Does Skewness Matter? Evidence from the Index Options Market

Transcription

1 Does Skewness Maer? Evidence from he Index Opions Marke Madhu Kalimipalli School of Business and Economics Wilfrid Laurier Universiy Waerloo, Onario, Canada NL 3C5 Tel: (ex. 87) Ranjini Sivakumar Cenre for Advanced Sudies in Finance School of Accounancy Universiy of Waerloo Waerloo, Onario, Canada NL 3G Tel: (ex. 5703) (preliminary version, April 00) Absrac We model he emporal properies of he firs hree momens of asse reurns and examine if incorporaing ime varying skewness in underlying asse reurns leads o profiable sraegies using a-he-money S&P 500 index opions. We devise rading rules ha incorporae he skewness forecass o rade in a-he-money dela-neural srips, sraps and sraddles. We find ha a simulaed rading sraegy using he GARCHS (skewness) model ouperforms he GARCH model boh before and afer adjusing for ransacion coss. The empirical evidence indicaes ha index opion prices for a-he-money opions do no reflec ime varying skewness. Our resuls sugges ha mispricing of opions causes he negaive skewness in he implici risk-neural disribuion in opion prices. Key Words: condiional volailiy and skewness, opion pricing biases, a-he-money dela-neural srips, sraps and sraddles JEL Classificaion: G0, G4

2 Does Skewness Maer? Evidence from he Index Opions Marke. Inroducion: Exising lieraure has documened significan ime varying skewness in sock index reurns (Harvey and Siddique, 999 and 000 and Hansen, 994). The naural developmen of skewness persisence models is an exension of volailiy persisence models and a direc consequence of asse pricing equaions ha conain hird cenral reurn momens. Harvey and Siddique (999) find srong evidence for ime varying variance and skewness in monhly and weekly sock index daa. Inclusion of condiional skewness is found o aenuae asymmeric variance and seasonaliy effecs in condiional momens and lead o lower persisence in he variance equaion. There is a significan empirical evidence (see e.g. Baes, 996b for a summary) ha he Black-Scholes valuaion model exhibis pricing biases across moneyness and mauriy. Baes (99) shows ha he ou-of-he-money (OTM) pus became very expensive relaive o OTM money calls during he year preceding he sock marke crash in Ocober 987 as skewness premium implici in OTM money opions on S & P 500 fuures became significanly negaive. The negaive skewness premium resuls in a smirk paern in index volailiies. In addiion, Baes (000) documens significan ime varying skewness in sock index opion daa. The ineresing quesion is how does he condiional skewness in he asse reurns affec he underlying risk neural pricing disribuion used in opion valuaion? Jackwerh and Rubinsein (996) documen ha in he pre 987 period, boh he risk-neural disribuion (opion implied disribuion) and he acual disribuion of S&P 500 reurns

3 are abou lognormal. However in he pos 987 period, while he acual disribuion looks abou lognormal again, he risk-neural disribuion is lef-skewed and lepokuric. Baes (000) suggess hree explanaions for he negaive skewness in he implici riskneural disribuion. The firs is ha invesors view he underlying sochasic process for S&P 500 reurns has changed, he second is a change in invesor s risk aversion and he hird reason being a mispricing of pos-crash opions. Bakshi, Cao and Chen (997) and Baes (000) among ohers look a he firs explanaion and propose opion valuaion models ha incorporae he asymmery in he risk neural pricing disribuion. Jackwerh (000) looks a he second explanaion. He empirically derives risk aversion funcions implied by opion prices and realized reurns on he SP500 index for he period In he pos 987 period, he finds negaive risk aversion funcions ha are inconsisen wih economic heory and concludes ha he marke misprices he opions. Bakshi e al. (997) examine opions on he S&P 500 index during he period Their empirical evidence suggess ha overall a model wih sochasic volailiy and random jumps is superior o he Black-Scholes model. Ineresingly, hey find ha for a-he-money (ATM) opions, he Black Scholes model is superior o he more complex models ha include he sochasic volailiy model wih jumps (Bakshi e al. 997 and Baes, 000). Specifically, in he ou-of-sample cross-secional performance, hey find ha ATM call opions (moneyness beween 0.97 and.03), valued using he BS model do no show any mauriy relaed bias. In his paper, we invesigae wheher i is mispricing ha causes he negaive skewness in he implici risk-neural disribuion. We model he emporal properies of he firs hree momens of asse reurns following Hansen (994) and Harvey and

4 Siddique (999) and examine if incorporaing ime varying skewness in underlying asse reurns leads o profiable opion based sraegies. We examine S&P 500 index opions daa during he period November 998 o March 000. Based on he Bakshi e al. (997) findings, we assume ha he Black-Scholes model is he appropriae opion valuaion model and ask wheher embedding skewness in spo pricing models leads o profiable sraegies using ATM opions. We use a framework proposed by Noh, Engle and Kane (994) o esimae he profis from he opions rading sraegies. Noh e al. (994) show ha simple GARCH models (ha incorporae ime varying volailiy) ouperform implied volailiy models for invesors rading in a-he-money sraddles, afer accouning for ransacion coss. We use he GARCHS (GARCH wih condiional skewness) model as in Harvey and Siddique (999) and obain he laen volailiy and skewness from spo daa. The GARCHS rading sraegy leads o rading in a srip or a srap. When condiional skewness is indeed consan, he GARCHS reduces o a GARCH model and boh models should yield similar reurns. We find ha a simulaed rading sraegy using he GARCHS (skewness) model ouperforms he GARCH model boh before and afer adjusing for ransacion coss. The empirical evidence suggess ha index opion prices for ATM opions do no reflec ime varying skewness. Our resuls sugges ha mispricing of opions causes he negaive skewness in he implici risk-neural disribuion. This paper is organized as follows. In secion, we provide a brief lieraure review. In secion 3, we describe he daa and provide he sample descripion saisics. In secion 3

5 4, we discuss he empirical mehodology and presen he resuls on he volailiy models. In he nex secion we presen he resuls on he rading sraegies. Secion 6 concludes.. Background and Lieraure Review: Wha causes skewness or asymmery in reurns? There are a leas four possible explanaions in he lieraure. The firs explanaion is he leverage effec whereby a drop in sock price leads o higher operaing and financial leverage and hence high volailiy in subsequen reurns (Black, 976). The second is based on he volailiy feedback mechanism whereby he direc effec of a posiive shock on volailiy is miigaed by an increase in risk premium, while in he presence of negaive shock boh direc and indirec effecs work o increase he risk premium (Campbell and Henschel, 99). The hird explanaion is based on a possible bursing of a bubble, a low probabiliy scenario wih large negaive consequences (Blanchard and Wason, 98). Finally invesor heerogeneiy and shor sale consrains of invesors explain skewness (Hong and Sein, 999). Hansen (994) provides a model of skewness evoluion in he conex of condiional densiy esimaion using a skewed Suden- disribuion. He proposes a model of skewness ha evolves much like a GARCH process in squares of cubed residuals and applies he approach o he esimaion of US Treasury securiies and he US dollar/swiss Franc exchange rae. He finds evidence of skewness persisence. Harvey and Siddique (999) adap Hansen's approach o a wide number of daily and monhly equiy reurn series. Harvey and Siddique (000) inroduce skewness in he CAPM framework by expressing he sochasic discoun facor or iner-emporal marginal rae of subsiuion as 4

6 a quadraic funcion of he marke reurn. They find ha he coskewness facor (defined as ha par of an asse s skewness ha is relaed o marke porfolio s skewness) has value in cross-secional CAPM regressions across asses. This is in addiion o size and book-o-marke facors ha were proposed by Fama and French (99). The momenum effec in porfolios is found o be relaed o he sysemaic skewness facor. The quesion ha follows is wha does a negaively skewed empirical disribuion imply for he implici risk-neural disribuion in opion prices. We nex review some of he opions relaed lieraure ha looks a his issue. Baes (99) shows ha he ou-of he money pus became very expensive during he laer half of 986, remained so unil early 987 and again during Augus of 987 as skewness premium implici in ou-of- he money opions on S & P 500 fuures became significanly negaive. No such effecs were found during he monhs immediaely preceding he Ocober 987 crash. Following he 987 crash, he negaive skewness premium coninued o be significan ill he end of 987. Ciing he specificaion of he underlying sochasic process as a possible reason for he skewness premium, he paper inroduces a diffusion model wih sysemic jump risk o capure he ime varying skewness in he daa. Using a jump-diffusion model, Baes (996a) finds a significan posiive implici skewness in currency opions on Deusche mark during he period , bu no from The paper shows ha a sochasic volailiy (SV) model wih jumps can explain high kurosis and skewness across differen opion mauriies. Bakshi e al. (997) propose an opion pricing model wih sochasic volailiy, sochasic ineres raes and random jumps. Their empirical evidence suggess ha a model wih sochasic volailiy 5

7 and random jumps is superior o he Black-Scholes model. Baes (000) again considers a SV model now wih ime varying jumps o explain he skewness implici in he S & P 500 fuures opion markes. The paper shows ha models wih SV or a negaive correlaion beween reurns and volailiy alone are no sufficien o generae he high negaive skewness or high volailiy of volailiy in he daa. In relaed research on he underlying sochasic process, Heson and Nandi (000) poin ou ha a GARCH opion valuaion model ha capures he negaive correlaion of spo reurns wih volailiy and he hisorical informaion in volailiy model resuls in reduced moneyness and mauriy biases in opion valuaion. They also show ha he GARCH opion valuaion model is superior o an ad-hoc (smoohed) Black-Scholes model proposed by Dumas, Fleming and Whaley (998). Chen, Hong and Sein (999) using a panel daa of U.S firms, find ha negaive skewness is mos pronounced in socks wih high pas rading volume and reurns and for larger sized socks. Bakshi, Kapadia and Madan (000) show ha risk-neural disribuions for individual socks differ from ha of he marke index by being far less negaively skewed and subsanially more volaile. Jackwerh (000) rules ou changes in invesor risk aversion as a reason for he negaive skewness and suggess mispricing as a possible reason. We explore his explanaion in his paper. 3. Daa and Sampling Procedure: In his sudy, we use S&P 500 daily opions daa and daily index levels from Ocober 998 o March 000. We examine he S&P500 index opions daa because hese opions 6

8 are widely raded. For each day, we use he closing opion price and he closing index level as repored in he Daasream Inernaional daabase. We assume ha he S&P 500 daily dividend yield inerpolaed o mach he mauriy of he opion conrac is a reasonable proxy for he dividends paid on each opion conrac. We use he six-monh Treasury-bill rae as a proxy for he risk-free rae in he Black-Scholes valuaion model. Only opions wih moneyness (srike price/ index level) in he range 0.80 o.0 are included. Opions wih mauriy less han fifeen days and greaer han 80 days are excluded. Only opions wih daily volumes greaer han 00 are reained. For a given exercise price and mauriy, only opions ha have boh pu and call prices are reained. Opions ha violae he pu-call pariy relaionship are excluded. Since he opion marke closes afer he sock marke, he opion holder has a wildcard opion. As in Noh e al. (994), we ignore he wildcard opion, undersaing he profis from he rading rules. Based on hese crieria, our sample consiss of,74 call-pu opions pairs in 7 rading days. Figure presens he weekly S & P 500 index daa for he period We see ha he index surged from mid 90 s onwards and peaked in he year 000 followed by a decline. Table presens he summary saisics of he weekly S & P 500 index daa for he period In general we see ha volailiy, skewness and kurosis have been varying over ime and have been high during he periods of oil shocks in he 70s, he 987 crash period and more recenly during 00. The sample period for our index opions (Nov 998-Mar 00) seems o be characerized by paricularly high volailiy compared o he hisorical average. 7

9 Table presens he summary saisics of he daily S & P 500 index daa for he period In general we see ha volailiy, skewness and kurosis vary over he week and are usually high on Mondays compared o he res of he week. Panel A in Table 3 presens he Augmened Dickey-Fuller uni roo ess for he daily and weekly ime series daa. We canno rejec he uni roo null hypohesis for he index daa. The firs differencing however seems o gives us he saionary reurn series. Panel B presens he Ljung Box saisics for he squared AR() reurn residuals. They indicae high auo-correlaions in he daily and weekly daa ha imply ime dependence in higher order momens such as GARCH effecs. Figures and 3 presen he densiy funcions of he weekly and daily ime series index daa. We see large negaive skewness and fa ails in he daa. 4. Resuls from Condiional Volailiy Models: In his secion we describe he condiional volailiy models and heir resuls based on he ime series index daa. We use he GARCH(,)-M wih leverage condiional skewness and degrees of freedom referred o as GARCHS(,) model- as he omnibus specificaion. Hansen (994) obains a densiy funcion for a random variable driven by is skewness and degrees of freedom in addiion o he firs wo momens (deails in he Appendix).This specificaion is very general and i reduces o several known disribuions as special cases. The GARCHS(,) specificaion is described below. 8

10 9 The above specificaion is he GARCH (,) in mean model wih leverage effec and ime varying condiional skewness and degrees of freedom (df). We refer o his model as Model 6 in our ables. Model has he usual GARCH(,) specificaion and is obained by seing df in Model 6 o a high number above 30 and by consraining skewness, leverage and lagged variance effec in he mean o 0. Model is he GARCH(,)-M wih leverage effec and is obained by seing df in Model 6 o a high number above 30 and by consraining skewness o 0. Model 3 is he EGARCH(,)-M model. Model 4 is obained by consraining he condiional df equaion in Model 6 o have inercep only and is skewness o zero. Model 5 is obained by consraining he condiional df and skewness equaions in Model 6 o only have inerceps. Table 4 presens he resuls for weekly index daa for he period From he Panel A in able 4 we see ha here is a high persisence in variance equaion and a srong evidence for leverage and skewness in he daa. Fa ails are driven by large (predominanly) negaive shocks o he reurns as evidenced by significan coefficien on lagged squared residual in he df equaion. The evidence for he risk premium in he mean equaion (he GARCH in-mean effec) is raher weak. Panel B ells us ha he ), ( ~ ) (, ˆ ˆ 6 Model < = + + = + + = = Ω = = u if u if d u Sk u df u d u h h Z g h u u h r r δ ε δ δ γ ε γ γ β β β β λ η ε ε α α α

11 Model 6 ouperforms ohers in erms of highes likelihood, AIC and SBC values. Model and 3 come ou as winners in erms of Jarque- Bera meric. The likelihood raio meric for nesed specificaions confirms ha Model 6 is a definie improvemen over models, and 4. However here is no much improvemen over Model 5. Table 5 presens he resuls for daily index daa for he period From he Panel A in able 5 we see ha here is a high persisence in variance equaion and a srong evidence for leverage and skewness in he daa. The evidence for he risk premium in he mean equaion (he GARCH in-mean effec) is raher weak. Panel B ells us ha he Model 6 ouperforms ohers in erms of highes likelihood, AIC and SBC values. Model and 3 come ou as winners in erms of Jarque- Bera meric. The likelihood raio meric for nesed specificaions confirms ha Model 6 is a definie improvemen over models, 4 and 5. Figure 4 plos reurns, laen condiional volailiy, skewness and degrees of freedom from he condiional skewness model Model 6- for he S & P 500 index weekly series and figure 5 has a similar plo for he daily index daa for he period In general we find ha periods of high volailiy are also periods of high negaive skewness and faness in he reurn disribuions. 5. Resuls for Opion Trading Sraegies: Table 6 presens he summary saisics of S & P 500 index opions daa for he period Nov 998-Mar 000. In general pus are cheaper relaive o calls and rade more heavily. A-he-money opions (ATMs) also seem o have a shorer mauriy (abou.3 monhs) compared o ou-of he money opions - OTMs (abou -.5 monhs). 0

12 Table 7 presens he resuls for dela-neural sraddles based on compeing models for S & P 500 index opions daa for he period Nov 998-Mar 000. Panel A shows us ha he pu prices are much higher relaive o he call prices. Model (GARCH (,) M wih normal disribuion for he error erm) comes closes o he acual marke prices of calls and pu ands sraddles, while he Model 4 (GARCH (,) M wih uncondiional skewness for he error erm) gives us he lower bounds. In general he model prices are much lower compared o he opion prices implying ha opions are over priced. Panel B (able 7) gives us he number of buys and sells of he dela- neural sraddles for compeing models. We find ha in general sraddles are sold in 74% of he rades and purchased in he remaining 6%. Models 3 and 4 (boh wih disribuions for he error erms) involve larger shor posiions in sraddles han oher models. Panel C (able 7) presens percenage reurns on rading in he dela-neural sraddles for compeing models. We find ha Model beas he simple uncondiional volailiy model. Moreover he condiional skewness model (Model 6) ouperforms all oher models boh before and afer 0.5% ransacion coss. Panels D and E (able 7) replicae Panels B and C resuls using a $ 0.50-filer rule for sock price changes. The filer represens he rading coss per sraddle. Trading akes place only if he absolue price deviaion is greaer han $0.50. We find ha he numbers of rades are now lower because of ariion due o he filer rule; he sraddles are sill sold more ofen han hey are bough. Model now ouperforms all ohers before and afer 0.5% ransacion coss. Figure 7 shows he % reurns from sraddle based on uncondiional volailiy ploed over each day during he sample period. We find ha reurns are more or less saionary

13 around zero excep for a few (four or five) exreme posiive ouliers ha would induce posiive skewness in reurns. Nex we urn o dela-neural srips, sraps and sraddles. Figure 6 shows us he differences beween he sraddles only sraegy and ha based on all srips, sraps and sraddles. Table 8 presens he resuls for dela-neural srips and sraps and sraddles based on compeing models for S & P 500 index opions daa. Panel A shows us ha Model comes closes o he acual marke prices of all he hree opion sraegies sraddles, while he Model 4 gives us he lower bounds. In general he model prices are much lower compared o he opion prices implying ha opions are over priced. Panel B (able 8) gives us he number of buys and sells of he dela- neural sraegies for compeing models. There is ariion in he acual number of rades from 69- his corresponds o hose rades ha do no saisfy he rading decisions laid ou in figure 6. In general we find ha srips, sraps and sraddles are sold in 74% of he rades and purchased in he remaining 6%. The buys and sells are now spread over srips, sraps and sraddles unlike sraddles only in able 7. Panel C (able 8) presens percenage reurns on rading in he dela-neural sraegies for he compeing models. We find ha reurns from boh condiional skewness models,- Models 5 and 6, ouperform all he resuls repored in able 7 boh before and afer 0.5% ransacion coss. The -saisics indicae ha he reurns from he sraegy are significanly differen from zero. Panels D and E (able 8) replicae Panels B and C resuls using a $ 0.50-filer rule for sock price changes. We find ha he numbers of rades are now lower because of

14 ariion due o he filer rule; he number of sells sill overwhelms he number of buys. Reurns from boh skewness models sill ouperform all ohers repored in able 7 before and afer 0.5% ransacion coss. 6. Summary and Conclusions: We invesigae wheher i is mispricing ha causes he negaive skewness in he implici risk-neural disribuion in S&P 500 index opion prices. We model he emporal properies of he firs hree momens of asse reurns following Hansen (994) and Harvey and Siddique (999) and examine if incorporaing ime varying skewness in underlying asse reurns leads o profiable sraegies using a-he-money opions. We find ha a simulaed rading sraegy using he GARCHS (skewness) model ouperforms he GARCH model boh before and afer adjusing for ransacion coss. The empirical evidence suggess ha index opion prices for ATM opions do no reflec ime varying skewness. Our resuls sugges ha mispricing of opions causes he negaive skewness in he implici risk-neural disribuion. 3

15 Appendix: Condiional Skewness Model: The GARCHS (,) specificaion for he condiional mean, condiional variance and condiional skewness, where he error erm in he mean has a skewed condiional suden disribuion wih changing degrees of freedom, is as follows: Condiional mean: where, = ˆ α + ˆ 0 αr + ˆ α h r + u u = ε and ε Ω ~ g( z η, λ) h where g ( ) is as described below. Condiional variance: h = β 0 + βh + β u + β3d u where, 0 if u 0 d = if u < 0 Condiional skewness Sk = δ + 0 δ ε + δ u Degrees of freedom: df = + 0 γ ε + γ u γ where < df < The likelihood funcion for he skewed disribuion (Hansen 994) is: g bz + a b c + η λ bz + a b c + η + λ η+ ( z η, λ) = η+ z < z a b a b where he ea sands for he degrees of freedom and is bounded as < η < and lambda is he skewness parameer and is bounded as < λ <. Furher he consans a, b and c are as defined below. 4

16 5 Γ + Γ = + = = ) ( 3 4 η η π η λ η η λ c a b c a Hansen (994) show ha his densiy funcion has a zero mean and uni variance. Seing lambda o zero gives us a regular -disribuion and seing ea o a high number over 30 and lambda o zero gives us a regular sandard normal disribuion.

17 Bibliography: Bakshi, G, Kapadia, N, and D. Madan, 000, Sock reurn characerisics, skew laws and he differenial pricing of individual equiy opions, Review of Financial Sudies forhcoming. Bakshi, G., C. Cao and Z. Chen, 997, Empirical performance of alernaive opion pricing models, Journal of Finance 5, Baes, David S., 99, The crash of 87: Was i expeced? The evidence from opions markes, Journal of Finance 46, Baes, David S., 996a, Jumps and sochasic volailiy: exchange rae processes implici in PHLX deusche mark opions, Review of Financial Sudies 9, Baes, David S., 996b, Tesing opion pricing models, in G. S. Maddala, C. R. Rao, eds., Saisical Mehods in Finance, Amserdam, Elsevier, Baes, David S., 000, Pos- 87 crash fears in he S&P 500 fuures opion marke, Journal of Economerics 94, Black, Fisher, 976, Sudies of sock price volailiy changes, Proceedings of he 976 meeings of he American Saisical Associaion, Business and Economical Saisics Secion, 77-8 Blanchard, O and M. Wason, 98, Bubbles, raional expecaions and financial markes, Paul Wachel ed., Crisis in economic and financial srucure, Lexingon MA: Lexingon Books, Campbell, J and L. Henschel, 99, No news is good news: an asymmeric model of changing volailiy in sock reurns, Journal of Financial Economics, 3,

18 Chen, J, H. Hong and J. Sein. 999, Forecasing crashes: Trading volume, pas reurns and condiional skewness in sock prices, NBER Working Paper, W 7687 Universiy Dumas, B., J. Fleming, and R. E. Whaley, 997, Implied volailiy funcions: Empirical ess, Journal of Finance 53, Fama, E and K. French, 99, The cross secion of expeced reurns, Journal of finance, 47, Harvey, C. R. and A. Siddique, 999, Auoregressive condiional skewness, Journal of Financial and Quaniaive Analysis 34, Hansen, B. E., 994, Auoregressive condiional densiy esimaion, Inernaional Economic Review 35, Harvey, C. R. and A. Siddique, 000, Condiional skewness in asse pricing ess, Journal of Finance 55, Heson, S., L., and S. Nandi, 000, A closed-form GARCH opion valuaion model, Review of Financial Sudies 3, Jackwerh, J. C., and M. Rubinsein, 996, Recovering probabiliy disribuions from opion prices, Journal of Finance 5, Jackwerh, J. C., 000, Recovering risk aversion from and realized reurns, Review of Financial Sudies 3, Noh, J., R. Engle, and A. Kane, 994, Forecasing volailiy and opion prices of he S&P index, Journal of Derivaives, Fall 984,

19 Table Summary saisics based on weekly S & P 500 index reurns Year Number of obs. % mean weekly reurns % median weekly reurns % annualized sandard deviaion skewness kurosis Whole period Time series daa Opions daa Nov 998-Mar

20 Table Summary saisics based on daily S & P 500 index reurns Number of obs. % mean daily reurns % median daily reurns % annualized sandard deviaion skewness kurosis Monday Tuesday Wednesday Thursday Friday Whole period:

21 Table 3 Uni roo and GARCH ess based on he weekly S & P 500 index daa and daily S & P 500 index daa Panel A: ADF ess based on regressions wih inercep y = α + γy + β y + ε 0 8 i= i i+ weekly daa daily daa Index Reurns We repor he ADF es saisics for he gamma coefficien for he following regression. The null of uni roo is represened as γ=0. The criical value is.86 a 95% confidence level. Panel B: Ljung-Box es saisic value for he squared AR() residuals from reurn series Lag Ljung-Box Saisic Weekly daa Ljung-Box Saisic Daily daa χ (lag) saisic ( 95% confidence level) We repor he Ljung-Box saisic for he squared residuals from he AR() reurn process a differen lags. The Ljung-Box saisic for squared residuals is significan for daily daa and weekly daa up o lag 0. 0

22 Table 4 Esimaes of compeing condiional volailiy and skewness models based on weekly S & P 500 index daa Panel A: Model esimaes Model Model Model 3 Model 4 Model 5 Model 6 inercep 0. (0.046) r (0.06) h (0.0) Mean- Equaion: (0.088) (0.074) (0.084) (0.06) (0.0) (0.039) (0.09) (0.09) 0.07 (0.076) (0.06) 0.03 (0.07) 0.9 (0.077) (0.05) 0.00 (0.06) inercep 0.9 (0.078) h (.6) U - 0. (0.40) I - *U - Variance Equaion: (0.073) (0.0) (0.079) (0.5) (0.05) (0.36) (0.5) (0.7) (0.039) (0.04) ε - -sqr(/π) 0.99 (0.07) ε (0.0) 0.6 (0.078) (0.39) (0.65) 0.3 (0.037) 0.36 (0.075) (0.364) (0.69) 0.46 (0.04) Degrees of freedom Equaion: inercep (0.657) (0.477) 0.43 (0.56) U (0.05) U (0.005) DF: Skewness Equaion: inercep (0.086) (0.09) U U - (0.0) -0.0 (0.006) SK 0.73 # of parameers Log likelihood T : 680. Sandard errors in parenheses

23 Panel B: Model Comparisons Model Model Model 3 Model 4 Model 5 Model 6 skewness kurosis JB saisic AIC SBC LR saisic 89.6*** 48.*** - 7.4*** *** all significan a % significance level

24 Table 5 Esimaes of compeing condiional volailiy and skewness models based on daily S & P 500 index daa Panel A: Model esimaes Model Model Model 3 Model 4 Model 5 Model 6 inercep (0.08) r (0.03) h (0.036) Mean- Equaion: (0.09) (0.03) (0.03) (0.05) (0.04) (0.04) (0.034) (0.044) (0.09) (0.0) 0.05 (0.033) 0.03 (0.03) (0.03) 0.0 (0.04) inercep (0.363) h (..5) U (0.65) I - *U - Variance Equaion: (0.55) (0.004) (0.08) (0.4) (0.007) (0.309) (0.09) (0.03) (0.04) (0.0) ε - -sqr(/π) 0.30 (0.07) ε (0.06) 0.04 (0.06) 0.90 (0.76) (0.0) 0.34 (0.0) 0.04 (0.06) 0.94 (0.64) (0.03) 0.46 (0.04) Degrees of freedom Equaion: inercep (0.54) (0.7) (0.9) U (0.38) U (0.09) DF: Skewness Equaion: inercep (0.07) (0.074) U U - ( (0.0) SK # of parameers Log likelihood T : 083. Sandard errors in parenheses 3

25 Panel B: Model Comparisons Model Model Model 3 Model 4 Model 5 Model 6 skewness kurosis JB saisic AIC SBC LR saisic 0.8*** 8.*** ***.9*** - *** all significan a % significance level 4

26 Table 6 Summary saisics of S & P 500 index opion daa Nov 998-Mar 000 Panel A: Averages average mauriy (days) average prices $ average moneyness (X/S) average volume ATM calls ATM pus OTM calls OTM pus Panel B: Medians median mauriy (days) median prices $ median moneyness (X/S) median volume ATM calls ATM pus OTM calls OTM pus

27 Table 7 Dela-neural sraddles based on compeing models for S & P 500 index opion daa Nov 998-Mar 000 Panel A: Average prices of he dela-neural sraddles for compeing models call pu sraddle marke price Uncondiional volailiy Model GARCH(,)-M Model 3 EGARCH(,)-M Model 4 GARCH(,)-M+ df Model 6 GARCH(,)-M +cdf+cskew Number of observaions 70. Average moneyness and mauriy of he dela-neural sraddles are.044, and days respecively Panel B: Number of buys and sells of dela-neural sraddles for he compeing models oal rades buys sells Uncondiional volailiy Model GARCH(,)-M Model 3 EGARCH(,)-M Model 4 GARCH(,)-M (-disn) Model 6 GARCH(,)-M +cdf+cskew Panel C: % Reurns on rading in he dela-neural sraddles for compeing models Before ransacion coss Afer ransacion coss of 0.5% % daily reurn % daily reurn # of mean median sd. -sa mean median sd. dev -sa obs dev Uncondiional volailiy Model : GARCH(,)-M Model 3: EGARCH(,)-M Model 4: GARCH(,)-M+df Model 6: GARCH(,)-M +cdf+ cskew

28 Panel D: Number of buys and sells of dela-neural sraddles for he compeing models wih $0.50 filer for sock prices oal rades buys sells Uncondiional volailiy Model GARCH(,)-M Model 3 EGARCH(,)-M Model 4 GARCH(,)-M (-disn) Model 6 GARCH(,)-M +cdf+cskew Panel E: % Reurns on rading in he dela-neural sraddles for compeing models wih $0.50 filer for sock prices Before ransacion coss Afer ransacion coss of 0.5% % daily reurn % daily reurn # of mean median sd. -sa mean median sd. dev -sa obs dev Uncondiional volailiy Model : GARCH(,)-M Model 3: EGARCH(,)-M Model 4: GARCH(,)-M+df Model 6: GARCH(,)-M +cdf+ cskew

29 Table 8 Dela- neural srips and sraps based on he condiional skewness model for S & P 500 index opion daa Nov 998-Mar 000 Panel A: Average prices of he dela-neural srips, sraps and sraddles for compeing models call pu sraddle marke price Uncondiional volailiy Model GARCH(,)-M Model 3 EGARCH(,)-M Model 4 GARCH(,)-M+ df Model 5 GARCH(,)-M+ df+ skew Model 6 GARCH(,)-M +cdf+cskew Number of observaions 70. Average moneyness and mauriy of he dela-neural sraddles are.044, and days respecively Panel B: Number of buys and sells of dela-neural srips, sraps and sraddles for he compeing models rading in sraddles rading in srips rading in sraps oal rades buys (%) sells(%) buys (%) sells(%) buys (%) sells(%) Model 5: GARCH(,)-M +df+skew Model 6: GARCH(,)-M +cdf+cskew Panel C: % Reurns on rading in he dela-neural srips, sraps and sraddles for compeing models Before ransacion coss Afer ransacion coss of 0.5% % daily reurn % daily reurn # of mean median sd. -sa mean median sd. dev -sa obs dev Model 5: GARCH(,)-M+df + skew Model 6: GARCH(,)-M +cdf+ cskew

30 Panel D: Number of buys and sells of dela-neural srips, sraps and sraddles for he compeing models wih $0.50 filer for sock prices Model 5: GARCH(,)-M +df+skew Model 6: GARCH(,)-M +cdf+cskew rading in sraddles rading in srips rading in sraps oal rades buys (%) sells(%) buys (%) sells(%) buys (%) sells(%) Panel E: % Reurns on rading in he dela-neural srips, sraps and sraddles for compeing models wih $0.50 filer for sock prices Before ransacion coss Afer ransacion coss of 0.5% % daily reurn % daily reurn # of mean median sd. -sa mean median sd. dev -sa Model 5: GARCH(,)-M+df + skew Model 6: GARCH(,)-M +cdf+ cskew obs dev

31 Figure S & P 500 index weekly series

32 Figure Densiy funcion for he S & P 500 index weekly series Figure 3 Densiy funcion for he S & P 500 index daily series

33 Figure 4 Plos of reurns, laen condiional volailiy, skewness and degrees of freedom from he condiional skewness model for he S & P 500 index weekly series

34 Figure 5 Plos of reurns, laen condiional volailiy, skewness and degrees of freedom from he condiional skewness model for he S & P 500 index daily series

35 Figure 6 Trading sraegies involving opions 34

36 Figure 7 Plos of he S & P 500 index ATM sraddle prices, reurns and mauriy for he period Nov 998-Mar