A new class of Bayesian semi-parametric models with applications to option pricing

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1 Quantitative Finance, 2012, 1 14, ifirst A new class of Bayesian semi-arametric models with alications to otion ricing MARCIN KACPERCZYKy, PAUL DAMIEN*z and STEPHEN G. WALKERx yfinance Deartment, Stern School of Business, New York, NY 10012, USA zdeartment of Information, Risk, and Oerations Management, McCombs School of Business, University of Texas in Austin, TX 78712, USA xinstitute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK (Received 5 Aril 2011; in final form 3 July 2012) This aer develos a new family of Bayesian semi-arametric models. A articular member of this family is used to model otion rices with the aim of imroving out-of-samle redictions. A detailed emirical analysis is made for Euroean index call and ut otions to illustrate the ideas. Keywords: Dirichlet rocess; Beta distribution; Scale mixtures; Otions 1. Introduction Reliable redictions lie at the heart of most investment decisions. But in economic alications, as is well documented, reliable redictions are often difficult to obtain. In this context, stochastic volatility with jums models lead to useful forecasting models; however, their arametric nature exoses them to some standard criticisms. Consider the following quote of Engle and Gonzalez (1991): If we assume that the mean and variance equations are well secified but we do not know to which robability function they belong, then the closest aroximation to the true generating mechanism we can think of should come from the data itself. A nonarametric density resonds to this concern. In this aer, a new class of Bayesian models a Semiarametric Scale Mixture of Betas (SSMB) is develoed. The roosed setting allows one jointly to model skewness and kurtosis while accounting for the dynamics of the volatility in the time series. Modelling these comonents is ractically imortant, given substantial emirical evidence that distributions of financial data tend to exhibit such features. For examle, Cambell et al. (1997) document the existence of fat tails in monthly returns of the S&P500 index or skewness in the daily data of the same index. *Corresonding author. Paul.Damien@mccombs. utexas.edu As is well known, arametric model missecification might roduce inconsistent estimators and these models require a fully secified distribution for the error term. Bollerslev et al. (1992) and Bates (1996) rovide summaries of the error secifications commonly used in financial data. Desite their well-served urose, non-arametric aroaches are not so well reresented in the otion ricing literature. Imortantly, due to technical difficulties, and often different focus, none of the existing models comrehensively tackles all the issues relevant for efficient otion valuation. Secifically, most of the models do not address jointly the resence of jums and stochastic volatility in the data, i.e. there is no clear counterart for the arametric Stochastic Volatility with Jums models. Several studies offer valuable insights into the behaviour of underlying returns using stochastic volatility models with jums (e.g. Bates 2000, Duffie et al. 2000, Eraker et al. 2003, Maheu and McCurdy 2004). Among the more oular models, Stutzer (1996) assumes i.i.d. structure on the data but does not introduce any dynamics into volatility. On the other hand, non-arametric studies by Derman and Kani (1994) and Rubinstein (1994) are less flexible in modelling skewness and excess kurtosis. Also, since some of the methods rely on both the return and otion data they require the resence of a liquid otion market (as examles, Ait-Sahalia and Lo 1998, Ait-Sahalia and Duarte 2003). Additionally, to achieve reasonable convergence roerties the models need to rely on a long time series of data (Ait-Sahalia and Lo 1998); Quantitative Finance ISSN rint/issn online ß 2012 Taylor & Francis htt:// htt://dx.doi.org/ /

2 2 M. Kacerczyk et al. this, in turn, results in oor estimates of the tails of the distribution. The modelling roblem confronting us in this aer is: How can one reliably rice different otions given that the cross-section of otions is so large and the otions are substantially different from each other? For examle, some models might rice in-the-money, long-term call otions adequately, but may fail to do so with dee-outof-the-money, short-term call otions. Alying the SSMB model to ricing otions leads to significant imrovements in accuracy comared to other nonarametric and arametric models. On average, rediction errors are reduced by u to three times for a broad range of otion contracts. Overall, the significant reduction in ricing errors suggests that jointly modelling the variance, skewness, and kurtosis better describes the riskneutral redictive distribution of otions. The rest of the aer roceeds as follows. In section 2, we describe the financial modelling aims of this aer. In section 3 a new family of semi-arametric Bayesian models is introduced. Section 4 discusses the construction and roerties of the data. Section 5 rovides out-of-samle emirical results for Euroean index otions written on the S&P500, using both nonarametric models and the arametric Black Scholes model. Some conclusions are rovided in section Modelling aims The two major goals of this section are to: (a) describe the time series, while also motivating the reason to find redictive distributions for such data; (b) describe the risk neutral evaluation method emloyed in ricing the otions The time series Let S t denote the value (rice) of an asset at time t. Let D t be the dividend value aid by the same comany over the eriod t 1tot. From a statistical modelling ersective, it is better to consider the natural log of asset returns, denoted r t, and defined as r t ¼ ln S t S t 1 þ D t : S t 1 It is well known that the value of a Euroean call (C) and ut otion (P) can be obtained using the following identities: C ¼ e E Q ðs T KÞ þ, P ¼ e E Q ðk S T Þ þ ð1þ, where E Q stands for the exectation oerator under the risk-neutral measure Q, S T is the rice of the underlying asset at maturity T, is the annualized sot interest rate, is the time to maturity (in years), K is the otion strike rice, and s þ max(s, 0). In our setting is assumed to be fixed and known, hence S T is the only unknown random quantity in the above equations. Consequently, to calculate the otion rice we need to construct the redictive distribution of S T. In our analysis, we will first estimate the redictive distribution of the log return on the underlying asset at time t, namely r t, and then obtain the redictive distribution of the asset rice at maturity (S T ) via the following recursive identity: S t ¼ S t 1 e r t, ð2þ where t ¼ 1,..., T. Note, by convention, S 0 is the observed value of the asset rice at the time of making redictions. Thus, the data which we model are the logreturn of the underlying asset Risk-neutral valuation In the last subsection, we motivated the need to obtain redictive distributions of asset rices. These distributions, however, are derived under a hysical robability measure. But to make use of our method for otion ricing, the hysical measure must be converted into a risk-neutral measure. The main benefit of using a riskneutral measure stems from the fact that once the riskneutral robabilities are found, every asset can be riced by simly calculating its exected ayoff (that is, discounting as if investors were risk neutral). If we used hysical robabilities, every security would require a different adjustment (as they differ in riskiness). Converting the hysical robability measure to the riskneutral one requires the absence of arbitrage. To this end, we follow the framework roosed by Huang and Litzenberger (1988) and Stutzer (1996). Secifically, the canonical valuation method of Stutzer (1996) allows one to convert the redictive distribution from the revious subsection into a risk-neutral distribution. The method, described in detail by Stutzer, utilizes the maximum entroy rincile to estimate the unknown martingale measure. Imortantly, we can rice otions under no arbitrage. We start with a sequence of M draws from the redictive distribution of the index rice. Using these values we can construct the -eriod gross returns frg M i¼1 fs T=S t g M i¼1, where ¼ T t denotes time to maturity of a given contract. The true risk-adjusted density, (i), has to satisfy the following: X M i¼1 ðiþ R i ¼ 1: As ointed out by Stutzer, the quantity can be obtained by solving the following convex minimization roblem: ^ ¼ argmin ðiþ 4 0, P M i¼1 ðiþ¼1 Ið, ^Þ XM i¼1 ð3þ ðiþ lnð ðiþ= ^ðiþ Þ ð4þ subject to (3) holding. The objective function Ið, ^Þ is the Kullback Leibler information criterion distance of the ositive robabilities to the emirical robabilities ^, and the equality follows from the fact

3 A new class of Bayesian semi-arametric models with alications to otion ricing 3 that ^ðiþ ¼1=M imlies that minimizing I is equivalent to constrained maximization of the Shannon entroy. It can be shown that the solution to the above otimization roblem is given by ^ ðiþ ¼ exð R i =Þ P M i ¼ 1,..., M: ð5þ i¼1 exð R i =Þ The Lagrange multilier,, is obtained from the following unconstrained convex roblem: ¼ argmin X M i¼1 ex ðr i = 1Þ : ð6þ Having obtained the simulated values of the risk-neutral density we will be able to rice call and ut otions using the following formulae: and C t ¼ XM i¼1 P t ¼ XM i¼1 ðs T,i KÞ þ ^ ðiþ, ðk S T,i Þ þ ^ ðiþ: Here t denotes the time of the call/ut and T the time of maturity, with t5t. 3. Methodology The rimary goal of this section is to introduce a new class of semi-arametric Bayesian models that is subsequently used to obtain redictive distributions of index returns Semi-arametric scale mixture of betas (SSMB) There are four ideas that we link together to develo the class of models we call the semi-arametric scale mixture of betas (SSMB). (1) Scale mixture reresentation; see, for examle, Feller (1971). (2) A non-arametric family of rior distributions, namely the Dirichlet rocess; see aendix A and Ferguson (1973). (3) Variance regression; and (4) Gibbs samling; see, for examle, Smith and Roberts (1993) Scale mixture reresentations. Since the scale mixture of uniform reresentation is central to the construction of the SSMB family of models, this is first exlained in detail. The other features of the construction are then readily tagged on to the scale mixture aroach. Since we want a unimodal density, we use uniforms and beta distributions in the mixture distributions, ensuring unimodality. With normal kernels we could get a mulitmodal density, which does not make sense in our ð7þ ð8þ context since outliers will be modelled incorrectly. Also, we want a heavier tail rather than another mode. It is for these reasons that we refer an SSMB rather than taking a normal kernel. It is intuitively easier to understand our model by first addressing the kurtosis in the underlying data distribution. With r denoting observed data, and U denoting a latent mixing random variable, Feller s formulation of the conditional distribution of r is given by f ðrju ¼ uþ Uniformð ffiffi ffiffiffi u, þ u Þ, ð9þ u F, for some distribution function F with suort on (0, 1). As F ranges over all such distribution functions, the density of r ranges over all unimodal and symmetric density functions. Consequently, with a flexible rior on F, such a model can cature wide ranges of kurtosis in the data. To ensure maximum flexibility we will model F nonarametrically in the next subsection. We use u rather ffiffiffi than u in the formulation above because we can exress higher moments for r in terms of lower moments for U. It is easier to understand the imact of this by setting ¼ 0 for now. In aendix B, the entire comutational form of the model is constructed with 6¼ 0. Now, we have Var(r) ¼ 2 E(U)/3 and E(r 4 ) ¼ 4 E(U 2 )/5. Hence, we can rewrite the model as f ðrjuþ ¼ ffiffiffiffi U ð1 2betað1, 1ÞÞ which will suggest the form of generalizations to asymmetric or skewed densities. From a simulation ersective, the notation f ðrjuþ ¼ ffiffiffiffi U ð1 2betað1, 1ÞÞ means that f(rju) can be generated as a ffiffiffiffi U ð1 2betað1, 1ÞÞ random variable. An interesting fact that is used later on is noted here: if F is distributed Gamma(3/2, 1/2), then the distribution of r is Normal(0, 2 ). Aart from endogenizing kurtosis (as we did above), in our otion ricing alication we are also interested in modelling the extent of skewness. The extension to asymmetric densities is quite straightforward. Since the uniform density is a beta(1, 1) density, we can introduce skewness by having instead a beta(1, a) density, for some arameter a40. The first equation of the model in (9) becomes f ðrju, aþ ¼a 1 ð1 þ aþ ffiffiffiffi U 1=ð1 þ aþ betað1, aþ : ð10þ We recover (9) when a ¼ 1. We will change (1 þ a)/a to. There is no loss in doing this because it is only a scaling issue; moreover, as a result, later on, the analysis is simlified considerably. We will shorten the writing of the model to f ðrju, aþ ¼ ffiffiffiffi U BðaÞ; ð11þ BðaÞ ¼1=ð1 þ aþ betað1, aþ: The density function for W ¼ B(a) is given by a 11ð a=ð1 f W ðwþ ¼aa=ð1þaÞþw þ aþ 5 w 5 1=ð1 þ aþþ:

4 4 M. Kacerczyk et al. The moments of this density are readily obtained. To see this, first note that, deending on whether we have left or right skewness, the above aroach allows us to have a ¼ 1 þ or a ¼ 1 for some small 40. Now it is easy to verify that EW 2 a ¼ ð1 þ aþ 2 ð2 þ aþ and EW 3 2 ¼ ð1 þ aþ 3 þ 6 ð1 þ aþ 2 ð2 þ aþ 6 ð1 þ aþð2 þ aþð3 þ 1Þ : Defining as usual the skewness as SkðaÞ ¼ EW3 ðew 2 Þ 3=2, we see that for a ¼ 1 þ we have Sk(a) ¼ þo( 2 ) and for a ¼ 1 we have Sk(a) ¼ þo( 2 ), where is a constant; for comleteness, one could calculate, which is actually 12 3/2 / Hence, it does not matter whether we use beta(1, a) (a41) or beta(a, 1)(a51). To elaborate on this oint, if X is beta(1, a) then 1 X is beta(a, 1). Combine this with the notion that for beta(1, a) we have essentially symmetric skewness (to first order). Hence, we could address first-order skewness using either beta(1, a) or beta(a, 1). Of course this relies on small skewness. On the other hand if skewness is large and in one direction, the choice of beta(1, a) or beta(a, 1) would be obvious in order to get the skewness on the correct side. Thus, the mixture of beta distributions catures ositive skewness via the arameter a Bayesian non-arametrics and semiarametrics. To make use of the scale mixture of betas, recall that we need to secify the distribution function F in (9). In doing so, we will follow a nonarametric Bayesian aroach. As noted reviously, such an aroach allows us to introduce greater flexibility into the choices of F; this is critical since the data analysed in this aer have different levels of kurtosis. A non-arametric scale mixture model is obtained by assigning F a stochastic rocess rior; here we use the well-known Dirichlet rocess; see aendix A and Ferguson (1973). F Dir(c, F 0 ) means that F is assigned a Dirichlet rocess rior with exectation F 0 and scale arameter c40. Here c is a measure of strength of belief in the rior choice of F 0. Note, as an examle, it is ossible to centre the location arameter, F 0, on any member of the exonential ower family. This imlies that our scale mixture of beta reresentation encasulates all ranges of kurtosis. We use the Dirichlet rocess for two reasons: (a) the theoretical roerties of the rocess are very aealing; see Ferguson (1973) and aendix A; (b) imlementing the overall model is highly simlified; see MacEachern (1998). We note here that the scale mixture reresentation is such that we actually byass simulating from the osterior distribution over the unknown F, i.e. the comutational burden is substantially reduced; see also Brunner and Lo (1989) and aendix B. With F t denoting all the information u to and including time t, we now have the following hierarchical modelling framework: f ðr t jf t 1 Þ¼ ffiffiffiffiffi t B t ðaþ, where U t U 1 U t ðu 1,..., u t Þ: ð12þ Here t is the volatility and it will be described later how this will deend on the ast. Finally, B t (a) are indeendent and identically distributed coies of B(a). We take (u 1,..., u t ) to be based on a Dirichlet rocess rior; see aendix A for details on this rior. Having modelled the kurtosis via the variance of U t, we next model t. A variance regression model In the above, t can be modelled so that it deends on context-secific regressors via the following variance regression: t ¼ ex 0 þ XM k¼1 k Z k,t 1!, ð13þ where 0,..., M are arameters to be estimated and the {Z k,t 1 } are observed information (indeendent variables) u to and including time t 1. For the emirical analysis in this aer, we use the squared ast log returns calculated as Z t ¼ {ln(s t /S t 1 )} 2, where S denotes the value of the S&P500 index. The motivation for using this variable rimarily comes from other studies in finance. For examle, it would be similar to a secification of ARCH-tye models; see Engle and Gonzalez (1991). Similarly, Ghysels et al. (2006) use lagged squared ast returns as their volatility redictor. We emhasize that our model formulation and comutational algorithms are not significantly deendent on this articular choice of regressor. An imortant feature of stock markets is the resence of leverage effects, i.e. volatility is larger when returns are negative. In the current version of the model, we do not exlicitly account for such effects, largely because we want to focus on the most novel asects of our new SSMB model and illustrate their emirical imortance. But in general one could model such leverage effects by setting the mean of arameter k to be less than zero. Overall, the class of models in equations (12) and (13) where t is modelled as a regression and F is modelled via a Dirichlet rocess is what is termed as SSMB in this aer. The redictions of future rices, S T, from this class of models are the the key inuts needed to rice otions; see equations (7) and (8). To comlete the Bayesian secification, rior distributions are assigned to F,, c and a Prior distributions. The following describes the various riors used in the emirical analysis. Where necessary, a conjugate hyer-rior is used. Given recent advances in Bayesian comutation, the ractitioner

5 A new class of Bayesian semi-arametric models with alications to otion ricing 5 can readily emloy non-conjugate rior distributions if needed; for details, see MacEachern (1998) and Mira et al. (2001). The scale arameter of the Dirichlet rocess, c, is assigned a Gamma(a 0, b 0 ) hyer-rior distribution. The second arameter is the rior guess at F 0, which we will take to be a Gamma(3/2, 1/2) distribution. The reason for this is that under this distribution, and in the symmetric case with a ¼ 1, the marginal distribution for r t will be Normal with mean 0 and variance t 2. From a finance ersective, the zero-mean excess stock returns assumtion may be somewhat restrictive, but is consistent with the common notion in finance that exected returns are difficult to redict on average. Also, it fits well with the standard no-arbitrage argument that exected returns are equal to the risk-free rate. Prior studies have used such secifications, including Merton (1976). From a statistical ersective, in the symmetric case, note that even if one centred the rior for F 0 around zero, the variance could be taken to be very large, thus alleviating the zero-mean assumtion. In ractice, in the absence of strong rior information, it is advisable to set the rior variance to be very large. We assign a rior distribution to each of the k in the variance regression (13), which are assumed to be indeendent normal distributions with zero means and variances 2 k. Finally, the skewness arameter a is assigned a Gamma distribution. A strength of the Bayesian aroach is its ability to incororate context-secific knowledge in the modelling rocess. However, for illustrative uroses, all of our rior settings were chosen to reflect vague rior knowledge. Denoting to be a rior distribution, ( 0 ) ¼ N(0, 10), ( 1 ) ¼ N(0, 10). We take (c), the scale arameter of the Dirichlet rocess, to be Gamma(a 0, b 0 ) with a 0 ¼ b 0 ¼ The skewness arameter a is assigned (a) ¼ Gamma(c 0, d 0 ), c 0 ¼ d 0 ¼ The comutational asect of SSMB is imlicit in its formulation and oints to a Markov chain Monte Carlo (MCMC) scheme that could be imlemented to obtain osterior and redictive distributions. The Gibbs samler for the above model is detailed in aendix B. We ran the samler for one million iterations. Using well-known convergence diagnostics, having burned-in the first iterates, aroximate indeendence in the samled variates was obtained by using every 1000th iterate from the chain. The algorithm could take a few days to execute if one analyses thousands of contracts. In ractice, one would seldom, if ever, use data going back to 1983 to rice an otion in the year, say Tyically, the ricing of otions, as is well documented, relies on data for no more than one year, and interest is usually on redicting rices of the current otions, in which case the results from the analysis in this aer would be quite fast indeed. 4. The data The data consist of Euroean call and ut otions, written on the S&P500 index, traded on the Chicago Board Otions Exchange. The samle covers a eriod of 20 years from January 1983 through December The selection of this eriod has been dictated by the availability of the data. By selecting such a long series of data, covering most of the sikes in the time series of asset returns, the emirical results should not be significantly driven by secific features of the market data. Details of the adjustments and exclusion criteria made to the data are resented in Bakshi et al. (1997). Here, in table 1, we rovide some key features of the data, namely the moneyness and the maturity date for each class of call and ut otions as well as the average otion rices, which is meant to hel interret the subsequently reorted ricing errors. In Panel A (B) of table 1, we reort the summary statistics for the call (ut) otions with the average rice and the total number of observations (in arentheses) for each moneyness maturity category. For examle, a call otion is considered to be out-of-the-money (OTM) if its moneyness, defined as S/K, does not exceed 0.97, and atthe-money (ATM) if its moneyness falls between 0.97 and All other contracts are in-the-money (ITM). In addition, we divide all contracts into three grous of maturities: those with not more than 40 days to maturity, those with maturities between 41 and 70, and those with maturity longer than 70 days. Our samle includes a total of otion observations with calls and uts. In the call (ut) grou, OTM and ATM otions make u aroximately 40.7 (50.9) ercent and 31.7% (28.8%) of the total samle, resectively. The average call rice ranges from $3.07 for short-term, dee OTM otions to $ for long-term, dee ITM contracts. In contrast, the smallest rice for a ut otion equals $2.97 while the largest one is $ Since the method used in this aer requires the history of ast stock index returns, we use one year of daily observations as an estimation eriod. The time range of the estimation window changes as we move forward in time. Also, in order to comare the rices obtained from the model to real-time rices, the maturity date of any otion cannot exceed December In order to evaluate the accuracy of our otion ricing model, we select the following two oular nonarametric benchmarks for comarison: the canonical valuation model of Stutzer (1996), and the constrained non-arametric estimator of Ait-Sahalia and Duarte (2003). The reason for working with these non-arametric benchmarks is mainly due to their close relationshi to the class of models advocated in this aer. We also rovide a comarison to a arametric model, namely the widely used Black Scholes otion ricing model. 5. Emirical results In section 2, we described the rocedure to obtain otion rices, noting there that the end oint of the analysis was to evaluate equations (7) and (8) for call and ut otions, resectively. The only random, unknown quantity needed to evaluate these equations was the redictive distribution

6 6 M. Kacerczyk et al. Table 1 Samle roerties of S&P500 index otions. Days-to-exiration Moneyness, S/K Subtotal Panel A: Call otions $3.07 $5.94 $17.82 OTM (16,638) (15,714) (46,478) (78,830) $7.30 $15.05 $37.62 (20,342) (12,999) (22,295) (55,636) $15.14 $25.31 $47.94 ATM (23,902) (12,976) (21,871) (58,749) $28.91 $37.79 $58.05 (20,440) (9,369) (17,237) (47,046) $45.50 $51.29 $68.44 ITM (14,628) (6,029) (11,346) (32,003) $99.92 $ $ (25,951) (13,381) (21,260) (60,592) Subtotal (121,901) (70,468) (140,487) (332,856) Panel B: Put otions $ $ $ ITM (10,411) (7,897) (18,843) (37,151) $42.35 $45.07 $63.89 (11,478) (6,327) (14,566) (32,371) $23.23 $31.64 $46.78 ATM (21,190) (10,961) (20,423) (52,574) $12.79 $21.10 $35.22 (22,695) (12,505) (20,581) (55,781) $7.51 $14.07 $25.26 OTM (19,462) (10,835) (16,796) (47,093) $2.97 $5.03 $11.96 (44,514) (34,358) (64,902) (143,774) Subtotal (129,750) (82,883) (156,111) (368,744) This table reorts the summary of the data used in the study. The cross-section of the call otions has been divided into 18 categories: with resect to exiration date ( 40 days, (40,704) days, and470 days) and moneyness (out of the money (OTM); at the money (ATM); and in the money (ITM)). Each cell reresents the average otion rice in each maturity-moneyness category along with the number of contracts which were used to calculate the averages (in arentheses). The samle covers the eriod of 1 January December Otions with maturity less than 6 days, rice lower than and those violating arbitrage conditions have been excluded from the samle. of the index returns. The SSMB model rovides this distribution. In the following discussion, and corresonding tables and grahs, the following abbreviations are used: Black Scholes model (BS); canonical valuation model of Stutzer (CV); the constrained non-arametric estimator of Ait-Sahalia and Duarte (SC); and SSMB denotes our model Predictions We begin with the resentation of our estimation results, which considers the evolution, with resect to maturity, of redicted rices for six different otion contracts, namely three calls and three uts. To cature the time-series variation in volatility we consider contracts sarsed over three different eriods, 1992, 1997 and These three eriods arguably have had very different volatility atterns, which allows us to assess the robustness of our model. We comare the redictions from the SSMB model to those from the SC model and the market rices. Figure 1 resents the results. In most contract configurations, the SSMB model erforms better than the SC model. The BS and CV alternatives erform uniformly worse for these contracts, and hence were omitted from the grahs. The suerior outerformance of SSMB is articularly visible for shorter maturities and out-of-the-money contracts. While somewhat informative, figure 1 does not give a recise estimate of the observed imrovements. Hence, we now turn to a thorough resentation of our results. To comare the full-samle efficiency of the SSMB method relative to other benchmarks, as well as across various strike rices and maturities, each contract in the samle is assigned into 18 grous sorted by maturity and moneyness. Secifically, all call and ut otions are divided into six moneyness classes. Next, within each such class, three maturity grous are formed. As a result, each contract is assigned to one of 18 bins. Within each bin the average dollar and ercentage errors are calculated. For each grou, the accuracy of the SSMB ricing model to that of the CV, SC and BS models are comared. The metric for comarison is the root mean squared errors (RMSEs), defined as the square root of the mean squared deviations of the model rice from the observed rice. To calculate ercentage errors, the RMSEs are further scaled by the average rice of the otion in the samle. Tables 2 and 3 resent the results for call and ut otions, resectively.

7 A new class of Bayesian semi-arametric models with alications to otion ricing 7 Figure 1. Evolution of rices for SSMB and SC models. This figure deicts the evolution of redicted otion rices as a function of time to maturity for six different otion contracts: three calls and three uts for the years 1992, 1997 and The rices are obtained from the SSMB and SC models and are lotted against the market otion rice. The results in table 2 indicate several beneficial asects of the SSMB aroach comared to the two nonarametric aroaches (CV and SC): it significantly imroves the fit of the volatility structure into the crosssection of otion contracts. First, observe that the average ricing errors across the different maturity/moneyness classes are less variable comared to CV and lower comared to SC. The ercentage ricing errors for SSMB vary between and 76.29%, while for CV the sread ranges between and %. Likewise, the resective range for the SC method is %. Second, observe that SSMB roduces errors of a similar magnitude irresective of the maturity date of the contract. This is in contrast esecially to the CV method, which tends to roduce very high errors esecially for contracts with longer time to maturity. Thus, for examle, for contracts with low moneyness (less than 0.94) and long maturity (more than 70 days), the average ricing error decreases from to 51.06% under the SSMB model. An exlanation for this is that the CV aroach recludes the existence of volatility clustering; thus the data converge very quickly to normality, which is at odds with observed rice atterns. Similar trends are noted across all moneyness series. The SC method does much better than the CV aroach in this regard; this oint has been detailed in Ait-Sahalia and Lo (1998). Third, within the various maturity contracts, the ricing errors generally decrease with moneyness both for the CV and SSMB models, but they decrease more for the latter, esecially for longer maturities. This is in contrast to the SC model for which the ricing errors form a U-shae attern. Such a attern is consistent with the intuition rovided by Ait-Sahalia and Duarte (2003) who argue that non-arametric methods, like the ones they develoed, fail to cature the tails of the crosssection of otion contracts. Consequently, the ricing errors under the SC method always increase at extreme moneyness values. This seems to be the case for all maturity classes where the ricing errors are lowest for at-the-money contracts. Along this dimension of comarison, the SSMB method rovides redictions with significantly lower errors for both out-of-the-money and in-the-money contracts. The differences are esecially

8 8 M. Kacerczyk et al. Table 2 Out-of-samle ricing errors of the samle of S&P500 index otions: call otions. Days-to-exiration Moneyness, Model S/K RMSE % Error RMSE % Error RMSE % Error BS $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ CV $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SC $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SSMB $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ This table reorts the average out-of-samle ricing errors for call otions with different maturities and moneyness levels for the eriod The root mean squared errors (RMSEs), and ercentage errors have been calculated for the Black Scholes model (BS), the canonical valuation (CV) of Stutzer (1996), the kernel estimation with shae constraint (SC) of Ait-Sahalia and Duarte (2003), and the Semi-arametric Scale Mixture of Betas (SSMB) model defined in this aer. All otions have been divided with resect to their maturities and moneyness levels (defined as the ratio of sot rice to strike rice) into eighteen grous. RMSE has been calculated as a root of the average mean squared error, while the ercentage rice error further scales the RMSE by the average rice of the otion. ronounced for dee in-the-money contracts, where the SC method results in errors which are aroximately three times larger than the errors comared to the SSMB method. Finally, we observe the highest average misricing for short-term out-of-the-money contracts. In this grou, the CV aroach roduces the highest ricing errors (over 137%), followed by the SC aroach, roducing errors of about 100%, while SSMB has errors of about 76%. The results for ut otions, resented in table 3, are generally in line with the findings documented for call otions. While SC and CV methods are based on a nonarametric aroach, the BS model is a arametric aroach. It is aarent from both tables that the main drawback of the BS model is its oor erformance for the very short-term out-of-the-money contracts. At the same time, the BS model does quite well for the in-the-money contracts, esecially relative to the SC and CV methods. This result is not surrising in light of the emirical literature that has found that arametric methods, in general, largely fail for the dee out-of-the-money contracts (e.g. Bakshi et al. 1997). To facilitate additional comarisons, figures 2 and 3 summary evidence of the ricing errors for the different tyes of ricing models. In these grahs, the x-axis indicates three different categories of otion maturity, and the y-axis indicates six different categories of otion moneyness, consistent with those in tables 2 and 3. To understand the aarent imrovement in the otion ricing consider figure 4, which deicts (randomly chosen) redictivedistributions of the underlying asset (S&P500 index) for four different time eriods in the otions data. What is striking is that the variance, skewness and kurtosis of these distributions are markedly different. The SSMB model thus nicely accounts for these differences. This flexibility in distributions is very imortant for ricing call and ut contracts with a sectrum of different strike rices. In fact, this is one of the main reasons that the ricing errors under the SSMB aroach are smaller than under other non-arametric aroaches. One could argue that the erformance of our method may be sensitive to a articular choice of the samle eriod. We feel this is not the case because our samle covers a eriod with tremendous market fluctuations (January 1983 through December 2002). In fact,

9 A new class of Bayesian semi-arametric models with alications to otion ricing 9 Table 3 Out-of-samle ricing errors of the samle of S&P500 index otions: ut otions. Days-to-exiration Moneyness, Model S/K RMSE % Error RMSE % Error RMSE % Error BS $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ CV $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SC $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SSMB $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ This table reorts the average out-of-samle ricing errors for ut otions with different maturities and moneyness levels for the eriod The root mean squared errors (RMSEs), and ercentage errors have been calculated for the Black Scholes model (BS), the canonical valuation (CV) of Stutzer (1996), the kernel estimation with shae constraint (SC) of Ait-Sahalia and Duarte (2003), and the Semi-arametric Scale Mixture of Betas (SSMB) model defined in this aer. All otions have been divided with resect to their maturities and moneyness levels (defined as the ratio of sot rice to strike rice) into eighteen grous. RMSE has been calculated as a root of the average mean squared error, while the ercentage rice error further scales the RMSE by the average rice of the otion. Figure 2. Pricing errors for call otions. This figure deicts the ercentage ricing errors for the cross-section of call otions as a function of time to maturity and moneyness for four different models: Black Scholes (BS), canonical valuation (CV), shae constraint (SC), and Semi-arametric Scale Mixture of Betas (SSMB). Maturity is divided into three bins: 540 days, (40,704) days, and 470 days. Moneyness is divided into six bins:50.94, (0.94, 0.97), (0.97, 1.00), (1.00, 1.03), (1.03, 1.06) and

10 10 M. Kacerczyk et al. Figure 3. Pricing errors for ut otions. This figure deicts the ercentage ricing errors for the cross-section of ut otions as a function of time to maturity and moneyness for four different models: Black Scholes (BS), canonical valuation (CV), shae constraint (SC), and Semi-arametric Scale Mixture of Betas (SSMB). Maturity is divided into three bins:540 days, (40, 704) days, and 470 days. Moneyness is divided into six bins: 50.94, (0.94, 0.97), (0.97, 1.00), (1.00, 1.03), (1.03, 1.06) and Figure 4. Predictive distributions of market index rices. This figure deicts the redictive distributions of market index rices based on daily index data for four different time eriods in the otion data. Each grah includes the resective mean, standard deviation, skewness and kurtosis of the distribution. All distribution functions have been obtained from simulated data using a kernel smoothing aroach. we include most of the imortant sikes in the returns of the S&P500 index. 6. Concluding remarks In this aer, we develo a new class of Bayesian Semi-arametric Scale Mixture of Beta (SSMB) models, and aly it to ricing the S&P500 index call and ut otions. The emirical results in this aer show that the SSMB structure offers significant benefits in describing the atterns of volatility in the cross-section of otions data when comared to non-bayesian non-arametric methods. For the short-term, dee out-of-the-money otions the arametric Black Scholes model does very

11 XML Temlate (2012) [ :28am] [1 14] A new class of Bayesian semi-arametric models with alications to otion ricing 11 oorly, but, consistent with revious findings, the arametric model does well for long-term, in-the-money contracts. Our model has substantial reduction in ricing errors across various moneyness/maturity classes for both call and ut otions. In many cases, the ricing errors are at least 50% lower than those obtained using comarable arametric and non-arametric alternatives. It aears that the SSMB model shows romise for ricing otions. But like all statistical models, the SSMB model has its limitations. Our model is basically roviding full suort on unimodal densities. If the model is bi-modal or multimodal, which is unlikely in our otions ricing alication, then SSMB would do badly. But in otions ricing, it is better to model outliers with heavy tails than ut another density at such oints, which may haen if an MDP is used. We also run into trouble if the arametric structure for the variance is wrong, but then this would be true for everyone using any semi-arametric model. In this aer, basically, we relace arametric unimodal densities with non-arametric unimodal densities, icking u more ossibilities via very high degrees of kurtosis. That such high levels of kurtosis need to be better modelled is exemlified via the redictive densities for both call and ut otions. While the statistical estimation of our model offers significant imrovements relative to cometing alternatives, certain context-secific limitations aly. For examle, we do not exlicitly model the feedback from the level of returns to volatility, a so-called leverage effect, which is known to hel ricing otions. We also use a very arsimonious lag structure in the evolution of both the mean and variance rocesses; adding other redictors could roduce more accurate moments estimates. Finally, we do not directly incororate information from otion markets which some models consider useful. These extensions will be reorted in subsequent research. Damien, P., Wakefield, J.C. and Walker, S.G., Gibbs samling for Bayesian nonconjugate and hierarchical models using auxiliary variables. J. Roy. Statist. Soc. B, Statist. Methodol., 1999, 61, Derman, E. and Kani, I., Riding on the smile. Risk, 1994, 7, Duffie, D., Pan, J. and Singleton, K., Transform analysis and asset ricing for affine jum-diffusions. Econometrica, 2000, 68, Engle, R.F. and Gonzalez, G., Semi-arametric ARCH models. J. Bus. & Econ. Statist., 1991, 9, Eraker, B., Johannes, M. and Polson, N., The imact of jums in volatility and returns. J. Finan., 2003, 58, Escobar, M.D. and West, M., Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc., 1995, 90, Feller, W., An Introduction to Probability Theory and its Alications, Vol. II, 1971 (Wiley: New York). Ferguson, T.S., A Bayesian analysis of some nonarametric roblems. Ann. Statist., 1973, 1, Ghysels, E., Santa-Clara, P. and Valkanov, R., Predicting volatility: How to get most out of returns data samled at different frequencies. J. Econometr., 2006, 131, Huang, C.F. and Litzenberger, R.H., Foundations for Financial Economics, 1988 (Prentice Hall: Englewood Cliffs, NJ). MacEachern, S.N., Comutational methods for mixture of Dirichlet rocess models. In Practical Nonarametric and Semiarametric Bayesian Statistics, edited by D. Deys, P. Muller, and D. Sinha, 1998 (Sringer: New York). Maheu, J.M. and McCurdy, T.H., News arrival, jums dynamics, and volatility comonents for individual stock returns. J. Finan., 2004, 59, Merton, R.C., Otion ricing when underlying stock returns are discontinuous. J. Finan. Econ., 1976, 3, Mira, A., Moller, J. and Roberts, G.O., Perfect slice samlers. J. Roy. Statist. Soc. B, Statist. Methodol., 2001, 63, Rubinstein, M., Imlied binomial trees. J. Finan., 1994, 69, Smith, A.F. and Roberts, G.O., Bayesian comutation via the Gibbs samler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. B, Statist. Methodol., 1993, 55, Stutzer, M., A simle nonaramteric aroach to derivative security valuation. J. Finan., 1996, 51, References Ait-Sahalia, Y. and Duarte, J., Nonarametric otion ricing under shae restrictions. J. Econometr., 2003, 116, Ait-Sahalia, Y. and Lo, A.W., Nonarametric estimation of state-rice densitites imlicit in financial assets. J. Finan., 1998, 53, Bakshi, G., Cao, C. and Chen, Z., Emirical erformance of alternative otion ricing models. J. Finan., 1997, 52, Bates, D.S., Testing otion ricing models. In Handbook of Statistics, edited by G.S. Maddala and C.R. Rao, Vol. 14, 1996 (Elsevier Science B.V: New York). Bates, D.S., Post- 87 crash fears in the S&P 500 futures otion market. J. Econometr., 2000, 94, Black, F. and Scholes, M.S., The ricing of otions and cororate liabilities. J. Polit. Econ., 1973, 81, Bollerslev, T., Chou, R.Y. and Kroner, K.F., ARCH modeling in finance. J. Econometr., 1992, 52, Brunner, L.J. and Lo, A.Y., Bayes methods for a symmetric and unimodal density and its mode. Ann. Statist., 1989, 17, Cambell, J.Y., Lo, A.W. and MacKinlay, C.A., The Econometrics of Financial Markets, 1997 (Princeton University Press: Princeton, NJ). Aendix A: The Dirichlet Process (DP) To motivate the DP, consider a simle examle. Suose X is a random variable which takes the value 1 with robability and the value 2 with robability 1. Uncertainty about the unknown distribution function F is equivalent to uncertainty about ( 1, 2 ), where 1 ¼ and 2 ¼ 1. A Bayesian would ut a rior distribution over the two unknown robabilities 1 and 2. Of course here we essentially have only one unknown robability since 1 and 2 must sum to one. A convenient rior distribution is the Beta distribution given, u to roortionality, by f ð 1, 2 Þ/ , where 1, 2, 1, 2 0, and 1 þ 2 ¼ 1. It is denoted Beta( 1, 2 ). Different rior oinions can be exressed by different choices of 1 and 2. Set i ¼ cq i with q i 0 and q 1 þ q 2 ¼ 1. We have, Eð i Þ¼q i

12 XML Temlate (2012) [ :28am] [1 14] 12 M. Kacerczyk et al. and Varð i Þ¼ q ið1 q i Þ : ða1þ c þ 1 If q i ¼ 0.5 and c ¼ 2 we obtain a non-informative rior. We denote our rior guess (q 1, q 2 )byf 0. The interretation is that the q i centre the rior and c reflects our degree of belief in the rior: a large value of c imlies a small variance, and hence strong rior beliefs. The Beta rior is convenient in the examle above. Why? Suose we obtain a random samle of size n from the distribution F. This is a binomial exeriment with the value X ¼ 1 occurring n 1 times (say) and the value X ¼ 2 occurring n 2 times, where n 2 ¼ n n 1. The osterior distribution of ( 1, 2 ) is once again a Beta distribution with arameters udated to Beta( 1 þ n 1, 2 þ n 2 ). Since the osterior distribution belongs to the same family as the rior distribution, namely the Beta distribution, such a rior-to-osterior analysis is called a conjugate udate, with the rior being referred to as a conjugate rior. The above examle is the well-known Beta-Binomial model for. We now generalize the conjugate Beta-Binomial model to the conjugate Multinomial-Dirichlet model. Now the random variable X can take the value X i with robability i, i ¼ 1,..., K, with i 0, and P K i¼1 i ¼ 1. Now, uncertainty about the unknown distribution function F is equivalent to uncertainty about ¼ ( 1,..., K ). The conjugate rior distribution in this case is the Dirichlet distribution (not to be confused with the Dirichlet rocess) given, u to roortionality, by f ð 1,..., K Þ/ K 1 K, ða2þ where i, i 0 and P K i¼1 i ¼ 1. If we set i ¼ cq i then we obtain the same interretation of the rior, and, in articular, the mean and variance are again given by equation (A1). As before, (q 1,..., q K ) reresents our rior guess (F 0 ) and c the certainty in this guess. A random samle from F now constitutes a Multinomial exeriment and when this likelihood is combined with the Dirichlet rior, the osterior distribution for is once again a Dirichlet distribution with arameters i þ n i, where n i is the number of observations in the ith category. We now make a jum from a discrete X to a continuous X (by imagining K!1). In traditional arametric Bayesian analysis, the distribution of X, say F, would be assumed to belong to a articular family of continuous robability density functions. For examle, if X can take on any real value the family of distributions is often assumed to be the Normal distribution, denoted N(, 2 ). A Bayesian analysis would then roceed by first lacing a rior distribution on and 2, and then obtaining the resultant osterior distributions of these two finitedimensional arameters. We enter the realm of Bayesian non-arametrics when F (in the last aragrah) itself is treated as a random variable; that is, one must now assign a rior distribution to F. Since F is infinite-dimensional, we need a stochastic rocess whose samle aths index the entire sace of distribution functions. In the main text in this aer, we noted that our focus is on ensuring greater levels of kurtosis in the conditional distribution of the asset. It is now easy to see that by treating this conditional distribution, F, itself as a random quantity, we allow the rocess that is driving the data to take on any degree of kurtosis. Parametric models imose a fixed degree of kurtosis, including fat-tailed distributions like the Student-t that is used in GARCH models. But the Bayesian non-arametric model, loosely seaking, allows the data to determine the level of kurtosis in the distribution of the asset, which could be much larger than under a arametric model. Indeed, since the focus of this aer is on the redictive distribution of the asset, allowing for larger degrees of kurtosis is critical because uncertainty in forecasts increases over time. Thus, a model that makes minimal assumtions about the distribution of the asset is likely to cature this greater uncertainty over time better. We are now ready to define the Dirichlet rocess, but first some notation. A artition B 1,..., B k of the samle sace is such that S K i¼1 B i ¼ and B i \ B j ¼; for all i 6¼ j. That is, we have a grou of sets that are disjoint and exhaustive. Stated differently, the sets cover the whole samle sace and are non-overlaing. Definition A.1: A Dirichlet rocess rior with arameter generates random robability distributions F such that, for every k ¼ 1, 2, 3,... and artition B 1,..., B k of, the distribution of (F(B 1 ),..., F(B k )) is the Dirichlet distribution with arameter (B 1 ),..., (B k )). Here is a finite measure on and so we can ut () ¼ cf 0 (), where c40 and F 0 is a robability distribution on. Examle A.2: Consider a random variable X with distribution function F defined on the real line. Now consider the robability ¼ r(x5x ) and suose we secify a DP rior with arameter for F. If we ut B 1 ¼ ( 1, x ] and B 2 ¼ (x, 1), then we see from the above definition that, a riori, has a Beta distribution with arameters 1 ¼ (B 1 ) ¼ cf 0 (x ) and 2 ¼ (B 2 ) ¼ c(1 F 0 (x )), where c ¼ ( 1, 1). This rior is such that E() ¼ F 0 (x ) and var() ¼ F 0 (x )(1 F 0 (x ))/(c þ 1), thus showing the link to equation (A1). The variance of and hence the level of fluctuation of about F 0 (x ) deends on c. In articular, if c is large then we have strong belief in F 0 (x ) and var() is small. Note that this is true for all artitions B 1, B 2 of the real line, or, equivalently, all values of a. Now consider observations X 1,..., X n from F. Let F be assigned a DP rior, denoted Dir(c, F 0 ). Then Ferguson showed that the osterior rocess F has arameters given by cf o þ nf n c þ n and, c þ n where F n is the emirical distribution function for the data, namely the ste function with jums of 1/n at each X i. The classical maximum likelihood estimator is given by F n.