Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model

Journal of Banking & Finance 29 (2005) 2947–2969 www.elsevier.com/locate/jbf

Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model An-Sing Chen a

a,1

, Mark T. Leung

b,*

Department of Finance, National Chung Cheng University, Ming Hsiung, Chia Yi 621, Taiwan, ROC b Department of Management Science and Statistics, College of Business, University of Texas, San Antonio, TX 78249, USA Received 8 September 2003; accepted 29 October 2004 Available online 19 January 2005

Abstract This paper compares the empirical performances of statistical projection models with those of the Black–Scholes (adapted to account for skew) and the GARCH option pricing models. Empirical analysis on S&P500 index options shows that the out-of-sample pricing and projected trading performances of the semi-parametric and nonparametric projection models are substantially better than more traditional models. Results further indicate that econometric models based on nonlinear projections of observable inputs perform better than models based on OLS projections, consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. The econometric option models presented in this paper should prove useful and complement mainstream mathematical modeling methods in both research and practice.
2004 Elsevier B.V. All rights reserved.

*

Corresponding author. Tel.: +1 210 458 5776; fax: +1 210 458 6350. E-mail addresses: fi[email protected] (A.-S. Chen), [email protected] (M.T. Leung). 1 Tel.: +886 5 272 0411x34201; fax: +886 5 272 0818.

0378-4266/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2004.10.005

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JEL classification: C53; G13 Keywords: Index options; Pricing and trading; Black–Scholes and GARCH option pricing; Statistical projections; Nonparametric; Semi-parametric; Response surface mapping

1. Introduction Since the seminal work by Black–Scholes (1973, henceforth BS) and Merton (1973), option pricing models have become increasingly complex in an attempt to better fit the reported structures of option prices and to relax some of the restrictive BS assumptions. In spite of the vast numbers of alternative option pricing models, 2 a common weakness tying the BS-based models is that these models cannot capture time series information without ad hoc adjustments. In addition, most of the empirical literature in option pricing has focused on the cross-sectional fit of models, ignoring the consistency of models and parameters over time. One approach to map time series information into option prices is to do so indirectly during the estimation, specifically, by jointly estimating the option pricing models across options (cross-sectional) and across time (e.g., Pan, 2002; Bates, 2000; Liu et al., 2003). Another approach is to incorporate time series information directly into the option pricing model (e.g., Heston and Nandi, 2000). For example, Heston and Nandi developed a closed-form option valuation formula for spot asset whose variance follows a Generalized Autoregressive Conditional Heteroskedastic (GARCH) process that can be correlated with the returns of the spot asset. In contrast to the BS-based models, GARCH models have the inherent advantage that volatility is readily observable from the history of asset prices. As a result, a GARCH option model allows one to value an option using spot volatilities computed directly from the history of asset returns without necessarily using the implied volatilities inferred from other contemporaneous options. Because the model describes option values as functions of the current spot price and the observed path of historical spot prices, one can readily combine the cross-sectional information in options with the time series information of the underlying asset. Experiments have shown that outof-sample valuation errors from the GARCH option model are much lower than those from the BS-based models, including heuristic rules that are used by market makers to fit to the variations in implied volatilities across exercise prices and maturities.

2

A list of models often cited by recent academic research papers include: (i) the stochastic-interest-rate option models of Merton (1973) and Amin and Jarrow (1992); (ii) the jump-diffusion/pure jump models of Bates (1991), Madan and Chang (1996), and Merton (1976); (iii) the constant-elasticity-of-variance model of Cox and Ross (1976); (iv) the Markovian models of Rubinstein (1994) and Ait-Sahalia and Lo (1996); (v) the stochastic-volatility models of Heston (1993), Hull and White (1987), Melino and Turnbull (1990, 1995), Scott (1987), Stein and Stein (1991), and Wiggins (1987); (vi) the stochastic-volatility and stochasticinterest-rates models of Amin and Ng (1993), Bailey and Stulz (1989), Bakshi and Chen (1997a,b), and Scott (1997); (vii) the stochastic-volatility jump-diffusion models of Bates (1996, 2000), and Scott (1997); and (viii) the stochastic-interest stochastic-volatility jump-diffusion models of Bakshi et al. (1997).

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An alternative way to map time series information into option prices is to employ a direct statistical projection of time series information onto the corresponding option prices. The use of econometric techniques to price options, however, was largely abandoned in the early 70s after the introduction of the BS model. 3 In the past thirty so years, however, econometric techniques have made many advances. Applying these newer econometric techniques to option pricing may prove useful and complement mainstream mathematical modeling methods. Specifically, this paper compares an array of statistical projection models with the BS and the GARCH option pricing models. To facilitate a more balanced comparison, the empirical examination adopts a variation of the BS model fitted to the skew (abbreviated by BSS). Empirical analysis on S&P500 index options shows that the out-of-sample pricing and projected trading performances of semi-parametric and nonparametric projection models are better than the BSS and the GARCH option models. Results further indicate that econometric models based on nonlinear projections of observable inputs perform better than models based on OLS projections, consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. Similar findings are obtained when the call and the put options are considered collectively and separately. Moreover, the results of secondary experiment show that the level of moneyness and the time to option expiration do not play a significant role in altering the relative performances of various models. This paper is organized as follows. In the next section, the foundation of the GARCH option pricing model and various statistical projection techniques are explained. These econometric models include semi-parametric projection, spanning polynomial, and multivariate nonparametric projection. In Section 3, we provide a description of the data set and some assumptions regarding the empirical methodology. Empirical results with respect to out-of-sample pricing and projected trading performances are then reported and analyzed in Section 4. Section 5 concludes the paper.

2. The models 2.1. GARCH option model In the current study, we use the Heston and Nandi (2000) GARCH option model as the benchmark for comparison. The GARCH option model is definitely a more challenging benchmark than, for example, the simple BS model for any competing option valuation model. Furthermore, Heston and Nandi show that a single lag 3

There are many reasons: the BS framework is theoretically elegant, option prices are nonlinear mappings of conditioning information, the logical strength of the no arbitrage argument, etc. Mainstream option pricing is now the mathematical modeling approach and follows basically the spirit of BS; in particular, assume a stochastic process and then develop an option pricing model to fit the assumed stochastic process via a no-arbitrage hedge scenario.

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version of the GARCH option model outperforms even the tough to beat ad hoc BS model often used by practitioners, in which each option has its own implied volatility depending on the exercise price and time to maturity. The equations of the GARCH option model for a European call option, at time t, with an exercise price X that expires at time T can be written as follows: C ¼ erðT tÞ Et ½MaxðSðT Þ  X ; 0Þ  i/   Z 1 erðT tÞ 1 X f ði/ þ 1Þ ¼ SðtÞ þ Re d/ 2 i/ p 0   i/    Z 1 1 1 X f ði/Þ þ  X erðT tÞ Re d/ ; 2 p 0 i/

ð1Þ

where f ð/Þ ¼ SðtÞ/ exp Aðt : T ; /Þ þ

p X

Bi ðt; T ; /Þhðt þ 2D  iDÞ

i¼1

þ

q1 X

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C i ðt; T ; /Þðzðt þ D  iDÞ  ci hðt þ D  iDÞÞ2

! ð2Þ

i¼1

and Aðt; T ; /Þ ¼ Aðt þ D; T ; /Þ þ /r þ B1 ðt þ D; T ; /Þx 1  lnð1  2a1 b1 ðt þ D; T ; /ÞÞ; 2

ð3aÞ 2

1 1=2ð/  c1 Þ b1 ðt; T ; /Þ ¼ /ðk þ c1 Þ  c21 þ b1 ðt þ D; T ; /Þ þ 2 1  2a1 B1 ðt þ D; T ; /Þ

ð3bÞ

and these coefficients can be calculated recursively from the terminal conditions: AðT ; T ; /Þ ¼ 0;

ð4aÞ

BðT ; T ; /Þ ¼ 0:

ð4bÞ

Et ½  denotes the expectation under the risk-neutral distribution. Put option values can be calculated using the put-call parity. In contrast to the Black–Scholes formula, this formula is a function of the current asset price, S(t), and the conditional variance, h(t + D). S(t) denotes the underlying asset price at time t and follows a process over time steps of length D. Since h(t + D) is a function of the observed path of the asset price, the option formula is effectively a function of current and lagged asset prices. In contrast to continuous-time models, volatility is a readily observable function of historical asset prices and need not be estimated with other procedures. In this model, one can directly observe h(t + D), at time t, as a function of the spot price as follows:

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hðt þ DÞ ¼ - þ b1 hðtÞ þ a1

ðlogðSðtÞÞ  logðSðt  DÞÞ  r  khðtÞ  c1 hðtÞÞ : hðtÞ ð5Þ

a1 determines the kurtosis of the distribution. c1 controls the skewness or the asymmetry of the distribution of the log-returns. A more detailed explanation of the model and its underlying process can be found in Appendix A. 2.2. Semi-parametric statistical projection This section posits several implementable semi-parametric statistical projections that can be used to project currently observable information along with time series information into an option price forecast. Firstly, we fit the BS model to the reported structure of option prices using a separate implied volatility for each option (specific to its exercise price and time to maturity) extracted from market prices along with a conditional standard deviation forecast from a first order GARCH process. This is designed to produce a very close fit to the shape of the implied volatilities across exercise prices and maturities and, at the same time, capture time series information. Specifically, the spot volatility of the asset that enters the BS option formula is specified to be a deterministic function of the exercise price, time to maturity, GARCH conditional standard deviation, or combinations of the terms: Model 1: rsm ¼ maxð0:01;a0 þ a1 X þ a2 X 2 þ rg Þ: ð6aÞ Model 2: rsm ¼ maxð0:01;a0 þ a1 X þ a2 X 2 þ a3 T þ a4 T 2 þ rg Þ: ð6bÞ Model 3: rsm ¼ maxð0:01;a0 þ a1 X þ a2 X 2 þ a3 T þ a4 T 2 þ a5 XT þ rg Þ: ð6cÞ Model S: switches among the volatility functions given by Models 1; 2; and 3; depending on whether the number of different option expiration dates in a given cross  section is one; two or three: X is the exercise price for the option. rg is the conditional standard deviation forecast from a first order GARCH process estimated from previous n-days of time series data. Model 1 attempts to capture variation in volatility attributable to asset price, and Models 2 and 3 capture additional variation attributable to time. Model S is introduced because some cross-sectional data have fewer expiration dates available, undermining our ability to estimate precisely the relation between the local volatility rate and time. A minimum value of the local volatility rate is imposed to prevent negative values. Quadratic forms for the volatility function are used because the BS implied volatilities for S&P500 Index options tend to have a parabolic shape. To operationalize this procedure, we first generate the conditional variance, rg, using a first order GARCH process. We then fit the BS model to the reported structure of option prices using Model S to describe the BS implied volatility. It follows that the particular functional form of rsm we select on a given day depends on the number of distinct option maturities in the sample on that day. The unknown

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coefficients are then estimated via nonlinear least squares, minimizing the sum of squared errors between the BS implied volatilities across different exercise prices (and maturities) and the model
s functional form of the implied volatility. In theory, applying the BS formula in this context is internally inconsistent because the BS formula is based on the assumption of constant volatility. Nonetheless, this procedure is a variation of what is adopted in practice as a means of predicting option prices. In estimating the semi-parametric model, we infer the current underlying index level from each cross-section of option prices simultaneously, along with the parameters of the volatility function. In this way, our empirical procedure relies only on observations from a single market, with no auxiliary assumptions of market integration. m-days ahead option prices can be computed either directly or indirectly. To generate direct option price forecasts, we can use the following ordinary least squares (OLS) projections: OLS1:

Otþm ¼ b0 þ b1 rsmt þ b2 S imt þ b3 rt þ b4 T t þ b5 X t ;

ð7Þ

where Ot+m is the S&P500 index option price (e.g. m-days ahead), rsm is the computed semi-parametric implied volatility from Model S, Sim is the implied S&P500 index level from Model S, r is the risk-free interest rate, T is the time to maturity for the option, and X is the exercise price for the option. Alternatively, m-days ahead option prices can also be computed indirectly using: OLS2:

rsmðtþmÞ ¼ b0 þ b1 rsmt þ b2 S imt þ b3 rt þ b4 T t þ b5 X t :

ð8Þ

Here, OLS is first used to project the m-days ahead implied volatility, rsm(t + m). The desired option prices corresponding to the future time period are then obtained by substituting the predicted value from Eq. (8) back into the BS model. It is worthwhile to note that the OLS1 and OLS2 models described above can be treated as a class of BS framework using OLS models to fit the skew. In other words, they can be considered in a general sense as variations of the BSS model with OLS skew fitting. With this notion in mind, we label the two OLS models as BSS-OLS1 and BSS-OLS2 throughout the paper. 2.3. Spanning polynomial projection Because option pricing models are essentially nonlinear, it is reasonable to expect that a nonlinear time-projection of the vector of independent variables to a desired target in the future may be a more accurate representation of the unknown underlying process than the OLS projection described previously. A viable method of modeling nonlinear functions is to assume that the unknown target function is the weighted sum of several simpler curved functions. Usually, there is no better choice for simple basis functions other than polynomials. According to the Weierstrass approximation theorem, any continuous function defined on a compact interval can be approximated to any degree of accuracy by a polynomial. This means that, in principle at least, it is always possible to find the desired approximating polyno-

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mial. Additional notes regarding the foundation of this technique can be found in Appendix A. A spanning polynomial projection (SPP) based on these notions can be used for least-squares fitting of nonlinear functions via polynomial regression. In concept, the SPP is similar to the OLS projections described previously. However, the spanning polynomial projection uses a mathematically more sophisticated nonlinear method to predict the desired dependent variables. First, all of the input and output data are scaled to lie as closely as possible within the (1, 1) range. To do this, let the variable
s maximum and minimum values in the estimation set be designated Vmax and Vmin respectively. The following formula is used to scale the original value V to the adjusted value A:   1  ð1Þ A¼ ð9Þ ðV  V min Þ þ Amin : V max  V min Each of the scaled input variables is then converted into a set of low-order Chebyshev spanning polynomials. Ordinary least squares is then used to fit the generated spanning polynomials to the scaled dependent variable. The predicted values from the nonlinear spanning polynomial regression is then unscaled using the following formula:   1  ð1Þ V ¼ ðA  ð1ÞÞ þ V min : ð10Þ V max  V min The spanning polynomial projection can, therefore, be thought of as a nonlinear version of the OLS projection. To make models comparable, we configure the SPP to use the same vector of independent variables as in the previous OLS projections. Specifically, we have: 2

3

O0tþm ¼ b0 þ b11 ðr0smt Þ þ b12 ð2ðr0smt Þ  1Þ þ b13 ð4ðr0smt Þ  3ðr0smt ÞÞ 4

2

2

þ b14 ð8ðr0smt Þ  8ðr0smt Þ þ 1Þ þ b21 ðS 0imt Þ þ b22 ð2ðS 0imt Þ  1Þ 3

4

2

þ b23 ð4ðS 0imt Þ  3ðS 0imt ÞÞ þ b24 ð8ðS 0imt Þ  8ðS 0imt Þ þ 1Þ þ b31 ðr0t Þ þ b32 ð2ðr0t Þ2  1Þ þ b33 ð4ðr0t Þ3  3ðr0t ÞÞ þ b34 ð8ðr0t Þ4  8ðr0t Þ2 þ 1Þ 2

3

þ b41 ðT 0t Þ þ b42 ð2ðT 0t Þ  1Þ þ b43 ð4ðT 0t Þ  3ðT 0t ÞÞ þ b44 ð8ðT 0t Þ 2

2

4

3

 8ðT 0t Þ þ 1Þ þ b51 ðX 0t Þ þ b52 ð2ðX 0t Þ  1Þ þ b53 ð4ðX 0t Þ  3ðX 0t ÞÞ 4

2

þ b54 ð8ðX 0t Þ  8ðX 0t Þ þ 1Þ;

ð11Þ

where the ( 0 ) notation denotes the adjusted values of the respective variables. 2.4. Multivariate nonparametric projection Multivariate nonparametric projection (MNP) is capable of performing a nonlinear projection and functional mapping in N-dimensional space. In marked contrast to standard parametric models, the nonparametric projection does not require the

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pre-specification of functional forms prior to estimation. Hence, this model may be superior to traditional parametric models in generating an unknown conditional mean in the absence of knowledge regarding the functional forms for the conditional mean. Also, the nonparametric model is especially useful when the unknown functional form describing the process of interest is nonlinear. Given the highly nonlinear nature of option pricing models, a detailed examination of the nonparametric projection and its nonlinear functional mapping capabilities is warranted. When we apply the nonparametric projection model to option pricing, the assumption of a particular functional form for the conditional mean is thus replaced with a less restrictive assumption that the conditional mean comes from a general class of functional forms consisting of twice-continuously differentiable functions. In the following, we briefly describe the foundation of the nonparametric projection technique adopted in this paper. Interested readers should refer to Casdagli and Eubank (1992) for a meticulous exposition of the methodology. The concept of MNP used in our empirical evaluation builds upon the generalized kernel estimation of an unknown density function f(x). This generalized kernel estimator has the form of: t 1 X f^ ðxÞ ¼ KðH 1 ðx  xi ÞÞ; tjH j i¼1

ð12Þ

where x is a p-dimensional vector, t is the number of observations, K(Æ) represents the relevant kernel function, and H is a bandwidth or smoothing parameter matrix. In words, a given arbitrary row or vector, x, from a data set is differenced from all the other xi rows of the data set with each of these differences scaled by H and assigned a probability mass via K. The scaled average of these masses is the estimate of the joint density at the given x. For this study, we set x to be the vector of input variables defined by the alternative ad hoc projection specifications. Intuitively, the kernel estimate places a scaled probability mass of size 1/n in the shape of the kernel function centered on each data point. The variable H, the smoothing factor or bandwidth, is used to scale the kernel function. Adding each of these scaled probability masses gives the kernel estimate. A more in-depth explanation of kernel can be found in Appendix A. Let f(y, x) denote the joint density of a set of random variables of interest (Y, X), where Y is a scalar random variable, X is a p-dimensional vector of inputs (explanatory factors) and let (yi, xi) be a realization of (Y, X). The density of Y conditional of (X = x) will be denoted by g(x) = f(y, x)/f1(x) where f1(x) denotes the marginal density of X. The conditional mean of Y with respect to a vector X of explanatory factors is defined as: EðY j X ¼ xÞ ¼

Z y

f ðy; x1 ; . . . ; xp Þ dy ¼ f1 ðy; x1 ; . . . ; xp Þ

Z y

f ðy; xÞ dy ¼ f1 ðxÞ

Z ygðxÞ dy:

ð13Þ

Substituting the appropriate kernel estimators and simplifying the terms yield the following estimator:

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Pt

j¼1 KðH

^ mðxÞ ¼ EðY j X ¼ xÞ ¼ Pt

1

i¼1 KðH

ðx  xi ÞÞY i

1

ðx  xi ÞÞ

;

2955

ð14Þ

where H denotes the appropriate bandwidth parameter matrix for the covariate vector, x. In our pilot study, we experiment with various methods in the selection of kernel function and bandwidth. Results suggest the use of the asymptotically optimal bandb ¼ diagðX1=2  t1=ðnþ4Þ Þ where X1/2 width for independent multivariate normal: H is the variance–covariance matrix of covariate X, t is the number of observations, and n is the number of variables. Interested readers should refer to Appendix A for more technical details and justifications. Another econometric issue pertinent to the MNP estimation is dimensionality. In our multivariate setting, this issue is basically reduced to the selection of variables in the model. In order to make our MNP model on comparable basis to the other models, we configure the MNP model with the same vector of independent variables used in the previous parametric models. Specifically, the form for general estimation is: C tþm ¼ f ðrsmt ; S imt ; rt ; T t ; X t Þ:

ð15Þ

3. Data selection and empirical methodology Our sample contains reported transaction prices of S&P500 Index options traded on the Chicago Board Options Exchange (CBOE) over the period January 1996– November 2000. This converts to 257 weeks of observations. Specifically, our study uses intra-day data, sampled every Wednesday between 2:30 P.M. and 3:15 P.M., and uses the mid-point of the bid–ask quote as the option price. Wednesdays are chosen for these estimations because fewer holidays fall on Wednesday than on any other trading days. S&P500 Index options are European-style and expire on the third Friday of the contract month. As noted earlier, we estimate the various models once each week during the sample. As many of the stocks in the S&P500 index pay dividends, one needs a time series of dividends for the index. The daily cash dividends for the S&P500 Index portfolio are collected from Datastream. For the risk free rate, we use the T-bill rates implied by the average of the bid and ask discounts reported in the Wall Street Journal. The ti-period interest rate is obtained by interpolating the rates for the two T-bills whose maturities straddle ti. Several exclusionary criteria are applied to the data. First, we eliminate options with fewer than six or more than 100 days to expiration. Second, we eliminate options whose absolute ‘‘moneyness’’, jX/F  1j, is greater than 10%, where X is the exercise price and F is the forward index level. Finally, we infer the index level simultaneously with the other parameters in the estimation procedure to lessen the imperfect synchronization problem of using both options data and underlying data in the same experiment. This allows our empirical procedure to rely only on observations from a single market, with no auxiliary assumption of market integration.

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The necessary implied coefficients are estimated once a week during the sample period by fitting Model S to the BS implied volatilities (for BSS model). We estimate the OLS (for BSS-OLS1 and BSS-OLS2), the spanning polynomial projection, and the multivariate nonparametric projection models using a 52-week rolling window regression in which the model parameters are updated weekly.

4. Empirical results Based on the S&P500 Index option data described previously, we now present the empirical results. First, we present the descriptive statistical results of our sample data. Then, we provide and discuss the prediction and projected trading results using the different models. 4.1. Descriptive statistics Our data set consists of 8915 call options and 9435 put options during the sample period. The average price of call options is 31.0385 with a standard deviation of 29.69 whereas the average price of put options is 28.1825 with a standard deviation of 25.43. Other descriptive statistics of the S&P500 index options are similar for both call and put options. The average exercise price of call options during sample period is 1071.475 (standard deviation is 288.45) while that of put options is 1020.325 (standard deviation is 278.64). The descriptive statistics with respect to the time to expiration and the T-bill rate are also similar between the call and the put options. Furthermore, using the method outlined in Bakshi et al. (2003), we computed the model-free risk-neutral skewness and kurtosis for our sample and found them to vary substantially over our sample. Specifically, the risk-neutral skewness varied from approximately 2.1 to 1.3, with an average value of approximately 0.95. The risk-neutral kurtosis varied from approximately 1.6 to 10.9, with an average of approximately 5.6. These results suggest the possibility that nonparametric time-series approaches may be better than traditional methods in modeling and capturing these time-varying effects. 4.2. Prediction and performance results To assess the quality of the various prediction models, we adopt the mean absolute percentage valuation error (MAPVE) as the primary evaluation criterion. Root mean squared valuation errors (RMSVE) for the models have also been computed. Since both criteria yield almost identical relative performance ranks for the models tested, our following discussions focus on the MAPVE results only. Technically, the MAPVE is the average of the absolute percentage deviations of the predicted option prices from the actual prices whereas the RMSVE is the square root of the average squared differences between the predicted and the actual option prices. All semi-parametric BSS-OLS and nonlinear projection models are estimated using a 52-week rolling window with model parameters updated weekly. For exam-

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ple, we use the first 52 weeks in the in-sample period to estimate the required model parameters. Then, the estimated parameters are used to project the expected option prices for the 53rd week. Likewise, the second to 53rd weeks are used to estimate the parameters that are used to project the expected option prices for the 54th week. Based on this rolling procedure, we obtain option price forecasts for the 53rd to 257th weeks (January 1997 to November 2000). The prediction errors are computed accordingly by comparing the predicted option prices with the actual option prices. Table 1 contains the MAPVEs across the 205 prediction days (one day each week) for the out-of-sample evaluation period from January 1997 to November 2000. The results are separated into three categories – call options, put options, and all options (which combine both call and put options). The average MAPVEs over the three categories are 1.99%, 1.88%, and 2.12%, respectively. The MAPVE for BSS model is 2.72%, which is the worst among all other models in the all-options category. The average MAPVE of the GARCH model is 2.02%, a better out-of-sample prediction performance than the constant volatility BSS model at the 10% significance level. Nevertheless, the two semi-parametric (BSS-OLS1 and BSS-OLS2), the SPP, and the MNP models outperform the GARCH model. All these models generate more accurate forecasts than the benchmark BSS model at the 5% significance level. Among the examined models, MNP produces the lowest MAPVE. The relative MAPVE performances as well as the statistical significances of the gaps for calloptions and put-options categories are the same as those for the all-options category, although it seems that there is a general deterioration of strength in predicting the put option prices. Tables 2 and 3 report the MAPVEs when the options are stratified by time to maturity and moneyness. Without any loss of generality, results for in-the-money options are not reported in order to avoid redundancy and overlapping of information with other tables. Most of the trading volume is concentrated on the out-of-themoney options. In all cases, the BSS model is the worst among all models included in this study. From Table 2, it can be seen that the GARCH model yields better prediction performance than the BSS model except the case where the time to maturity is over 70 days and the call options are deep out-of-the-money. 4 On the other hand, the general performance advantage of the GARCH model reduces drastically when we examine the MAPVEs for the put options, especially in the cases where the time to maturity is lengthy and the moneyness is positive. The prediction errors between the GARCH and the BSS models are not significantly different from each other. On the contrary, the BSS-OLS, SPP, and MNP models outperform the benchmark BSS model at a minimum of 10% significance level. Like the aggregate results in Table 1, the stratified results in Tables 2 and 3 suggest that the MNP is better than the others. This superiority is probably due to the MNP
s ability to incorporate more complex nonlinear relationships among its input variables and its relaxation of the more stringent parametric requirement. Despite its performance, the MNP suffers incapacity of direct interpretation of its estimated

4

Same results also hold for the deep in-the-money call options.

2958

Models

Aggregate results All options

Call options

Put options

MAPVE (%)

DMAPVE (%)

t stat

N

MAPVE (%)

DMAPVE (%)

t stat

N

MAPVE (%)

DMAPVE (%)

t stat

N

BSS GARCH BSS-OLS1 BSS-OLS2 SPP MNP

2.72 2.02 1.86 1.83 1.79 1.74

0.70 0.86 0.89 0.93 0.98

1.90* 2.33** 2.42** 2.52** 2.66**

14,655 14,655 14,655 14,655 14,655 14,655

2.55 1.94 1.75 1.74 1.69 1.60

0.61 0.80 0.81 0.86 0.95

1.76* 2.30** 2.33** 2.48** 2.73**

7230 7230 7230 7230 7230 7230

2.86 2.23 1.95 1.95 1.88 1.83

0.63 0.91 0.91 0.98 1.03

1.62* 2.34** 2.34** 2.52** 2.64**

7425 7425 7425 7425 7425 7425

Average

1.99

1.88

2.12

MAPVE is the mean absolute percentage dollar valuation errors averaged across all days in the out-of-sample period from January 1997 to November 2000. DMAPVE is the difference in MAPVE between the corresponding model and the BSS model. The t stat indicates the significance of that difference. BSS is the Black–Scholes with S skew model. GARCH is the GARCH option model. BSS-OLS1 and BSS-OLS2 are the two semi-parametric empirical projection models (using OLS fit to the skew). SPP is the spanning polynomial projection model. MNP is the multivariate nonparametric projection model. * Different from BSS at 10% significance level. ** Different from BSS at 5% significance level.

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Table 1 Mean absolute percentage valuation errors for the S&P500 Index options

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Table 2 Valuation errors for out-of-money call options with respect to moneyness and days to expiration Moneyness (%)

Lower 0

5

Models

Upper 5

10

Days to expiration Less than 40

40–70

MAPVE (%)

MAPVE (%)

BSS GARCH

2.79 1.99

BSS-OLS1

1.84

BSS-OLS2

1.81

SPP

1.81

MNP

1.68

BSS GARCH

2.81 2.00

BSS-OLS1

1.71

BSS-OLS2

1.55

SPP

1.54

MNP

1.61

DMAPVE (t stat) 0.80% (2.18)* 0.95% (2.34)* 0.98% (2.35)* 0.98% (2.35)* 1.11% (2.42)* 0.81% (2.16)* 1.10% (2.38)* 1.26% (2.41)* 1.27% (2.41)* 1.20% (2.32)*

2.50 1.92 1.78 1.93 1.91 1.62 3.37 1.96 1.72 1.73 1.70 1.59

More than 70 DMAPVE (t stat) 0.58% (1.94)** 0.72% (2.04)* 0.57% (1.93)** 0.59% (1.96)* 0.88% (2.24)* 1.41% (2.37)* 1.65% (2.49)* 1.64% (2.47)* 1.67% (2.50)* 1.78% (2.52)*

MAPVE (%) 2.62 1.99 1.85 1.81 1.81 1.68 2.30 1.93 1.80 1.90 1.89 1.58

DMAPVE (t stat) 0.63% (1.92)** 0.77% (2.01)* 0.81% (2.13)* 0.81% (2.13)* 0.94% (2.29)* 0.37% (1.51) 0.50% (1.75)** 0.40% (1.68)** 0.41% (1.66)** 0.72% (2.08)*

Notes: Moneyness is defined as X/F  1 where X is the exercise price and F is the forward index level.The numbers in parentheses indicate the corresponding t statistics for pairwise comparisons of the differences in MAPVEs between the BSS and each of the models.Without any loss of generality, results for in-themoney options are not reported here in order to avoid redundancy and overlapping of information with other tables. Most of the trading volume is concentrated at the out-of-the-money options. The best model in each category is highlighted. * Different from BSS at 5% significance level. ** Different from BSS at 10% significance level.

model parameters. This also leads to a problem for those researchers who would like to obtain the exact functional form of the estimated model. With this notion in mind, we compare the performances of MNP with SPP, another nonlinear projection model, and BSS-OLS2, its semi-parametric counterpart based on the skew fitting of BS model. From the MAPVEs in Table 1, the experimental results indicate that the general differences among the three models are marginal. The stratified results in Tables 2 and 3 confirm this observation. The finding is meaningful because it shows SPP and BSS-OLS2 can serve as not only competing alternatives but also independent verifications to the results from MNP. Further, the capability of these models in providing estimated parameters (i.e., exact functional form of the estimated model) may prove to be useful to many researchers in empirical pricing.

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Table 3 Valuation errors for out-of-money put options with respect to moneyness and days to expiration Moneyness (%)

Lower

Upper

10

5

5

0

Models

BSS

Days to expiration Less than 40

40–70

MAPVE (%)

MAPVE (%)

DMAPVE (t stat)

2.95

GARCH

1.94

BSS-OLS1

1.91

BSS-OLS2

1.82

SPP

1.81

MNP

1.81

BSS

2.99

GARCH

2.26

BSS-OLS1

2.04

BSS-OLS2

1.96

SPP

1.93

MNP

1.81

More than 70 DMAPVE (t stat)

3.24 1.01% (2.04)* 1.04% (2.03)* 1.13% (2.19)* 1.14% (2.25)* 1.14% (2.27)*

2.34 1.92 1.94 1.88 1.84

2.30 2.10 2.13 2.04 1.90

DMAPVE (t stat)

2.87 0.90% (1.79)** 1.32% (2.41)* 1.30% (2.35)* 1.36% (2.37)* 1.40% (2.50)*

2.92 0.73% (1.69)** 0.95% (2.01)* 1.03% (2.11)* 1.06% (2.12)* 1.18% (2.22)*

MAPVE (%)

2.36 2.05 2.00 1.98 1.91

0.51% (1.57) 0.82% (1.97)* 0.87% (2.01)* 0.89% (2.03)* 0.96% (2.06)*

2.79 0.62% (1.76)** 0.82% (2.08)* 0.79% (1.98)* 0.88% (2.16)* 1.02% (2.32)*

2.37 2.04 2.00 1.96 1.87

0.42% (1.45) 0.75% (2.18)* 0.79% (2.21)* 0.83% (2.27)* 0.92% (2.38)*

Notes: Moneyness is defined as X/F  1 where X is the exercise price and F is the forward index level.The numbers in parentheses indicate the corresponding t statistics for pairwise comparisons of the differences in MAPVEs between the BSS and each of the models.Without any loss of generality, results for in-themoney options are not reported here in order to avoid redundancy and overlapping of information with other tables. Most of the trading volume is concentrated at the out-of-the-money options. The best model in each category is highlighted. * Different from BSS at 5% significance level. ** Different from BSS at 10% significance level.

4.3. Projected trading results In this section, we evaluate the performances of the models based on forming a portfolio on day t and unwinding it one week later. Because our focus is on model performance and not on the issues raised by hedging and continuous rebalancing through time, we propose the following test on projected trading: freeze the model and re-price the options one week later using the model with updates for only observable variables. Specifically, the calculations examine model fit projected ahead by one week. The evaluation criterion for our projected trading test is the mean absolute percentage trading error (MAPTE), an adaptation of the MAPVE criterion used in the previous section. MAPTE is a percentage measured in dollar terms weighted by the price of the underlying option.

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Table 4 contains a summary of the results. It can be seen that the average projected trading errors for all-options, call-options, and put-options categories are 1.55%, 1.49%, and 1.60%, respectively. BSS is still the worst performer. However, unlike the results with respect to valuation error, the projected trading errors of GARCH and BSS-OLS1 models are not significantly different from those of the benchmark BSS model. With the exception of the call options, the projected trading guided by BSS-OLS2 cannot generate an error level significantly lower than that by the BSS. On the other hand, the two statistical projection models perform reasonably well relative to others. Excluding the put options, both MNP and SPP creates a smaller projection error than BSS at 5% significance level. For the put-options category, MNP yields a smaller average projection error than BSS at 10% significance level whereas SPP is about on the same error level as BSS. It is interesting to point out that, by comparing the results in Tables 1 and 4, the difference between the average valuation errors of call and put options is much larger than the difference between the average projection errors of call and put options. Further, the percentage differences across the various models are much smaller for the projection error than for the valuation error. This observation is true for both call and put options categories. We suspect that this outcome is possibly a result of a disproportionate deterioration of the statistical projection models and/or an improvement of the BSS and the GARCH models in trading guidance. Our conjecture is partially substantiated by the relatively poor performance of the projection models in the put-options category in Table 4.

Table 4 Mean absolute percentage projected trading errors for the S&P500 Index options Models

Aggregate results All options

Call options

Put options

MAPTE DMAPTE t stat (%) (%)

MAPTE DMAPTE t stat (%) (%)

MAPTE DMAPTE t stat (%) (%)

BSS GARCH BSS-OLS1 BSS-OLS2 SPP MNP

1.59 1.56 1.58 1.54 1.52 1.50

1.58 1.51 1.49 1.47 1.45 1.42

1.61 1.60 1.64 1.61 1.58 1.56

Average

1.55

0.03 0.01 0.05 0.07 0.09

0.86 0.29 1.43 2.01* 2.58*

1.49

0.07 0.09 0.11 0.13 0.16

1.26 1.63 1.99* 2.35* 2.89*

0.01 0.03 0.00 0.03 0.05

0.36 1.09 0.00 1.09 1.81**

1.60

MAPTE is the mean absolute percentage projected trading errors averaged across all days in the out-ofsample period from January 1996 to November 2000. It is measured in dollar terms weighted by the price of the underlying option. DMAPTE is the difference in MAPTE between the corresponding model and the BSS model. The t stat indicates the significance of that difference. BSS is the Black–Scholes with S skew model. GARCH is the GARCH option model. BSS-OLS1 and BSS-OLS2 are the two semi-parametric empirical projection models (using OLS fit to the skew). SPP is the spanning polynomial projection model. MNP is the multivariate nonparametric projection model. * Different from BSS at 5% significance level. ** Different from BSS at 10% significance level.

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Tables 5 and 6 tabulate the results with respect to the time to maturity and the moneyness factors. Overall ranks of model performances are very close to those reported in Tables 2 and 3. However, the gaps among the MAPTEs of the various models are much narrower, especially for the put options. Focusing on the calloptions category, MNP is the best performer except in the cases where moneyness is more than 10%. When moneyness is between 5% and 10%, SPP performs marginally better than MNP although both statistical projection models generate lower average percentage error than the BSS model at 5% significance level. The GARCH and BSS-OLS1 models obtain statistically smaller errors than the benchmark BSS model in some cases. The BSS-OLS2 model provide better results than the GRACH and BSS-OLS1 models but its good performance is not as consistent and versatile as Table 5 Projected trading errors for out-of-money call options with respect to moneyness and days to expiration Moneyness (%)

Lower 0

5

Models

Upper 5

10

BSS

Days to expiration Less than 40

40–70

MAPTE (%)

MAPTE (%)

DMAPTE (t stat)

1.56

GARCH

1.49

BSS-OLS1

1.49

BSS-OLS2

1.46

SPP

1.47

MNP

1.45

BSS

1.58

GARCH

1.53

BSS-OLS1

1.58

BSS-OLS2

1.52

SPP

1.46

MNP

1.46

More than 70 DMAPTE (t stat)

1.58 0.07% (1.68)* 0.07% (1.68)* 0.10% (2.16)** 0.09% (2.02)** 0.11% (2.23)**

1.51 1.51 1.46 1.43 1.40

1.50 1.48 1.47 1.42 1.43

DMAPTE (t stat)

1.59 0.07% (1.41) 0.07% (1.42) 0.12% (1.75)* 0.15% (1.98)** 0.18% (2.11)**

1.60 0.05% (1.16) 0.00% (0.00) 0.06% (1.39) 0.12% (2.32)** 0.12% (2.32)**

MAPTE (%)

1.49 1.47 1.46 1.44 1.41

0.10% (1.80)* 0.12% (2.06)** 0.13% (2.13)** 0.15% (2.24)** 0.18% (2.31)**

1.59 0.10% (1.66)* 0.11% (1.79)* 0.13% (2.00)** 0.18% (2.27)** 0.17% (2.19)**

1.45 1.46 1.46 1.43 1.45

0.14% (2.32)** 0.13% (2.21)** 0.13% (2.32)** 0.16% (2.45)** 0.14% (2.32)**

Notes: Moneyness is defined as X/F  1 where X is the exercise price and F is the forward index level.The numbers in parentheses indicate the corresponding t statistics for pairwise comparisons of the differences in MAPTEs between the BSS and each of the models.Without any loss of generality, results for in-themoney options are not reported here in order to avoid redundancy and overlapping of information with other tables. Most of the trading volume is concentrated at the out-of-the-money options. The best model in each category is highlighted. * Different from BSS at 10% significance level. ** Different from BSS at 5% significance level.

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Table 6 Projected trading errors for out-of-money put options with respect to moneyness and days to expiration Moneyness (%)

Lower

Upper

10

5

5

0

Models

BSS

Days to expiration Less than 40

40–70

MAPTE (%)

MAPTE (%)

DMAPTE (t stat)

1.60

GARCH

1.59

BSS-OLS1

1.61

BSS-OLS2

1.57

SPP

1.57

MNP

1.55

BSS

1.60

GARCH

1.59

BSS-OLS1

1.62

BSS-OLS2

1.58

SPP

1.60

MNP

1.57

More than 70 DMAPTE (t stat)

1.59 0.01% (0.71) 0.01% (0.64) 0.03% (1.32) 0.03% (1.37) 0.05% (2.02)**

1.61 1.63 1.60 1.56 1.56

1.61 1.64 1.61 1.59 1.57

DMAPTE (t stat)

1.63 0.02% (1.04) 0.04% (1.37) 0.01% (0.56) 0.03% (1.31) 0.03% (1.39)

1.61 0.01% (0.88) 0.02% (1.18) 0.02% (1.33) 0.00% (0.00) 0.03% (1.72)*

MAPTE (%)

1.60 1.62 1.61 1.61 1.58

0.03% (1.76)* 0.01% (0.73) 0.02% (1.04) 0.02% (1.25) 0.05% (2.04)**

1.62 0.00% (0.00) 0.03% (1.47) 0.00% (0.00) 0.02% (1.16) 0.04% (1.91)*

1.64 1.66 1.65 1.60 1.59

0.02% (0.76) 0.04% (1.58) 0.03% (1.04) 0.02% (0.86) 0.03% (1.13)

Notes: Moneyness is defined as X/F  1 where X is the exercise price and F is the forward index level.The numbers in parentheses indicate the corresponding t statistics for pairwise comparisons of the differences in MAPTEs between the BSS and each of the models.Without any loss of generality, results for in-themoney options are not reported here in order to avoid redundancy and overlapping of information with other tables. Most of the trading volume is concentrated at the out-of-the-money options. The best model in each category is highlighted. * Different from BSS at 10% significance level. ** Different from BSS at 5% significance level.

the two nonlinear projection models. Also, it does not seem like there exist any patterns in describing possible interactions with the moneyness and/or the time to maturity factors. These experimental findings are roughly the same between outof-the-money and in-the-money options (not reported here). The performance rank of each model is about the same for the call and put options categories. Nevertheless, the significance of the difference between a model
s MAPTE and that of the BSS model is much lower. For most models including SPP, the average percentage errors are not significantly different from the BSS model. We can also observe a fall in the performance of the MNP model as some of its errors are not significantly different from the benchmark BSS
s or are significant at only 10% level (compared with 5% significance for the call options). Given

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the closeness in performances of the two statistical projection models, it is likely that these nonlinear models may actually complement with each other and a synergy of two may enhance the performance in more versatile settings.

5. Conclusions This paper applies several newer econometric techniques to the option pricing problem and compares their out-of-sample valuation and projected trading performances to the BS with S skew and the GARCH models. We find the following conclusions. First, both the semi-parametric and nonlinear econometric models offer better pricing performance than the BSS model. Even when the models are benchmarked against the GARCH option model, which can incorporate time series information, the nonparametric statistical projection models show better performance. This is consistent with our hypothesis that accounting for nonlinear time series information is important in option valuation and projected trading. Second, for several option classifications, even linear OLS projections of observable inputs via semiparametric specifications (in the BS fitted with skew framework) can outperform the BSS and GARCH option models. Third, models based on nonlinear projections of observable inputs perform the best overall across categories. This is consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. Finally, the various econometric option models analyzed in study should be of interest to not only academic researchers but also practitioners, who may not be so interested in the mathematical arguments underlying option pricing but rather in the out-of-sample performance and implementability. The econometric option models presented in this paper should prove useful and complement mainstream mathematical modeling methods.

Acknowledgments The authors wish to thank Ji-Ming Ho of National Chung Cheng University for his exploratory computer work and data collection. Also, the authors appreciate the encouraging support from Jerry Keating and Ram Tripathi of University of Texas and the valuable comments from Hazem Daouk of Cornell University. Any errors in this paper are our own. Partial research support from Taiwan National Science Council (Grant number NSC93-2416-H-194-025) is gratefully acknowledged.

Appendix A A.1. Additional explanation of the GARCH option model In the equations presented in Section 2.1, S(t) denotes the underlying asset price at time t and is assumed to follow the following process over time steps of length D:

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logðSðtÞÞ ¼ logðSðt  DÞÞ þ r þ khðtÞ þ hðtÞ ¼ - þ

p X

bi hðt  iDÞ þ

i¼1

pffiffiffiffiffiffiffiffi hðtÞzðtÞ;

q  X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ai zðt  iDÞ  ci hðt  iDÞ ;

2965

ðA:1Þ ðA:2Þ

i¼1

where r is the continuously compounded interest rate for the time interval D and z(t) is a standard normal disturbance. h(t) is the conditional variance of the log returns between t  D and t and is known from the information set at time t  D. The conditional variance h(t) appears in the mean as a return premium. This allows the average spot return to depend on the level of risk. k is the risk premium parameter and the functional form of this risk premium, kh(t), prevents arbitrage by ensuring that the spot asset earns the riskless interest rate when the variance equals zero. Eq. (A.1) assumes that the expected spot return exceeds the riskless rate by an amount proportional to the variance h(t). Since volatility equals the square root of h(t), this implies the return premium per unit of risk is also proportional to the square root of h(t), as in the Cox et al. (1985) model. In particular limiting cases the variance becomes constant. As the ai and bi parameters approach zero, it is equivalent to the BS model observed at discrete intervals. In this model, one can directly observe h(t + D), at time t, as a function of the spot price as follows: 2

hðt þ DÞ ¼ - þ b1 hðtÞ þ a1

ðlogðSðtÞÞ  logðSðt  DÞÞ  r  khðtÞ  c1 hðtÞÞ : hðtÞ ðA:3Þ

a1 determines the kurtosis of the distribution. c1 controls the skewness or the asymmetry of the distribution of the log-returns. This paper will focus on the first-order case (p = q = 1). The first-order GARCH process remains stationary with finite mean and variance if b1 þ a1 c21 < 1. The variance process h(t) and the spot return are correlated as follows: CovtD ½hðt þ D; logðSðtÞÞ ¼ 2a1 c1 hðtÞ:

ðA:4Þ

Given a positive a1, a positive value for c1 results in negative correlation between spot returns and variance. This is consistent with the postulate of Black (1976) and the leverage effect documented by Christie (1982) and others. A.2. Additional notes on spanning polynomial projection Technically speaking, it is possible to model the training data to any desired accuracy by finding a high-degree polynomial that passes through a large number of points in the training set, and hope that the determined polynomial function will resemble the data that are not in the training set. Unfortunately, this approach may not work well if the objective is out-of-sample projection or forecasting. In other words, it is generally the case that the more accurately fit a high-degree polynomial is to a training set, the worse the fit becomes out-of-sample (in areas that are

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not included in the training set). Moreover, a theoretical n-dimensional hyperplane plot of the polynomial found this way would almost invariably go wildly off scale, shooting out the top of the plot between one pair of training points, then plunging into the ground between the next pair. Of course, the polynomial dutifully passes through each and every point that went into its definition but is virtually worthless everywhere else. To circumvent the problem of high-order polynomial generalizability for out-ofsample data, instead of using a high-order polynomial to exactly fit the set of representative training points, a set of low-order polynomials can be computed from the independent variable for each training point to provide a least-squares fit to the training points. Selective use of relatively low-order polynomials can ensure smooth interpolation across the domain. These low-order polynomials, however, should be relatively independent of each other, that is, polynomials that behave very differently within some reasonable domain. No matter what the value of the independent vector is within that domain, we hope that when the independent vector changes, some of the polynomials go up and some go down. Furthermore, we want this pattern to change throughout the domain. The resulting variety of extracted features is required to induce stability. Technically, each of the scaled input variables is converted into a set of low-order Chebyshev spanning polynomials. The first k spanning polynomials can be computed from the following recursive formula: T k ðAÞ ¼ 2AT k1 ðAÞ  T k2 ðAÞ;

ðA:5Þ

where T0(A) = A and T1(A) = 2A2  1. A.3. Additional notes on multivariate statistical projection Heuristically, applying the kernel estimator is analogous to constructing a histogram where H is the bin width of the histogram. The kernel estimate can therefore be thought of as the limiting case of an averaged shifted histogram where the kernel is actually centered on the data, whereas a standard histogram places the data in a rigid grid. On the other hand, one can think of a kernel estimate in terms of a standardized distance between a point and each data point which is then converted into a probability based on this distance. Points close to the data receive relatively more probability mass than points farther away. The accuracy of the MNP estimation primarily depends on the kernel function K(Æ) and the bandwidth H. Theoretical studies by Scott (1992) and Hardle (1990) have shown that the choice of kernel function is not nearly as important as the choice of bandwidth. The kernel density estimate does inherit the properties of the kernel, but due to the high frequency if the averaging process, various kernel functions can be shown to be almost equivalent. Despite this rather general finding, Scott (1992) and Hardle (1990) still recommend the use of a smooth and uni-modal kernel for estimation. This study uses the product Epanechnikov kernel for the kernel function K(Æ). The Epanechnikov kernel is the optimal kernel based on a calculus of variations

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solution to minimize the integrated mean square error of the kernel estimator. The multivariate Epanechnikov kernel is given by the formula: p Y

KðzÞ ¼

Kðzj Þ

ðA:6Þ

j¼1

for z = (z1, . . . , zp) where   8 3 1 2 < p ffiffiffi 1  zj 5 Kðzj Þ ¼ 4 5 : 0

if z2j < 5:0

:

otherwise

As mentioned earlier, the choice of bandwidth is far more critical than the choice of the underlying kernel function. Casdagli and Eubank (1992, p. 77) stated that models for MNP estimation ‘‘can be very sensitive to the actual choice of widths, and there are no simple rules for determining optimal values’’. In our pilot study, we experiment with various methods in the selection of bandwidth and find out the asymptotically optimal bandwidth for independent multivariate performs reasonably well, a finding which is in general agreement with Scott
s study (1992). The expression of the asymptotically optimal bandwidth for independent multivariate normal is: b ¼ diagðX1=2  t1=ðnþ4Þ Þ; H

ðA:7Þ

1/2

where X is the variance–covariance matrix of covariate X, t is the number of observations, and n is the number of variables. A.4. Additional notes on computing t statistics The t-statistics in the tables were computed in a GMM setting. Specifically, we let ft be the (2 · 1) vector of mean absolute percentage prediction errors at time t corresponding to the pair of option pricing models to be compared. Let Ef be the population value, and let f be the sample average. Then, if we know the population values for all the parameters, f  Ef is asymptotically normal with the variance– covariance matrix: S ff ¼

1 X

Eðft  Ef Þðftj  Ef Þ0 :

ðA:8Þ

j¼1

On the basis of this observation and a generalized method of moments reasoning, we determine m, a (2 · 1) constant vector, such that X minm ðft  mÞ0 Xðft  mÞ; ðA:9Þ t

where X is the Newey–West heteroskedasticity-consistent, (2 · 2) variance–covariance matrix. In this way, we obtain asymptotic t-statistics for the mean absolute prediction errors.

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