Interpreting the Volatility Smile: An Examination of the Information Content of Option Prices

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 706 August 2001 INTERPRETING THE VOLATILITY SMILE: AN EXAMINATION OF THE INFORMATION CONTENT OF OPTION PRICES Steven A. Weinberg NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

INTERPRETING THE VOLATILITY SMILE: AN EXAMINATION OF THE INFORMATION CONTENT OF OPTION PRICES Steven A. Weinberg* Abstract: This paper evaluates how useful the information contained in options prices is for predicting future price movements of the underlying assets. We develop an improved semiparametric methodology for estimating risk-neutral probability density functions (PDFs), which allows for skewness and intertemporal variation in higher moments. We use this technique to estimate a daily time series of riskneutral PDFs spanning the late 1980's through 1999, for S&P 500 futures, U.S. dollar/Japanese yen futures and U.S. dollar/deutsche mark futures, using options on these futures. For the foreign exchange futures, we find little discernable additional information contained in the estimated PDFs beyond the information derived from the Black-Scholes model, a fully parametric specification. For S&P 500 futures, we find that the riskneutral distribution implied by the volatility smile better fits the realized returns than the Black-Scholes model, although this better overall fit is not exhibited in the second and third moments. JEL Classification: F31, G13, G15 Keywords: foreign exchange; derivative asset pricing; probability density functions * The author is an economist in the International Finance Division of the Federal Reserve Board and can be reached by e-mail at steven.a.weinberg@frb.gov. Helpful comments were provided Doug Bernheim, Alain Chaboud, Neil Ericsson, Linda Kole, Mico Loretan and Will Melick. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1 Introduction Part of the job of risk management is assessing the likelihood of a large adverse price move of a financial asset. Itwouldbeusefultomarketparticipantsinmakinginvestmentdecisionstoknowhowtheirindividual assessmentof the risk ofholding an assetcompareswith the market’sassessmentof this risk. Policy makers would also benefit from knowing the market’s assessment of risks. The Black-Scholes option pricing model (1973) allows for the computation of an estimate of market expectations of the volatility of the assets underlyingthe options. Theseestimateshavegenerallybeenfoundto be goodpredictorsoffuture volatility. However, researchers have noted that a maintained assumption of the Black-Scholes model, that the price path of the underlying assetis fully described by geometric Brownianmotion, tends not to hold in practice. This is reflected in the presence of a “volatility smile,”1 the empirical fact that implied volatilities for optionswithstrikepricesthatarefaroutofthe moneytendtoexceedthoseofat-the-moneyoptions. These differencesinimplied volatilitiesacrossstrikepricescanbe exploitedtoobtainestimatesofthe impliedriskneutral probability density functions (PDF) of expected returns. The purpose of this paper is to examine the information content of the PDFs estimated from options prices, in particular to determine how helpful risk-neutral PDFs are in predicting the future distribution of returns. A unique element of this study is that we not only examine how the risk-neutral distributions implied by options prices compare with realized distributions, but also test the information content of the second and third moments of the risk-neutral distribution against the corresponding moments of daily returns. In practice, a risk manager is generally most concerned about large price movements – high volatility – and is likelytobeevenmoresensitivetolargemovesthatdecreasethetotalvalueofassetsunderhiscontrol–high skewness. Having reliable measures for predicting volatility and skewness would help a risk manager better assess when his portfolio is exposed to adverse risks, as he then might mitigate this exposure. Soon after publication of the Black-Scholes option pricing model (1973), stock return volatility derived from option prices was compared with realized volatility. Early on, Latan´e and Rendelman (1976) used prices of options listed on the Chicago Board Options Exchange to compare the realized volatility of a cross-section of stocks with the volatility implied by the Black-Scholes model. They found that forecasts using implied volatility did better in predicting realized volatility than those based on historical volatility. Later studies examining the time-series properties of Black-Scholes implied volatilities confirmed, by and large, that these are more useful than historical volatility for forecasting realized future volatility. More 1SeeDermanandKani(1994)foranoverviewofthevolatilitysmile. 1

recently, economic and financial researchers have developed a number of techniques to estimate the optionimplied probability density function (PDF) of an underlying security’s price movements. Much of the body ofresearchinthis fieldhas,thus far,focusedondevelopingandcomparingestimationtechniquesforimplied PDFs. However,onlyahandfulofstudieshaveexaminedwhethertheinformationderivedfromoptionprices can be used to forecast more than just the realized volatility, i.e., to predict further characteristics of the realized distribution. This study attempts to address this gap. Financial assets were selected for this study based upon their liquidity, range of strike prices, and the existence of sufficiently long series of historical data. Options on futures traded on the Chicago Mercantile Exchange (CME) were chosen because they provide synchronized daily settlement prices that are matched with the settlement price for the underlying futures contract. We formalize the range of strikes criterion by requiring that for each day at least ten strike prices be actively traded. The three underlying financial assets that best met the selection criterion were the S&P 500 index futures (SP), the U.S. dollar/Japanese yen exchange rate futures (JY) and the U.S. dollar/deutsche mark futures (DM).2 For each day in the sample, we estimate a risk-neutral PDF of the underlying futures price. To obtain our PDF estimates, we extend a well-known nonparametric estimation technique by applying a variation of the generalizedcross-validationmethodtodeterminethesmoothingparameterforacubicsmoothingspline. We initiate the investigation by utilizing a number of different tests of the goodness of fit to determine whether the risk-neutralPDF obtainedby the volatilitysmile smoothingmethod is areasonablerepresentation of the true stochastic distribution. We also compare the fit of the PDF obtained by the volatility smile smoothing method against a lognormal distribution obtained by the parametric Black-Scholes model. We find that both models fit the realizedreturns of the JY and DM series reasonablywell, but surprisingly,the lognormaldistributionoutperformsthemoreflexibilesemiparametricdistributionintermsofgoodnessoffit. Neithermodel,however,fitrealizedreturnsoftheSPserieswell,asriskneutralityisinadequateformodeling equitypricemovements. Tocorrectforriskneutralityweshiftthe distributionsbytheaverageexcessreturn oftheS&P500futuresovertheperiod. Fortherisk-adjustedSPcase,wefindthatthedistributionobtained usingthe volatilitysmile smoothingmethoddominatedthe lognormaldistribution. Infact,wecannotreject at the 10% level the hypothesis that the distribution of realized returns is statistically equivalent to that derived from the volatility smile smoothing method. 2TheinstrumentontheCMEwiththehighestoptionsandfuturesvolumesisthethree-montheurodollarcontract. However, formuchofthetimethatoptionsonthiscontracthastraded,thenumberofactivelytradedstrikepriceshasbeenquitelimited. 2

WethenutilizetheestimatedPDFstorevisittheliteratureontestingtheinformationcontentofimplied volatility. We examine the predictive ability of three measures of volatility: a moving average of historical volatility, the implied volatility from the at-the-money option, and the implied volatility of the distribution obtained using the smoothed volatility smile method. We find that both measures of implied volatility containsignificantinformationfor predicting realizedvolatility beyondthat of the movingaveragemeasure. However, we also find that implied volatility of the distribution obtained using the complete set of traded strike prices is highly correlated with the implied volatility of just the at-the-money option, with neither measure of implied volatility dominating the other for the three data series we examine. This paperthen examinesthe next-highermoment, utilizing two measuresofskewnessto determine how well implied skewness performs in predicting realized skewness. We find that implied skewness does not help in predicting realized skewness, though historical skewness is not a good predictor of realized skewness either. We find that in situations of asymmetric PDFs, we could have profited from a strategy of selling an option with greater implied volatility than on the option on the opposite side of the PDF, and buying the less expensive option. This suggests that investors either consistently misassessed the probability of large moves, or more likely, that investors were willing to pay more than the actuarial cost for protection against large adverse price movements. The remainder of this paper is organized as follows. We review previous studies that examined the role of options prices in predicting various characteristics of the distribution of price movements in section 2. Our methodology for estimating risk-neutral PDFs is presented in section 3. Section 4 discusses the data and gives summary statistics for the data and estimated PDFs. Section 5 examines the goodness of fit of the overalldistribution. The information content of the second moment estimates is presented in Section 6. The information content of the third moment is addressed in Section 7, and the final section offers some conclusions. 2 Literature Review We reviewpreviousstudies thathaveempiricallyexaminedthe informationcontentofoptionspricessothat theresultsofourstudycanbemorereadilycompared. Thisalsohelpsdemonstratethemorenovelempirical contributions of this paper. 3

2.1 Implied Volatility Studies Research examining the information content of implied volatility addresses only a subset of what we study in this paper, since option-implied volatility is merely the second moment of the PDF. Findings of previous empiricalstudies havebeen mixed. Canina and Figlewski(1993)examinedthe most actively tradedoptions contractintheUnitedStates,ontheS&P100index.3 CaninaandFiglewski(CF)regressedrealizedvolatility overthe remaining life ofthe contractonimplied volatility andfound thatimplied volatility containedlittle information for predicting realized volatility. In contrast, CF reported that a moving average of volatility overthe previous60tradingdayshadmuchgreaterpredictivepower. Whenrealizedvolatilitywasregressed on both implied volatility and historical volatility, CF found that the contribution of implied volatility was insignificant. Christensenand Prabhala(1998)reexaminedthe findings of CF, using a longer sample period that included nearly eight years of data after the 1987 stock market crash, as well as the four years of pre-crashdatausedbyCF. ChristensenandPrabhala’sempiricalfindings differedfromthoseofCF,inthat impliedvolatilitywasfoundtocontainsubstantiallymoreinformationandhistoricalvolatilityhadmuchless explanatory power. Jorion (1995) also addressed the CF findings, examining implied volatility from options on foreign exchange futures traded on the CME. Jorion’s main finding also ran counter to that of CF; when realized volatility is regressedon implied volatility4, historical volatility becomes insignificant and implied volatility hassubstantialpredictivepower. Thisinformationalsuperiorityofimpliedvolatilityoverhistoricalvolatility in predicting future volatility was called into question by Neuhaus (1995), who considered options on German Bund futures traded on the London International Financial Futures and Options Exchange (LIFFE). Neuhausconcludedthatwhile impliedvolatilitywasusefulinpredictingrealizedvolatility,historicalvolatility outperformed implied volatility in prediction. 2.2 Information Content of the Probability Density Function There have only been a few studies until now that have examined the predictive capabilities of the wealth of information containedin time series estimates of PDFs. One reasonis that the density function obtained from options prices are based on a risk-neutral pricing kernel which may differ from the subjective density thatmarketparticipantsplaceonfuturereturns. Sincethesubjectivedensitycannotbederivedfromoptions 3Our data set differs somewhat, as we used options on S&P 500 futures, whereas Canina and Figlewski used options on a cashindex. 4Joriondefinedhistoricalvolatilityastheaveragevolatilityoverthepreceeding20tradingdays. 4

prices without additional assumption,5 the risk-neutral density has been used instead. Bates (1991, 1997) links the risk neutral stochastic process to the actual stochastic process generating returns and derives a measure to assess abnormalexpectations of return skewnessfrom the risk-neutralPDF. Bates (2000)found thatthehighdegreeofnegativeskewnessimpliedinS&P500indexfuturesoptionpricesisinconsistentwith two option pricing models that extend the Black-Scholesmodel — stochastic volatility models, such as Hull and White (1987) and Heston (1993), and jump diffusion models such as Merton (1976) and Bates (1991). Thus far, models that attempt to representthe underlying stochastic processin assetprice returns havenot been successful in generating options prices that match market prices. This lack of fit offers the potential for methods that simply use observed option prices to extract risk-neutral PDFs to have better predictive power than methods based on estimating the underlying stochastic process. Fackler and King (1990) developed a method, which they called the empirical calibration function, to evaluate the fit between the option-based risk-neutral distribution and the actual returns. Their method does not explicitly differentiate between the risk-neutraldensity and the subjective density. The calibration function is based on transforming a series of independent observations of realized prices to cumulative distribution function values under the hypothesized distribution. Fackler and King applied their calibration functiontovariousagriculturalcommodityoptionsandfoundthatthelognormaldistributionobtainedfrom parametric Black-Scholes model fit most of the price distributions they assessed. Silva and Kahl (1993) looked at more recent data for some of the same markets that Fackler and King investigated, and found that the fit of the lognormal risk-neutral distribution improved in the later years, suggesting that prices in immature or illiquid markets tend to be less efficient. Dumas, Fleming and Whaley (1998) defined a statistical measure, the hedge portfolio error,to compare the forecasting performance of various option pricing techniques. The hedge portfolio error is the difference betweenthe changeofthe marketquotedoptionpriceandthe changeinits model-derivedtheoreticalvalue. For S&P 500 index options from June 1988 to December 1993, Dumas, Fleming and Whaley examined a series of one week intervals and found that the Black-Scholes model had a smaller hedge portfolio error than techniques that took into account the shape of the volatility smile. However, in examining the hedge portfolio error on FTSE-100 index options trading on the LIFFE over one day intervals between 1987 and 1997, Gemmill and Saflekos(1999) found, using an implied distribution based on a mixture of lognormals, that the error was smaller when taking into account the volatility smile, as compared to the Black-Scholes 5Jackwerth (1996), A¨ıt-Sahalia and Lo (2000) and Coutant (1999) have studied the relationship between the risk neutral density,thesubjectivedensityandtheriskaversionfunction. 5

model. However, the smaller hedge portfolio errors only translated to improved pricing of about the same magnitude as the bid/ask spread on the options. Most of the quantitative applications of PDFs have related the skewness of the risk-neutral distribution to the returns ofthe underlying asset. A number of studies have noted the highdegree of negativeskewness implied by equity options. Bates (1991) found that the PDF of the S&P 500 index futures was negatively skewed between October 1986 and August 1987, which he interpreted as caused by crash fears. However, the implied distribution was not negatively skewed for the two months before the October 1987 crash. Jackwerth and Rubinstein (1996) examined options on the S&P 500 index from 1986 through 1993 and foundthat sincethe 1987crashthe risk-neutralprobabilityofa largedecline inthe index is far greaterthan underthe assumptionoflognormality. NakamuraandShiratsuka(1999)lookedatoptionsonthe Nikkei225 index between 1989 and 1996 and found that skewness moved in the opposite direction to changes in the index, which they interpretedas a lag by marketparticipants in adjusting their expectations to new market levels. A risk-neutral PDF may better represent the actual distributions for foreign exchange returns than for equityindexreturns. Oneofthereasonspositedfortheexcessskewnessfoundinrisk-neutralPDFsofequities isthe highermarginalutility ofwealthunderastatewherethemarketexperiencedalargedownturn. While the probabilityofa crashmaybe low,the marketwill value aunit ofwealthmuchmore highly inthis state, blurring the ability to distinguish between changes in market’s perception of the probability of a crash and the value of wealth in this state. Following this line of reasoning, a change in a foreign exchange rate is not as likely to generate the same change in the marginal utility of wealth as a change in an equity index. A change in the exchange value of a currency is less likely to have a noticeable impact on an individual’s wealth, given the home bias in asset allocation. In addition, movements in exchange rates among the major currencies likely generate little net change in aggregate global wealth. Campa, Chang and Reider (1998) looked at implied skewness in one- and three-month over-the-counter options on a number of different exchange rates between April 1996 and March 1997. They found that the direction of skewness was positively correlated with returns over the remaining length of the option. They interpreted this as indicating positive momentum, i.e., a strong currency is likely to get stronger, rather than revert to an earlier level. Malz (1997b) also examined over-the-counter foreign exchange rates, using one-month options from April 1992 through June 1996. Malz found that investors can earn excess returns (in a CAPM sense) by holding currencies whose option prices indicate positive skewness. 6

The implied skewnessof PDFs fromforeignexchangeoptions has alsobeen usedin event studies. Leahy and Thomas (1996) examined the PDF of the U.S. dollar-Canadian dollar exchange rate around the time of the Quebec independence referendum in 1995. Galati and Melick (1999) looked at the effect of market perceptionsofforeigncurrencyinterventionoperationsbytheFederalReserveandtheBankofJapanonthe distribution of the dollar/yen exchange rate. Galati and Melick found that while there was no statistically significant effect on the skewness of the PDFs, the market perception of intervention was associated with a higher variance of future spot rates. 3 Estimation of Implied PDFs 3.1 Review and Comparison of Estimation Methods Employing the taxonomy of methods presented in Chang and Melick (1999), the techniques used to recover the risk-neutralprobabilitydistributionfunction of anassetonthe date the optioncontractexpires fallinto two broadcategories. The first semiparametriccategoryis basedonthe observation– presentedby Breeden andLitzenberger(1978)–thattherisk-neutralPDF,f(S),oftheassetprice,S,canbedeterminedbytaking the second derivative of the price of a European-style call, C, with respect to its strike price, X: ∂2C(S,T,X) f(S)=erT , (1) ∂X2 where T is the time to expiry of the option. Putting this technique into operation has been far from straightforward,as the range of available strikes generally falls far short of the continuity in C(:,X) needed to produce reasonably smooth PDFs. Basedon a resultshownby Cox andRoss (1976)that the price ofa call optionis equalto its discounted value under a risk-neutral basis: (cid:1) ∞ C(T,X)=e−rT (S−X)f(S)dS, (2) X a second category of techniques imposes some structure on the function f(S) in order to guarantee that a proper probability distribution will be found. In empirical work, the most frequently used functional form for f(S) has been the mixture of lognormaldensities.6 Some other functional forms that have been applied 6MelickandThomas(1997) 7

include the Burr III polynomial,7 a Hermite polynomial expansion around the lognormal density,8 and a Bayesian maximum entropy estimate using the lognormal density as the prior.9 A number of papers have compared the performance of some of the functional forms used to estimate the risk-neutralPDFbasedonequation(2). Coutant,JondeauandRockinger(1998)comparedthe mixture of lognormals with the Hermite polynomial expansion and the method of Bayesian maximum entropy, and found that the three methods produced similar PDFs, though the Hermite expansionwas found to be more robust. McManus (1999) also compared the mixture of lognormals, the Hermite polynomial expansion and the Bayesian maximum entropy method in terms of their accuracy in fitting the options data. McManus found that the mixture of lognormals performed best. BlissandPanigirtzoglou(2000)comparedthe mixture oflognormalstechnique withamethod, tobe discussedbelow,basedonequation(1). ThebasisforBlissandPanigirtzoglou’scomparisonwastherobustness and stability of the estimated PDFs to small errors in the quoted option prices. Bliss and Panigirtzoglou randomly perturbed the quoted prices by up to one-half of the contract’s tick size,10 so that the perturbed prices would still round to the quoted prices if they were constrained (as they would be when traded on the exchange) to only move in tick increments. Bliss and Panigirtzoglou found strong evidence in favor of greater robustness of the volatility smoothing method over a mixture of lognormals method. GivenBlissandPanigirtzoglou’sfindingthatthevolatilitysmoothingmethodproducedmorerobustPDF estimates than those based on equation (2), an extension of the volatility smoothing method was developed for this paper. 3.2 Volatility Smoothing Method The idea of smoothing the “volatility smile” to obtain continuous option prices was introduced in Shimko (1993). The volatility smile is obtained by applying the Black-Scholes options pricing formula to listed options at each strike price to calculate the implied volatility. Shimko then fit the implied volatilities as a parabolic function of strike price; outside the range of listed strikes, Shimko assumed that implied volatility was constant. By obtaining continuous call option prices, the PDF can be directly found by twice differentiating call prices with respect to strike price, as given by equation (1). It should be noted that this technique does not require that the maintained assumptions of the Black-Scholes option price model 7Sherrick,GarciaandTirupattur(1996) 8MadanandMilne(1994) 9BuchenandKelley(1996) 10Theticksizeisthesmallestincrementthatanexchange quotedpricemaymove. 8

hold;theBlack-Scholesmodelmerelyservesasameansoftransformingoptionpriceinformationintoamore tractable form. Campa, Chang and Reider (1997) improved upon the fit of the implied volatility function by employing a cubic smoothing spline instead of the parabolic function. While exchange-listed options are priced in terms of specific strike prices, the pricing convention in over-the-counter (OTC) foreign exchange options is represented in terms of delta, where δ ≡ ∂C/∂S is the derivative of the Black-Scholes call option price with respect to the underlying asset price.11 Utilizing the OTCpricingconvention,Malz(1997a)smoothedBlack-Scholesimpliedvolatilitiesasaparabolicfunctionof delta. ThemethoddevelopedbyBlissandPanigirtzoglouentailssmoothingBlack-Scholesimpliedvolatilities via a cubic spline using delta rather than strike price as the independent variable. There are some operational advantages to using delta rather than the strike price as the independent variable. First, as shown in Clews, Panigirtzoglou and Proudman (2000), working in delta space, where possible values range from 0 to e−rT, is more conducive to examining the term-structure of volatility as opposed to working in strike space, where the distribution of strike prices widens as time to maturity, T, increases. Second,the transformationfromstrikespaceinto delta space givesprominenceto the mostliquid contracts, the at-the-money-contracts, which trade near the current spot price of the asset. Near the spot price, a given change in the strike price translates into a relatively large change of delta, while a given change in the strike price farther from spot translates into a smaller change in delta. This property of the transformationfromstrike space into delta space playedanimportantrole in the greaterrobustnessof Bliss andPanigirtzoglou’sPDFestimates overthose obtainedusing a mixture-of-lognormalstechnique. Bliss and Panigirtzoglou further emphasized the more liquid contracts by weighting, in the smoothing spline, each observation in terms of the parameter vega, ν, where ν ≡ ∂C/∂σ is defined as the derivative of the Black- Scholes call price with respect to the Black-Scholes implied volatility. The value of an option becomes less sensitive to volatility the further an option’s strike price is from the spot price. While the cubic smoothing spline technique used by Bliss and Panigirtzoglou produced more robust results than the mixture of lognormals, for general usage, it remained unresolved how to optimally choose the appropriate amount of smoothing to apply. For a given λ, the smoothing spline is determined by minimizing: (cid:2)N (cid:3) (cid:4) (cid:5)(cid:6) 2 (cid:1) e−rT min w(i) y(i)−f δ(i) +λ f(cid:3)(cid:3) (δ) 2dδ, (3) f(δ) i=1 0 11Itiseasilyshownthat0≤δ≤e−rT. 9

whereN is the numberofunique strikepricesandf is apiecewisecubic polynomialwithknotpointsδ(i)at the observed deltas. The weight of each observation,w(i), is set equal to ν(i), and y(i) is the Black-Scholes implied volatility for observation i. The cubic polynomial f is constrained so that it has continuous first and second derivatives at the knot points. The smoothness is determined by the parameter λ ≥ 0, which controls how much to penalize departures from smoothness in the spline function, f: (cid:1) e−rT (f(cid:3)(cid:3) (δ)) 2dδ. (4) 0 Varying the value of λ changes the effective parameters of the curve, which can vary from a linear function (with a least-squares fit) as λ approaches infinity, to an interpolation spline when λ is zero. In their study, Bliss and Panigirtzoglou set λ so that the mean-squared fitted option price error across dates and strikes approximately equaled the errors from those using the method of equation (2), with a mixture of two lognormals as the underlying functional form. Oneproblemwithusingafixedvalueforλisthatthenumberofstrikescanvaryacrossdaysforthesame option contract. Even holding constant the number of strikes, the amount of smoothing that a fixed value ofλinduceswilldependontheobserveddata. Therefore,byallowingλtovary,wecantreatobservationsan a more evenhandedmanner. A number of techniques have been developed to allow a data-drivenmethod of choosing the amount of smoothing. One such technique that has been widely used in scientific applications, is the method of generalized cross validation (GCV), described in Wahba (1990). The idea behind GCV is to find a value of λ that minimizes the error: (cid:2)N (cid:3) (cid:4) (cid:5)(cid:6) 2 w(i) y(k)−f[k] δ(k) (5) λ k=1 wheref[k] istheminimizationofequation(3),foragivenλ,withdatapointk omitted. GCVineffectserves λ to find an optimal λ by lowering the influence of outlying data points on the curve.12 Fisher, Nychka and Zervos(1994)usedGCVtofindthesmoothingparametereachdaywhenfittingthetermstructureofinterest rates. Fisher, Nychka andZervos(1994)foundin a comparisonwith other interpolationmethods that GCV smoothing splines produced the most stable, accurate and least-biased results when fitting forward interest rates. 12If a point on the curve differs too much from its immediate neighbors, it will be eliminated because of no-arbitrage considerations,aswillbediscussedinSection4. 10

In the process of developing the PDF estimation method for this study, we found that spline methods usingtheGCValgorithmtendedtocompute avalueofλthatresultedinchoppyPDFsthatoftencontained negative probabilities. While the splines are optimal in a GCV sense in delta-implied volatility space, after transformation into strike space and then after twice-differentiation to extract the underlying PDF, the resulting PDFs tended not to be particularly smooth. Apparently, the GCV-determined smoothness penalty was too low. However, our GCV computational framework also allowed us to compute a value of λ corresponding to a specified number of effective parameters. For example, fixing λ = 0 would give as many effective parameters as there are strikes, while a value of λ near infinity would give only two effective parameters. For this study we want the semiparametrically estimated PDFs to have roughly the same number of effective parameters as in the parametric methods.13 Previous studies using the mixture of lognormal distribution usedeither a mixture oftwo lognormals,whichhas five free parameters14, ora mixture ofthree lognormals, which has eight. While allowing more than eight effective parameters allows a closer fit of the observedimpliedvolatilities,thisgenerallycomesatacostoflesssmoothnessandhighersensitivitytonoisy observations. Data limitations led us to fix the effective number of parameter at six. We want to ensure that there are sufficient degrees of freedom (equal to the number of strikes minus the number of effective parameters) to still have some capability to mitigate the effects of data errors. We were not able to use foreign exchange options data in the initial years of exchange trading (1984 and 1985) because the degrees offreedom,evenforsixeffective parameters,werecloseto zeroonmanydays,asittook severalyearsbefore enough interest in these options developed to allow a relatively large number of strikes to be traded. In Figure 1 we graph the PDF of the SP series on June 24, 1998 using 6 effective parameters. For comparision we also plot the same data using the average value of λ for the SP series, which gave 9.2 effective parameters. The additionaleffective parameters placedfewer constraintson the shape of the PDF, butinthepresenceofnoisydatathiscanleadtocurvesthatseemlesslikelytorepresentthetrueunderlying risk-neutral distribution. The sharp wiggles of the dashed PDF in Figure 1 suggest that extra effective parameters can amplify rather than dampen the effects of noisy data. 13In practice, for the same number of effective parameters, the semiparametric volatility smoothing method tends to have theabilitytoreflectawidervarietyofshapesthanamixtureoflognormals. 14The free parameters for a mixture of two lognormals are the mean and variance parameters for each of the lognormal distributionsaswellasamixingparameter. 11

4 Data 4.1 Procedures for Data Selection Given the purpose of this paper – to examine the information content of option-implied PDFs – a premium is placed on utilizing observations with the most information content. For most exchange-listed futures and their associated options, volume and open interest begin to increase shortly after a particular contract becomes the second-to-expiration contract, i.e., with an expiry of approximately six months. The next-toexpiration futures contract, with an expiry of less than three months, is generally the most actively traded. The pattern of trading activity for exchange-listedoptions on futures closely follows the futures themselves. With exchange-listed option strike prices available only at fixed intervals, (usually whole numbers), there is a tradeoff between liquidity and information. As the time to expiry gets closer, options with strike prices farther away from the current futures price become both less valuable and less liquid. Besidestheliquidityconsiderations,whichfavorusingdatesclosetoexpiration,acompetingconsideration is to allow as much time as possible until expiry, so that realized values of volatility and skewness would be less influenced by any single day’s return. In addition, as the time to expiration diminishes, the number of effective strike prices decreases since options relatively far out of the money become essentially worthless.15 Therefore,theselectionruleforeachtradingdateistoselectthenext-to-expirationcontractuntilfewerthan 45 calendar days remained until expiration, at which point the second-to-expiration contract was selected. Hence, option contract times to expiry in this study range from 45 to 139 calendar days. Data on settle prices16 for futures and options on futures were obtained from the CME. The contracts examined are on the S&P 500 futures (SP), the Japanese yen/U.S. dollar futures (JY), and the deutsche mark/U.Sdollarfutures(DM). Foreachseries,theoptionsonfutureswereAmerican-style. FortheSPseries, we chose to study post-crashobservations,in view of the results of Canina andFiglewski(1993). Therefore, the sample period of daily observations went from January 4, 1988 to September 30, 1999, covering all quarterly futures contracts from 1988 through 1999. Futures trading in the JY began in early 1986. We use daily observations covering the quarterly futures contracts that expired from September 1986 through December 1999. Unfortunately, for the JY options, the CME was unable to provide data for 1987, as some ofthedatathatyearwascorrupted. Forthe DM,weselectedfutures contractsbetween1986and1998,with 15Optionsfarinthemoneyalsohavelittleinformationcontent,astheirvalueessentiallybecomesthedifferencebetweenthe strikepriceandthefuturesprice. 16Settle pricesaredeterminedattheendofeachdaybyasettlement committee comprisedofmarketparticipants. Theuse ofsettlepricesservestoensurethesynchronizationoftheoptionpricesandtheassociatedfuturesprice. 12

the exception of 1993, for which the CME was unable to provide uncorrupted data. With the introduction ofthe euroatthe beginning of1999,DM futures trading activity beganto dry up, in favorofactivity in the euro.17 The raw option price data arefiltered to eliminate observationsthat allowedfor arbitrageopportunities: ∂C(T,X) −X ≤ ≤0 (6) ∂X ∂P(T,X) X ≥ ≥0, (7) ∂X where P is the price of a put. The first constraint states that the price of a call option should not become more expensivefor higher strikeprices,while the secondconstraintstatesthat a put shouldnotbecome less expensivefor higherstrikeprices. The firstconstraintalsostates,forexample,thata callwithastrike price of $45 cannot be worth more than $5 above a call option with a strike price of $50, all else constant. For a given strike price, both a put option and a call option may be available. For our data series, put-call parity did not generally hold, thus the contracts were not redundant. An examination of the CME options data confirms the conventional wisdom that for a given strike price, an out-of-the-money option is more liquid than its in-the-money counterpart. Therefore,in the case where both puts and calls were traded for a given strike price, the out-of-the-money option was selected. Themethodologyofequation(1),describedintheprevioussection,isbasedondifferentiatingEuropeanstyle call options. Because of the value of early exercise, an American-style option is always at least as valuable as a European-style option. If volatility is extracted from American option prices, this implied volatility is biased upward.18 Thus, all American-style option prices are converted to European-style prices by applying the quadratic approximation of Barone-Adesi and Whaley (1987). 4.2 Variable definitions Foreachdayofoptionsdataweestimatearisk-neutralPDF19 ofthefuturesforthedateonwhichtheoption contractsexpire. Aswe movealongtime, the PDFforaparticularoptioncontractis forafixedexpirydate. For each date, we compute the implied volatility of the estimated PDF. To convert the moments of the volatility-smoothed PDF into an implied volatility measure, we use the following equations, presented in 17Theconversionratebetweenthedeutschemarkandeurowaspermanentlyfixedattheendof1998,givingtheeurocontract thesamecapabilitiesforhedgingandspeculatingonthedeutsche mark/U.S.dollarexchange rateasDMcontracts. 18Mostotherresearchershaveignoredthisissue. 19For the remainder of the paper we will use “PDF” as our shorthand notation for the distribution obtained by using the smoothedvolatility-smilemethod. 13

Jarrow and Rudd (1982), to compute σ, the implied volatility: q2 =m2/µ2 (8) (cid:7) σ = log(q2+1)/T (9) wherem2isthesecondmomentofthePDF,µisthefirstmomentofthePDFandT isthetimetoexpiration. We also calculate the Black-Scholes implied volatility by averaging the implied volatilities of the least-outof-the-moneyput andcall. The realizedfuture volatilityis computedoverthe remainingtrading daysofthe contract, τ, by (cid:8) (cid:9) (cid:9) (cid:2)t+τ (cid:10)1 σˆft = τ (R i −R¯)2, (10) i=t whereR i isthe returnondayiandR¯ isthe meanreturnovertheremainingdaysofthecontract. Historical volatility is computed as (cid:8) (cid:9) (cid:9) (cid:2)60 (cid:10) 1 σˆht = (R t−i+1 −R¯)2, (11) 60 i=1 a moving averageof returns over the 60 trading days preceeding date t. Historicalandrealizedfutureskewnessnecessitatescomputingthethirdcentralmoment,withtherealized third moment given by (cid:2)t+τ(cid:4) (cid:5) mˆ 3 ft = τ 1 R i −R¯ 3 , (12) i=t and the historical third moment by (cid:2)60 (cid:4) (cid:5) mˆ 3 ht = 1 R i −R¯ 3 . (13) 60 i=t Skewnessisthenmˆ3/σˆ3. FromthePDFweextracttwomeasuresofskewness. Thefirstmeasureiscomputed bydirectlycalculatingthesecondandthirdmomentsfromthePDF. Becausethereisgenerallylessliquidity as strike prices move away from the current futures price, the resulting imprecision in estimating the shape of the extreme tails of the PDF may decrease the reliability of the first measure of implied skewness. For this reason we also compute, from the PDF, the Pearson median skewness measure: µˆ−Zˆ 50, (14) σˆ 14

where µˆ is the mean ofthe PDF (which equals the currentfutures price) andZˆ 50 is the medianof the PDF. As will be discussed in Section 7, we adjust the two skewness measures by removing the skewness that is already inherent in the lognormal distribution to obtain measures of excess skewness. 4.3 Descriptive statistics Table 1 presents descriptive statistics for the three data series. Comparing one-day returns, averagereturns on the SP are substantially larger than the foreign exchange series, but only slightly more volatile. For each series we see that Black-Scholes implied volatility is less than the PDF-implied volatility, indicative of implied volatility forming either a smile pattern, where implied volatility increases in either direction away from the center of the curve, or a smirk/sneer pattern, where volatility rises on one side of the curve. For DM and JY, average implied volatility is reasonably close to realized volatility. For SP, implied volatility is too high, which indicates that SP options are relatively expensive. Figures 2a–2c plot the realized future volatility, σ ft, against implied volatility for each of the time series. Observing the summary statistics for realized skewness provides some insights into the difficulty of accurately forecasting skewness. For each of the three data series, the standard deviation of realized skewness greatly exceeds its mean. Estimates based on the two measures of implied excess skewness show that the rangeofmarketexpectationsofskewnessisgenerallymuchnarrowerthanwhatisactuallyrealized,consistent with the view thatpredicting the timing ofa largemoveis quite difficult. The directionofskewnessimplied by the PDFs is, onbalance,the same asthatofrealizedskewness. For DMandJY, impliedexcessskewness tendedtoremainnearzero. ForSP,bothimpliedskewnessmeasuresconsistentlypredictednegativerealized skewness. The time series of implied skewness is plotted in Figure 3. The kurtosis implied by the risk-neutral PDFs exceeds the level of kurtosis found in a lognormal distribution, whichindicates thatmarketparticipantsmay expectexcesskurtosisorthey may be willing to paya higherpremiumforprotectionagainstlargepricechanges. Thedaily returnsshowthatthe realizedkurtosis is far greater than would be observed under geometric Brownian motion. 5 Goodness of Fit We beginourinvestigationofthe informationcontentofrisk-neutralPDFs bytesting whether asequenceof realizedvaluesoffuturesprices—takenattheexpirationofthecorrespondingoptioncontract—comefrom the risk-neutral distribution implied by option prices. We compare the goodness of fit of PDFs obtained 15

from the volatility smoothing method with the goodness of fit of a lognormal distribution with the same mean andstandarddeviation(i.e., as implied by the Black-Scholesmodel with matching implied volatility). Our methodology for measuring the goodness of fit uses statistics based on the empirical distribution function(EDF),asdescribedinStephens(1974). ThestandardEDFtestcomparesanempiricaldistribution functionF n(x)obtainedfromarandomsampleofobservedvaluesx 1 ,x 2 ,...,x n,withahypothesizeddistribution function F(x). In our framework, daily changes in option prices are transformed into a new F(t,x) each day. Because EDF tests require independent realizations, we are restricted to using only one realized value x, per contract, with n equal to the total number of contracts in the sample for the financial variable inquestion. Wealsomusteliminatethedistributions,F(t,x)havinggreaterthan90daysuntilexpirationto ensure that there is no overlapbetween contracts. Stephens showedthat the Anderson-DarlingA2 statistic, the Cram´er-vonMises W2 statistic and the Watson U2 statistic are the most powerful goodness of fit tests. The Anderson-Darling and Cram´er-von Mises test are more powerful in detecting differences in location betweenthe hypothesizedandrealizeddistributions. TheW2 statistic is sensitiveto the observationsacross the entire distribution, while the A2 statistic is more sensitive to the tails (cumulative probabilities near 0 and 1). The U2 statistic is more powerful in detecting differences in variance. Using the same framework as in EDF tests, Fackler and King (1990) developed a graphical technique, the empirical calibration function, that helps in assessing differences between F(x) and F n(x). Both the calibration function and the EDF tests utilize a sequence of cumulative probabilities of the realized futures valueatmaturityforthevarioushypothesizeddistributionfunctionsF(t,x). TheEDFtestsandtheempirical calibrationfunctioncomparethissequenceofcumulativeprobabilitieswiththeuniformdistributionfunction onthe unit interval[0,1]. Silva andKahl(1993)succinctly describehow toformthe calibrationfunction for options on futures: [T]he empirical calibration function, C(U T), is specified as C(U T)=T/n for T =1,2,...,n where U T =F(P T a) (T =1,2,...,n) = the cumulative probability assessment for P T a, the actual futures price at the maturity date T, and the values of U T have been placed in ascending order (i.e., U 1 ≤ U 2 ≤ ··· ≤ U n); and T/n = the estimator of the “real” cumulative probability. (p. 768) The calibrationfunction can be viewedby plotting (U,C(U)) onthe unit square. The closerthe calibration curve lies to the 45◦ line, the better the fit between the hypothesized distribution function F(x) and the realizeddistributionfunction F n(x). The calibrationfunction alsohelps in detecting deviationsbetween 16

the hypothesized distribution function and the realizeddistribution function. If a calibrationcurve lies consistentlybelowthe diagonal,thisimplies thatthe locationofthe hypothesizeddistributionfunctionisbelow that of the realized distribution. This could occur, for example, if futures prices tended to be downwardbiasedpredictorsofspotprices,asthemeanoftherisk-neutralPDFwouldthenbetoolow. Iffuturesprices tended to exceed realized prices, then the mean of the PDF would be too high and the calibration function woud lie above the diagonal. An S-shaped calibrationcurve suggests that the hypothesized distribution has too much dispersion; this would be the case, for example, if implied volatility consistenly exceeded realized volatility. A reverse-S-shapedcurve is consistent with too little dispersion in the hypothesized distribution. We createcalibrationcurvesforeachofthe financialvariablesby obtaining the CDF valueF(x,t) forall contracts, holding time to expiry, t, constant. We plot calibration curves for the minimum and maximum values of t. In the upper panels of Figure 4 we examine the calibration curves for the DM contract. At 45 days to expiration, the volatility smoothing method generates a calibration curve that is slightly closer to the 45◦ line than the lognormaldistribution. At 90 days to expirationthe lognormalappears to be closerto the diagonal, with both curves displaying the reverse-S-shapeconsistent with smaller dispersion in either of the risk-neutral distributions (RND’s) than in the realized distribution. FortheJYcontract(lowerpanelsofFigure4),thecalibrationcurveforthelognormaldistributionappears to be closer to the diagonal at both maturities. The reverse-S-shape again suggests that the dispersion is underestimated for both of the RND’s. ThecalibrationfunctionsoftheSPcontractareexaminedintheupperpanelsofFigure5. Bothcurveslie primarilybelowthe45◦ line,whichindicatesthatmeansofbothRND’sarelocatedtotheleftoftherealized distribution. By a no-arbitrageargument,the meanofa risk-neutraldistribution is tied to the risk-freerate of return. Historically, the return on equities has tended to exceed the risk-free rate. Hence, it comes as no surprise that the mean of the risk-neutral distribution lies below the true distribution. To correct for this effect, a second set of distributions was computed, with the same shape as the RND’s, but with the means adjusted to equal the average excess return of equities over the risk-free rate for the years 1988 through 1999. The risk-adjusted SP calibration curves are plotted in the lower panels of Figure 5. At 45 days to expiration, the volatility smoothing method clearly dominates the lognormal. The difference between the two calibration curves is less at 90 days until expiration. At the longer maturity, both calibration curves are clearly S-shaped, suggesting that the hypothesized distributions overestimated dispersion (i.e., implied volatility exceeded realized volatility). This is consistent with the results in Table 1, which showed for SP that implied volatility, on net, exceeded realized volatility. 17

Table 2 displays the results of the EDF goodness of fit tests for the financial instruments in our study. We compute the goodness of fit at 45 and 90 days until expiration, to quantitatively measure the fit shown by the calibrationcurves in Figures 4 and 5. We also compute EDF statistics for the entire sample by using a Monte-Carlo technique. For each contract, we randomly selecting one date (allowing time to expiry to randomly vary from 45 and 90 days). We repeat this exercise 1000 times, taking the average of the EDF statistics over the 1000 trials. In addition to the EDF statistics, we want to compute a measure that would give some insight into whether the volatility-smoothing method is a better predictor of realized values than the Black-Scholes model. Therefore we also calculate the log-likelihood of the realized value, x, for each distribution, F(t,x), weighted by the standard deviation of the distribution. In Table 2, we report the averagevalue overthe 1000iterations of the four statistics,for both the PDF using the volatility-smoothing method and the matching lognormal distribution. Underneath the sample means we report the number of times that one distribution fits the data better than the other for the 1000 iterations. We first examine the goodness of fit for the DM contract. At the shortest time horizon, 45 days to expiration, the volatility smoothing method slightly outperforms the Black-Scholes method. At 90 days to expiration, the Black-Scholes outperforms the volatility smoothing method by a greater margin than at 45 daystoexpiry. Forbothmethods thefitismuchbetter attheshortermaturity. Thesestatisticalresultscan be visuallyverifiedin Figure4. For the full sample,the lognormaldistribution clearlyoutperforms,winning more than 90% of the time when measured by the three EDF statstics. For the A2 and W2 statistics, we cannotrejectthehypothesisatthe10%confidencelevelthattherealizeddatacamefromeitherdistribution. ForthemorestringentU2statisticwecanrejectthevolatility-smoothingmethodatthe90%confidencelevel. ForJY,thelognormaldistributionoutperformsthevolatility-smoothingmethodforboththeshorterand longer maturities. Both methods have a better fit at 90 days to expiry than at 45 days. For the full sample, thelognormaldistributionwinsmorethan95%ofthetimeusingthethreeEDFmeasures. We cannotreject the hypothesis that the realized data came from either distribution at the 10% confidence level. Using the U2 statistic, we can reject the volatility-smoothing method at the 95% confidence level and the lognormal distribution at the 90% confidence level. As discussed above and also shown in Figure 5, for the SP series the risk-neutral mean understates the realizedmean. Notsurprisingly,theEDFstatisticsthataremostsensitivetolocationofthedistribution,A2 and W2, reject both methods at the 99% confidence level. Even the U2 test, which is less sensitive to first momentdifferences,rejectsbothmethodsatthe99%confidencelevelforthefullsample,althoughat45days toexpiration,thevolatilitysmoothingmethodisrejectedatthe95%confidencelevel. Whenthedistribution 18

meansareadjustedtocompensateforrisk-neutrality,theimprovementintheEDFstatisticsisstriking. The volatility-smoothingmethod performs much better at 45 days to expirationthan at 90 days. At 45 days,by the A2 and W2 statistics, we cannot rejectthe hypothesis at the 10%confidence levelthat the realizeddata came from either distribution. For the more stringent U2 statistic we can reject the volatility-smoothing method at only the 90% confidence level. At 90 days, we can reject the volatility-smoothing method at the 99%confidencelevelbytheU2 statistic. Forthefullsample,thevolatility-smoothingmethodwoneverytrial versus the lognormal distribution. Whereas we can reject the lognormal distribution at the 95% confidence level for the A2 and W2 tests, we cannot reject the distribution given by the volatility-smoothing method at the 10% confidence level. The log-likelihoodmeasure clearly favorsthe volatility-smoothingmethod over the Black-Scholes model. In summary, the Black-Scholes model appears to fit the realized data very well for the DM series and nearly as well for the JY series. The volatility-smoothing method slightly underperforms for both series in terms of statistical fit. This underperformance is consistent across days to expiration, since the lognormal wins over 90% of the time in head-to-head comparisons. For the adjusted-SP series, in contrast, we obtain the opposite the result. The volatility-smoothing method seems to fit the realized data well in terms of location,assuggestedbythe A2 andW2 statistics. Inhead-to-headcomparisons,thelognormaldistribution is dominated by the volatility smoothing method. The results of these goodness of fit tests are quite surprising. For the two series that, a priori, seemed to better fit the assumption of risk-neutrality, the DM and JY, the extra information contained in the distribution obtained from smoothing implied volatilities was expected to lead to a better fit than the lognormal distribution. For the SP, in which richly-valued out-of-the-money puts generate a consistently negative skew in the volatility-smoothing method, this distribution was not expected to do as well as the lognormal distribution (which is unencumbered by the negative skew), especially after both distributions were adjusted for excess returns. This result is puzzling, and suggests that for the SP contract, a set of options contained information about the realized distribution of price movements beyond what was found from just utilizing at-the-money options. 6 Implied Volatility Inthe previoussectionwesawthatthelognormaldistributionfitthe realizeddataalittle betterfortheDM and JY series, while the distributions generated by the volatility smile smoothing method fit the SP series 19

better. In this sectionwe investigatehowwellthe models do inpredictingrealizedvolatility;we relatethese results to the qualitative implications we found from the calibration curves. To measure the ability of our models to predict volatility, we use the same regressionframework that was used in many earlier studies: σ ft =α+βσˆt+e t , (15) where σˆt is either the Black-Scholes implied volatility for day t, the implied volatility of the smoothed volatility smile distribution (labeled as PDF in Table 3), or the historical volatility. If the estimate of β is significant and positive, then implied volatility contains some useful predictive information about future volatility. Thehypothesisthatimpliedvolatilityisanunbiasedforecastoffuturevolatilitymaybeexpressed as α=0 and β =1. The efficiency of implied volatility, which Canina and Figlewiski found to be particularly poor, is tested with σ ft =α+β 1 σˆIt+β 2 σˆht+e t , (16) regressingrealized volatility on both implied and historical volatility. If implied volatility contains all information conveyed by historical volatility, β 2 should be 0. Giventhatrelativelyfewquarterlyobservationscouldoccuroverthesampleperiod,andsincewewantto takeadvantageofalloftheinformationavailable,everydailyobservationisused. However,thisinducesserial correlationintheerrorterm. Furthermore,thereisnoreasontobelievethattheerrortermishomoscedastic. Therefore,thestandarderrorswerecorrectedforserialcorrelationandheteroscedasticityusingtheprocedure of Newey and West (1987), using a lag length of 10. The results of these information content regressions are shown in Table 3. In the information content regressions for the three series, β for both Black-Scholes and PDF implied volatility is significantly greater than zero, confirming that implied volatility provides useful information in forecasting future volatility. Comparing the Black-Scholes β coefficient for the DM and JY series with the results of Jorion (1995), the magnitude and the statistical significance of the regression coefficient is greater in our study20 – we benefit from having sevenyears of additional data. Another difference between the studies is that we convertedthe American-styleoptionpricesto European-styleprices,whichprovidesabetter estimateofimpliedvolatility. Given that using American-style option prices will produce an upward biased estimate of volatility, it is 20For DM wehad a slopecoefficient of 0.592 with standard errorof (0.061), whereas Jorionhad a slopecoefficient of 0.547 withstandarderrorof(0.138). ForJYwehadaslopecoefficient of0.667withstandarderrorof(0.080)vs. 0.496and(0.181) forJorion. 20

not surprising that the coefficient in our study is larger. The 60-day moving average also provides a better estimate of volatility than the 20-day moving average that Jorion used. For the SP series, the findings of Canina and Figlewski (1993) are not confirmed. The Black-Scholes β coefficient is similar in magnitude to the coefficient found by Christensen and Prabhala (1998).21 Eachofthe regressioninTable3 canbe consideredtobe nestedfromamoregeneralmodelwhichhasas regressors the Black-Scholes implied volatility, the implied volatility from the PDF, and the 60-day moving average. The right-most column of Table 3 provides results of specification tests for the nested regressions versus the general model. For all three series, we can reject the null hypothesis that the 60-day moving averageencompassesallthe informationavailableintheimpliedvolatilityobtainedfrombothoptions-based methods. FortheDMseries,thespecificationtestsindicatethatthereisstatisticallysignificantindependent informationfrom eachof the three independent regressors,as we canreject all the nested models in favorof thefullmodel. ForJY,iftheBlack-Scholesestimateofimpliedvolatilityisincluded,wecannotrejectthenull hypothesis that the other two measures of volatility have no statistically significant additional explanatory power. If the Black-Scholes regressor is not included, we can reject the hypothesis that the Black-Scholes implied volatility measure does not have incremental explanatory power at the 95% confidence level. For SP, if either options-based implied volatility measure is included, we cannot reject the null hypothesis that the other measure is irrelevant. Theefficiencyregressionsshow,fortheSPandJYseries,thatthecoefficientonMA(60)isnotsignificantly differentfromzero. FortheDMseriesthecoefficientonMA(60)remainssignificant,butshrinksinmagnitude by a factor of two relative to its standalone value, while the implied volatility coefficient remains far more significant than the historical volatility regression coefficient. These results suggest that the predictive informationinoption-impliedvolatilitytendstosubsumetheinformationcontainedfromhistoricalvolatility, which is consistent with results found by Jorion (1995) and Christensen and Prabhala (1998). Table 3b shows the daily correlations for the measures of volatility. For all three series, the correlation of implied volatilities between the two options-based methods is very close to 1, which helps explain the similarityofmagnitudeandsignificanceoftheregressioncoefficientsforthesemeasuresinthenestedspecifications. Thesecorrelations,andthecomparablemagnitudesofthetwoimpliedvolatilitymeasuresdisplayed in Table 1, providean indicatationof how little implied volatility differs whenonly the at-the-moneyoption isused,versuswhenitiscomputedusingtheentirerangeofoptionsprices. Thecorrelationsbetweenvolatilities obtained from the options-based methods and the 60-day moving average is about 0.8, which suggests 21ChristensenandPrabhalausednon-overlappingmonthlyoptioncontracts ontheS&P100. 21

that past volatility is an important eliminate in markets assessment of future volatility. The option-based measures are more highly correlated with realized volatility for SP than the two foreign exchange series. In addition,forSP,theoption-basedmeasuresoutperformthemoving-averagemeasurebyawidermarginthan for the foreign exchange series. This is consistent with the possibility that market participants can better anticipateeventsthatwouldcauseswingsinU.S.equitymarketsthanthey cananticipateeventsthatwould lead the exchange rate of the dollar to fluctuate. 7 Implied Skewness The volatility regressions suggest that option prices do contain a great deal of useful information for forecasting future (realized)volatility. However,this informationappears to be concentratedaroundthe at-themoney options, as the volatility forecastingperformance degradessomewhat(for the foreignexchangedata) when the full risk-neutral PDF is used. We now investigate whether the PDFs generated by the volatility smoothing methodcontaininformationusefulin forecastingskewness. Because ofconcernsoverthe reliabilityofoptionpricesforfar-out-of-the-moneystrikes,formeasuringimpliedskewness,wealsousethePearson median skewness measure, which is less sensitive to the tails of the distribution. For volatility,it is straightforwardusing equations (8) and(9) to obtainanestimate of implied volatility from the first two moments of the risk-neutral distribution. However, for skewness, the linkage between daily returns and the risk-neutral distribution is more complex. The Black-Scholes model is based on the assumption that returns follow geometric Brownianmotion, which has a skewness of zero. It can be readily shown that returns generatedby geometric Brownianmotion gives rise to a lognormaldistribution of prices for the underlying variable. The skewness of the lognormal distribution is always greater than zero, and when generated by Brownian motion, the skewness of the resulting lognormal is an increasing function of the implied volatility. CorradoandSu(1996)developedamodelthatlinksreturnsthatdiffer fromgeometricBrownianmotion to explicitly price options. Using their model, we can use the volatility smoothing method to obtain the risk-neutralPDFimpliedbytheirmodel,thuslinkingthestochasticdistributionofreturnstotheriskneutral distribution at expiry of the options contract. Corrado and Su extended the Black-Scholes pricing model by using a Gram-Charlier series expansion of the normal density function, which allows non-zero skewness in returns, as well as kurtosis that can differ from the normal distribution’s kurtosis of 3. In Table 4 we use Corradoand Su’s model to generate option prices for various combinations of implied volatility,implied 22

skewness and implied kurtosis. We tabulate both of our skewness measures from the resulting volatilitysmoothing method PDF, arranging them by implied skewness and implied volatility. Implied skewness of 0 corresponds to the assumptions of the Black-Scholes model, and the center column of Table 4a shows how skewnessincreasesasafunctionofimpliedvolatility. InTable4bwetabulateexcessskewness,bysubtracting the skewness of a lognormal distribution with the same implied volatility from the skewness of the PDF. For the median skewness measure, skew2, most of the variation caused by changes in implied volatility is removed. For larger negative values of implied skewness, this operation increases the sensitivity to implied volatility when measured by skew1. In Table 4c we report excess skewness when implied kurtosis is 4. In this situation, skew1 varies less for large negative values of implied skewness, but at implied skewness of 0, there is again a positive relationship between skew1 and implied volatility. Across kurtosis measures, the relationship between skew2 and implied volatility remains more constant for a given implied skewness. Overall, we find that excess skewness better corresponds to implied skewness than total skewness. Table 4 also suggests that skew2 is less sensitive to variation in implied volatility than skew1, when measured using excess skewness. In our regressionanalysis, we will use both measures of skewness, as skew2 does not dominate skew1 in all conditions in terms of sensitivity to implied volatility and kurtosis. We use the same information content framework for implied skew as we did for implied volatility. We regress realized skew on implied skewness, m3 ( σ3 )ft =α+βω It+e t , (17) where ω It is our measure of implied skew. The results are presented in Table 5. Examining the sign and t-statistic of the skew1 slope coefficient, we see that it is not significantly different from zero for any of the series. The skew2 slope coefficient is significantly different from zero for all three series. However, its sign is negative. The negative slope coefficients for the moving-averagemeasure suggest that for the foreign exchange series, there may be some mean reversion, where a large appreciation would tend to follow soon after a large depreciation, and vice-versa. It is possible that only when implied skewness is highly positive or negative does the market signal a clear expectation of future activity, and that implied skewness values closer to zero have little information content. To test this hypothesis we discretize the data by assigning a value (equal to 1) to the largest 10% of the implied skewness observations and a value (equal to -1) to the smallest 10% of the implied skewness observations. The remaining implied skewness observations are set equal to zero. The realized skewness 23

measure is dichotomized around zero. In essence, we are testing whether the strongest implied skewness signalshaveanyabilitytopredictthedirectionofskewnessinreturns. Theresults,presentedinTable6,are similar to those of regression 17. Only for measure skew1 on the JY series does implied skewness correctly predict the direction of realized skewness. Our findings on skewness suggest that profit opportunities may exist. If the options market strongly indicates skewness in a particular direction without a real underlying tendency for large price movements in that direction, one could use the options market to place a bet against a large price movement in that direction. Such a bet is called a risk reversal. In a risk reversalone buys an out-of-the-money call option in one directionand sells anequally out-of-the money put optionin the other direction. Atypical riskreversal traded on the over-the-counter market is the 25-delta risk reversal – the price is equal to the price of a 25-delta call minus a 25-delta put, where delta is the derivative of the Black-Scholes option with respect to the underlying asset price. The strike price for the 25-delta call is obtained by solving 0.25 = ∂C/∂S, and the strike price for the 25-delta put is obtained by solving 0.25 = ∂P/∂S. There is no payoff from a risk reversal unless the underlying asset moves outside the range of either the put or the call strike price. For a 25-delta risk reversal, to move outside the zero payoff range requires a move of about 0.8 standard deviations. A 10-delta risk reversal only has a non-zero payoff after relatively large net price moves, as the call and strike prices are farther out of the money than for the 25-delta risk reversal. To test whether apparent inefficiencies in option prices are exploitable, we compute daily prices of a 10and25-deltariskreversal. The previousskewnessdatasuggeststhatovertime weshouldbe abletoearnnet profits from shorting risk reversals when the premium (the difference between the call price and put price) is positive and being long when the premium is negative. In essence, we are expecting to keep the premium while taking our chances that movements in price outside of the zero-payoff area will balance out. Table 7 shows the profit result of this exercise, where we use a sampling framework similar to what we used for the EDFtestsinTable2. Foronerandomlyselecteddaypercontractperiod,wesellorbuyenoughriskreversals so that our total premium is $100 for the contract. We always transact unless the risk reversal premium is less than $.0001. For the foreign exchange series, this strategy is quite risky, as our Monte-Carlo simulation shows that average loss of the 25-delta risk reversal well exceeds the $5000 total premium we would take in from executing our strategy over the entire period. The 10-delta risk reversal also shows profits for the JY and losses for the DM. The large magnitudes of the payoffs indicate that the size of the risk reversal premiumisdwarfedbythemagnitudeofriskfromlargepricemovements. Thelargevariationinpayofffrom executingourstrategyatdifferingfixeddaystoexpirationprovidessomeindicationoftheriskiness,assmall 24

premiums entailexposuretoalargenumber ofriskreversalcontractstoearn$100intotalpremiums forthe day. For the SP series, the 25-delta risk reversal strategy was consistently profitable, as we earned both the risk reversal premium as well as gains from large positive moves that more than offset losses from large negativepricemovements(becauseofthesteadynegativeskewness,thestrategyledustoalwaysbelongthe call option and short the put). There were consistent positive returns, primarily from the premium, for the 10-deltariskreversals. The profits fromthe 10-deltariskreversalare consistentwith Figure 5 – the S-shape resulted from too few realized values falling into the extreme tails of the option-implied distribution. Most ofthe profitofthe 10-deltariskreversalresultedfromthe premium,asthere wererelativelyfew caseswhere the bounds givenby the 10-delta put and call strike prices were breached,and for those cases with a payoff, the gains or losses were comparatively small. The results in Table 7 suggestthat eventhe most heavily skewedPDFs, which correspondto the largest premiums on risk reversal contracts, do not correctly predict skewness. To pursue this, we ran 1000-trial Monte Carlo simulations for two more selective cases: 1) We only transact if the absolute value of the risk reversal premium is above the median and 2) We only transact if the absolute value of the premium is abovethe thirdquartile. Tomakeallthe payoffscomparable,weearn$200ofpremiuminthe above-median trials and $400 in the above-third quartile trials. The bottom rows of Table 7 display the results of these simulations. For the foreign exchange series, these more selective strategies produce different results than whenwealwaystransacted,asincreasingpositiveriskreversalreturnscorrespondingtothelargerpremiums. The magnitude of these returns greatly exceeded the total value of the premiums, indicating that over the entire period, a high value of skewness in one direction was more likely to lead to a large price movement in the opposite direction. For the SP series,our more selective strategydid not materially changethe previous results,astheriskreversalpremiumwasconsistentlylarge,andlargepricemovementsovertheentireperiod were, on balance, more likely to be in the direction of higher prices than lower prices. The ability to profit from the apparent mispricing of skewness in the options market indicates another financial markets puzzle. Transactions costs certainly would have reduced the gains from our risk reversal strategy,butitis particularlydifficult toreconcilethe factthatthe strongersignalfromthe excessskewness of the PDF, the more misleading it would have been to have used the information for prediction purposes. 25

8 Conclusions It has generally been thought that expectations incorporated in market prices are a good, if not the best, means of obtaining information about future outcomes for returns, volatilities and other properties of the distribution. This paper addressed the information content of options by using observed prices to form a risk-neutral probability density function. An improved methodology for estimating PDFs from options prices was introduced, then applied to estimate a time-series of risk-neutral PDFs for three widely-traded financial futures contracts. The amount of information on the future contained in the series of PDFs was then systematically measured. We began our investigationby examining how well the risk-neutralPDF fit the realized data. We found for the foreign exchange series, DM and JY, both the lognormal distribution and the PDF fit the realized data reasonablywell,thoughsurprisinglythe lognormaldistribution obtainedfromthe Black-Scholesmodel fit the data better than the PDF. The results from the A2 and W2 tests suggests that for JY and DM, the risk neutrality assumption inherent in options-based methods is not unreasonable. For SP, we rejected the hypothesis that the realizeddata came from a risk-neutraldistribution. Therefore,we shifted the the PDFs by their averageexcessreturnoverthe entireperiodandre-ranthe goodnessoffit tests onthe risk-adjusted PDFs. For the risk-adjustedSP case,we found thatthe PDF dominated the equivalentlognormalwhen the two distributions were pitted against each other. For the A2 and W2 tests, we could not reject at even the 10% level the hypothesis that the realized data may have come from the risk-adjusted PDF. Weproceededwithfurtherinvestigationsofhowwellthericherinformationsetavailablefromutilizingall the thefullrangeofoptionstranslatedintoimprovedpredictionsofmovementsintheunderlyingassets. We began by following an established line of research, comparing the information content of implied volatility from the PDF obtained from the complete set of option prices against the standard measure of implied volatility from the at-the-money option. Finally, we tested the information content of the implied skewness found only in the PDF, as the Black-Scholes model is based on geometric Brownian motion. OurresultsconformwiththefindingsofJorion(1995)andChristensenandPrabhala(1998)thatimplied volatility predicts future realized volatility. Howerver, we found that the implied volatility from the distribution obtained using the volatility smile smoothing method performed slightly worse in predicting future volatilitywhencomparedto the at-the-moneyBlack-Scholesimplied volatility. Thetwomeasuresofimplied volatility were highly correlated, and neither measure dominated the other in specification tests. 26

For the information content of the third moment we used two different measures of skewness to provide varyingsensitivitytothetailsofthedistribution. Inourregressions,wecreatedameasureofexcessskewnes by subtracting the skewness inherent in the lognormal distribution from the raw skewness in the PDF. In general, the ability of the PDF-implied skewness measures to predict realized skewness were poor. The forecastingabilityofimpliedskewnessdoesnotimprovewhenwerestrictedoursampletoonlythestrongest signals of implied skewness. In fact, we found that there were profit opportunities available from betting against the market in situations where pronounced skewness existed. By selling the relatively “overvalued” optionwiththehigherimpliedvolatilityandbuyingtheundervaluedoptionatthe oppositeendofthe PDF, we could profit from the apparent inability of implied skewness to translate into realized price movements. Wecandrawafewconclusionsfromtheseresults. First,wehavenotbeenabletodeterminewhetherthe generallackof predictive ability of the implied skew is due to a lack of informationcontent in option prices, particularly for strike prices relatively far from the current futures prices. Another possible explanation for the lack of implied skew to help predict realized skew is that market participants may be willing to pay a price higher than the true statistical risk to hedge against price movements, causing portfolio insurance needstomaskmarketassessmentsoflargepricemovements. Athirdpossibilitymaybethatskewnessisjust exceedinglydifficulttopredict–wefoundverylargechangesinrealizedskewnessoccuredoverrelativelyshort periods of time, whereas implied skewness moved in a far narrower range. Looking for further clarification from the goodness of fit results blurs the picture, unfortunately, although the goodness of fit results do add insight into the U.S. equity index crash-premium puzzle. For the options on the S&P 500 futures, we found the negatively-skewed distribution generated by the volatilitysmilesmoothingmethoddidabetterjoboffittingtherealizeddatathanthelognormaldistribution. Given the poor performance of the negatively-skewed SP PDFs in predicting realized skewness and the comparable performance in terms of implied volatility of the PDF and the lognormal, the better fit of the volatility smile smoothing method is difficult to reconcile. Apossibledirectionforfutureinvestigationmightbetoexaminesourcesofinefficienciesinoptionmarkets in an attempt to understand why the use of the extra degrees of freedom from the full gamut of options strikes(forforeignexchangeoptions)failstoaddpredictiveinformationbeyondthatcontainedinthe Black- Scholesmodel. Itwouldalsobe ofinterestto performfurtherstudies to determine whether, forotherequity markets, the option-implied PDF maintains the goodness of fit superiority over the lognormal distribution that we found for the S&P 500 futures. 27

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Table 1: Descriptive Statistics Mean Std. Dev. Skewness Kurtosis DM (N=3006) 1-Day Return .00012 .007 -.007 5.27 Realized Volatility .108 .027 Black-Scholes IV .114 .020 PDF IV .118 .020 MA(60) Volatility .109 .027 Realized Skew .064 .605 Implied Skew 1 .033 .102 Implied Skew 2 -.0017 .019 MA(60) Skew .074 .574 Implied Kurtosis 3.95 .370 JY (N=3114) 1-Day Return .00005 .008 .612 10.3 Realized Volatility .114 .033 Black-Scholes IV .113 .026 PDF IV .119 .028 MA(60) Volatility .116 .035 Realized Skew .364 .788 Implied Skew 1 .084 .165 Implied Skew 2 .003 .024 MA(60) Skew .308 .750 Implied Kurtosis 4.14 .606 SP (N=2959) 1-Day Return .00043 .0098 -.774 12.43 Realized Volatility .143 .0545 Black-Scholes IV .180 .0549 PDF IV .190 .0552 MA(60) Volatility .147 .0718 Realized Skew -.208 .658 Implied Skew 1 -1.252 .268 Implied Skew 2 -.194 .045 MA(60) Skew -.280 .716 Implied Kurtosis 4.72 .973 31

Table 2: Goodness of Fit A-D (A2) CVM (W2) WAT (U2) Likelihood BS PDF BS PDF BS PDF BS PDF DM 45 Days to .427 .353 .149 .136 .148 .131 -71.1 -71.5 Maturity 90 Days to .748 1.09 .209 .270 .164* .227* -75.7 -77.4 Maturity Monte Carlo .535 .740 .156 .193 .141 .182* -73.6 -74.4 906 94 926 74 953 47 776 224 JY 45 Days to 1.48 1.85 .337 .416* .232* .289*** -85.5 -83.6 Maturity 90 Days to .759 1.16 .188 .253 .181* .246** -77.0 -80.4 Maturity Monte Carlo .663 .941 .172 .226 .160* .209** -79.0 -80.8 965 35 982 18 976 24 865 135 SP 45 Days to 6.00** 3.82** 1.35** .803** .549* .220** -61.4 -55.6 Maturity 90 Days to 7.27** 3.76** 1.63** .823** .797* .358*** -57.0 -48.7 Maturity Monte Carlo 6.76** 3.83** 1.52** .828** .664* .268*** -59.8 -53.1 0 1000 0 1000 0 1000 0 1000 SP (riskadjusted) 45 Days to 2.82** .886 .598** .210 .428* .174* -58.0 -51.2 Maturity 90 Days to 3.43** 2.33* .646** .450* .618* .411*** -52.5 -47.9 Maturity Monte Carlo 3.03** 1.32 .608** .274 .511* .266*** -55.8 -50.2 0 1000 0 1000 0 1000 0 1000 * Reject Null at 10% ** Reject Null at 5% ***Reject Null at 1% A-D represents the Andersen-Darling A2 statistic. CVM represents the Cramér-von Mises W2 statistic. WAT represents the Watson U2 statistic. Likelihood is the log likelihood. For each series, the means from 1000 trials of randomly selecting one observation per contract are reported in the Monte-Carlo row. The next row reports the number of wins for each respective model in the head-to-head goodness of fit competition for each test statistic. 32

Table 3: Information Content of Implied Volatility Regressions Test of Nested Regressions: Intercept Black-Scholes PDF# MA(60) Excluded regressors = 0 DM .041 .592 F(2,3006) = 12.62 (.0069) (.061) P-val = 0.00 .041 .564 F(2,3006) = 16.21 (.0072) (.062) P-val = 0.00 .064 .409 F(2,3006) = 17.23 (.0054) (.049) P-val = 0.00 .044 .384 .189 F(1,3006) = 18.98 (.0069) (.106) (.084) P-val = 0.00 .046 .322 .222 F(1,3006) = 23.30 (.0073) (.106) (.084) P-val = 0.00 JY .037 .667 F(2,3114) = 1.93 (.008) (.080) P-val = 0.146 .040 .616 F(2,3114) = 3.67 (.008) (.075) P-val = 0.026 .063 .426 F(2,3114) = 15.48 (.006) (.056) P-val = 0.00 .038 .552 .102 F(1,3114) = 1.17 (.008) (.105) (.067) P-val = 0.280 .040 .503 .108 F(1,3314) = 4.14 (.008) (.100) (.067) P-val = 0.042 SP .025 .654 F(2,2959) = 1.98 (.008) (.046) P-val = 0.138 .019 .652 F(2,2959) = 1.65 (.008) (.046) P-val = 0.193 .089 .368 F(2,2959) = 51.58 (.012) (.082) P-val = 0.00 .022 .728 -.072 F(1,2959) = 2.22 (.008) (.071) (.05) P-val = 0.136 .014 .754 -.098 F(1,2959) = 0.02 (.008) (.073) (.050) P-val = 0.8823 OLS estimates of specifications (15) and (16) in the paper. Standard errors corrected for overlapping sample using Newey-West (1987) procedure. Asymptotic Newey-West standard errors in parentheses. R2 for full regression model: DM = 0.254, JY = 0.281, SP = 0.460. # Estimated using smoothed implied volatility smile method 33

Table 3b : Correlations of Volatility Measures (Daily Data) DM Realized Black-Scholes PDF# MA(60) Volatility Realized Volatility 1 0.449 0.433 0.428 Black-Scholes 1 0.997 0.794 PDF 1 0.794 MA(60) 1 JY Realized Black-Scholes PDF# MA(60) Volatility Realized Volatility 1 0.524 0.520 0.464 Black-Scholes 1 0.996 0.811 PDF 1 0.815 MA(60) 1 SP Realized Black-Scholes PDF# MA(60) Volatility Realized Volatility 1 0.674 0.675 0.508 Black-Scholes 1 0.996 0.801 PDF 1 0.815 MA(60) 1 # Estimated using smoothed implied volatility smile method 34

Table 4: Skewness in a Distribution as a function of Implied Skewness, Implied Volatility and Implied Kurtosis Implied Skewness -1 -.5 0 .5 1 Total Skewness; kurtosis=3 Implied (skew1= third moment skewness measure) .05 -0.9409 -0.4249 0.0721 0.5511 1.0129 .10 -0.9449 -0.3770 0.1487 0.6362 1.0889 A .20 -0.9304 -0.2659 0.3010 0.7877 1.2078 .30 -0.0445 -0.1643 0.4555 0.9474 1.3426 .40 -1.0445 -0.0765 0.6139 1.1196 1.4977 (skew2 = Pearson’s median skewness measure) .05 -0.1399 -0.0668 0.0119 0.0881 0.1575 .10 -0.1389 -0.0583 0.0246 0.1043 0.1793 .20 -0.1204 -0.0351 0.0494 0.1291 0.2005 .30 -0.0988 -0.0108 0.0740 0.1514 0.2186 .40 -0.0766 -0.0137 0.0979 0.1726 0.2352 Excess Skewness; kurtosis=3 (skew1 ) .05 -1.0160 -0.5005 -0.0039 0.4746 0.9359 .10 -1.0918 -0.5258 -0.0020 0.4836 0.9344 B .20 -1.2166 -0.5601 -0.0010 0.4781 0.8907 .30 -1.3757 -0.6024 -0.0007 0.4738 0.8521 .40 -1.5905 -0.6575 -0.0005 0.4733 0.8208 (skew2 ) .05 -0.1525 -0.0794 -0.0008 0.0754 0.1447 .10 -0.1633 -0.0831 -0.0005 0.0789 0.1536 .20 -0.1676 -0.0836 -0.0004 0.0781 0.1483 .30 -0.1673 -0.0822 -0.0002 0.0746 0.1391 .40 -0.1645 -0.0794 -0.0001 0.0699 0.1282 Excess Skewness; kurtosis=4 (skew1) .05 -0.9775 -0.4630 0.0326 0.5102 0.9706 .10 -1.0077 -0.4463 0.0735 0.5553 1.0025 C .20 -1.0242 -0.3888 0.1526 0.6165 1.0162 .30 -1.0418 -0.3247 0.2339 0.6746 1.0260 .40 -1.0661 -0.2539 0.3196 0.7332 1.0360 (skew2) .05 -0.1393 -0.0722 -0.0006 0.0686 0.1321 .10 -0.1465 -0.0743 -0.0005 0.0704 0.1373 .20 -0.1492 -0.0742 -0.0006 0.0688 0.1318 .30 -0.1478 -0.0726 -0.0010 0.0644 0.1219 .40 -0.1441 -0.0703 -0.0021 0.0584 0.1099 Skewness of PDF with given implied parameters, using model of Corrado and Su (1996). 35

Table 5: Information Content of Implied Skew DM JY SP 500 intercept slope intercept slope intercept slope Skew 1 .071 -.172 .318 .094 -.448 -.192 (2.1) (0.6) (5.8) (0.4) (2.9) (1.6) Skew 2 .062 -2.22 .339 -5.13 -.837 -3.23 (1.8) (2.1) (7.7) (3.4) (5.3) (4.6) MA(60) .075 -.138 .398 -.244 -.189 .066 (2.0) (2.2) (7.7) (4.3) (4.8) (1.5) OLS estimates of specification (17) in the paper. Standard errors corrected for overlapping sample using Newey-West (1987) procedure. Asymptotic Newey-West t-statistics in parentheses. 36

Table 6: Extreme Values of Implied Skew as a Predictor of Skewness Sign DM JY SP 500 intercept slope intercept slope intercept slope Skew 1 .781 -.111 .428 .137 .502 -.046 (8.4) (2.5) (4.7) (3.3) (4.8) (0.9) Skew 2 .856 -.148 .811 -.054 .671 -.130 (13.2) (4.8) (10.2) (1.4) (6.4) (2.6) MA(60) .622 -.03 .861 -.08 .252 .08 (5.3) (0.6) (7.9) (1.5) (2.2) (1.4) OLS estimates of specification (17) in the paper, with dependent variable dichotomized by sign of realized skewness. Independent variables are discrete dummy variables representing the extreme 10% of implied skewness of the daily PDFs. Standard errors corrected for overlapping sample using Newey-West (1987) procedure. Asymptotic Newey-West t-statistics in parentheses. 37

Table 7: Profit Potential from Implied Skewness DM JY SP 25-Delta 10-Delta 25-Delta 10-Delta 25-Delta 10-Delta Risk Risk Risk Risk Risk Risk Reversal Reversal Reversal Reversal Reversal Reversal 45 Days to 23,900 5,540 8,330 -33,410 8,970 4,550 Maturity 90 Days to -58,930 -20,420 57,280 38,730 7,870 4,640 Maturity 135 Days to -31,500 -18,160 -96,040 -18,960 13,140 6,650 Maturity Monte Carlo: Full -18,360 -7,650 -15,160 4,330 9,630 5,000 Sample |premium| > 1,560 2,490 10,320 4,070 9,600 5,740 median |premium| > 47,250 21,860 24,670 16,650 8,700 4,660 .75 percentile Each cell represents the realized profit over the sample period from a strategy of buying the relatively less expensive out of the money call (put) and selling the more expensive out of the money put (call). The means from 1000 trials of randomly selecting one observation per contract are reported in the Monte-Carlo cells. In the Monte Carlo rows, the strategies are grouped by increasing selectivity before a transaction is entered. Transactions costs are not factored into these results. 38

Figure 1 S&P 500 Density Functions for June 24, 1998 Fixed l 6 Effective Parameters 800 900 1000 1100 1200 1300 1400 1500 1600 39

Figure 2 PDF #-Implied Volatility vs. Realized Volatility * Deutsche mark futures 0.5 0.4 0.3 0.2 Implied volatility 0.1 Realized volatility 0.0 1986 1988 1990 1992 1994 1996 1998 Japanese yen futures 0.5 0.4 0.3 0.2 Implied volatility 0.1 Realized volatility 0.0 1986 1988 1990 1992 1994 1996 1998 S&P futures 0.5 0.4 0.3 Implied volatility 0.2 0.1 Realized volatility 0.0 1986 1988 1990 1992 1994 1996 1998 * 5 day moving average # PDF from smoothed implied volatility smile method 40

Figure 3 Measures of Implied Skewness# Deutsche mark futures 0.50 Skew1 0.25 0.00 Skew2* -0.25 1986 1988 1990 1992 1994 1996 1998 Japanese yen futures 0.75 Skew1 0.50 0.25 0.00 Skew2* -0.25 -0.50 1986 1988 1990 1992 1994 1996 1998 S&P 500 futures 0.0 -0.5 Skew2* -1.0 -1.5 Skew1 -2.0 1986 1988 1990 1992 1994 1996 1998 Skew1 = Third moment skewness Skew2 = Pearson’s median skewness # 15 day moving average * Skew2 multiplied by a factor of 6 41

1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C 1 Lognormal PDF 0.8 Perfect Calibration 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C Lognormal PDF Perfect Calibration 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C 1 Lognormal PDF 0.8 Perfect Calibration 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C DM 45 Days Ahead DM 90 Days Ahead JY 45 Days Ahead JY 90 Days Ahead Lognormal PDF Perfect Calibration Figure 4: Calibration Functions for DM and JY 42

1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C 1 Lognormal PDF 0.8 Perfect Calibration 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C Lognormal PDF Perfect Calibration 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C 1 Lognormal PDF 0.8 Perfect Calibration 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 U )U(C SP 45 Days Ahead SP 90 Days Ahead SP (risk-adjusted) 45 Days Ahead SP (risk-adjusted) 90 Days Ahead Lognormal PDF Perfect Calibration Figure 5: Calibration Functions for SP and Risk-adjusted SP 43