American Option Pricing using GARCH models ... - Semantic Scholar

American Option Pricing using GARCH models and the Normal Inverse Gaussian distribution∗ Lars Stentoft Department of Finance HEC Montréal Email: [email protected] November 12, 2007

Abstract In this paper we propose a feasible way to price American options in a model with time varying volatility and conditional skewness and leptokurtosis using GARCH processes and the Normal Inverse Gaussian distribution. We show how the risk neutral dynamics can be obtained in this model using the Generalized Local Risk Neutral Valuation Relationship of Duan (1999) and we derive approximation procedures which allows for a feasible implementation of the model. A Monte Carlo study shows that there are important pricing differences compared to the Gaussian case. When the model is estimated the results indicate that compared to the Gaussian case the extensions are important. A large scale empirical examination confirms this finding and shows that our model outperforms the Gaussian case for pricing options on three large US stocks. In particular, improvements are found for out of the money options and short term options. These are among the most traded and the suggested model is therefore important. JEL Classification: C22, C53, G13 Keywords: GARCH models, Normal Inverse Gaussian distribution, American Options, Least Squares Monte-Carlo. ∗A

previous version of this paper was entitled “Modelling the Volatility of Financial Asset Returns Using the Normal Inverse

Gaussian Distribution: With an Application to Option Pricing”. The author thanks participants at the 3 rd Nordic Econometric Meeting and the Montréal Actuarial and Financial Mathematics Day, participants and discussants at the 2006 SCSE and NFA meetings, as well as seminar participants at HEC Montréal and Brock University. Financial support from The Danish Social Science Research Council (Grant 95015328) and Institut de Finance Mathématique de Montréal (IFM2) is gratefully appreciated.

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Electronic copy available at: http://ssrn.com/abstract=946002

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Introduction

A recent application of the generalized autoregressive conditional heteroskedasticity, or GARCH, models of Engle (1982) and Bollerslev (1986) has been in the context of derivative pricing. The obvious reason for this is that for this particular type of assets correct estimation and modelling of the volatility of the underlying asset return process is of paramount importance. Since the GARCH models offer a very flexible framework for this it constitutes an obvious extension to the constant volatility framework of Black & Scholes (1973) and Merton (1973). In terms of option pricing the added flexibility comes at a cost since with time varying volatility the market is no longer complete. However, in Duan (1995) a GARCH option pricing model is derived under the assumption of conditionally Gaussian distributed innovations and under some familiar assumptions on investor preferences. The paper also shows that the implied volatility backed out from option prices generated with a Gaussian GARCH model shows patterns similar to those found empirically, and the Gaussian GARCH option pricing model could thus potentially explain the mispricings found empirically when e.g. constant volatility models are used. When the Gaussian GARCH models are compared to e.g. the constant volatility model smaller pricing errors are obtained empirically. In particular this is found for European style options on the Standard & Poor’s 500 Index in Bollerslev & Mikkelsen (1996), Bollerslev & Mikkelsen (1999), Heston & Nandi (2000), Christoffersen & Jacobs (2004), and Hsieh & Ritchken (2005). In addition to this the GARCH models are found to perform well for European style options on the German DAX index in Härdle & Hafner (2000), on the Hang Seng Index in Duan & Zhang (2001), and on the FTSE 100 Index in Lehar, Scheicher & Schittenkopf (2002). More recently the GARCH option pricing model has been used to price American style options on a US index and three major individual stocks using an application of the Least Squares Monte-Carlo method of Longstaff & Schwartz (2001) to take account of the early exercise feature. This was done in Stentoft (2005), and again the finding is that the GARCH models provides smaller pricing errors than does the constant volatility alternative. However, in terms of empirical option pricing the latter paper in particular also shows that systematic mispricing does still occur when Gaussian GARCH models are used empirically for option pricing. In particular, it is found that short term deep out of the money options remains underpriced by the model even when time varying volatility is considered. One obvious suggestion is to consider models which are driven by non Gaussian innovations. In the vast empirical finance literature such models are well known within the GARCH framework where alternative assumptions on the conditional distribution have been suggested and extensively analyzed. The best known examples are probably the use of the Student’s t-distribution in Bollerslev (1987) and of the Generalized Error distribution in Nelson (1991). Both of these distributions can potentially account for the excess kurtosis often found in the standardized residuals from the GARCH models, and a general finding in these studies is that financial return distributions indeed do have fatter tails than the Gaussian distribution. In addition to these leptokurtic distributions various skewed distributions, like e.g. the Skewed Student’s t-distribution, 2

Electronic copy available at: http://ssrn.com/abstract=946002

have also been used (see e.g. Lambert & Laurent (2001) which also lists a number of different contributions in this area). While the empirical results are less abundant it does appear that skewness is potentially important for financial asset return processes. The theoretical foundation for option pricing in this more general framework was provided in Duan (1999) which extends the Gaussian GARCH option pricing model to situations with conditional leptokurtic distributions. In particular the choice of distribution in that paper is the Generalized Error Distribution and the paper examines the effect on the model predicted option prices compared to what is obtained in the Gaussian case. The results of the paper indicate that conditional leptokurtosis plays an important role in determining the option prices. These findings are confirmed in Hafner & Herwartz (2001) which uses a Student’s t-distribution as the conditional distribution in the GARCH framework and which obtain qualitatively the same results. However, empirical applications of such models has been very limited most likely because of the added complexity of the framework. In this paper we propose a feasible way to price American options in a framework with time varying volatility and conditional skewness and leptokurtosis based on the framework of Duan (1999), we analyze the properties of the model, and we provide a large scale empirical application to individual stock options. In particular, the first objective of the present paper is to extend the application of the GARCH option pricing model from Duan (1999) to the situation where the conditional distribution is allowed to be skewed in addition to being leptokurtic. To achieve this we chose to work with the Normal Inverse Gaussian distribution which can accommodate both of these features and which has been introduced in the finance literature recently and used together with GARCH models in e.g. Barndorff-Nielsen (1997), Andersson (2001), and Jensen & Lunde (2001). One particularly appealing property of the Normal Inverse Gaussian distribution is that its use can be motivated through a Mixing Distribution Hypothesis (see Clark (1973)). In particular, if asset returns are assumed to be normally distributed conditional on the variance, and if the conditional variance of the returns follows an Inverse Gaussian distribution, then the unconditional distribution of the returns is a Normal Inverse Gaussian distribution. Thus, this gives a theoretical justification for the use of the suggested distribution. The second objective of the paper is to discuss in detail how the GARCH option pricing model may be implemented numerically in a computationally efficient way since the model involves some additional complexities. The method we proposed extends and simplifies the methods suggested in Duan (1999). In particular, the proposed method avoids the use of Monte Carlo simulation to evaluate particular functions and it exploits the use of simple linear approximations. This ensures that the method is relatively simply to implement and because of the approximations also relatively fast. We also reiterate how a simulation approach can be used to price options with early exercise features using the approach suggested in Longstaff & Schwartz (2001) which was used in a Gaussian GARCH context in Stentoft (2005). Since this style of options occurs in many situation and since both the most liquid index options and all individual options

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traded on the Chicago Board Options Exchange are American style options this is of great importance. Based on the Normal Inverse Gaussian GARCH option pricing framework and the procedure for the numerical implementation the third objective of the paper is to analyze the properties of the proposed model in terms of the calculated option prices. In particular, since the parameters of the distribution can be linked to the conditional skewness and leptokurtosis in an intuitive way it becomes possible to gauge the importance of either of these distributional features for option pricing purposes by comparing the model prices to those obtained under a Gaussian alternative. The analysis indicates that by allowing the model to be governed by a skewed and leptokurtic conditional distribution added flexibility is provided compared to that of the Gaussian alternative. Finally, since it seems plausible that the model may be able to explain some of the shortcomings alternative models have been shown to have empirically the final part of the paper applies the model empirically to analyze the pricing performance for options written on three major US stocks. The results obtained shows that improvements are found compared to the Gaussian special case. In particular, we find large improvements in the pricing performance for out of the money options in general and for short maturity options in particular when compared to a Gaussian alternative. Since options with these characteristics are among the most traded the extension suggested in this paper is clearly important. It should be noted that while we use historical data on the underlying asset to obtain the parameters to be used in option valuation these parameters could potentially be backed out from historical option data also. In particular, this procedure which consists of calibrating the option pricing model to existing data is used in Heston & Nandi (2000), Christoffersen & Jacobs (2004), and Hsieh & Ritchken (2005) among others for the Gaussian case. In Christoffersen, Heston & Jacobs (2006) and Lehnert (2003) GARCH type models with skewed and leptokurtic innovations are used although neither of these use the exact framework of Duan (1999). However, in all these cases the options under consideration are European style and the models considered is limited to those which allow for approximately closed form pricing formulas. Such formulas are unfortunately not available for the American style options and for this reason the calibration procedure would be much more computationally complex. Finally, we note that in addition to the proposed discrete time GARCH models other models with time varying volatility have been proposed. In particular, within the context of derivatives pricing the continuous time Stochastic Volatility (SV) models are important and models with non Gaussian distributions have been derived recently (see e.g. Barndorff-Nielsen & Shephard (2001), Carr & Wu (2004), Eberlein (2001), and Huang & Wu (2004)). One of the main advantages the continuous time SV framework offers over the discrete time GARCH framework is that in some cases analytical formulas exist for European style derivatives. However, again this is unfortunately not the case for the American style derivatives considered here. Furthermore, while the GARCH framework only depends on the observable return process the application of any SV model would, in addition, require the unobserved volatility as a state variable.

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This complicates not only the estimation procedure but also the actual option pricing procedure. For these reasons in the present paper the focus is on the discrete time GARCH framework for modelling time varying volatility in a setting with conditional skewness and leptokurtosis. The rest of the paper is organized as follows: Section 2 presents the discrete time Normal Inverse Gaussian GARCH model which allows for skewness and excess kurtosis in addition to time varying volatility and the appropriate model to be used for option pricing is derived. In Section 3 we discuss how the model may be implemented numerically using linear approximations and we provide maximum likelihood estimation results. Section 4 explains how options may be priced using simulation techniques and using a Monte Carlo study the potential gains obtained with the Normal Inverse Gaussian GARCH model compared to the Gaussian GARCH benchmark in terms of option pricing are analyzed. Section 5 presents the results of our empirical application and documents the improvements obtained with the proposed model. Finally, Section 6 offers concluding remarks and some directions for future research.

2

The skewed and leptokurtic generalized GARCH framework

In this paper a discrete time economy is considered. We denoted by St the asset prices, by dt dividends, and ³ ´ t by Rt = ln SSt−1 + dt the continuously compounded return process. We assume that the return process

can be modelled using what we will refer to as a generalized GARCH framework. In particular, we specify the dynamics for Rt as Rt = mt (·; θm ) +

p ht εt

ht = g (hs , εs ; −∞ < s ≤ t − 1; θh ) εt | Ft−1 ∼ D (0, 1; θD ) ,

(1) (2) (3)

where Ft−1 is the information set containing all information up to and including time t − 1. In equation (1) we use mt (·; θm ) to denote the conditional mean, which is allowed to be governed by a set of parameters θm provided that the process is measurable with respect to the information set Ft−1 . Likewise, in (2) the parameter set θh governs the variance process. This process is allowed to depend on lagged values of the innovations to the return process, lagged values of the volatility itself, as well as transformations hereof. Finally, in (3) we use D (0, 1; θD ) to denote a zero mean and unit variance distribution function which is also allowed to depend on a set of parameters θD . For notational convenience if the following we let θ denote the set of all parameters in θm , θh , and θD . The above framework is very general and nest many well known GARCH specifications. First of all, we observe that the specification in (1) allows for quite general forms of the conditional mean. In particular, for the GARCH in Mean specification suggested in e.g. Duan (1995) the parameter set is given by θm =

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© ª r, λ, − 12 and the functional form can be specified as

p 1 mt (·; θm ) = r + λ ht − ht , 2

(4)

where r is the risk free interest rate and − 12 ht is the well known correction for continuously compounded return. A particular nice feature of this specification is that the parameter λ may be interpreted as the risk premium. Secondly, in order to relate the flexible variance specification in (2) to some existing models we may start by noting that for the well known GARCH specification θh would consist of the set of parameters {ω, β, α} and (2) would have the following functional form: ht = ω + βht−1 + αht−1 ε2t−1 .

(5)

Obviously, using more lags can be considered as simple extensions. Alternatively, we may whish to consider specifications which can potentially accommodate asymmetric responses to negative and positive return innovations. Such models are generally said to allow for a leverage effect, which refers to the tendency for changes in stock prices to be negatively correlated with volatility. One such extension to the GARCH model considered in (5) is the non-linear asymmetric GARCH model, or NGARCH, of Engle & Ng (1993). The particular specification of the variance process for this model is given by ht = ω + βht−1 + αht−1 (εt−1 + γ)2 .

(6)

Thus, for this specification we would have θh = {ω, α, β, γ}. In the NGARCH model the leverage effect is modelled through the parameter γ, and if γ < 0 leverage effects are said to be found. It is clear that this model nests the ordinary GARCH specification, which obtains when γ = 0, and the model thus allows us to compare the contribution of asymmetry directly by comparison to the GARCH specification.

2.1

The NIG GARCH model

While the generalized GARCH framework can be used with a host of conditional distributions like the Generalized Error Distribution or the Student’s t-distribution in the present paper we chose to use the Normal Inverse Gaussian, or NIG for short, distribution. Following Jensen & Lunde (2001) the N IG (μ, δ, a, b) distribution can be defined in terms of the so called location and scale invariant parameters as fN IG (x; μ, δ, a, b) = where q (z) =

µ µ µp ¶ µ ¶−1 ¶¶ x−μ a (x − μ) x−μ a2 − b2 + b K1 aq exp q , πδ δ δ δ

(7)

√ 1 + z 2 and K1 (·) is the modified Bessel function of third order and index 1. For the

distribution to be well defined we obviously need to ensure that 0 ≤ |b| ≤ a and 0 < δ.

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In (7), μ is a location parameter and δ is a scale parameter, and if we define ρ = b/a the mean and variance of a N IG (μ, δ, a, b) distributed variable are given as ρδ

E (X) = μ + p

1 − ρ2

V ar (X) =

δ2

a (1 − ρ2 )3/4

, and

(8)

.

(9)

Thus, since the distribution is a location-scale family, that is X ∼ N IG (μ, δ, a, b) ⇔

X−μ δ

∼ N IG (0, 1, a, b),

a zero mean and unit variance NIG distributed variable can be obtained by restricting μ and δ to have the following form: −ρδ μ= p , and 1 − ρ2 q 3/4 δ = a (1 − ρ2 ) .

(10) (11)

In the following we will refer to this standardized distribution as the N IG(a, b) distribution and this distribution will be used as the zero mean unit variance distribution in (3). From Jensen & Lunde (2001) we also have the following expressions for the skewness and kurtosis of the standardized N IG(a, b) distribution: 3ρ , and √ p 4 a 1 − ρ2 ! Ã 4ρ2 + 1 . Kurt (X) = 3 1 + p a 1 − ρ2

Skew (X) =

(12) (13)

Thus, it is is clear that neither skewness nor kurtosis is affected by the location and scale parameters which explains why the a and b parameters are referred to as being location and scale invariant. From (12) and (13) it is also seen that we may interpret a and b as shape parameters with a determining the leptokurtosis and b the asymmetry. In particular the distribution is symmetric when b = 0. Furthermore, as much leptokurtosis as desired may be accommodated within the standardized N IG(a, b) distribution as a tends to zero. It should be noted that the NIG distribution has been used in a GARCH framework previously in e.g. Forsberg & Bollerslev (2002) and Jensen & Lunde (2001). However these applications can all be thought of as special cases of the generalized framework above. In particular, with b = 0 the restrictions in (10) √ and (11) simplify to yield μ = 0 and δ = a. If it is further assumed that the mean specification in (1) 2 is identically equal to zero and the volatility specification in (2) is given by ht = ω + βht−1 + αRt−1 we

essentially obtain the model used for the ECU/USD exchange rate in Forsberg & Bollerslev (2002). Also, the model used in Jensen & Lunde (2001) is obtained by restricting the mean equation in (1) to be given by √ √ √ Rt = μ + λ ht + ht εt which is obtained in the above framework when mt (·; θm ) = μ + λ ht .

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2.1.1

The restriction to conditional normality

When using the N IG (a, b) distribution in (3) it should be noted that the original Gaussian GARCH model of Bollerslev (1986) is in fact obtained in the limit as a tend to infinity. Thus, the Gaussian distribution becomes a natural benchmark to be used when analyzing the effect of skewness and leptokurtosis in the generalized NIG GARCH framework. While the Gaussian distribution is also a special case of both the Generalized Error Distribution and the Student’s t-distribution, neither of these distribution can accommodate skewness like it is possible when using the above NIG GARCH model. Thus, one further argument in favour of the use of the NIG distribution is the added flexibility.

2.2

Option pricing with the NIG GARCH model

In Duan (1999) the Local Risk Neutral Valuation Relationship, or LRNVR for short, of Duan (1995) is generalized to situations where the innovations to the asset return process are potentially non Gaussian. In particular, the Generalized LRNVR, or GLRNVR for short, applies if this distribution is skewed and leptokurtic. In our setting this corresponds to the general situation when using the N IG (a, b) distribution in (3) with b different from zero and a finite. However, the relationship obviously also holds when we consider the limiting case of a Gaussian distribution. Sufficient conditions for the GLRNVR to hold are stated in Proposition 3 in Duan (1999) and involves restrictions on the utility function of the representative agent. Under these assumptions Proposition 4 in the paper states that the risk neutralized dynamics of the system in (1) − (3) may be specified as Rt = mt (·; θm ) +

p −1 ht FNIG [Φ (Zt − λt )]

(14)

ht = g (hs , εs ; −∞ < s ≤ t − 1, θh )

(15)

εs = FN−1IG [Φ (Zs − λs )] if s ≥ t,

(16)

where Zt , conditional on Ft−1 , is a standard Gaussian variable under the risk neutral measure Q, and where λt is the solution to ´¯ h ³ i p ¯ E Q exp mt (·; θm ) + ht FN−1IG [Φ (Zt − λt )] ¯ Ft−1 = exp (rt ) .

(17)

In the above equation rt denotes the one period risk free interest rate at time s, and although this rate has to be deterministic it may in fact be time varying. −1 In the pricing system described by (14) − (17) above FNIG denotes the inverse cumulative distribution

function associated with the standardized N IG (a, b) distribution, whereas Φ denotes the standard Gaussian cumulative distribution function. Thus, in the limit when a tends to infinity such that the underlying distribution corresponds to the Gaussian it follows that FN−1IG [Φ (z)] = z, for any z, and in this situation the innovations in the risk neutral world remain Gaussian although with non zero mean. However, when

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−1 the underlying distribution departs from the Gaussian the transformation in FNIG [Φ (z)] yields innovations

under the risk neutral measure with the appropriate properties to be used when pricing e.g. options. In particular, note that when λs = 0 the innovations in (16) corresponds to random draws from the N IG (a, b) distribution. 2.2.1

The restriction to conditional normality

In the limiting special case of conditional Gaussianity it may be observed immediately that the restriction in (17) simplifies to

´¯ h ³p i ¯ mt (·; θm ) + ln E Q exp ht (Zt − λt ) ¯ Ft−1 = rt ,

since mt (·; θm ) is measurable with respect to the time t−1 information set by assumption. Furthermore, since Zt is standard Gaussian variable conditional on the information set Ft−1 under the risk neutral probability ¡ ¢ √ measure Q, the expectation above equals exp −λt ht + 12 ht . Thus, upon rearranging it follows that λt =

rt − mt (·; θm ) + 12 ht √ , ht

(18)

is the solution required in (17), and it is immediately seen that λt is a simple function of the mean specification.

2.3

A feasible specification of the NIG GARCH option pricing model

In principle the system above is completely self-contained. However, when it comes to actually implementing it problems may occur. In particular, problems may result from the requirement that λt be the solution to (17) since this may be extremely difficult to solve. In particular, a simple functional relationship like the one obtained in (18) for λt , given rt , mt (·; θm ), and ht , in the Gaussian special case will not be available. In the present paper we turn the problem slightly around and note that even in this situation there is a one to one correspondence between potential λt ’s and potential mean specifications mt (·; θm ) even though this is no longer simple. In particular, since mt (·; θm ) is measurable with respect to the time t−1 information set by assumption it follows that (17) may be rewritten as ´¯ h ³p i ¯ mt (·; θm ) = rt − ln E Q exp ht FN−1IG [Φ (Zt − λt )] ¯ Ft−1 .

(19)

Thus, from (19) it follows that instead of solving (17) for λt given rt , mt (·; θm ), and ht , we may specify λt directly and “imply” the corresponding mean specification, mt (·; θm ). It is immediately clear that this suggestion potentially complicates the estimation procedure since (19) would have to be evaluated numerically in search of the parameter estimates when estimating the system in (1) − (3). In fact, in the numerical part of Duan (1999) it is argued that this procedure becomes extremely demanding in terms of computing time. In the present paper we provide results showing that the procedure of implying the solution to (19) is in fact feasible and the estimation procedure does not necessarily 9

become computationally excessively demanding since this expectation can be calculated easily following the procedures outlined below in Section 3.1. While it is possible to specify λt in a quite flexible manner generally a choice have to be made about the specific structure of λt . In the present paper we will assume that λt = λ for any t, that is constant through time. Furthermore, we will assume that the interest rate is constant, that is rt = r for any t also. These two assumptions clearly simplifies the implementation of the model significantly and serve as obvious first benchmarks. 2.3.1

The restriction to conditional normality

While the restriction to a constant λ and r not only serves as a relevant first approximation in the general framework it also implies that we obtain the specification suggested in Duan (1995) in the Gaussian case. In particular, from (18) we see that when λ is specified an explicit form the mean may be derived under the assumption of Gaussianity. To be specific, what is obtained is the relationship in (4) which may be substituted directly into the system in (1) − (3). In this case the system to be estimated simply becomes p p 1 Rt = r + λ ht − ht + ht εt (20) 2 ht = g (hs , εs ; −∞ < s ≤ t − 1, θh ) (21) εt | Ft−1

∼ N (0, 1)

(22)

and from this we see that in this situation λ may be easily interpreted as the price of risk. Moreover, by substituting (4) into (14) − (16) it follows that the system to be used for pricing purposes becomes p 1 Rt = r − ht + ht Zt 2

(23)

ht = g (hs , εs ; −∞ < s ≤ t − 1, θh )

(24)

εs = (Zs − λ) if s ≥ t,

(25)

where Zt , conditional on Ft−1 , is a standard Gaussian variable under the risk neutral measure Q. When the P volatility dynamics are assumed to be of the simple GARCH form corresponding to ht = ω+ qi=1 αi ht−i ε2t−1 + Pp i=1 β i ht−i this limiting specification in fact corresponds exactly to the specification used in Duan (1995). Thus, this model will serve as an obvious benchmark for this paper.

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Implementation and estimation of the NIG GARCH option pricing model

One of the important properties of the GLRNVR framework of Duan (1999) is that a close link is provided between the observed asset return process and the process which has to be used for valuation of the corre10

sponding options. To be specific, note that by substituting (19) into the system in (1) − (3) it is possible to obtain the following specification for the return process to be used for estimation: ´¯ h ³p i p ¯ ht FN−1IG [Φ (εt − λ)] ¯ Ft−1 + ht εt Rt = r − ln E Q exp ht

εt | Ft−1

(26)

= g (hs , εs ; −∞ < s ≤ t − 1, θh )

(27)

∼ N IG (a, b) .

(28)

Comparing this system to the one used for pricing in (14)−(16) it is immediately clear that it is in fact possible to estimate all the necessary parameters from the historical returns. Thus, one of the major strengths of the proposed NIG GARCH framework is that cumbersome calibration procedures involving matching model option prices to observed prices to derive the model parameters may be avoided. However, before we can actually implement the NIG GARCH option pricing model we need to ob−1 tain procedures for evaluating the transformation of the random variables Zt through FNIG [Φ (Zt − λ)]

as well as for evaluating the logarithm of the expectation of the scaled exponential value of this, that is ¢¯ £ ¡√ ¤ −1 ln E Q exp ht FNIG [Φ (Zt − λ)] ¯ Ft−1 . Note that such procedures would be needed even if we were to use a calibration based method. In the following sections we detail first how each of these expressions can be calculated and approximated and we provide estimation results for the suggested models.

3.1

Approximation procedures

While numerical methods for calculating the necessary functionals in the symmetric situation are provided in Duan (1999) the paper claims that especially for estimation purposes these become extremely demanding in terms of computational time. In this section we extend the methods of Duan (1999) to a situation with conditional skewness and in each case we explore the transformations for the standardized N IG (a, b) distribution used in this paper. More importantly, however, we improve on the methods and we suggest methods for how to approximate the two functions efficiently. When it comes to actually implementing the algorithm these approximations are important not only because of simplicity and but also because they increase the computational speed. 3.1.1

Computing and approximating FN−1IG [Φ (Zt − λ)]

−1 [x] is proposed, where D represent the chosen In Duan (1999) a numerical scheme for calculating FD

distribution. This method involves approximating the Cumulative Distribution Function, or CDF, FD [xi ] at a sequence of points i = 1, 2, ... using the Probability Density Function, or PDF, fD . The approximation is piecewise linear in the x’s and the inverse value can therefore be found using the two values of xi ’s surrounding any value of x. For any finite number of points the calculated value is an approximation to −1 the true value of FD [x] although it should converge to the true value as the number of points used in the

approximation tends to infinity. 11

In the following we suggest to use a further approximation step which speeds up the calculations. This step is based on the observation that FN−1IG [Φ (Zt − λ)] is a smooth function of Zt and that it may therefore be well approximated using a low order polynomial in Zt . The method we propose consists of the following three steps: 1. Calculate an approximation to the CDF of the chosen distribution using an extension of the procedure outlined in Duan (1999). In particular, since the distribution under consideration is skewed a choice has to made as to how the algorithm should be initialized. In the present paper this is done by selecting a large negative value and building the approximation from this. If the value chosen is large enough the CDF should be approximately zero. In order to improve on the approximation, however, a numerical integration of the PDF to this point is performed. 2. Evaluate the expression FN−1IG [x] at a number of points equidistantly distributed between zero and one in probability according to the piecewise linear approximation obtained above. This corresponds to evaluating the function FN−1IG [Φ (y)] at the corresponding quantiles, y, of the Gaussian distribution. −1 Hence it may be realized that it also corresponds to evaluating the expression FNIG [Φ (Z − λ)] at

Z = y + λ. −1 3. Calculate an approximation to FNIG [Φ (Z − λ)] which may be realized to be a simple smooth function

of Z. In particular, this can be done by regressing the values of FN−1IG [Φ (Z − λ)] from the step above

on a set of transformations of the values of Z. −1 In Figure 1 the approximations to FNIG [Φ (Z − λ)] from step two above are plotted against the values

of Z. In the procedure of Duan (1999) we use 5, 000 points each spaced with 0.005 symmetrically around zero thus covering the interval from −12.5 to 12.5 to approximate the CDF. The sample of Z equals the 9, 999 quantile values from the Gaussian distribution and λ = 0.05. The top panel of the figure shows the one to one relation between the Z’s and FN−1IG [Φ (Z − λ)] in the Gaussian case. However, the figure also shows that the relationship remains smooth and well behaved even when leptokurtic or skewed distributions are used as is indicated by the middle and bottom panel which shows the relationship between the Z’s and −1 FNIG [Φ (Z − λ)] with the symmetric N IG (2, 0) and the N IG (2, 0.2) distributions respectively.

With these findings in mind we suggest that the function FN−1IG [Φ (Z − λ)] can be approximated well

by a low order polynomial in Z. In our experience good approximations can be obtained with a fifth order polynomial approximation in the Z’s, and this particular choice of approximation is used in the following. In particular, when performing this simple regression the ensuing R2 is 100% indicating that all the variation is explained. The resulting predicted values are superimposed as a line in the two bottom plots. When compared to the suggestion in Duan (1999) we note that the use of this approximation should speed up the computational work significantly because it involves less computational complexity. In particular, the approximation will have to be calculated only once for given values of the parameters of the chosen 12

distribution. For the rest of the procedure simple linear operations are used. Thus, if a large number of evaluations is needed the computational complexity is essentially that of evaluating a five degree polynomial. This is, however, not the case if the approximation is not used since each evaluation involves a search among the values of the approximated CDF constructed in step 2. Since the computational complexity of searching among n elements is approximately proportional to log (n) this method will be dominated by that of using the approximation unless n is very low and hence the approximation to the CDF very coarse. 3.1.2

¢¯ £ ¡√ ¤ Computing and approximating ln E Q exp ht FN−1IG [Φ (Zt − λ)] ¯ Ft−1

In Duan it is suggested that ¶¯ one¸can calculate the k’th element of the conditional expectation ∙ µ(1999) q ¯ (k) −1 Q E ht FN IG [Φ (Zt − λ)] ¯¯ Ft−1 using a “vector” Monte Carlo simulation with N independent exp © ªN standard Gaussian random variables, Zti i=1 , for which the inverse is calculated. Thus, the k’th element µq ¶ ¢¤ PN (k) −1 £ ¡ i should be approximated by N1 i=1 exp ht FNIG . However, since this procedure will Φ Zt − λ

have to be repeated for each value of ht it will be extremely time consuming, and this even so if the above −1 approximation to FNIG [Φ (Zt − λ)] is used.

In the following we suggest an approximation which speeds up the calculations. The procedure exploits the fact that ht is measurable with respect to the current information set and therefore the logarithm of the ¢¯ £ ¡√ ¤ expectation, ln E Q exp ht FN−1IG [Φ (Zt − λt )] ¯ Ft−1 , may be considered as a function of this variable. In √ particular, the expression turns out to be a relatively simple function of ht which can be easily approximated √ using a low order polynomial in ht . The method we propose consists of the following three steps: 1. Select a number of appropriate values for h. This can be done either by selecting random numbers from an appropriate distribution or by simply choosing a range of potential variances. In particular, if we believe that variances are IG distributed a random draw from this distribution may be used or a number of quantiles may be used. Secondly, select appropriate numbers for Z. While these may also be selected as a random draw from the Gaussian distribution a more sensible choice is to use the quantiles from the approximation procedure in the previous section. −1 2. Calculate the product of each h and the vector of FNIG [Φ (Z − λ)] based on the approximation scheme

above and exponentiate the values. The logarithm of the mean of this is an estimate of the value of ´¯ h ³√ i ¯ −1 ln E Q exp hFNIG [Φ (Z − λ)] ¯ Ft−1 for each h. In other words, we estimate the logarithm of the expectation by the logarithm of the mean of the exponential values of the product of the particular h

−1 and the approximation FNIG [Φ (Z − λ)]. ´¯ h ³√ i ¯ 3. Calculate an approximation to ln E Q exp hFN−1IG [Φ (Z − λ)] ¯ Ft−1 which may be realized to be √ a relatively simple smooth function of h. In particular, this can be done by regressing the values of ´¯ h ³√ i ¯ −1 ln E Q exp hFNIG [Φ (Z − λ)] ¯ Ft−1 from the step above on a set of transformations of the values √ of h.

13

´¯ h ³√ i ¯ In Figure 2 the approximations to the function ln E Q exp hFN−1IG [Φ (Z − λ)] ¯ Ft−1 from step two √ above are plotted against values of h. The sample of h equals the 999 quantile values from the Inverse Gaussian distribution with a = 2 and FN−1IG [Φ (Z − λ)] is calculated as specified in the previous section.

The top panel of the figure shows the approximation calculated in step two above for the Gaussian case together with the true relationship which is superimposed as a solid line. The panel shows that the approx√ imation is very close to the exact expectation which is given as 12 h − λ h. However, the figure also shows that the relationship remains smooth and well behaved even when leptokurtic or skewed distributions are √ used as is indicated by the middle and bottom panels which shows the relationship between the h’s and ´¯ h ³√ i ¯ −1 ln E Q exp hFNIG [Φ (Z − λ)] ¯ Ft−1 with the symmetric N IG (2, 0) and the N IG (2, 0.2) distributions respectively.

´¯ h ³√ i ¯ With these findings in mind we suggest that the function ln E Q exp hFN−1IG [Φ (Z − λ)] ¯ Ft−1 can be √ approximated well by a low order polynomial in h. In our experience good approximations can be obtained √ with a third order polynomial approximation in the h’s, and this particular choice of approximation is used in the following. In particular, when performing this simple regression the ensuing R2 is 100% indicating that all the variation is explained. The resulting predicted values are superimposed as a line in the two bottom plots. Again we note that compared to the suggestion in Duan (1999) the use of the above approximation should speed up the computational work significantly. In particular, as it was the case with the inverse transformations the approximation will need to be calculated only once and for the rest of the procedure simple linear operations are used. This is, however, not the case if the approximation is not used since each evaluation would involve repeating step 2 above. Although these calculations could be based on the same values of the inverse approximations in FN−1IG [Φ (Z − λ)] they would still involve the use of the computationally expensive

logarithmic and exponential functions a large number of times which will be computationally inefficient.

3.2

Estimation Procedure

With the approximations outlined above it becomes possible to estimate the system in (26) − (28) using a Maximum Likelihood based approach. In this section we provide estimation results for three major US stocks those being General Motors (GM), International Business Machines (IBM), and Merck & Company Inc (MRK) in Tables 1 through 3 respectively. The data used consists of time series of continuously compounded returns from 1976 through 1995 which have been corrected for dividend payments. The source of the data is the 1997 Stock File from CRSP (see CRSP (1998)). As the short rate, r, we take the 1 month LIBOR rate on December 29, 1995, at which time is was 5.4% annualized and continuously compounded. The data used corresponds to what was used in Stentoft (2005). In terms of estimation results, the Gaussian case serves as an appropriate benchmark for comparison since we know the exact form of the transformations. The results presented shows that it is in fact possible to obtain

14

parameter estimates which are statistically equal to those from the exact procedure in a matter of minutes. The estimation procedure is obviously more time consuming for the general case with N IG (a, b) innovations. However, it remains feasible and results can be obtained in a matter of minutes on a standard laptop computer. To speed up the estimation process we apply the concept of variance targeting originally proposed in Engle & Mezrich (1996) which ensures that the implied unconditional level of the variance corresponds to the historical volatility actually observed. The procedure can be implemented in our framework simply ¡ ¡ ¢ ¢ by setting ω = V ar (Rt ) ∗ 1 − α 1 + γ 2 − β , with γ = 0 when the volatility is of the GARCH type and

α = β = γ = 0 in the constant volatility case. 3.2.1

Estimation results for the Gaussian GARCH model

Panel A of the three tables provides estimation results for the Gaussian special case using the estimation procedure derived in the present paper. We start by comparing the results reported here to the results of Stentoft (2005) which allows us to gauge the validity of the approximations suggested above. When doing so it is observed that for all the volatility specifications similar results are obtained which validates the approximation procedures. In terms of the actual estimates is suffices to say that all the parameters are significant and that the asymmetry parameters in the NGARCH specification has the expected sign. We also note that allowing for GARCH volatility specifications is important and does lower the test statistics for ARCH and serial dependency in the standardized returns as well as the squared standardized returns. However, Panel A of the three tables also show that a major problem is that the assumption of normality can be rejected for all series. This is evident from the J-B test statistics which are all significant at any reasonable level. More detailed tests, although not reported here, indicate that it is particularly in terms of excess kurtosis that departures are found although significant skewness is also found to be present in the standardized residuals for most of the models. The problem with non Gaussianity is also evident from the left hand panels of Figure 3 which plots the log density of the standardized residuals for the Gaussian NGARCH models with the Gaussian distribution superimposed as the dotted line. These plots clearly indicate that the standardized residuals are skewed and leptokurtic. 3.2.2

Estimation results for the NIG GARCH model

Panel B of the three tables provides the corresponding estimation results for the NIG GARCH models. Examining first the actual estimates of the parameters related to the volatility specifications the tables show that the leverage effect remains important. For all series the γ parameter in the NGARCH specification is estimated significantly different from zero and with the expected sign as it was the case in the Gaussian case. Although the point estimates are generally of the same order of magnitude as in the Gaussian case this is not so for IBM where the γ estimate with a NIG distribution is approximately half that of the corresponding estimate assuming a Gaussian distribution. For the other volatility parameters however the conclusion holds

15

that these are quite similar when compared across distribution. We next examine the parameters related to the N IG (a, b) distribution for which the tables clearly show that departures from the Gaussian distribution are important. In particular, the point estimate for the a parameter which governs the degree of leptokurtosis implies a conditional distribution which is quite leptokurtic for all the series. Furthermore, for all the series the b parameter, which governs the skewness of the conditional distribution, is estimated significantly different from zero. These results also allow us to conclude that the symmetric although leptokurtic version of the NIG model which has b = 0 would be too simple a model for the series considered here. This conclusion does in fact also hold when the estimation results are compared to what would be obtained with e.g. the Generalized Error distribution although the results are not reported here. Furthermore, although the GED models and the symmetric N IG (a, 0) models have equal characteristics in terms of estimates of the volatility parameters as well as residual characteristics the latter model dominates the GED model in terms of likelihood values. This finding further provides evidence favouring the use of the NIG distribution in the generalized GARCH framework. The importance of the distributional flexibility may also be analyzed by plotting the distribution of the standardized residuals. This is done in the right hand panels of Figure 3 for the NIG NGARCH which also shows the NIG distribution corresponding to the estimated parameters superimposed as a dotted line. These plots may be readily compared in terms of fit to what was obtained when Gaussianity was assumed. When this is done it is seen that the fit is much improved. In particular, the improvement is pronounced in the tails of the conditional distributions. When comparing the different NIG GARCH specifications to the equivalent Gaussian specifications the tables show large increases in the likelihood values when allowing for skewness and leptokurtosis indicating the importance of the extra modelling flexibility. However, a better comparison may be performed by comparing the Schwarz Information Criteria which is reported in the last line of each panel in each table. This criteria takes into consideration the additional number of parameters and the preferred model is the one with the minimum value. The panels show that for all the return series the model achieving the lowest value is in fact the NIG NGARCH model. We note that when only Gaussian models are considered the NGARCH specification is also the preferred one for all series. On the other hand when either of the volatility specifications is compared across distribution the NIG version is always the preferred one. Thus, in all respects our estimation results indicate a consistent ranking of the volatility specifications across the different distributions and of the distributions across different volatility specifications in favor of the N IG (a, b) distribution and the NGARCH volatility specification. To be specific, we obtain parameter estimates for the N IG (a, b) distribution which are significantly different from the Gaussian special case, the fit for the standardized residuals is greatly improved, and when comparing the Schwarz Information Criteria the NIG GARCH models are preferred to the Gaussian alternatives. Since these conclusions hold for all the returns series we may conclude that the skewed and leptokurtic NIG GARCH model is an important

16

extension to the Gaussian GARCH model for the return series considered in the present paper.

4

Pricing procedure for and properties of the NIG GARCH model

By substituting (19) into (14) − (16) it is possible to derive the following specification for the risk neutralized return process to be used for pricing purposes: ´¯ h ³p i p ¯ −1 ht FN−1IG [Φ (Zt − λ)] ¯ Ft−1 + ht FNIG [Φ (Zt − λ)] , Rt = r − ln E Q exp

(29)

ht = g (hs , εs ; −∞ < s ≤ t − 1, θh ) ,

(30)

−1 [Φ (Zs − λ)] if s ≥ t. εs = FNIG

(31)

With the approximation schemes outlined above and with the appropriate parameter values this system yields the dynamics to be used to price options in a NIG GARCH framework. However, one potential problem with the framework is that no closed form solutions for the option prices exist even in the case where European options are considered. The reason is that in this general framework it is difficult if not impossible to obtain an analytical expression for the distribution of the future value of the underlying asset. Therefore alternative numerical procedures are needed for option pricing applications. In the following we describe a numerical procedure which can be readily used with the NIG GARCH option pricing model. The proposed method uses a simulation framework along the lines of that used in Stentoft (2005) which we generalize to the case of non Gaussian distributions. The framework is then used to perform a Monte Carlo study of the properties of the NIG GARCH model. The results of this study is compared to previous results in the Gaussian case in Stentoft (2005) and to results from Duan (1999) when applicable.

4.1

Pricing Procedure

While it is possible to apply different approaches to obtain option prices given the structure of the system in (29) − (31) the obvious choice in the present setting is to use Monte Carlo simulation. In particular, random future paths can be easily simulated under the appropriate risk neutral dynamics according to this system. Once we have obtained a sample of paths the Least Squares Monte Carlo method, or LSM method, of Longstaff & Schwartz (2001) can be used to obtain the American option prices. To be specific, the numerical procedure to be used involves a simulation step and a pricing step. 4.1.1

Simulation step

In the first step a large number of future paths for the underlying and the conditional variances are simulated. In particular this is done by successively applying the system in (29) − (31) until the maturity date of the 17

option. Such a simulation can be easily performed for a large number of paths at each step with minimum added computational complexity. Although the simulation involves transforming the Gaussian innovations, the Z’s, at every step along all paths this is in fact feasible when the approximations from Section 3.1 are used. In particular, because the approximations need only be calculated once at the beginning of the simulation the computational complexity remains approximately linear in the number of paths and the number of steps in the simulation. 4.1.2

Pricing step

Given the full sample of random paths the pricing step is initiated at the maturity of the option. At this time it is possible to decide along each path if the option should be exercised since the future value trivially equals zero. Working backwards through time, however, this is not so. In particular, assuming exact knowledge of the future value of the underlying asset and thereby perfect foresight would lead to biased estimates of the option prices. The LSM method offers a solution to this which uses all the cross sectional information at a given time step in the simulation to estimate the expected value of holding the option for one more period conditional on the state. In particular, this is done by regressing the pathwise future values on appropriate transformations of the current state variables for all the paths for which early exercise is potentially optimal. The fitted values from this regression can now be used as estimates of the conditional expected value of holding the option for one more period and a decision of wether to exercise or not can be made along each path. Since the estimates used are based on all the paths no assumption of pathwise perfect foresight is made and this mitigates the potential high bias. The validity of using the LSM simulation approach in a GARCH framework is established in Stentoft (2005) in which the method is compared to the Markov Chain method of Duan & Simonato (2001) and to the lattice based algorithm of Ritchken & Trevor (1999). That paper shows that appropriate American option prices can be obtained by using a simple second order polynomial in the stock price and in the one step ahead level of the volatility together with the cross product and a constant term in the cross sectional regression in the LSM method. Since the extension from the Gaussian GARCH model to the NIG GARCH model involves no extra complexity in terms of the dynamics to be used for pricing this simulation based approach should remain equally valid in this setting.

4.2

Pricing Properties

To illustrate the pricing properties we perform a Monte Carlo study using the same basic setup as was used in Stentoft (2005). In particular, we use the same set of artificial options and the same parameter values. However, we augment the sample studied in two directions. First of all we include options with an ultra short maturity of only 7 trading days. The reason for doing this is that we expect that these options are mostly 18

effected by a differences in the underlying distribution. Secondly, we price both American style put options and call options since the introduced asymmetries might have different implications for the different styles of options. For the American feature to be interesting for the call options dividends need to be included. Thus, we include dividend payments in the simulation at a rate of 3% annually paid out continuously. Since the primary goal of this study is to examine the effect on option prices when the underlying distribution is skewed and leptokurtic we report results for the Gaussian distribution which constitutes a natural benchmark, for a symmetric NIG distribution where the parameter b is restricted to zero, and for the general NIG distribution with non zero b and finite a. Thus, while the latter distribution allows for both skewness and leptokurtosis we also consider a leptokurtic but symmetric distribution in order to assess the importance of either of these departures from a Gaussian distributions. With respect to the parameter values we chose these in order to correspond to the empirical values and set a = 2 and b = 0.2. We note that because of the consistency of the QMLE results for the two volatility specifications under consideration we are ensured that the same variance parameter values should in fact be used in the simulations for the different distributions. 4.2.1

Pricing results for GARCH models

Columns five and six of Panel A in Table 4 reports the American put price estimates and the standard errors obtained for a symmetric NIG distribution with a = 2 and b = 0 with a GARCH volatility process. In addition, column seven reports the difference between the Gaussian GARCH estimate and the symmetric NIG GARCH estimate relative to the latter. This column, which is headed “Rel Diff”, can be interpreted as the mispricing that would occur if a Gaussian GARCH model had been used when the true model was in fact a NIG GARCH model with N IG(2, 0) distributed errors. The table shows that when the Gaussian GARCH model is compared to the symmetric NIG GARCH model it severely underprices the short term out of the money options. On the other hand, the at the money options are slightly overpriced by the Gaussian GARCH model. Turning the attention to the corresponding columns in Panel B of the table we see that equivalent results are found for the estimated American call prices which confirms the findings in Duan (1999) where the leptokurtic but symmetric GED distribution is used to price European style options. Columns eight and nine of Panels A and B in Table 4 reports the American option price estimates and the standard errors when conditional skewness is allowed for in the NIG distribution by setting b = 0.2. In addition, column ten compares these price estimates with the Gaussian GARCH model for the puts and calls respectively in the same manner as it was done for the symmetric NIG model. This column shows that although the underpricing of the out of the money options is smaller than for the symmetric N IG(2, 0) for the put options in Panel A it is much larger for the call options in Panel B. Thus, the results indicate that there are in fact important differences in terms of option pricing when skewness in introduced. Note though, that as the term to maturity increases eventually the effects of skewness and excess kurtosis will vanish as a

19

central limit starts to apply for the combined innovations. 4.2.2

Pricing results for NGARCH models

For the NGARCH model, Table 5 shows that the overall results are in line with those for the GARCH specification. In particular, if the Gaussian GARCH model had been used large negative pricing errors would be observed for out of the money options in general and out of the money short term options in particular. However, if the volatility specification has asymmetries the amount of mispricing by the Gaussian model is reduced somewhat for the put options relative to the various alternative distributions, whereas it is increased significantly for the call options. This holds for both types of non Gaussian distributions, and again it is seen that there are important differences in the effect depending on the style of the options. However, while the effect on option pricing of introducing leverage in the volatility specification is in the same direction for all distributions the relative mispricing which would be observed with a Gaussian model is significantly different when the distribution is also allowed to be conditionally skewed compared to what is seen if the true distribution although leptokurtic is symmetric. In particular, when comparing the two panels of Table 5 it is seen that if the true model is based on the symmetric N IG(2, 0) distribution the mispricing by the Gaussian model almost becomes symmetric between put and call options. This is not the case for the asymmetric model based on the N IG (2, 0.2) distribution for which the tables show that severely asymmetric mispricing is observed for the Gaussian model. Thus, there is in fact important effects from allowing the conditional distribution to be skewed when the variance process has leverage as is possible in the NGARCH specification.

5

Empirical performance of option pricing with the NIG GARCH model

In Stentoft (2005) the Gaussian GARCH model was used to price options on a sample of individual stocks. The paper showed that these models reduced the pricing errors compared to the constant volatility BlackScholes-Merton model. However, that paper also showed that some pricing errors remain. To be specific, the results indicated that for the Gaussian NGARCH specification the pricing errors were somewhat smaller for the put options than for the call options. We note that the results of our Monte Carlo study would indicate that the asymmetric NIG NGARCH model could be the most appropriate specification. Thus, in this section we use the same sample with the NIG GARCH option pricing model. To be specific, the sample used for our empirical application consists of weekly observation on options traded between 1991 and 1995 on the three individual stocks considered in Section 3.2 that is General Motors (GM), International Business Machines (IBM), and Merck & Company Inc. (MRK). These options are American style, traded on the Chicago Board Options Exchange (CBOE), and were made available 20

through the Berkeley Options Data Base (BODB). At any particular day the sample contains an end of day option quote observed immediately before 15.00 for all existing contracts, that is for each combination of strike price and maturity. The data used for estimation of the NIG GARCH models as well as the estimation procedure corresponds to that used for estimation previously in Section 3.2. However, at any given day where options are priced only the available historical data is used. This means that when options on a given Wednesday are to be priced the sample period used for estimation is January 2, 1976, to the Tuesday immediately before this Wednesday. As we move forward in time this sample will increase. The estimated parameters are then used in the pricing procedure where we for simplicity assume that future dividend payments are known in advance and spill over fully on the simulated option prices. As the risk free interest rate we take the LIBOR rate on the last day used for estimation. Thus, although the same constant interest rate is used both in the estimation and in the simulation at any given day, in fact the interest rate does vary from one Wednesday to the next Wednesday. We note that both the assumption on dividends and on interest rates could be changed as long as they remain known at the time of pricing. In the actual pricing step 20, 000 paths are used and the LSM method is implemented assuming exercise once a day. In the cross sectional regressions second order polynomials in the stock price and in the one step ahead level of the volatility together with the cross product and a constant term are used as it was the case in the Monte Carlo study. We note that the calculated prices may have some approximation error due to the finite choices of simulated paths and regressors. However, since we use the same procedure to price options for all specifications when comparing the methods this poses little or no problem.

5.1

Overall pricing results for the NIG GARCH models

The actual pricing results for the NIG GARCH are presented in Table 6 whereas Table 7 report the pricing errors relative to those obtained with corresponding Gaussian specifications. In both tables Panel A provides the results when all the options are combined whereas Panel B through D provides the pricing results for the individual companies. Each panel provides pricing results for all the options as well as for the put options and the call options separately. As the Monte Carlo results indicated that the skewed and leptokurtic NIG specification was the most appropriate we refrain from reporting results for the symmetric NIG specification. These results do however confirm that skewness is important and may be obtained from the author upon request. 5.1.1

Pricing performance of the NIG GARCH model

To gauge the pricing performance we use two of the relevant metrics from the literature. In particular, letting Pk and P˜k denote the k’th observed price respectively the k’th model price, these are the relative mean bias, PK (P˜k −Pk ) PK (P˜k −Pk )2 RBIAS ≡ K −1 k=1 Pk , and the relative mean squared error, RSE ≡ K −1 k=1 . The P2 k

21

reason for using these relative metrics is that the actual option prices varies significantly between the in the money and out of the money options. Thus, if absolute metrics were used much more weight would be put on pricing the in the money options correctly than on pricing the out of the money options correctly. We consider first Panel A of Table 6 from which it is observed that large improvements are found for the NIG GARCH model when compared to the CV model with NIG errors. Thus, the first conclusion is that allowing the volatility to be time varying is important when pricing American options on the individual stocks considered in this paper. From the panel it is seen that these results hold true when the put options and the call options are considered individually also, and the panel shows that the CV model is the one with the largest pricing errors in all cases using either of the chosen metrics. In addition to this, the panel shows that not only are GARCH features important but so are asymmetries. In particular, the NGARCH model is the best performing one in half the cases, and this specification works particularly well for the put options. Next consider the individual results in Panels B through D. These results again show that using the CV model yields the largest pricing errors irrespective of which underlying stock, which metric, or which style of options are considered. However, the three panels also show that while there are improvements in the pricing performance for all the stocks the magnitude of the improvements varies. Thus, while the panels show that the overall pricing errors of the CV model is about five times that of either of the GARCH specifications for GM, the improvements is much less pronounced for MRK, with the results of IBM somewhat in between. However, when the models are ranked it turns out that the best performing model on an individual basis is the NGARCH model for the majority of the cases. In particular, when using the RSE metric the panels show that when both puts and calls are considered asymmetries in the volatility process appear to be quite important. 5.1.2

Pricing performance relative to the Gaussian GARCH model

The results reported above on the absolute pricing performance of the NIG GARCH model corresponds in general to what was found for the Gaussian case in Stentoft (2005). Thus it makes sense to compare these results directly and consider the improvements obtained with the NIG GARCH model over the corresponding Gaussian GARCH model as it is done in Table 7. In particular, this table reports the difference in pricing performance metrics for each volatility specification, specifically that of the Gaussian model minus that of the NIG model, relative to the metrics value for the Gaussian model. Thus, in the table a negative cell value indicates that the NIG version has smaller pricing error for this model whereas a positive cell value indicates that the Gaussian model has in fact the smallest pricing error. The actual cell value may be interpreted as the decrease in error, respectively the increase in error, obtained by using the NIG model instead of the Gaussian equivalent. We consider first Panel A of the table from which it is observed that the numbers overall favour the NIG version of the GARCH models. In particular, when considering all the options the cell values are negative for

22

both metrics and for all volatility specifications. Moreover, the results indicate that the NIG model performs relatively better than the Gaussian model when the volatility process is allowed to be time varying with the largest improvements for the NGARCH specification. Panel A also indicates that this relative improvement is consistent when put and call options are considered individually except for the put options when the RBIAS metric is considered and this particularly so for the NGARCH specification. However, this is more than anything a result of the exceptional pricing performance of the Gaussian model in this situation. In particular, for the put options the RBIAS for the Gaussian NGARCH specification is approximately 8% whereas it is more than 12% for the call options. These results are not shown but available on request. Thus, although the average RBIAS for both puts and calls remains at the same level as with the GARCH specification this clearly indicates a severe asymmetry in the Gaussian NGARCH pricing performance. As mentioned above once skewness and leptokurtosis is allowed for in the conditional distribution this asymmetry in the pricing performance is significantly mitigated. Next consider Panels B through D of Table 7 showing the results for the individual stocks. From these panels it becomes clear that while there are differences in size of the relative improvements such improvements are found in the RSE metric for the three stocks and in the RBIAS metric for all but MRK. The panels also show that for some of the specifications large decreases in the pricing errors occur when allowing the conditional distribution to be of the NIG type. In particular, the panels show that for GM the RBIAS is reduced in excess of 48% for the GARCH and 33% for the NGARCH specifications when the NIG model is used instead of the Gaussian model. Furthermore, for the RSE the improvements are in excess of 10% for GM and in excess of 15% for IBM for the two specifications when the NIG model is compared to the Gaussian alternative. When considering the results for put and call options individually in Panels B through D the same pattern is observed for the three individual stocks. In particular, the panels shows that the only specification for which the Gaussian model provides the smallest pricing errors is for the NGARCH specification when the RBIAS metric is used and when put options are considered. For the call options, which constitutes the most important part of the sample, the NIG model consistently provides smaller pricing errors across volatility specifications and metrics. The relatively poor performance according to the RBIAS metric for the put options is once again related to the large asymmetries found with this specification when the Gaussian distribution is used for particularly GM and IBM. For MRK this asymmetry is less pronounced as is the difference in performance.

5.2

Moneyness and maturity effects

From the Monte Carlo study in Section 4 we know that introducing skewness and excess kurtosis generates a virtual smile or smirk in the mispricing which would be observed with a Gaussian GARCH model. Thus, in this section the pricing results are reported for different categories of moneyness and maturity.

23

Throughout we measure moneyness as the ratio between the asset price and the discounted strike price, M = S/ (K ∗ exp (−r ∗ T )). This puts long term options and short term options on a more equivalent basis. As the cut of point we choose 2.5% and thus in the money call options are options with a moneyness in excess of 102.5% whereas out of the money call options have M < 97.5%. For the put options this is reversed. When it comes to maturity the Monte Carlo study showed that the pricing differences were related to the time to maturity. In particular, the study showed that the pricing differences between e.g. a NIG GARCH model and the Gaussian counterpart are relatively larger for short term options than for long term options which is to be expected as a central limit theorem starts to apply for the combined innovations. Thus, in this section we also report results from partitioning the sample of options into short term options, denoted ST, with less than T = 42 trading days to maturity, middle term options, denoted MT, with between T = 42 and T = 126 trading days to maturity, and the long term options, denoted LT, with more than T = 126 trading days to maturity. 5.2.1

Moneyness effects

Panel A of Table 8 shows the overall results in terms of moneyness. From this panel it is clear that as we move towards the out of the money region the pricing errors from the NIG GARCH option pricing model increase. This holds both for the RBIAS and the RSE metrics and for all the NIG GARCH models. It remains true though that the CV model is outperformed by the specifications with time varying volatility and in most of the partitions an asymmetric model is the best performing one. In particular, when the RSE metric is used the NIG NGARCH model has the lowest pricing errors for all partitions. In Panel B of Table 8 the overall results of the NIG GARCH models form Panel A are compared to those from the Gaussian GARCH models. From this panel it is clear that in spite of the worsening in absolute performance of the NIG GARCH model there is in fact no worsening of the performance relative to the Gaussian GARCH model as we move away from being in the money. In fact there is a tendency for the NIG GARCH model to perform even better relative to the Gaussian GARCH models for the OTM option than for the ITM options. Furthermore, when the RSE metric is used the panel shows that the NIG specifications actually outperforms the Gaussian specifications for all options for the GARCH and NGARCH models, and for the OTM options the relative improvement is as much as approximately 14% and 19% respectively. We note that one of the conclusions from Stentoft (2005) was that it was particularly for the OTM options that pricing errors persisted when the Gaussian GARCH models were applied empirically. Thus, the results above indicate that the NIG GARCH models provide an important extension to the corresponding Gaussian models. This confirms our conjecture based on the results from Section 4 on the model properties.

24

5.2.2

Maturity effects

Panel C of Table 8 shows the overall results in terms of maturity. From this panel it is observed that there is no clear relationship between the pricing performance in absolute terms and maturity. In particular, while the RSE decrease the RBIAS errors increase slightly for all volatility specifications when time to maturity is increased. In Panel D of Table 8 the overall results of the NIG GARCH models from Panel C are compared to those from the Gaussian GARCH models. From this panel it can be seen that while the RSE pricing errors decrease in absolute terms with maturity this is in fact not so in relative terms. In particular, for the ST options the NIG GARCH specification improves on the Gaussian alternative with approximately 16%. For the LT options the improvement is less pronounced at approximately 9%. The same pattern may be found for the CV specification and confirms that as the time to maturity increases the effects of having excess kurtosis and skewness in the return distributions vanishing as a central limit theorem starts to apply in the long run. For the NGARCH specification there does not appear to be any relationship between the time to maturity and the improvements which for all categories remains approximately 17% to 18%. We note that another conclusion from Stentoft (2005) was that it was particularly for the short term options that large pricing errors persisted when the Gaussian GARCH models were applied empirically. Thus, the results above again indicate that the NIG GARCH models provide an important extension to the corresponding Gaussian models. Again this confirms our conjecture based on the results from Section 4 on the model properties. 5.2.3

Predictive regressions

The discussion above suggests that there might be a systematic relationship between the pricing errors of the various GARCH models and variables such as the maturity and the moneyness of the option for both of the assumed underlying distributions. In order to analyze this further we performed the following regression of the pricing errors for each of the individual stocks and each of the two distributions using the price estimates obtained with the GARCH or NGARCH volatility specifications: RBIAS

= α + β T T + β T 2 T 2 + β M M + β M 2 M 2 + β T ∗M T ∗ M + αP + β P,T TP + β P,T 2 TP2 + β P,M MP + β P,M 2 MP2 + β P,T ∗M TP ∗ MP + β σ h2 + ε.

(32)

Here the α’s are constant terms, a subscript P denotes variables which are non-zero only for the put options, and h2 is the one step ahead variance at the time the option is priced. The intuition behind the regression in (32) is that since all the right hand side variables are in the information set at the time the option is priced there should be no relationship between these variables and the ensuing pricing error if the model is correct. Thus, when the correct model is used for option pricing 25

one should find that all estimated coefficients are statistically insignificant and that there is no explanatory power of either of the regressors. Instead of tabulating the parameter estimates we chose to report only the R2 although the actual regression results are available from the author on request. In general we find that when performing the regression in (32) for a NIG specification the predictive power of the regressors is always less than for a similar Gaussian model indicating that the latter model produce pricing errors with somewhat more systematic variation that can not be accounted for. In particular, for GM the actual adjusted R2 are 21% compared to 24% for the NIG and Gaussian GARCH models respectively and 18% compared to 24% for the NIG and Gaussian NGARCH models respectively. For IBM the values are in the same order 25% compared to 26% and 26% compared to 32% whereas for GM they are 28% compared to 28% and 30% compared to 33%. Thus it is seen that the conclusions are consistent across the stocks and all indicate that the NIG GARCH models provides an important extension of the Gaussian counterparts.

6

Conclusion

In this paper we propose a feasible way to price American options in a framework with time varying volatility and conditional skewness and leptokurtosis. To be specific, we use a Generalized Autoregressive Conditional Heteroskedasticity framework where the conditional distribution is the Normal Inverse Gaussian. We show how the Generalized Local Risk Neutral Valuation Relationship of Duan (1999) can be used in this framework Because of the departure from the Gaussian framework the application of the NIG GARCH model is computationally more complex. In particular, in order to implement the model it is necessary to derive methods for computing functions of normal variates and conditional first moments of these. In the present paper computationally easy and efficient approximations for doing this are suggested which rely only on the potential for efficient approximations of smooth functions with low order polynomials. We provide estimation results which support the use of the Normal Inverse Gaussian distribution as an important extension of the Gaussian model for modelling asset returns. Through a Monte Carlo experiment we show that the ability of the NIG GARCH model to accommodate conditional skewness and leptokurtosis can explain some of the systematic pricing errors found in recent empirical work on option pricing with time varying volatility. In particular, the NIG GARCH model has the ability to generate higher prices than the Gaussian GARCH model for out of the money put and call options particularly of short maturity. Since such findings have been documented empirically the results support the use of the Normal Inverse Gaussian distribution as an important extension of the Gaussian model for option pricing purposes. Finally, we perform an empirical evaluation of the NIG GARCH model in order to evaluate its potential when it comes to pricing American options on a number of individual US stocks. The results provide

26

evidence that favours the NIG GARCH model over the Gaussian alternative. In particular, improvements are found for out of the money options and for short maturity options. Since these options are among the most actively traded the proposed extensions to the classical Gaussian framework should be considered very important. In the empirical application the importance of the systematic component in the pricing errors are also examined, and it is found that these are generally smaller for the NIG NGARCH model than for the Gaussian NGARCH model. In terms of future research we note that this paper uses a very flexible estimation and pricing framework which can accommodate many potential extensions of and alternatives to the NIG GARCH models considered here. Examining such extensions within the framework should definitely be the focus of future research and we are currently conducting a large scale comparison of alternative conditional distributions.

References Andersson, J. (2001), ‘On the Normal Inverse Gaussian Stochastic Volatility Model’, Journal of Business and Economic Statistics 19(1), 44—54. Barndorff-Nielsen, O. E. (1997), ‘Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling’, Scandinavian Journal of Statistics 24, 1—13. Barndorff-Nielsen, O. E. & Shephard, N. (2001), ‘Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics’, Journal of the Royal Statistical Society, Series B 63, 167—241. Black, F. & Scholes, M. (1973), ‘The Pricing of Options and Corporate Liabilities’, Journal of Political Economy 81, 637—654. Bollerslev, T. (1986), ‘Generalized Autoregressive Conditional Heteroskedasticity’, Journal of Econometrics 31, 307—327. Bollerslev, T. (1987), ‘A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return’, Review of Economics and Statistics pp. 542—547. Bollerslev, T. & Mikkelsen, H. O. (1996), ‘Modelling and Pricing Long Memory in Stock Market Volatility’, Journal of Econometrics 73, 151—184. Bollerslev, T. & Mikkelsen, H. O. (1999), ‘Long-Term Equity Anticipation Securities and Stock Market Volatility Dynamics’, Journal of Econometrics 92, 75—99. Carr, P. & Wu, L. (2004), ‘Time-changed Lévy processes and option pricing’, Journal of Financial Economics 71, 113—141.

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Christoffersen, P., Heston, S. & Jacobs, K. (2006), ‘Option valuation with vonditional skewness’, Journal of Econometrics 131, 253—284. Christoffersen, P. & Jacobs, K. (2004), ‘Which GARCH Model for Option Valuation?’, Management Science 50(9), 1204—1221. Clark, P. K. (1973), ‘A subordinated stochastic process model with finite variance for speculative prices’, Econometrica 41(1), 135—155. CRSP (1998), CRSP Stock File Guide, Center for Research in Security Prices, Graduate School of Business, University of Chicago. Duan, J.-C. (1995), ‘The GARCH Option Pricing Model’, Mathematical Finance 5(1), 13—32. Duan, J.-C. (1999), ‘Conditionally Fat-Tailed Distributions and the Volatility Smile in Options’, Mimio, Hong Kong University of Science and Technology . Duan, J.-C. & Simonato, J.-G. (2001), ‘American Option Pricing under GARCH by a Markov Chain Approximation’, Journal of Economic Dynamics and Control 25, 1689—1718. Duan, J.-C. & Zhang, H. (2001), ‘Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach’, Journal of Banking and Finance 25, 1989—2014. Eberlein, E. (2001), Application of generalized hyperbolic Lévy motions to finance, in O. E. BarndorffNielsen, T. Mikosch & S. Resnick, eds, ‘Lévy Processes: Theory and Apllications’, Birkhauser Verlag, Boston, pp. 319—336. Engle, R. F. (1982), ‘Autoregressive Conditional Heteroscedasticity With Estimates of The Variance of United Kingdom Inflation’, Econometrica 50(4), 987—1007. Engle, R. F. & Ng, V. K. (1993), ‘Measuring and Testing the Impact of News on Volatility’, Journal of Finance 48(5), 1749—1778. Engle, R. & Mezrich, J. (1996), ‘GARCH for Groups’, Risk 9(8), 36—40. Forsberg, L. & Bollerslev, T. (2002), ‘Bridging the Gap Between the Distribution of Realized (ECU) Volatility and ARCH Modeling (of the EURO): The GARCH Normal Inverse Gaussian Model’, Journal of Applied Econometrics 17, 535—548. Hafner, C. M. & Herwartz, H. (2001), ‘Options pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis’, Journal of Empirical Finance 8, 1—34. Härdle, W. & Hafner, C. M. (2000), ‘Discrete time option pricing with flexible volatility estimation’, Finance and Stochastics 4, 189—207. 28

Heston, S. L. & Nandi, S. (2000), ‘A Closed-Form GARCH Option Valuation Model’, Review of Financial Studies 13(3), 585—625. Hsieh, K. C. & Ritchken, P. (2005), ‘An Empirical Comparison of GARCH Option Pricing Models’, Review of Derivatives Research 8, 129—150. Huang, J.-Z. & Wu, L. (2004), ‘Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes’, Journal of Finance LIX(3), 1405—1439. Jensen, M. B. & Lunde, A. (2001), ‘The NIG-S&ARCH model: a fat-tailed, stochastic, and autoregressive conditional heteroskedastic volatility model’, Econometrics Journal 4, 319—342. Lambert, P. & Laurent, S. (2001), ‘Modelling Financial Time Series Using GARCH-Type Models with a Skewed Student Distribution for the Innovations’, Mimeo, Université de Liège . Lehar, A., Scheicher, M. & Schittenkopf, C. (2002), ‘GARCH vs. stochastic volatility: Option pricing and risk management’, Journal of Banking and Finance 26, 323—345. Lehnert, T. (2003), ‘Explaining Smiles: GARCH Option Pricing with Conditional Leptokurtosis and Skewness’, Journal of Derivatives 10(3), 27—39. Longstaff, F. A. & Schwartz, E. S. (2001), ‘Valuing American Options by Simulation: A Simple Least-Squares Approach’, Review of Financial Studies 14, 113—147. Merton, R. C. (1973), ‘Theory of Rational Option Pricing’, Bell Journal of Economics and Management Science 4, 141—183. Nelson, D. B. (1991), ‘Conditional Heteroskedasticity In Asset Returns: A New Approach’, Econometrica 59(2), 347—370. Ritchken, P. & Trevor, R. (1999), ‘Pricing Options under Generalized GARCH and Stochastic Volatility Processes’, Journal of Finance 59(1), 377—402. Stentoft, L. (2005), ‘Pricing American Options When the Underlying Asset follows GARCH Processes’, Journal of Empirical Finance 12(4), 576—611.

29

Figures and Tables

Gaussian_inverted_values × Z

2.5 0.0 −2.5 −3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

NIG(2,0)_inverted_values × Z NIG(2,0) inverted values (fitted) × Z

5

0

−5 −3.5

−3.0

−2.5

−2.0

−1.5

NIG(2,0.2)_inverted_values × Z NIG(2,0.2) inverted values (fitted) × Z

5

0

−5 −3.5

−3.0

−2.5

−2.0

−1.5

−1 Figure 1: Cross-plots of the approximated values of FNIG [Φ (Z − λ)] as a function of Z equal to the 9, 999 quantile values from the Gaussian distribution with λ = 0.05. The top panel shows the results for the Gaussian distribution, the middle panel shows the results for the symmetric NIG distribution with a = 2 and b = 0, and the bottom panel shows the results for the NIG distribution with a = 2 and b = 0.2. Superimposed on the two bottom plots as a solid line is the estimated relationship from the suggested 5 degree polynomial approximation.

30

−0.0005

−0.0010 0.005

Gaussian expectation × sqrt(h_t) Gaussian expectation (true) × sqrt(h_t)

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.020

0.025

0.030

0.035

0.040

0.045

0.025

0.030

0.035

0.040

0.045

−0.0005

−0.0010 0.005

NIG(2,0) expectation × sqrt(h_t) NIG(2,0) expectation (fitted) × sqrt(h_t)

0.010

0.015

−0.0005

−0.0010 0.005

NIG(2,0.2) expectation × sqrt(h_t) NIG(2,0.2) expectation (fitted) × sqrt(h_t)

0.010

0.015

0.020

¢¯ £ ¡√ ¤ Figure 2: Cross-plots of the approximations to the function ln E Q exp ht FN−1IG [Φ (Zt − λ)] ¯ Ft−1 as a √ function of ht with ht equal to the 999 quantile values of the Inverse Gaussian distribution with a = 2 and FN−1IG [Φ (Zt − λ)] calculated according to the suggested approximation. The top panel shows the results for the Gaussian distribution, the middle panel shows the results for the symmetric NIG distribution with a = 2 and b = 0, and the bottom panel shows the results for the NIG distribution with a √= 2 and b = 0.2. Superimposed as a solid line on the Gaussian plot is the true relation given by 12 ht − λ ht . For the NIG distributions the solid line indicates the estimated relationship from the suggested 3 degree polynomial approximation.

31

0.0

GM Std. Res. Normal

GM St d. Res. NIG(2.34,0.193)

-2.5

-2.5

-5.0 -5.0

-7.5

-4 0.0

-2

0

2

4

-4

IBM Std. Res. Normal

-2

0

2

4

0

2

4

0

2

4

IBM St d. Res. NIG(1.73,0.133)

-2.5

-2.5

-5.0 -5.0

-7.5

-4 0.0

-2

0

2

4

-4 0.0

M RK Std. Res. Normal

-2.5

-2 M RK Std. Res . NIG(2.09,0.221)

-2.5

-5.0 -5.0

-7.5

-4

-2

0

2

4

-4

-2

Figure 3: Plot of residual log-densities from a NGARCH volatility specification for General Motors (GM), International Business Machines (IBM), and Merck & Company Inc. (MRK). Left hand column shows plots for the Gaussian model and right hand column for the general NIG model. Superimposed as a dotted line on all plots is the corresponding distribution.

32

Table 1: Estimation results for General Motors (GM) Panel A: Gaussian models Model Likelihood

λ β α γ

J-B Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-9690.36

-9361.31

-9337.59

Estimate 0.0174

Std.Err. (0.0141)

Estimate 0.0323 0.9328 0.0552

Std.Err. (0.0144) (0.0307) (0.0226)

Estimate 0.0209 0.9393 0.0403 -0.5659

Std.Err. (0.0149) (0.0194) (0.0135) (0.1298)

Statistic 24490.1210 41.6229 635.0186 116.3819 3.8340

P-value [0.0000] [0.0031] [0.0000] [0.0000]

Statistic 1862.6173 34.2966 38.1142 4.5505 3.7042

P-value [0.0000] [0.0242] [0.0037] [0.0004]

Statistic 1289.7245 32.8846 37.1974 3.5890 3.6949

P-value [0.0000] [0.0347] [0.0049] [0.0031]

Panel B: Normal Inverse Gaussian models Model Likelihood

λ β α γ a b

Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-9429.06

-9242.62

-9232.56

Estimate 0.0192

Std.Err. (0.0146)

Estimate 0.0212 0.9584 0.0355

Std.Err. (0.0139) (0.0096) (0.0075)

1.2527 0.1229

(0.1354) (0.0465)

2.1890 0.1760

(0.3004) (0.0779)

Statistic 41.6229 634.8770 116.3197 3.7309

P-value [0.0031] [0.0000] [0.0000]

Statistic 33.7275 65.9905 10.7021 3.6574

P-value [0.0280] [0.0000] [0.0000]

Estimate 0.0144 0.9572 0.0303 -0.5080 2.3358 0.1934

Std.Err. (0.0132) (0.0087) (0.0071) (0.1426) (0.3273) (0.0803)

Statistic 32.6596 55.8605 8.0495 3.6535

P-value [0.0368] [0.0000] [0.0000] (*)

Notes: This table reports Quasi Maximum Likelihood Estimates (QMLE) for the daily returns from 1976 through 1995 assuming a risk-free interest rate of 5.4% corresponding to the value on December 29, 1995. Robust standard errors are reported in parentheses. J-B is the value of the usual Jarque-Bera normality test for the standardized residuals. Q(20) is the Ljung-Box portmanteau test for up to 20’th order serial correlation in the standardized residuals, whereas Q2 (20) is for up to 20’th order serial correlation in the squared standardized residuals. In square brakets next to all test statistics p-values are reported. The last row reports the Schwarz Information Criteria, with an asterix denoting the minimum value.

33

Table 2: Estimation results for International Business Machines (IBM) Panel A: Gaussian models Model Likelihood

λ β α γ

J-B Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-9112.36

-8800.54

-8770.65

Estimate 0.0098

Std.Err. (0.0141)

Estimate 0.0318 0.9388 0.0485

Std.Err. (0.0181) (0.0273) (0.0202)

Estimate 0.0190 0.9262 0.0463 -0.5517

Std.Err. (0.0188) (0.0277) (0.0144) (0.1328)

Statistic 121373.1300 48.2966 207.8221 33.5182 3.6053

P-value [0.0000] [0.0004] [0.0000] [0.0000]

Statistic 14629.4250 24.9561 8.9064 0.4835 3.4823

P-value [0.0000] [0.2031] [0.9619] [0.7889]

Statistic 7355.8608 24.9746 9.9343 0.4234 3.4706

P-value [0.0000] [0.2024] [0.9340] [0.8327]

Panel B: Normal Inverse Gaussian models Model Likelihood

λ β α γ a b

Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-8715.89

-8570.31

-8566.76

Estimate 0.0133

Std.Err. (0.0154)

Estimate 0.0149 0.9549 0.0353

Std.Err. (0.0142) (0.0095) (0.0067)

1.0679 0.0827

(0.1051) (0.0448)

1.6791 0.1332

(0.2050) (0.0676)

Statistic 48.2966 207.7460 33.4917 3.4487

P-value [0.0004] [0.0000] [0.0000]

Statistic 26.3204 9.6944 0.9332 3.3914

P-value [0.1555] [0.9414] [0.4581]

Notes: See the notes to Table 1.

34

Estimate 0.0079 0.9512 0.0356 -0.2933 1.7313 0.1326

Std.Err. (0.0143) (0.0105) (0.0067) (0.1150) (0.2070) (0.0666)

Statistic 26.0620 9.7321 0.7903 3.3901

P-value [0.1638] [0.9403] [0.5565] (*)

Table 3: Estimation results for Merck & Company Inc. (MRK) Panel A: Gaussian models Model Likelihood

λ β α γ

J-B Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-9040.23

-8858.54

-8849.72

Estimate 0.0386

Std.Err. (0.0141)

Estimate 0.0513 0.9073 0.0596

Std.Err. (0.0140) (0.0274) (0.0152)

Estimate 0.0406 0.9100 0.0535 -0.3504

Std.Err. (0.0140) (0.0204) (0.0115) (0.1216)

Statistic 2459.8188 38.6187 507.3857 35.3236 3.5767

P-value [0.0000] [0.0074] [0.0000] [0.0000]

Statistic 524.8064 38.8368 20.0081 1.4434 3.5053

P-value [0.0000] [0.0070] [0.3324] [0.2052]

Statistic 431.7495 39.0098 22.8488 1.8072 3.5019

P-value [0.0000] [0.0066] [0.1965] [0.1078]

Panel B: Normal Inverse Gaussian models Model Likelihood

λ β α γ a b

Q(20) Q2 (20) ARCH1-5 Schwarz

CV

GARCH

NGARCH

-8863.94

-8762.45

-8757.27

Estimate 0.0391

Std.Err. (0.0143)

Estimate 0.0470 0.9188 0.0521

Std.Err. (0.0140) (0.0220) (0.0116)

1.4120 0.1297

(0.1469) (0.0448)

2.0535 0.2194

(0.2555) (0.0700)

Statistic 38.6187 507.4184 35.3190 3.5073

P-value [0.0074] [0.0000] [0.0000]

Statistic 38.5589 21.7117 1.8726 3.4674

P-value [0.0076] [0.2450] [0.0956]

Notes: See the notes to Table 1.

35

Estimate 0.0408 0.9170 0.0495 -0.3348 2.0858 0.2207

Std.Err. (0.0139) (0.0168) (0.0093) (0.1135) (0.2567) (0.0700)

Statistic 38.8096 23.3027 2.0132 3.4654

P-value [0.0070] [0.1792] [0.0736] (*)

Table 4: American price estimates in a model with GARCH volatility processes

Gaussian T

Panel A: American put price estimates Symmetric NIG

NIG

Strike

Price

Std.

Price

Std.

Rel Diff

Price

Std.

Rel Diff

7 7 7

85 100 115

0.000 1.604 15.000

(0.0002) (0.0097) (0.0007)

0.001 1.580 15.000

(0.0005) (0.0086) (0.0012)

-72.701% 1.544% 0.000%

0.001 1.574 15.000

(0.0004) (0.0088) (0.0007)

-53.363% 1.929% 0.000%

21 21 21

85 100 115

0.042 2.719 15.006

(0.0031) (0.0144) (0.0089)

0.053 2.686 15.008

(0.0040) (0.0167) (0.0103)

-20.971% 1.234% -0.010%

0.042 2.673 15.010

(0.0034) (0.0157) (0.0112)

-0.789% 1.719% -0.023%

63 63 63

85 100 115

0.504 4.538 15.368

(0.0136) (0.0243) (0.0360)

0.522 4.484 15.335

(0.0144) (0.0292) (0.0393)

-3.361% 1.201% 0.220%

0.470 4.453 15.359

(0.0134) (0.0292) (0.0376)

7.307% 1.907% 0.063%

126 126 126

85 100 115

1.364 6.214 16.187

(0.0227) (0.0305) (0.0509)

1.369 6.134 16.127

(0.0241) (0.0328) (0.0504)

-0.347% 1.299% 0.376%

1.279 6.075 16.149

(0.0220) (0.0336) (0.0513)

6.656% 2.287% 0.237%

Gaussian T

Panel B: American call price estimates Symmetric NIG

NIG

Strike

Price

Std.

Price

Std.

Rel Diff

Price

Std.

Rel Diff

7 7 7

115 100 85

0.002 1.684 15.072

(0.0005) (0.0116) (0.0091)

0.004 1.660 15.071

(0.0009) (0.0104) (0.0080)

-55.247% 1.409% 0.005%

0.005 1.656 15.072

(0.0011) (0.0105) (0.0085)

-68.408% 1.700% 0.000%

21 21 21

115 100 85

0.103 2.955 15.228

(0.0060) (0.0199) (0.0116)

0.113 2.926 15.235

(0.0070) (0.0237) (0.0119)

-9.143% 0.987% -0.048%

0.129 2.915 15.227

(0.0079) (0.0241) (0.0127)

-20.241% 1.341% 0.006%

63 63 63

115 100 85

0.981 5.232 16.015

(0.0249) (0.0393) (0.0250)

0.972 5.183 16.033

(0.0260) (0.0460) (0.0273)

0.919% 0.958% -0.108%

1.012 5.158 15.987

(0.0268) (0.0464) (0.0277)

-3.039% 1.451% 0.175%

126 126 126

115 100 85

2.558 7.552 17.364

(0.0467) (0.0621) (0.0483)

2.524 7.498 17.376

(0.0427) (0.0600) (0.0427)

1.338% 0.716% -0.068%

2.561 7.452 17.298

(0.0457) (0.0617) (0.0445)

-0.118% 1.331% 0.381%

Notes: This table shows American option prices for a GARCH variance specification with different underlying distributions for a set of artificial options. The interest rate is fixed at 6% with a dividend yield of 3% both of which are annualized using 252 days a year. T denotes the time to maturity in days, Strike denotes the strike price, and for all the options the stock price is 100. The parameter values for the different GARCH processes corresponds to those from Stentoft (2005) whereas the distrbutional parameters are the ones specified in the text. In the cross-sectional regressions powers of and cross-products between the stock level and the volatility level of total order less than or equal to two were used. Exercise is considered once every trading day. Prices reported are averages of 100 calculated prices using 20.000 paths and different seeds in the random number generator with standard errors reported in parenthesis. The columns headed Rel Diff reports the difference between the Gaussian GARCH specification and corresponding NIG GARCH specification, and indicate the mispricing which would occur by the Gaussian GARCH volatility model if the true model is the corresponding NIG GARCH model. 36

Table 5: American price estimates in a model with NGARCH volatility processes

Gaussian T

Panel A: American put price estimates Symmetric NIG

NIG

Strike

Price

Std.

Price

Std.

Rel Diff

Price

Std.

Rel Diff

7 7 7

85 100 115

0.001 1.576 15.000

(0.0002) (0.0103) (0.0013)

0.002 1.553 15.000

(0.0005) (0.0093) (0.0008)

-65.469% 1.462% 0.000%

0.001 1.549 15.000

(0.0004) (0.0093) (0.0010)

-43.696% 1.739% 0.000%

21 21 21

85 100 115

0.061 2.694 15.001

(0.0041) (0.0157) (0.0043)

0.070 2.666 15.002

(0.0049) (0.0177) (0.0038)

-12.782% 1.048% -0.002%

0.059 2.658 15.002

(0.0042) (0.0177) (0.0044)

3.907% 1.347% -0.004%

63 63 63

85 100 115

0.657 4.580 15.237

(0.0165) (0.0284) (0.0339)

0.661 4.536 15.214

(0.0168) (0.0342) (0.0384)

-0.730% 0.963% 0.151%

0.616 4.520 15.234

(0.0170) (0.0334) (0.0397)

6.603% 1.323% 0.015%

126 126 126

85 100 115

1.679 6.380 16.050

(0.0296) (0.0344) (0.0511)

1.667 6.307 16.010

(0.0271) (0.0388) (0.0545)

0.758% 1.147% 0.248%

1.593 6.275 16.035

(0.0269) (0.0378) (0.0556)

5.413% 1.669% 0.093%

Gaussian T

Panel B: American call price estimates Symmetric NIG

NIG

Strike

Price

Std.

Price

Std.

Rel Diff

Price

Std.

Rel Diff

7 7 7

115 100 85

0.000 1.655 15.073

(0.0002) (0.0105) (0.0092)

0.002 1.633 15.071

(0.0006) (0.0097) (0.0087)

-70.333% 1.330% 0.009%

0.002 1.630 15.072

(0.0007) (0.0096) (0.0081)

-80.823% 1.512% 0.007%

21 21 21

115 100 85

0.056 2.929 15.241

(0.0038) (0.0176) (0.0105)

0.066 2.905 15.249

(0.0047) (0.0217) (0.0102)

-15.221% 0.828% -0.052%

0.078 2.899 15.239

(0.0053) (0.0217) (0.0099)

-28.147% 1.007% 0.008%

63 63 63

115 100 85

0.803 5.275 16.163

(0.0200) (0.0340) (0.0177)

0.808 5.236 16.169

(0.0203) (0.0398) (0.0195)

-0.548% 0.749% -0.037%

0.849 5.225 16.129

(0.0216) (0.0402) (0.0204)

-5.373% 0.959% 0.212%

126 126 126

115 100 85

2.404 7.715 17.671

(0.0391) (0.0545) (0.0376)

2.390 7.671 17.670

(0.0367) (0.0523) (0.0346)

0.589% 0.564% 0.007%

2.439 7.652 17.608

(0.0383) (0.0532) (0.0355)

-1.412% 0.823% 0.354%

Notes: This table shows American put prices and early exercise premiums for a NGARCH variance specification with different underlying distributions for the set of artificial options in Table 4 using parameter values from the same sources.

37

Table 6: Overall performance for the NIG GARCH option pricing model Panel A: All stocks (8424) Put (3009)

Model

All

CV GARCH NGARCH

RBIAS -17.397% -9.203% -9.541%

RSE 9.630% 6.500% 6.416%

RBIAS -19.978% -11.950% -10.221%

2 1

3 1

3 1

Best Worst

Model

All

(1835)

CV GARCH NGARCH

RBIAS -15.619% -2.680% -3.281%

RSE 6.727% 3.212% 2.995%

2 1

3 1

Best Worst

Model

All

(4745)

CV GARCH NGARCH

RBIAS -20.182% -11.262% -11.760%

RSE 11.437% 8.255% 8.326%

2 1

2 1

Best Worst

Model

All

(1844)

CV GARCH NGARCH

RBIAS -12.001% -10.399% -10.059%

RSE 7.867% 5.255% 4.908%

3 1

3 1

Best Worst

Panel B: GM Put

Call

(5415)

RSE 10.869% 7.241% 6.653%

RBIAS -15.963% -7.677% -9.162%

RSE 8.941% 6.088% 6.285%

3 1

2 1

2 1

(629)

Call

(1206)

RBIAS -18.783% -4.545% -2.659%

RSE 9.014% 4.297% 3.802%

RBIAS -13.969% -1.706% -3.605%

RSE 5.534% 2.646% 2.574%

3 1

3 1

2 1

3 1

Panel C: IBM Put

(1827)

Call

(2918)

RBIAS -21.684% -13.799% -12.245%

RSE 12.126% 8.572% 8.103%

RBIAS -19.241% -9.673% -11.456%

RSE 11.006% 8.056% 8.465%

3 1

3 1

2 1

2 1

Panel D: MRK Put

(553)

Call

(1291)

RBIAS -15.703% -14.265% -12.137%

RSE 8.827% 6.193% 5.103%

RBIAS -10.415% -8.744% -9.169%

RSE 7.456% 4.854% 4.824%

3 1

3 1

2 1

3 1

Notes: This table reports results on the overall pricing performance of the NIG GARCH option pricing model using the metrics in Section 5. In particular, letting Pk and P˜k denote the k’th observed price respectively the k’th model price these are the relative S (P˜k −Pk ) mean bias, RBIAS ≡ K −1 K , and the relative mean squared error, RSE ≡ k=1 Pk ˜k −Pk )2 S P ( K . K −1 k=1 P2 k

38

Table 7: Pricing performance for the NIG GARCH option pricing model relative to the Gaussian GARCH model Panel A: All stocks (8424) Put (3009)

Model

All

CV GARCH NGARCH

RBIAS -0.732% -10.546% -11.239%

RSE -1.926% -13.595% -17.666%

RBIAS 3.118% 2.415% 28.425%

Max Min

-0.732% -11.239%

-1.926% -17.666%

28.425% 2.415%

Model

All

(1835)

CV GARCH NGARCH

RBIAS -0.536% -48.017% -33.127%

RSE -2.233% -11.026% -15.179%

Max Min

-0.536% -48.017%

-2.233% -15.179%

Model

All

(4745)

CV GARCH NGARCH

RBIAS -1.159% -8.895% -11.896%

RSE -2.124% -15.906% -20.294%

Max Min

-1.159% -11.896%

-2.124% -20.294%

Model

All

(1844)

CV GARCH NGARCH

RBIAS 0.895% 3.334% 1.866%

RSE -0.913% -4.685% -5.782%

3.334% 0.895%

-0.913% -5.782%

Max Min

Call

(5415)

RSE 1.494% -6.718% -4.473%

RBIAS -3.245% -19.373% -25.502%

RSE -4.109% -17.610% -23.852%

1.494% -6.718%

-3.245% -25.502%

-4.109% -23.852%

(629)

Call

(1206)

RBIAS 3.631% -23.596% 30.387%

RSE 2.584% -4.616% -0.627%

RBIAS -3.264% -64.002% -43.680%

RSE -5.983% -15.819% -23.778%

30.387% -23.596%

2.584% -4.616%

-3.264% -64.002%

-5.983% -23.778%

(1827)

Call

(2918)

RBIAS 2.678% 3.911% 32.631%

RSE 1.283% -9.092% -6.933%

RBIAS -3.698% -17.930% -28.058%

RSE -4.343% -19.905% -26.609%

32.631% 2.678%

1.283% -9.092%

-3.698% -28.058%

-4.343% -26.609%

(553)

Call

(1291)

RBIAS 4.458% 11.008% 15.756%

RSE 1.205% 3.884% 6.839%

RBIAS -1.280% -1.428% -4.623%

RSE -1.953% -8.796% -10.569%

15.756% 4.458%

6.839% 1.205%

-1.280% -4.623%

-1.953% -10.569%

Panel B: GM Put

Panel C: IBM Put

Panel D: MRK Put

Notes: This table reports the difference in pricing metrics for each volatility specification, specifically that of the Gaussian model minus that of the NIG model, relative to the value for the Gaussian model using both of the metrics in Section 5. Thus, in the table a negative cell value indicates that the NIG version has smaller pricing error for this model whereas a positive cell value indicates that the Gaussian model has smalles pricing error. The actual cell value may be interpreted as the decrease in error, respectively the increase in error, obtained by using the NIG model instead of the Gaussian equivalent.

39

Table 8: Pricing performance for the NIG GARCH option pricing model, both absolutely and relative to the Gaussian GARCH model, in terms of moneyness and maturity Panel A: Overall performance in terms of moneyness of the NIG GARCH model Model ITM (2419) ATM (2190) OTM (3815)

CV GARCH NGARCH Best Worst

RBIAS -5.60% -2.85% -2.58%

RSE 0.84% 0.49% 0.45%

RBIAS -7.97% -5.05% -5.15%

RSE 3.12% 2.31% 2.25%

RBIAS -30.29% -15.61% -16.48%

RSE 18.94% 12.72% 12.59%

3 1

3 1

2 1

3 1

2 1

3 1

Panel B: Relative performance in terms of moneyness of the NIG GARCH model Model ITM (2419) ATM (2190) OTM (3815)

CV GARCH NGARCH Max Min

RBIAS 1.06% -4.54% 6.09%

RSE 0.79% -9.38% -5.83%

RBIAS 4.90% -5.14% -4.79%

RSE 0.54% -8.61% -9.43%

RBIAS -1.73% -12.12% -13.69%

RSE -2.23% -14.18% -18.66%

6.09% -4.54%

0.79% -9.38%

4.90% -5.14%

0.54% -9.43%

-1.73% -13.69%

-2.23% -18.66%

Panel C: Overall performance in terms of maturity of the NIG GARCH model Model ST (4687) MT (3208) LT (529)

CV GARCH NGARCH Best Worst

RBIAS -17.54% -9.38% -9.70%

RSE 10.81% 7.67% 7.66%

RBIAS -17.41% -8.75% -9.09%

RSE 8.32% 5.17% 4.99%

RBIAS -16.01% -10.37% -10.83%

RSE 7.08% 4.17% 4.08%

2 1

3 1

2 1

3 1

2 1

3 1

Panel D: Relative performance in terms of maturity of the NIG GARCH model Model ST (4687) MT (3208) LT (529)

CV GARCH NGARCH Max Min

RBIAS -0.56% -6.16% -6.63%

RSE -2.40% -16.29% -18.07%

RBIAS -0.85% -16.09% -17.04%

RSE -1.08% -7.76% -16.73%

RBIAS -1.63% -13.73% -14.27%

RSE -1.42% -9.16% -17.80%

-0.56% -6.63%

-2.40% -18.07%

-0.85% -17.04%

-1.08% -16.73%

-1.63% -14.27%

-1.42% -17.80%

Notes: This table reports the pricing errors of the NIG GARCH model for different categories of moneyness in Panel A and maturity in Panel C using the appropriate categories from Section 5. Thus, the reported results have the same interpretation as those in Table 6. The table also reports the pricing errors relative to those obtained when using the Gaussian GARCH model for different categories of moneyness in Panel B and maturity in Panel D. For these panels the reported results have the same interpretation as those in Table 7.

40