Mandelbrot and the Smile

Transcription

1 Mandelbro and he Smile Thorsen Lehner * Limburg Insiue of Financial Economics (LIFE), Maasrich Universiy, P.O. Box 616, 600 MD Maasrich, The Neherlands Firs version: January 006 Absrac I is a well-documened empirical fac ha index opion prices sysemaically differ from Black- Scholes prices. However, previous research provides inconclusive resuls wheher he observed volailiy smile could be explained by a discree-ime dynamic model of sock reurns wih skewed, lepokuric innovaions. The improvemens in pricing errors are paricularly pronounced for ou-of-he money pu opions, while he models parly underperform a Gaussian alernaive for near-he-money opions. Moivaed by heses empirical evidence, I develop a new GARCH opion-pricing model wih a more flexible innovaion srucure. In an applicaion of he model o DAX index opions, I es he relaive performance of he approach agains a sandard nesed GARCH specificaion and he wellknown praciioners Black-Scholes model. I show ha he performance of he runcaed Lévy GARCH opion pricing model is superior o exising approaches. JEL code: Keywords: G1 GARCH, opion pricing, ou-of-sample, runcaed Levy disribuion. * Correspondence o: Tel: ; Fax: lehner@berfin.unimaas.nl. The auhor would like o hank Deusche Börse Group for providing he daa.

2 1. Inroducion There is exensive empirical evidence ha observed marke prices of raded opions sysemaically differ from Black-Scholes prices. Ou-of-he-money calls and pus are relaively overpriced compared o a-he-money opions. A fac ha is ofen represened by he well-known volailiy smile implied ou from observed opion prices. For index opions, he smile is skewed owards ou-of-he money pus. As a resul, he implied risk-neural densiy funcion is lepokuric and heavily skewed o he lef. The difference beween he acual reurn disribuion of he underlying and he risk-neural disribuion implied ou from opion prices can only be explained by exremely high levels of risk aversion. Marke paricipans seem o overesimae he probabiliy of exreme downward movemens and are willing o purchase overpriced ou-of-he money pu opions. A he empirical level, several sudies have shown ha opion valuaion models wih condiional heeroskedasiciy and negaive correlaion beween volailiy and spo reurns capure he paricular paern and significanly improve upon he performance of he Black- Scholes model. The discree-ime GARCH opion pricing model has shown o be a flexible, empirically successful model (see among ohers Heynen, e al (1994), Duan (1996), Heson and Nandi (000), and Richken and Trevor (1999)). Recenly, an increasing number of numerical mehods for his class of opion pricing models become available (see Hanke (1997), Richken and Trevor (1999), Duan and Simonao (1998), Duan e al. (001) and Heson and Nandi (000)). Heson and Nandi (000) developed a closed form soluion of a GARCH opion-pricing model. They show ha he single lag version of heir model conains Heson s (1993) model as a coninuous-ime limi, bu he discree-ime counerpar is much easier o apply wih available daa. However, despie he raher sophisicaed modeling approach, a Gaussian model canno adequaely accoun for he paricular paern observed in opion prices. Using he generalized GARCH opion-pricing framework of Duan (1999), Lehner (003) showed ha condiional lepokurosis and skewness reinforces he effecs of condiional heeroskedasiciy and asymmery in he volailiy process. His GARCH opion pricing model driven by skewed generalized error disribued innovaions ouperforms he closed-form GARCH opion pricing model of Heson and Nandi in-sample as well as ou-of-sample. The improvemens in pricing errors are paricularly pronounced for ou-of-he money pu and call opions, while he model

3 parly underperforms he Gaussian model for near-he-money opions. The resuls are parly in line wih recen resuls obained by Chrisoffersen e al. (003). They developed and empirically es a GARCH opion pricing model wih condiional skewness. While hey demonsrae he imporance of condiional skewness and umps for he pricing of ou-of-he-money pus, heir closed-form Inverse Gaussian GARCH opion pricing model significanly underperforms a sandard Gaussian model for several oher ypes of opions. In conras o Lehner (003), Chrisoffersen e al. (003) conclude ha he overall pricing performance is inferior o he sandard Gaussian model. Therefore, he empirical evidence does no necessarily sugges ha modeling umps in reurns and volailiy in addiion o sochasic volailiy is he appropriae approach for he purpose of opion valuaion. In his paper, I argue ha mos is o be gained by modeling deviaions from normaliy. In order o invesigae he research quesion furher, I empirically invesigae wheher he use of a more flexible innovaions disribuion is able o improve he performance for ou-of-he-money opions wihou worsening he performance for near-he-money opions. A possible candidae migh be he runcaed Lévy disribuion ofen sudied in physics (see Lehner and Wolff (004)). Mandelbro (1963) firs proposed he idea ha price changes are disribued according o a Lévy sable law. This model was frequenly criicized, because he ails are now oo fa from a financial modeling perspecive and he infinie variance makes i impossible o apply he Cenral Limi Theorem. The problems wih hese kinds of disribuions are he power law ails, which decay oo slowly. This problem can be overcome by aking he Lévy disribuion in he cenral par and inroducing a cuoff in he far ails ha is faser han he Lévy power law ails. The Lévy disribuion wih a cuoff and exponenially declining ails was inroduced in he physics lieraure by Manegna and Sanley (1994) and is known as a runcaed Lévy disribuion. The exponenial decay in he ails ensures ha all relevan momens are finie. In an in-sample and ou-of-sample analysis, I compare my model wih wo benchmarks: a sandard nesed Gaussian alernaive specificaion and he ad-hoc Black-Scholes model of Dumas, Fleming and Whaley (1998) (DFW) and conclude which mehod is superior in describing he observed marke smile in DAX index opions. The paper is arranged as follows. Secion describes he GARCH opion-pricing framework. Secion 3 discusses he daa and mehodology. In Secion 4, I provide he empirical resuls and finally Secion 5 concludes. 3

4 . Economeric Framework.1. Opion Pricing under GARCH In a Gaussian discree-ime economy he value of he index a ime, S, can be assumed o follow he following dynamics (see e.g. Duan (1995)): S r = ln S 1 + d = r f + λσ + σ e e Ω 1 ~ N(0,1) under probabiliy measure P ln ( σ ) ω + ω ln( σ ) + ω ( e γ e ) = (1) where d is he dividend yield of he index porfolio, r f is he risk-free rae, λ is he price of risk; Ω -1 is he informaion se in period -1 and he combinaion of ω and γ capures he leverage. 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 r f: [ r ) ] exp( r ) Q E exp( Ω 1 = f, () Therefore, he LRNVR ransforms he physical reurn process o a risk-neural dynamic. The risk-neural Gaussian GARCH process reads: r = r f 1 σ + σ ε ε Ω 1 ~ N(0,1) under risk-neuralized probabiliy measure Q ln ( σ ) ω + ω ln( σ ) + ω ε λ γ ( ε λ) ( ) = (3) where he erm gives addiional conrol for he condiional mean. In Equaion (3), 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. 4

5 .. GARCH opion pricing wih condiional skewness and lepokurosis Lévy flighs have been observed experimenally in physical sysems and have been used very successfully o describe for insance he specral random walk of a single molecule embedded in a solid. In all hese cases an unavoidable cuoff in he ails of he disribuion is always presen. One possible cuoff is he exponenial funcion, for which he characerisic funcion (CF) of he so-called runcaed Lévy disribuion (TLD) has been developed (Koponen (1995)). For financial daa he cuoff region is early in he ails, which ensures he finieness of all relevan momens. The sandardized CF wih locaion parameer equal o zero and scale parameer c equal o one reads (Nakao (000)): ψ TLD α / α δ k + δ ( k, α, δ, β ) = cos ( πα / ) cos α arcan k δ 1 + i sgn () k (4) k β an α arcan δ where α is he characerisic exponen deermining he shape of he disribuion and especially he faness of he ails (0 < α, bu α 1 ) and δ is he cuoff parameer, which deermines he speed of he decay in he ails and as a resul he cuoff region. The parameer β (β [-1,1]) deermines he skewness when β 0, he disribuion is skewed o he righ when -1 < β < 0 and skewed o he lef when 0 < β < 1. For δ + 0 he TLD reduces o he Lévy disribuion and for δ + 0, β=0 and α= he TLD reduces o he Normal disribuion wih scale parameer c. For comparison purposes, Figure 1 shows he densiy of a runcaed Lévy disribuion wih reasonable parameer values for financial reurn daa and he special case of a Gaussian densiy. Boh densiies are sandardized, such ha he scale parameer c equals one. Accurae numerical values for he densiy ψ TLD can be calculaed by Fourier-ransforming he CF and evaluaing he inegral numerically. I use Romberg inegraion, which allows ex-ane specificaion of he oleraed error and in fac a calculaion of he densiy as precise as necessary (see Lamber and Lindsey (1999)). The analogue of he sandard deviaion σ in he family of Lévy disribuions is he scale parameer c. If I replace he sandard deviaion σ by he scale parameer c, I allow he condiional 5

6 scale parameer c o be serially correlaed and o vary over ime. If e, condiional on Ω -1 is a skewed runcaed Lévy disribued random variable, hen he risk-neural GARCH process reads 1 : r Q ( E ( exp ( c η ) Ω )) + c η = r f ln 1 ε 1 ~ ( = 0, c = 1) under risk-neuralized probabiliy measure Q. Ω N Lévy µ ( φ ( ε λ ); µ = 0, c = 1, α, δ β ) 1 η = TLD, ln Lévy ( c ) + ω ln( ) + ω ( η γ η ) = 0 1 c ω, (5) where φ Lévy [.] sands for a sandardized normal cumulaive disribuion wih zero mean and scale parameer c equal o 1, TLD -1 [.] sands for he inverse skewed runcaed Lévy cumulaive disribuion wih sandardized mean equal o 0, scale parameer c equal o 1, ail parameer α, skewness parameer β and δ conrols for he exponenial decay in he ails. The erm E Q (. Ω) gives addiional conrol for he condiional mean and can be evaluaed numerically. The ω 0 + ω E uncondiional volailiy level is equal o exp 1 ω1 [( η γ η) ] and can be evaluaed numerically. The parameer ω 1 measures he persisence of he variance process. A European call opion wih exercise price X and ime o mauriy T has a ime price equal o: Q ( rt ) E [ max( S X,0) Ω ] c (6) = exp 1 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. I use he recenly proposed simulaion adusmen 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 1 See Lehner (003) for deails regarding GARCH opion pricing wih condiional lepokurosis and skewness. 6

7 simulaion error. Mone Carlo simulaions are also frequenly and successfully used in various risk managemen applicaions (e.g. see Bams, Lehner and Wolff (00))..3. Alernaive opion valuaion approach I compare boh GARCH opion-pricing models wih he so-called praciioners Black-Scholes model (see e.g. Dumas, Fleming and Whaley (1998) or Chrisoffersen and Jacobs (00b)). I allow each opion o have is own Black-Scholes implied volailiy depending on he exercise price X and ime o mauriy T and use he following funcional form for σ: σ = π + M T, (7) i 0 + π 1M i + π M i + π 3T + π 4T π 5 i where σ i denoes he implied volailiy and M i he moneyness, X i, (F is he forward price) of F an opion for he i-h exercise price and -h mauriy. For every exercise price and mauriy I can compue he implied volailiy and derive opion prices using he Black-Scholes model. I calibrae he parameers of he various model by minimizing he roo mean squared absolue pricing error beween he marke prices and he heoreical opion prices: RMSE = 1 N min n mi ( cˆ i, ci, ) i= 1 = 1 n mi 1 or RMSE = min ( pˆ i, pi, ) N i= 1 = 1, (8) where N is he oal number of call opions evaluaed, he subscrip i refers o he n differen mauriies and subscrip o he m i differen srike prices in a paricular mauriy series i. Depending on he moneyness of he opion, pu or call prices are used. Moivaed by he acual rading volume, I use pu prices for opion wih moneyness of less han 1 and call prices for opions wih moneyness of more han 1. As saring values for he calibraion, I make use of he ime-series esimaes from he augmened GARCH model using approximaely hree years (75 rading days) of hisorical reurns. Addiionally, I use he ime-series parameer esimae of he price of risk parameer δ for he opion calibraion (see e.g. Chrisoffersen and Jacobs (00a)). 7

8 3. Daa and Mehodology I 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 a good 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 on he exchange-rading day immediaely prior o ha Friday. I exclude opions wih less han one week and more han 5 weeks unil mauriy and opions wih a price of less han Euro 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 K/F-1 of more han 10% (see DFW), I exclude opions wih a daily urnover of less han Euro (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 I 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 calculaing opion prices, heoreically I don have o adus 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, one observe empirically a differen implied dividend adused underlying for differen mauriies. For his reason, I always work wih he adused underlying index level implied ou from fuures or opion marke prices. 8

9 In paricular I m using he following procedure for one paricular day o price opions on he following rading day: Firs, I compue he implied ineres raes and implied dividend adused index raes from he observed pu and call opion prices. I m using a modified pu-call pariy regression proposed by Shimko (1993). The pu-call pariy for European opions reads: c i, pi, = [ S PV ( D )] i r ( T ) X e (9) where c i, and p i, are he observed call and pu closing prices, respecively, wih exercise prices X i and mauriy (T -), PV(D ) denoes he presen value of dividends o be paid from ime unil he mauriy of he opions conrac a ime T and r is he coninuously compounded ineres rae ha maches he mauriy of he opion conrac. Therefore, I can infer a value for he implied dividend adused index for differen mauriies, S -PV(D ), and he coninuously compounded ineres rae for differen mauriies, r. In order o ensure ha he implied dividend adused index value is a non-increasing funcion of he mauriy of he opion, I occasionally adus he sandard pu-call pariy regression. Therefore, I conrol and ensure ha he value for S -PV(D ) is decreasing wih mauriy, T. Since I m using 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, I esimae he parameers of he paricular models by minimizing he loss funcion (8). Given reasonable saring values, I price European call opions wih exercise price X i and mauriy T. Using well-known opimizaion mehods (e.g. Newon-Raphson mehod), I obain he parameer esimaes ha minimize he loss funcion. The goodness of fi measure for he opimizaion is he mean squared valuaion error crierion. Third, having esimaed he parameers in-sample, I 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. I 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 marke are closely inegraed, herefore i can also be assumed ha he fuures price is more informaive for opion pricing han us using he value of he index. For every observed fuures 9

10 closing price I can derive he implied underlying index level and evaluae he opion. Given a fuures price F wih ime o mauriy T, spo fuures pariy is used o deermine S -PV(D ) from S -r T PV ( D ) = F e (10) where PV(D ) denoes he presen value of dividends o be paid from ime unil he mauriy of he opions conrac a ime T and r 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 S -PV(D ) 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 S -PV(D ) 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 = (11) (T T ) 3 Hence, he quaniy S -PV(D) = S e dt associaed wih he opion ha expires a ime T in he fuure can be compued by S e dt F e ( ( r1 d ) T1 dt ) = 1. (1) See e.g. he appendix in Poeshman (001) for deails. 10

11 This mehod allows us o perfecly mach he observed opion price and he underlying dividend adused spo rae. Given he parameer esimaes and he implied dividend adused underlying I 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 predicion performance of he various models are evaluaed and compared by using he roo mean squared valuaion error crierion. I compare he prediced opion values wih he observed prices for every raded opion. I repea he whole procedure over he ou-of-sample period and conclude, which model minimizes he ou-of-sample pricing error. 4. Empirical resuls In his secion, I presen esimaion and evaluaion resuls using 54 days of opion daa from he year 000. Each rading day, on average 85 opion prices are used for he calibraion and evaluaion of he models, wih a minimum of 6 and a maximum of 155. The numbers of opion conracs in he in-sample and ou-of-sample daa se are repored across moneyness and mauriy in Table 1. [Table 1] In oal more han 1400 opion conracs have been evaluaed. The number conracs are nearly equally disribued over he five differen moneyness caegories, excep he fewer number opions wih moneyness of less han 0.9. Mos opions are shor erm (<1 rading days unil mauriy) or long erm (>63 days unil mauriy), bu here is also a subsanial number opions raded in he medium erm. All models are calibraed using he Euro roo mean squared error loss funcion (8). For he GARCH models, heoreical opion prices are obained using (5), and he local scale parameer c +1 is esimaed ogeher wih he oher parameers. I addiionally use a ime-series parameer esimae for he opion calibraion: he price of risk parameer λ. The oin idenificaion of λ and γ is possible bu no reasonable, since boh parameers conrol for asymmery by modifying he news impac curve. 11

12 Table shows he averages of daily parameer esimaes of he hree models analyzed, obained by minimizing he squared Euro pricing error, using daa for 53 rading days in he year 000. [Table ] For he models ha are invesigaed, esimaing he parameers using a single day of opion prices is a convenien approach. The obecive funcion is always well behaved and no numerical problems have been encounered. I is well-known fac ha he parameers of he ad-hoc Black- Scholes model can be significanly esimaed, bu subsanially vary over ime, already on a dayo-day basis. Similar resuls are obained for he paricular opion daa se under invesigaion (resuls no repored). For he GARCH opion pricing models, parameer esimaes are obained ha are more sable and in line wih he ones expeced from a ime-series calibraion. The value of 0.98 for ω 1 suggess srong mean reversion of he volailiy process. The significan esimaes for ω and γ sugges ha he volailiy process is asymmeric, meaning ha reurns and volailiy are negaively correlaed and resuling in negaive skewness in he simulaed muliperiod index reurns. However, in he case of he runcaed Lévy GARCH opion pricing model, his effec is augmened by negaive skewness in he innovaions disribuion. Addiionally, also he oher parameer esimaes sugges ha he daa dicae a non-normal innovaions disribuion: he ail faness parameer α is differen from and he cuoff parameer δ is differen from 0. A value of around 0.5 for δ implies ha he exponenial decay is inroduced earlier in he ails, which reecs he exremely fa-ailed Lévy disribuion as a possible alernaive. The in-sample relaive pricing errors are in he range of 1 Euros; usually slighly smaller for he runcaed Lévy GARCH opion pricing model (on average around 1.5) and he ad-hoc Black-Scholes model (on average around ) compared o he Gaussian GARCH opion valuaion model (on average around.). Table 3 repors he resuls for he in-sample pricing errors of he various models. [Table 3] 1

13 In general, he in-sample resuls are in line wih he empirical findings of Chrisoffersen e al. (003) and Lehner (003): The more flexible innovaions srucure improves he pricing performance of he GARCH opion pricing model significanly. The resuls are consisen over differen levels of moneyness and for various mauriies. The good pricing performance of he GARCH model wih condiional skewness and lepokurosis canno be confirmed for he Gaussian alernaive: on average he model underperforms he ad-hoc Black-Scholes model. However, i is a well-known fac ha he ad-hoc Black-Scholes model ypically overfis he daa in-sample, bu when evaluaed ou-of-sample, i ypically underperforms GARCH-ype opion pricing approaches (Heson and Nandi (000)). Addiionally Chrisoffersen e al. (003) show ha a GARCH model wih condiional skewness and umps overfis he daa in-sample, bu underperforms a Gaussian model when evaluaed ou-of-sample. Therefore, we canno rely on he in-sample resuls, bu an ou-of-sample analysis has o be conduced. The ou-of-sample valuaion errors are presened in Table 4. Resuls srongly sugges ha he findings of Chrisoffersen e al. (003) canno be confirmed for a more recen daa se of DAX opions: he ou-of-sample valuaion errors of he runcaed Lévy GARCH model are on average lower compared o he Gaussian alernaive and he ad-hoc Black-Scholes model. [Table 4] The improvemens in he pricing performance are in paricularly pronounced for shor-erm ouof-he-money pu opions. Chrisoffersen e al. (003) presened empirical evidence ha he pricing performance of he GARCH model wih skewness and umps is inferior o he Gaussian alernaive especially for longer-erm opions. Neverheless, i is remarkable ha also wih he approach proposed in his sudy, he pricing performance worsens for longer-erm opions, bu sill remains superior compared he Gaussian alernaive. One migh conclude ha he more flexible innovaions srucure of he runcaed Lévy GARCH opion pricing model does no seem o overfi he daa in-sample, bu resuls in significan valuaion improvemens for all ypes of opions. Therefore, one migh argue ha mos is o be gained from modeling deviaions from normaliy, bu less is o be gained from modeling umps in reurns and volailiy in addiion o sochasic volailiy. In addiion, he findings of Heson and Nandi (000) can be confirmed: he 13

14 alernaive Gaussian GARCH model ouperforms he ad-hoc Black-Scholes model ou-ofsample. As expeced, he alernaive approach of modeling he scale parameer c insead of he variance does no resul in a differen pricing performance of he Gaussian GARCH opion valuaion model. However, one resul is in paricularly ineresing: for all approaches he pricing performance dramaically worsens when he mauriy of he opion conrac increases. This is in line wih he resuls of Chrisoffersen e al. (003) and shows ha sill no even he more sophisicaed GARCH approach adequaely capures he volailiy dynamics underlying opion prices. 5. Conclusions This paper presens a new opion valuaion model ha is based on a reurn dynamic ha conains condiional skewness and lepokurosis as well as condiional heeroskedasiciy and a leverage effec. The runcaed Lévy GARCH opion pricing model ness a sandard GARCH model, which conains Gaussian innovaions, and he empirical comparison beween our new model and he sandard GARCH model invesigaes he imporance of modeling condiional skewness and lepokurosis. Our empirical resuls are srongly in favor of he new modeling approach: he runcaed Lévy GARCH opion pricing model achieves a beer fi han sandard models insample and ou-of-sample. The improvemens in he pricing performance are paricularly pronounced for shor-erm deep ou-of-he-money pus, bu i also performs beer han sandard models for longer erms and for several oher ypes of opions. 14

15 References Bams, D., T. Lehner and C.C.P. Wolff (00), An Evaluaion Framework for Alernaive VaR Models, forhcoming Journal of Inernaional Money and Finance. Chrisoffersen, P. and K. Jacobs (00a), Which Volailiy Model for Opion Valuaion?, Working Paper, McGill Universiy. Chrisoffersen, P. and K. Jacobs (00b): The Imporance of he Loss Funcion in Opion Valuaion, forhcoming Journal of Financial Economics. Chrisoffersen, P., S. Heson and K. Jacobs (003), Opion Valuaion wih Condiional Skewness, Working Paper, McGill Universiy. 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, Duan, J.-C. (1999), Condiional Fa-Tailed Disribuions and he Volailiy Smile in Opions, Roman School of Managemen, Universiy of Torono, Working Paper. Duan, J.-C. Gauhier G. and J.-G. Simonao (001), Asympoic Disribuion of he Empirical Maringale Simulaion Opion Price Esimaor, o appear in Managemen Science. Dumas, B., J. Fleming and R.E. Whaley (1998): Implied Volailiy Funcions: Empirical Tess, The Journal of Finance 53, Heson, S.L. (1993), A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, The Review of Financial Sudies, 6, Heson, S.L. and S. Nandi (000), A Closed-Form GARCH Opion Valuaion Model, The Review of Financial Sudies 3, Heynen, R., A. Kemma and T. Vors (1994), Analysis of he erm srucure of implied volailiies, Journal of Financial and Quaniaive Analysis, 9, Koponen, (1995), Analyic approach o he problem of convergence of runcaed Lévy flighs owards he Gaussian sochasic process, Physical Review E, Volume 5, Number 1, Lamber P. and J.K. Lindsey (1999), Analysing financial reurns by using regression models based on non-symmeric sable disribuions, Journal of Applied Saisics, 48,

16 Lehner, T. (003), Explaining Smiles: GARCH Opion Pricing wih Condiional Lepokurosis and Skewness, Journal of Derivaives, 10, 3, Lehner, T. and C.C.P. Wolff (004), Scale-Consisen Value-a-Risk, forhcoming Finance Research Leers. Mandelbro, B. (1963), The Variaion of Cerain Speculaive Prices, Journal of Business, 36, Manegna, R. N. and H. E. Sanley (1994), Sochasic process wih ulraslow convergence o a Gaussian: he runcaed Lévy fligh, Phys. Rev. Le., 73, Nakao, H., (000), Muli-scaling properies of runcaed Lévy flighs, Manuscrip, Graduae School of Mahemaical Sciences, Universiy of Tokyo. Poeshman, A.M. (001), Underreacion, Overreacion, and Increasing Misreacion o Informaion in he Opion Marke, Journal of Finance, 56, 3, Richken, P. and R. Trevor (1999), Pricing opions under generalized GARCH and sochasic volailiy processes, Journal of Finance, 54, Shimko, D. (1993), Bounds of probabiliy, RISK, 6,

17 Figure 1: Truncaed Lévy Densiy The graph depics a comparison of a runcaed Lévy densiy and a normal densiy. Boh densiies are sandardized, such ha he scale parameer c is equal o one. 17

18 Table 1: Number of Observaions Days o Expiraion < 1 [1,63] > 63 Moneyness Toal Daily Median Daily Max Toal Daily Median Daily Max Toal Daily Median Daily Max Toal < [0.9,0.96) [0.96,1.00) [1.00,1.04) > Toal Noes: This Table repors he number of observaions for differen moneyness and mauriy during he period January 000 unil December 000. Moneyness is defined in he following way: a pu opion is said o be near- or a-he-money if K/F [0.96,1.00] or K/F [1.00,1.04), ou of he money if K/F [0.9,0.96), deep ou-of-he-money if K/F < 0.9 and in he money if K/F > 1.04, where K is he srike price and F is he forward price. Similar erminology is defined for calls by replacing K/F by F/K. Days o Expiraion is he number of rading days unil mauriy. Toal is he oal number of opions priced during he period Augus 000 unil March 001 of he paricular mauriy and/or exercise price. Median and Max are he median and he maximum number of opions priced on one rading day of he paricular moneyness and/or mauriy. 18

19 Table : Parameer Esimaes Ad-hoc Black-Scholes Gaussian GARCH Truncaed Lévy GARCH Parameer Mean Parameer Mean Mean π ω π ω π ω π γ π c τ π α δ β Noes. The able presens he average parameers esimaes of he daily esimaions of he various models during he period January 000 unil December

20 Table 3: In-Sample Analysis: Average Valuaion Errors across moneyness and mauriy Ad-hoc Black-Scholes EGARCH-Normal EGARCH-runcaed Lévy Days o Expiraion Days o Expiraion Days o Expiraion Moneyness < 1 [1,63] > 63 Toal < 1 [1,63] > 63 Toal < 1 [1,63] > 63 Toal < [0.9,0.96) [0.96,1.00) [1.00,1.04) > Toal Noes: This Table repors he average absolue in-sample pricing errors of he alernaive models for differen moneyness and mauriy during he period January 000 unil December 000. Moneyness is defined in he following way: an opion is said o be near- or a-he-money if K/F [0.96,1.00] or K/F [1.00,1.04), ou of he money if K/F [0.9,0.96), deep ou-of-he-money if K/F < 0.9 and (deep) in he money if K/F > 1.04, where K is he srike price and F is he forward price. Days o Expiraion is he number of rading days unil mauriy. 0

21 Table 4: Ou-of-Sample Analysis: Average Valuaion Errors across moneyness and mauriy Ad-hoc Black-Scholes EGARCH-Normal EGARCH-runcaed Lévy Days o Expiraion Days o Expiraion Days o Expiraion Moneyness < 1 [1,63] > 63 Toal < 1 [1,63] > 63 Toal < 1 [1,63] > 63 Toal < [0.9,0.96) [0.96,1.00) [1.00,1.04) > Toal Noes: This Table repors he average absolue ou-of-sample pricing errors of he alernaive models for differen moneyness and mauriy during he period January 000 unil December 000. Moneyness is defined in he following way: an opion is said o be near- or a-he-money if K/F [0.96,1.00] or K/F [1.00,1.04), ou of he money if K/F [0.9,0.96), deep ou-of-he-money if K/F < 0.9 and (deep) in he money if K/F > 1.04, where K is he srike price and F is he forward price. Days o Expiraion is he number of rading days unil mauriy. 1