sudies

23 Huchinson, J., A. Lo, T. Poggio A nonparameric approach o pricing and hedging derivaive securiies via learning neworks. Journal of Finance Lehner, T Explaining Smiles: GARCH Opion Pricing wih Condiional Lepokurosis and Skewness. Journal of Derivaives 10(3) Pan, J. 00. How imporan is he correlaion beween reurns and volailiy in a sochasic volailiy model? Empirical evidence from pricing and hedging in he S&P 500 index opions marke. Journal of Financial Economics Poeshman, A.M Underreacion, Overreacion, and Increasing Misreacion o Informaion in he Opion Marke. Journal of Finance 56(3) Shimko, D Bounds of Probabiliy. RISK

24 Table 1: Roo Mean Squared Errors, Ad-hoc Black-Scholes Model Panel A: In-Sample RMSE Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE 0.7 (100%) 3.6 (0%) 7.7 (0%) RMSE 1.0 (0%) 1.91 (100%) 8.1 (0%) % RMSE 1.9 (0%) 8.0 (0%).6 (100%) Panel B: Ou-of-Sample Average RMSE Esimaion Days Evaluaion IV RMSE RMSE % RMSE IV RMSE 1 1. (50%) RMSE (46%) 7.8 (16%) 6.76 (80%) 13.9 (15%) 11.4 (3%) % RMSE 1.1 (4%) (4%) 7.6 (6%) IV RMSE (47%) RMSE (48%) (10%) (84%) 17.5 (13%) 15.6 (38%) % RMSE 5. (5%) (6%) 13.0 (49%) IV RMSE 0.7 (46%) 1.57 (9%) 5.0 (8%) RMSE 0.7 (5%) 0.53 (90%).1 (45%) % RMSE (%) 3.89 (1%) 1.9 (47%) Noes. The able presens he average in- and ou-of-sample Roo Mean Squared Errors from he daily esimaion and evaluaion of he ad-hoc Black-Scholes model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on a following day (1, 5 and 10 days ou-of-sample). A he evaluaion sage, we also compue he resuls for 3

25 he alernaive loss funcions. The able presens he differences in implied volailiies in percenages (IV RMSE), he absolue price differences in Euros ( RMSE) and he relaive pricing errors in percenages (% RMSE), respecively. To arrive a more sensible unis we have muliplied he enries in boh he IV RMSE and in he % RMSE by 100. For he combinaions where he same loss funcion was used a he evaluaion sage, we show in parenheses how frequenly he paricular combinaion resuled in he lowes RMSE. 4

26 Table : Asymmeric Selecion Crierion, Ad-hoc Black-Scholes Model Panel A: In-Sample ASC Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE (81%) 0.34 (18%) 1.9 (10%) RMSE (18%) 0.55 (79%) (%) % RMSE 3.36 (1%) 0.1 (3%) 1.38 (68%) Panel B: Ou-of-Sample Average ASC Esimaion Days Evaluaion IV RMSE RMSE % RMSE IV RMSE (5%) RMSE (44%) % RMSE (4%) 0.15 (5%) 0.4 (68%) 0.09 (7%) (16%) (30%) (54%) IV RMSE (43%) RMSE (50%) % RMSE (7%) 0.17 (16%) 0.3 (79%) 0.08 (5%) 1.71 (13%) 14.1 (39%) (48%) IV RMSE (45%) RMSE (5%) 0.13 (13%) 0.9 (81%) (3%) (49%) % RMSE (3%) 0.11 (6%) (48%) Noes. The able presens he average values of he Asymmeric Selecion Crierion (ASC) for he in- and ou-of-sample Roo Mean Squared Errors from he daily esimaion and evaluaion of he ad-hoc Black-Scholes model. Each day he model is esimaed using one paricular loss funcion and evaluaed using he hree differen loss funcions. The ou-of-sample evaluaion is 5

27 performed over differen ime horizons. The able presens he figures for he asymmeric selecion crierion (ASC) defined in Equaion (3). For he combinaions where he same loss funcion was used a he evaluaion sage, we show in parenheses how frequenly one paricular combinaion resuled in he highes ASC. 6

28 Table 3: Roo Mean Squared Errors, GARCH Opion Pricing Model Panel A: In-Sample RMSE Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE 0.8 (100%) 3.6 (0%) 7. (0%) RMSE 0.9 (0%).34 (100%) 6.9 (0%) % RMSE 1.7 (0%) 7.59 (0%) 3.9 (100%) Panel B: Ou-of-Sample Average RMSE Esimaion Days Evaluaion IV RMSE RMSE % RMSE IV RMSE RMSE 1 1. (5%) (45%) 7.33 (18%) 6.05 (8%) 1.8 (1%) 9.3 (9%) % RMSE (3%) 9.83 (0%) 7. (59%) IV RMSE RMSE (47%) (5%) 14.0 (15%) 1.46 (85%) 16. (11%) 14.4 (4%) % RMSE 5.3 (1%) (1%) 1.8 (47%) IV RMSE 0.5 (46%) 0.98 (9%) 3.9 (5%) RMSE 0.4 (46%) (89%) 0. (45%) % RMSE (8%) 4.3 (%) 19.9 (50%) Noes. The able presens he average in- and ou-of-sample Roo Mean Squared Errors from he daily esimaion and evaluaion of he GARCH Opion Pricing model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on a following day (1, 5 and 10 days ou-of-sample). A he evaluaion sage, we also compue he resuls for 7

29 he alernaive loss funcions. The able presens he differences in implied volailiies in percenages (IV RMSE), he absolue price differences in Euros ( RMSE) and he relaive pricing errors in percenages (% RMSE), respecively. To arrive a more sensible unis we have muliplied he enries in boh he IV RMSE and in he % RMSE by 100. For he combinaions where he same loss funcion was used a he evaluaion sage, we show in parenheses how frequenly he paricular combinaion resuled in he lowes RMSE. 8

30 Table 4: Asymmeric Selecion Crierion, GARCH Opion Pricing Model Panel A: In-Sample ASC Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE (78%) 0.39 (10%) (9%) RMSE 89.8 (%) 0.6 (89%).98 (3%) % RMSE (0%) 0.4 (1%) 5.09 (59%) Panel B: Ou-of-Sample Average ASC Esimaion Days Evaluaion IV RMSE RMSE % RMSE IV RMSE (49%) 0.3 (17%) 17.9 (18%) RMSE (45%) 0.33 (78%) 19.5 (40%) % RMSE (6%) 0.16 (5%) 0.05 (4%) IV RMSE (46%) 0.4 (15%) (1%) RMSE (46%) 0.35 (79%) (43%) % RMSE (8%) 0.19 (6%) (45%) IV RMSE (44%) 0.5 (7%) (18%) RMSE (51%) 0.39 (87%) (4%) % RMSE (5%) 0.1 (6%) 19. (40%) Noes. The able presens he average values of he Asymmeric Selecion Crierion (ASC) for he in- and ou-of-sample Roo Mean Squared Errors from he daily esimaion and evaluaion of he GARCH Opion Pricing model. Each day he model is esimaed using one paricular loss 9

31 funcion and evaluaed using he hree differen loss funcions. The ou-of-sample evaluaion is performed over differen ime horizons. The able presens he figures for he asymmeric selecion crierion (ASC) defined in Equaion (3). For he combinaions where he same loss funcion was used a he evaluaion sage, we show in parenheses how frequenly one paricular combinaion resuled in he highes ASC. 30