Comparing Discrete-Time and Continuous-TimeOption Valuation Models

Peter Christoffersen Kris JacobsKarim Mimouni

Faculty of Management, McGill University∗

April 18, 2005

Abstract

This paper provides an empirical comparison of four option valuation models. The firstof these models is the benchmark affine stochastic volatility model in the continuous-timeoption valuation literature. This model yields closed-form solutions for European optionprices. The second model is a discrete-time affine option valuation model that also allows fora closed form solution. The third is a non-affine discrete-time model and the fourth is a non-affine stochastic volatility model. The latter two models do not yield closed-form solutions.Using a root mean squared dollar error criterion, the non-affine models outperform the affinestochastic volatility model by approximately 15% in- and out-of-sample. The affine discrete-time model outperforms the affine continuous-time model by 10% in-sample and 6% out-of-sample The non-affine discrete time model slightly outperforms the non-affine continuoustime model. These findings may suggest that the distinction between continuous-time anddiscrete-time models is not very relevant from an empirical perspective. The distinctionbetween non-affine and affine volatility models is more important and non-affine modelsneed to be studied more extensively. At the methodological level, the paper presents anew method for estimating continuous-time option valuation models that can be used in avariety of applications.

JEL Classification: G12

Keywords: stochastic volatility; GARCH; option valuation; filtering; out-of-sample.

∗Christoffersen and Jacobs are also affiliated with CIRANO and CIREQ and want to thank FQRSC, IFM2

and SSHRC for financial support. Mimouni was supported by a grant from IFM2. Any remaining inadequaciesare ours alone. Correspondence to: Peter Christoffersen, Faculty of Management, McGill University, 1001 Sher-brooke Street West, Montreal, Quebec, Canada, H3A 1G5; Tel: (514) 398-2869; Fax: (514) 398-3876; E-mail:peter.christoffersen@mcgill.ca.

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1 Introduction

Following the finding that Black-Scholes (1973) model prices systematically differ from marketprices, the literature on option valuation has formulated a number of theoretical models designedto capture these empirical biases.1 One particularly popular modeling approach has attemptedto correct the empirical biases in the Black-Scholes model by modifying the Black-Scholes as-sumption that volatility is constant across maturity and moneyness. Studies using returns dataas well as options data have demonstrated that return volatility is time-varying, and that im-portant improvements in the performance of option pricing models can be made by modelingvolatility clustering in the dynamic of the underlying asset return. Moreover, it is now alsowell established that it is necessary to model a leverage effect in the volatility process, whichcaptures the negative correlation between returns and volatility. Another way of thinking aboutthe leverage effect is that it generates negative skewness in the distribution of the underlyingasset return.2

Most of the existing literature has modeled volatility clustering and the leverage effect usingcontinuous-time stochastic volatility models. In particular, the Heston (1993) model, which ac-counts for time-varying volatility and a leverage effect, has been implemented in a large numberof empirical studies. In order to address the limitation of continuous sample paths that is inex-tricably linked with the continuous-time framework, the Heston (1993) model is often combinedwith models of jumps in returns and volatility.3

There exists a smaller literature that values options using GARCH processes to describe theunderlying return dynamic, building on the results in Duan (1995) and Amin and Ng (1993). Theliterature on GARCH processes is too voluminous to cite in full here,4 but it is almost exclusivelyfocused on the statistical fit of the underlying returns. A limited number of studies empiricallytest the performance of GARCH option valuation models.5

The objective of this paper is to document differences in performance between discrete-timeand continuous-time option valuation models. This seems like a natural question to ask, but thereare a number of reasons why this issue has not yet been addressed in the literature. First, manyfinancial economists believe that continuous-time methods are preferable to model options. Themost often heard motivation for this opinion is that continuous-time methods are mathematicallyelegant and lead to closed form solutions for option prices. However, apart from the Heston(1993) model, the literature contains few if any other volatility dynamics that lead to closed-form solutions. The discrete-time literature also contains a volatility dynamic that yields a

1See Bakshi, Cao and Chen (1997), Dumas, Fleming and Whaley (1998), Eraker (2004) and the referencestherein.

2The leverage effect was first characterized in Black (1976). For empirical studies that emphasize the im-portance of volatility clustering and the leverage effect for option valuation see among others Benzoni (1998),Chernov and Ghysels (2000), Eraker (2000), Heston and Nandi (2000) and Nandi (1998).

3For empirical studies that implement the Heston (1993) model by itself or in combination with different typesof jump processes, see for example Andersen, Benzoni and Lund (2002), Bakshi, Cao and Chen (1997), Bates(1996, 2000), Chernov and Ghysels (2000), Huang and Wu (2004), Nandi (1998), Pan (2002), Eraker (2004) andEraker, Johannes and Polson (2003).

4The classical references are Engle (1982) and Bollerslev (1986). See Bollerslev, Chou and Kroner (1992) andDiebold and Lopez (1995) for reviews.

5See Amin and Ng (1993), Bollerslev and Mikkelsen (1996), Engle and Mustafa (1992), Heston and Nandi(2000), Christoffersen and Jacobs (2004) and Duan, Ritchken and Sun (2002).

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closed-form solution for option prices, namely the dynamic in Heston and Nandi (2000).A second reason for the lack of attention to empirical comparisons between discrete-time and

continuous-time models may be the belief that the performance of selected discrete-time andcontinuous-time models ought to be very similar when the continuous-time dynamic is the limitof the discrete-time dynamic. The first such limit result was demonstrated by Nelson (1990). Arelated result is given by Heston and Nandi (2000), who show that the continuous-time model inHeston (1993) can be obtained as the limit of the GARCH dynamic they suggest. It has becomeclear, however, that while these limit results are theoretically intriguing, their practical relevancemay be modest. A given discrete-time model can have several continuous-time limits and viceversa, a given continuous-time model can be the limit for more than one discrete-time model.6

A third reason for the limited amount of work done comparing discrete and continuous modelsfor option valuations is likely to be methodological. The two classes of models are typicallyimplemented using very different econometric methods which renders fair comparisons difficult.We contribute to the literature by suggesting a newmethodology which allows for straightforwardcomparisons of latent volatility continuous time models with discrete time GARCHmodels. Ourmethod makes joint use of option prices and underlying returns and it allows for a fair comparisonby ensuring that each model is implemented using the same objective function and the sameinformation set.7

We argue that a comparison of the performance of discrete-time and continuous-time modelsis of substantial interest. The link between continuous- and discrete-time models, while relevant,is not sufficiently unambiguous to justify ignoring one class of models. This also means that onecannot simply ascribe empirical results obtained for one class of models to their mathematicalequivalent in the other class. Despite the fact that the limit results suggest that there may besimilarities in the performance of certain continuous- and discrete-time models, it may well bethe case that one class of models systematically outperforms the other. Moreover, a comparisonof discrete-time and continuous-time models will help to establish a benchmark for the modelingof option prices in discrete time, much like the Heston (1993) model currently fulfills this rolein the continuous-time literature. In other words, by comparing the empirical performance of acandidate discrete-time benchmark model with the continuous-time Heston (1993) benchmarkmodel, we attempt to establish a benchmark for judging the performance of future discrete-timemodels. For example, if an existing paper improves over a given benchmark in the GARCHoption valuation literature, it may be difficult to interpret the significance of this result for thecontinuous time option valuation literature, because the performance of the benchmark GARCHmodel may not be satisfactory vis-a-vis the performance of the continuous-time Heston (1993)model. The empirical results in this paper thus aim to facilitate comparisons between differentclasses of models.The paper proceeds as follows. In Section 2 we introduce the discrete and continuous time

volatility models, and we discuss their implementation. In Section 3 we present and discuss theempirical results. Section 4 concludes.

6See for example Corradi (2000) for results along these lines.7See Chernov and Ghysels (2000) and Bollen and Rasiel (2003) for other comparisons of discrete-time and

continuous-time models.

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2 Volatility Dynamics and Model Implementation

In this section we introduce four volatility models. The most popular volatility model in theoption valuation literature is the continuous-time stochastic volatility model of Heston (1993).This model has been estimated and tested in a number of influential studies. The existingcontinuous-time literature does arguably not contain modifications of the stochastic volatilitydynamic that outperform this model out-of-sample, and therefore it is a good benchmark. Theunderlying reason for this finding is that the Heston (1993) model captures two important stylizedfacts that are needed to model option prices: volatility clustering and the leverage effect.8 Afteraccounting for these two stylized facts, additional modifications of the volatility dynamic do notresult in significant out-of-sample improvements in fit. An important advantage of the Heston(1993) model is that it yields closed-form solutions for European option prices. The closed-formsolution is due to the affine structure of this model. Henceforth we refer to the Heston (1993)model as AF-SV, indicating that it is an affine stochastic volatility model.There is an extensive and growing literature on the use of jumps in returns and volatility to

improve the performance of the Heston model.9 Extending our comparison to models of this typeis interesting, but beyond the scope of this paper. Our paper investigates 1) whether formulatingthe volatility model in discrete or continuous time affects the performance of the option valuationmodel, and 2) whether the affine structure imposes empirically important limitations on themodel.Following the theoretical developments in Duan (1995) and Amin and Ng (1993), the discrete-

time GARCH option valuation literature has resulted in fewer empirical studies.10 Interestingly,the choice of a benchmark model in this literature is somewhat harder, because the number ofcompeting volatility models is far greater. Following the work of Engle (1982) and Bollerslev(1986), volatility modeling using returns data has proceeded in discrete time, and this literaturehas spawned a large number of competing models. For practical reasons, we limit the numberof models to two. The first discrete-time model we investigate is the one by Heston and Nandi(2000). This model is a natural choice, as it was designed with option valuation in mind. Also,like the Heston (1993) model, it has a closed-form solution, it contains a leverage effect and itallows for volatility clustering. Heston and Nandi (2000) have demonstrated that it performssatisfactorily vis-a-vis ad-hoc benchmarks for the purpose of option valuation. Because thismodel also has an affine structure, we refer to it as AF-GARCH.The second discrete-time model we investigate is the non-affine NGARCH model of Engle

and Ng (1993), henceforth referred to as NA-GARCH. This is the simplest model in the GARCHliterature that contains both volatility clustering and a leverage effect. It is also the modelconsidered by Duan (1995). Moreover, Christoffersen and Jacobs (2004A) demonstrate that

8See among others Benzoni (1998), Chernov and Ghysels (2000), Christoffersen and Jacobs (2004A), Eraker(2004), Eraker, Johannes and Polson (2003), Heston (1993), Heston and Nandi (2000) and Nandi (1998) for theimportance of volatility clustering and the leverage effect for option valuation.

9See Andersen, Benzoni and Lund (2002), Bakshi, Cao and Chen (1997), Bates (1996, 2000), Chernov, Gallant,Ghysels and Tauchen (2003), Eraker, Johannes and Polson (2003), Eraker (2004), Pan (2002), Broadie, Chernovand Johannes (2004), Carr and Wu (2004) and Huang and Wu (2004).10Amin and Ng (1993), Bollerslev and Mikkelsen (1996), Engle and Mustafa (1992), and Duan, Ritchken and

Sun (2002) estimate model parameters using the underlying asset returns and subsequently value options. Hestonand Nandi (2000) and Christoffersen and Jacobs (2004A) provide an integrated analysis of equity option pricesand the underlying returns.

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several richer GARCH parameterizations do not improve on the option valuation performanceof the NA-GARCH model. An additional motivation for investigating the NA-GARCH modelis that, in contrast with the Heston (1993) and Heston and Nandi (2000) models, it has a non-affine volatility specification. Affine models are extremely popular in option valuation becausethey yield closed-form pricing results.11 It is therefore of interest to verify whether this focuson closed-from valuation results comes at the cost of a deterioration in the model’s empiricalperformance compared with non-affine classes of volatility dynamics.Finally, to complete the comparison between continuous-time and discrete-time models on

the one hand and affine and non-affine models on the other hand, we also analyze a non-affinecontinuous-time stochastic volatility model. We refer to this model as NA-SV.We now turn to a description of these four volatility models and a discussion of their imple-

mentation for the purpose of option valuation. For the two discrete-time GARCH models, weconsider the GARCH(1,1) representation because it is most closely related to the Heston (1993)continuous-time model. We start with the discrete-time Heston and Nandi (2000) model becauseits econometric implementation is simpler than that of the continuous-time Heston (1993) model.We subsequently discuss our implementation of the Heston (1993) model, which is different fromthe implementation available in the literature, and designed to facilitate the comparison withthe discrete-time models, as well as to provide the best possible fit for the model. We concludewith the specification of the non-affine NA-GARCH and NA-SV models, which rely on numericaltechniques for option valuation.

2.1 The Affine GARCH(1,1) Model (AF-GARCH)

Heston and Nandi (2000) propose a class of affine GARCH models (AF-GARCH) that allow fora closed-form solution for the price of a European call option. We investigate the GARCH(1,1)version of this model, which is given by

ln(St+1) = ln(St) + r + λht+1 +pht+1zt+1 (2.1)

ht+1 = ω + bht + a³zt − c

pht´2

(2.2)

where St+1 denotes the underlying asset price, r the risk free rate, λ the price of risk and ht+1the daily variance on day t+ 1 which is known at the end of day t. The zt+1 shock is assumedto be i.i.d. N(0, 1). The Heston-Nandi model captures time variation in the conditional variancein ways similar to Engle (1982) and Bollerslev (1986). The parameter c represents the leverageeffect, which captures the negative relationship between returns and volatility (Black (1976))and results in a negatively skewed conditional distribution of multi-day returns. Note also thatusing the conventional GARCH notation, the conditional variance for day t+ 1 denoted ht+1 isknown at the end of day t.Variance persistence can be computed via

b+ ac2 ≡ 1− κ

11Affine models are also very popular in the term structure literature for exactly the same reason. See forinstance Duffie and Kan (1996) and Dai and Singleton (2000).

5

and the unconditional variance can be computed via

(ω + a)/¡1− b− ac2

¢= (ω + a)/κ ≡ θ

Now we can rewrite the variance process as

ht+1 − ht = κ(θ − ht) + a³¡z2t − 1

¢− 2cztpht´

which suggests the model’s relationship with the diffusion volatility models considered below.The risk-neutral dynamics for the GARCH(1,1) model (2.1)-(2.2) are given by12

ln(St+1) = ln(St) + r − 12ht+1 +

pht+1z

∗t+1 (2.3)

ht+1 = ω + bht + a(z∗t − c∗pht)

2

with c∗ = c+ λ+ 0.5 and z∗t ∼ N(0, 1) under the risk neutral measure.We provide an integrated analysis of this model, using data on equity option prices as well

as the time series of underlying equity returns. In order to value options at each date t, we needan estimate of the conditional volatility ht on that particular date. This is often referred toas the filtering problem. One of the appealing aspects of discrete-time GARCH models is thatthis filtering problem is extremely simple, and that it is therefore straightforward to implementan integrated analysis of option prices and the underlying equity returns. Indeed, the filteringproblem is solved by noting that from (2.1) we have

zt+1 = (Rt+1 − r − λht+1) /pht+1 (2.4)

where Rt = ln(St/St−1). Substituting (2.4) in (2.2), it can be seen that the updating from ht toht+1 is done using an updating function that exclusively involves observables

ht+1 = ω + bht + a((Rt − r) /pht − (λ+ c)

pht)

2 (2.5)

Model parameters are obtained by using the nonlinear least squares (NLS) estimation tech-niques to minimize

$MSE =1

NT

Xt,i

(Ci,t − Ci (ht+1))2 (2.6)

where NT =TPt=1

Nt, T is the total number of days included in the options sample and Nt is the

number of options included in the sample at date t, Ci,t is the market price of option i quotedon day t and Ci (ht+1) is the model price.The implementation is therefore relatively simple: the NLS routine is called with a set of

parameter starting values. The variance dynamic in (2.5) is then used to update the variancefrom day to day and the GARCH(1,1) option valuation formula from Heston and Nandi (2000)

12For the underlying theory on risk neutral distributions in discrete time option valuation see Rubinstein (1976),Brennan (1979), Amin and Ng (1993), Duan (1995), Camara (2003), Heston and Nandi (2000) and Schroder(2004).

6

is used to compute the model prices. At time t, a European call option with strike price K thatexpires at time T can be calculated from

C (ht+1) = e−r(T−t)E∗t [Max(ST −K, 0)] (2.7)

= StP1 −Ke−r(T−t)P2

where

P1 =1

2+

1

πSter(T−t)

Z ∞

0

Re

·K−iφf∗(t, T ; iφ+ 1)

iφ

¸dφ (2.8)

P2 =1

2+1

π

Z ∞

0

Re

·K−iφf∗(t, T ; iφ)

iφ

¸dφ

and where f∗(t, T ; iφ) is the conditional characteristic function of the logarithm of the spot priceunder the risk neutral measure, which is characterized by a set of difference equations withterminal conditions. See Heston and Nandi (2000) for these equations.

2.2 The Affine Stochastic Volatility Model (AF-SV)

The Heston (1993) continuous-time stochastic volatility model (AF-SV) is defined by the fol-lowing two equations

dSt = µStdt+pVtStdw

St (2.9)

dVt = κ(θ − Vt)dt+ σpVtdw

Vt (2.10)

with corr(dwSt , dw

Vt ) = ρ. This model also allows for volatility clustering through the autore-

gressive component of volatility, as well as for the leverage effect through a negative correlationcoefficient ρ, which translates into negative skewness of the return distribution. Under the as-sumption that the volatility risk premium λ(St, Vt, t) is equal to λVt, the risk neutral dynamicexpressed in terms of the physical parameters is

dSt = rStdt+pVtStdw

∗St (2.11)

dVt = (κ+ λ)(κθ/(κ+ λ)− Vt)dt+ σpVtdw

∗Vt (2.12)

with corr(dw∗St , dw∗Vt ) = ρ. Heston (1993) demonstrates that this model admits a closed formsolution, which is presented here in terms of the physical parameters κ, θ, λ, ρ and σ in order tofacilitate the description of our estimation procedure below.

C(Vt) = StP1 −Ke−r(T−t)P2 (2.13)

where

Pj =1

2+1

π

Z ∞

0

Re

·exp(−iφ log(K))fj(x, Vt, T ;φ))

iφ

¸dφ, j = 1, 2

and

7

fj(x, Vt, T ;φ) = exp(C(T − t;φ) +D(T − t;φ)Vt + iφx)

C(τ ;φ) = rφiτ +κθ

σ2

µ(bj − ρσφi+ d)τ − 2 log

·1− g exp(dτ)

1− g

¸¶D(τ ;φ) =

bj − ρσφi+ d

σ2

µ1− exp(dτ)1− g exp(dτ)

¶

d =q(ρσφi− bj)

2 − σ2(2µjφi− φ2)

g =bj − ρσφi+ d

bj − ρσφi− d

µ1 =1

2, µ2 = −

1

2, b1 = κ+ λ− ρσ, b2 = κ+ λ

The Heston model has been investigated empirically in a large number of studies. Oftenit is used as a building block together with models of jumps in return and volatility. For ourpurpose, it is important to note that the model can be estimated and investigated empiricallyusing a number of different techniques. First, the model’s parameters can be estimated using asingle cross-section of option prices (for example see Bakshi, Cao and Chen (1997)). A secondtype of implementation of the Heston model uses multiple cross sections of option prices butdoes not combine this with a fully integrated analysis of the underlying asset returns. Instead,for every cross section a different initial volatility is estimated, leading to a highly parameterizedproblem (see for instance Bates (2000) and Huang and Wu (2004)). A number of other papersprovide a likelihood-based analysis of the stochastic volatility model. Eraker (2004) provides aMarkov Chain Monte Carlo analysis. Finally, Chernov and Ghysels (2000) use an analysis usingthe efficient method of moments and Pan (2002) uses a method of moments technique as well.In this paper we implement the Heston model in a novel way. This is mainly motivated by

the objective of this paper, which is to compare the performance of discrete-time and continuous-time methods. As such, we implement the model using a method based on the same objectivefunction (2.6) used to implement the Heston-Nandi (2000) GARCH model. In our opinion, thismethod guarantees the best possible performance for the Heston model in- and out-of-sample.This is motivated by the insights of Granger (1969), Weiss (1996) and Weiss and Andersen(1984) who demonstrate that the choice of objective function (also labeled loss function) is anintegral part of model specification. It follows that estimating a model using one objectivefunction and evaluating it using another one amounts to suboptimal choice of objective function.Christoffersen and Jacobs (2004B) demonstrate that this issue is empirically relevant for theestimation of the deterministic volatility functions in Dumas, Fleming and Whaley (1998). Wetherefore implement the Heston model in a way that is consistent with these insights.Our implementation uses the Auxiliary Particle Filter (APF) algorithm along with the ob-

served stock price to filter the volatility.13 As shown by Pitt and Shephard (1999) the APF offersa convenient filtering algorithm for non-linear models such as the stochastic volatility model weconsider here.13We have also implemented the Sampling-Importance-Resampling (SIR) particle filter as a robustness check,

and this yields similar results.

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2.2.1 Volatility Transformation and Discretization

To prevent Vt from becoming negative, we work with f(Vt) = log(Vt). The dynamic of interestis therefore

d log(Vt) =∂f

∂VtdVt +

1

2

∂2f

∂V 2t

d hV itwhere hV it is the quadratic variation of Vt. Using Ito’s lemma yields

d log(Vt) =1

Vt

µκ(θ − Vt)− 1

2σ2¶dt+ σ

1√VtdwV

t (2.14)

In order to compute option prices according to (2.13) within the iterative search, we need thestructural parameters, but also the volatility path which is not observed.Note that equations (2.9) and (2.10) specify how the unobserved state is linked to observed

stock prices. This relationship allows us to infer the volatility path using the returns data. Wefirst need to discretize equations (2.9) and (2.14) . There are different discretization methods andevery scheme has certain advantages and drawbacks. We use the Euler scheme which is easy toimplement and has been found to work well for this type of applications.14 Discretizing equation(2.9) (using log(St) instead of St and applying Ito’s lemma again) and (2.14) gives

log(St+1) = log(St) +

µµ− 1

2Vt

¶+pVtε

St+1 (2.15)

log(Vt+1) = log(Vt) +1

Vt

µκ (θ − Vt)− 1

2σ2¶+ σ

1√VtεVt+1 (2.16)

We implement the discretized model in (2.15) and (2.16) using daily returns, and all parameterswill be expressed in daily units below. The model is characterized by six structural parameters:µ, κ, θ, σ, λ and ρ for which we have to choose a set of starting values. Subsequently, we haveto choose an initial variance V0 (the starting value for the variance path). We set the initialvolatility equal to the unconditional variance, V0 = θ.Our optimization algorithm minimizes (2.6) using an iterative procedure. At each iteration,

the volatility is filtered using the information embedded in observed returns. Since the min-imization is performed relative to option prices, option data also indirectly contribute to thedetermination of the volatility path. Finally, using the filtered volatility and the structural pa-rameters option prices are computed according to Heston’s formula and the MSE is calculated.This procedure is repeated until the optimum is reached. Because this procedure is relativelynew in finance, we now describe it in more detail.

2.2.2 Filtering the volatility path using the APF algorithm

Although the choice of the initial variance V0 is well-motivated, we still want to mitigate itsimpact on the valuation exercise. For that reason, we start iterating using the volatility dynamic252 days (1 year) before the first option price is observed. This implementation is identical to

14See e.g. Johannes and Polson (2003).

9

that followed in the implementation of the discrete-time Heston-Nandi model. In what followsV0 therefore corresponds to the volatility 252 days before the first option price is observed.The idea underlying the APF technique is to infer the volatility path from the observed

returns data. V0 is propagated one day ahead using equation (2.14) into N possible states (orparticles).15 Subsequently, we use an auxiliary variable ι and the available data to decide whichparticles to keep in order to simulate the one day ahead volatility.Assume that we are at date t and we have an initial set of particles

©V jt ,W

jt

ªNj=1

with V jt

the volatility at day t for the state j, W jt the weight associated with state j at date t and

j = 1, .., N.16 We want to propagate Vt one day ahead into©V jt+1,W

jt+1

ªNj=1

using the APF . Thistask requires the following steps:

Step 1: Selecting the particlesThe weight W j

t reflects the information available at time t only and does not include ourexpectations about (t+1). So even if state j is very likely according to the realization of the stockprice at date t, it is possible that the realization of the stock at date (t+1) suggests a certain re-adjustment of the probability that state j has occurred. By combining the information availableat t and our expectations about (t + 1), we can eliminate many states with a low realizationprobability right before the propagation step. This is achieved by:I) Computing a summary location statistic for (t+ 1) that reflects the information at t. We

use the mean µjt+1 given by:µjt+1 = E

¡log(Vt+1)|V j

t

¢II) Simulating the auxiliary variable ιj

ιj ∝W jt p¡log(St+1)|µjt+1

¢where p

¡log(St+1)|µjt+1

¢is the conditional density of log(St+1) which can be easily inferred

from (2.15). This auxiliary variable is simply an index that tells us which particle to keep andwhich particle to discard. After this selection exercise we obtain N new particles which areimplicitly functions of the auxiliary variable ι,

{V (ι)t,W (ι)t}Nj=1In order to keep the notation simple we will omit ι below.Step 2: Simulating the state forward (Sampling)This is done by computing Vt+1 using equation (2.16) and taking the correlation into account.

We have

log

µSt+1St

¶=

µµ− 1

2Vt

¶+pVtε

St+1

which gives

εSt+1 =log³St+1St

´− ¡µ− 1

2Vt¢

√Vt

15We set N = 500 in the initial search. Once a candidate optimum is identified we confirm it by increasing Nto 5, 000. The results change very little when N is increased.16At time 0, the initial set is constructed by setting each particle equal to the unconditional variance θ and

giving all particles equal weight, 1/N.

10

SinceεVt+1 = ρεSt+1 +

p1− ρ2εt+1

where corr(εSt+1, εt+1) = 0, we get

log(Vt+1) =

log(Vt) +1

Vt

µκ (θ − Vt)− 1

2σ2¶+ σ

1√Vt

ρlog³St+1St

´− ¡µ− 1

2Vt¢

√Vt

+p1− ρ2εt+1

We simulate N states (N particles) which describe the set of possible values of Vt+1.

Step 3: Computing and normalizing the weights (Importance Sampling)At this point, we have a vector of N possible values of Vt+1 and we know according to

equation (2.15) that given the other available information, Vt+1 is sufficient to generate log(St+2).Therefore, equation (2.15) offers a simple way to evaluate the likelihood that the observation St+2has been generated by Vt+1. Hence, we have to compute the vector W whose elements representthe weight given to each particle (or the likelihood or probability that the particle n has generatedSt+2). The likelihood is computed as follows:

Wnt+1 =

1pV nt+1

exp

−12

³log³St+2St+1

´− ¡µ− 1

2V nt+1

¢´2V nt+1

This is to be repeated for n = 1, .., N. Finally, because nothing guarantees that

PNn=1W

nt+1 = 1,

we have to normalize and set Wn

t+1 =Wnt+1PN

n=1Wnt+1

. In summary therefore, at the end of step 2, we

obtain a set of N particles describing the density of Vt+1. This procedure (Steps 1, 2 and 3) isrepeated for t = 1, ...T . To obtain the filtered volatility path, we then compute

V̄t+1 =NXn=1

Wn

t+1Vnt+1

for each t.

2.2.3 Computing option prices and evaluating the loss function

We are now in a position to evaluate option prices Ci

¡V̄t¢based on the filtered volatility path

using Heston’s closed form solution according to equation (2.13). We subsequently evaluate theloss function

$MSE =1

NT

Xt,i

¡Ci,t − Ci

¡V̄t¢¢2

(2.17)

as before.We use a standard numerical optimization routine to update the model parameters anditerate until convergence is achieved.Notice that the methodology we have suggested here for estimating the continuous time

stochastic volatility model relies on the same information set and the same objective function asthose used for the discrete time Heston-Nandi GARCH model. This will allow for a fair empiricalcomparison.

11

2.3 The Non-Affine GARCH Model (NA-GARCH)

One objective of this paper is to compare the performance of discrete-time and continuous-timeoption valuation models. The comparison between the Heston (1993) and Heston and Nandi(2000) models is a natural one, because both models belong to the affine class. A second objectiveof the paper is to investigate whether non-affine models can outperform affine models. The optionvaluation literature has focused almost exclusively on affine models because they yield closed-form solutions. This paper investigates if there is a price to be paid for this computationalconvenience in terms of empirical fit.To investigate this further, we compare the two models introduced above with the non-affine

NGARCH model of Engle and Ng (1993). We could have equivalently introduced a non-affinecontinuous-time model to investigate the importance of the affine restriction. However, we chosethe NGARCH model because it is relatively easy to analyze. We will refer to it as NA-GARCH.The model is given by

ln(St+1) = ln(St) + r + λpht+1 − 0.5ht+1 +

pht+1zt+1 (2.18)

ht+1 = ω + bht + aht(zt − c)2 (2.19)

The variance persistence can be computed via

b+ a¡1 + c2

¢ ≡ 1− κ

and the unconditional variance can be computed via

ω/¡1− b− a

¡1 + c2

¢¢= ω/κ ≡ θ

Now we can rewrite the variance process as

ht+1 − ht = κ(θ − ht) + aht¡¡z2t − 1

¢− 2czt¢which again suggests the GARCH models’ relationship with the diffusion volatility models.Note that this model differs in some subtle ways from the Heston-Nandi model in (2.1)-

(2.2). The Heston-Nandi model was engineered with the specific purpose of yielding closed-fromoption prices. The specification in (2.18)-(2.19) does not yield closed form option prices, but wasdesigned to provide a good fit to the underlying equity returns. The question of interest is if therestrictions built into affine models such as (2.1)-(2.2) reduce the ability of the model to fit thedata.The risk-neutral dynamics for the NA-GARCH model (2.18)-(2.19) can be obtained using the

same theoretical arguments underlying the Heston-Nandi model

ln(St+1) = ln(St) + r − 0.5ht+1 +pht+1z

∗t+1 (2.20)

ht+1 = ω + bht + aht(z∗t − c∗)2

with c∗ = c + λ and z∗t ∼ N(0, 1). We can then estimate the model by minimizing (2.6), usingthe updating rule

ht+1 = ω + bht + aht³h(Rt − r + 0.5ht−1) /

pht−1

i− (c+ λ)

´2(2.21)

12

Option prices are computed numerically according to

C (ht+1) = e−r(T−t)E∗t [Max(ST −K, 0)]

where the expectation is calculated by Monte Carlo simulation of the daily returns from (2.20).We use 1000 simulated paths and a number of numerical techniques to increase numerical ef-ficiency: the empirical martingale method of Duan and Simonato (1999), stratified randomnumbers, antithetic variates and a control variate technique. The parameters are again chosento minimize

$MSE =1

NT

Xt,i

(Ci,t − Ci (ht+1))2 (2.22)

2.4 The Non-Affine Stochastic Volatility Model (NA-SV)

In order to complete our comparison of discrete versus continuous time volatility dynamics onthe one hand and affine versus non-affine models on the other, we need a non-affine continuoustime stochastic volatility model (NA-SV). While several sensible models are available, we focuson a continuous time limit of the non-affine NA-GARCH model considered above. Duan (1996,1997) shows that using the physical measure the discrete time NA-GARCH model convergesweakly to the bivariate continuous time process defined as

d log (St) =

µr + λ

pht − 1

2ht

¶dt+

phtdw1t

dht =¡ω +

¡b+ a(1 + c2)− 1¢ht¢ dt+ vtρdw1t + vt

p1− ρ2dw2t

where w1t and w2t are independent Brownian motions under the physical measure, and where

vt =p2a2(2c2 + 1)ht

ρ = − cp(c2 + 0.5)

so that −1 < ρ < 0 when c > 0.Notice that if we define persistence as 1 − κ = b + a(1 + c2), and unconditional variance as

θ = ω/κ we can write

dht = κ (θ − ht) dt+ vtρdw1t + vtp1− ρ2dw2t

which shows the similarly with the affine SV model in terms of volatility drift. But of coursethe innovations are now scaled by the conditional variance rather than by the square root of theconditional variance as is the case in the AF-SV model.Under the risk neutral measure, the continuous time limit of the NA-GARCH model is

d log (St) =

µr − 1

2ht

¶dt+

phtdw1t

dht =¡ω +

¡b+ a(1 + c2)− 1 + 2λac¢ht¢ dt+ vtρdw

∗1t + vt

p1− ρ2dw∗1t

13

where w∗1t = w1t+λt and w∗2t = w2t are two independent Brownian motions under the risk neutralmeasure.The NA-SV model presented here has an unobserved variance factor and no closed form

option valuation formula. Thus we need to implement it using the auxiliary particle filter toconstruct the variance path (as in AF-SV) and Monte Carlo simulation to calculate the optionprices (as in NA-GARCH). It is therefore the most computationally intensive of the four modelsconsidered.

3 Empirical Results

This section presents the empirical results. We first discuss the data, followed by an empiricalevaluation of the four models under investigation and a detailed discussion of the differences inperformance of these models in- and out-of-sample.

3.1 Data

We conduct our empirical analysis using six years of data on S&P 500 call options, for theperiod 1990-1995. We apply standard filters to the data following Bakshi, Cao and Chen (1997).We only use Wednesday and Thursday options data. For the in-sample analysis, we use theWednesday data. Wednesday is the day of the week least likely to be a holiday. It is also lesslikely than other days such as Monday and Friday to be affected by day-of-the-week effects. Forthose weeks where Wednesday is a holiday, we use the next trading day. The decision to pick oneday every week is to some extent motivated by computational constraints. The optimizationproblems are fairly time-intensive, and limiting the number of options reduces the computationalburden. Using only Wednesday data allows us to study a fairly long time-series, which is usefulconsidering the highly persistent volatility processes. An additional motivation for only usingWednesday data is that following the work of Dumas, Fleming andWhaley (1998), several studieshave used this setup (see for instance Heston and Nandi (2000)).We obtain six sets of parameter estimates in the in-sample analysis. We simply split the six

years of data in six datasets, one for each calendar year, and perform annual estimation exercises.For each estimation sample, we use a volatility updating rule starting from the model impliedunconditional variance on January 1, 1989.Table 1 presents descriptive statistics for the options data for the 1990-1995 Wednesday in-

sample data by moneyness and maturity. Panels A and B indicate that the data are standard.Panel C displays the volatility smirk in the data. The slope of the smirk clearly differs acrossmaturities. We summarize the data for all six estimation samples in one set of tables to savespace. Descriptive statistics for the separate samples different sub-periods (not reported here)reveal similar stylized facts. The slope of the smirk changes over time, but the smirk is presentthroughout the sample. The top panel of Figure 1 gives some indication of the pattern of impliedvolatility over time. For the 313 Wednesdays of options data used in the empirical analysis,we present the average implied volatility of the options on each Wednesday. It is evident fromFigure 1 that there is substantial clustering in implied volatilities. It can also be seen thatvolatility is higher in the early part of the sample. The bottom panel of Figure 1 presents atime series for the 30-day at-the-money volatility (VIX) index from the CBOE for our sample

14

period. A comparison with the top panel clearly indicates that the options data in our sampleare representative of market conditions, although the time series based on our sample is of coursea bit more noisy due to the presence of options with different moneyness and maturities.After conducting six in-sample estimations, we proceed to conduct separate out-of-sample

analyses for each of the six sample years using the trading day following each in-sample Wednes-day. We refer to this as Thursday data. Table 2 presents descriptive statistics for the out-of-sample data. The patterns in the data are clearly similar to those in the in-sample data in Table1.

3.2 Parameter Estimates and Option Mean-Squared-Errors

Table 3 presents the parameter estimates for each of the four models and for each of the sixannual estimation samples. Some of the parameters vary considerably over time. However,time-variation in individual parameters does not necessarily indicate time-variation in model fitand model performance, because the key properties of the models are determined by nonlinearcombinations of the individual model parameters. An exception to this is the AF-SV model,where parameters are more easily interpreted individually; for example, κ denotes variance meanreversion and θ denotes the unconditional variance. These parameters appear to be relativelystable over time, even though the mean reversion increases over time. Finally, note that thecorrelation parameter ρ in the AF-SV model hits the prespecified boundary of -0.999 in 1990 and1991. The parameter that determines the size of the leverage effect c is also higher for the othermodels in 1990 and 1991, but the fact that the parameter ρ hits the boundary seems to indicatethat it is difficult for the AF-SV model to match this stylized fact in the data in this period.Table 4 complements Table 3 by focusing on key characteristics of the models, which are

given by nonlinear combinations of the parameters in Table 3. It reports variance persistence (1minus variance mean reversion) and the unconditional variance for each model, under both thephysical and risk neutral measures. The variance persistence is close to one for all models, whichis consistent with other findings in the literature. It is generally larger under the risk neutralmeasure and often largest for the AF-SV specification. The unconditional variance displaysconsiderable variation over time. In 1991 the persistence in the NA-GARCH model is very closeto 1, leading to an unrealistically large estimate of the unconditional variance. Keeping in mindthe difficulties of the AF-SV model to capture the leverage effect mentioned above, we thereforeconclude that it is challenging for most models to provide a satisfactory fit to the 1991 data.The models yield different results for some of these key characteristics. For instance, while

the models display high persistence for all sample years, the physical persistence is always morethan 99 percent for the AF-SV model, while in the AF-GARCH model it drops to 92.10 percentin one of the samples. Also, the unconditional physical volatility in the AF-SV model is usuallyconsiderably higher than that of the other models, but this is not the case for the unconditionalrisk neutral volatility.The in- and out-of-sample RMSEs from the four models are reported in Table 5 for each of

the six samples. First note that the NA-GARCH model is best overall and the AF-SV model isworst overall both in- and out-of-sample. The differences between the best and the worst modelsare around 16% in sample and 15% out of sample. The performance of the NA-GARCH andNA-SV models is very similar in- and out-of-sample with the NA-GARCH model performingslightly better overall. The fit of the AF-GARCH model falls in between the AF-SV and the

15

non-affine models. The AF-GARCH is around 10% better than the AF-SV in sample and about6% better out of sample. Looking across the six samples, it is clear that the non-affine modelssubstantially outperform the affine models in every year, both in- and out-of-sample.Figure 2 provides further perspective on the similarities and differences between the four

models by providing volatility sample paths for the models. The figure plots volatility pathsfor the four models using two sets of parameters estimates for each, the estimates for the 1990sample (left column) and the estimates for the 1995 sample (right column). It can be seen thatdespite the fact that there are significant differences in the estimated long-run volatility betweenthe model estimates for the 1990 sample, the sample paths for the four models are quite similar.For the 1995 estimates, the models all display increases in volatility around the same period, butthe overall sample paths are rather different for the four models. When comparing the samplepaths for the 1990 and 1995 estimates for a given model, it is clear that the 1995 estimatesdisplay lower persistence. The larger volatility of volatility estimates in the 1995 sample arealso apparent.

3.3 Pricing Errors Over Time and Across Moneyness and Maturity

We now analyze the performance of the four models in more detail. Figure 3 addresses theperformance of the models over time. The top four panels present the RMSE on a week-by-weekbasis. It can clearly be seen that the four models display important similarities in terms of thepricing patterns that they can and cannot explain. This observation is confirmed by inspectingthe week-by-week bias in Figure 4.What is even more striking in Figures 3 and 4 are the similarities in RMSE and bias over

time for the two affine models on the one hand and the two non-affine models on the other hand.Whether the model is affine or not seems to be much more important for its performance thanwhether it is formulated in continuous-time or not. This confirms the message from the overallRMSEs in Table 5, but the message is much more striking when delivered visually on a week-by-week basis as in Figures 3 and 4. It is also interesting to visually inspect the relationshipbetween the level of the volatility in the bottom panel of Figures 3 and 4 and the RMSE and biasfor the different models. It is clear that in periods of high volatility, the RMSE for all modelsincreases.Tables 6 and 7 present an analysis of the in- and out-of-sample RMSE by moneyness and

maturity. This table allows for some important conclusions. For example, consider the differencebetween the AF-SV and AF-GARCH models. While the overall RMSE difference between thetwo models in Table 5 is approximately 10% in sample and 6% out-of-sample, there are importantvariations in the relative performance of the models across maturity. The overall RMSE of theAF-SV model is larger than that of the AF-GARCH model, but for short-maturity options theAF-SV model performs significantly better. This finding is perhaps somewhat surprising. Whilewe believe that the limit arguments have some empirical value, our prior was that continuous-time models would prove to be somewhat restrictive because of the assumption of a continuoussample path. This restriction is well recognized in the continuous-time option valuation literature,and stochastic volatility models are augmented with jump models to improve their performance.However, jump models are believed to help the performance of stochastic volatility models mainlyfor short-maturity options. We therefore expected that if the Heston (1993) AF-SV model would

16

underperform the AF-GARCH model, it would be for short-maturity options. Instead, the AF-SV model seems to underperform the AF-GARCH model mainly for longer maturities.Finally, an important conclusion from Tables 6 and 7 is that the two non-affine models

outperform the corresponding affine models for almost every cell in the moneyness-maturitymatrix, in-sample as well as out-of-sample.

3.4 Conditional Density Dynamics

Ultimately, the option prices for the different models are determined by the model-implied con-ditional density dynamics. Therefore, we now discuss model differences by focusing on variousaspects of the conditional density.In order to asses the different models’ ability to generate time-variation in the asymmetry of

the return distribution, Figure 5 plots the conditional covariance between returns and variancesfor each model. We refer to this as the conditional leverage path, which for the four models isgiven by

AF-GARCH: covt (log(St+1), ht+2) = −2acht+1AF-SV: covt (log(St+1), Vt+1) = σρVt

NA-GARCH: covt (log(St+1), ht+2) = −2ach3/2t+1

NA-SV: covt (log(St+1), Vt+1) = −2acV 3/2t

Notice that critical differences between the affine and non-affine models show up in theseconditional moments. The 3/2 term on the volatility of the non-affine models suggests thatthese models may be able to exhibit more variation in the conditional leverage paths. Figure5 confirms this intuition. For each model we plot the daily conditional leverage path during1990-1995, annualized by multiplying by 252. The left column uses the 1990 estimates fromTable 3 and the right column uses the 1995 estimates. The four rows of panels correspond tothe AF-GARCH, AF-SV, NA-GARCH and NA-SV models respectively. Notice that the scalingis different between the two columns, because the 1995 estimates imply a much larger level of(and variation in) the leverage effect.The main conclusion is that for both set of estimates the non-affine models imply more

substantial leverage, as well as more substantial variation over time in the leverage effect. Giventhe importance of the leverage effect for option valuation, this may be a very important factorin explaining the differences in fit between affine and non-affine models documented in Table 5.Option prices are a function of the conditional variance, and therefore the variation in option

prices over time is related to the conditional variance of variance. Figure 6 plots the square rootof the conditional variance of variance of returns for the four models, which is given by

AF-GARCH: V art(ht+2) = 2a2 + 4a2c2ht+1

AF-SV: V art(Vt+1) = σ2Vt

NA-GARCH: V art(ht+2) = 2a2(1 + 2c2)h2t+1

NA-SV: V art(Vt+1) = 2a2(1 + 2c2)V 2

t

Notice again that these conditional moments indicate important differences between affine andnon-affine models. The conditional variance shows up in levels in the affine models and in squared

17

form in the non-affine models, which again suggests that non-affine models will display morevariation in the conditional volatility of variance.17 Figure 6 reports the empirical results. Foreach model, we plot the daily conditional volatility of variance path during 1990-1995 annualizedby multiplying by 252. The left column uses the 1990 estimates from Table 3 and the rightcolumn uses the 1995 estimates. The four rows of panels correspond to the AF-GARCH, AF-SV,NA-GARCH and NA-SV models respectively, and again the scaling differs between the columnsbecause the 1995 estimates imply higher conditional volatility of variance. Figure 6 indicatesthat for both sets of estimates the non-affine models also display much more time-variation inthe volatility of variance. These differences between the models further help us understand thesuperior fit of the non-affine models.

3.5 State Price Densities

Figures 7 and 8 provide additional intuition for the differences in performance between the fourmodels. Using estimates for the 1990 and 1995 samples respectively, these figures depict thesimulated state price densities for a one-month, three-month and one-year horizon. Each rowof panels reports the risk neutral distribution of the index return according to the AF-GARCH,AF-SV, NA-GARCH and NA-SV models respectively. The normal distribution correspondingto the Black-Scholes model is superimposed for reference. The left column reports the 1-monthhorizon, the center column the 3-month horizon distribution and the right column shows the 1-year distribution. The distributions are constructed by simulating daily returns from each modelsetting the initial spot variance equal to the unconditional variance. Kernel density estimatesare then constructed from the standardized simulated returns.It can clearly be seen that deviations from normality are critical, and that the estimated

parameters for the four models imply different deviations from normality. It is interesting tonote that the leverage effects present in each model generate a substantial amount of skewnessin the risk-neutral return distributions, even at the 1-year horizon. This finding is consistentwith the nonparametric evidence in Ait-Sahalia and Lo (1998) that skewness persists at longhorizons, contradicting many financial economists’ intuition that deviations from normality tendto disappear at longer horizons.

4 Conclusions and Directions for Future Work

This paper provides an empirical comparison of four option valuation models. The first ofthese models is the benchmark affine stochastic volatility model in the continuous-time optionvaluation literature (AF-SV), due to Heston (1993). The second model is a discrete-time GARCHaffine option valuation model that allows for a closed form solution (AF-GARCH). The thirdmodel is a non-affine discrete-time model (NA-GARCH), and the fourth model is a non-affinecontinuous-time stochastic volatility model (NA-SV). We find that the NA-GARCH model veryslightly outperforms the NA-SV model, in-sample as well as out-of-sample. The improvement inperformance of the AF-GARCH model over the AF-SV model is more substantial, approximately10% in-sample and 6% out-of-sample. The NA-GARCH model outperforms the AF-GARCH

17Notice also that in the affine GARCH model the variance of variance will be constant when c = 0 whereasthis is not the case in the non-affine models nor in the AF-SV model.

18

model by approximately 7% in-sample and 9% out-of-sample, and the NA-SV model outperformsthe AF-SV model by approximately 15% in-sample and 13% out-of-sample .These empirical results allow us to draw two conclusions. First, regarding the relationship

between continuous-time and discrete-time models, the difference in fit is not very large, and thismay indicate that these models are largely similar. However, the differences in fit are differentfor the affine versus the non-affine case. Also, the differences in fit depend on the option ma-turity. It would therefore be overly simplistic to conclude that there are no differences betweenthe empirical performance of continuous-time and discrete-time models. From a theoretical per-spective, limiting results similar to those in Nelson (1990) suggest that the continuous-time anddiscrete-time models in this paper may yield a similar empirical performance, but Corradi (2000)has demonstrated that these limiting results have to be interpreted cautiously. To provide someintuition for the problems with the interpretation of the mathematical results in Nelson (1990),note that the Heston (1993) model is effectively a model with two stochastic shocks, while theHeston and Nandi (2000) model, like other GARCH models, only contains one stochastic shock.We therefore conclude that differences in performance are not large, but that the relationshipbetween discrete-time models and continuous-time models merits further empirical investigation.The second conclusion concerns the distinction between affine and non-affine models. The

focus of the option valuation literature on affine models is well motivated, because the resultingclosed-form solutions are extremely convenient. However, our results suggest that this analyticalconvenience comes at a price. At the very least we believe therefore that non-affine models needto be studied more extensively. In fact, we surmise that the distinction between non-affine andaffine volatility models may be more important than the distinction between discrete-time andcontinuous-time volatility models.At the methodological level, the paper presents a new method to estimate continuous-time

option valuation models that can be used in a variety of applications. We plan to study theperformance of this method in more detail in future work.

19

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Figure 1: Average Implied Volatility in S&P500 Option Data and the VIX

1990 1991 1992 1993 1994 1995 19960

0.1

0.2

0.3

0.4Average Implied Volatility

1990 1991 1992 1993 1994 1995 19960

0.1

0.2

0.3

0.4VIX Volatility Index

Notes to figure: The top panel plots the average implied Black-Scholes volatility each Wednesdayduring 1990-1995. The average is taken across maturities and strike prices using the call optionsin our data set. For comparison, the bottom panel shows the one-month, at-the-money VIXvolatility index retrieved from the CBOE website.

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Figure 2: Spot Volatility Paths from Each Model. 1990-1995.Using 1990 and 1995 Estimates

0

0.10.20.30.4

AF-S

V

0

0.10.20.30.4

1990 Estimates

AF-

GAR

CH

0

0.10.20.30.4

NA-

GA

RC

H

1990 1991 1992 1993 1994 1995 19960

0.10.20.3

0.4

NA-

SV

0

0.10.20.30.4

AF-S

V

0

0.10.20.30.4

1995 Estimates

AF-

GAR

CH

0

0.10.20.30.4

NA-

GA

RC

H

1990 1991 1992 1993 1994 1995 19960

0.10.20.3

0.4

NA-

SV

Notes to figure: For each model we plot the daily spot volatility path (annualized) during 1990-1995. The left column uses the 1990 estimates from Table 3 and the right column uses the1995 estimates. The top row of panels show the volatility paths from the Heston-Nandi AffineGARCH model, the second row shows the Heston affine stochastic volatility model, the third rowshow the Duan non-affine GARCH model and the bottom row shows the non-affine stochasticvolatility model.

24

Figure 3: Weekly In Sample Root Mean Squared Error (RMSE) for each Model. 1990-1995.Average Weekly Implied Volatility (IV) is Shown for Reference

1990 1991 1992 1993 1994 1995 19960.1

0.2

0.3

IV

1990 1991 1992 1993 1994 1995 19960

1

2

AF-

GA

RC

H

1990 1991 1992 1993 1994 1995 19960

1

2

AF-S

V

1990 1991 1992 1993 1994 1995 19960

1

2

NA

-GA

RC

H

1990 1991 1992 1993 1994 1995 19960

1

2

NA-

SV

Notes to figure: The four top panels show the weekly root mean squared error (RMSE) for theAF-GARCH, AF-SV, NA-GARCH and NA-SV models respectively. The bottom panel showsthe weekly average implied Black-Scholes volatility from Figure 1 for reference.

25

Figure 4: Weekly In Sample Bias for each Model. 1990-1995.Average Weekly Implied Volatility (IV) is Shown for Reference

1990 1991 1992 1993 1994 1995 19960.1

0.2

0.3

IV

1990 1991 1992 1993 1994 1995 1996-2

0

2

AF-

GA

RC

H

1990 1991 1992 1993 1994 1995 1996-2

0

2

AF-S

V

1990 1991 1992 1993 1994 1995 1996-2

0

2

NA

-GA

RC

H

1990 1991 1992 1993 1994 1995 1996-2

0

2

NA-

SV

Notes to figure: The top four panels show the weekly bias for the AF-GARCH, AF-SV, NA-GARCH and NA-SV models respectively. The bottom panel shows the weekly average impliedBlack-Scholes volatility from Figure 1 for reference.

26

Figure 5: Conditional Leverage Paths from Each Model. 1990-1995.Using 1990 and 1995 Estimates

-1.5

-1

-0.5

0x 10-4

AF-

GAR

CH

90 Estimates

-1.5

-1

-0.5

0x 10-3 95 Estimates

AF-

GAR

CH

-1.5

-1

-0.5

0x 10-4

AF-S

V

-1.5

-1

-0.5

0x 10-3

AF-S

V

-1.5

-1

-0.5

0x 10-4

NA-

GA

RC

H

-1.5

-1

-0.5

0x 10-3

NA-

GA

RC

H

1990 1991 1992 1993 1994 1995 1996-1.5

-1

-0.5

0x 10-4

NA-

SV

1990 1991 1992 1993 1994 1995 1996-1.5

-1

-0.5

0x 10-3

NA-

SV

Notes to figure: For each model we plot the daily conditional leverage path defined as the con-ditional covariance between shocks to returns and shocks to variance. The paths are annualizedby multiplying by 252 and plotted during 1990-1995. The left column uses the 1990 estimatesfrom Table 3 and the right column uses the 1995 estimates. Note that the scales are different inthe two columns. The top row of panels show the volatility paths from the Heston-Nandi AffineGARCH model, the second row shows the Heston affine stochastic volatility model, the third rowshow the Duan non-affine GARCH model and the bottom row shows the non-affine stochasticvolatility model

27

Figure 6: Conditional Volatility of Variance Paths from Each Model. 1990-1995.Using 1990 and 1995 Estimates

0

0.02

0.04

0.0695 Estimates

AF-

GAR

CH

0

2

4

6x 10-3 90 Estimates

AF-

GAR

CH

0

0.02

0.04

0.06

AF-S

V

0

2

4

6x 10-3

AF-S

V

0

2

4

6x 10-3

NA-

GA

RC

H

0

0.02

0.04

0.06

NA-

GA

RC

H

1990 1991 1992 1993 1994 1995 19960

2

4

6x 10-3

NA-

SV

1990 1991 1992 1993 1994 1995 19960

0.02

0.04

0.06

NA-

SV

Notes to figure: For each model we plot the daily conditional volatility of variance path defined asthe square root of the conditional variance of the conditional variance of returns. The paths areannualized by multiplying by 252 and plotted during 1990-1995. The left column uses the 1990estimates from Table 3 and the right column uses the 1995 estimates. Note that the scales aredifferent in the two columns. The top row of panels show the volatility paths from the Heston-Nandi affine GARCH model (AF-GARCH), the second row shows the Heston affine stochasticvolatility model (AF-SV), the third row show the Duan non-affine GARCHmodel (NA-GARCH)and the bottom row shows the non-affine stochastic volatility model (NA-SV).

28

Figure 7: Model Implied State Price Densities. 1990 Estimates1-Month, 3-Month and 1-Year Horizon

-2 0 20

0.2

0.4

1-Month Horizon

AF-G

AR

CH

-2 0 20

0.2

0.4

AF-S

V

-2 0 20

0.2

0.4

NA-

GAR

CH

-2 0 20

0.2

0.4

NA

-SV

-2 0 20

0.2

0.4

3-Month Horizon

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

1-Year Horizon

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

Notes to figure: Each row of panels reports the risk neutral distribution of the index returnaccording to the AF-GARCH, AF-SV, NA-GARCH and NA-SVmodels respectively. The normaldistribution corresponding to the Black-Scholes model is superimposed for reference. The leftcolumn reports the 1-month horizon, the center column the 3-month horizon distribution and theright column shows the 1-year distribution. The distributions are constructed by simulating dailyreturns from each model setting the initial spot variance equal to the unconditional variance.Kernel density estimates are then constructed from the standardized simulated returns. 1990estimates from Table 3 converted to the risk neutral measure are used to simulate the risk neutralreturns.

29

Figure 8: Model Implied State Price Densities. 1995 Estimates1-Month, 3-Month and 1-Year Horizon

-2 0 20

0.2

0.4

1-Month Horizon

AF-G

AR

CH

-2 0 20

0.2

0.4

AF-S

V

-2 0 20

0.2

0.4

NA-

GAR

CH

-2 0 20

0.2

0.4

NA

-SV

-2 0 20

0.2

0.4

3-Month Horizon

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

1-Year Horizon

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

-2 0 20

0.2

0.4

Notes to figure: Each row of panels reports the risk neutral distribution of the index returnaccording to the AF-GARCH, AF-SV, NA-GARCH and NA-SVmodels respectively. The normaldistribution corresponding to the Black-Scholes model is superimposed for reference. The leftcolumn reports the 1-month horizon, the center column the 3-month horizon distribution and theright column shows the 1-year distribution. The distributions are constructed by simulating dailyreturns from each model setting the initial spot variance equal to the unconditional variance.Kernel density estimates are then constructed from the standardized simulated returns. 1995estimates from Table 3 converted to the risk neutral measure are used to simulate the risk neutralreturns.

30

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 101 1,884 1,931 1,765 5,681

0.975<S/X<1 283 1,272 706 477 2,7381<S/X<1.025 300 1,212 726 523 2,761

1.025<S/X<1.05 261 1,167 654 406 2,4881.05<S/X<1.075 245 1,039 582 390 2,256

S/X>1.075 549 2,345 1,679 1,242 5,815All 1,739 8,919 6,278 4,803 21,739

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.88 2.30 6.25 11.92 6.61

0.975<S/X<1 2.29 6.83 15.19 27.50 12.121<S/X<1.025 8.35 13.60 22.48 34.34 19.29

1.025<S/X<1.05 17.57 22.00 30.11 42.03 26.941.05<S/X<1.075 27.11 30.84 38.14 48.83 35.43

S/X>1.075 50.67 52.78 58.98 68.30 57.69All 24.32 23.66 28.68 36.03 27.89

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.1625 0.1266 0.1348 0.1393 0.1341

0.975<S/X<1 0.1308 0.1296 0.1448 0.1562 0.13871<S/X<1.025 0.1527 0.1459 0.1558 0.1607 0.1522

1.025<S/X<1.05 0.1914 0.1647 0.1665 0.1657 0.16831.05<S/X<1.075 0.2429 0.1828 0.1775 0.1739 0.1875

S/X>1.075 0.3878 0.2353 0.1961 0.1868 0.2347All 0.2639 0.1751 0.1639 0.1618 0.1780

Table 1: In-Sample S&P500 Index Call Option Data 1990-1995.

Panel B. Average Call Price

Panel C. Average Implied Volatility from Call Options

Panel A. Number of Call Option Contracts

Notes: The sample contains European call options on the S&P500 index. We use quotes within 30 minutes from closing on every Wednesday during the January 1, 1990 to December 31, 1995 period. We apply the moneyness and maturity filters used by Bakshi, Cao and Chen (1997). Implied volatilities are computedusing the Black-Scholes formula.

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 91 1,797 1,870 1,705 5,463

0.975<S/X<1 274 1,231 723 452 2,6801<S/X<1.025 279 1,183 679 487 2,628

1.025<S/X<1.05 264 1,093 614 350 2,3211.05<S/X<1.075 197 941 519 258 1,915

S/X>1.075 357 1,407 1,003 707 3,474All 1,462 7,652 5,408 3,959 18,481

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.86 2.28 6.18 11.98 6.62

0.975<S/X<1 2.21 6.65 15.09 27.46 11.981<S/X<1.025 8.17 13.42 22.45 33.85 18.98

1.025<S/X<1.05 17.07 21.80 29.82 41.64 26.381.05<S/X<1.075 26.67 30.34 37.53 47.75 34.26

S/X>1.075 47.60 49.90 56.25 69.09 55.40All 20.33 19.70 24.39 31.59 23.67

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.1636 0.1260 0.1353 0.1400 0.1344

0.975<S/X<1 0.1293 0.1282 0.1444 0.1566 0.13791<S/X<1.025 0.1531 0.1442 0.1554 0.1608 0.1513

1.025<S/X<1.05 0.1943 0.1609 0.1651 0.1649 0.16671.05<S/X<1.075 0.2477 0.1815 0.1769 0.1763 0.1875

S/X>1.075 0.3903 0.2396 0.2116 0.2074 0.2461All 0.2481 0.1670 0.1629 0.1629 0.1728

Table 2: Out-of-Sample S&P500 Index Call Option Data 1990-1995.

Panel A. Number of Call Option Contracts

Panel B. Average Call Price

Panel C. Average Implied Volatility from Call Options

Notes: The sample contains European call options on the S&P500 index. We use quotes within 30 minutes from closing on every Thursday during the January 1, 1990 to December 31, 1995 period. We apply the moneyness and maturity filters used by Bakshi, Cao and Chen (1997). Implied volatilities are computedusing the Black-Scholes formula.

ω a b c λ µ κ θ σ ρ λ1990 1.50E-07 1.98E-07 0.2315 1959.9 3.533 3.53E-04 1.75E-03 1.10E-04 6.09E-04 -0.9990 1.53E-031991 5.86E-08 2.02E-07 0.5090 1553.7 1.382 4.80E-04 3.12E-03 9.02E-05 7.26E-04 -0.9990 5.15E-041992 1.02E-06 1.98E-06 0.2098 599.8 15.280 4.27E-04 4.49E-03 1.27E-04 8.12E-04 -0.8819 1.86E-031993 3.64E-07 1.65E-06 0.2575 652.1 1.159 3.98E-04 9.39E-03 9.59E-05 9.22E-04 -0.8175 5.54E-031994 4.72E-07 1.43E-06 0.1332 761.7 0.964 4.04E-04 8.00E-03 8.86E-05 8.31E-04 -0.8377 3.88E-031995 5.76E-07 9.43E-07 0.3264 822.0 4.655 4.20E-04 7.55E-03 9.01E-05 8.15E-04 -0.8596 4.02E-03

ω b a c λ ω b a c λ1990 4.66E-07 0.7793 0.0081 4.9179 0.1727 4.73E-07 0.7806 0.0081 4.8909 0.17651991 2.08E-07 0.7781 0.0058 6.1037 0.0017 2.10E-07 0.7756 0.0058 6.1221 0.00171992 1.14E-06 0.8261 0.0730 1.0466 0.0723 1.19E-06 0.8237 0.0740 1.0435 0.06951993 1.69E-06 0.5824 0.0759 2.0088 0.0467 1.69E-06 0.5847 0.0759 1.9928 0.04751994 8.67E-07 0.7763 0.0518 1.6318 0.1395 8.91E-07 0.7759 0.0519 1.6314 0.13731995 1.66E-06 0.3877 0.0452 3.3814 0.0896 1.67E-06 0.3864 0.0455 3.3629 0.0918

Notes: For each model, we perform six estimation exercises using Nonlinear Least Squares on the valuation errors. We use Wednesday option prices in each of the years 1990, 1991, 1992, 1993, 1994 and 1995 to conduct separate estimation exercises.

Table 3: Parameter Estimates

AF-GARCH AF-SV

NA-GARCH NA-SV

AF-GARCH AF-SV NA-GARCH NA-SV AF-GARCH AF-SV NA-GARCH NA-SV1990 0.9938 0.9983 0.9834 0.9829 1990 0.9969 0.9967 0.9974 0.99721991 0.9972 0.9969 0.9998 0.9993 1991 0.9984 0.9964 0.9999 0.99941992 0.9210 0.9955 0.9792 0.9783 1992 0.9590 0.9936 0.9906 0.98941993 0.9601 0.9906 0.9644 0.9621 1993 0.9637 0.9851 0.9788 0.97671994 0.9641 0.9920 0.9659 0.9660 1994 0.9673 0.9881 0.9905 0.99021995 0.9637 0.9925 0.9495 0.9468 1995 0.9717 0.9884 0.9773 0.9753

Average 0.9666 0.9943 0.9737 0.9726 Average 0.9762 0.9914 0.9891 0.9881

AF-GARCH AF-SV NA-GARCH NA-SV AF-GARCH AF-SV NA-GARCH NA-SV1990 0.1190 0.1664 0.0842 0.0836 1990 0.1694 0.1216 0.2143 0.20681991 0.1524 0.1508 0.4886 0.2800 1991 0.1998 0.1397 0.7247 0.30951992 0.0978 0.1787 0.1173 0.1175 1992 0.1356 0.1502 0.1748 0.16821993 0.1128 0.1554 0.1093 0.1060 1993 0.1183 0.1233 0.1416 0.13511994 0.1155 0.1494 0.0801 0.0813 1994 0.1211 0.1226 0.1516 0.15161995 0.1027 0.1506 0.0912 0.0890 1995 0.1164 0.1217 0.1359 0.1306

Average 0.1167 0.1586 0.1618 0.1262 Average 0.1434 0.1298 0.2572 0.1836

Notes: Using the parameter estimates reported in Table 3, we compute unconditional volatility and persistence for each of the four models using the formulas given in the paper. We compute physical as well as risk-neutral estimates for each of the six estimation exercises

Table 4: Persistence and Annual Volatility

Panel A. Physical Persistence

Panel C. Unconditional Physical Volatility (Annualized)

Panel B. Risk Neutral Persistence

Panel C. Unconditional Risk Neutral Volatility (Annualized)

AF-GARCH AF-SV NA-GARCH NA-SV1990 0.9711 1.0495 0.9667 0.96021991 0.8455 0.9431 0.7956 0.86291992 0.7758 0.7824 0.7602 0.73631993 0.8118 0.9532 0.7015 0.72001994 0.9591 1.1179 0.8761 0.92671995 0.8728 0.9218 0.7654 0.7549

Overall 0.8768 0.9696 0.8108 0.8272Normalized 0.9043 1.0000 0.8361 0.8531

AF-GARCH AF-SV NA-GARCH NA-SV1990 0.9784 1.0665 0.9730 0.95391991 0.9005 0.9805 0.8370 0.92441992 0.7806 0.7738 0.7370 0.71791993 0.6925 0.8187 0.5728 0.60261994 0.9577 1.0516 0.7686 0.85701995 0.9449 0.9340 0.8214 0.8350

Overall 0.8837 0.9410 0.7966 0.8247Normalized 0.9391 1.0000 0.8465 0.8764

models, for each of the six in-sample and out-of-sample periods. RMSE refers to the square root of the mean-squared valuation errors.

Table 5: RMSE In- and Out-of-Sample

Panel A. RMSE In-Sample

Panel B. RMSE Out-of-Sample

Notes: Using the parameter estimates from Table 3, we compute RMSEs for all four

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4760 0.7316 0.7652 1.0177 0.8380

0.975<S/X<1 0.8628 0.9601 0.8416 1.0448 0.93701<S/X<1.025 0.7844 0.9141 0.8056 0.9276 0.8760

1.025<S/X<1.05 0.5865 0.8202 0.7347 0.9607 0.80261.05<S/X<1.075 0.7884 0.7565 0.7618 1.1451 0.8411

S/X>1.075 0.8022 0.8393 0.8197 1.2265 0.9274All 0.7634 0.8354 0.7903 1.0753 0.8768

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.5046 0.9100 0.9550 1.0859 0.9775

0.975<S/X<1 0.6890 0.9971 0.9086 1.0133 0.95011<S/X<1.025 0.6151 0.8404 0.8081 0.9717 0.8377

1.025<S/X<1.05 0.5706 0.7872 0.8242 1.0761 0.83211.05<S/X<1.075 0.7721 0.8316 0.9645 1.2283 0.9402

S/X>1.075 0.7728 0.9687 1.0596 1.4105 1.0881All 0.6919 0.9061 0.9519 1.1717 0.9696

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4581 0.6600 0.7317 0.9872 0.7956

0.975<S/X<1 0.6150 0.7800 0.7657 1.0183 0.80841<S/X<1.025 0.6508 0.7603 0.7831 0.9111 0.7863

1.025<S/X<1.05 0.6031 0.7306 0.7490 0.9153 0.75681.05<S/X<1.075 0.6001 0.7303 0.8082 1.0359 0.7992

S/X>1.075 0.7439 0.7693 0.8070 1.1160 0.8631All 0.6534 0.7382 0.7712 1.0156 0.8108

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4656 0.6955 0.7730 1.0168 0.8298

0.975<S/X<1 0.6116 0.8007 0.7799 1.0438 0.82651<S/X<1.025 0.6294 0.7617 0.7754 0.9358 0.7885

1.025<S/X<1.05 0.5894 0.7241 0.7521 0.9674 0.76401.05<S/X<1.075 0.5978 0.7256 0.8199 1.0893 0.8123

S/X>1.075 0.7438 0.7612 0.8125 1.1579 0.8734All 0.6473 0.7448 0.7871 1.0515 0.8272

Panel D. NA-SV

Table 6: RMSE by Moneyness and Maturity. 1990-1995. In Sample

Panel A. AF-GARCH

Panel B. AF-SV

Panel C. NA-GARCH

Notes: We use the NLS estimates from Table 3 to compute the root mean squared option valuationerror (RMSE) for various moneyness and maturity bins for the four models. The option prices used in the table are for the 1990-1995 in-sample period, which consists of Wednesday option prices.

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4946 0.7190 0.8009 1.0651 0.8647

0.975<S/X<1 0.8428 1.0015 0.9155 1.0993 0.98151<S/X<1.025 0.8022 0.9587 0.8885 0.9792 0.9294

1.025<S/X<1.05 0.5880 0.8434 0.7947 0.9241 0.81871.05<S/X<1.075 0.7508 0.7510 0.7959 1.0395 0.8077

S/X>1.075 0.5763 0.8319 0.9121 1.0392 0.8800All 0.7005 0.8504 0.8484 1.0408 0.8837

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.5075 0.9032 0.9937 1.1412 1.0085

0.975<S/X<1 0.6774 1.0611 0.9853 1.0660 1.00881<S/X<1.025 0.5925 0.8619 0.8548 1.0116 0.8661

1.025<S/X<1.05 0.5396 0.7496 0.8193 0.9510 0.78211.05<S/X<1.075 0.7438 0.7938 0.9253 1.0692 0.8675

S/X>1.075 0.5696 0.9103 1.0516 1.1028 0.9673All 0.6127 0.8931 0.9625 1.0900 0.9410

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4398 0.6308 0.7394 1.0048 0.7977

0.975<S/X<1 0.5818 0.7935 0.8196 1.0346 0.82801<S/X<1.025 0.6390 0.7495 0.8198 0.9083 0.7891

1.025<S/X<1.05 0.5843 0.7086 0.7852 0.8298 0.73631.05<S/X<1.075 0.5547 0.7269 0.8008 1.0061 0.7755

S/X>1.075 0.5509 0.7774 0.8953 0.9248 0.8255All 0.5750 0.7276 0.8022 0.9685 0.7966

DTM<20 20<DTM<80 80<DTM<180 DTM>180 AllS/X<0.975 0.4583 0.6741 0.7989 1.0436 0.8434

0.975<S/X<1 0.5974 0.8354 0.8622 1.0586 0.86411<S/X<1.025 0.6438 0.7776 0.8545 0.9441 0.8185

1.025<S/X<1.05 0.5793 0.7184 0.8030 0.8797 0.75401.05<S/X<1.075 0.5534 0.7260 0.8211 1.0160 0.7825

S/X>1.075 0.5503 0.7728 0.9181 0.9487 0.8364All 0.5787 0.7489 0.8402 1.0015 0.8247

Notes: We use the NLS estimates from Table 3 to compute root mean squared option valuationerrors (RMSEs) for various moneyness and maturity bins for the four models. The option prices used are for the 1990-1995 out-of-sample period, which consists of Thursday option prices.

Panel D. NA-SV

Table 7: RMSE by Moneyness and Maturity. 1990-1995. Out of Sample

Panel A. AF-GARCH

Panel B. AF-SV

Panel C. NA-GARCH