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Electronic copy available at: http://ssrn.com/abstract=1875847

American Option Pricing with Discrete and Continuous Time

Models: An Empirical Comparison∗

Lars Stentoft†

HEC Montréal, CIRANO, CIRPEÉ, and CREATES

June 30, 2011

Abstract

This paper considers discrete time GARCH and continuous time SV models and uses these

for American option pricing. We perform a Monte Carlo study to examine their differences in

terms of option pricing, and we study the convergence of the discrete time option prices to their

implied continuous time values. Finally, a large scale empirical analysis using individual stock

options and options on an index is performed comparing the estimated prices from discrete time

models to the corresponding continuous time model prices. The results indicate that, while the

differences in performance are small overall, for in the money options the continuous time SV

models do generally perform better than the discrete time GARCH specifications.

JEL Classification: C22, C53, G13

Keywords: American Options, Augmented GARCH, Least Squares Monte Carlo, Stochastic

Volatility.

∗The author thanks participants and the discussant at the 2008 Northern Finance Association Meeting as well as

seminar participants at CREATES and HEC Montréal for valuable comments and suggestions. Financial support

from IFM2 is gratefully appreciated.

†Address correspondance to Lars Stentoft, HEC Montreal, 3000 chemin de la Cote-Sainte-Catherine, Montreal

(Quebec) Canada H3T 2A7, or e-mail: lars.stentoft@hec.ca. Phone: (+1) 514 340 6671. Fax: (+1) 514 340 5632.

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Electronic copy available at: http://ssrn.com/abstract=1875847

1Introduction

In the seminal paper by Black & Scholes (1973) a closed form solution for the price of an European

option is derived. The Black-Scholes formula has been celebrated as one of the major successes

of modern financial economics, although empirical analysis has pointed towards several systematic

pricing errors when compared to observed option prices. To be specific, numerous studies have

documented smiles in the implied volatility as a function of moneyness as well as a tendency for

the constant volatility model to underprice in particular short term out of the money options. In

response to these “empirical regularities”, a number of alternative models have been developed. In

particular, the assumptions underlying the Black-Scholes model have been widely criticized, and

much effort has been put into extending the valuation framework. Apart from the assumption of

continuous trade, a crucial assumption in the Black-Scholes model is that of constant volatility

and lognormality. However, the constant volatility lognormal model fails to explain a number of

empirical regularities found in asset return series, the most important of which are leptokurtosis

and the volatility clustering phenomenon (see Bollerslev, Engle & Nelson (1994)).

In the option pricing literature, some of the earliest extensions to the Black-Scholes model are the

continuous time stochastic volatility, or SV, models of Hull & White (1987), Wiggins (1987), Scott

(1987), Stein & Stein (1991), and Heston (1993). More recent studies include, to name a few, Bakshi,

Cao & Chen (1997) and Bates (2000). In these papers volatility is modelled as a separate stochastic

process allowing for a large degree of flexibility in the specification. A particularly appealing feature

of these models is that under certain assumptions elegant solutions can be derived for the price

of European options. In some cases, the pricing formulas are approximately closed form solutions

of the same general type as the Black-Scholes formula with integrals of the stochastic volatility.

However, in real applications a problem with the continuous time SV models is that volatility

is unobservable and hence estimation of this type of models is rather complicated. Examples

on feasible estimation methods include the Efficient Method of Moments, or EMM, procedure of

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Gallant & Tauchen (1996) and more recently procedures using Markov Chain Monte Carlo, or

MCMC, has been applied by Jacquier, Polson & Rossi (1994) and Jacquier, Polson & Rossi (2004)

(see also Johannes & Polson (2010)). Moreover, when pricing claims for which numerical procedures

are needed, as it is the case with e.g. American options, future values of the unobservable volatility

is needed. This variable is latent and hence potentially complicated to predict, and thus such

procedure may require e.g. the full reprojection machinery associated with the EMM procedure

(see Gallant & Tauchen (1998)).

In the time series literature, several competing discrete time asset return models have been

developed which can take account of the empirical regularities observed for financial returns. A

large number of these has been within the framework of autoregressive conditional heteroskedastic,

or ARCH, processes suggested by Engle (1982) and the generalized ARCH, or GARCH, models

introduced by Bollerslev (1986). A particular appealing feature with these models is that data is

readily available for estimation and this can be done with simple maximum likelihood procedures.

These models have been successfully applied to financial data such as stock return data as seen

from Bollerslev, Chou & Kroner’s (1992) survey article. However, in applications to option pricing,

the GARCH framework is somewhat complicated because generally no closed form solutions exist,

although see Heston & Nandi (2000) for an exception. Thus, although the appropriate dynamics

were derived in Duan (1995), numerical methods have to be used for the actual pricing. Options on

the Standard and Poor’s 500 Index have been the focus of a large part of this research, and various

GARCH specifications have been used by Heston & Nandi (2000), Hsieh & Ritchken (2005), and

Christoffersen & Jacobs (2004). Also, Bollerslev & Mikkelsen (1996) and Bollerslev & Mikkelsen

(1999) have successfully used the GARCH option pricing framework together with fractionally inte-

grated GARCH processes to price long-term European style equity anticipation securities (LEAPS)

on this particular Index.

In spite of the recent progress in terms of available estimation techniques for continuous time

SV models and numerical methods for option pricing in the discrete time GARCH models, there is

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still a gap between the two strands of the literature. In particular, it is probably fair to say that

in applications it is far from the standard to report estimation results of both discrete time and

continuous time return models. Likewise, when pricing options with a continuous time SV model,

the results are usually only compared to other models formulated in continuous time and likewise

for discrete time GARCH models the results are usually only compared to other models formulated

in discrete time. Thus, the important question of which models are preferable remains open. In this

paper, we bridge the gap between the discrete time approach and the continuous time approach

using the Augmented GARCH model of Duan (1997) as the basic framework. To be specific, we

choose a number of GARCH models within this framework, for which stochastic volatility models

are obtained in the limit. In doing so, it becomes possible to compare in a consistent way option

prices calculated in discrete time to those calculated from continuous time models.

The contribution of this paper is threefold. First of all, we provide estimation results for the

GARCH models as well as for the diffusion limit SV models using simple maximum likelihood

techniques. The estimates for the SV models are obtained by appropriately re-parametrizing the

discrete time specifications and hence are essentially based on the same procedures. Secondly,

we compare the option price estimates from the discrete time models to their continuous time

counterparts through a Monte Carlo study. We also examine the convergence in terms of time

discretization and in terms of potential early exercise times. The results show that there are in

fact differences between the GARCH models and their continuous time counterparts. In terms

of convergence properties, we find that the discrete time GARCH prices converge quickly to the

continuous time SV values, and that allowing for multiple intraday early exercises is important for

in the money options. Thirdly, the paper contains a large scale empirical analysis using options on

a number of individual stocks and a stock index. The analysis shows that in actual applications of

the models the differences in overall performance are small. This generally holds through time and

across maturity and moneyness. The exception is for in the money put options on the individual

stocks for which the continuous time specifications outperforms the models formulated in discrete

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time. For this subsample of options, we show that allowing for multiple intraday early exercises

also improves on the performance.

The rest of the paper is organized as follows: In Section 2 the framework is introduced and

estimation results are provided. In Section 3 the risk neutral dynamics are derived and we describe

how option pricing can be performed using simulation methods. In Section 4 the results of an

extensive Monte Carlo study of the properties of option pricing models are reported. In Section 5

the option pricing models are applied to the data, and Section 6 concludes. Tables and Figures can

be found in the corresponding appendix.

2Asset return model

In this paper we consider a discrete time economy with the price of an asset denoted and

the dividends of that asset denoted . We assume that the continuously compounded return

process, = ln(+ ) − ln−1, can be modelled using the GARCH framework. The specific

parametrization we use is

= +

p

−1

2+

p

with (1)

= + −1(−1+ )2+ −1(2)

where |F−1∼ (01), with F−1denoting the information set containing all information up to

and including time − 1. It follows from lognormality that one plus the conditional expected rate

of return equals exp¡ + √

when is the continuously compounded risk-free rate of return. The model corresponds to the

¢, and hence in (1) is readily interpreted as the unit risk premium

NGARCH model proposed by Engle & Ng (1993), which allows for the well known leverage effect

through the parameter , and if 0 such an effect is said to be found. It is clear that this model

nests the ordinary GARCH specification which obtains when = 0.

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2.1Diffusion limits for the NGARCH model

The NGARCH model as well as the GARCH model are special cases of the Augmented GARCH

framework of Duan (1997). This framework has been shown to contain as its limit several of the

bivariate diffusion processes that are used as building blocks in various stochastic volatility models.

See Nelson (1990) for similar results for the GARCH model. In this section we explain how this

framework can be used to obtain the diffusion limits of the NGARCH model. We also discuss how

the results allow estimation of the parameters of the resulting continuous time models.

2.1.1The diffusion limit of Duan (1997)

In order to study the diffusion limits, we follow Duan (1997) and rewrite the discrete time daily

NGARCH model from above as

()

=

µ

+¡ + ¡1 + 2¢− 1¢()

+

q

()

−1

2()

¶

+

q

+ ()

()

√ (3)

()

(+1)− ()

=

³

(+ )2−¡1 + 2¢´√(4)

where = 1 is the length of the daily subintervals and ∼ (01). Note that the specification

in (1) and (2) obtains when = = 1. The limit is now considered as the length of the daily

subintervals, , tends to zero and from Duan (1997, Theorem 3) it can be shown that the limiting

diffusion model of the system in (3) and (4) is characterized by

ln

=

µ

+

p

−1

2

¶

+

p

1 (5)

= +¡ + ¡1 + 2¢− 1¢ + 21+√22(6)

where 1and 2are two independent Wiener processes. This model corresponds to the bivariate

diffusion model used in Hull & White (1987). In the following, we will refer to it as the C-NGARCH

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specification. Furthermore, the C-GARCH model, which obtains when = 0, is given by

= + ( + − 1) +√22 (7)

This model corresponds to the special case of the Hull & White (1987) model where the volatility

process is independent of the return process.

It should be noted, that the diffusion limits of Duan (1997) and Nelson (1990) may not be unique.

For example, Corradi (2000) shows that for some parametrizations one obtains a degenerate limit

for the GARCH model. Moreover, Heston & Nandi (2000) shows that for their affine NGARCH

model the limiting behavior is very different from that of the classical GARCH model as the same

process drives both the spot asset and the variance dynamics. However, since the limiting results

depend on the parameterization used to obtain the limits we are to a certain degree free to choose

which limits to consider. The benefit of considering the results of Duan (1997) is that the derived

limits correspond to models which are very popular in the continuous time option pricing literature.

Thus, by picking these limits carefully we can study in a simple way the relationship between the

GARCH process and these limits when it comes to option pricing.1Moreover, though the derived

limits in Corradi (2000) and Heston & Nandi (2000) allow options to be valued solely using the

hedging arguments of Black & Scholes (1973) and Merton (1973), they are much less appealing from

an empirical perspective. In particular, if we were to use the limiting results of e.g. Corradi (2000),

where the resulting diffusion is degenerate, the obtained results would differ greatly. However,

this model is much less flexible and clearly not appropriate as it does not allow for time varying

volatility.

1Ritchken & Trevor (1999) also consider the limits derived by Duan (1997). However, they report results for

European options only and do not conduct an empirical exercise like we do.

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2.1.2Implementation and parameter estimation

The immediate benefit of working with the model above is that the parameters of the diffusion

model can be implied from the discretely observed data. In particular, the diffusion limit dynamics

in (5) and (6) only depend on the parameters of the original discrete time system in (1) and (2).

Thus, since the discrete time parameters are readily available with simple maximum likelihood

estimation techniques so are the implied parameters of the diffusion limits. However, an alternative

to this procedure is to estimate the diffusion parameters directly. To do this, we first re-parametrize

the diffusion limit of the NGARCH specifications as

ln

=

µ

0 + 1 + 21+ 32

+

p

−1

2

¶

+

p

1(8)

=(9)

where 0= , 1=¡ + ¡1 + 2¢− 1¢, 2= 2, and 3=√2. Next, we rewrite the model

in (1) and (2) in terms of these parameters to obtain

= +

p

³

−1

3√2

2+

´

p

³

with (10)

=0+−1

−1+

³

2√23

´´2+

³

1 + 1− 3√2 + 2

2

√

233

´

−1(11)

The parameters in the model in (10) and (11) can also be estimated using simple maximum likeli-

hood techniques, though compared to the parametrization in (1) and (2) the approach yields directly

the implied continuous time parameters. Note that, using this parametrization the GARCH model

obtains when 2= 0.

A few comments are relevant with respect to the outlined estimation approach. First of all, our

method relies on the availability of diffusion limits for the selected discrete time model. Though

the Augmented GARCH framework of Duan (1997) contains many other discrete time models, it is

possible that the diffusion limit of the discrete time process selected in a careful empirical analysis

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of the series to be analyzed is not known. Alternatively, one may wish to consider a continuous

time model which does not correspond to the diffusion limit of a discrete time model, and in these

situations one would have to use an alternative estimation procedure. Secondly, though the models

in (1) and (2) and in (10) and (11) are estimated on the same data and using the same approach,

there is no guarantee that they fit the data equally well. In particular, the model in (10) and (11)

is highly nonlinear and this could lead to differences. Finally, the estimation procedure outlined

above may not be fully efficient for the parameters of the diffusion limit as it does not allow for

an additional shock.2Fully efficient estimates could be obtained with e.g. the EMM or MCMC

methods, though these methods are computationally much more complex. However, since these

methods are often implemented using daily data our approach may nevertheless give a good idea

about the model performance.

2.2Empirical results

In this section we use the above framework to study a sample of financial assets. The next section

introduces the data which has been previously studied in Stentoft (2005) and Stentoft (2008). We

then provide estimation results for the discrete time NGARCH and GARCH models and we discuss

the implied parameters of the diffusion models.

2.2.1 Data

For the empirical work we will use data for General Motors (GM), International Business Machines

(IBM), Merck and Company Inc. (MRK), as well as for the Standard and Poor’s 100 Index (OEX).

The reason for choosing these three stocks is that for the period under consideration options on

these three stocks were the most traded in terms of actual trades as well as in terms of total volume.

The reason for choosing the Standard and Poor’s 100 index (OEX) is that this is the broadest index

for which options are traded on the CBOE and it has been the focus of much research.

2In this respect, the proposed approach is similar to that used in e.g. Fleming & Kirby (2003) for estimating

discrete time stochastic volatility models using GARCH filters.

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The return series for the individual stocks were obtained from the Center for Research in Security

Prices (CRSP). We use return data beginning January 2, 1976, since this is as far back data on

the individual stock return and dividend are available to us on a daily basis. The continuously

compounded return series in percentage terms for the Standard and Poor’s 100 Index was calculated

from the return index supplied by Datastream. Since our data on the corresponding options ends

December 29, 1995, this date also marks the end of the sample which as a result contains 5055

daily observations.

Table 1 shows sample statistics and Figure 1 provides time series plots for the four return series.

From the table it is seen that returns are generally negatively skewed and leptokurtic, although

for MRK the skewness is insignificantly different from zero. From the figure it is seen that the

returns are clearly not independently and identically distributed through time. On the contrary,

periods of low volatility are followed by high volatility periods and vice versa, a finding known

as volatility clustering. The GARCH framework has been successfully applied to data with these

characteristics, see e.g. Bollerslev et al.’s (1992) survey article.

2.2.2Estimation results

Tables 2 to 5 report the Quasi Maximum Likelihood (QML) estimation results for the model in (1)

and (2). First of all, column three in the tables shows the estimation results for the GARCH models,

that is with = 0. Compared to the simpler CV model in column two the tables show that allowing

for time varying volatility leads to large increases in the Log-Likelihood values. Furthermore, for

all series, both extra parameters, and , are significantly different from zero and the estimates

are in line with what is usually found in the literature. In terms of serial correlation in the squared

standardized residuals, the 2(20) statistics show that for all but GM the null of no correlation

cannot be rejected. Thus, it seems that modelling volatility as a GARCH process goes quite a way

in terms of eliminating the ARCH effects for IBM MRK, and OEX. Furthermore, for three of the

four series the (20) statistics are now insignificant at a one percent level.

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Secondly, column four of the tables presents estimation results for the NGARCH model and

shows that in all cases adding the leverage parameter also leads to large increases in the Log-

Likelihood value. Furthermore, for all the return series the estimated leverage parameter, , is

significantly different from zero and has the expected sign. In terms of the diagnostic tests, however,

adding the asymmetry parameter does not change a lot except for OEX where the (20) statistics

is now also insignificant at a five percent level. The Schwarz Information Criteria, SIC, value

is smaller for the asymmetric models than for the symmetric GARCH model, which indicates

that asymmetries in the volatility specification are important features of the return data under

consideration and that this type of models should be preferred as an appropriate model.

Next, we consider the diffusion limits of the GARCH and NGARCH models above. In columns

six and seven of Tables 2 to 5 we report the Quasi Maximum Likelihood (QML) estimation results

for the model in (10) and (11), that is the model re-parametrized directly in terms of the diffusion

limit parameters. Using this specification for estimation instead of simply implying the parameters

allows for direct testing of the parameters of the diffusion limits.3We first of all note that overall

the estimated parameters seem very reasonable. In particular, the estimated parameters of the

diffusion limits show sign of strong persistence in the shocks to the variance process. Secondly,

when allowed for, asymmetries are found to be highly significant.

Finally, when comparing the results for the discrete time and continuous time models in Tables

2 to 5 it is seen that the highly nonlinear parameter transformations imposed in the continuous

time specification do not lead to any decrease in the likelihood values. Thus, we would obtain the

same parameters if we were to imply these from the discrete time model in (1) and (2) instead of

estimating them directly using (10) and (11). On the other hand, since the estimation procedure

does not allow explicitly for an additional shock in the diffusion model, it is not surprising that

the statistical fit of the discrete time models and the continuous time models is virtually identical.

However, a more detailed econometric analysis of the results is beyond the scope of the present

3If the parameters are implied instead, the delta method would have to be used to obtain the appropriate standard

errors.

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paper, and at this time we refrain from commenting further on the implied parameters. Our metric

is after all one of option pricing performance, one which we turn towards now.

3Risk neutral dynamics

The discrete time models used in this paper can all be written in the following general form:

= +

p

−1

2+

p

with (12)

=(; ≤ − 1)(13)

where |F−1 ∼ (01) under measure P. In (13), denotes the set of parameters used to

specify the variance dynamics. For example, for the NGARCH specification in (2) we have =

{}. To use this model for option pricing purposes we use the Locally Risk-Neutral Valuation

Relationship (LRNVR), derived in Duan (1995).

The LRNVR can be shown to hold under some familiar assumptions on preferences and assumed

conditional lognormality and invoking this the dynamics to by used for option pricing are easily

shown to be given by

= −1

2+

p

∗

with(14)

=(∗

− ; ≤ − 1)(15)

where ∗

|F−1 ∼ (01) under measure Q. Thus, the risk neutral dynamics depend only the

parameters in the original volatility specification and the unit risk premium . Since all of

the necessary parameters can be estimated from returns obtaining the risk neutral dynamics is

straightforward.

Equations (14) and (15) show that the dynamics remain Gaussian but with a shifted mean. The

shift in mean corresponds to what is needed to compensate investors for holding the risky assets.

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Note that, while we may be lured into believing that we have successfully eliminated all preference

related parameters this is not the case. However, the LRNVR is sufficient to reduce the preference

considerations to the constant unit risk premium present in the variance equation.

3.1 Option pricing with the limiting diffusion

As it is the case under the data generating process, the Augmented GARCH process under the

risk-neutralized pricing measure can be shown to converge to a bivariate diffusion system. This

was shown in Duan (1996, Theorem 2), and this general theorem can be used to derive a system

corresponding to the risk-neutralized version of the GARCH variance specifications used above.

Alternatively, it is possible to show that the risk-neutral dynamics can be derived directly from the

corresponding diffusion limits under the data generating process.

To fix ideas, assume that the diffusion limit of the Augmented GARCH system in (12) and (13)

has been derived under measure P. With a slight abuse of notation we specify this as

ln

=

µ

(12)

+

p

−1

2

¶

+

p

1 (16)

= (17)

where 1 and 2 are the two independent Wiener processes from above, and where we again

let denote the set of parameters used to specify the variance dynamics. It then follows from

Duan (1996) that the diffusion limit under the corresponding risk-neutralized pricing measure, Q,

becomes

ln

=

µ

(∗

−1

2

¶

+

p

∗

1(18)

=

1− ∗

2)(19)

where ∗

1and ∗

2are two independent Wiener processes.

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We note that, as it was the case in the discrete time system in (14) and (15), the dynamics

remain Gaussian but with a shifted mean in one of the innovation terms in the variance process. On

the one hand, the result in (18) and (19) confirms the result of the original paper by Hull & White

(1987). In particular, they imply that when the two innovations are independent the premium for

volatility risk is zero. On the other hand, they extend the results and provides the risk neutral

dynamics for the case of correlated innovations. For further discussion of these issues see Duan

(1996).

3.2Implementation of the GARCH option pricing model using simulation

Although the pricing system in (14) and (15), or for that sake the system in (18) and (19), is com-

pletely self-contained, actual application of the pricing system to even the simple European option

is difficult because of the lack of a closed form expression for the time value of the underlying

asset. However, it is immediately clear that using the system in (14) and (15), respectively in (18)

and (19), a large number of paths of the risk-neutralized asset prices can be generated, possibly

by using a type of discretization scheme. From this sample of paths, an estimate of the European

option value can be obtained as a simple average of the discounted pathwise final payoff. For option

pricing purposes this method has been used at least since Boyle (1977).

For the American options, things are not as simple since an optimal exercise strategy has to be

determined simultaneously. However, the work by Carriere (1996), Longstaff & Schwartz (2001),

and Tsitsiklis & Van Roy (2001) among others has shown how this can be done using a simulation

approach and by now these methods have become standard tools. The most important of these

contributions in terms of their use is probably the Least Squares Monte Carlo (LSM) method of

Longstaff & Schwartz (2001). In a GARCH context the LSM method was used successfully in

Stentoft (2005) and Stentoft (2008). The method relies only on being able to generate simulated

paths from the appropriate risk neutral system. It can therefore be equally well used to price

American options for the discrete time case described in (14) and (15) and for the continuous time

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case described in (18) and (19). Thus, in the present paper we use this particular algorithm to

price American options in both of these cases.

The LSM method for pricing American style options proceeds as follows. First of all, given the

full sample of random paths, the pricing step is initiated at the maturity date 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. Hence, the pathwise payoffs may be easily determined. Next, working

backward through time at any point in time where early exercise is to be considered, a cross-

sectional regression is performed. In the regression, the future pathwise values in the simulation

are regressed on transformations of the current stock prices and volatility levels. The fitted values

from this regression are then used as estimates of the pathwise conditional expected value of holding

the option for one more period along. The decision of whether to exercise or not along each path

can now be made by comparing the estimated conditional expected value of continuing to hold the

option to the value of immediate exercise. If immediate exercise yields superior payoff, this is the

optimal choice along this particular path. Once the decision has been recorded along each path,

we can move back through the simulation to the previous early exercise point and perform a new

cross-sectional regression with the appropriate pathwise values based on the previous choices. With

the optimal early exercise strategies along each path an estimate of the American option value can

be obtained as a simple average of the discounted pathwise payoff, as it is the case for the European

option.

4Option pricing properties

In this section we conduct a Monte Carlo study in order to compare the results on American and

European option pricing using the discrete time models with what is obtained with the corre-

sponding diffusion limits. We report results on the pricing from the discrete time GARCH and

NGARCH specifications as well as the pricing results from their corresponding diffusion limits.

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