Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices

Transcription

1 THE JOURNAL OF FINANCE VOL LIII, NO. 2 APRIL 1998 Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices YACINE AÏT-SAHALIA an ANDREW W. LO* ABSTRACT Implicit in the prices of trae financial assets are Arrow Debreu prices or, with continuous states, the state-price ensity (SPD). We construct a nonparametric estimator for the SPD implicit in option prices an we erive its asymptotic sampling theory. This estimator provies an arbitrage-free metho of pricing new, complex, or illiqui securities while capturing those features of the ata that are most relevant from an asset-pricing perspective, for example, negative skewness an excess kurtosis for asset returns, an volatility smiles for option prices. We perform Monte Carlo experiments an extract the SPD from actual S&P 500 option prices. ONE OF THE MOST IMPORTANT theoretical avances in the economics of investment uner uncertainty is the time-state preference moel of Arrow (1964) an Debreu (1959) in which they introuce elementary securities each paying one ollar in one specific state of nature an nothing in any other state. Now known as Arrow Debreu securities, they are the funamental builing blocks from which we erive much of our current unerstaning of economic equilibrium in an uncertain environment. In a continuum of states, the prices of Arrow Debreu securities are efine by the state-price ensity (SPD), which gives for each state x the price of a security paying one ollar if the state falls between x an x x. The existence an characterization of an SPD can be obtaine either in preference-base equilibrium moels, as in Lucas (1978), Rubinstein (1976), or in the arbitrage-base moels of Black an Scholes (1973) an Merton (1973). Each stran of the literature has aopte its own lexicon to enote closely relate concepts. In the equilibrium framework, the SPD can be expresse in terms of a stochastic iscount factor or pricing kernel such that asset prices are martingales uner the actual istribution of aggregate consumption after multiplication by the stochastic iscount factor (see, for example, Hansen an Jagannathan (1991) an Hansen an Richar (1987)). *Aït-Sahalia is from the University of Chicago an the NBER, an Lo is from MIT an the NBER. We receive helpful comments from George Constantinies, Eric Ghysels, Martin Haugh, John Heaton, Jens Jackwerth, Mark Rubinstein, René Stulz (the eitor), Jiang Wang, two referees, an especially Lars Hansen, as well as seminar participants at Duke University, the Fiels Institute, Harvar University, MIT, Tilburg University, the University of Chicago, ULB, Washington University, the Econometric Society Winter Meetings (1995), the NBER Asset Pricing Conference (1995), the NNCM Conference (1994) an the WFA Meetings (1996). A portion of this research was conucte uring the first author s tenure as an IBM Corp. Faculty Research Fellow an the secon author s tenure as an Alfre P. Sloan Research Fellow. 499

2 500 The Journal of Finance Among the no-arbitrage moels, the SPD is often calle the risk-neutral ensity, base on the analysis of Ross (1976) an Cox an Ross (1976) who first observe that the Black Scholes formula can be erive by assuming that all investors are risk neutral, implying that all assets in such a worl incluing options must yiel an expecte return equal to the risk-free rate of interest. The SPD also uniquely characterizes the equivalent martingale measure uner which all asset prices iscounte at the risk-free rate of interest are martingales (see Harrison an Kreps (1979)). Given the enormous informational content that Arrow Debreu prices possess an the great simplification they provie for pricing complex state-contingent securities such as options an other erivatives, it is unfortunate that pure Arrow Debreu securities are not yet trae on any organize exchange. However, they may be estimate or approximate from the prices of trae financial securities, as suggeste by Banz an Miller (1978), Breeen an Litzenberger (1978), an Ross (1976), an three methos have been propose to o just this. In the first metho, sufficiently strong assumptions are mae on the unerlying asset-price ynamics for the SPD to be obtaine in close form. For example, if asset prices follow geometric Brownian motion an the risk-free rate is constant, the SPD is log-normal this is the Black Scholes0Merton case. For more complex stochastic processes, the SPD cannot be compute in close-form an must be approximate by numerically intensive methos. 1 The secon metho requires specifying the SPD irectly in some parametric form. 2 An the thir metho begins by specifying a prior parametric istribution as a caniate SPD, typically the Black Scholes log-normal ensity. The SPD is then estimate by minimizing its istance to the prior parametric istribution uner the constraints that it correctly prices a selecte set of erivative securities. 3 In this paper, we propose an alternative to these three methos, in which the SPD is estimate nonparametrically, that is, with no parametric restrictions on either the unerlying asset s price ynamics or on the family of istributions that the SPD belongs to, an no nee for choosing any prior istribution for the SPD. 4 Although parametric approaches are clearly pref- 1 See, for example, Golenberg (1991) for formulas in integral form for the case of iffusions other than geometric Brownian motion, Heston (1993) for an implicit characterization of the SPD in a stochastic volatility moel, an Bates (1995) for a moel with stochastic volatility an jumps. 2 See Jarrow an Ru (1982), Shimko (1993), Longstaff (1995), an Maan an Milne (1994). 3 See Rubinstein (1994) an Jackwerth an Rubinstein (1996), who experiment with ifferent istance criteria. 4 In several other contexts the potential benefits of nonparametric methos for asset-pricing applications have recently begun to be explore. Aït-Sahalia (1996a) constructs nonparametric estimators of the iffusion process followe by the unerlying asset return, as a basis to price interest rate erivatives nonparametrically. Aït-Sahalia (1996b) tests parametric specifications of the spot interest rate process against a nonparametric alternative. The estimators all use iscrete ata, an require no iscrete approximation even though the estimate moel is in continuous time. Hutchinson, Lo, an Poggio (1994) show how a neural network can approximate the Black Scholes formula an other erivative-pricing moels. Bououkh et al. (1995) price mortgages, an Stutzer (1996) estimates the empirical istribution of stock returns.

3 Nonparametric Estimation of State-Price Density 501 erable when the unerlying asset s price process is known to satisfy particular parametric assumptions, for example, geometric Brownian motion, nonparametric methos are preferre when such assumptions are likely to be violate. Because recent empirical evience seems to cast some oubt on the more popular parametric specifications, 5 a nonparametric approach to estimating the SPD may be an important alternative to the more traitional methos. In particular, our nonparametric SPD estimator can yiel valuable insights in at least four contexts. First, it provies us with an arbitrage-free metho of pricing new, more complex, or less liqui securities, such as OTC erivatives or nontrae flexible options, given a subset of observe an liqui funamental prices, such as basic call-option prices, that are use to estimate the SPD. 6 From a risk management perspective, we provie information that is crucial to unerstaning the nature of the fat tails of assetreturn istributions implie by options ata. Volatility cannot be use as a summary statistic for the entire istribution when typical return series isplay events that are three stanar eviations from the mean approximately once a year. Our approach yiels an estimate of the entire return istribution, from which single points, such as value-at-risk, can easily be erive. Secon, our nonparametric estimator captures those features of the ata which are most salient from an asset-pricing perspective an which ought to be incorporate into any successful parametric moel. It also helps us unerstan what features are misse by tightly parametrize moels. For example, in our empirical application to S&P 500 inex options in Section III, the nonparametric SPD estimator naturally captures the so-calle volatility smile (see Figure 3 in that section) because this is a prominent feature of the ata. But we also ocument changes in the shape of the volatility smile over ifferent maturities which parametric moels so far have not incorporate. The nonparametric SPD estimator also exhibits persistent negative skewness an excess kurtosis (see Figure 7 later) because these too are features of the ata. Inee, a nonparametric analysis can often be avocate as a prerequisite to the construction of any parsimonious parametric moel, precisely because important features of the ata are unlikely to be misse by nonparametric estimators. In contrast, typical parametric stochasticvolatility moels have ifficulty eliminating biases in short-term option prices because they o not generate enough kurtosis, an typical jump moels encounter ifficulties with longer-term options because they revert too fast to the Black Scholes prices as the maturity ate grows. Thir, an perhaps most importantly, the nonparametric estimator highlights the empirical features of the ata in a way that is robust to the classical joint hypothesis problem. Our estimator is free of the joint hypotheses on asset-price ynamics an risk premia that are typical of parametric ar- 5 See, for example, the iscussion in Campbell, Lo, an MacKinlay (1997, Chapter 2). 6 Of course, markets must be ynamically complete for such prices to be meaningful (see, for example, Constantinies (1982), an Duffie an Huang (1985)). This assumption is almost always aopte, either explicitly or implicitly, in parametric erivative-pricing moels, an we shall aopt it here as well.

4 502 The Journal of Finance bitrage moels, or on preferences in the equilibrium approach to erivative pricing. Of course, nonparametric techniques o require certain assumptions on the ata-generating process itself, but these are typically weaker than those of parametric moels an are less likely to be violate in practice. Fourth, if we make the aitional assumption that unerlying asset prices follow iffusion processes, our estimator of the SPD can be use to estimate nonparametrically the instantaneous volatility function of the unerlying asset-return process. 7 Thus, if we restrict attention to iffusions, we obtain the continuous-state analog to the implie binomial trees propose by Rubinstein (1994) (see also Derman an Kani (1994), Shimko (1993), an the implie volatility functions in Dupire (1994), an Dumas, Flemming, an Whaley (1995)). In Section I we review briefly the relation between SPDs an the pricing of erivative securities, introuce our nonparametric SPD estimator, an compare it to alternative approaches. We present the results of extensive Monte Carlo simulation experiments in Section II in which we generate simulate price ata uner the Black Scholes assumptions an show that our SPD estimator can successfully approximate the Black Scholes SPD. In Section III, we apply our SPD estimator to the pricing an elta-heging of S&P 500 inex options. Options with ifferent times-to-expiration yiel a family of SPDs over ifferent horizons. We ocument several empirical features of the SPD over time, incluing the term structures of mean returns, volatility, skewness, an kurtosis, that are implie by these istributions. Moreover, unlike many parametric option-pricing moels, we show that the SPDgenerate option-pricing formula is capable of capturing persistent volatility smiles an other empirical features of market prices. We then point to parametric moeling strategies that woul incorporate these features. We conclue in Section IV, an give technical results an etails in the Appenix, incluing the asymptotic istributions of our SPD estimators an corresponing specification test statistics. I. Nonparametric Estimation of SPDs The intimate relation between SPDs, ynamic equilibrium moels, an erivative securities is now well known, but for completeness an to evelop notation we shall provie a brief summary here (see, for example, Huang an Litzenberger (1988, Chapters 5 8) for a more etaile iscussion). In a ynamic equilibrium moel such as those of Lucas (1978) an Rubinstein (1976), the price of any financial security can be expresse as the expecte net present value of its future payoffs, where the present 7 This complements the approach in Aït-Sahalia (1996a) where a nonparametric estimator of the same volatility function is obtaine from the time series of the unerlying asset returns. We can also infer the volatility function of the unerlying price process from erivative prices. Girsanov s Theorem implies that they shoul be the same, which is a testable implication of the no-arbitrage pricing paraigm.

5 Nonparametric Estimation of State-Price Density 503 value is calculate with respect to the riskless rate r an the expectation is taken with respect to the marginal-rate-of-substitution-weighte probability ensity function (PDF) of the payoffs. This PDF, which is istinct from the PDF of the payoffs, is the SPD or risk-neutral PDF (see Cox an Ross (1976)) or equivalent martingale measure (see Harrison an Kreps (1979)). More formally, the ate-t price P t of a security with a single liquiating ate-t payoff ~S T! is given by: P t e r t,tt E t T!# e r t,t t 0 ~S T!f t * ~S T! S T, (1) where S T is a state variable (e.g., aggregate consumption in Lucas (1978)), r t,t is the constant risk-free rate of interest between t an T [ t t, an f t * ~S T! is the ate-t SPD for ate-t payoffs. The ynamic equilibrium approach illustrates the enormous information content that SPDs contain an the enormous information reuction that SPDs allow. For example, if parametric restrictions are impose on the atagenerating process of asset prices (e.g., geometric Brownian motion), the SPD estimator may be use to infer the preferences of the representative agent in an equilibrium moel of asset prices (see, e.g., He an Lelan (1993)). Alternatively, if specific preferences are impose, such as logarithmic utility, the SPD may be use to infer the ata-generating process of asset prices. Inee, any two of the following imply the thir: (1) the representative agent s preferences; (2) asset-price ynamics; an (3) the SPD. From a pricing perspective, SPDs are sufficient statistics in an economic sense they summarize all relevant information about preferences an business conitions for purposes of pricing financial securities. A. SPDs an Derivative Securities In contrast to the PDF of the payoffs, the SPD cannot be easily estimate from the time series of payoffs because it is also influence by preferences, that is, the marginal rate of substitution. However, the SPD can be estimate from the time series of prices because prices represent the amalgamation of payoffs an preferences in an equilibrium context. In fact, builing on Ross s (1976) insight that options can be use to create pure Arrow Debreu state-contingent securities, Banz an Miller (1978) an Breeen an Litzenberger (1978) provie an elegant metho for obtaining an explicit expression for the SPD from option prices: The SPD is the secon erivative, normalize to have an integral of one, of a call option-pricing formula with respect to the strike price. For example, recall that uner the hypotheses of Black an Scholes (1973) an Merton (1973), the ate-t price H of a call option maturing at ate T [

6 504 The Journal of Finance t t, with strike price X, written on a stock with ate-t price S t when the ivien yiel is t,t an the stock s volatility is s, is given by 8 H BS ~S t, X,t, r t,t, t,t ;s! e r t,tt 0 * max~s T X,0!f BS, t ~S T! S T where S t e t,tt ~ 1! Xe r t,tt ~ 2!, (2) 1 [ ln~s t0x! ~r t,t t,t 1 _ 2 s 2!t, s!t 2 [ 1 s!t. (3) In this case the corresponing SPD is a lognormal ensity with mean ~~r t,t t,t! s 2 02!t an variance s 2 t for ln~s T 0S t!: * f BS, t ~S T! e r t,tt?2 H?X 2 X S T 1 S T % 2ps 2 t T 0S t! ~r t,t t,t s 2 02!t# 2 2s 2. (4) t This expression shows that the SPD can epen on many quantities in general (although for simplicity we write it explicitly as a function of S T only), an is istinct from but relate to the PDF of the terminal stock price S T. Arme with equation (4), the price of any other erivative security can be calculate trivially, as long as the hypotheses that lea to the option-pricing formula (2) are satisfie. 9 For example, the ate-t price of a igital call option a security that gives the holer the right to receive a fixe cash payment of one ollar if S T excees a prespecifie level X is given by D t e r t,tt E t T X!# e r t,t t 0 I~S T X!f t * ~S T! S T e r F t * ~X!#, (5) 8 Let F t,t enote the value at t of a futures contract written on the asset, with the same maturity t as the option. At the maturity of the futures, the futures price equals the asset s spot price. Thus a European call option on the asset has the same value as a European call option on the futures contract with the same maturity. As a result, we will often rewrite the Black Scholes formula as H BS ~F t,t, X,t, r t,t ;s! e rt,tt ~F t,t ~ 1! X ~ 2!!, with 1 [~ln~f t,t 0X! ~s 2 02!t!0~s!t! an 2 [ 1 s!t. 9 That is, frictionless markets, unlimite riskless borrowing an lening opportunities at the same instantaneous constant rate r, geometric Brownian motion ynamics for S t with a known an constant iffusion coefficient. See Merton (1973) for further iscussion.

7 Nonparametric Estimation of State-Price Density 505 where I~S T X! is an inicator function that takes on the value one if S T X an zero otherwise, an F t * is the cumulative istribution function (CDF) corresponing to the SPD. We shall return to this example in our empirical analysis of S&P 500 options in Section III. B. Nonparametric Option Prices an SPDs It is clear from equation (4) that the SPD is inextricably linke to the parametric assumptions unerlying the Black Scholes0Merton optionpricing moel. If those parametric assumptions o not hol, for example, if the ynamics of $S t % contain Poisson jumps or the volatility of the asset returns varies with the stock price, then the SPD in equation (4) will yiel incorrect prices, prices that are inconsistent with the ynamic equilibrium moel or the hypothesize stochastic process riving $S t %.Anonparametric estimator of the SPD an estimator that oes not rely on specific parametric assumptions such as geometric Brownian motion ynamics can yiel prices that are robust to such parametric specification errors. Breeen an Litzenberger (1978) first observe that f * t ~S T! exp~r t,t t!? 2 H~{!0?X 2. We propose to estimate the SPD nonparametrically in the following way: use market prices to estimate an option-pricing formula H~{! nonparametrically, then ifferentiate this estimator twice with respect to X to obtain? 2 H~{!0?X 2. Uner suitable regularity conitions, the convergence (in probability) of H~{! to the true option-pricing formula H~{! implies that? 2 H~{!0?X 2 will converge to? 2 H~{!0?X 2, which is proportional to the SPD. But how o we obtain H~{! nonparametrically? Given a set of historical option prices $H i % an accompanying characteristics $ i [@S ti X i t i r ti,t i ti,t i # ' %,we seek a function H~{! not a set of parameters that comes as close to $H i % as possible. In particular, using mean-square-error as our measure of closeness, we wish to solve: min n H~{! ( i i H~ i!# 2, (6) where G is the space of twice-continuously-ifferentiable functions. It is well known that the solution is given by the conitional expectation of H given. To estimate this conitional expectation, we employ a statistical technique known as nonparametric kernel regression. Nonparametric kernel regression prouces an estimator of the conitional expectation of H, conitione on, without requiring that the function H~{! be parametrize by a finite number of parameters (hence the term nonparametric). Kernel regression requires few assumptions other than smoothness of the function to be estimate an regularity of the ata use to estimate it, an it is robust to the potential misspecification of any given parametric call-pricing formula (see the Appenix). On the other han, kernel regression tens to be ata intensive. Financial applications are a natural outlet for kernel regression because typical parametric assumptions (e.g., normality or geometric Brownian motion)

8 506 The Journal of Finance have been rejecte by the ata, yet large sample sizes of high quality ata are not uncommon. Another motivation for kernel regression that is particularly insightful is the local averaging or smoothing interpretation. Suppose we wish to estimate the relation between two variables i an H i that satisfy the following nonlinear relation: H i H~ i! e i, i 1,...,n, (7) where H~{! is an unknown but fixe nonlinear function an $e i % is white noise. Consier estimating H~{! at a specific point i0 z 0 an suppose that for this one particular observation of i, we are able to obtain repeate observations of the variable H i0, say H ~1! i0,...,h ~q! i0. In this case, a natural estimator of the function H~{! at the point z 0 is simply q H~z 0! 1 q ( j 1 H i0 ~ j! H~z 0! 1 q ( j 1 q ~ e j! i0, q an, by the law of large numbers, ~10q!( j 1 ~ e j! i0 becomes negligible for large q. Of course, we o not have the luxury of repeate observations for a given i0 z 0. However, if we assume that the function H~{! is smooth, then for time series observations i near the value z 0, the corresponing values of H i shoul be close to H~z 0!. In other wors, if H~{! is smooth, then in a small neighborhoo aroun z 0, H~z 0! will be nearly constant an may be estimate by taking an average of the H i s that correspon to those i s near z 0. The closer the i s are to the value z 0, the closer an average of the corresponing H i s will be to H~z 0!. This argues for a weighte average of the H i s, where the weights ecline as the i s get farther away from the point z 0. Such a weighte average must be compute for each value of z in the omain of H~{! to estimate the entire function, hence computational consierations become important. This weighte-average proceure of estimating H~z! is the essence of smoothing. To implement such a proceure, we must efine what we mean by near an far. If we choose too large a neighborhoo aroun z to compute the average, the weighte average will be too smooth an will not exhibit the genuine nonlinearities of H~{!. If we choose too small a neighborhoo aroun z, the weighte average will be too variable, reflecting noise as well as nonlinearities in H~{!. Therefore, the weights must be chosen carefully to balance these two consierations. The choice of weighting function typically given by a probability ensity function (because such functions integrate to one), though the particular ensity function plays no probabilistic role here etermines the egree of local averaging (see Härle (1990) an Wan an Jones (1995) for a more etaile iscussion of nonparametric regression). To specify a particular kernel regression moel, we start with the natural assumption that the option-pricing formula H we seek to estimate is a func-

9 Nonparametric Estimation of State-Price Density 507 tion of a vector of option characteristics or explanatory variables, t X t r t,t t,t # ', so that each option price H i, i 1,...,n, containe in our ataset is paire with the vector i [@S ti X i t i r ti,t i ti,t i # '. 10 Because we have five explanatory variables, we select a five-imensional weighting or kernel function K~! that integrates to one. The ensity K~ i!, as a function of, has a certain sprea aroun the ata point i. We can change the sprea of the kernel K aroun, using a banwith h, to form the new ensity function ~10h!K~~ i!0h!. An estimator of the conitional expectation of H conitione on is then given by the following expression, calle the Naaraya Watson kernel estimator, where h becomes closer to zero as the sample size n grows: H~! n ( i 1 n ( i 1 K~~ i!0h!h i. (8) K~~ i!0h! Intuitively, the estimate of the conitional expectation at a point, that is, the price of an option with characteristics, is given by a weighte average of the observe prices H i s with more weight given to the options whose characteristics i s are closer to the characteristics of the option to be price. The closer h is to zero, the more peake is this new ensity function aroun i, an hence the greater is the weight given to realizations of the ranom variable i that are close to. To illustrate kernel regression in our context, consier a sample of options with ifferent strike prices an the same timeto-expiration. Suppose we are intereste in learning how the implie volatilities of options of that maturity vary with the option s moneyness the now familiar volatility smile. The choice of the kernel function typically has little influence on the en result, but the choice of the banwith h is the etermining factor because the sprea of the ensity aroun i is much more sensitive to the choice of h than that of K, provie of course that the vari- 10 Our approach assumes that the vector of state variables is correctly specifie, an is esigne to be very flexible in the way it accommoates the influence of inclue state variables. Plausible omitte variables inclue stochastic market volatility, option an market traing volumes, relative supply an eman for particular options ue to portfolio consierations, etc. These variables will be partially reflecte in the estimate SPD to the extent that some of the variation in these variables can be accounte for by the inclue variables; for example, if, as the empirical evience suggests, market volatility covaries negatively with the market price level. Note however that, unlike the classical regression case, our estimator is not mae inconsistent by the omission of relevant variables. Suppose that the true vector contains two sets of variables, 1 an 2, but that we only inclue 1 to construct our estimator. In that case, we woul be estimating 1 # instea of 1, 2 #; but by the law of iterate expectations 1 1 #, an so our estimator woul still be consistent for the reuce-form call-pricing function as a function of 1. In a classical linear regression, the reuceimension estimator (on 1 only) woul be inconsistent (unless # 0!.

10 508 The Journal of Finance ance of the class of potential kernel functions is restricte. Figure 1A emonstrates the effect of unersmoothing: h is 0.1 times the optimal value. The estimator is correctly centere aroun the true function (the ashe line), that is, it has low bias but is highly variable. Figure 1B is optimally smoothe, accoring to the formula we erive in Appenix A. Figure 1C is oversmoothe: h is four times the optimal value. This estimator exhibits little variance, but is substantially biase. In Appenix A we etermine the banwith h that achieves the optimal trae-off between bias an variance. C. Practical Consierations: Dimension Reuction We also show in Appenix A that obtaining accurate estimators of the regression function is more ifficult when the number of regressors is large (we start here with 5as i [@S ti X i t i r ti,t i ti,t i # '!, an then erivatives of the function H~{! nee to be compute. Recall that to obtain the SPD we must ifferentiate the call-pricing function twice. 11 To be fully nonparametric, all five regressors in [@S t Xtr t,t t,t # ' must be inclue in the kernel regression (8) of call option prices H on. To reuce the number of regressors, we examine the following possibilities. First, we coul assume that the option-pricing formula (an hence the SPD) is not a function of the asset price S t, the risk-free rate r t,t, an the ivien yiel t,t separately, but only epens on these three variables through the futures price F t,t S t e ~r t,t t,t!t an the risk-free rate; that is, H~S t, X,t, r t,t, t,t! H~F t,t, X,t, r t,t!. By no-arbitrage, the mean of the SPD epens only on (an, in fact, is equal to) F t,t an the assumption here woul be that the entire istribution has this property. It is satisfie by the Black Scholes SPD equation (4). Uner this assumption, the number of regressors is reuce from 5to 4. Note that the value of a European option written on a futures contract with the same maturity as the option is ientical to the value of the corresponing option written on the asset. Further, the rift of the futures process is zero uner the riskneutral measure (hence inepenent of t,t!, so it is quite reasonable to expect that the ivien yiel oes not enter the option-pricing formula other than through the futures value. Theoretically, it may still enter the function, for instance if the volatility of the asset s returns epens on t,t, but this woul most likely be a fairly remote situation. 11 As the sample size increases, the estimator (8), as well as its erivatives with respect to, converge to the true function H~{! an its erivatives at every point. Therefore, when the true function satisfies certain shape restrictions, such as monotonicity an convexity with respect to certain variables, the estimator H~{! will also have these properties. However, because the asymptotic convergence of the higher-orer erivatives may in practice be slow, it can be useful in small samples to moify the estimator to force it to satisfy these restrictions. One simple way to enforce monotonicity is to run an isotonic regression after the kernel regression, thereby guaranteeing that the resulting estimator H~{! is monotonic. To enforce convexity in small samples, we can exten this proceure one step further: ifferentiate the isotonic kernel estimator to obtain H ' ~{!, an then run an isotonic regression on H ' ~{!. We are then guarantee that H ' ~{! also is monotonic, an hence H~{! is convex.

11 Nonparametric Estimation of State-Price Density Implie Volatility Moneyness A: Kernel Regression with Unersmoothe Banwith Implie Volatility Moneyness B: Kernel Regression with Optimal Banwith Implie Volatility Moneyness C: Kernel Regression with Oversmoothe Banwith Figure 1. Kernel Regression an Banwith Selection. In each panel, the soli line is the kernel-regression estimator of the theoretical relation given by the ashe line. This figure exemplifies the practical importance of banwith selection. In Panel A the banwith is too small by a factor of ten, in Panel B the banwith is optimal (see Appenix A), an in Panel C the banwith is too large by a factor of four. The sample size equals 1,000.

12 E E E D 510 The Journal of Finance Secon, we coul assume that the option-pricing function is homogeneous of egree one in F t,t an X, as in the Black Scholes formula. This assumption woul also reuce the number of regressors from 5to 4. Combining this assumption with the previous one, the imension of the problem woul be further reuce to 3. It can be shown, however, that the callpricing function is homogeneous of egree one in the asset price an the strike price when the istribution of the returns is inepenent of the level of the asset price (see Merton (1973, Theorem 9) who also provies a counterexample showing that the homogeneity property can fail if the istribution of the returns is not inepenent of S t!. An example of a pricing formula satisfying the homogeneity property woul be the one generate by a stochastic volatility moel where the rift an iffusion functions of the stochastic volatility process epen on the volatility itself but not on the asset price (see Renault (1995)). Although this assumption may not be too restrictive in practice, this is nevertheless the type of assumption on the asset-price ynamics that we wish to avoi in the first place by using a nonparametric estimator. Our thir propose approach to imension reuction is semiparametric. Suppose that the call-pricing function is given by the parametric Black Scholes formula (2) except that the implie volatility parameter for that option is a nonparametric function s~f t,t,x,t!: H~S t, X,t, r t,t, t,t! H BS ~F t,t, X,t, r t,t ;s~f t,t,x,t!!. (9) In this semiparametric moel, we woul only nee to estimate nonparametrically the regression of s on a subset of the vector of explanatory variables. The rest of the call-pricing function H~{! is parametric, thereby consierably reucing the sample size n require to achieve the same egree of accuracy as the full nonparametric estimator. We partition the vector of explanatory variables [@E ' F t,t r t,t # ', where contains nonparametric regressors. As a result, the effective number of nonparametric regressors is given by. D In our empirical application, we will consier both E [@XF t,t t# ' ~ D 3! an [@X0F t,t t# ' ~ D 2, by combining this imension-reuction technique with the previous one). Given the ata $H i,s ti, X i,t i, r ti,t i, ti,t i %, the full-imensional SPD estimator takes the following form. We construct the fully nonparametric callpricing function as H~S t, X,t, r t,t, t,t! t, X,t, r t,t, t,t # (10) using a multivariate kernel K in equation (8), forme as a prouct of five univariate kernels where subscripts refer to regressors (so, for example, k t ~{! is the kernel function use for time-to-expiration as a regressor, an h t is the corresponing banwith value):

13 E E H~S t, X,t, r t,t, t,t! n ( i 1 n ( i 1 Nonparametric Estimation of State-Price Density 511 k S S t S ti h S k S S t S ti h S k X X X i h X k X X X i h X k t t t i h t k t t t i h t k r r t,t r ti,t i h r k r r t,t r ti,t i h r k t,t ti,t i H h i. k t,t ti,t i h For the reuce-imension SPD estimators, we substitute the appropriate list of regressors for ~S t, X,t, r t,t, t,t! in equation (11). For example, in the semiparametric moel with [@XF t,t t# ', we form the three-imensional kernel estimator of t,t, X,t# as s~f [ t,t,x,t! n ( i 1 n ( i 1 k F F t,t F ti,t i h F k F F t,t F ti,t i h F k X X X i h X k X X X i h X k t t t i s h i t k t t t i h t (11), (12) where s i is the volatility implie by the price H i. We then estimate the call-pricing function as H~S t, X,t, r t,t, t,t! H BS ~F t,t, X,t, r t,t ; s~f [ t,t,x,t!!. (13) In what follows, we will refer to any one of the above estimators H~S t,x,t,r t,t, t,t! as nonparametric estimators an specify in each case the appropriate vector. The option s elta an the SPD estimators follow by taking the appropriate partial erivatives of H~{!: t? H~S t, X,t, r t,t, t,t! (14)?S t f t * ~S T! e r t,tt?2 H~S t, X,t, r t,t, t,t!?x 2 6X S T. (15) D. Comparisons with Other Approaches Several other approaches to fit erivative prices have been propose in the recent literature, hence a comparison of their strengths an weaknesses to those of the nonparametric kernel estimator is appropriate before turning to the Monte Carlo analysis an empirical application of Sections II an III. D.1. Learning Networks Hutchinso, Lo, an Poggio (1994) apply several nonparametric techniques to estimate option-pricing moels that they escribe collectively as learning networks: artificial neural networks, raial basis functions, an projection

14 512 The Journal of Finance pursuit. Although they show that these techniques can recover optionpricing moels such as the Black Scholes0Merton moel, they o not consier extracting the SPD from their nonparametric estimators. Of course, SPDs can be extracte from many nonparametric option-pricing estimators incluing learning-network moels simply by taking the secon erivative of the nonparametric estimator with respect to the strike price (see Section II.A). However, the secon erivative of a nonparametric estimator nee not be a goo estimator for the secon erivative of the function to be estimate. This concern is particularly relevant for estimating nonlinear functions that are not smooth, that is, infinitely ifferentiable, or where the egree of smoothness is unknown. Because the focus of our paper is the SPD an not the option-pricing function, we have constructe our estimator to reflect this focus. For example, Appenix A outlines our choices for the kernel function, the orer of the kernel, an the banwith parameter each choice etermine to some egree by our interest in the secon erivative of the estimator. Therefore, the properties of our estimator are likely to be quite ifferent from an superior to the secon erivatives of other nonparametric option-pricing estimators. Also, Hutchinson, Lo, an Poggio (1994) o not provie any formal statistical inference to gauge the accuracy of their estimators. This is a problem enemic to learning networks an other recursive estimators an is extremely ifficult to resolve as White (1992) has shown. In contrast, nonparametric regression an other nonrecursive smoothing methos are better suite to hypothesis testing an other stanar types of statistical inference. If such inferences are important in empirical applications, our estimator is preferable. In particular, we erive both pointwise an global confience intervals for our estimator in Appenices B an C respectively, as well as stability tests in Appenix D. D.2. Implie Binomial Trees Perhaps the closest alternative to our approach is Rubinstein s (1994) implie binomial tree, in which the risk-neutral probabilities $p * i % associate with the binomial terminal stock price S T are estimate by minimizing the sum of square eviations between $p * i % an a set of prior risk-neutral probabilities $ pi * i %, subject to the restrictions that $p * i % correctly price an existing set of options an the unerlying stock, in the sense that the optimal riskneutral probabilities yiel prices that lie within the bi ask spreas of the options an the stock (see also Jackwerth an Rubinstein (1996) for smoothness criteria). This approach is similar in spirit to Jarrow an Ru (1982) an Longstaff s (1995) metho of fitting risk-neutral ensity functions using a four-parameter Egeworth expansion (however, Rubinstein (1994) points out several important limitations of Longstaff s metho when extene to a binomial moel, incluing the possibility of negative probabilities; see also Derman an Kani (1994) an Shimko (1993)). Although Rubinstein (1994) erives his risk-neutral probabilities within a binomial tree moel a parametric family his approach can be interprete

15 Nonparametric Estimation of State-Price Density 513 as nonparametric in the sense that virtually any SPD can be approximate to any egree of accuracy by a binomial tree. But there are two main ifferences between his approach an ours: (1) he requires a prior $ pi * i % for the risk-neutral probabilities; an (2) he fits one set of risk-neutral probabilities $p * i % for each cross section of options, whereas kernel SPDs aggregate option ata over time to get a single SPD. The first ifference is not substantial because Rubinstein (1994) shows that the prior has progressively less influence on the implie binomial tree as the number of options use to constrain the probabilities increases. However, the secon ifference is significant. In particular, an n-step implie binomial tree calculate at ate t 1 will, in general, be ifferent from an n-step implie binomial tree calculate at ate t 2. In contrast, because the nonparametric kernel estimator is base on both cross-sectional an timeseries option prices, the resulting n-step SPD is the same for t 1 an t 2. This ifference is the result of a ifference in the moeling strategies of the two approaches. The implie binomial tree is an attempt to obtain the risk-neutral probabilities that come closest to correctly pricing the existing options at a single point in time. As such, it can an will change over time. The nonparametric kernel estimator is an attempt to estimate the risk-neutral probabilities as a fixe function of certain economic variables, such as current stock price, riskless rate, etc. If it is successful, the functional form of the estimate SPD shoul be relatively stable over time, though the risk-neutral probabilities for any particular event, say S t t X, can change over time (in a specific way) if the economic variables on which the SPD is base change over time (we test this hypothesis below in Section III.F). Both approaches have avantages an isavantages. The implie binomial tree is completely consistent with all option prices at each ate, but is not necessarily consistent across time. The nonparametric kernel SPD estimator is consistent across time, but there may be some ates for which the SPD fits the cross section of option prices poorly an other ates for which the SPD performs very well. Whether or not consistency over time is a useful property epens on how well the economic variables use in constructing the kernel SPD can account for time variation in risk-neutral probabilities. Aitionally, the kernel SPDs take avantage of the ata temporally surrouning a given ate. Tomorrow s an yesteray s SPDs contain information about toay s SPD information that is ignore by the implie binomial trees but not by kernel-estimate SPDs. Implie binomial trees are less ata-intensive; by construction, they require only one cross section of option prices whereas the kernel SPD requires many cross sections. However, kernel SPDs are smooth functions by construction an easily (an optimally) interpolate between strike prices, maturities, an other kins of iscreteness. As we show, it is also possible to conuct statistical inference with kernel-estimate SPDs, something not easily hanle by implie binomial trees. In summary, although the two approaches have similar objectives, they also have important ifferences in their implementation. We examine below

16 Table I Summary Statistics for S&P 500 Inex Options Data Summary statistics for the sample of all trae CBOE aily call an put option prices on the S&P 500 inex in the perio January 4, 1993 to December 31, 1993 (14,431 observations). Implie s enotes the implie volatility of the option, an Implie ATM s enotes the implie volatility of at-the-money options. t enotes the options time-to-expiration, r the riskless rate, X the options strike price, an F the S&P 500 futures value implie from the at-the-money call an put prices. St. Dev. enotes the sample stanar eviation of the variable. During this perio, the sample aily mean an stanar eviation of continuously compoune returns of the S&P 500 inex were 7.95 an percent (annualize with a 252-ay year), respectively. The average aily value of the S&P 500 inex was Percentiles Variable Mean St. Dev. Min 5% 10% 50% 90% 95% Max Call Price H ($) Put Price G ($) Implie s (%) Implie ATM s (%) t (ays) X (inex points) F (inex points) r (%) The Journal of Finance

17 Nonparametric Estimation of State-Price Density 515 in Section III.G how these methos perform on actual ata, by comparing both their in-sample an out-of-sample forecast errors. II. Monte Carlo Analysis To examine the practical performance of the nonparametric SPD estimator, we perform several Monte Carlo simulation experiments uner the assumption that call-option prices are truly etermine by the Black Scholes formula. Our nonparametric approach shoul be able to approximate Black Scholes prices, from which the SPD may be extracte accoring to Breeen an Litzenberger (1978). The nonparametric pricing formula an SPD may then be compare to the Black Scholes formula an theoretical SPD, respectively, to gauge the accuracy of the nonparametric approach. Naturally, the avantage of our nonparametric approach lies in its robustness. If the options were price by another formula, the nonparametric approach shoul be able to approximate it as well because, by efinition, it oes not rely on any parametric specification for the unerlying asset s price process. Therefore, similar Monte Carlo simulation experiments can be performe for alternative option-pricing moels. However, we choose to perform the simulation experiments uner the Black Scholes assumptions as this is the leaing case from which most applications an extensions are erive. A. Calibrating the Simulations Our empirical application involves S&P 500 inex options, so we perform Monte Carlo simulation experiments to match the basic features of our ataset (see Section III an Table I for further etails). We start by simulating one year (252 ays) of aily inex prices generate by a geometric Brownian motion with constant rift an iffusion parameters that match the moments of the ata. Specifically, the values of the initial inex level, the interest rate, an the inex return mean an stanar eviation are fixe at 455, 3 percent, 7.95 percent, an percent, respectively. Throughout this stuy we follow the common convention of reporting returns an their means an stanar eviations at an annual frequency. For the purpose of calculating option prices, annual parameter values are converte to a aily frequency by assuming that a year consists of 252 traing ays. At the start of this one-year sample of simulate aily prices, we create call options with strike prices an times-to-maturity that follow the Chicago Boar Options Exchange (CBOE) conventions for introucing options to the market. As the inex price changes from one simulate ay to the next, existing options may expire an new options may be introuce, again accoring to CBOE conventions, with strike prices that bracket the inex in five-point increments. Therefore, on any given simulate ay, the number of options is an enogenous function of the prior sample path of the inex

18 516 The Journal of Finance price there are approximately 80 ifferent call options on any one simulation ay, incluing both existing an new ones. We then price these options by the Black Scholes formula using the actual inex volatility, an a a small white-noise term to those prices. By oing so, we seek to reflect the existence in real ata of a bi ask sprea an other possible sources of error in the recore prices. 12 By construction, the option prices in the simulation satisfy the Black Scholes formula on average; therefore, when we apply our nonparametric pricing function to the simulate ata, we shoul be able to recover the Black Scholes formula. By this, we mean that H~S t, X,t, r t,t, t,t! shoul approximate the Black Scholes formula (2) numerically, not necessarily algebraically in practice, the functional form of H~{! may be quite ifferent from the Black Scholes formula (2), but both expressions will prouce similar prices over the range of input values in the ata. The two objectives of our simulation experiments are to etermine how close H is to formula (2), an how close the corresponing nonparametric SPD is to the theoretical SPD in equation (4). Specifically, we take the Black Scholes prices in the simulate ataset an the option characteristics as the inputs $H i,s ti, X i,t i, r ti,t i, ti,t i % of our proceure, an then compute the smooth nonparametric call-pricing function of Section I.B given by equation (13). We then construct the option elta estimator an SPD estimator f * accoring to equations (14) an (15), respectively. This entire proceure is repeate 5,000 times to provie an inication of the accuracy of our estimators. B. Accuracy of Prices, Deltas, an SPDs To assess the performance of the nonparametric option-pricing formula an its corresponing elta an SPD, we consier the percentage ifferences between the nonparametric option-pricing formula, elta, an SPD, an their theoretical Black Scholes counterparts, respectively. In Figure 2, the theoretical values for prices, eltas, an SPDs are plotte in Panels A, B, an C, an the average ifferences between the theoretical values an the nonparametric ones, average over the 5,000 replications, are plotte in Panels D, E, an F. The figures show that the nonparametric quantities are within 1 percent of their theoretical counterparts; the estimators are virtually free of any bias. The ispersion of the estimates across the simulation runs yiels the sampling istribution of the estimator an allows us to confirm the accuracy of the asymptotic istribution erive in Appenix B for the sample size relevant for our empirical stuy in Section III. These simulations also illustrate empirically the fact that higherorer erivatives SPD relative to eltas, eltas relative to prices are estimate at a slower rate of convergence, which we emonstrate theoretically in Appenix A, commonly calle the curse of ifferentiation. 12 The white-noise term is Gaussian with stanar eviation equal to either one or two price ticks, epening on whether the option characteristics mae it a high- or low-volume option.

19 Nonparametric Estimation of State-Price Density 517 III. Estimating SPDs from S&P 500 Options Data To assess the empirical relevance of our nonparametric option-pricing formula an the corresponing SPD estimator, we present an application to the pricing an heging of S&P 500 inex options using ata obtaine from the CBOE for the sample perio January 4, 1993 to December 31, A. The Data Table I escribes the main features of our ata set. The beginning sample contains 16,923 pairs of call an put option prices we take averages of bi an ask prices as our raw ata. Observations with time-to-maturity of less than one ay, implie volatility greater than 70 percent, an price less than 1 _ 8 are roppe, which yiels a final sample of 14,431 observations an this is the starting point for our empirical analysis. During 1993, the mean an stanar eviation of continuously compoune aily returns of the S&P 500 inex are 7.95 percent an percent, respectively. During this perio short-term interest rates exhibit little variation: they range from 2.85 percent to 3.21 percent. The options in our sample vary consierably in price an terms; for example, the time-to-maturity varies from 1 ay to 350 ays, with a meian of 66 ays. Given the volatility an movement in the inex uring this perio an CBOE rules for introucing new options to the market, our sample contains a fairly broa cross section of options. S&P 500 Inex Options (symbol: SPX) are among the most actively trae financial erivatives in the worl. The average total aily volume uring the sample perio was 65,476 contracts. The minimum tick for series traing below 3 is _ 1 16 an for all other series 1 _ 8. Strike price intervals are 5 points, an 25 for far months. The expiration months are the three near-term months followe by three aitional months from the March quarterly cycle (March, June, September, December). The options are European, an the unerlying asset is an inex, the most likely case for which a lognormal assumption (with continuous ivien stream) can be justifie. By the simple effect of iversification, jumps are less likely to occur in inices than in iniviual equities. In other wors, this market is as close as one can get to satisfying the assumptions of the Black Scholes moel. This is, therefore, a particularly promising context to test our approach: How ifferent is our estimate SPD from the Black Scholes SPD? Even though the options are European an o not have a wilcar feature, the raw ata present three challenges that must be aresse. First, because in-the-money options are very infrequently trae relative to at-themoney an out-of-the-money options, in-the-money option prices are notoriously unreliable. For example, the average aily volume for puts that are 20 points out-of-the-money is 2,767 contracts; in contrast, the volume for puts that are 20 points in-the-money is 14 contracts. This reflects the strong eman by portfolio managers for protective puts (a phenomenon that starte in late 1987 for obvious reasons).

20 518 The Journal of Finance A: Black Scholes Option Price Option Price Days to Expiration Current Inex Price B: Black Scholes Option Delta Option Delta Days to Expiration Current Inex Price C: Black Scholes State-Price Density Density Days to Expiration Inex Price of Expiration Figure 2. Pricing an Heging Errors of Nonparametric Estimator of the Black Scholes Formula. Theoretical values for prices, eltas, an SPDs are plotte in Panels A, B, an C, an the average ifferences (in percent) between the theoretical values an the nonparametric estimators, average over the 5,000 replications, are plotte in Panels D, E, an F.

21 0 1 Nonparametric Estimation of State-Price Density 519 D: Option Price % Error % Error Days to Expiration Current Inex Price E: Option Delta % Error 0 1 % Error Days to Expiration Current Inex Price F: State-Price Density % Error 0 1 % Error Days to Expiration Inex Price of Expiration Figure 2. Continue.

22 520 The Journal of Finance Secon, it is ifficult to observe the unerlying inex price at the exact times that the option prices are recore. In particular, there is no guarantee that the closing inex value reporte is recore at the same time as the closing transaction for each option. S&P 500 inex futures are trae on the Chicago Mercantile Exchange (CME), not the CBOE, an time-stampe reporte quotes may not necessarily be perfectly synchronize across the two markets. Even slight mismatches can lea to economically significant but spurious pricing anomalies. Thir, the inex typically pays a ivien an the future rate of ivien payment is ifficult, if not impossible, to etermine. Stanar an Poor s oes provie aily ivien payments on the S&P 500, but by nature these ata are backwar-looking, an there is no reason to assume that the actual iviens recore ex post correctly reflect the expecte future iviens at the time the option is price (see, for example, Harvey an Whaley (1991)). We propose to aress these three problems by the following proceure. Because all option prices are recore at the same time on each ay, we require only one temporally matche inex price per ay. To circumvent the unobservability of the ivien rate t,t, we infer the futures price F t,t for each maturity t. By the spot-futures parity, F t,t an S t are linke through 13 F t,t S t e ~r t,t t,t!t. (16) To erive the implie futures, we use the put call parity relation, which must hol if arbitrage opportunities are to be avoie, inepenently of any parametric option-pricing moel 14 : H~S t, X,t, r t,t, t,t! Xe r t,tt G~S t, X,t, r t,t, t,t! F t,t e r t,tt, (17) where G enotes the put price. To infer the futures price F t,t from this expression, we require reliable call an put prices (prices of actively trae options) at the same strike price X an time-to-expiration t. To obtain such reliable pairs, we must use calls an puts that are closest to at-the-money (recall that in-the-money options are illiqui relative to out-of-the-money ones, hence any matche pair that is not at-the-money woul have one potentially unreliable price). The average aily volume for at-the-money calls an puts is 4,360 contracts an we are therefore very confient in both prices. On every ay t, we o this for all available maturities t to obtain for each maturity the implie futures price from put call parity. 13 This relationship requires that interest rates o not covary with the futures price. During 1993, short term rates i not exhibit much variation at all (see Table I). 14 Because any violation of put call parity woul give rise to a pure arbitrage opportunity, it can be expecte to hol with some egree of confience. CBOE floor traers of S&P 500 options have confirme that put call parity is almost never violate in practice. See also Black an Scholes (1972), Harvey an Whaley (1992), Kamara an Miller (1995), an Rubinstein (1985).

23 D E Nonparametric Estimation of State-Price Density 521 Table II Banwith Values for the SPD Estimator Banwith selection for the SPD estimator with kernels k F, k X, an k t accoring to the relation h j c j s j n 10@2p # for regressor j, where n is the sample size, p is the number of continuous erivatives of the function to be estimate, is the number of regressors, s j is the unconitional stanar eviation of the regressor, an c j c j0 0ln~n!, epens on the particular regressor an the kernel function, with c j0 a constant. m j is the orer of the partial erivative with respect to the regressor that we wish to estimate, an s j is the orer of the corresponing kernel (see Appenix A for further etails). The nonparametric regression correspons to the semiparametric moel with 3 an [@XF t,t t# '. The sample size n equals 14,431. The coefficient of etermination R measures the gooness-of-fit of the nonparametric kernel regression s~f [ t,t,x,t!. Kernel s j p m j c j s j h j k F k ~4! k X k ~2! k t k ~4! Given the erive futures price F t,t, we then replace the prices of all illiqui options, that is, in-the-money options, with the price implie by put call parity at the relevant strike prices. Specifically, we replace the price of each in-the-money call option with G~S t, X,t, r t,t, t,t! F t,t e r t,tt Xe r t,tt where, by construction, the put with price G~S t, X,t, r t,t, t,t! is out-of-themoney an therefore liqui. After this proceure, all the information containe in liqui put prices has been extracte an resies in corresponing call prices via put call parity; therefore, put prices may now be iscare without any loss of reliable information. B. A Nonparametric S&P 500 Inex Option Pricing Formula We use price ata on every option trae uring 1993, for a total of n 14,431 options after applying the filters escribe in the previous section. We estimate the SPD using each of the imension reuction techniques iscusse in Section I.C. In the interest of saving space, we report below only the results for the estimator given by equation (13). Note from Table I that the recore interest rate ata exhibit little significant variation uring 1993 an thus we coul reasonably have exclue r t,t from the regressor list that is, treating it as constant given our sample. This woul have further reuce the imensionality of the regression. In the semiparametric approach, this turns out to be unnecessary because r t,t oes not enter the semiparametric regression. Naturally a ifferent time perio where interest rates are more volatile woul require that r t,t be kept in for the full nonparametric moel. The kernel an banwith values are given in Table II. The estimator s~f [ t,t,x,t! necessary in equation (13) generates a strong volatility smile with respect to moneyness (see Figure 3). We follow this

24 522 The Journal of Finance Implie Volatility per Year Moneyness = Strike/Futures Days-to-Expiration Figure 3. Implie Volatility Surface of the Nonparametric Estimator. Plot of the nonparametric estimator of the implie volatility as a function of time-to-maturity an moneyness. This correspons to the semiparametric moel where we partition the vector of explanatory variables E ' F t,t r t,t # ' with E [@X0F t,t t# '. The nonparametric estimator generates a strong volatility smile, efine as the relationship between the option s moneyness X0F t,t an its implie volatility s~!. E market s convention of quoting (an heging) the options in terms of the futures rather than the cash inex an therefore efine moneyness as the ratio of strike X to futures prices F t,t. The implie volatility at a fixe maturity is a ecreasing nonlinear function of moneyness. Note in particular that our estimate smile is strongly asymmetric, pointing to the anecotal evience that out-of-money put prices have been consistently bi up since the crash of 1987 by investment managers looking for protection against future ownwar inex movements. In contrast, stochastic volatility moels the class of parametric moels most commonly use to generate smile effects typically prouce symmetric smiles (see Bates (1995) an Renault (1995)). A further result of our multiimensional approach is the changing shape of the smile as time-to-maturity increases, that is, the variation of s~f [ t,t,x,t! with t. This is best seen in Figure 4. The one-month smile is the steepest. We fin that the implie volatility curves are generally flatter for longer times-to-maturity, but we ocument a persistence in the smile over longer maturities that is not capture by existing stochastic volatility moels. In

25 E Nonparametric Estimation of State-Price Density Implie Volatility per Year ays ays ays ays Moneyness = Strike/Futures Figure 4. Implie Volatility Curves for the Nonparametric Estimator. Implie volatility curves for the nonparametric option estimator for various times-to-maturity as a function of the option s moneyness, corresponing to the function X0F t,t s~! E for the four expirations t, where [@X0F t,t t# '. these moels, mean reversion in stochastic volatility typically inuces a rapi isappearance of the smile as the time horizon increases unlike longmemory effects. Our results therefore suggest that moeling long-term memory in stochastic volatility, for instance along the lines of Harvey (1995), coul be a promising empirical approach. Note also that the curves for all maturities intersect at approximately the same level of moneyness (0.975). In other wors, options with moneyness of are price at about the same volatility for all maturities. At-the-money options (moneyness equals one) have an implie volatility that increases slightly with maturity. The implie volatility of out-of-the-money puts (calls) ecreases (increases) with maturity. This suggests that it may be misleaing to focus, as is often the case in practice, on the term structure of at-themoney volatilities as a way of fitting the Black Scholes moel to the ata for ifferent maturities. Our nonparametric approach ocuments that the small ifferences in at-the-money implie volatilities across maturities, where all the curves are close together, unerstate the overall variation of implie volatilities over the full range of trae strikes. We report in Table III the nonparametric prices an eltas (with respect to the futures) for a sample of calls an puts for maturities of four months, price for a current futures price of 455. To compute eltas with respect to the stock, note that?h0?s ~?H0?F!~?F0?S! ~?H0?F!e ~r!t!. We give the price of every option with a elta greater or equal to 0.05 in absolute value.

26 D 524 The Journal of Finance Table III Nonparametric Estimates of S&P 500 Inex Option Prices Estimate nonparametric call option, put option, normalize butterfly-sprea, an igital option prices on the S&P 500 inex for a four-month maturity (84 ays) an strike prices with eltas greater than or equal to 0.05 in absolute value, price for a current inex value of The nonparametric kernel estimator is base on a sample of 14,431 CBOE aily call an put option prices on the S&P 500 inex from January 4, 1993 to December 31, The nonparametric regression correspons to the semiparametric moel with 3 an E [@XF t,t t# '. The Black Scholes prices are compute at the actual at-the-money implie volatility. Option eltas are compute with respect to the futures price, not the spot price. Butterfly prices are stanarize by the square of the strike price increment (five points), to provie a iscrete approximation to the SPD value. SPD Value enotes the exact nonparametric estimate of the SPD. Both values are multiplie by 100. Digital Price an BS Digital Price enote the nonparametric an Black Scholes prices of the igital option, respectively. Their ifferences illustrate the ifferences between the nonparametric an Black Scholes estimates of the cumulative istribution functions corresponing to the two SPDs. Futures: $458.04, Interest Rate: 3.05%, Time-to-Maturity: 84 Days Strike Price Call Price Call Delta Put Price Put Delta Implie Vol. Butterfly Price SPD Value Digital Price BS Digital Price Not surprisingly, compare to Black Scholes prices, our prices are consistent with the features of the actual market prices. We also report the prices of the stanarize butterflies over five-point strike spreas, which, in light of the result of Breeen an Litzenberger (1978), yiel a iscrete approximation to the value of the SPD at that strike level. C. S&P 500 Inex Option SPDs In Figure 5 the nonparametric SPDs are overlai with the corresponing Black Scholes SPDs at the same maturities (the Black Scholes log-normal SPDs are evaluate at the at-the-money implie volatility for that maturi-

27 Nonparametric Estimation of State-Price Density 525 Density: Probability Percent Density: Probability Percent Maturity = ays Nonparametric SPD 95% Upper 95% Lower Black Scholes Cash Price at Expiration Maturity = ays Nonparametric SPD 95% Upper 95% Lower Black Scholes Cash Price at Expiration Density: Probability Percent Density: Probability Percent Maturity = ays Nonparametric SPD 95% Upper 95% Lower Black Scholes Cash Price at Expiration Maturity = ays Nonparametric SPD 95% Upper 95% Lower Black Scholes Cash Price at Expiration Figure 5. Nonparametric SPD Estimates. Nonparametric SPD estimates (soli lines) are overlai with the corresponing Black Scholes SPDs (ashe lines) at the same maturities. Dotte lines aroun each SPD estimate are 95 percent confience intervals constructe from the asymptotic istribution theory erive in Appenix B. These estimates of the SPD are erive from the semiparametric moel where E [@XF t,t t# '. The secon erivative of the callpricing function is evaluate at X S T, an we plot the functions S T f * t ~! an * S T f BS,t ~! for values of the other elements in fixe at their sample means. The volatility for the Black Scholes SPD is the average at-the-money implie volatility from the market prices of options with the corresponing maturity. ty). Figure 5 also reports the 95 percent confience intervals aroun each estimate SPD. The confience interval is constructe from the asymptotic istribution theory erive in Appenix B. Figure 6 shows the estimate nonparametric ensity of the continuously compoune t-perio return, u t [ ln~s T 0S t!, that is compatible with our nonparametric SPD estimate. We compute the ensity of u t by noting that: Pr ln S S T S t u t e u Pr~S T S t e u! f t 0 * ~S T! S T. (18)

28 526 The Journal of Finance Density Probability Percent Nonparametric Black Scholes Maturity = ays Log-Return Density Probability Percent Nonparametric Black Scholes Maturity = ays Log-Return Density Probability Percent Maturity = ays Nonparametric Black Scholes Log-Return Density Probability Percent Nonparametric Black Scholes Maturity = ays Log-Return Figure 6. Nonparametric Estimate of SPD-Generate Densities for Continuously Compoune Returns. Estimate nonparametric ensity of the continuously compoune t-perio returns, u t [ ln~s T 0S t!, that is compatible with our nonparametric SPD estimator, for the same four maturities as in Figures 4 an 5. The corresponing Black Scholes ensities, evaluate at the average at-the-money implie volatility for each maturity, are overlai. The ensity of continuously compoune t-perio returns equivalent to the SPD for prices is then??u Pr ln S T S t u S t e u f t * ~S t e u!. (19) We compare this ensity to the Gaussian Black Scholes ensity N~~r t,t t,t s 2 02!t,s 2 t! for each maturity. Not surprisingly, the ifferences in Figure 6 between the nonparametric an Black Scholes continuously compoune return ensities are quite similar to those of the estimate SPDs for prices in Figure 5. However, computing the ensities for returns allows us to illustrate the magnitue of the ifferences by plotting in Figure 7 the term structures of implie mean,

29 Nonparametric Estimation of State-Price Density 527 Implie Mean Nonparametric Black Scholes Days-to-Expiration A: Term Structure of Implie Mean of Returns Implie Volatility Nonparametric Black Scholes Days-to-Expiration B: Term Structure of Implie Volatility of Returns Implie Skewness Nonparametric Black Scholes Days-to-Expiration C: Term Structure of Implie Skewness of Returns Implie Kurtosis Nonparametric Black Scholes Days-to-Expiration D: Term Structure of Implie Kurtosis of Returns Figure 7. Term Structure of Implie Moments of SPD-Generate Return Densities. The term structures of implie mean, stanar eviation, skewness, an kurtosis of the SPDgenerate continuously compoune return ensities along with their Black Scholes counterparts. All the moments of the returns are annualize. This figure ocuments the increase in the levels of skewness in absolute value an kurtosis that are implie by the nonparametric SPDgenerate return ensities as a function of the options maturities. stanar eviation, skewness, an kurtosis of the SPD-generate return ensities along with their Black Scholes counterparts. We efine kurtosis as the value in excess of the stanar Gaussian ensities. All the moments of the returns in Figure 7 are annualize. Denote by U t the annual continuously compoune return obtaine by summing ½ inepenent t-perio return an observe that t # t # Var@U t # ~10t! Var@u t # Skew@U t #!t Skew@u t # Kurt@U t # t Kurt@u t #.

30 D E 528 The Journal of Finance Table IV Moments of Nonparametric Return Densities Annualize moments of nonparametric an Black Scholes ensities of the continuously compoune t-perio return, u t [ ln~s T 0S t!, that are compatible with the SPD estimates for prices. The estimates are base on a sample of 14,431 CBOE aily call an put option prices on the S&P 500 inex from January 4, 1993 to December 31, The nonparametric regression for the semiparametric moel has 3 an [@XF t,t t# '. This table quantifies the ifferences between the nonparametric an Black Scholes return ensities in terms of their first four moments, for maturities of one, two, four, an six months. Note that the mean of the price SPD is given by the futures price F t,t, an is the same for the nonparametric an Black Scholes price SPDs. However, as a result of Itô s Lemma, the means of the return ensities are not equal because the estimate volatility of the nonparametric iffusion for the S&P 500 inex is not constant. SPD Estimator Mean Stanar Deviation Skewness Kurtosis Time-to-maturity: 21 ays Nonparametric Black Scholes Time-to-maturity: 42 ays Nonparametric Black Scholes Time-to-maturity: 84 ays Nonparametric Black Scholes Time-to-maturity: 126 ays Nonparametric Black Scholes Figure 7 highlights the ifferences between the nonparametric an Black Scholes return ensities. Although the nonparametric return ensities in Figure 6 have comparable means an stanar eviations to those obtaine from the Black Scholes formula (we estimate the Black Scholes SPD at the actual at-the-money implie volatility), they exhibit consierably ifferent skewness an kurtosis. Specifically, for all four maturities the nonparametric SPDs are negatively skewe, have fatter tails an the amount of skewness an kurtosis both increase with maturity. Table IV quantifies the ifferences in the nonparametric an Black Scholes SPDs for the four maturities that are the focus of our empirical application. The skewness an kurtosis of the 21-ay nonparametric ensity are an , respectively, becoming progressively more severe as the maturity ate increases, reaching an , respectively, at the 126-ay horizon. D. An Application to Digital Options These results point to important ifferences between the nonparametric SPD an the Black Scholes SPD, which lea to important ifferences in the pricing implications of the two. To illustrate these ifferences, we compare the prices of igital call options uner the nonparametric an Black Scholes SPDs in the last two columns of Table III.

31 Nonparametric Estimation of State-Price Density 529 Recall from Section I.A that a igital call option is a security that gives the holer the right to receive a fixe cash payment of one ollar if S T excees a prespecifie level X. Its price is irectly relate to the CDF F t * of the SPD in the simple manner given in equation (5). This represents a typical application of our estimator: we infer the SPD from the prices of highly liqui calls an puts, an then price an over-the-counter contract consistently using the liqui prices. The nonparametric SPD yiels a higher price for the igital calls at strikes of 415 an 420: 0.06 versus 0.04 an 0.08 versus 0.06, respectively. The prices are then similar at a strike of 425: But as the strike price increases, the Black Scholes igital price increases more rapily than the nonparametric; the at-the-money nonparametric igital is price at 0.39, whereas the corresponing Black Scholes price is 0.46, an 18 percent ifference. However, these ifferences ecline steaily until the 480 strike where the two prices are 0.79 an 0.80 respectively. The behavior of the igital prices is symptomatic of the ifferences between the nonparametric an Black Scholes log-returns SPDs: the Black Scholes SPD is Gaussian an obviously cannot accommoate fat tails or skewness, whereas the nonparametric SPD can an oes. In other wors, the ifferences in Table III between the nonparametric an Black Scholes prices are the translation at the level of prices (i.e., integrals of the SPDs) of the ifferences in SPDs, or their skewness an kurtosis, that were ientifie in Table IV. E. Specification Tests: Parametric versus Nonparametric SPDs Figures 5 an 6 suggest that the SPDs we estimate are substantially ifferent from the benchmark Black Scholes case. The 95 percent confience interval reporte in Figure 5 is pointwise, an therefore oes not provie a global answer to the question: Coul the nonparametric family of SPDs have been generate by the Black Scholes moel? To answer this question, we erive in Appenix C a test base on the integrate square istance between the two ensities, nonparametric an Black Scholes, an erive the asymptotic istribution of the test statistic. Empirical results are in Table V. We strongly reject the null hypothesis that the ensities are equal. We show in Appenix C that because the alternative moel is nonparametric, it is not equivalent to test the Black Scholes moel at the level of prices, eltas, or SPDs. The curse of ifferentiation that we iscuss in Appenix A plays a role here as well, just as in Figure 2. We test each one of these three null hypotheses, an fin that in each case the Black Scholes moel is rejecte with p-values equal to F. Is the SPD Stable over Time? Our approach assumes that the SPD is a constant function of a vector of state variables over the time perio that is use to estimate it. A natural question that arises is whether the ata actually valiate this hypothesis.

32 E E Table V Nonparametric Specification Test of the Black Scholes Moel Nonparametric specification tests of the Black Scholes option-pricing moel base on the option-pricing formula H BS an its corresponing elta BS an SPD f * BS, base on the sample of 14,431 CBOE aily call an put option prices on the S&P 500 inex from January 4, 1993 to December 31, The nonparametric regression for the semiparametric moel has D 2 an [@X0F t,t t# '. The banwiths are chosen to be h 1 h 01 s 1 n 101 with 1 2 2m an h 2 h 02 s 2 n 102, where s j, j 1,2 is the unconitional stanar eviation of the jth regressor. The banwiths to estimate the conitional variance of the nonparametric regression are h 1,cv an h 2,cv The Black Scholes volatility value, estimate as the mean of the nonparametric regression of implie volatility on, iss BS percent. The weighting functions v H, v, an v f * are trimming inices, that is, only observations with estimate ensity above a certain level, an away from the bounaries of the integration space, are retaine. The two numbers in the column Trim refer respectively to the trimming level (as a percentage of the mean estimate ensity value), an the percentage trimme at the bounary of the integration space when calculating the test statistics. For instance, if the latter is 5 percent, the trimming inex retains the values between 1.05 times the minimum evaluation value an 0.95 times the maximum value. Integral refers to the percentage of the estimate ensity mass on the integration space that is kept by the trimming inex. Test Statistic refers to the stanarize istance measure between the nonparametric an Black Scholes estimates (remove the bias term, ivie by the stanar eviation). The istance measure for each of the three null hypotheses consiere is D~ H, H BS!, D~, BS!, an D~ f *, f * BS! respectively. The integrals D are calculate over the integration space given by the rectangle [0.85, 1.10] [10, 136] in the moneyness ays-to-expiration space. The kernel weights are constructe using the binning metho with 30 bins in the moneyness imension an 20 in the ays-to-expiration imension. The test is escribe in Appenix C. 530 The Journal of Finance Null m s k 1 2 h 01 h 02 h 1 h 2 Trim Integral Test Statistic p-value H H BS k ~2! BS k ~2! f * * f BS k ~2!

33 Nonparametric Estimation of State-Price Density 531 We erive in Appenix D a iagnostic test base on the integrate square ifference between the two SPDs estimate over two ifferent time perios f * l ~!, l 1,2. The intuition behin our test is simple: Uner the null, the expecte value of ~ f * 1 ~! f * 2 ~!! 2 shoul be small, whereas it will be large if the SPD is not the same over the two subperios. We compute the asymptotic istribution of the test statistic an apply the test to pairwise comparisons of the SPD estimate separately over each of the four quarters of the year We report the results in Table VI. The tests o not reject the null hypothesis for any pair of quarters: the p-value for the pairwise quarterly comparisons are 0.41, 0.42, an 0.27 (recall in interpreting the values of the stanarize S~ f * 1, f 2 *! that the test is one-sie; we only reject when the test statistic is large an positive, that is, when f * 1 is sufficiently far away from f * 2!. These results therefore provie strong evience in favor of the stability of the SPD function f * ~! over subperios of our sample. G. In-Sample an Out-of-Sample Forecasts As we iscusse in Section I.D, one of the potential avantages of our approach to estimating SPDs lies in its ability to use a set of option prices with ajacent option characteristics (i.e., close values of! to estimate the SPD, resulting in smooth an potentially more stable estimates. Relative to methos that rely exclusively on the current cross section of option prices to infer the SPD, we woul therefore expect our estimator to result in lower out-of-sample forecasting errors. This woul typically be achieve at the expense of eteriorating the in-sample fit. We will verify this intuition by comparing the in-sample an out-of-sample fits of four moels: (1) Jackwerth an Rubinstein s (1996) (JR) metho of extracting ensities 15 by minimizing a criterion function that penalizes for unsmoothness; (2) an extension of Hutchinson, Lo, an Poggio (1994) (HLP) approach in which we estimate a single-hien-layer feeforwar neural network with a logistic activation function an ifferentiate it twice numerically; (3) the Black an Scholes (1973) (BS) moel; an (4) our SPD estimator (AL). We present in Table VII two types of evience that confirm empirically the arguments mae in Section I.D. In Panel A, we examine the ability of the SPDs estimate on ay t for the trae maturity closest to six months to preict the SPD for the same six-month maturity on ay t t, for t 0 (in-sample), an t 1, 5, 10, 15, an 20 traing ays (out-of-sample) corresponing to forecast ates that are getting progressively farther away. We use the options ata from the first three quarters of the year to construct the AL an HLP estimators on ay t, the corresponing information on ay t to construct the JR ensity, an the implie volatility from the at-the-money option on ay t to estimate the BS ensity. When calculating the ensities, 15 We thank Jens Jackwerth an Mark Rubinstein for graciously proviing us with the empirical ensities from their estimation metho.

34 D E Table VI Stability Tests for the SPD Estimator Tests of stability of the SPD estimator across subperios corresponing to each quarter of Each row reports a test of the null hypothesis that the SPDs in quarters Q i an Q j are ientical, where Q i an Q j are nonajacent quarters of 1993 to reuce the effects of epenence. Each quarter contains n 3,607 observations an the banwiths are chosen to be h 1 h 01 s 1 n 101 with 1 2 2m an h 2 h 02 s 2 n 102, where s j, j 1,2 is the unconitional stanar eviation of the jth regressor. The banwiths to estimate the conitional variance of the nonparametric regression are h 1,cv an h 2,cv The weighting function v f * is a trimming inex. The two numbers in the column Trim refer respectively to the trimming level (as a percentage of the mean estimate ensity value), an the percentage trimme at the bounary of the integration space when calculating the test statistics. For instance, if the latter is 5 percent, the trimming inex retains the values between 1.05 times the minimum evaluation value an 0.95 times the maximum value. Integral refers to the percentage of the estimate ensity mass on the integration space that is kept by the trimming inex. The estimates f * l, l 1,2, of the SPD are calculate for the semiparametric moel where the nonparametric regression has 2 an [@X0F t,t t# '. The integrals S are calculate over the integration space given by the rectangle [0.85, 1.10] [10, 94] in the moneyness ays-to-expiration space. This rectangle has the highest concentration of options observe in each of the four quarters. The kernel weights are constructe using the binning metho with 30 bins in the moneyness imension an 20 in the ays-toexpiration imension. The test is escribe in Appenix D. Null m s k 1 2 h 01 h 02 h 1 h 2 Trim Integral Test Statistic p-value Q 1 Q k ~2! Q 1 Q k ~2! Q 2 Q k ~2! The Journal of Finance

35 Nonparametric Estimation of State-Price Density 533 Table VII In-Sample an Out-of-Sample Forecasts Comparison of average forecast errors of the ensities prouce by four moels: (JR) Jackwerth an Rubinstein s (1996) metho of extracting ensities by minimizing a criterion function that penalizes for unsmoothness; (HLP) an extension of Hutchinson, Lo, an Poggio (1994) where we use their learning network (with five variables an four hien units) an ifferentiate it twice numerically; (BS) the Black Scholes ensity with the volatility taken as the at-the-money implie volatility of the corresponing options; an finally our SPD estimator (AL) where the nonparametric regression correspons to the semiparametric moel with D 3 an E [@XF t,t t#. In Panel A, the ensity forecast error is expresse as a percentage of the moe value of the comparison ensity. In Panel B, the price forecast error is reporte as a percentage of the options market price over the comparison interval. When forecasting prices, we consier the 400, 425, 450, an 475 strike options. (MKT) refers to the actual CBOE market price. Optionpricing moels an their corresponing SPDs are estimate with aily ata from January 4 to September 30, 1993 an out-of-sample forecasts are generate for various forecast horizons t on a aily rolling basis from October 1 to December 31, Panel A: Density Forecast Error Forecast of JR-Density t Traing Days Ahea Forecast of AL-Density t Traing Days Ahea t JR HLP AL BS JR HLP AL BS Panel B: Market-Price Forecast Error t JR HLP AL BS MKT we annualize the returns to ajust for the ifferences in the actual numbers of ays-to-maturity of the six-month options on ay t an t t. We then examine the out-of-sample forecasting properties of these estimators by repeating the computations for each ay t in the fourth quarter of the year, an rolling the estimation perio to retain the most recent nine months. We calculate the square root of the mean-square ifference between the ensity preicte by each metho for ay t t, base on the market information available on ay t, an the ensity that is realize on ay t t. The realize ensity on ay t t is represente by either the JR or the AL estimators (see the two parts of Panel A). For scaling purposes, we report

36 534 The Journal of Finance these ifferences as a percentage of the comparison ensity value at its moe. In Panel B, we run the same comparison, but focus this time on ifferences between the option prices preicte on ay t by each moel, obtaine by integrating the call payoff function against the moel s ensity, an the realize market prices on ay t t (again ajusting for ifferences in aysto-maturity). We also examine how the simplest possible proceure using the implie volatility of market price at ate t to preict the market price at ate t t, that is, treating the implie volatility smile as a martingale woul have performe uring the same perio (this is the last column of Panel B, labele MKT). We forecast the prices of the 400-, 425-, 450-, an 475-strike options, an report the forecast error as a percentage of the market price over the comparison interval. Both panels lea to similar conclusions. Comparing ensities, the JR ensities prouce a better fit in-sample an over short horizons (for t up to 15 ays), but only when the forecaste ensity is estimate by the JR metho (the left part of Panel A). However the AL estimates are uniformly better over every horizon at preicting the future ensity, when it is measure by the AL metho (the right part of Panel A), but also if it is measure by the JR ensity for t 20 ays (the left part of Panel A). The HLP estimator, esigne to fit prices rather than their erivatives, prouces comparatively larger errors, an how to optimize it for that purpose is unclear in the absence of further asymptotic guiance. Comparing market prices, the JR ensity estimates lea to a better fit in-sample or for less than a week ~t 0, 1, 5 ays) than the AL ensities; simply using the market prices woul have one better however (see the last column in Panel B). Over longer forecasting horizons, using the AL ensities leas to better forecasts of the market prices on ay t t than using either the JR ensities ~t 10, 15, 20 ays) or the market prices on ay t (for t 15 an 20 ays). In each scenario, the BS ensities lea to large forecasting errors relative to the JR an AL methos. This suggests that the features of the implie volatility smile that these methos ientify are sufficiently persistent to be of some value as we forecast the future SPDs an market prices. The spee of mean reversion of the implie volatility smile to its average pattern is sufficiently slow that over short horizons the most recent ata remain the best preictor of the future ata (a martingale approach). However, there is enough mean reversion for a metho base on estimating a stable function over time (AL) to forecast better over a longer horizon than one that bases the forecast exclusively on the most recent ay s ata (JR). IV. Conclusion In this paper, we propose a nonparametric technique for estimating stateprice ensities base on the relation between state-price ensities an option prices. We also erive a number of statistical properties of the estimator,

37 Nonparametric Estimation of State-Price Density 535 incluing pointwise asymptotic istributions, specification test statistics, an a test for stability across subsamples. Although this approach is ata intensive, generally requiring several thousan atapoints for a reasonable level of accuracy, it offers a promising alternative to stanar parametric pricing moels when parametric restrictions are violate. This trae-off between parametric restrictions an ata requirements lies at the heart of the nonparametric approach. Although parametric formulas are surely preferable to nonparametric ones when the unerlying asset s price ynamics are well unerstoo, this is rarely the case in practice. Because they o not rely on restrictive parametric assumptions such as lognormality or sample-path continuity, nonparametric alternatives are robust to the specification errors that plague parametric moels. A nonparametric approach is particularly valuable in option-pricing applications because the typical parametric restrictions have been shown to fail, sometimes ramatically. For example, Figure 7 shows that the primary failure of the Black Scholes moel in pricing S&P 500 inex options is its inability to account for the skewness an kurtosis apparent in the nonparametric estimates of the returns ensity, an even more so for longer-maturity options. Parametric extensions of the Black Scholes moel, whether they capture stochastic volatility, jumps, or are multi-parameter extensions of the basic formula, shoul focus on capturing these empirical facts. The amount of persistence in the smile is such that parametric moels incorporating long-term memory in stochastic volatility may be the most promising. Also, by their very nature nonparametric methos are aaptive, responing to structural shifts in the ata-generating processes in ways that parametric moels o not. An finally, they are flexible enough to encompass a wie range of erivative securities an funamental asset-price ynamics, yet relatively simple an computationally efficient to implement. Appenix In this Appenix we escribe in more etail our proceure for selecting the kernel functions an the banwiths in our option-pricing context. We also iscuss the rates of convergence of our kernel estimator of the call-pricing function to the true function, the partial erivative with respect to the asset price to the true option elta, an the secon erivative with respect to the strike price to the true SPD. Finally, we propose a specification test for our kernel estimator, an a test of its stability across ifferent time perios. A. Kernel an Banwith Selection We have to choose both the univariate kernel function k an the banwith parameters in h for each regressor. We nee to ifferentiate twice the right-han-sie of equation (8) with respect to the strike price X to obtain an estimate of? 2 H0?X 2, or once with respect to S t to estimate the option

38 536 The Journal of Finance elta. We therefore require that k X ~k S! be at least twice (once) continuously ifferentiable. Four elements etermine the choice of the kernel an banwith: the sample size n, that is, the number of options use to construct the estimator H~{!; the total number of regressors inclue in the nonparametric regression; the number p j of existing continuous partial erivatives of the true option-pricing function H~{! with respect to the jth regressor j ; an finally the orer m j of the partial erivative with respect to the jth regressor that we wish to estimate (with the convention that m j 0 when no partial ifferentiation is require with respect to j!. We naturally assume that the call-pricing function H~{! to be estimate is sufficiently smooth, that is, p j m j for all j 1,...,. In our problem, the highest erivative that we shall estimate is the secon partial erivative of the call price with respect to the strike price. We assume that the function H~{! amits four continuous erivatives with respect to each of its regressors. 16 Define the orer s of a kernel function k as the even integer that satisfies the following relation: * z l k~z! z 0 for l 1,...,s 1, an * 6z6 s k~z! z,. For example, the popular Gaussian kernel k ~2! ~z! 1 %2p e z 2 02 (A1) is of orer s 2; the following kernel is of orer s 4: k ~4! ~z! 3 %8p 1 z 2 3 e z (A2) Using higher-orer kernels has the effect of accelerating the spee of convergence of the estimator to the true function as the sample size increases, in a mean-square sense. We set the orer of the kernel for each regressor j to be s j p j m j. When estimating the m j th partial erivative of H~{! with p respect to the jth regressor, we obtain a bias term of orer h 2 m j j an a variance term of orer n 1 ~2m h j! j. We are minimizing the mean-square error of the estimator, which consists of the sum of the square bias an variance terms. Because the banwith h j goes to zero as the sample size n grows, the larger the orer of the kernel, the lower the bias term (the curve 16 It is possible to make primitive assumptions on the ata-generating process that imply the necessary smoothness of the true call-pricing function. In particular, if we assume that the unerlying asset price followe a stochastic ifferential equation with iffusion function s 2 which amitte at least p continuous erivatives, p 2, then the call-pricing function woul also amit at least p continuous erivatives. This follows by writing the call-pricing function as the solution of the generalize Black Scholes partial ifferential equation erive from using the stanar ynamic replicating strategy (see Merton (1973)). It is a parabolic partial ifferential equation satisfying all the hypotheses in Frieman (1964, Chapter I) whose solution is then known to have at least p erivatives.

39 Nonparametric Estimation of State-Price Density 537 fit improves) but the larger the variance term (the curve estimator becomes noisier). The choice of the banwith given in equation (A3) optimally balances these two effects. However, higher-orer kernels can be cumbersome to use in practice they are no longer uniformly positive, for example. Experience suggests that it is esirable to limit the choice of kernels to orers no larger than s 4. Therefore, to estimate the call-pricing function where m j 0 for each regressor we set in the full nonparametric moel k X k S k t k r k k ~4!, an in the semiparametric moel, equation (9), k X k F k t k ~4!. To estimate the option elta where m j 0 for each regressor except S t (for which m j 1!, we set: in the full nonparametric moel, k X k t k r k k ~4! an k S k ~2!, an in the semiparametric moel, k X k t k ~4! an k F k ~2!. Finally, to estimate the SPD where m j 0 for each regressor except X (for which m j 2!, in the full nonparametric moel we set k S k t k r k k ~4! an k X k ~2!, an in the semiparametric moel we set k F k t k ~4! an k X k ~2!. Monte Carlo evience in Section II shows that for the choices of kernel functions above, H~{! is very accurate for the large sample size n consiere in our empirical application. For each of the regressors in, we set the corresponing banwith parameter h j accoring to the relation h j c j s~ j!n 10~ 2p!, (A3) where p p 1... p an s~ j! is the unconitional stanar eviation of the regressor j, j 1,...,. 17 This banwith choice is such that our estimator H~{! achieves the optimal rate of convergence in the mean-square sense among all possible nonparametric estimators of H~{!: n ~ p m!0~ 2p!, (A4) where m [ ( j 1 m j. The parameter c j epens on the choice of the kernel an the function to be estimate; it is typically of the orer of one an small eviations from the exact value have no large effects. It can be selecte by cross valiation (see Härle (1990), for example), a technique that ensures that we minimize the mean-square error of our estimator H~{!. Any choice of banwith going to zero at a strictly faster rate than that given by equation (A3) will center the asymptotic istribution at zero (see Appenix B). In practice, we select c j c j0 0ln~n!, with c j0 constant (see Table II). This results in a rate of convergence for the estimator that is slightly slower than the rate given in equation (A4), but in exchange we eliminate asymptotically the bias term. 17 To estimate the option s elta, we set p S 3 an p x p t p n p 4 so equation (A3) shoul be ajuste in the obvious manner to reflect the assumption that the p j s are not common across regressors.

40 538 The Journal of Finance Note from the rate in equation (A4) that the lower the imensionality of the regression function, (i.e., the smaller the number of regressors in!, the faster the convergence of the estimator H~! an its erivatives to the true function. This is the technical motivation for our propose approaches to imension reuction in Section I.C. Note also from the rate in equation (A4) that higher-orer erivatives converge at slower spees, hence the SPD estimator (for which m 2! converges slower than the elta estimator (for which m 1!, which in turn converges slower than the price estimator (for which m 0!. One aitional orer of ifferentiation slows own the rate of convergence by as much as two aitional regressors; that is, the curse of ifferentiation (the ecrease in rate of convergence as m increases) is twice as amning as the curse of imensionality (the ecrease in rate of convergence as increases). As a theoretical matter, we can still get arbitrarily close to the parametric rate of convergence n 102 if the call-pricing function has enough continuous erivatives, an we use a kernel of sufficiently high orer p m: Aspincreases, the rate of convergence n ~ p m!0~2p! converges to the parametric rate n 102. Suppose now that we have estimate the option-pricing function an extracte the SPD. One of the main practical reasons for oing so is the ability to price other erivative securities consistently. If the payoff function of the erivative is smooth, that is, if c~s T! is twice continuously ifferentiable in S T everywhere on the range of integration, an its erivatives are boune at the bounary of that range, then the erivative price can be rewritten as: e r t,tt 0 c~s T! f t * ~S T!S T c~s T! 0?2 H 2 t S?X T X ST 0? 2 c~s T! 2 H?S t ~S T!S T T where H t ~S T![ H~!. Integrating a smooth function against H~{! spees up the convergence rate of the estimator relative to the pointwise rate for H~!: H~! converges at spee n p0~ 2p!, but its integral against a smooth function of S T converges at the faster rate n p0~~ 1! 2p!. We woul therefore gain a factor n p0$~ 2p!~ 1 2p!% in terms of rate of convergence when integrating the secon erivative of the payoff function against H, that is, when computing the price of another erivative security with smooth payoff function to be consistent with the pricing function H~{!. B. Asymptotic Distributions We collect option prices in the form of panel ata, an assume strict stationarity of the ata. Methos base on local averaging can be sensitive to epartures from stationarity in the form of a unit root. In such cases, the same region of the support of the istribution fails to be revisite by the

41 D Nonparametric Estimation of State-Price Density 539 sample path often enough. One way to impose stationarity is to erive the asymptotics with respect to the cross-sectional imension, holing the time series imension fixe. We obtain the following from the general results in Aït-Sahalia (1995), where precise regularity conitions are state: For the full nonparametric moel in which t tr t,t t,t # ' (an we orer the regressors in so that the first entry is the variable with respect to which H~{! is to be ifferentiate m times), we have n 102 h )j 2 h 102 H~! H~!# & N~0,s H 2! (A5) n 102 h )j 2 h 102 ~! ~!# & N~0,s 2! (A6) n 102 h )j 2 h 102 f t * ~! f * t ~!# & N~0,s f* 2!, (A7) where s H 2 [ k 2 ~4! ~v! v s 2 ~!0p~ i! s 2 [ k '2 ~2! ~v! v k 2 ~4! ~v! v 1 s 2 ~!0p~! s f* 2 [ '' k 2 ~2! ~v! v k 2 ~4! ~v! v 1 s 2 ~!0p~!, p~! is the joint ensity function of the vector (i.e., the marginal istribution of option characteristics) an s 2 ~! is the conitional variance of the nonparametric regression. The banwiths are chosen as iscusse in Appenix A an the kernel constants are given in Table AI. In the semiparametric moel, we partition the vector of explanatory variables [@ E ' F t,t r t,t # ', where E contains nonparametric regressors. In our specific case, we consier both E [@XF t,t t# ' ~ D 3! an E [@X0F t,t t# ' ~ D 2!, an we estimate the implie volatility function s~! E nonparametrically. This yiels the following asymptotic istributions: n 102 h n 102 h n 102 h D )j 2 D )j 2 D )j 2 h 102 s~ [! s~ E!# E h j 102? s[?x ~! E?s?X ~! E h j 102?2 s[?x 2 ~! E?2 s?x 2 ~ E! & N~0,s s 2! & N~0,s 2 s! & 2 N~0,s 2 s!, (A8)

42 540 The Journal of Finance Table AI Kernel Constants for Asymptotic Distribution of SPD Estimator, Specification, an Stability Tests Kernel constants that characterize the asymptotic variances of the nonparametric an semiparametric estimators for option prices, eltas, an SPDs, which are given in Appenix B, as well as the specification an stability tests in Appenices C an D, respectively. The kernel functions k ~2! an k ~4! are efine in Appenix A. k ' an k '' enote the first an secon erivative of the kernel function, respectively. ' Functional k k ~2! k k ~2! '' k k ~2! ' k k ~4! k k ~4! '' k k ~4! k 2 ~v! v 1 2!p 1 4!p 3 8!p 27 32!p !p !p k~ v!k~v J v! J v 2 J v 1 2%2p 3 32%2p 105%2 1024!p 7881% !p where s s 2 [ D k 2 ~4! ~v! v s 2 ~!0p~ E! E 2 s s 2 s 2 s [ [ k '2 ~2! ~v! v '' k 2 ~2! ~v! v k ~4! k ~4! 1 D 2 ~v! v 1 D 2 ~v! v s 2 ~!0p~ E! E s 2 ~!0p~ E!, E p~! E is the joint ensity function of the vector, E an s 2 ~! E is the conitional variance of the nonparametric regression s~!. E In Table AI we provie the values of the kernel integrals that appear in the expressions above. The asymptotic istributions of the semiparametric estimators for prices, eltas, an SPDs then follow from the elta metho; that is, H~! H BS ~ s~ [!,! E H BS ~s~!,! E?H BS?s ~s~!,!@ E s~ [! s~ E!# E O p~7 s~ [! s~ E!7 E 2 L 2!; so H BS ~ s~ [!,! H E BS ~s~!,! E behaves asymptotically as?h BS 0?s~ s~ [! s~ E!!; E an because erivatives of s~ [! E converge at a slower rate (see the rates of convergence in equation (A8)) the asymptotic istribution of?h BS 0?F~ s~ [!,! E?H BS 0?F~s~!,! E is that of?h BS 0?s ~?[ s0?f?s0?f!, an the asymptotic istribution of? 2 H BS 0?X 2 ~ s~ [!,! E? 2 H BS 0?X 2 ~s~!,! E is that of?h BS 0?s ~? 2 s0?k [ 2? 2 s0?k 2!.

43 Nonparametric Estimation of State-Price Density 541 C. Specification Tests We propose to test the null hypothesis H 0, H 0 :Pr~H~S,X,t,r,! H BS ~S,X,t, r,!! 1, against the alternative hypothesis H A, H A :Pr~H~S,X,t,r,! H BS ~S,X,t, r,!! 1. A natural test statistic is the sum of square eviations between the two option-pricing formulas: D~H, H BS![E@~H~! H BS ~!! 2 v H ~!#, (A9) where v H ~! is a weighting function. The intuition behin our propose test is simple: If the null Black Scholes moel is correctly specifie, then its call-pricing formula shoul be close to the formula estimate nonparametrically. An estimator for D~H, H BS! is either the sample analog D~ H, n H BS![ 1 n ( i 1 ~ H~ i! H BS ~ i!! 2 v H ~ i!, (A10) or any other evaluation of the integral on the right-han-sie of equation (A9). In practice, we evaluate numerically the integral on a rectangle of values of representing the support of the ensity p, an use the binning metho to evaluate the kernels (see, e.g., Wan an Jones (1995) for a escription of the binning metho). To test the hypotheses that ~! BS ~! an f * ~! f * BS ~!, we efine D~, BS! an D~ f *, f * BS! in a similar fashion. To form a test, we erive the asymptotic istributions of these test statistics. Uner the null hypothesis of the Black Scholes moel H 0, we can show that nh 1 h j )j 2 D~ H, H BS! j 1 ) h 102 j B H & N~0, 2 H! (A11) nh 1 h j )j 2 D~, BS! j 1 ) h 102 j B & N~0, 2! (A12) nh 1 h j )j 2 D~ f *, f BS *! j 1 ) h 102 j B f * & N~0, 2 f *!, (A13) where is the number of inclue nonparametric regressors (the imension of!. We orer the regressors in so that the first one is the variable with respect to which the call-pricing function is to be ifferentiate m times. For simplicity, we use a common banwith for the variables j 2,..., (if these

44 542 The Journal of Finance regressors have ifferent scales, they shoul first be stanarize). The banwiths h j, j 1,..., are given in each case m 0,1,2, corresponing respectively to prices, eltas, an SPDs, by: h 1 O~n 10 1!, h j O~n 10 2!, j 2,...,, (A14) with 1 2 2m an 2 satisfying: 1 m 2 2m _ 2 2m r 2 2 2m ~ 1!02 2 (A15) an 3_ 2 ~ 1!02 2 2m, (A16) 2 where we use a common kernel k of orer r on every variable. Equation (A16) ensures that erivatives of lower orer o not affect the istribution of the highest-orer erivative term. Note that these banwith selection inequalities result in banwith values that unersmooth relative to the values in Appenix AI. The bias an variance terms are given by: B H B B f * k 2 ~w! s w 2 ~! vj H ~! k '2 ~w! w k ''2 ~w! w 1 k 2 ~w! w s 2 ~! vj ~! 1 k 2 ~w! w s 2 ~! vj f *~! ; H k~w!k~w v! w s v 4 ~! vj 2 H ~! f * 2 2 k ' ~w!k ' ~w v! w v 2 k~w!k~w v! w s v 1 4 ~! vj 2 ~! 2 k '' ~w!k '' ~w v! w v 2 k~w!k~w v! w s v 1 4 ~! vj 2 f * ~!,

45 E E Nonparametric Estimation of State-Price Density 543 where vj v for the full nonparametric moel. As for the pointwise asymptotic istributions, we report in Table AI the values of the kernel integrals that appear in the expressions above. To estimate consistently the conitional variance of the regression, s 2 ~! y M~!! 2 6# y 2 6# y6# 2, where y is the epenent variable an M~![E@y6#, we calculate the ifference between the kernel estimator of the regression of the square epenent variable y 2 on an the square of the regression M~!. The two regressions appearing in s 2 ~! are estimate with banwith h cv O~n 10 cv 0ln~n!! where cv 2r (recall that we only nee a consistent estimator of s 2 ~!, so there is no nee to unersmooth as in equations (A15) an (A16)). The test statistics are forme by stanarizing the asymptotically normal istance measures D; we estimate consistently the asymptotic mean B an variance 2, then subtract the mean an ivie by the stanar eviation. The test statistic then has an asymptotic N~0,1! istribution. Because the test is one-sie (we only reject when D is too large, hence when the test statistic is large an positive), the 10 percent critical value is 1.28, an the 5 percent value is For the semiparametric case, we partition into the nonparametric regressors an the remaining variables, hence we seek to test H~! H BS ~s~!,!. E The test statistic for prices is given by D~H, H BS![E@~H BS ~s~ E!,! H BS ~s BS,!! 2 v H ~!#, which behaves asymptotically like E?H BS~s BS,! 2 ~s~! E s?s BS! 2. For eltas an SPDs, the corresponing test statistics test the specification of the first an secon partial erivatives, respectively, of s~! E with respect to X, which is containe in the first argument of. E They also behave asymptotically like their leaing terms, which are E?H BS~s BS,!?s 2?s~! E? E 1? E 1?F 2, E?H er t,tt BS~s BS,!?s 2?2 s~! E 2? E 1? E 2 1?X 2, where 1 is the first component of the vector. E Therefore, in the formulas above, we set vj H ~! [?H BS~s BS,! 2 v?s H ~! (A17)

46 E D E 544 The Journal of Finance for prices an vj ~! [?H BS~s BS,!? E 2 1 v?s?f ~!, vj f *~![ e?h r t,tt BS~s BS,!? E?s 2 1?X 2 v f *~! (A18) for eltas an SPDs. To construct the test, we fix the variables in that are exclue from at their sample means an let [@X0F t,t t# '. Thus M~! above is s~x0f t,t,t!,? E 1 0?F X0F 2, an? E 1 0?X 10F. Then we can apply the results above with 2 instea of the full value of, p 4, an m 0,1,2 for prices, eltas, an SPDs, respectively. Values of j, j 1,2, corresponing banwith parameters an test results can be foun in Table V, Section III. D. Testing the Stability of the SPD across Time Perios We now construct a test of the null hypothesis that the pricing function, elta, an SPD in one subsample are the same in another subsample; hence the null hypothesis is an the alternative hypothesis is H 0 :Pr~H 1 ~! H 2 ~!! 1 H A :Pr~H 1 ~! H 2 ~!! 1 uner the maintaine assumption that the marginal istribution of option characteristics, p~!, is ientical over the two subsamples. Let n be the sample size of the two subsamples (assume to be the same across subsamples for simplicity). In what follows, the subscript l 1or2 enotes the subsample that was use to estimate the function, so H 1 enotes the call-pricing function estimate on subsample 1, etc. Relying on the same intuition as in Appenix C, we form the (suitably normalize) sum of square eviations between the two option-pricing formulas estimate on the two subsamples: S~H 1, H 2![E@~H 1 ~! H 2 ~!! 2 v H ~!#. (A19) The intuition behin our propose test is straightforwar: If the null Black Scholes moel is correctly specifie, then its call-pricing formula shoul be close to the formula estimate nonparametrically. To test the hypotheses that 1 ~! 2 ~!an f 1 * ~! f 2 * ~!, we efine S~ 1, 2! an S~ f 1 *, f 2 *! in a similar fashion. Uner the null hypothesis of equality of the pricing function, elta, an SPD over the two subsamples, we can show that:

47 Nonparametric Estimation of State-Price Density nh 1 h j )j 2 S~ H 1, H 2! j 1 ) h 102 j C H & N~0, 2 H! (A20) nh 1 h j )j 2 S~ 1, 2! j 1 ) h 102 j C & N~0, 2! (A21) nh 1 h j )j 2 S~ f 1 *, f 2 *! j 1 ) h 102 j C f * & N~0, 2 f *!, (A22) where the banwiths h j, j 1,..., (common to both subsamples) are given in each case m 0,1,2 (corresponing to prices, eltas, an SPDs respectively) by h 1 O~n 10 1!, h j O~n 10 2!, j 2,..., with 1 2 2m an 2 satisfying: 1 m 2 2m _ 2 2m 3r ~ 1!02 2 2~ 2 2m! 2 2 (A23) (note that the right-han-sie of this inequality is ifferent from that of equation (A15)), an 3_ 2 ~ 1!02 2 2m. (A24) 2 We again use a common kernel k of orer r for every variable. The bias an variance terms are given by: C H k 2 ~w! $s 1 w 2 ~! s 2 2 ~!% vj H ~! C k '2 ~w! w k 2 ~w! $s 1 w 1 2 ~! s 2 2 ~!% vj ~! C f * k ''2 ~w! w k 2 ~w! $s 1 w 1 2 ~! s 2 2 ~!% vj f *~! H 2 2 k~w!k~w v! w 2 $s 1 v 2 ~! s 2 2 ~!% 2 vj 2 H ~! 2 2 k ' ~w!k ' ~w v! w 2 v k~w!k~w v! w 2 v 1 $s 1 2 ~! s 2 2 ~!% 2 vj 2 ~!

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