Loss Functions in Option Valuation: A Framework for Model Selection

Transcription

1 Loss Funcions in Opion Valuaion: A Framework for Model Selecion Dennis Bams, Thorsen Lehner, Chrisian C.P. Wolff * Limburg Insiue of Financial Economics (LIFE), Maasrich Universiy, P.O. Box 616, 600 MD Maasrich, The Neherlands 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. Our empirical resuls sugges ha differen loss funcions lead o uncerainy abou he pricing error iself. A he same ime, i provides 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 pricing error, bu a confidence inerval. Keywords: JEL code: opion pricing, loss funcions, esimaion risk, GARCH, implied volailiy G1 * Corresponding auhor. Tel.: ; fax address: c.wolff@berfin.unimaas.nl.

2 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 (00) show ha he evaluaion loss can be minimized by aking he same loss funcion for in-sample esimaion and ou-of-sample evaluaion. In conras, empirical researchers are inconsisen in heir choice of he loss funcions. They do no align he esimaion and evaluaion loss funcion and herefore he resuls of hese sudies may be misguiding. For example he saisics lieraure 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 may ouperform 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 sage; examples are among ohers 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-of-sample performance of heir model using idenical loss funcions in he esimaion and evaluaion sage. While Chrisoffersen and Jacobs (00) 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 he empirical analysis characerizes he model specificaion under consideraion. Therefore, i is sill possible ha even if he loss funcions are aligned, a misspecified model may ouperform a correcly specified model when he inappropriae loss funcion is used. They correcly sugges ha he alignmen is more a rule-of-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 loss funcions, he user pus more or less weigh on he correc pricing of opions

3 wih differen moneyness. The purpose of his paper is o provide an objecive measure o evaluae differen loss funcions. 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, as refleced by he sandard errors of he parameer esimaes. Uncerainy in he parameer esimaes leads o uncerainy in he forecased fuure price process and, hence, uncerainy in he ou-of-sample pricing error. 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 average pricing error and he precision of repored pricing errors provides a firs yardsick o evaluae he adequacy of a paricular loss funcion or opion pricing model. 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. The mehod was proposed by Bams e al. (00) in order o evaluae Value-a-Risk models and in his paper we apply i o he problem of model 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 and secion 4 provides he empirical resuls for he sandard model. In secion 5 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 6 concludes.. Economeric framework 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 1 In a laer secion, we generalize he analysis using a srucural model. 3

4 opion o have is own Black-Scholes implied volailiy depending on he exercise price K and ime o mauriy T and use he following funcional form for he opions implied volailiy: IV ω 0 + ω1m + ω M + ω 3T j + ω 4T j + ω 5 = M T, (1) j where IV denoes he implied volailiy and M he moneyness of an opion for he i-h exercise price and j-h mauriy. T j denoes he ime o mauriy of an opion for he j-h mauriy. For every exercise price and mauriy we can compue he implied volailiy and derive opion prices using he Black-Scholes model. We esimae he parameers of he ad-hoc Black-Scholes model by minimizing he n li 1 paricular loss funcion: (1) he implied volailiy error IVMSE = ( IVˆ IV ) N i = 1 j = 1, n li 1 () he absolue pricing error MSE = ( cˆ c ) N i = 1 j = 1 and (3) he relaive pricing error n li 1 cˆ c % MSE =. ĉ denoes he heoreical call price, c is he observed N i = 1 j = 1 c call price, IV ˆ is he implied volailiy associaed wih he heoreical call price, IV is he implied volailiy associaed wih he observed call price, n is he number of exercise prices, l i represens he number of prices available ou of all mauriies for he i-h exercise price and N is he oal number of conracs included on he paricular rading day. Le p denoe he vecor of unknown parameers ω i and le L ( p) denoe he associaed likelihood funcion. The covariance marix of parameer esimaes follows: ( p) 1 L C = p p' () p= pˆ We inroduce he following noaion: Moneyness is defines as forward price of an opion for he i-h exercise price and j-h mauriy, respecively. K i, where Ki is he exercise price and F j is he F j 4

5 where pˆ denoes he vecor of maximum-likelihood poin esimaes for he unknown parameers. In he ou-of-sample algorihm ha we propose, his covariance marix plays a crucial role since i reflecs parameer uncerainy. Parameer uncerainy may be incorporaed by sampling from he parameer disribuion. Asympoic disribuion heory leads o he following disribuion for he parameer esimaes: ( p C) p ˆ ~ N, (3) where pˆ are he parameer values ha maximize he likelihood funcion, and C denoes he associaed covariance marix of he parameer esimaes. In a Bayesian framework we sample from: ( pˆ C) p ~ N,. (4) Consider H samples 3, which are denoed wih ( 1) ( H ) p, K, p. For all hese parameer values, we calculae heoreical call opion prices for all raded opions. This leads o H values for he pricing error, e.g. denoed wih ( 1) ( H ) ( p ), MSE p ( ) MSE K, for he absolue pricing error. Therefore, insead of arriving a one MSE, we now have an enire sample of MSEs. The uncerainy in he MSE may be quanified by calculaing confidence inervals of he MSE. The expression of he MSE shows ha also he parameers may be reaed as random variables ha have impac on he size and he uncerainy in he esimaed MSE. In order o es he adequacy of he assumed loss funcion, we propose an ou-of-sample analysis. 3. Daa and Mehodology 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 opionpricing models. 3 We conduc he empirical analysis using H=5000 samples. 5

6 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 exchange-rading 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 volailiies. Insead of using a saic rule and exclude opions wih absolue moneyness 4 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 ouof-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 5. For his reason, we always work wih he underlying index level implied ou from fuures or opion prices. In paricular we are using he following procedure for one paricular day o price opions on he following rading day: 4 In our noaion, absolue moneyness is defined as K/F-1, where K is he exercise price and F is he forward price. 5 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. 6

7 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 (5) 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 opions 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 conrol and 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 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 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 insample calibraion. The fuures marke is he mos liquid marke, and he opions and he fuures markes are closely inegraed. Therefore i 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 7

8 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 (6) 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 a delivery monh for he fuures conrac. When an opion expires on a dae oher han he delivery dae of he fuures conrac, hen 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 = (7) (T T ) 3 Hence, he quaniy X fuure can be compued by 6 PV D) = X e dt ( associaed wih he opion ha expires a ime T in he X e dt ( ( rf1 d ) T1 dt ) = F1 e. (8) 6 See e.g. he appendix in Poeshman (001) for deails. 8

9 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. For he ou-of-sample par he same loss funcions for call opions are used. The predicive performance of he various models are evaluaed and compared by using he roo mean squared valuaion error crierion. We compare he prediced opion values wih he observed prices for every raded opion. We repea he whole procedure over he ou-of-sample period and conclude which loss funcion minimizes he uncerainy in he ou-of-sample pricing error. 4. Empirical resuls For each rading day of he year 000, we esimaed he model using closing prices of raded opions ha fulfill 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 of he Table corresponds o he loss from using he same loss funcion a he esimaion and evaluaion sage. The off-diagonal enries repor he losses from using differen loss funcion in- and ou-of-sample. As expeced, calibraing he model using one loss funcion also resuls in a minimum pricing error (in bold) for ha paricular loss funcion. The esimaed parameers when calibraing he model using one loss funcion are only sub-opimal when looking a he pricing error regarding a differen loss funcion. [Table 1] In a nex sep, we use he model calibraed on one rading day o price all raded opions on he following day (closing prices). Again, we esimae he model using one loss funcion, bu also evaluae he model using anoher loss funcion. Therefore, we change he specificaion a he evaluaion sage. Table 1, Panel B repors he average RMSEs for he ou-of-sample pricing 9

10 exercise. Sill, as expeced, using he same loss funcion a he esimaion and evaluaion sages minimizes he pricing error a he evaluaion sage (in bold). We confirm he resuls of Chrisoffersen and Jacobs (00)). Therefore, we canno conclude which loss funcion is he preferred one, because he individual resuls are no direcly comparable. An objecive measure in opion valuaion could be he precision of he paricular pricing error in he ou-of-sample analysis. Therefore, we use our simulaion approach in order o derive a whole sample of ou-of-sample pricing errors by incorporaing he uncerainy in he parameer esimaes from he calibraion. Using he mos appropriae loss funcion should resul in he smalles confidence inerval around he mean pricing error; acually for all loss funcions. The resuls show ha he disribuion of pricing errors is heavily skewed o he righ, meaning ha significan pricing errors may occur, because, even when all parameer esimaes are significan, here is a lo of uncerainy in he esimaes. In order o make he uncerainy of differen pricing errors comparable, we define a crierion ha measures he acual uncerainy relaive o he mean: UpperBound LowerBound precision =, (9) Mean where upper/lower bound refers o he empirical upper/lower bounds of he 95% confidence inerval around he mean, respecively and mean refers o he average pricing error. The choice of a paricular crierion is of course arbirary, bu our precision measure is a leas plausible, because of he following logical characerisics: he measure goes o zero if here is no uncerainy in he parameer esimaes and hence, in he pricing error and we obain posiive values in case of uncerain parameer esimaes. This measure is a sandardized crierion and allows o objecively compare he differen loss funcion under consideraion. Table, Panels A and B repor he average values of he measure for he in- and ou-of-sample pricing exercise. The resuls show ha here are obvious differences in he resuls for he in- and ou-of-sample analyses. [Table ] 10

11 From he in-sample analysis, we see ha he conclusions of Chrisoffersen and Jacobs (00) are also valid for our ype of analysis: aligning he loss funcion a he esimaion and evaluaion sage also reduces he relaive uncerainy in he pricing error. However, he resuls do no allow us o recommend one or he oher loss funcion. While we canno draw a conclusion from he insample analysis, we can conclude from he ou-of-sample analysis ha using he absolue Euro pricing error a he esimaion sage, minimizes he uncerainy in he pricing error a he evaluaion sage for all loss funcions (in bold). Therefore, i migh be considered as he mos appropriae, objecive loss funcion in opion valuaion, because i minimizes he relaive uncerainy in he ou-of-sample pricing error regardless which loss funcion is used o evaluae he model. 5. Comparison wih a GARCH Opion Pricing Model So far he empirical analysis has focused on esing he impac of differen loss funcions on he precision of he pricing error 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 follow he following dynamics (see e.g. Duan (1995)): r X = ln X 1 + d = µ + σ e e Ω 1 ~ N(0,1) under probabiliy measure P, ln ( σ ) ω + α ln( σ ) + β ( e γ e ) = (10) where he condiional mean is defined as µ = +, d is he dividend yield of he index rf λσ porfolio, 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. 11

12 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 : [ r ) ] exp( rf ) Q E exp( Ω 1 =, (11) Therefore, he LRNVR ransforms he physical reurn process o a risk-neural dynamic. The risk-neural Gaussian GARCH process reads 7 : r = rf 1 σ +σ ε ε Ω 1 ~ N(0,1) under risk-neuralized probabiliy measure Q, ln ( σ ) ω + α ln( σ ) + β ε λ γ ( ε λ) ( ) = (1) where he erm gives addiional conrol for he condiional mean. In Equaion (1), ε is 1 σ no necessarily normal, bu o include he Black-Scholes model as a special case we ypically assume ha ε is a Gaussian random variable. The uncondiional volailiy level is equal o ω + β E exp [( ε λ γ ( ε λ) )] 1 α measures he persisence of he variance process. and can be evaluaed numerically. The parameer α 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. (13) A European call opion wih exercise price K i and mauriy T j has a ime price equal o: Q ( rf T ) E max( X K,0) [ Ω ] c (14) = exp j i 1 7 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)). 1

13 For his kind of derivaive valuaion models wih a high degree of pah dependency, compuaionally demanding Mone Carlo simulaions are commonly used for valuing derivaive securiies. We use he recenly 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 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-ofsample average pricing errors 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 funcion a he esimaion and evaluaion sage minimizes he in- and ou-of-sample pricing error a he evaluaion sage (in bold). Addiionally, when comparing he wo pricing models under consideraion, he resuls are consisen wih previous findings (e.g. see Heson and Nandi (000)). The ad-hoc Black-Scholes model ypically overfis he daa in-sample, bu when 13

14 evaluaed ou-of-sample, i ypically underperforms GARCH-ype opion pricing approaches. The resuls are consisen and do no depend on he paricular loss funcion used a he esimaion or evaluaion sage. [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 precision of he pricing error, again he findings of he previous secion can be confirmed. Aligning he esimaion and evaluaion loss funcions minimizes he in-sample precision of he pricing error a he evaluaion sage (Panel A), bu using he absolue Euro pricing error a he esimaion sage minimizes he uncerainy in he pricing error a he evaluaion sage regardless which loss funcion is used o evaluae he model (Panel B). Therefore, he resuls from he previous secion are robus and seem o be independen of he underlying opion pricing model under consideraion. 6. 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 funcion. The approach allows us o promoe a paricular loss funcion. Using he absolue pricing error crierion a he esimaion sage minimizes he uncerainy in he parameer esimaes and herefore maximizes he precision of he ou-of-sample pricing error regardless which loss funcion is used a he evaluaion sage. We confirm he empirical resuls of Chrisoffersen and Jacobs (00) and find srong evidence for heir conjecure ha he absolue pricing error crierion may serve as a general purpose loss funcion in opion valuaion applicaions. The resuls are far-reaching for he opion valuaion lieraure, because researchers are ypically 14

15 inconsisen in heir choice of he loss funcions and resuls are herefore incomparable. Of course, he choice of he loss funcion is subjecive, bu he framework proposed in his paper allows idenificaion of he mos appropriae loss funcion for he purpose a hand. 15

16 References Bakshi, C., C. Cao and Z. Chen (1997): Empirical performance of alernaive opion pricing models, Journal of Finance 5, Bams, D., T. Lehner and C.C.P. Wolff (00): An Evaluaion Framework for Alernaive VaR Models, forhcoming Journal of Inernaional Money and Finance. Black, F. and M. Scholes (1973): The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 81, Blair, B.J., S.-H. Poon and S.J. Taylor (001), Forecasing S&P100 volailiy, The incremenal informaion conen of implied volailiies and high frequency index reurns, Journal of Economerics, 105, 5-6. Chernov, M. and 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 56, Chrisoffersen, P. and K. Jacobs (00): The Imporance of he Loss Funcion in Opion Valuaion, forhcoming Journal of Financial Economics. Derman, E. (1999), Regimes of Volailiy, RISK, 4, Duan, J.-C. (1995), The GARCH Opion Pricing Model, Mahemaical Finance 5, Duan, J.-C. (1995), The GARCH Opion Pricing Model, Mahemaical Finance 5, Duan, J.-C. (1996), Cracking he Smile, RISK 9, Duan, J.-C. and J.-G. Simonao (1998), Empirical Maringale Simulaion for Asse Prices, Managemen Science 44, Dumas, B., J. Fleming and R.E. Whaley (1998): Implied Volailiy Funcions: Empirical Tess, The Journal of Finance 53, Engle, R.F. (1993): A commen on Hendry and Clemens on he limiaions of comparing mean squared forecas errors, Journal of Forecasing, 1, Heson, S.L. and S. Nandi (000): A Closed-Form GARCH Opion Valuaion Model, The Review of Financial Sudies 3, Huchinson, J., A. Lo and T. Poggio (1994): A nonparameric approach o pricing and hedging derivaive securiies via learning neworks, Journal of Finance 49, Lehner, T. (003). Explaining Smiles: GARCH Opion Pricing wih Condiional Lepokurosis and Skewness. Journal of Derivaives, 10, 3,

17 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, 63, Poeshman, A.M. (001), Underreacion, Overreacion, and Increasing Misreacion o Informaion in he Opion Marke, Journal of Finance, 56, 3, Shimko, D. (1993): Bounds of Probabiliy, RISK 6,

18 Table 1: In- and Ou-of-Sample Pricing Errors, Ad-hoc Black-Scholes Model Panel A: In-Sample Pricing Errors Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Panel B: Ou-of-Sample Pricing Errors Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Noes. The able presens he average in- and ou-of-sample pricing errors from he daily esimaion and evaluaion of he model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on he following day. A he evaluaion sage, we also compue he resuls for he remaining 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. The figures in bold refer o he bes resul given a paricular loss funcion a he evaluaion sage. 18

19 Table : Precision of Pricing Errors, Ad-hoc Black-Scholes Model Panel A: Precision of Pricing Errors (In-Sample) Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Panel B: Precision of Pricing Errors (Ou-of-Sample) Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Noes. The able presens he average precision of he in- and ou-of-sample pricing errors from he daily esimaion and evaluaion of he model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on he following day. A he evaluaion sage, we also compue he resuls for he remaining loss funcions. The able presens he figures for he precision crierion defined in Equaion (9). The figures in bold refer o he bes resul given a paricular loss funcion a he evaluaion sage. 19

20 Table 3: In- and Ou-of-Sample Pricing Errors, GARCH Opion Pricing Model Panel A: In-Sample Pricing Errors Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Panel B: Ou-of-Sample Pricing Errors Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Noes. The able presens he average in- and ou-of-sample pricing errors from he daily esimaion and evaluaion of he model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on he following day. A he evaluaion sage, we also compue he resuls for he remaining 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. The figures in bold refer o he bes resul given a paricular loss funcion a he evaluaion sage. 0

21 Table 4: Precision of Pricing Errors, GARCH Opion Pricing Model Panel A: Precision of Pricing Errors (In-Sample) Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Panel B: Precision of Pricing Errors (Our-of-Sample) Evaluaion Esimaion IV RMSE RMSE % RMSE IV RMSE RMSE % RMSE Noes. The able presens he average precision of he in- and ou-of-sample pricing errors from he daily esimaion and evaluaion of he model. Each day he model is esimaed using one paricular loss funcion and evaluaed using raded opions on he following day. A he evaluaion sage, we also compue he resuls for he remaining loss funcions. The able presens he figures for he precision crierion defined in Equaion (9). The figures in bold refer o he bes resul given a paricular loss funcion a he evaluaion sage. 1