Approximating the GJR-GARCH and EGARCH option pricing … · 2015. 2. 7. · 01 PROOF 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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Approximating the GJR-GARCH andEGARCH option pricing models analytically

Jin-Chuan DuanRotman School of Management, University of Toronto, 105 St. George Street, Ontario,Canada M5S 3E6

Geneviève GauthierHEC Montréal, 3000 Côte-Sainte-Catherine, Montréal, Canada H3T 2A7

Jean-Guy SimonatoHEC Montréal, 3000 Côte-Sainte-Catherine, Montréal, Canada H3T 2A7

Caroline SassevilleKellogg School of Management, 2001 Sheridan Road, Evanston, IL 60208, USA

In Duan, Gauthier and Simonato (1999), an analytical approximation to priceEuropean options in the generalized autoregressive conditional heteroskedastic(GARCH) framework was developed. The formula is, however, restricted to thenon-linear asymmetric GARCH model. This paper extends the approximationto two other popular GARCH specifications, GJR-GARCH and EGARCH. Weprovide the corresponding formulas and study their numerical performance.

1 Introduction

During the past decade, researchers have begun to study generalized autoregres-sive conditional heteroskedastic (GARCH) models for option pricing because oftheir superior performances in describing asset returns. Duan (1995) developed atheory with which options can be priced when the evolution of the asset returnfollows a GARCH process. Empirically, Heynen et al (1994), Duan (1996),Heston and Nandi (2000), Hsieh and Ritchken (2000), Hardle and Hafner (2000),Duan and Zhang (2001), Lehar et al (2002), Lehnert (2003), Stentoft (2005), andChristoffersen and Jacobs (2004) have shown that the GARCH model can be usedto capture the pricing behavior of exchange-traded options. Analytically, pricingEuropean options requires the knowledge of the risk-neutral distribution of thecumulative return with respect to a given model. However, the analytical form ofthe distribution for the time-aggregated return (ie, cumulative return) is unknown

Duan, Gauthier and Simonato acknowledge the financial support from the Natural Sciencesand Engineering Research Council of Canada (NSERC), le Fonds Québécois de la Recherchesur la Société et la Culture (FQRSC) and from the Social Sciences and Humanities ResearchCouncil of Canada (SSHRC). Duan also acknowledges support received as the Manulife Chairin Financial Services.

1

Revised Proof Ref: Duan9(3)/32352e April 24, 2006

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2 J.-C. Duan et al

for all GARCH specifications, and thus computing option prices must rely onsome time-consuming numerical procedures.

In recent years, researchers have tried to speed up the valuation of Europeanoptions under GARCH by developing analytical solutions or approximations forspecific forms of the GARCH model. Heston and Nandi (2000) developed ananalytical formula to price European options when the dynamic of the conditionalvariance is given by a specific GARCH process.1 In contrast, Duan, Gauthierand Simonato (1999) (DGS hereafter) developed an analytical approximation toprice European options under GARCH. Their approach utilizes the idea of Jarrowand Rudd (1982) to find an approximate option price under general stochasticprocesses.

Specifically, DGS (1999) use an Edgeworth expansion to obtain an approxi-mate pricing formula for the linear GARCH specification of Bollerslev (1986)or the non-linear asymmetric GARCH specification of Engle and Ng (1993)(NGARCH). The resulting approximation formula is similar to a Black–Scholesformula adjusted for skewness and kurtosis of the cumulative return underGARCH. The DGS (1999) approximation performs well numerically, especiallyfor shorter-term options. In contrast to the approach of Heston and Nandi (2000),it is not limited to a specific form of GARCH. Indeed, comparable formulas canbe obtained for other GARCH specifications, even though the modifications mayentail significant algebraic work.

In this paper, we derive the various components needed for applying the DGS(1999) approach to the GARCH specification of Glosten, Jagannathan and Runkle(1993) (GJR-GARCH) and to the exponential GARCH specification of Nelson(1991) (EGARCH). The choice of these two models is justified by the fact thatthese two specifications and the NGARCH model all exhibit the leverage effect,an important feature of stock return data. In contrast to the NGARCH, these twospecifications incorporate the leverage effect not by shifting the minimum of thenews impact curve away from zero but by altering the shape. These differentspecifications have been shown in some empirical studies to better describe assetreturns and may thus be useful in explaining option prices. Our development ofthe analytical approximation formulas corresponding to the GJR-GARCH andEGARCH specifications may thus facilitate future empirical option research aswell as provide a practical tool for potential on-line applications of these models.

The remainder of this paper is organized as follows. In Section 2, we show howthe analytical approximation can be modified for the GJR-GARCH and EGARCHoption pricing models. We then examine, in Section 3, the numerical performanceof these approximation formulas. Finally, Section 4 concludes.

1More appropriately stated, Heston and Nandi’s (2000) approach is a numerical technique. Theyfirst derived a difference equation system for the characteristic function. They then solved thedifference equation system numerically. Finally, they relied on a numerical Fourier inversion toobtain the European option price.

Journal of Computational Finance


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Approximating the GJR-GARCH and EGARCH option pricing models analytically 3

2 The analytical approximation

2.1 General valuation framework

We start by assuming that the asset return dynamic, under the physical measureP , is

ln

(St+1

St

)= r + λ

√ht+1 − 1

2ht+1 + √

ht+1εt+1, for t = 0, 1, 2, . . . (1a)

where {εt : t ∈ {1, 2, . . . }} is a sequence of independent standard normal randomvariables with respect to measure P and

ht+1 = β0 + ht [β1 + β2ε2t + β3 max(0, −εt )

2] (1b)

for the GJR-GARCH process or

ln(ht+1) = β0 + β1 ln(ht ) + β2(|εt | + γ εt ) (1c)

for the EGARCH process.Note that r is the one-period continuously compounded risk-free rate, λ

is a constant unit risk premium, and ht+1 is the conditional variance of theasset return. While the GJR-GARCH specification imposes the typical parameterrestrictions β0 > 0, β1 ≥ 0, β2 ≥ 0, β3 ≥ 0 to ensure that the conditional variancestays positive, the EGARCH model does not require such restrictions becausethe conditional variance is guaranteed positive by the exponential operation. Theparameter β3 (or γ ) determines how the GJR-GARCH (EGARCH) incorporatesthe leverage effect. It is important to note that the conditional variance is apredictable process under both specifications because ht is expressed only in termsof variables known at time t − 1.

Invoking Duan’s (1995) locally risk-neutral pricing result with respect to thepricing measure Q, the asset return dynamic can be written as

ln

(St+1

St

)= r − 1

2ht+1 + √

ht+1εt+1, for t = 0, 1, 2, . . . (2a)

where {εt : t ∈ {1, 2, . . . }} is a sequence of independent standard normal randomvariables with respect to measure Q and

ht+1 = β0 + ht [β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2] (2b)

for the GJR-GARCH process or

ln(ht+1) = β0 + β1 ln(ht ) + β2[|εt − λ| + γ (εt − λ)] (2c)

for the EGARCH process.In the above, the random innovation term εt = εt + λ becoming a standard nor-

mal random variable under measure Q is in fact the essence of the GARCH optionpricing theory of Duan (1995). For the GJR-GARCH model, there is variance

Volume 9/Number 3, Spring 2006


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4 J.-C. Duan et al

weak stationarity2 if β1 + (β2 + β3N(λ))(1 + λ2) + β3λn(λ) < 1, where N(•)

and n(•) stand for the standard normal distribution and density functions. In thecase of the EGARCH, the condition β1 < 1 ensures that the conditional variancedoes not explode with time. Note that the stationarity conditions are fundamentallydifferent under the two models. For the GJR-GARCH, the stationarity conditionunder measure Q differs from that under P , which is known to be β1 + β2 +β3/2 < 1, and the difference arises from the risk premium parameter λ. Forthe EGARCH, the stationarity condition remains unchanged when moving frommeasure P to Q. This dichotomy is driven by the different nature of the volatilityshocks in the two models. According to Duan (1997), the volatility shock enteringinto the time series dynamic in the case of the EGARCH is additive whereas thevolatility shock for the GJR-GARCH is multiplicative.

2.2 Analytical approximation

Following the approach of DGS (1999), which is in turn rooted in Jarrow and Rudd(1982), we use the Edgeworth expansion to derive the two approximate pricingformulas. As stated in DGS (1999), the price of a European call with strike priceK and a maturity T can be approximated with

Capp = C + κ3A3 + (κ4 − 3)A4 (3a)

where

C = S0 eδσρT N(d) − K e−rT N(d − σρT) (3b)

A3 = 1

3!S0 eδσρT σρT[(2σρT

− d)n(d) + σ 2ρT

N(d)] (3c)

A4 = 1

4!S0 eδσρT σρT[(d2 − 1 − 3σρT

(d − σρT))n(d) + σ 3

ρTN(d)] (3d)

d = d + δ (3e)

d = ln(S0/K) + (rT + 12σ 2

ρT)

σρT

(3f)

δ = µρT− rT + σ 2

ρT/2

σρT

(3g)

where ρT = ln(ST /S0) is the continuously compounded cumulative asset return,µρT

and σρTare the mean and standard deviation of ρT under the locally risk-

neutralized measure Q, and κ3 and κ4 are the skewness and kurtosis coefficients ofthe standardized cumulative asset return, under measure Q. Again, the functionsn(•) and N(•) represent the standard normal density and distribution functions.If we adopt the distribution assumption of the Black–Scholes model, the mean of

2That is, EQ[ht ] = h∗ for any t , where h∗ is the stationary variance.



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the cumulative asset return is rT − σ 2ρT

/2, which implies that δ = 0. Moreover,κ3 = 0 and κ4 = 3. Hence, Equation (3a) reduces to the Black–Scholes formula.

The analytical approximation formula requires expressions for the first fourmoments of the cumulative return for an arbitrary maturity T ; that is,

EQ0

[(ln

ST

S0

)k]for k = 1, 2, 3, 4 (4)

Because ST is a function of all conditional variances from period 1 to T ,expressions for these moments are naturally specific to a given GARCH model.

2.3 Moments of the cumulative asset return

DGS (1999) provided some general moment formulas in Appendix C. Theseformulas, however, require inputs specific to a GARCH specification. We followthe approach suggested in Section 6 of DGS (1999) to compute the first fourmoments of the cumulative asset return under the GJR-GARCH and EGARCHspecifications.

For the GJR-GARCH, we need to compute µ1, µ2, µ3,ν1, ν2, ν3, ζ1, ζ2, andξ1 defined as

µk = EQ0 {[β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2]k} (5a)

νk = EQ0 {εt [β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2]k} (5b)

ζk = EQ0 {ε2

t [β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2]k} (5c)

ξk = EQ0 {ε3

t [β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2]k} (5d)

Appendix A provides the specific formulas for these quantities.3 Assuming thatµ1 �= 1, µ2 �= 1 and µ3 �= 1, we can plug these quantities into the expressionsgiven in Appendix C of DGS (1999) to evaluate the first four moments. We notethat the terms involving non-integer moments of ht are approximated by a Taylorseries expansion around E

Q0 [ht ], which means that the expressions obtained for

the moments are sometimes approximated. Furthermore, we have dropped theterms SQ1, SQ3, SQ4 (except for terms 8 and 12 of SQ4) and SQ5 (except forterms 2, 3, 6, 7 and 8 of SQ5), as defined in Appendix C of DGS (1999) becausethese omitted terms have negligible effects on the quality of approximation. Theiromission reduces the computation time significantly since they involve quadruplesums.

For the EGARCH, we again derive analytical expressions for the terms inAppendix C of DGS (1999). The specific results are provided in Appendix B.Although we are able to obtain an exact expression for the expected value of ha

t ,where a is any real number, we approximate E[ha

t ] by a Taylor series expansion

3The derivation of the results given in the appendix is available from the authors upon request.



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6 J.-C. Duan et al

around E[ht ]a . This helps in reducing the computation time because the exactformula for E[ha

t ] involves lengthy recursive operations. The specific detailsfor this Taylor series expansion are given in Appendix D. To further improvethe computation speed, we approximate the triple sums in the third and fourthmoments by making the index of the third sum equal to a single value. Morespecifically, we fix the index value of the third sum at its middle value, rounded tothe nearest integer.4 Finally, as in the case of the GJR-GARCH, we have droppedthe terms SQ1, SQ3, SQ4 (except for terms 8 and 12 of SQ4) and SQ5 (except forterms 2, 3, 6, 7 and 8 of SQ5).

The formulas developed above are quite complex but can be used to quicklycompute the required quantities. Some formulas in the appendices were derivedmanually whereas others were obtained using Maple, a symbolic algebra packageincorporated in Scientific Workplace, a typesetting program. For example, the for-mulas in Appendix A.1 have been derived manually, and then used in conjunctionwith Maple to produce the results in Appendix A.2. It should be noted that all ofthese formulas have been verified by Monte Carlo simulations.

Because the first four moments of the cumulative return are sometimes approx-imated, we need to first ascertain the approximation quality before plugging theminto the approximate option pricing formula in Equation (3a). Needless to say, wewill examine the performance of the approximate option pricing formula as well.

3 Numerical performance

In this section, we assess the numerical performance of the analytical approxima-tions developed for the GJR-GARCH and EGARCH option pricing models. Thenumerical analysis is conducted with realistic parameter values obtained from atime series return estimation. We analyze the quality of the analytic moments andoption pricing formulas for both models. Finally, we assess the approximationquality using a test pool of options.

For both GARCH specifications, the parameter values are obtained from amaximum likelihood time series estimation of the Standard & Poor’s (S&P) 500daily index returns in excess of the risk-free rate from 2nd January 1991 to 15thMay 1998. In each panel of the subsequent tables, the first set of numbers alwayscorresponds to setting the initial conditional variance, h1, equal to the stationaryvariance under measure Q, ie, h∗, a quantity given in Appendix C. The second andthird sets, on the other hand, correspond to h1 being fixed at 20% above and belowthe stationary level. Finally, the analytical values are compared to high-precisionMonte Carlo estimates. Whenever necessary, we assume a risk-free interest rateof 5% per annum and an initial asset price of US$50. Finally, the numbers arepresented for four different maturities: 10, 30, 90 and 270 days.

4For example, expressions similar to∑

i

∑j

∑nk=1 xi,j,k are approximated by

n∑

i

∑j xi,j,n where n is the quantity n/2 rounded down to the nearest integer.



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3.1 GJR-GARCH

Tables 1–4 present the results obtained with the historical parameter estimatesβ0 = 9.61 × 10−7, β1 = 0.93, β2 = 0.024, β3 = 0.059 and λ = 0.065. Theseparameter values imply a volatility persistence of 0.9835 (ie, β1 + β2 + β3/2)under measure P , a level consistent with typical empirical findings. With respectto measure Q, the volatility persistence is increased to 0.9868 (ie, β1 + β2 +β3N(λ)(1 + λ2) + β3λn(λ)).

Table 1 compares the analytical values of two non-integer moments(E

Q0 [h1/2

t ] and EQ0 [h3/2

t ]) from a Taylor expansion to their Monte Carlo coun-terparts. The results indicate that the Taylor series expansion is fairly accuratein approximating the non-integer moments, especially for shorter maturities.In contrast, the approximation does a poorer job for longer maturities.

Table 2 examines the mean and variance of the cumulative return. Thesequantities are defined as µT = µρT

/T and σ 2T = σ 2

ρT/T , where µρT

and σ 2ρT

arethe mean and variance of the cumulative asset return under measure Q, and T

is the number of days underlying the cumulative return. The analytical mean isan exact quantity, but the analytical variance is an approximate value becausesome of the terms in the second moment expression have been approximatedvia a Taylor expansion. This, however, does not seem to affect the quality of theapproximation because both quantities are close to their Monte Carlo counterparts,even for maturities as long as 270 days.

Table 3 examines the skewness and kurtosis of the standardized cumulativereturn. Even though the third and fourth moments are not exact quantities, theresults show that the analytical values match closely those obtained throughMonte Carlo simulation. The quality of the approximation, however, seems todeteriorate as the maturity becomes longer. The values for skewness and kurtosisclearly indicate a departure from the usual normality assumption for cumulativeasset returns. For instance, the cumulative return exhibits negative skewness, amanifestation of the leverage effect in the volatility dynamic under measure Q.Owing to the central limit theorem, one would expect a “reversion to normality”to take place for the standardized cumulative return as the maturity increases. Theexpected reversion to normality kicks in rather slowly, however. For maturities aslong as 270 days, the standardized cumulative return is still away from normality.This slow reversion to normality is not surprising owing to the high level ofvolatility persistence.

Table 4(a) compares the prices obtained for European calls using our analyticalapproximation with those obtained by Monte Carlo simulation. For each maturity,we consider three moneyness ratios (1.1, 1.0 and 0.9), defined as strike-to-priceratios. For the 30-day maturity, the pricing error never exceeds 1 US cent. Forlonger maturities, however, the pricing error increases significantly and it cango up to nearly 10 US cents for the 270-day maturity. These differences are,nonetheless, reasonable when we consider them relative to the level of the optionprice.



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8 J.-C. Duan et al

TABL

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TABL

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onte

Car

lo9.

9087

e−00

57.

2872

e−00

51.

0180

e−00

47.

3290

e−00

59.

8710

e−00

57.

4006

e−00

51.

0125

e−00

47.

4626

e−00

5St

d2.

6995

e−00

61.

1363

e−00

71.

5630

e−00

61.

2222

e−00

79.

0680

e−00

71.

3675

e−00

75.

2573

e−00

71.

4001

e−00

7h

1=

h∗ ×

1.20

Ana

lytic

al9.

3761

e−00

58.

6642

e−00

59.

4580

e−00

58.

5385

e−00

59.

6348

e−00

58.

2675

e−00

59.

8635

e−00

57.

9130

e−00

5M

onte

Car

lo9.

9551

e−00

58.

6654

e−00

59.

2544

e−00

58.

5300

e−00

59.

6022

e−00

58.

2782

e−00

59.

8783

e−00

57.

9040

e−00

5St

d2.

9437

e−00

61.

3520

e−00

71.

6862

e−00

61.

4286

e−00

79.

5906

e−00

71.

5314

e−00

75.

4105

e−00

71.

5528

e−00

7h

1=

h∗ ×

0.80

Ana

lytic

al1.

0747

e−00

45.

9132

e−00

51.

0666

e−00

46.

0996

e−00

51.

0489

e−00

46.

5120

e−00

51.

0260

e−00

47.

0733

e−00

5M

onte

Car

lo1.

0361

e−00

45.

9195

e−00

51.

0585

e−00

46.

0773

e−00

51.

0518

e−00

46.

5174

e−00

51.

0285

e−00

47.

0762

e−00

5St

d2.

4330

e−00

69.

2146

e−00

81.

4233

e−00

61.

0061

e−00

78.

5097

e−00

71.

1938

e−00

75.

1194

e−00

71.

3343

e−00

7



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

10 J.-C. Duan et al

TABL

E3

Perf

orm

ance

ofth

ean

alyt

ical

appr

oxim

atio

nfo

rth

esk

ewne

ssan

dku

rtos

isof

the

cum

ulat

ive

retu

rnfo

rth

eG

JR-G

AR

CH

.β

0=

9.61

×10

−7,β

1=

0.93

,β

2=

0.02

4,β

3=

0.05

9an

dλ

=0.

065.

“Ana

lytic

al”

isth

ean

alyt

ical

appr

oxim

atio

n,“M

onte

Car

lo”

isth

eM

onte

Car

loes

timat

eba

sed

on1,

000,

000

sam

ple

path

san

d“S

td”

isth

est

anda

rdde

viat

ion

ofth

eM

onte

Car

loes

timat

e.

T=

10da

ysT

=30

days

T=

90da

ysT

=27

0da

ys

κ3,

Tκ

4,T

κ3,

Tκ

4,T

κ3,

Tκ

4,T

κ3,

Tκ

4,T

h1

=h∗ ×

1.00

Ana

lytic

al−2

.279

5e−0

013.

4262

e+00

0−4

.006

5e−0

013.

7219

e+00

0−6

.001

9e−0

014.

2303

e+00

0−6

.725

1e−0

014.

4028

e+00

0M

onte

Car

lo−2

.284

3e−0

013.

4315

e+00

0−3

.979

7e−0

013.

7811

e+00

0−5

.998

5e−0

014.

4142

e+00

0−6

.357

0e−0

014.

5201

e+00

0St

d4.

8725

e−00

31.

6767

e−00

25.

9683

e−00

32.

9442

e−00

28.

4281

e−00

36.

4149

e−00

28.

6645

e−00

36.

4224

e−00

2h

1=

h∗ ×

1.20

Ana

lytic

al−2

.295

2e−0

013.

4289

e+00

0−4

.070

6e−0

013.

7357

e+00

0−6

.186

6e−0

014.

2872

e+00

0−6

.906

9e−0

014.

4690

e+00

0M

onte

Car

lo−2

.287

3e−0

013.

4344

e+00

0−4

.060

7e−0

013.

8050

e+00

0−6

.105

4e−0

014.

4222

e+00

0−6

.708

5e−0

014.

8597

e+00

0St

d4.

8473

e−00

31.

6324

e−00

25.

9775

e−00

32.

7162

e−00

28.

1568

e−00

35.

6164

e−00

21.

2192

e−00

21.

6339

e−00

1h

1=

h∗ ×

0.80

Ana

lytic

al−2

.257

6e−0

013.

4220

e+00

0−3

.920

7e−0

013.

7020

e+00

0−5

.782

6e−0

014.

1616

e+00

0−6

.545

3e−0

014.

3384

e+00

0M

onte

Car

lo−2

.282

7e−0

013.

4231

e+00

0−3

.867

5e−0

013.

7408

e+00

0−5

.816

5e−0

014.

3551

e+00

0−6

.374

5e−0

014.

5554

e+00

0St

d4.

8051

e−00

31.

5831

e−00

26.

0912

e−00

33.

7740

e−00

28.

1676

e−00

35.

7685

e−00

28.

7717

e−00

36.

0002

e−00

2



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45


TABL

E4

Perf

orm

ance

ofth

ean

alyt

ical

appr

oxim

atio

nfo

rth

eEu

rope

anca

llop

tion

pric

efo

rth

eG

JR-G

AR

CH

.(a)

β0=

9.61

×10

−7,β

1=

0.93

,β

2=

0.02

4,β

3=

0.05

9an

dλ

=0.

065;

(b)β

0=

2.0

×10

−6,β

1=

0.93

,β

2=

0.02

4,β

3=

0.05

9an

dh

1=

h∗ .

(a)

Mat

urity

=10

days

Mat

urity

=30

days

Mat

urity

=90

days

Mat

urity

=27

0da

ys

K/S

01.

101.

000.

901.

101.

000.

901.

101.

000.

901.

101.

000.

90

h1

=h∗ ×

1.00

Ana

lytic

al0.

0002

0.56

375.

0617

0.01

741.

0120

5.20

800.

2381

1.86

465.

7747

1.36

633.

6467

7.30

22M

onte

Car

lo0.

0001

0.56

545.

0619

0.01

441.

0200

5.20

490.

2487

1.89

705.

7384

1.45

623.

7117

7.25

16St

d0.

0000

0.00

030.

0004

0.00

000.

0004

0.00

070.

0003

0.00

080.

0011

0.00

090.

0014

0.00

18Pr

icin

ger

ror

0.00

01−0

.001

7−0

.000

20.

0029

−0.0

080

0.00

30−0

.010

7−0

.032

30.

0363

−0.0

899

−0.0

650

0.05

06h

1=

h∗ ×

1.20

Ana

lytic

al0.

0005

0.61

125.

0619

0.02

661.

0829

5.22

040.

2815

1.94

595.

8158

1.42

123.

7057

7.34

48M

onte

Car

lo0.

0002

0.61

365.

0627

0.02

361.

0921

5.21

580.

2966

1.98

295.

7753

1.51

783.

7756

7.29

20St

d0.

0000

0.00

030.

0005

0.00

010.

0005

0.00

080.

0003

0.00

080.

0012

0.00

100.

0015

0.00

19Pr

icin

ger

ror

0.00

02−0

.002

4−0

.000

80.

0030

−0.0

091

0.00

45−0

.015

1−0

.037

00.

0405

−0.0

966

−0.0

699

0.05

29h

1=

h∗ ×

0.80

Ana

lytic

al0.

0000

0.51

165.

0616

0.01

010.

9350

5.19

770.

1953

1.77

845.

7332

1.30

923.

5852

7.25

89M

onte

Car

lo0.

0000

0.51

345.

0619

0.00

760.

9428

5.19

810.

2019

1.80

615.

7019

1.39

053.

6417

7.20

69St

d0.

0000

0.00

020.

0004

0.00

000.

0004

0.00

070.

0002

0.00

070.

0011

0.00

090.

0014

0.00

18Pr

icin

ger

ror

0.00

00−0

.001

8−0

.000

20.

0025

−0.0

078

−0.0

005

−0.0

066

−0.0

277

0.03

13−0

.081

3−0

.056

50.

0519

(b)

Mat

urity

=90

days

Mat

urity

=18

0da

ys

K/S

01.

201.

101.

000.

900.

800.

701.

201.

101.

000.

900.

800.

70

λ=

0.0

Ana

lytic

al0.

0835

0.54

082.

3282

5.96

2510

.555

215

.431

50.

4357

1.37

693.

4788

6.91

1211

.217

315

.896

9M

onte

Car

lo0.

0776

0.55

972.

3521

5.93

5110

.545

315

.433

20.

4502

1.41

963.

5096

6.88

2711

.177

215

.891

0St

d0.

0002

0.00

050.

0010

0.00

140.

0016

0.00

160.

0005

0.00

100.

0015

0.00

190.

0022

0.00

23Pr

icin

ger

ror

0.00

58−0

.018

9−0

.023

90.

0274

0.00

98−0

.001

7−0

.014

5−0

.042

8−0

.030

80.

0285

0.04

010.

0060

λ=

0.1

Ana

lytic

al0.

1738

0.78

222.

6831

6.23

8110

.692

915

.455

60.

6568

1.72

793.

9074

7.31

7011

.533

816

.058

7M

onte

Car

lo0.

1706

0.83

742.

7513

6.20

1910

.643

315

.458

00.

7322

1.88

624.

0497

7.31

0311

.413

915

.988

4St

d0.

0003

0.00

070.

0012

0.00

160.

0018

0.00

190.

0008

0.00

120.

0017

0.00

220.

0025

0.00

26Pr

icin

ger

ror

0.00

32−0

.055

2−0

.068

20.

0362

0.04

96−0

.002

4−0

.075

4−0

.158

3−0

.142

30.

0068

0.11

990.

0703

λ=

0.2

Ana

lytic

al0.

5934

1.58

713.

6767

7.03

7611

.273

415

.800

01.

0590

2.34

294.

6348

8.04

0512

.317

016

.928

6M

onte

Car

lo0.

7010

1.81

333.

9152

7.09

6811

.126

315

.661

91.

8053

3.31

035.

5700

8.60

8512

.318

116

.519

5St

d0.

0008

0.00

120.

0017

0.00

220.

0025

0.00

270.

0015

0.00

200.

0025

0.00

290.

0033

0.00

36Pr

icin

ger

ror

−0.1

076

−0.2

261

−0.2

385

−0.0

592

0.14

710.

1381

−0.7

463

−0.9

674

−0.9

352

−0.5

681

−0.0

011

0.40

91



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

12 J.-C. Duan et al

Table 4(b) also compares the prices obtained for European calls usingour analytical approximation and those obtained by Monte Carlo simulation.This table is based on more extreme strike prices and different risk premiumparameters. Three different levels of risk premium are examined: 0, 0.1 and0.2. It should be noted that changing the risk premium value alters the per-sistence of the volatility process. These risk premia, in conjunction with theother parameter values, correspond to the following values for the three criticalconstants µ1, µ2 and µ3 governing the GJR-GARCH variance process: {µ1 =0.9835, µ2 = 0.9756, µ3 = 0.9788}, {µ1 = 0.9888, µ2 = 0.9876, µ3 = 0.9999},{µ1 = 0.9951, µ2 = 1.0022, µ3 = 1.0257}. As discussed in DGS (1999), µ1, µ2

and µ3 determine the behavior of the relevant moments of the cumulative assetreturn in relation to maturity. The analytical approximation will not experienceexplosive behavior when these constants are less than one. Thus, the third case(λ = 0.2) corresponds to a violation of the condition whereas the second (λ = 0.1)is a near violation. Finally, it should be noted that the three risk premia inducethree different stationary variances under measure Q, ie, h∗. The three stationarystandard deviations (annualized) are 0.2103, 0.2547 and 0.3867, respectively.

The results in Table 4(b) show that the approximation’s performance decreasesas the value of λ increases. This is especially true for extreme strike prices. Forλ equal to 0 or 0.1, the pricing errors (relatively to the Monte Carlo prices) arecomparable to those presented in the first panel. For λ = 0.2, however, the resultsquickly deteriorate as the maturity increases, a reflection of the explosive behaviorunder such a parameter value. The numerical performance of the GJR-GARCHoption pricing is thus consistent with the conclusion of DGS (1999) in that thequality of the analytical approximation for the NGARCH model decreases as thematurity increases, particularly when there is a high level of volatility persistence.

3.2 EGARCH

Tables 5–8 present results using the historical parameter estimates: β0 = −0.3057,β1 = 0.98, β2 = 0.1223, λ = 0.0586 and γ = −0.5057. Table 5 is used to checkthe mean and variance of the cumulative asset return, where the analytical mean isan exact quantity but the analytical variance is an approximate value. Again, thevariance is an approximate value because a Taylor expansion has been employed.The results show that the analytical values for both mean and variance are closeto the Monte Carlo counterparts for a maturity as long as 270 days.

Table 6 is used to examine the quality of the approximation for skewness andkurtosis of the standardized cumulative return. The results indicate that the thirdand fourth analytical moments match closely the Monte Carlo values. The qualityof approximation for the fourth moment again decreases as the maturity becomeslonger. Skewness and kurtosis reveal a clear departure of the GARCH model fromthe usual normality assumption. Negative skewness is again due to the leverageeffect of the EGARCH model. As with GJR-GARCH, reversion to normality forthe standardized cumulative return is evident.



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45


TABL

E5

Perf

orm

ance

ofth

ean

alyt

ical

appr

oxim

atio

nfo

rth

em

ean

and

vari

ance

ofth

ecu

mul

ativ

ere

turn

for

the

EGA

RC

H.

β0=

−0.3

057,

β1=

0.98

,β

2=

0.12

23,λ

=0.

0586

and

γ=

−0.5

057.

“Ana

lytic

al”

isth

ean

alyt

ical

appr

oxim

atio

n,“M

onte

Car

lo”

isth

eM

onte

Car

loes

timat

eba

sed

on1,

000,

000

sam

ple

path

san

d“S

td”

isth

est

anda

rdde

viat

ion

ofth

eM

onte

Car

loes

timat

e.

T=

10da

ysT

=30

days

T=

90da

ysT

=27

0da

ys

µT

σ2 T

µT

σ2 T

µT

σ2 T

µT

σ2 T

h1

=h∗ ×

1.00

Ana

lytic

al1.

1593

e−00

44.

2196

e−00

51.

1567

e−00

44.

2881

e−00

51.

1561

e−00

44.

3331

e−00

51.

1589

e−00

44.

3027

e−00

5M

onte

Car

lo1.

1482

e−00

44.

2188

e−00

51.

1697

e−00

44.

2908

e−00

51.

1439

e−00

44.

3407

e−00

51.

1628

e−00

44.

2929

e−00

5St

d2.

0540

e−00

66.

6469

e−00

81.

1959

e−00

67.

2758

e−00

86.

9448

e−00

77.

6908

e−00

83.

9874

e−00

77.

1378

e−00

8h

1=

h∗ ×

1.20

Ana

lytic

al1.

1211

e−00

44.

9862

e−00

51.

1252

e−00

44.

9253

e−00

51.

1369

e−00

44.

7228

e−00

51.

1514

e−00

44.

4565

e−00

5M

onte

Car

lo1.

1367

e−00

44.

9892

e−00

51.

1203

e−00

44.

9218

e−00

51.

1362

e−00

44.

7355

e−00

51.

1523

e−00

44.

4573

e−00

5St

d2.

2336

e−00

67.

8471

e−00

81.

2809

e−00

68.

3104

e−00

87.

2537

e−00

78.

4127

e−00

84.

0631

e−00

77.

4160

e−00

8h

1=

h∗ ×

0.80

Ana

lytic

al1.

1982

e−00

43.

4404

e−00

51.

1898

e−00

43.

6222

e−00

51.

1769

e−00

43.

9097

e−00

51.

1672

e−00

44.

1328

e−00

5M

onte

Car

lo1.

1942

e−00

43.

4496

e−00

51.

1752

e−00

43.

6238

e−00

51.

1750

e−00

43.

9121

e−00

51.

1709

e−00

44.

1380

e−00

5St

d1.

8573

e−00

65.

4248

e−00

81.

0991

e−00

66.

1802

e−00

86.

5930

e−00

76.

9906

e−00

83.

9148

e−00

76.

9213

e−00

8



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

14 J.-C. Duan et al

TABL

E6

Perf

orm

ance

ofth

ean

alyt

ical

appr

oxim

atio

nfo

rth

esk

ewne

ssan

dku

rtos

isof

the

cum

ulat

ive

retu

rnfo

rth

eEG

AR

CH

.β

0=

−0.3

057,

β1=

0.98

,β

2=

0.12

23,λ

=0.

0586

and

γ=

−0.5

057.

“Ana

lytic

al”

isth

ean

alyt

ical

appr

oxim

atio

n,“M

onte

Car

lo”

isth

eM

onte

Car

loes

timat

eba

sed

on1,

000,

000

sam

ple

path

san

d“S

td”

isth

est

anda

rdde

viat

ion

ofth

eM

onte

Car

loes

timat

e.

T=

10da

ysT

=30

days

T=

90da

ysT

=27

0da

ys

κ3,

Tκ

4,T

κ3,

Tκ

4,T

κ3,

Tκ

4,T

κ3,

Tκ

4,T

h1

=h∗ ×

1.00

Ana

lytic

al−3

.048

7e−0

013.

4745

e+00

0−5

.062

3e−0

013.

8026

e+00

0−6

.478

2e−0

013.

9713

e+00

0−5

.575

1e−0

013.

4418

e+00

0M

onte

Car

lo−3

.044

4e−0

013.

4823

e+00

0−5

.044

5e−0

013.

8752

e+00

0−6

.445

5e−0

014.

1393

e+00

0−5

.550

8e−0

013.

7646

e+00

0St

d4.

9767

e−00

31.

7309

e−00

26.

0993

e−00

32.

8509

e−00

26.

7870

e−00

33.

3734

e−00

25.

7435

e−00

32.

3514

e−00

2h

1=

h∗ ×

1.20

Ana

lytic

al−3

.041

1e−0

013.

4718

e+00

0−5

.030

4e−0

013.

7898

e+00

0−6

.406

3e−0

013.

9397

e+00

0−5

.516

5e−0

013.

4226

e+00

0M

onte

Car

lo−3

.015

0e−0

013.

4738

e+00

0−4

.997

6e−0

013.

8510

e+00

0−6

.439

3e−0

014.

1560

e+00

0−5

.541

1e−0

013.

7682

e+00

0St

d4.

9504

e−00

31.

7271

e−00

25.

9964

e−00

32.

6028

e−00

26.

9093

e−00

33.

6889

e−00

25.

8399

e−00

32.

5308

e−00

2h

1=

h∗ ×

0.80

Ana

lytic

al−3

.058

0e−0

013.

4777

e+00

0−5

.100

5e−0

013.

8179

e+00

0−6

.562

8e−0

014.

0092

e+00

0−5

.643

8e−0

013.

4646

e+00

0M

onte

Car

lo−3

.037

8e−0

013.

4730

e+00

0−5

.146

6e−0

013.

9086

e+00

0−6

.569

9e−0

014.

1930

e+00

0−5

.683

8e−0

013.

7977

e+00

0St

d4.

9331

e−00

31.

6941

e−00

26.

2417

e−00

32.

9421

e−00

26.

9234

e−00

33.

4738

e−00

25.

8286

e−00

32.

4010

e−00

2



PRO

OF

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45


TABL

E7

Perf

orm

ance

ofth

ean

alyt

ical

appr

oxim

atio

nfo

rth

eEu

rope

anca

llop

tion

pric

efo

rth

eEG

AR

CH

.(a)

β0=

−0.3

057,

β1=

0.98

,β

2=

0.12

23,λ

=0.

0586

and

γ=

−0.5

057.

(b)

β0=

−0.2

75,β

1=

0.98

,β

2=

0.12

23,γ

=−0

.505

7an

dh

1=

h∗ .

“Ana

lytic

al”

isth

ean

alyt

ical

appr

oxim

atio

n,“M

onte

Car

lo”

isth

eM

onte

Car

loes

timat

eba

sed

on10

,000

,000

sam

ple

path

san

d“S

td”

isth

est

anda

rdde

viat

ion

ofth

eM

onte

Car

loes

timat

ean

d“P

rici

nger

ror”

isth

edi

ffere

nce

betw

een

the

two

estim

ates

.

(a)

Mat

urity

=10

days

Mat

urity

=30

days

Mat

urity

=90

days

Mat

urity

=27

0da

ys

K/S

01.

101.

000.

901.

101.

000.

901.

101.

000.

901.

101.

000.

90

h1

=h∗ ×

1.00

Ana

lytic

al0.

0000

0.43

725.

0616

0.00

250.

8009

5.18

850.

0776

1.53

285.

6427

0.88

283.

1394

6.94

99M

onte

Car

lo0.

0000

0.43

895.

0616

0.00

100.

8086

5.18

890.

0759

1.54

925.

6279

0.88

513.

1369

6.92

94St

d0.

0000

0.00

020.

0003

0.00

000.

0003

0.00

060.

0001

0.00

060.

0009

0.00

060.

0011

0.00

15Pr

icin

ger

ror

0.00

00−0

.001

7−0

.000

00.

0015

−0.0

078

−0.0

005

0.00

17−0

.016

40.

0149

−0.0

023

0.00

250.

0205

h1

=h∗ ×

1.20

Ana

lytic

al0.

0000

0.47

205.

0616

0.00

390.

8499

5.19

160.

0946

1.58

435.

6580

0.91

653.

1740

6.96

54M

onte

Car

lo0.

0000

0.47

425.

0619

0.00

200.

8582

5.19

240.

0937

1.60

075.

6409

0.91

693.

1687

6.94

30St

d0.

0000

0.00

020.

0004

0.00

000.

0004

0.00

060.

0001

0.00

060.

0009

0.00

060.

0011

0.00

15Pr

icin

ger

ror

0.00

00−0

.002

2−0

.000

30.

0019

−0.0

084

−0.0

009

0.00

09−0

.016

40.

0171

−0.0

003

0.00

540.

0224

h1

=h∗ ×

0.80

Ana

lytic

al0.

0000

0.39

855.

0616

0.00

140.

7456

5.18

630.

0605

1.47

445.

6265

0.84

523.

1005

6.93

30M

onte

Car

lo0.

0000

0.40

025.

0617

0.00

040.

7535

5.18

800.

0580

1.49

075.

6141

0.84

793.

0976

6.91

14St

d0.

0000

0.00

020.

0003

0.00

000.

0003

0.00

050.

0001

0.00

060.

0009

0.00

060.

0011

0.00

15Pr

icin

ger

ror

0.00

00−0

.001

8−0

.000

10.

0011

−0.0

079

−0.0

017

0.00

25−0

.016

30.

0124

−0.0

027

0.00

290.

0216

(b)

Mat

urity

=90

days

Mat

urity

=18

0da

ys

K/S

01.

201.

101.

000.

900.

800.

701.

201.

101.

000.

900.

800.

70

λ=

0.0

Ana

lytic

al0.

1331

0.74

592.

6435

6.16

6210

.630

815

.441

90.

6535

1.78

383.

9438

7.23

6711

.369

615

.948

9M

onte

Car

lo0.

1340

0.76

132.

6662

6.14

1610

.611

215

.445

40.

6607

1.79

083.

9444

7.21

0811

.336

415

.941

3St

d0.

0002

0.00

060.

0011

0.00

160.

0018

0.00

190.

0007

0.00

120.

0017

0.00

210.

0024

0.00

26Pr

icin

ger

ror

−0.0

009

−0.0

154

−0.0

227

0.02

460.

0196

−0.0

035

−0.0

072

−0.0

069

−0.0

005

0.02

590.

0332

0.00

76λ

=0.

1A

naly

tical

0.26

891.

0746

3.06

646.

4704

10.7

856

15.4

856

1.04

492.

3475

4.54

737.

7171

11.6

675

16.0

926

Mon

teC

arlo

0.27

421.

1010

3.10

286.

4486

10.7

433

15.4

838

1.05

392.

3563

4.54

967.

6898

11.6

182

16.0

648

Std

0.00

040.

0008

0.00

130.

0018

0.00

210.

0022

0.00

100.

0014

0.00

200.

0024

0.00

280.

0030

Pric

ing

erro

r−0

.005

3−0

.026

5−0

.036

40.

0217

0.04

240.

0018

−0.0

091

−0.0

088

−0.0

023

0.02

730.

0494

0.02

78λ

=0.

2A

naly

tical

0.53

561.

5660

3.64

486.

9170

11.0

553

15.6

113

1.67

113.

1459

5.37

348.

4079

12.1

460

16.3

759

Mon

teC

arlo

0.55

151.

6098

3.69

826.

9068

10.9

880

15.5

790

1.68

533.

1583

5.37

758.

3819

12.0

861

16.3

200

Std

0.00

060.

0011

0.00

160.

0021

0.00

240.

0026

0.00

130.

0018

0.00

240.

0029

0.00

320.

0035

Pric

ing

erro

r−0

.016

0−0

.043

9−0

.053

40.

0102

0.06

720.

0324

−0.0

142

−0.0

124

−0.0

041

0.02

600.

0599

0.05

59



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19

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21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

16 J.-C. Duan et al

TABL

E8

Pric

ing

diffe

rent

ials

due

toth

eus

eof

mom

enta

ppro

xim

atio

nsin

the

anal

ytic

alEG

AR

CH

appr

oxim

atio

nfo

rmul

afo

rth

eEu

rope

anca

llop

tion

pric

e.β

0=

−0.3

057,

β1=

0.98

,β

2=

0.12

23,λ

=0.

0586

and

γ=

−0.5

057.

“Ana

lytic

al”

isth

ean

alyt

ical

appr

oxim

atio

npr

ice

base

don

appl

ying

aTa

ylor

expa

nsio

nto

ha t

and

usin

gap

prox

imat

etr

iple

sum

s,“E

xact

”is

the

anal

ytic

alap

prox

imat

ion

pric

ew

ithou

tus

ing

the

Tayl

orex

pans

ion

and

appr

oxim

ate

trip

lesu

ms

and

“Pri

cing

erro

r”is

the

diffe

renc

ebe

twee

nth

etw

opr

ice

estim

ates

.

Mat

urity

=10

days

Mat

urity

=30

days

Mat

urity

=90

days

Mat

urity

=27

0da

ys

K/S

01.

101.

000.

901.

101.

000.

901.

101.

000.

901.

101.

000.

90

h1

=h∗ ×

1.00

Tayl

or0.

0000

0.43

725.

0616

0.00

250.

8009

5.18

850.

0776

1.53

285.

6427

0.88

283.

1394

6.94

99Ex

act

0.00

000.

4372

5.06

160.

0025

0.80

085.

1885

0.07

881.

5306

5.64

340.

8736

3.12

536.

9536

Pric

ing

erro

r0.

0000

−0.0

000

0.00

00−0

.000

00.

0000

0.00

00−0

.001

20.

0022

−0.0

007

0.00

920.

0141

−0.0

036

h1

=h∗ ×

1.20

Tayl

or0.

0000

0.47

205.

0616

0.00

390.

8499

5.19

160.

0946

1.58

435.

6580

0.91

653.

1740

6.96

54Ex

act

0.00

000.

4720

5.06

160.

0039

0.84

985.

1916

0.09

591.

5816

5.65

880.

9067

3.15

936.

9689

Pric

ing

erro

r0.

0000

−0.0

000

0.00

00−0

.000

00.

0000

0.00

00−0

.001

30.

0027

−0.0

008

0.00

980.

0147

−0.0

035

h1

=h∗ ×

0.80

Tayl

or0.

0000

0.39

855.

0616

0.00

140.

7456

5.18

630.

0605

1.47

445.

6265

0.84

523.

1005

6.93

30Ex

act

0.00

000.

3985

5.06

160.

0014

0.74

575.

1863

0.06

161.

4726

5.62

700.

8366

3.08

726.

9368

Pric

ing

erro

r0.

0000

−0.0

000

0.00

00−0

.000

0−0

.000

00.

0000

−0.0

011

0.00

17−0

.000

50.

0086

0.01

34−0

.003

8



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The analytical prices and their Monte Carlo counterparts for European callsare presented in Table 7. In Table 7(a), for each maturity, we consider threemoneyness ratios. For maturities less than or equal to 90 days, the pricing errorsare very small and do not exceed 2 US cents. For a maturity of 270 days, thedifferences become larger but do not exceed 2.5 US cents. Table 7(b) also com-pares the prices obtained for European calls using our analytical approximationand the prices obtained by Monte Carlo simulation. In this table, two maturities(90 and 180 days) and six moneyness ratios (1.2, 1.1, 1.0, 0.9, 0.8 and 0.7) areexamined. Three different levels of the risk premium are examined: 0, 0.1 and0.2. They lead to annualized stationary volatilities of 0.2405, 0.2866 and 0.3500,respectively. The results reveal patterns similar to those obtained under the GJR-GARCH specification; that is, a lower precision for a large value of λ or extrememoneyness ratios. Performance is always better for shorter-maturity options.

In the implementation of the formula related to the EGARCH model, twoadditional approximations (when compared with the GJR-GARCH case) wereused to further reduce the computing time. First, we used a Taylor expansion toobtain an approximate value for E0[ha

t ], where a is any real number. Second, weapproximated the triple sums in the third and fourth moments by making the indexof the third sum equal to a single value. Even though we are able to obtain exactexpressions for E0[ha

t ] and the third moment of the cumulative return under theEGARCH, their evaluation is rather computer intensive. Their exact expressionsare nevertheless useful for gauging the quality of the approximations. Table 8compares the prices of options with and without these two further approximations.It should be noted that we continue to drop the terms SQ1, SQ3, SQ4 (exceptfor terms 8 and 12 of SQ4) and SQ5 (except for terms 2, 3, 6, 7 and 8 of SQ5)in computing the option prices. For options with maturity less than or equal to90 days, there is no significant difference between the two prices. When thematurity reaches 270 days, however, the difference can go as high as 1.47 UScents, suggesting that the quality of these approximations deteriorates slightly asmaturity becomes longer. Also note that there seems to be a smaller differencebetween the two prices for deep in-the money options and a bigger difference forat-the-money options.

3.3 A large-scale test on a randomly generated pool of options

The numerical analysis thus far suggests that our analytical approximation workswell for both the GJR-GARCH and EGARCH option pricing models, particularlyfor shorter-maturity options. This conclusion is obviously limited to the parametervalues used. We now study the performance on a test pool of options to ascertainthe general performance of the analytical approximation, an approach first intro-duced in Broadie and Detemple (1996).

Our test pool consists of 1,000 options. Each option corresponds to a set ofparameter values obtained from independent random drawings from predeter-mined distributions. For both the GJR-GARCH and EGARCH models, we assume



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18 J.-C. Duan et al

S0 equals 100. Other parameters are chosen from the following distributions: T

is uniformly distributed between 0.1 and 1.0 year with a probability of 0.75, anduniformly between 1.0 and 3.0 years with a probability of 0.25; K is uniformlydistributed between 70 and 130; r is uniformly distributed between 0 and 0.1 witha probability of 0.8, and equal to 0 with probability 0.2.

The GJR-GARCH parameter values are drawn from the following distribu-tions: β0 is uniformly distributed between 0 and 1 × 10−4; β1 is uniformlydistributed between 0 and 1; β2 is uniformly distributed between 0 and 1; β3 isuniformly distributed between 0 and 1; λ is uniformly distributed between 0 and 1.It is important to note that parameter sets violating the condition µ1 < 1, µ2 < 1,and µ3 < 1 are discarded. This criterion is introduced to prevent cases with anexplosive GARCH system dynamic.

The distributions used to draw the EGARCH parameter values are based onempirical results obtained in an (unreported) empirical analysis. The distributionsare as follows: β0 is uniformly distributed between −0.9230 and −0.6460; β1

is uniformly distributed between 0.9251 and 0.9550; β2 is uniformly distributedbetween 0.1741 and 0.2025; γ is uniformly distributed between −0.3248 and−0.2784; λ is uniformly distributed between 0 and 1. Parameter sets resultingin an annualized stationary standard deviation in excess of 0.60 are discardedbecause these cases are not particularly relevant.

The analytical approximate option price and its corresponding Monte Carlovalue are computed and compared. In order to assess the overall quality of theanalytical approximation, we measure the aggregate pricing error by computing,for each GARCH model, the root mean squared error (RMSE), defined as

RMSE =√√√√ 1

m

m∑i=1

e2i (6)

where ei = |Ci(a) − Ci |/Ci is the ith relative pricing error, Ci is the ith MonteCarlo price using 200,000 sample paths with the empirical martingale adjustmentof Duan and Simonato (1998), Ci(a) is the ith analytical approximation price andm is the number of option prices. In our actual sampling procedure, the optionswith a Monte Carlo price below 0.50 are discarded to avoid a large pricing errordue to a small divisor. We perform the sampling procedure until a test pool of1,000 options is obtained.

The test results are reported in Table 9. For the GJR-GARCH, the overallRMSE is 0.0062, indicating that the analytical approximation works quite well.If we separate short-maturity (ie, T < 150 days) and long-maturity (ie, T ≥ 150days) options, the RMSE measures become 0.0071 and 0.006, respectively. Thissuggests that the analytical approximation is more accurate, in terms of percentagepricing errors, for longer-maturity options. This finding does not contradictour earlier conclusion that the approximation formula works better for shorter-maturity options in terms of US dollar accuracy because shorter-maturity optionstend to have lower option values and thus magnify their percentage pricing errors.



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TABLE 9 Percentage pricing errors based on a randomly generated test pool ofoptions. Prop is the proportion of cases for which the absolute percentage pricingerror is greater than 5% in the test pool of 1,000 options; whole sample is the RMSEfor the whole test pool of 1,000 options; T < 150 is the RMSE for the subsample ofoptions with a maturity less than 150 days whereas T ≥ 150 is for the remainingoptions in the test pool; NTM is the RMSE of the subsample of near-the-moneyoptions satisfying 0.9 < S0/K < 1.1 whereas AFTM is for the remaining optionsdesignated as away-from-the-money; vol < 0.2 and vol ≥ 0.2 are the RMSEs for thesubsamples of options with low and high stationary volatilities.

Prop Wholesample

T < 150 T ≥ 150 NTM AFTM Vol < 0.2 Vol ≥ 0.2

GJR-GARCH 0.0040 0.0062 0.0071 0.0060 0.0054 0.0067 0.0028 0.0084EGARCH 0.0060 0.0101 0.0095 0.0103 0.0119 0.0088 0.0057 0.0136

If we classify the test pool into near-the-money (0.9 < S0/K < 1.1) and away-from-the-money groups (other cases), the results indicate a better performance fornear-the-money options (RMSE of 0.0054 versus 0.0067). If options are groupedinto low- and high-volatility categories (volatility < 0.2 or ≥ 0.2), one will find aRMSE of 0.0028 for the low-volatility group compared with 0.0084 for the high-volatility group. The fact that the approximation quality for the GJR-GARCHoption pricing model is sensitive to the volatility level is hardly surprising. Finally,0.4% of the cases have a relative pricing error exceeding 5% using the GJR-GARCH approximation formula.

For the EGARCH, the overall RMSE is 0.0101, suggesting that the analyticalapproximation is less accurate than that of the GJR-GARCH approximation for-mula, even though its overall performance is still quite satisfactory. If we separateoptions into short- and long-maturity categories, the RMSE measures become0.0095 and 0.0103, respectively. This suggests that the analytical approximationis more accurate in pricing shorter-maturity options even in terms of percentagepricing errors. If options are grouped into near-the-money and away-from-the-money, we will have 0.0119 and 0.0088 as their respective RMSE, suggesting thatthe approximation formula is more accurate for away-from-the-money options.Using volatility to classify options, the RMSE becomes 0.0057 and 0.0136 for thelow- and high-volatility groups, implying that the approximation formula worksbetter when the underlying asset has a lower volatility. For EGARCH, there are0.6% of the cases where the relative pricing error exceeds 5%.

In summary, the results of the performance analysis on a test pool of 1,000options indicate that our analytical approximations are adequate for the purposeof pricing options in the GJR-GARCH and EGARCH frameworks.

3.4 Computation speed

The approximations developed in this paper involve double and triple sums,and could have used quadruple sums. A natural concern is the computing time



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TABLE 10 Number of computed prices per second. The programs are coded inMatlab and computation was performed on a desktop PC with a 3066 Mhz processor.

T = 10 days T = 30 days T = 90 days T = 270 days

Black–Scholes 121,655 121,655 121,655 121,655GJR-GARCH 11,442 4,638 676 77E-GARCH 4,505 1,333 293 50

associated with these approximate formulas. Table 10 reports the number ofoption prices computed per second for different maturities. For comparison wealso report the numbers for the Black–Scholes formula. All models have beenimplemented in the Matlab programming language on a personal computer witha 3066 MHZ processor. For both the GJR-GARCH and EGARCH, compiledc-programs have been linked to Matlab in order to avoid bottlenecks caused bythe extensive looping required for the double and triple sums.

It is important to note that, for a given maturity, the formula evaluation iscomposed of two steps: (1) computing the required moments of the cumulativereturn; and (2) plugging in the computed moments into the approximate pricingformula. Also note that option values for different strike prices can share thesame moments of the cumulative return. Since computing the moments is a moredemanding step, a meaningful assessment of computing speed should account forthe fact that exchange-traded options typically have multiple strike prices for agiven maturity. On 25th August 2005, for example, the Chicago Board OptionsExchange listed the S&P 500 index options with 67 different strike prices for asingle maturity date of 5th September 2005. To roughly reflect the fact of multiplestrike prices, we have adopted 50 strike prices for a single maturity in generatingthe results in Table 10.

Unlike the Black–Scholes formula, the computing time for the approximationformula will increase with maturity since the loops required for the GARCHapproximation are indexed over time. The number of prices completed persecond is clearly much smaller than that of the Black–Scholes formula for allmaturities. Even so, approximate option prices can be computed very quickly;for example, the approximate GJR-GARCH option pricing formula can compute4,638 option prices per second if the maturity is 30 days. It should be noted that,for all maturities, the approximate formulas are much faster than the availablealternatives such as Monte Carlo simulation, the lattice technique of Ritchken andTrevor (1999) and the Markov chain method of Duan and Simonato (2001).

4 Conclusion

Following the approach developed by DGS (1999), we derive analytical approxi-mations for European option pricing in the GJR-GARCH and EGARCH frame-works. The analytical approximations target the skewness and kurtosis of thecumulative asset return under these two GARCH option pricing models. In each



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case, the resulting analytical pricing formulas is the Black–Scholes formula plusadjustment terms accounting for skewness and kurtosis. Consistent with DGS(1999), our results suggest that the approximations are adequate, particularly forshorter-maturity options.

In the literature, empirical studies based on the GARCH option pricing modelhave been conducted using Monte Carlo simulation as a mean for computingoption prices. The Monte Carlo method is typically computer intensive. For alarge-scale empirical study, computation time can be a serious concern. Ouranalytical approximation formulas can be useful in this regard. Our results extendthe existing set of approximation formulas for the GARCH family of models.Presumably, one can begin to analyze empirically the exchange-traded Europeanoptions using several GARCH models without being unduly constrained by thecomputing consideration. Our approximation formulas can also be potentiallyuseful if one chooses to apply the GARCH option pricing model to on-lineapplications because European option prices can be computed rather quickly withthe approximation formulas.

Appendix A Analytical expressions for the GJR-GARCH

A.1 Some intermediate results

Let n(•) and N(•) denote respectively the density and the distribution functionsof a standard normal random variable. The following formula are required toderive the analytical expressions for µ1, µ2, µ3,ν1, ν2, ν3, ζ1, ζ2 and ξ1. Since∫ λ

−∞ n(z) dz = N(λ) and∫ λ

−∞ zn(z) dz = −n(λ), then∫ λ

−∞ zkn(z) dz is obtainedby recursion:∫ λ

−∞zkn(z) dz = −λk−1n(λ) + (k − 1)

∫ λ

−∞zk−2n(z) dz, k ∈ {2, 3, 4, . . . }

For any integer k and m ≥ 0,

EQ[(max(0, −ε + λ))mεk] =∫ λ

−∞(λ − z)mzkn(z) dz

=m∑

i=0

(m

i

)(−λ)m−i

∫ λ

−∞zk+in(z) dz

A.2 Analytical formulas for µ1, µ2, µ3,ν1, ν2, ν3, ζ1, ζ2 and ξ1

The derivations for all formulas below utilize the results in the previous subsec-tion.

A.2.1 Formulas for µ1, µ2, µ3

µ1 = EQ0 [β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2]

= β1 + [β2 + β3N(λ)](1 + λ2) + β3λn(λ)



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22 J.-C. Duan et al

µ2 = EQ0 [(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)2]

= β21 + β2

2 (λ4 + 6λ2 + 3) + 2β1β2(1 + λ2) + 2β3β1[(λ2 + 1)N(λ) + λn(λ)]+ [β2

3 + 2β2β3][(λ4 + 6λ2 + 3)N(λ) + (λ3 + 5λ)n(λ)]µ3 = E

Q0 [(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)3]

= β31 + β3

2 (15 + λ6 + 15λ4 + 45λ2) + 3β21β2(1 + λ2)+3β1β

22 (3+λ4 +6λ2)

+ [3β23β1 + 6β1β2β3][(λ3 + 5λ)n(λ) + (λ4 + 6λ2 + 3)N(λ)]

+ 3β21β3[λn(λ) + (λ2 + 1)N(λ)] + [β3

3 + 3β22β3 + 3β2

3β2]× [(λ5 + 14λ3 + 33λ)n(λ) + (λ6 + 15λ4 + 45λ2 + 15)N(λ)]

A.2.2 Formulas for ν1, ν2, ν3

ν1 = EQ0 [ε(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)]

= −2β2λ − 2β3[n(λ) + λN(λ)]ν2 = E

Q0 [ε(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)2]

= −4β22 (3λ + λ3) − 4β1β2λ − [4β2

3 + 8β2β3]× [(λ2 + 2)n(λ) + (λ3 + 3λ)N(λ)] − 4β3β1[n(λ) + λN(λ)]

ν3 = EQ0 [ε(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)3]

= −6β32λ(15 + λ4 + 10λ2) − 6β2

1β2λ − 12β22β1(3λ + λ3)

− (6β33 + 18β3β

22 + 18β2

3β2)

× [(λ4 + 9λ2 + 8)n(λ) + (λ5 + 10λ3 + 15λ)N(λ)]− [12β2

3β1 + 24β3β1β2][(λ2 + 2)n(λ) + (λ3 + 3λ)N(λ)]− 6β3β

21 [n(λ) + λN(λ)]

A.2.3 Formulas for ζ1, ζ2, ξ1

ζ1 = EQ0 [ε2(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)]

= β1 + β2(3 + λ2) + β3[λn(λ) + (λ2 + 3)N(λ)]ζ2 = E

Q0 [ε2(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)2]

= β21 + β2

2 (15 + λ4 + 18λ2)

+ 2β1β2(3 + λ2) + 2β3β1[λn(λ) + (λ2 + 3)N(λ)]+ [β2

3 + 2β2β3][(λ3 + 17λ)n(λ) + (λ4 + 18λ2 + 15)N(λ)]Journal of Computational Finance


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ξ1 = EQ0 [ε3(β1 + β2(ε − λ)2 + β3 max(0, −ε + λ)2)]

= −6β2λ − 2β3(4n(λ) + 3λN(λ))

Appendix B Analytical expressions for the EGARCH

B.1 Some intermediate results

This subsection provides some intermediate results required to derive theanalytical approximation for the EGARCH model. For all k ∈ R, i ∈ N andYi = β0 + β2[|εi − γ | + γ (εi − γ )] we have:

EQ0 [εi exp(kYi)] = E

Q0 [εi exp(k(β0 + β2[|εi − λ| + γ (εi − λ)]))]

= 1

2

(A erf

(λ − A√

2

)exp

(−1

2A(−A + 2λ)

)

− B erf

(λ − B√

2

)exp

(−1

2(−B + 2λ)

)

+ A exp

(−1

2A(−A + 2λ)

)+ B exp

(−1

2B(−B + 2λ)

))

EQ0 [ε2

i exp(kYi)] = EQ0 [ε2

i exp(k(β0 + β2[|εi − λ| + γ (εi − λ)]))]

= 1

2

(−A

√2

πexp

(C − 1

2λ2

)

+ (A2 + 1)

[erf

(λ − A√

2

)+ 1

]exp

(C − λA + 1

2A2

)

+ B

√2

πexp

(C − 1

2λ2

)− (B2 + 1)

×[

erf

(λ − B√

2

)− 1

]exp

(C − λB + 1

2B2

))

EQ0 [ε3

i exp(kYi)] = EQ0 [ε3

i exp(k(β0 + β2[|εi − λ| + γ (εi − λ)]))]

= 1

2

(−A

√2

π(λ + A) exp

(C − 1

2λ2

)

+ (A3 + 3A)

(erf

(λ − A√

2

)+ 1

)exp

(C − λA + 1

2A2

)

+ B

√2

π(λ + B) exp

(C − 1

2λ2

)

− (B3 + 3B)

(erf

(λ − B√

2

)− 1

)exp

(C − λB + 1

2B2

))

where erf( 12

√2x) = 2N(x) − 1 and A = kβ2(γ − 1), B = kβ2(γ + 1) and C =

kβ0.



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24 J.-C. Duan et al

B.2 Analytical results for the terms in Appendix C of DGS (1999)

B.2.1 The value of EQ0 [hm

i ]For any real number m ∈ R, E

Q0 [hm

i ] is obtained using induction on i:

EQ0 [hm

i ] = 1

2i−1h

mβi−11

1 exp

(mβ0

βi−11 − 1

β1 − 1

)

×i−1∏k=1

(2N(λ − mβk−1

1 β2γ + mβk−11 β2)

× exp

(1

2β2m(γ − 1)(β2β

2k−21 mγ − β2β

2k−21 m − 2βk−1

1 λ)

)+ 2N(−(λ − mβk−1

1 β2 − mβk−11 β2γ ))

× exp

(1

2β2m(γ + 1)(β2β

2k−21 mγ + β2β

2k−21 m − 2βk−1

1 λ)

))(7)

B.2.2 The value of EQ0 [hihi+j ]

EQ0 [hihi+j ] = E

Q0

[hi × (hi)

βj

1 exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j

11

EQ0 [hβ

j1 +1

i ]EQ0 [hj+1]

B.2.3 The value of EQ0 [√hiεihi+j ]

EQ0 [√hiεihi+j ] = E

Q0

[√hiεi(hi)

βj

1 exp

( j∑k=1

βk−11 Yi+j−k

)]

= 1

hβ

j−11

1

EQ0 [(hi)

(βj

1 + 12 )]EQ

0 [εi exp(gYi)]EQ0 [hj ]

where g = βj−11 , A = gβ2(γ − 1), B = gβ2(γ + 1) and C = gβ0.

B.2.4 The value of EQ0 [hihi+jhi+j+k]

EQ0 [hihi+jhi+j+k]

= EQ0

[hi(hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

)(hi+j )

βk1 exp

( k∑w=1

βw−11 Yi+j+k−w

)]

= 1

hβ

j1 (1+βk

1 )+βk1

1

EQ0 [(hi)

βj1 +1+β

j+k1 ]EQ

0 [h1+βk1

j+1 ]EQ0 [hk+1]



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B.2.5 The value of EQ0 [h2

i hi+j ]

EQ0 [h2

i hi+j ] = EQ0

[h2

i (hi)β

j

1 exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j

11

EQ0 [(hi)

βj1 +2]EQ

0 [hj+1]

B.2.6 The value of EQ0 [hih

2i+j ]

EQ0 [hih

2i+j ] = E

Q0

[hi

((hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

))2]

= 1

h2β

j1

1

EQ0 [(hi)

2βj

1 +1]EQ0 [h2

j+1]

B.2.7 The value of EQ

0 [hi

√hi+j εi+jhi+j+k]

EQ0 [hi


= EQ0

[hi

((hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

))1/2

× εi+j

((hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

))βk1

exp

( k∑w=1

βw−11 Yi+j+k−w

)]

= 1

hβ

j

1 ( 12 +βk

1 )+βk−11

1

× EQ0 [εi+j exp(gYi+j )]EQ

0 [(hi)12 β

j

1 +1+βj

1 βk1 ]EQ

0 [h( 12 +βk

1 )

j+1 ]EQ0 [hk]

where g = βk−11 , A = gβ2(γ − 1), B = gβ2(γ + 1) and C = gβ0.


0 [√hiεihi+jhi+j+k]

EQ0 [√hiεihi+jhi+j+k]

= EQ0

[(hi)

1/2εi(hi)β

j1 exp

( j∑w=1

βw−11 Yi+j−w

)(hi+j )

βk1

× exp

( k∑w=1

βw−11 Yi+j+k−w

)]



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26 J.-C. Duan et al

= 1

hβ

j−11 (1+βk

1 )+βk1

1

× EQ0 [(hi)

12 +β

j

1 +βj

1 βk1 ]EQ

0 [εi exp(gYi)]EQ0 [h(1+βk

1 )

j ]EQ0 [hk+1]

where g = (1 + βk1 )β

j−11 , A = gβ2(γ − 1), B = gβ2(γ + 1) and C = gβ0.


0 [h3/2i εihi+j ]

EQ0 [h3/2

i εihi+j ] = EQ0

[h

3/2i εi(hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j−11

1

EQ0 [h

32 +β

j

1i ]EQ

0 [εi exp(gYi)]EQ0 [hj ]



0 [√hiεih2i+j ]

EQ0 [√hiεih

2i+j ] = E

Q0

[(hi)

12 +2β

j

1 εi exp

(2

j∑w=1

βw−11 Yi+j−w

)]

= 1

h2β

j−11

1

EQ0 [(hi)

12 +2β

j

1 ]EQ0 [εi exp(gYi)]EQ

0 [h2j ]

where g = 2βj−11 , A = gβ2(γ − 1), B = gβ2(γ + 1) and C = gβ0.

B.2.11 The value of EQ0 [√hiεi


EQ0 [√hiεi


= EQ0

[(hi)

1/2εi

((hi)

βj

1 exp

( j∑w=1

βw−11 Yi+j−w

))1/2

× εi+j (hi+j )βk

1 exp

( k∑w=1

βw−11 Yi+j+k−w

)]

= EQ0 [(hi)

( 12 + 1

2 βj

1 +βj+k

1 )]EQ0 [εi exp(gYi)]EQ

0 [εi+j exp(mYi+j )]

× 1

h( 1

2 +βk1 )β

j−11

1

EQ0 [h( 1

2 +βk1 )

j ] 1

hβk−1

11

EQ0 [hk]

where g = βj−11 ( 1

2 + βk1 ), A = gβ2(γ − 1), B = gβ2(γ + 1), C = gβ0 and m =

βk−11 .



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B.2.12 The value of EQ0 [hiε

2i hi+j ]

EQ0 [hiε

2i hi+j ] = E

Q0

[(hi)

(βj1 +1)ε2

i exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j−11

1

EQ0 [(hi)

(βj1 +1)]EQ

0 [ε2i exp(gYi)]EQ

0 [hj ]



0 [h3/2i ε3

i hi+j ]

EQ0 [h3/2

i ε3i hi+j ] = E

Q0

[h

3/2i ε3

i (hi)β

j

1 exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j−11

1

EQ0 [h

32 +β

j1

i ]EQ0 [ε3

i exp(gYi)]EQ0 [hj ]


B.2.14 The value of EQ0 [hiεihi+j ]

EQ0 [hiεihi+j ] = E

Q0

[hiεi(hi)

βj1 exp

( j∑w=1

βw−11 Yi+j−w

)]

= 1

hβ

j−11

1

EQ0 [h1+β

j1

i ]EQ0 [εi exp(gYi)]EQ

0 [hj ]


Appendix C Computing the stationary variance, h∗

C.1 GJR-GARCH

Starting from Equation (2b), we find

EQ[ht+1] = β0 + EQ[ht ]EQ[β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2]= β0 + EQ[ht ]µ1

because ht is Ft−1-measurable and hence independent of the term β1 + β2(εt −λ)2 + β3 max(0, −εt + λ)2. Recall that µ1, as defined in Equation (5a), equalsE

Q0 [β1 + β2(εt − λ)2 + β3 max(0, −εt + λ)2], which in turn equals EQ[β1 +

β2(εt − λ)2 + β3 max(0, −εt + λ)2]. For a variance-stationary GJR-GARCHprocess, we have EQ[ht ] = EQ[ht+1] for all t = 1, 2, . . . , which in turn impliesthat

h∗ = EQ[ht ] = β0

1 − µ1



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28 J.-C. Duan et al

C.2 EGARCH

The analytical expression for the expected value of ht conditional on the informa-tion available initially, ie, E

Q0 [ht ], is given in Equation (7). From that expression,

we can find the stationary variance of the EGARCH process by letting t go toinfinity. For practical purposes, we have set t = 500 in computing h∗.

Appendix D Taylor series expansion for hat

Using a Taylor expansion of hat at the point E

Q0 [ht ]a , we obtain

hat ≈ E

Q0 [ht ]a + ∂ha

t

∂ht

(ht − EQ0 [ht ]) + 1

2

∂2hat

∂h2t

(ht − EQ0 [ht ])2

+ 1

6

∂3hat

∂h3t

(ht − EQ0 [ht ])3 + 1

24

∂4hat

∂h4t

(ht − EQ0 [ht ])4

which then yields

EQ0 [ha

t ]≈ (E

Q0 [ht ]a(1 + 23

12a − 238 a2 + 13

12a3 − 18a4) + ( 19

4 a2 − 3a − 2a3 + 14a4)

× EQ0 [ht ]a−2E

Q0 [h2

t ] + ( 76a3 − 7

3a2 + 43a − 1

6a4)EQ0 [ht ]a−3

× EQ0 [h3

t ] + ( 124a4 − 1

4a3 + 1124a2 − 1

4a)EQ0 [ht ]a−4E

Q0 [h4

t ])

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