Journal of Econometrics 102 (2001) 67}110 Do option markets correctlyprice the probabilities of movement of the underlying asset? Yacine AmK t-Sahalia *, Yubo Wang, Francis Yared Department of Economics, Princeton University, Princeton, NJ 08544-1021, USA Fixed Income Research, J.P. Morgan Securities Inc., New York, NY 10017, USA Lehman Brothers International (Europe), London EC 2M7HA, UK Received 4 January1999; received in revised form 21 August 2000; accepted 2 October 2000 Abstract We answer this question bycomparing te risk-neutral densityestimated in complete markets from cross-section of S&P 500 option prices to the risk-neutral densityinferred from the time series densityof the S&P 500 index. If investors are risk-averse, the latter densityis di !erent from the actual densitythat could be inferred from the time series of S&P 500 returns. Naturally, the observed asset returns do not follow the risk-neutral dynamics, which are therefore not directly observable. In contrast to the existing literature, we avoid making anyassumptions on investors ' preferences, bycomparing two risk-adjusted densities, rather than a risk-adjusted densityfrom option prices to an unadjusted densityfrom index returns. Our onlymaintained hypothesisis a one-factor structure for the S&P 500 returns. We propose a new method, based on an empirical Girsanov's change of measure, to identifythe risk-neutral densityfrom the observed unadjusted index returns. We design four di!erent tests of the null hypothesis that the S&P 500 options are e$cientlypriced given the S&P 500 index dynamics,and reject it. Byadding a jump component to the index dynamics,we are able to partlyreconcile the di!erences between the index and option-implied risk-neutral densities, and propose a peso-problem interpretation of this evidence. 2001 Elsevier Science S.A. All rights reserved. JEL classixcation: C14; G13 * Corresponding author. Tel.: #1-609-258-4015; fax: #1-609-258-0719. E-mail address: [email protected] (Y. AmK t-Sahalia). 0304-4076/01/$ - see front matter 2001 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 9 1 - 9 68 Y. An(t-Sahalia et al. / Journal of Econometrics 102 (2001) 67}110 Keywords: State-price densities; Risk-neutral densities; Densitycomparison; Arbitrage relationships; Girsanov's Theorem; Implied volatilitysmile; Jump risk; Peso problem 1. Introduction Do investors rationallyforecast future stock price distributions when they price options? Provided that markets are dynamically complete, arbitrage arguments tie down option prices to the prices of primaryassets. We may therefore expect that option prices will accuratelyre #ect the risk implicit in the stochastic dynamics of their underlying assets } and nothing else. Is this the case empirically, at least within the framework of a given class of models? An arbitrage-free option pricing model can be reduced to the speci"cation of a densityfunction assigning probabilities to the various possible values of the underlying asset price at the option's expiration. This densityfunction is named the state-price density(SPD), due to its intimate relationship to the prices of Arrow}Debreu contingent claims. A number of econometric methods are now available to infer SPDs from option prices. These methods deliver SPD esti- mates which either relax the Black}Scholes (1973) and Merton (1973) log-normal assumption in speci"c directions, or explicitlyincorporate the deviations from the Black}Scholes model when estimating the option-implied SPD and pricing other derivative securities. One of the main attractions of the latter methods is that theyproduce direct estimates of the Arrow }Debreu state prices implicit in the market prices of options. Their main drawback is that theyempirically ignore the evidence contained in the observed time series dynamics of the underlying asset } which, in theory, should indeed be redundant information. The goal of this paper is to examine whether the information contained in the cross-sectional option prices and the information contained in the time series of underlying asset values are empirically consistent with each other, i.e., whether option prices are rationallydetermined, under the maintained hypothesisthat the onlysource of risk in the economyis the stochastic nature of the asset price. More concretely, does the S&P 500 option market correctly price the S&P 500 index risk? It is certainlytempting to answer this question simplybycomparing features of the SPD implied byS&P 500 option prices to features of the observable time series of the underlying asset price. See Ross (1976), Banz and Miller (1978), Breeden and Litzenberger (1978) and Harrison and Kreps (1979). See Cox and Ross (1976), Merton (1976), Jarrow and Rudd (1982), Bates (1991), Goldberger (1991), Longsta! (1995) and Bakshi et al. (1997). See Shimko (1993), Derman and Kani (1994), Dupire (1994), Rubinstein (1994), Stutzer (1996), Jackwerth and Rubinstein (1996), AmKt-Sahalia and Lo (1998) and Dumas et al. (1998). Y. An(t-Sahalia et al. / Journal of Econometrics 102 (2001) 67}110 69 In fact, previous studies in the literature have always compared the observed option data to the observed underlying returns data, i.e., a risk-neutral density to an actual density. Naturally, in the case where all the Black}Scholes assumptions hold, the comparison between the option-implied and the asset-implied SPDs, both log-normal with the same mean, reduces to a comparison of implied volatilities, assumed common across the cross-section of options, to the index's time series realized volatilityover the life of the option. More generally, the well-documented presence in option prices of an &implied volatilitysmile ',whereby out-of-the-moneyput options are more expensive than at-the-moneyoptions, directlytranslate into a negativelyskewed option-implied SPD. At the same time, there is evidence in the S&P 500 time series of time-varying volatility, generally negativelycorrelated with the index changes, as well as jumps; both of these e !ects can give rise to a negativelyskewed SPD. We argue in this paper that no conclu- sions regarding the rationalityof option prices should be drawn from putting these two pieces of evidence together. The distribution of the underlying asset values that can be inferred from its observed time series, and the SPD implied by option prices, are not comparable without assumptions on investors' preferences. In fact, it is possible to use the exact same data to determine the representative preferences that are implicit in the joint observations on option and underlying asset prices. In other words, the extra degree of freedom introduced byprefer- ences can &reconcile' anyset of option prices with the observed time series dynamics of the underlying asset (up to suitable restrictions). Thus it is always possible to conclude that the options are rationallypriced from a comparison of the option-implied SPD and the actual densityof index returns, when prefer- ences (albeit strange ones) can be found that legitimize the observed option prices. Yet verylittle is known about aggregate investors ' preferences } even within the class of isoelastic utilityfunctions, there is wide disagreement in the literature regarding what constitutes a reasonable value of the coe$cient of relative risk aversion: is the coe$cient of relative risk aversion equal to 5 or 250? Therefore there is no reason a priori to compare the dynamics of the underlying asset that are implied by the option data to the dynamics implied by the actual time series, unless for some reason one holds a strong prior view on preferences. Consequently, the combination of the facts that the observed S&P 500 returns are negativelyskewed and that there exists a persistent implied volatilitysmile, cannot be used as evidence either in favor or against the rationalityof option prices. See Chiras and Manaster (1978), Schmalensee and Trippi (1978), Macbeth and Merville (1979), Rubinstein (1985), Dayand Lewis (1988, 1990), Engle and Mustafa (1992), Canina and Figlewski (1993), Lamoureux and Lastrapes (1993), and for a studyof the e !ect of institutional features on these comparisons Harveyand Whaley(1992). See AmKt-Sahalia and Lo (2000) and Jackwerth (2000). See for example Mehra and Prescott (1985) and Cochrane and Hausen (1992). 70 Y. An(t-Sahalia et al. / Journal of Econometrics 102 (2001) 67}110 In fact, the two SPDs need not even belong to the same parametric family, meaning that comparisons of their variances alone, as would be appropriate under the Black}Scholes assumptions, can lead to type-II errors (if the variances happen to be equal but other features of the densities are distinct) and/or type-I errors (if the SPDs are in fact equal but the variance estimates are inconsistent due to misspeci"cation of the assumed parametric form). For instance, Lo and Wang (1995) give an example of misspeci"ed variance estimates in a simpler context. The di$cultyat this point is that we do not observe the data that would be needed to infer directlythe time-series SPD, since we onlyobserve the actual realized values of the S&P 500, not its risk-neutral values. The main noveltyin this paper is to show that it is nevertheless possible to use the observed asset prices to infer indirectlythe time-series SPD that should be equal to the option-implied cross-sectional SPD. We relyon Girsanov 's charac- terization of the change of measure from the actual densityto the SPD: the di!usion function of the asset's dynamics is identical under both measures, only the drift needs to be adjusted. This fact is of course not surprising from a theoretical point of view, but had not been exploited empirically. Our econo- metric strategyis to start byestimating the di !usion function from the observed S&P 500 index returns. Then byGirsanov 's Theorem, this is the same di!usion function as the di!usion function of the risk-neutral S&P 500 returns.