American option valuation: Implied calibration of GARCH pricing models

We thank Mark Rubinstein, Lars Stentoft, and participants of the 2010 FMA European Meeting, Hamburg,for valuable comments.

*Correspondence author, ICMA Centre–Henley Business School, University of Reading, WhiteknightsReading, RG6 6BA, UK. Tel: �44 (0)118 3784389, Fax: �44 (0)118 9314741, e-mail: [email protected]

Received October 2009; Accepted September 2010

� Michael Weber is with the Haas School of Business, University of California at Berkeley,Berkeley, California.

� Marcel Prokopczuk is with the ICMA Centre, Henley Business School, University of Reading,Whiteknights, Reading, United Kingdom.

The Journal of Futures Markets, Vol. 31, No. 10, 971–994 (2011)© 2010 Wiley Periodicals, Inc.Published online November 8, 2010 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/fut.20496

AMERICAN OPTION VALUATION:IMPLIED CALIBRATION OF

GARCH PRICING MODELS

MICHAEL WEBERMARCEL PROKOPCZUK*

This study analyzes the issue of American option valuation when the underlyingexhibits a GARCH-type volatility process. We propose the usage of Rubinstein’sEdgeworth binomial tree (EBT) in contrast to simulation-based methods beingconsidered in previous studies. The EBT-based valuation approach makes animplied calibration of the pricing model feasible. By empirically analyzing thepricing performance of American index and equity options, we illustrate the supe-riority of the proposed approach. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:971–994, 2011

INTRODUCTION

Following the seminal work of Engle (1982) and Bollerslev (1986), GARCH-based financial time series models have been shown to be capable of capturingmany stylized facts of financial time series, mainly the leptokurtosis of asset

972 Weber and Prokopczuk

Journal of Futures Markets DOI: 10.1002/fut

returns and volatility clustering.1 Moreover, motivated by the weak empiricalperformance of the Black–Scholes–Merton option pricing formula, producingthe well-known volatility smile phenomenon, GARCH models have been suc-cessfully introduced for option pricing. These approaches primarily build onthe local risk-neutral valuation principle of Rubinstein (1976) and Brennan(1979), which was later generalized for a GARCH framework by Duan (1995).

Numerous empirical studies analyzing the pricing performance ofEuropean options for various GARCH specifications exist.2 However, when itcomes to the option pricing of underlyings with GARCH volatility, one usuallyhas to rely on simulation-based valuation approaches.3 If the options underconsideration are of the American type, a simulation-based approach is mademore complicated by the fact that an optimal exercise policy has to be deter-mined simultaneously throughout the option’s life. As a result, almost all stud-ies have only considered European options, although most exchange-tradedoptions are American and, thus, exercisable early.

The first study we are aware of that considered the pricing of American-styleoptions in a GARCH volatility framework is Stentoft (2005). Relying on historicalmaximum-likelihood parameter estimates and the least squares Monte Carlo(LSM) method developed by Longstaff and Schwartz (2001) to approximate theearly exercise premium, Stentoft considered the pricing of (American) S&P 100index options (OEX) and stock options. The major disadvantage of this approachcomes from the fact that only historical information is used for parameter esti-mation, as an implied calibration of the options pricing model is not feasible in reasonable computer time for most real-world applications when relying onsimulation-based approaches. This is unfortunate, as it is well known that usingoption-implied parameter estimates improves the pricing performance significantly.4

In this study, we therefore empirically investigate the potential of impliedcalibration of GARCH options pricing models when dealing with Americanoptions. To the best of our knowledge, no empirical study on this issue cur-rently exists. To accomplish this goal, we apply an option valuation techniquebased on Edgeworth binomial trees (EBTs), proposed by Rubinstein (1998)and Duan, Gauthier, Sasseville, and Simonato (2003). This approach, restingon the ideas presented by Rubinstein (1994) and an Edgeworth expansion,allows us to build a recombining tree under general distribution functions.1See, for example, Bollerslev, Chou, and Kroner (1992) for an excellent overview.2Bollerslev and Mikkelsen (1999), Heston and Nandi (2000), Christoffersen and Jacobs (2004), Hsieh andRitchken (2005), Christoffersen, Heston, and Jacobs (2006), Duan, Ritchken, and Sun (2006b), Badescu,Kulperger, and Lazar (2008), and Christoffersen, Jacobs, and Ornthanalai (2008) have illustrated thatGARCH models are well capable of pricing (European) S&P 500 index options. Studies for other marketsinclude Härdle and Hafner (2000) and Lehnert (2003) for DAX index options, Duan and Zhang (2001) forHang Seng index options, and Lehar, Scheicher, and Schittenkopf (2002) for FTSE 100 index options.3One exception is the closed-form valuation formula for European options by Heston and Nandi (2000).4See, for example, Bakshi, Cao, and Chen (1997), Bates (1996), and also Barone-Adesi, Engle, and Mancini(2008).

American Option Valuation 973


Duan et al. (2003) demonstrated how to adopt this approach in a GARCHframework, but did not test the model empirically.

Thus, the main contribution of our study is twofold. First, we are the first toempirically analyze the pricing performance of an implicitly calibrated GARCHoptions pricing model when dealing with American options. As a benchmark, weconsider the historical LSM approach suggested by Stentoft (2005). Second, we study the empirical option valuation performance of Rubinstein’s Edgeworthtree in a GARCH volatility framework when pricing American-style index andstock options, which has not been conducted previously.

In the empirical study, we implement an NGARCH-based option pricingmodel for S&P 100 index options and options written on General Electric (GE)stocks.5 We estimate the structural parameters and value options in two differentways. (i) by employing the historical maximum-likelihood approach only usingasset returns and valuing options by the LSM algorithm. This procedure was sug-gested by Stentoft (2005) when valuing American options under GARCH volatili-ty and serves as a benchmark in our study. (ii) we implicitly estimate the GARCHparameters directly under the risk-neutral measure by valuing the options byemploying Rubinstein’s EBT using options prices and asset returns. The two con-sidered approaches thus differ with respect to the pricing algorithm and the esti-mation methodology. Our results show that the latter approach yields a superiorpricing performance, which can be mainly attributed to the possibility of animplied calibration.6 However, as noted above, this estimation approach becomesinfeasible when relying on simulation-based methods. We therefore provide thefirst empirical evidence on the suitability of the EBT-based pricing approach.

The rest of the study is organized as follows. The second section intro-duces GARCH option pricing in the European case, briefly describes the LSM-based pricing of American options, and introduces the pricing by EBTs. Thethird section introduces the data set, while the fourth section describes theestimation approaches. The fifth section provides the results of our empiricalstudy. The sixth section concludes.

GARCH OPTION PRICING

GARCH Volatility Dynamics and the Pricing ofEuropean Options

GARCH volatility models were developed by Engle (1982) and Bollerslev(1986). The pricing of options in a GARCH volatility framework usually buildson the local risk-neutral valuation relationship suggested by Duan (1995).

5As a robustness check, we also consider a GJR-GARCH dynamics and more underlyings.6Numerical tests showed that the two pricing approaches yield very similar precision when employing thesame set of parameters.



There exists extensive empirical evidence of negative return innovationshaving a stronger impact on conditional volatility than positive shocks of a sim-ilar magnitude. To capture this stylized fact, Engle and Ng (1993) proposed anasymmetric, but parsimonious extension of the standard GARCH dynamics.This type of GARCH model, frequently labeled NGARCH in the literature, hasbeen proven to perform well for pricing equity index options, as demonstratedby Christoffersen and Jacobs (2004) and Hsieh and Ritchken (2005).Therefore, we follow their approach and consider an NGARCH-type volatilityprocess.

The logarithm of asset returns under the data-generating probability meas-ure is assumed to follow the dynamics

(1)

(2)

where r and q denote the one-period, continuously compounded risk-free rateand dividend yield, respectively; ht represents the conditional variance of theasset returns and Pt represents a standard normal random variable. Conditionalon the information set �t�1 available at time t � 1 the log-normality impliesthat .

Hence, we can interpret l as the constant unit risk premium. The condi-tional variance follows an NGARCH process. We impose the typical restric-tions b0 � 0, b1 � 0, and b2 � 0 to guarantee a positive unconditional volatility.The parameter u determines the so-called leverage effect: a positive value of uinduces a negative correlation between the asset returns and the conditionalvolatility. Duan (1995), generalizing the results of Rubinstein (1976) andBrennan (1979), derived the locally risk-neutral valuation relationship whendealing with GARCH volatility dynamics. This is satisfied by a risk-neutralmeasure , if

(3)

(4)

Under the measure , we can derive the risk-neutral asset return process as:

(5)ln a St

St�1b � r � q �

12

ht � 2htet

�

Var� c lna St

St�1b ` �t�1 d �

a.s.Var� c lna St

St�1b ` �t�1d .

E� c St

St�1` �t�1 d � er�q

�

E�[St�St�1 ƒ �t�1] � er�q�l2ht

ht � b0 � b1ht�1 � b2ht�1(Pt�1 � u)2

lna St

St�1b � r � q � l2ht �

12

ht � 2htPt

�



(6)

where �t � Pt � l is a standard normal random variable under the locally risk-neutral measure .

When pricing (European) options in a GARCH-style framework, it is com-mon practice to rely on simulation-based valuation approaches. This is due tothe fact that, on the one hand, the final risk-neutral distribution is not knownin closed form, but, on the other hand, the discrete nature of the GARCHframework makes simulation-based approaches straightforward to implement.Using Equations (5) and (6), one can easily generate random draws of futurerisk-neutral stock prices. The value of an European option is then computed asthe mean of the discounted option payoff at maturity.

Pricing American Options with GARCH Volatility by Simulation

The valuation of American options is not as straightforward as the valuation ofEuropean options. Owing to the American feature, one needs not only the risk-neutral distribution of the underlying at maturity but also has to simultaneous-ly determine an optimal early exercise strategy. This boils down to comparingthe value of immediate exercise with the continuation value of the option atevery possible exercise point. Stentoft (2005) has suggested applying the LSMmethodology developed by Longstaff and Schwartz (2001) to value Americanoptions in the case of the underlying features a GARCH volatility process. Thisapproach has the appealing characteristic that it builds on the simulationapproach and is implemented by adding simple OLS regressions. Starting atthe maturity of the option contract and working backwards, one estimates thepath-specific continuation value at every possible exercise point by regressingdiscounted cash-flows on some basis functions of the current values of theunderlying and the volatility levels. Having approximated the continuationvalue, one can easily decide whether it is optimal to exercise the option or not.7

In our empirical implementation of the LSM algorithm, we use a total of 100,000 asset paths and assume one exercise possibility per day. We considerthe first three Laguerre polynomials for the underlying and the volatility, includ-ing the respective cross-terms.8 We model the local variance starting at the uncon-ditional variance implied by the maximum-likelihood-based GARCH parameters252 trading days before the valuation date and model the evolution of the

�

ht � b0 � b1ht�1 � b2ht�1(et�1 � u � l)2

7For a more detailed presentation of the algorithm and an analysis regarding different parameters, such asthe number of simulation paths or different kinds of basis functions, we refer to Longstaff and Schwartz(2001), Moreno and Navas (2003), and Stentoft (2004).8This choice has been proven to yield good results, see, for example, Longstaff and Schwartz (2001) andMoreno and Navas (2003).



conditional volatility, according to the proposed GARCH dynamics. Furthermore,we make use of antithetic random numbers and the empirical martingale methodof Duan and Simonato (1998) to enhance the computational efficiency.

Pricing American Options with GARCH Volatility by Binomial Trees

When working in a constant volatility environment, it is common practice to applylattice-based approaches for valuing American options. These are used becausethey can easily determine the optimal exercise strategy. However, the inherentpath dependence of GARCH processes implies that the standard lattice-based approaches grow exponentially in the number of discrete time steps.

Building on the ideas of Jarrow and Rudd (1982), Rubinstein (1998) sug-gested the usage of an Edgeworth expansion to construct a so-called EBT for thepricing of American options. This method mitigates the drawback of exponential-ly growing complexity. The density of a standard binomial distribution is trans-formed into a density with a mean of zero, a variance of one, and the desiredskewness and kurtosis. Thus, only the first four moments of the cumulative risk-neutral asset returns are necessary, which can be analytically approximated.

Based on this modified density, Rubinstein (1998) suggested modeling theentire stochastic return process using the implied binomial tree approach pre-sented in Rubinstein (1994).

In the following, we briefly describe the tree-building procedure for theEBT.9 Consider an underlying with price St and an option with maturity date T;the time to maturity is denoted t � T � t; the cumulative return and annual-ized volatility are given by RT � ln(ST �St) and . Thus, thestandardized return for the option’s maturity (in years) under the risk-neutralmeasure is given by .

In a first step, we generate an n-step standardized binomial density with n + 1 possible values yj and corresponding probabilities b(yj), according to

and b(yj) � (n!/(j!(n � j)!))(1�2)n, where j goes from 0 to n.In a second step, we transform this density, given the prespecified skewnessand kurtosis, into a density f(y) with desired third and fourth moments usingthe Edgeworth expansion:

(7)�

172

k23(y6 � 15y4 � 45y2 � 15) d b(y)

f(y) � c1 �16

k3(y3 � 3y) �124

(k4 � 3)(y4 � 6y2 � 3)

yj � ((2j) � n)�2n

zt � (RT � E�[RT])�s2t

s � 2Var[RT]�t

9See Rubinstein (1998), Duan et al. (2003) or also Staunton (2007).



where denotes the skewness and denotes the kurtosisof the cumulative standardized return for the options’s maturity (in years).

As the Edgeworth expansion is only an approximation, we have to re-scalethe probabilities:

(8)

to ensure that they sum to unity. The variable y is then no longer binomially dis-tributed. Standardizing this variable by subtracting mean anddividing by standard deviation yields a random variable xwith mean zero and variance one.10

Now, we can obtain the value of the underlying in the last time step andnode j as

(9)

with , where rann and qann denote the continu-ously compounded risk-free rate and dividend yield on an annual basis, respec-tively. The expected risk-neutral return equals the risk-free rate reduced forpossible dividend payments. Hence, m ensures risk-neutrality (see Rubinstein,1998).

Finally, one constructs the entire stochastic process working backwardsfrom the maturity of the option. Following Rubinstein (1998), we calculate theprobabilities pj of a single path to node j as

(10)

Then, invoking the principle of no-arbitrage, one can calculate the proba-bility and asset price pair (p, S) for the preceding node:

(11)

(12)

(13)S � [(1 � pu)Sj � puSj�1]e� (r�q) tn.

pu �pj�1

p

p � pj � pj�1

pj �Pj

a n!j!(n � j)!

b.

m � rann� qann � 1t lng

nj�0 Pje

s2txj

ST,j � Stemt�s2txj

V � 2gj Pj(yj � M)2

M � gj Pjyj

Pj �f(yj)

ajf(yj)

k4 � E�[z4t]k3 � E�[z3

t]

10As pointed out by Rubinstein (1998), another problem that can arise is the fact that f(y) may not be non-negative everywhere, and thus not a valid density function. This problem increases with the maturity of theoption. However, he also demonstrates that for maturities of up to 100 days, there exists a sufficiently largenumber of possible values for the third and fourth moments, where this problem does not arise. As we onlyconsider options with a maturity of up to 100 days, this issue is consequently of minor importance for ourstudy.



Following this recursive algorithm, we can construct the entire tree, whichprecludes arbitrage, as shown by Rubinstein (1994).

Finally, to be able to construct the EBT as described previously, one has tospecify the first four moments of the cumulative return distribution. Duan,Gauthier, and Simonato (1999), as well as Duan, Gauthier, Simonato, andSasseville (2006a), derived analytical expressions for GARCH, NGARCH,GJR-GARCH, and EGARCH processes. These moments can be calculated asfollows:

(14)

for T � {1, 2, . . .} and k � {1, 2, 3, 4}, where T is the number of discrete timesteps. We obtain the required moments by expanding the right-hand side ofEquation (14) and applying the expectation operator to the resulting terms. As theresulting formulas are algebraically cumbersome, we refrain from presentingthem in the study.11

RETURN AND OPTION DATA

The data used in the main empirical study include options written on the S&P100 index (OEX) as well as on the stocks of GE. The S&P 100 index optionsare an obvious choice, as they are one of the most liquid contracts available. In addition, they are the natural American counterpart to the options on theS&P 500, which is the subject of many empirical studies regarding the pricingperformance for European option contracts. GE was chosen, as it is a major UScompany, and options written on its stock belong to the options on dividendpaying underlings traded most heavily during our option sample period.

The return series of the two underlyings was sampled for the period fromJuly 2, 1991 to December 31, 2008, yielding a total of 4,412 observations foreach series. Table I provides an overview of diverse descriptive statistics andselected diagnostic tests. The table shows that GE has a substantially higherannualized volatility compared with OEX, whereas the skewness and kurtosisindicate that the index as well as the individual stock returns are slightly leftskewed and exhibit a high excess kurtosis. Not surprisingly, primarily due to thehigh kurtosis, the null hypothesis that the returns follow a Gaussian distribu-tion is rejected by the Jarque–Bera test of normality at any reasonable signifi-cance level.

The Ljung-Box statistic testing the hypothesis of zero autocorrelation upto lag 20 suggests, both for the returns and the squared returns, the existence

E�[RkT ƒ �0] � E� caT(r � q) �

12 a

T

s�1 hs � a

T

s�1 2hsesb

k d

11These formulas can be found in the appendices of Duan, Gauthier, and Simonato (1999) and Duan et al.(2006a).



of significant autocorrelation in the first and second moments of the returnprocesses. Besides, considering the ARCH-LM test statistic, we can reject thenull of independent and identically distributed return innovations for bothseries. This implies the existence of GARCH effects.

Fitting an NGARCH model according to Equation (2) to the two timeseries and repeating the Ljung–Box test for the squared residuals yields clearevidence that such a model seems to be appropriate to adequately capture thedependence structure of the return series. The null of independence cannot berejected with p-values of 0.82 and 0.58 (see line ).

The empirical study is conducted using 3 years of option data for optionstraded on the Chicago Board of Options Exchanges. The data were sampledweekly on Wednesdays for the period of January 2006 through December2008. When a particular Wednesday was a holiday, we used the following trad-ing day. The weekly sampling frequency enables us to analyze a comparablylong data sample in a reasonable amount of computer time. Moreover, follow-ing Dumas, Fleming, and Whaley (1998), such a procedure stands in the tradi-tion of many other empirical studies on option pricing (e.g., Christoffersen andJacobs, 2004; Stentoft, 2005, among others). We chose Wednesdays for ourstudy, as fewer holidays occur on a Wednesday; in our sample period, only oneholiday fell on Wednesday. For further possible advantages of this approach,refer to Dumas, Fleming, and Whaley (1998).

Q̂2(20)

TABLE I

Sample Statistics for Return Series

OEX GE

Statistic Estimate p-value Estimate p-value

m 1.9846E–04 – 0.1497E–04 –s ann 0.1865 – 0.2803 –Min �0.0919 – �0.1368 –Max 0.1066 – 0.1276 –Skewness �0.1919 – �0.1388 –Kurtosis 12.2235 – 10.1708 –JB 15,666.2687 [0.0000] 9466.9751 [0.0000]Q(20) 119.3203 [0.0000] 72.0299 [0.0000]Q2(20) 5,880.1405 [0.0000] 3836.1590 [0.0000]

14.2171 [0.8193] 18.1822 [0.5754]LM(5) 963.8075 [0.0000] 699.4798 [0.0000]

This table reports descriptive statistics for the daily continuously compounded returns for the S&P 100 (OEX) and General Electric(GE). The sample period is January 2, 1991 to December 31, 2008 for a total of N � 4,412 observations. For the Jarque-Bera statis-tic (JB), the brackets right of the statistic show the p-value for the significance of the difference of the third and fourth empiricalmoments and their theoretical values from the Gaussian distribution. Q (20) reports the value of the Ljung–Box test statistic for up to20th order autocorrelation in the returns, whereas Q 2(20) and show the statistic for the squared returns and GARCH residu-als, respectively. LM(5) reports the ARCH-LM test statistic for up to fifth order serial correlation in the returns. Next to these statistics,p-values are reported.

Q̂ 2(20)

Q̂ 2(20)



Before analyzing our data, we cleaned our sample for all contracts violat-ing the simple no-arbitrage condition, Vt � max{d (St � X), 0}, as the contin-uation value should not be negative. Here, Vt denotes the market price of the option at time t; d is a dummy variable equaling one when we consider acall option and minus one for a put option; St denotes the price of the underly-ing at time t; and X represents the exercise price.

Following Dumas, Fleming, and Whaley (1998), we excluded all optionswith less than six and more than 100 trading days to maturity. Moreover, onlyoptions on the S&P 100 whose prices lie between moneyness bounds of[�0.05; 0.05], with moneyness Mon defined as Mon � (d (St � X))�X wereused. For options on GE, however, we abstained from such a filter criterion dueto the discrete tick size. As the individual stock has a significantly smaller pricecompared with the OEX, we would only include a comparably small band ofdifferent strikes if using a relative moneyness filter.

In addition, we follow Lehnert (2003) and exclude all options with a trad-ing volume of less than 20 contracts on a specific Wednesday to circumventany liquidity-related pricing biases. As we apply a relative loss function in ourmodel comparison, we exclude all contracts with prices of less than $0.25.Following the study of Hsieh and Ritchken (2005), we divide each of our threesample years of option data into two halves. The first halves are denoted in-sample (is), whereas the second halves are denoted out-of-sample (oos)periods. We further subdivide the options sample with respect to maturity andmoneyness to get a more comprehensive picture of the data set and the subse-quent results.

Regarding moneyness, all options with moneyness between [�0.05; 0.02)are denoted as out-of-the-money (OTM), between [�0.02; 0.02) as at-the-money, and between [0.02; 0.05) as in-the-money. For GE, the moneyness binsare [�1.00; �0.05), [�0.05; 0.05], and [0.05; 1.00]. Option contracts withless than 41 trading days to maturity are considered to be short-term, from 41to 70 trading days as middle-term, and from 71 to 100 trading days as long-term contracts. To keep the presentation manageable, we do not report resultsfor call and put options separately.

Tables II (OEX) and III (GE) present selected sample statistics for theoption prices for the different moneyness and maturity bins, as well as for the entire data set. We report the number of observations, the average optionprice, the average trading volume, and the implied volatility, which enhances acomparison across moneyness. The implied volatility is calculated using a stan-dard binomial tree.

One can observe that the number of observations is of equal size for thein- and out-of-sample bins. Most contracts considered are classified as short-term, which is not unexpected, as trading increases when maturity approaches.



This can be directly observed from the volume statistics in Panel C of Tables IIand III.

It is remarkable that the average option price out-of-sample is higher thanthe in-sample for all moneyness and maturity bins. Looking at yearly data, wefind that this phenomenon only occurs in the years 2007 and 2008 and can beexplained by a higher volatility in the second halves of these years. The num-bers in Panel D of Tables II and III clearly indicate that the binomial treeimplied volatility is not constant, but rather seems to follow systematic money-ness and maturity patterns.

ESTIMATION METHODOLOGY

Historical Estimation Using Asset Returns Only

In a first step, we estimate the parameters under the physical probability measure using the method of maximum-likelihood. To be more specific,

TABLE II

Descriptive Statistics for the S&P 100 (OEX) Option Sample

5 DTM � 40 40 DTM � 70 70 DTM � 100 All

is oos is oos is oos is oos

Panel A. Number of Options Contracts

OTM 750 691 201 154 33 33 984 878ATM 1,074 980 224 198 38 49 1,336 1,227ITM 244 251 25 33 7 4 276 288All 2,068 1,922 450 385 78 86 2,596 2,393

Panel B. Average Price

OTM 3.57 5.57 7.13 9.56 11.61 14.99 4.57 6.63ATM 9.85 12.19 15.38 18.08 18.84 23.38 11.03 13.59ITM 24.37 27.36 27.99 35.73 31.07 35.88 24.87 28.44All 9.29 11.79 12.40 16.19 16.88 20.74 10.05 12.82

Panel C. Average Volume

OTM 937.89 770.17 212.36 209.85 164.64 93.33 763.76 646.45ATM 1,232.68 1,034.53 165.27 152.86 128.16 182.57 1,022.30 858.23ITM 209.84 223.07 232.32 244.79 76.00 97.75 208.49 223.82All 1,005.09 833.52 190.03 183.54 138.91 144.38 837.78 704.18

Panel D. Implied Volatility (%)

OTM 15.10 20.30 13.53 16.73 14.42 16.45 14.76 19.53ATM 14.62 19.38 14.35 17.47 13.58 16.44 14.54 18.95ITM 17.62 23.72 14.42 24.67 14.53 17.61 17.25 23.74All 15.15 20.28 13.99 17.79 14.02 16.50 14.91 19.74

This table reports descriptive statistics for options on the S&P 100 index quoted at the close of every Wednesday for the sample periodfrom January 2, 2006 to December 31, 2008.The implied volatilities are calculated using a standard binomial tree. DTM denotes days tomaturity, OTM, ATM, and ITM represent out-, at-, and in-the-money, respectively. is stands for in-sample, oos for out-of-sample.



we maximize the following log-likelihood function (conditioned on the firstobservation):

(15)

where denotes the vector of GARCH parameters; Rt represents the timeseries of continuously compounded returns of the underlying; and ht followsthe GARCH process described by Equation (2). According to the chosenGARCH-in-mean model, we have to estimate five parameters, namely b0, b1,b2, u, and l. To reduce this number and improve the reliability of the estima-tion procedure, we apply the so-called variance targeting methodology of Engle

�

ln L(�; Rt) � �12£(T�1) ln(2p)� a

T

t�2 ln(ht) �

aRt � r � q � l2ht �12

htb2

ht

§

TABLE III

Descriptive Statistics for the General Electric (GE) Option Sample

5 DTM � 40 40 � DTM � 70 70 DTM � 100 All


Panel A. Number of Options Contracts

OTM 20 84 51 91 46 90 117 265ATM 235 271 118 144 79 126 432 541ITM 275 288 135 167 98 123 508 578All 530 643 304 402 223 339 1057 1384

Panel B. Average Price

OTM 0.42 0.56 0.47 0.60 0.49 0.62 0.47 0.60ATM 0.93 1.15 1.27 1.44 1.42 1.78 1.11 1.37ITM 4.73 5.72 5.78 6.55 5.85 6.37 5.23 6.10All 2.88 3.12 3.14 3.38 3.18 3.14 3.02 3.20

Panel C. Average Volume

OTM 2,035.50 3,799.01 1,641.00 1,618.75 2,377.93 1,454.19 1,998.17 2,253.96ATM 3,857.15 4,340.93 1,948.35 1,987.94 1,873.32 1,645.83 2,972.98 3,086.93ITM 767.39 1,324.74 442.14 473.91 440.57 535.08 617.91 910.87All 2,185.23 2,919.18 1,227.91 1,275.40 1,347.77 1,191.94 1,733.22 2,018.65

Panel D. Implied Volatility (%)

OTM 27.78 52.06 24.42 42.49 21.52 23.45 23.47 39.06ATM 18.56 25.96 18.38 21.44 17.88 19.73 18.37 23.30ITM 23.26 29.96 21.68 27.51 21.09 23.27 22.37 27.83All 21.35 31.16 20.86 28.73 20.15 22.00 20.91 28.21

This table reports descriptive statistics for options on General Electric quoted at the close of every Wednesday for the sample periodfrom January 2, 2006 to December 31, 2008. The implied volatilities are calculated using a standard binomial tree. DTM denotes daysto maturity, OTM, ATM, and ITM represent out-, at-, and, in-the-money, respectively. is stands for in-sample, oos for out-of-sample.



and Mezrich (1996), demanding that the unconditional variance equal the sam-ple variance. Besides the usual non-negativity constraints of the b-parameters,we impose the restriction that the process is covariance stationary. This is equalto requiring a persistence n smaller than one, where n is given byn � b1 � b2(1 � u2).

We estimate these parameters for every year in our option sample over aperiod of 15 years, ending with the in-sample period of the specific year, e.g.,for 2006 from July 2, 1991 to June 30, 2006. We use a comparably long estima-tion window, as it is well known in the literature that it is difficult to estimateGARCH parameters precisely from return data with short samples. We use themean of the bid–ask price of the 1-month T-Bill rate at the end of the respec-tive estimation window as the risk-free interest rate, as most option contracts in our sample lie in the first maturity bin. The current dividend yield of theOEX of the specific year is taken as dividend yield in the estimation, both con-tinuously compounded and converted to a daily basis with a factor of 1/252.For GE, we account for the actual cash dividends.

Implied Estimation Using Option Prices and Asset Returns

The estimation of GARCH parameters using asset returns only draws on pastinformation. The values of options, however, are determined by the expectedfuture price movements of the underlying. As such, it seems likely that animplied calibration of GARCH parameters enlarges the available information setand ultimately results in a smaller pricing error. A further advantage of thisapproach is rooted in the fact that one directly obtains risk-neutral parameters,since an additional risk neutralization is obsolete. As pointed out by Barone-Adesi, Engle, and Mancini (2008), an estimation under and subsequent trans-formation to risk-neutral parameters under , leads to a rather poor pricing performance of European options and is dominated by an implied calibration.

Thus, implied model calibration can now be seen as the standard approachto value European options. However, efficient calculation schemes were lack-ing to translate this procedure to the issue of pricing American options.Therefore, we explain the methodology used subsequently in some detail.

We estimate the GARCH parameters applying a non-linear least squaresoptimization approach using 1 day of option data. For each optimization, weconsider the mean absolute percentage error (MAPE) as the objective function:

(16)MAPE �1N

aN

i�1

ƒ V~

i � Vi ƒVi

�

�



where and Vi denote the model price and market price of the ith option,respectively, and N denotes the number of option contracts on a particularWednesday. We apply a relative loss function to assign OTM options sufficientweight. However, in contrast to the majority of empirical studies, we do not usethe root mean-squared error, as the MAPE calculates a percentage error and is,in our opinion, easier to interpret. The model prices are computed using the EBT described in Section 2. The local volatility is determined by taking theunconditional volatility implied by the GARCH parameters 252 trading daysbefore the specific pricing day and updated by using the observed return data.To be more specific, we update the volatility from ht�1 to ht by substituting

(17)

into , resulting in an updating ruleconsisting of observables only:

(18)

Owing to estimation under the risk-neutral measure, we cannot identify uand l separately, as we can only estimate u*� u + l. This issue, however, is noproblem in an option pricing context, as we only require the sum of the twoparameters. We use the maximum-likelihood parameters as starting values forthe first optimization in every year and thereafter use the estimates of the pre-vious week as starting values.

For the implementation of the EBT, we apply a two-step procedure. First,we calculate the first four moments of the cumulative return over the maturityof the respective option. Second, we implicitly construct the tree, starting atthe maturity of the option and assuming one time step for every day to maturi-ty in the tree construction with one exercise possibility per day. The interestrate and the dividend yield are held constant over the maturity of the option.

For options on GE, we assume that we know the actual cash dividend aswell as the dividend day ex ante. According to the dividend history of GE andbearing in mind the fact that the majority of the options fall into the short-to-medium maturity bins, this assumption does not seem to be very restrictive andis in accordance with previous literature such as Stentoft (2005). Furthermore,we hypothesize that any cash dividend fully spills over to the stock price, ignor-ing tax aspects, transaction costs, or investor-specific preferences. We account

ht � b0 � b1ht�1 � b2ht�1 £ °Rt�1 � r � q �

12

ht�1

2ht�1

¢ � u � l §2

.

ht � b0 � b1ht�1 � b2ht�1(et�1 � u � l)2

et�1 � °Rt�1 � r � q �

12

ht�1

2ht�1

¢

V~

i



for these discrete dividends by first calculating the values of the underlying, asdescribed above, and then subtracting the sum of the compounded dividends.As the sole exception from the described construction of the tree for the index,whenever a dividend payment occurs, we add this amount back to the implicit-ly calculated stock price after the exercise decision has been made. Thismethodology ensures that the implicitly calculated asset price at time t0 corre-sponds to the actually observed stock price.

We re-calibrate this model every week and only use the options of the par-ticular Wednesday for the optimization. Heston and Nandi (2000) argued thatthe estimation of the required parameters using only 1 day of option datamight be problematic. First, such a procedure might lead to an over-fitting ofthe data, which would lead to a good in-sample pricing performance. The out-of-sample performance, however, might significantly deteriorate. Moreover,this approach could unnecessarily restrict the available information set, as itseems possible that the historical evolution of the underlying could containadditional valuable information over and above that contained in optionprices. In our opinion, the first objection should not gain any empirical validityas, for example, the average number of contracts for OEX is greater than 31options per Wednesday. The second argument is mitigated by the use of thevolatility updating rule used to determine the starting volatility. For robustnessreasons, we tested a model for OEX options, which calibrated the parametersover the option data over the last 5 weeks. The results, however, showed thatthis approach is, and is thus not reported.

Finally, we emphasize that the tree-based valuation is about 500 timesfaster than the simulation approach and thus much more suitable for real-world applications.

EMPIRICAL RESULTS

Parameter Estimates

Table IV summarizes the estimated parameters using the maximum-likelihoodapproach and provides a statistical overview. All GARCH parameters seem to bequite stable over time, which is, of course, due to the fact that the estimationwindows are largely overlapping. The l-parameter, though, is, especially forOEX, an exception, as it is quite volatile. This finding, however, is in accordancewith the previous literature and nothing special for this data sample. As l onlyenters the mean equation, it might be possible that this parameter is only poor-ly estimated. Furthermore, the leverage parameter u is positive for both under-lyings. This finding implies a negative correlation between the return and theconditional volatility and results in a left-skewed distribution when considering



multiple periods. The persistence of both processes is quite high for all periods,again in line with the existing literature. The annual volatility implied by theGARCH parameters is roughly 17% for OEX and 26% for GE.

Table V summarizes the parameter estimates of the implied estimationapproach using option and return data. We report the mean and the standard

TABLE IV

MLE Estimates of the GARCH Parameters

2006 2007 2008

Parameter OEX GE OEX GE OEX GE

r 1.7922E–04 1.7922E–04 1.6326E–04 1.6326E–04 5.6344E–05 5.6344E–05q 9.1400E–05 NA 8.3248E–05 NA 1.0224E–04 NA

l 2.7778E–02 3.4768E–02 3.3798E–02 3.9615E–02 4.5767E–02 4.3504E–02b0 1.0991E–06 8.3154E–07 1.3177E–06 7.8980E–07 1.5284E–06 1.2291E–06b1 0.8935 0.9423 0.8742 0.9421 0.8650 0.9323b2 0.0557 0.0358 0.0640 0.0377 0.0664 0.0466u 0.8571 0.7255 0.8826 0.6763 0.9148 0.5981

n 0.9901 0.9969 0.9880 0.9970 0.9870 0.9956sann 0.1672 0.2610 0.1663 0.2582 0.1723 0.2645LL 12,488.36 10,714.70 12,542.03 10,803.47 12,366.44 10,678.73

This table reports GARCH parameters using MLE to the daily log return series of the S&P 100 index (OEX) and General Electric (GE)over a period of 15 years. r and q represent the continuously compounded 1-month T-Bill rate and dividend yield respectively, at theend of the particular period, converted to a daily basis. n denotes the persistence and sann refers to the annualized unconditionalvolatility, both based on the respective GARCH parameters, whereas the last row contains the value of the Log-Likelihood function.

TABLE V

NLS Estimates of the GARCH Parameters

2006 2007 2008

Parameter OEX GE OEX GE OEX GE

b0 9.0686E–06 2.4310E–05 1.9439E–05 4.4206E–05 1.2140E–04 1.3670E–04[8.4230E–06] [1.3391E–05] [2.1527E–05] [3.2449E–05] [2.4929E–04] [1.8454E–04]

b1 0.5543 0.4571 0.4827 0.4351 0.4622 0.4390[0.2059] [0.2572] [0.1831] [0.2139] [0.2575] [0.2376]

b2 0.1323 0.1945 0.1147 0.1859 0.1292 0.1492[0.0628] [0.0919] [0.0372] [0.0826] [0.0515] [0.0586]

u* 1.2062 0.6868 1.5175 0.6687 1.3146 1.1667[0.3946] [0.3227] [0.4441] [0.3172] [0.6508] [0.2651]

n 0.8792 0.7434 0.8615 0.7041 0.8148 0.7914sann 0.1375 0.1545 0.1880 0.1940 0.4064 0.4064

This table reports the mean of the risk-neutral GARCH parameters using a non-linear least-squares routine minimizing the loss func-tion of the respective week with a weekly re-calibration of the model. Standard deviations of the estimates are shown in square brack-ets below. n refers to the risk-neutral volatility persistence, sann to the risk-neutral annualized unconditional volatility, implied by themean of the estimated GARCH parameters.



deviation of the weekly estimated parameters. It is noteworthy that b0 is theparameter with the highest fluctuation, followed by b1, whereas b2 and u* areestimated relatively stable, which is in line with the literature (see, e.g., Hestonand Nandi, 2000). The parameter b2 determines the volatility of volatility,whereas u* influences the skewness of the multi-period returns.

Comparing these estimates with those obtained using historical data andmaximum-likelihood, it is evident that b0 as well as b2 are partially estimatedsubstantially higher, whereas b1 takes on consistently smaller values. For u*,however, we cannot draw any clear-cut conclusions. Regarding the OEX data,the risk-neutral estimates are clearly higher than the sum of the maximum-likelihood-based parameters u and l. Considering the GE estimates, however,the implied calibration approach results in slightly lower estimates for theyears 2006 and 2007 and in a substantially higher parameter for the year2008.

Considering persistency, it becomes clear that the physically estimatedparameters are considerably more persistent. This might be at least partiallyexplained by the extreme amplitudes of the implied volatilities during theperiod of the option sample. For example, in March 2007, the implied volatil-ity of options on OEX more than doubled to quickly return to its base level.Engle and Mustafa (1993) showed, in an analysis of implied volatilities ofoptions on the S&P 500 around the stock market crash of 1987, that thevolatility dynamic changed structurally directly after the Black Mondayresulting in a significantly lower persistency of the process. For options onthe index, the annualized volatility implied by the mean parameters of thenon-linear least squares approach is lower for 2006, but higher for the following years, compared with the maximum-likelihood-based volatili-ties. In contrast, the volatility of GE is lower until 2007, but higher for 2008. As the volatility of the non-linear least squares approach is based on meanparameters, an entirely true explanation seems impossible. Looking at theimplied volatilities we see, however, that at least until the first half of 2007,the options implied volatility is considerably lower than the volatility impliedby the maximum-likelihood-based GARCH parameters. The latter fact is theresult of a quasi-average consideration during the last 15 years and implies assuch, through the inclusion of the very volatile markets around the burst ofthe dotcom bubble, itself a high volatility. Additionally, it seems possible that theparameters of the two different estimation methodologies are generally dif-ferent, as the risk-neutral and the physical estimation procedures mightemphasize different moments of the cumulative asset returns distribution.The variation of the implied estimated parameters over time indicates that aweekly implied re-calibrating might improve the pricing performance com-pared with the maximum-likelihood-based approach.



Option Valuation

Each option in the sample is valued using the two estimation and valuationapproaches described above, i.e., we consider the LSM-based method proposedand analyzed by Stentoft (2005) as a benchmark for the implicitly calibratedparameters employing the EBT.

In addition, we contrast these two models with GARCH volatility with thestandard model of constant volatility using a simple binomial tree. To be a com-petitive benchmark, we re-estimate the constant volatility for the options weekly.

Tables VI (OEX) and VII (GE) summarize the model performances overalland for different moneyness and maturity bins. The performance measureemployed is the MAPE, which was used for estimation. For the historical esti-mation and LSM valuation approach, in-sample pricing errors are based on theparameters of Subsection estimated with data until June 30 of the respectiveyear; out-of-sample errors are based on these same parameters. This meansthat the parameters are held constant for every specific year.

For the Edgeworth and the standard binomial tree, we calculate in-sampleerrors with the parameters and volatility of the week used for estimation; forout-of-sample, we use the parameters and volatility of the previous week.

TABLE VI

Model Performance for Options on the S&P 100 Index

5 DTM � 40 40 DTM � 70 70 DTM � 100 All


OTMLattice 0.4200 0.4933 0.2633 0.3225 0.2709 0.1937 0.3830 0.4521LSM 0.4078 0.3154 0.3430 0.2331 0.2399 0.1486 0.3889 0.2947EBT 0.1274 0.2374 0.1007 0.1486 0.1111 0.1999 0.1214 0.2204

ATMLattice 0.1087 0.1521 0.1362 0.1543 0.1844 0.1537 0.1155 0.1525LSM 0.1296 0.1081 0.1236 0.1133 0.1252 0.0769 0.1285 0.1077EBT 0.0614 0.1014 0.0691 0.0891 0.0998 0.1193 0.0638 0.1001

ITMLattice 0.0727 0.0865 0.1095 0.1130 0.1265 0.1376 0.0774 0.0902LSM 0.0657 0.0699 0.0829 0.1120 0.1095 0.0790 0.0684 0.0748EBT 0.0550 0.0669 0.0726 0.0568 0.0829 0.0475 0.0573 0.0654

AllLattice 0.2174 0.2662 0.1915 0.2181 0.2158 0.1683 0.2128 0.2549LSM 0.2229 0.1776 0.2194 0.1611 0.1723 0.1045 0.2208 0.1723EBT 0.0846 0.1458 0.0834 0.1101 0.1031 0.1469 0.0849 0.1401

This table shows the overall pricing performance of the different models for the years 2006 to 2008 using the mean absolute percent-age error (MAPE) as the loss function. DTM denotes days to maturity, OTM, ATM, and ITM represent out, at and in the money, respec-tively. Lattice denotes the binomial lattice, LSM the least squares Monte Carlo algorithm, and EBT the Edgeworth binomial tree. isstands for in-sample, oos for out-of-sample.



Contrasting the in-sample pricing performance of the implicitly estimatedEBT with the LSM approach for the two underlyings, we observe that the for-mer yields substantially smaller pricing errors. For the OEX, the MAPEamounts to 22.08% for the LSM approach compared with 8.49% when usingthe implied EBT, which is almost three times smaller. For the GE options, theoverall performance is better for both approaches; again, a much better per-formance of the EBT approach, which yields an in-sample MAPE of 3.24%compared with 12.84% for the LSM approach, is observed. Interestingly, thelatter is even outperformed by the simple binomial tree with constant volatility.

It is noteworthy that the result of the smaller pricing errors of the EBTapproach holds true for every subcategory with the largest differences being forOTM contracts with only a short time to maturity. Even the standard binomialtree with the assumption of constant volatility results for GE in smaller errorsfor every bin; for options on OEX, however, the LSM simulation and the bino-mial tree seem to be overall comparable with the first model, being superior forsome categories, whereas the latter one is so for others. The EBT, however, isclearly superior to the standard lattice.

TABLE VII

Model Performance for Options on General Electric

5 DTM � 40 40 DTM � 70 70 DTM � 100 All


OTMLattice 0.2764 0.4885 0.2419 0.4740 0.1960 0.3142 0.2298 0.4244LSM 0.5243 0.3477 0.4364 0.3452 0.3447 0.2875 0.4153 0.3264EBT 0.0585 0.2660 0.0335 0.3141 0.0293 0.2180 0.0361 0.2662

ATMLattice 0.0580 0.1140 0.0490 0.0896 0.0506 0.0856 0.0542 0.1009LSM 0.1466 0.1298 0.1485 0.1177 0.1834 0.1401 0.1538 0.1290EBT 0.0509 0.0908 0.0363 0.0957 0.0245 0.0940 0.0421 0.0928

ITMLattice 0.0271 0.0419 0.0281 0.0408 0.0278 0.0411 0.0275 0.0414LSM 0.0381 0.0391 0.0426 0.0362 0.0450 0.0488 0.0406 0.0403EBT 0.0254 0.0355 0.0211 0.0332 0.0205 0.0371 0.0233 0.0352

AllLattice 0.0502 0.1306 0.0721 0.1564 0.0706 0.1301 0.0608 0.1380LSM 0.1045 0.1176 0.1498 0.1354 0.1559 0.1461 0.1284 0.1297EBT 0.0379 0.0889 0.0291 0.1192 0.0237 0.1063 0.0324 0.1020

This table shows the overall pricing performance of the different models for the years 2006 to 2008 using the mean absolute percent-age error (MAPE) as the loss function. DTM denotes days to maturity, OTM, ATM, and ITM represent out, at and in the money, respec-tively. Lattice denotes the binomial lattice, LSM the least squares Monte Carlo algorithm, and EBT the Edgeworth binomial tree. isstands for in-sample, oos for out-of-sample.



These in-sample results are not too surprising, as the GARCH parametersfor the simulation are held constant, whereas for the other two models, therequired parameters are optimally determined by the option data. As correctlynoted by Bakshi, Cao, and Chen (1997), a true model comparison can only beperformed in an out-of-sample context, as for in-sample, the superior pricingresults might just be the result of an over-fitting of the data.

Looking at out-of-sample pricing errors, it is especially evident for thestandard binomial lattice that the good in-sample results are merely the resultof data fitting. The pricing performance worsens for nearly every bin. Althoughthis point is also true for the EBT, the latter approach does remain clearly supe-rior compared with the LSM-based approach, resulting in smaller pricingerrors for every bin. Overall, the implicit EBT yields a MAPE of 14.01% for theOEX options and 10.20% for GE. In contrast, the benchmark approach ofStentoft (2005) produces overall pricing errors of 17.23% and 12.97%, respec-tively.

Looking at the performance of the LSM-based approach, it seems surpris-ing that the out-of-sample pricing errors are smaller than the in-sample onesfor nearly every bin. The fact, however, that we can hardly talk of in-samplewhen estimating the GARCH parameter with maximum-likelihood using a historical return series of 15 years, can be seen as an indicator of why the in-sample pricing performance need not be higher than that of out-of-sample.

Analysis of Pricing Errors

Looking at the pricing performance of the different models, the results clearlysuggest that there might still be some systematic factors driving the pricingerrors. Therefore, we analyze the errors in more detail by performing a simpleregression analysis. We regress the MAPEs of the respective models on a con-stant (Con), days to maturity (DTM), squared days to maturity (DTM2), mon-eyness (Mon), squared moneyness (Mon2), trading volume (V), and a dummyvariable (P), which is set to one if the option is a put and zero otherwise. To beable to compare the results, we re-scale the moneyness to lie between zero andone. The estimated equation reads:

(19)

with . Table VIII summarizes the results of this regression for theOEX options. Robust t-statistics and partial R2 are reported in parenthesis andbrackets, respectively.

As all regressors except the trading volume are known in advance, poten-tial systematic pricing errors could be considered in the pricing models.

j � N(0, s2t )

MAPE � a0 � a1DTM � a2DTM2 � a3Mon � a4Mon2 � a5V � a6P � j



Looking at the results, we can conclude that nearly all coefficients are, at leastat the 5% level, significantly different from zero. Days to maturity for the LSMapproach and trading volume for the binomial tree, however, are exceptions.Options with longer maturities are priced better for the standard binomial treeand the EBT; this effect, however, weakens with a longer time to maturity. Allmodels price options that are in-the-money more precisely than the out-of-the-money options, whereas this effect also decreases with moneyness. Trading vol-ume seems to have only negligible effects on pricing performance, which mightbe explained by the fact that we have already disregarded less liquid contractswhen constructing the data set. The LSM and EBT-based prices for put optionsare more precise than for calls. This result is reversed for the standard binomi-al lattice. Contrasting the models with variable variance with the constantvolatility tree, it is clear that these models have a considerably lower R2. Thus,

TABLE VIII

MAPE Regression Results for Options on the S&P 100 Index

Lattice LSM EBT

is oos is oos is oos

Con 0.6014 0.7481 0.6136 0.4519 0.2063 0.5762(30.1920) (17.6476) (15.5766) (17.0630) (15.5374) (4.0686)

[0.2911] [0.2545] [0.1171] [0.1777] [0.1690] [0.0575]

DTM �0.0042 �0.0040 0.0005 �0.0003 �0.0023 �0.0169(�6.2870) (�4.5435) (0.4872) (�0.4104) (�4.7961) (�1.6601)

[0.0084] [0.0041] [0.0000] [0.0001] [0.0127] [0.0278]

DTM 2 3.73E–05 2.23E–05 �1.67E–05 �8.83E–06 2.67E–05 2.28E–04(5.0922) (2.2731) (�1.5358) (�1.0494) (4.9954) (1.4463)[0.0050] [0.0009] [0.0004] [0.0003] [0.0125] [0.0364]

Mon �1.4510 �1.5727 �1.3096 �1.0015 �0.3551 �0.7520(�23.2861) (�12.8778) (�11.8085) (�11.5277) (�10.3546) (�6.5442)

[0.1687] [0.1107] [0.0531] [0.0859] [0.0498] [0.0096]

Mon2 0.9937 1.0320 0.8524 0.6823 0.2535 0.4671(16.1873) (9.6736) (8.3307) (8.6905) (7.2497) (5.4129)

[0.0613] [0.0403] [0.0174] [0.0337] [0.0197] [0.0031]

V �2.45E–07 2.69E–06 1.77E–05 9.26E–06 3.91E–06 �8.15E–06(�0.0877) (0.6762) (3.0278) (2.3859) (2.1007) (�0.9117)

[0.0000] [0.0001] [0.0028] [0.0018] [0.0017] [0.0003]

P 0.1041 �0.0048 �0.1326 �0.0265 �0.0049 �0.0034(15.7446) (�0.4127) (�9.8472) (�2.9891) (�1.2901) (�0.1692)

[0.0501] [0.0001] [0.0315] [0.0031] [0.0006] [0.0000]

R2 0.4647 0.2768 0.1825 0.2022 0.1261 0.0625

This table reports the results from regressing the mean absolute percentage error (MAPE) on a constant, maturity (DTM), moneyness(Mon), volume as well as a put–call dummy. Heteroscedasticity-robust t-statistics are shown in parenthesis; in brackets, we report thepartial R2; is stands for in-sample, oos for out-of-sample.



maturity, moneyness, volume and the put–call dummy variable have a less sys-tematic impact on pricing errors. According to the partial R2, the smaller R2

can especially be attributed to a less systematic influence of the moneynesseffects. The results for GE are comparable and thus not reported but are avail-able on request.

Looking at the overall results, we can conclude that the EBT does not onlyresult in the smallest mean absolute pricing errors but also less exposed to sys-tematic pricing errors. However, days to maturity and moneyness still have asignificant impact.

Robustness Checks

To check the robustness of the results found in the previous sections, weextended the analysis for more underlyings and a different GARCH specifica-tion. More precisely, we repeated the study for two more major stocks, namelyIBM and Microsoft. Furthermore, we repeated our analysis employing theGJR-GARCH volatility dynamics of Glosten, Jagannathan, and Runkle (1993).The results of these analyses are very similar to the ones presented in the study.Owing to space constraints, they are not reported, but are available on request.

CONCLUSIONS

In this study, we consider the problem of calibrating option pricing modelswhen the underlying asset exhibit a GARCH type volatility and the option is ofthe American-type. This is of high relevance, as GARCH option pricing modelsare frequently employed by practitioners and academics alike. Although mostexchange-traded option contracts are early exercisable, previous empiricalresearch has mainly considered European options.

In a recent study, Stentoft (2005) suggested coping with the American fea-ture by using the LSM simulation approach. Although this algorithm is extreme-ly popular, we argue in this study that it might not be the preferable approachwhen a GARCH option pricing model needs to be calibrated. In contrast, wesuggest the use of Rubinstein’s EBT, which allows for an implicit model calibra-tion. The empirical results show that this method is indeed superior.

We conclude the study by outlining possible extensions of the presentedresearch. First, one might consider other lattice-based approaches such as theJohnson binomial tree suggested by Simonato (2010), or trinomial trees as out-lined by Ritchken and Trevor (1999), Cakici and Topyan (2000), and Lyuu andWu (2005) for pricing American options. Furthermore, Barone-Adesi, Engle,and Mancini (2008) have recently developed a filtered historical simulationapproach, which has been adopted for American options pricing by Tsao and



Hung (2009), which is also an interesting alternative for pricing Americanoptions.

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American option valuation: Implied calibration of GARCH pricing models

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