model risk of the implied garch-normal model
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Model risk of the implied GARCH-normal modelShih-Feng Huanga & Meihui Guob
a Department of Applied Mathematics, National University of Kaohsiung, Taiwan.b Department of Applied Mathematics, National Sun Yat-sen University, Taiwan.Published online: 01 Dec 2011.
To cite this article: Shih-Feng Huang & Meihui Guo (2014) Model risk of the implied GARCH-normal model, QuantitativeFinance, 14:12, 2215-2224, DOI: 10.1080/14697688.2011.630323
To link to this article: http://dx.doi.org/10.1080/14697688.2011.630323
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Model risk of the implied GARCH-normal model
SHIH-FENG HUANGy and MEIHUI GUO*z
yDepartment of Applied Mathematics, National University of Kaohsiung, TaiwanzDepartment of Applied Mathematics, National Sun Yat-sen University, Taiwan
(Received 30 July 2010; in final form 5 October 2011)
Model risk causes significant losses in financial derivative pricing and hedging. Investors mayundertake relatively risky investments due to insufficient hedging or overpaying implied byflawed models. The GARCH model with normal innovations (GARCH-normal) has beenadopted to depict the dynamics of the returns in many applications. The implied GARCH-normal model is the one minimizing the mean square error between the market option valuesand the GARCH-normal option prices. In this study, we investigate the model risk of theimplied GARCH-normal model fitted to conditional leptokurtic returns, an important featureof financial data. The risk-neutral GARCH model with conditional leptokurtic innovations isderived by the extended Girsanov principle. The option prices and hedging positions of theconditional leptokurtic GARCH models are obtained by extending the dynamic semipara-metric approach of Huang and Guo [Statist. Sin., 2009, 19, 1037–1054]. In the simulationstudy we find significant model risk of the implied GARCH-normal model in pricing andhedging barrier and lookback options when the underlying dynamics follow a GARCH-tmodel.
Keywords: Conditional leptokurtic model; Dynamic semiparametric approach; ExtendedGirsanov principle; GARCH model; Model risk; Option pricing
JEL Classification: C5, C51, G1, G13
1. Introduction
The use of financial models has become very prevalent
recently. Any model is a simplified version of reality, and
with any simplification there is model risk involved in
valuing financial securities. That is, the model that is used
to measure an asset’s market value does not perform the
tasks or capture the risks as designed. Investors may
undertake relatively risky investments due to insufficient
hedging or overpaying implied by flawed models. The
Long Term Capital Management debacle in 1997–1998 is
a well-known example attributed to model risk.
Therefore, the importance of model risk should not to
be overlooked in financial derivative pricing and hedging.In the literature, various models are proposed to
describe the dynamics and specific features of financial
data such as non-normality, non-constant volatility,
volatility clustering, etc. Among these models, conditional
heteroscedastic models such as ARCH and GARCH
models (Engle 1982, Bollerslev 1986) are the most
popular. GARCH models have been shown to
outperform the well-known Black–Scholes model (Blackand Scholes 1973) for market option pricing (see, forexample, Engle and Rosenberg (1994, 1995), Duan (1995),Hagerud (1996) and Heston and Nandi (2000)). Oneapproach is to fit the dynamic GARCH model byhistorical asset price data and compute the derivativeprices or hedging positions by its corresponding risk-neutral model (Duan 1995, Duan and Simonato 1998, Siuet al. 2004, Badescu and Kulperger 2008, Huang and Guo2009). However, this best fitted asset model is notnecessarily the optimal option pricing model minimizingthe mean square error between the model option pricesand the market values. To obtain the optimal optionpricing model the implied GARCH model is introducedby matching the GARCH option prices with the marketplain vanilla values (Fofana and Brorsen 2001, Yung andZhang 2003). This approach is analogous to the impliedvolatility function model in the Black–Scholes framework(Dupire 1994, Andersen and Brotherton-Ratcliffe 1998).Hull and Suo (2002) showed that there is little evidence ofmodel risk when the implied volatility function of theBlack–Scholes model is used to price and hedge plainvanilla and compound options while significant risk arisesfor barrier options when the underlying asset is assumed*Corresponding author. Email: [email protected]
� 2011 Taylor & Francis
http://dx.doi.org/10.1080/14697688.2011.630323
Quantitative Finance, 2014
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to be a stochastic volatility model. In this paper, we studythe model risk of the implied GARCH model with normalinnovations, henceforth called the implied GARCH-normal. Although the GARCH-normal model has beenwell studied in the literature and applied widely in practicedue to its superiority to the Black–Scholes model inderivative pricing and hedging (Duan 1995, Duan andSimonato 1998, Ritchken and Trevor 1999), it can onlydepict the unconditional but not the conditional lepto-kurtic property. Many empirical studies have identifiedthat financial returns exhibit the conditional leptokurticproperty, which is also known to be a factor determiningthe shape of the volatility smile in options (Duan 1999,Christoffersen et al. 2006). In this study, we assume thatthe true underlying asset follows a GARCH model withleptokurtic innovations (Bollerslev 1987, Nelson 1991,Duan 1999, Fan and Yao 2003, Siu et al. 2004,Christoffersen et al. 2006) and investigate the model riskof the impled GARCH-normal model fitted to suchconditional leptokurtic returns in option pricing and deltahedging for plain vanilla and exotic options.
To obtain the no-arbitrage option prices and hedgingpositions for conditional leptokurtic GARCH models weadopt the extended Girsanov principle (Elliott andMadan 1998) and the dynamic semiparametric approach(DSA) of Huang and Guo (2009). The extended Girsanovprinciple is a change of measure process for discrete timemodels. The no-arbitrage price derived by the extendedGirsanov principle is equal to the hedging cost minimizingthe variance of additional hedging capital (Elliott andMadan 1998). The DSA proposed by Huang and Guo(2009) is a backward iterative computation procedure forderivative prices such as European options, Americanoptions and convertible bonds. Huang and Guo (2009)apply the DSA to Black–Scholes, jump-diffusion andGARCH-normal models. Here we extend it to condi-tional leptokurtic GARCH models with the case ofdouble exponential innovations demonstrated as anexample. Simulation results show that, for conditionalleptokurtic GARCH models, the DSA gives accurateEuropean option prices and delta values compared withthe benchmark values obtained by the empirical martin-gale simulation (EMS) method of Duan and Simonato(1998). To access the model risk we assume the marketdynamics follow a GARCH-t model and compare thenominal option prices and hedging positions with thoseobtained by the implied GARCH-normal model.Simulation studies show that there exists significantmodel risk of the implied GARCH-normal model inpricing and hedging barrier and lookback options whileless discrepancy is found in Asian options.
This article is organized as follows. In section 2, theimplied GARCH-normal model is introduced. In section3, the risk-neutral GARCH models are derived by theextended Girsanov principle. In section 4, the DSA ofHuang and Guo (2009) is extended to compute theGARCH option prices with heavy-tailed innovations. Insection 5, simulation studies are performed to investigatethe DSA and the model risks of the GARCH-normalmodels. Discussions are given in section 6. All the proofs,
tables, figures and a brief illustration of the EMS methodare given in the appendix.
2. The implied GARCH-normal model
Let Rt¼ log (St/St�1) denote the log return, where St isthe stock price at time t. The following GARCH-normalmodel has been considered as the dynamic model of St inmany applications:
Rt ¼ �t � 0:5�2t þ �t"t, "t � i.i.d. Nð0, 1Þ,
�2t ¼ f�ð�k, "k, k � t� 1Þ,
(ð1Þ
where �t is the F t�1-measurable continuously com-pounded risk interest rate of the asset in time period[t� 1, t) and F t denotes the �-field generated by {Su, u� t,�u, u� tþ 1}. The evolution of the conditional volatility�2t is governed by the function f�(�) and the meancorrection factor 0:5�2t is included to ensure that equalityholds in Et�1ðStÞ ¼ St�1e
�t , where Et�1(�) denotes theconditional expectation given F t�1 under the dynamicmeasure P. Model (1) has been widely applied to financialderivative pricing and hedging. For example, Duan (1995)and Duan and Simonato (2001) considered the case of�t¼ rþ ��t, where r denotes the constant riskless interestrate and � the constant unit risk premium. Heston andNandi (2000) considered the case of �t ¼ rþ ��2t þ 0:5�2t .
If the parameters of model (1) are estimated byminimizing the mean square error between the model-implied plain vanilla option prices and the market optionprices, then we call the model the implied GARCH-normal model. Since the implied GARCH-normal modelis designed to accommodate the market plain vanillaoption pricing it usually has better performance in optionpricing and hedging than the GARCH-normal modelobtained from the maximum likelihood approach basedon historical asset price data (Bates 1996, Bakshi et al.1997, Dumas et al. 1998). Regardless of its prevalence theGARCH-normal model is deemed insufficient in depict-ing the conditional leptokurtic property found in manyempirical applications (Duan 1999, Christoffersen et al.2006). To remedy this shortcoming researchers extend itto the GARCH model with leptokurtic innovations. Forexample, Bollerslev (1987) considered the standardizedt-innovation, Nelson (1991) discussed the generalizedexponential innovation, Fan and Yao (2003) andMills (1999) considered the generalized Gaussian andStudent’s-t innovation, Shephard (1996) and Tong (1990)considered ARCH-type models with non-normal innova-tions, and Lee and Lee (2009) proposed the normalmixture quasi-maximum likelihood estimators forGARCH models with heavy-tailed or skewed innova-tions. For the above reasons we assume that, throughout,the underlying asset return follows a GARCH model withleptokurtic innovations.
It is of interest to investigate the effect of misfitting theimplied GARCH-normal model to a GARCH model withheavy-tailed innovations. In the famous Black–Scholes(B–S) framework, Hull and Suo (2002) studied the model
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risk problem for the implied volatility function when themarket dynamics actually follow a stochastic volatilitymodel. The implied volatility is solved by the B–S optionformula, which can be viewed as the market participants0
assessment of the volatility. Several studies have foundthat implied volatilities tend to differ across moneynessand time to expiration. Hull and Suo (2002) found that,interestingly, for a stochastic volatility underlying model,the misspecified B–S-implied volatility function modelcan still provide accurate pricing and hedging for plainvanilla instruments. The bias problem becomes relevant,however, for the consideration of exotic options. Here weinvestigate the model risk in pricing and hedging financialderivatives of the implied GARCH-normal model for theconditional leptokurtic GARCH underlying model. Toproceed, we adopt the following procedure.
(i) Assume that the returns follow a GARCH modelwith heavy-tailed innovations. The derivation ofthe associated risk-neutral model is introduced insection 3.
(ii) Compute the prices of plain vanilla options by therisk-neutral GARCH model and treat them as thenominal values. The details of the computation areillustrated in section 4.
(iii) Estimate the parameters of the implied GARCH-normal model by minimizing the mean squareerror between the plain vanilla nominal optionvalues and the GARCH-normal option prices.This criterion was also adopted by Bates (1996,Bakshi et al. (1997) and Dumas et al. (1998).
(iv) Evaluate the biases of the implied GARCH-normal model for exotic option pricing and deltahedging. A simulation example is given in section 5for the GARCH-t model.
3. Risk-neutral GARCH models
In this section, we illustrate the deviation of the risk-neutral GARCH models with leptokurtic innovations.Firstly, we extend the innovations in model (1) to have ageneralized distribution, that is the log return Rt satisfiesthe following model:
Rt ¼ �t � �t þ �t"t, "t � i:i:d: Dð0, 1Þ,
�2t ¼ f�ð�k, "k, k � t� 1Þ,
�ð2Þ
where the innovation "t follows a distribution function Dwith zero mean and unit variance, and �t denotes themean correction factor to ensure Et�1ðStÞ ¼ St�1e
�t .Since the GARCH model is a discrete time and infinite
state space model, the market is incomplete and theequivalent martingale measures are not unique (Duan1995, Siu et al. 2004, Christoffersen et al. 2006, Badescuand Kulperger 2008). Here we adopt the extendedGirsanov change of measure approach proposed byElliott and Madan (1998) to derive the risk-neutraldynamics of the conditional leptokurtic GARCH model.Assume that the discounted asset price process ~St ¼ Ste
�rt
satisfies the multiplicative decomposition ~St ¼ ~S0AtMt,
where
At �Ytk¼1
Ek�1
~Sk
~Sk�1
!is a predictable process and
Mt ¼~St
~S0At
is a positive martingale. Define the change of measure
density process {�t, t¼ 0, 1, . . . ,T} by
�t ¼Ytk¼1
�kð ~Sk= ~Sk�1Þeuk
�kðe�uk ~Sk= ~Sk�1Þ, ð3Þ
which is a P-martingale, where uk ¼ log Ek�1ð ~Sk= ~Sk�1Þ
denotes the risk premium of the underlying asset and �k(�)is the conditional density of Mk/Mk�1 given F k�1 under
the dynamic measure P. The equivalent martingale
measure Q under the extended Girsanov change of
measure (i.e. ~St is a martingale under Q) is defined by
dQ¼�T dP, that is the positive process �t is the Randon–
Nikodym derivative of Q with respect to P restricted
to F t�1.For instance, in model (2) we have the predictable
process At¼ exp{u1þ � � � þ ut}, the positive martingale
Mt ¼ ~St expf�ðu1 þ � � � þ utÞg= ~S0, and the risk-premium
ut ¼ logEt�1ðexpf�rþ �t � �t þ �t"tgÞ ¼ �t � r:
When the innovations {"t} are standard normal variables,
then by (3) we have
�t ¼Ytk¼1
exp1
2�2k½u2k � 2ukðlogð ~Sk= ~Sk�1Þ þ �kÞ�
� �,
since the random variable
Xt ¼Mt=Mt�1 ¼ e�ut ~St= ~St�1 ¼ e��tþ�t"t
given F t�1 is log-normally distributed with density
function
�tðxÞ ¼1
xffiffiffiffiffiffiffiffiffiffi2p�2t
p exp �1
2�2tðlog xþ �tÞ
2
� �under measure P. In the following proposition, the
extended Girsanov principle is utilized to derive the
equivalent risk-neutral model of model (2).
Proposition 3.1: Assume that the dynamic of the loga-
rithmic return process {Rt} under the dynamic measure P
satisfies model (2), then the risk-neutral model under the
measure Q is
Rt ¼ r� �t þ �t�t, �t � i.i.d. Dð0, 1Þ,
�2t ¼ f� �k, �k þr� �k
�k, k � t� 1
� �,
8<: ð4Þ
where the law of
�t ¼ "t �r� �t
�t
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under the risk-neutral measure Q is the same as that of "tunder the dynamic measure P.
Interestingly, the risk-neutral GARCH-normal model
derived by proposition 3.1 is the same as that of Duan
(1995) by an expected utility maximizer where the
parameter settings in model (1) are
�t ¼ r� 0:5�2t þ ��t, �2t ¼ 0 þ 1�2t�1"
2t�1 þ 1�
2t�1:
In model (2), the existence of the moment generating
function (mgf) of "t is required to ensure that
Et�1ðStÞ ¼ Et�1fSt�1 expð�t � �t þ �t"tÞg51,
which is not satisfied by the leptokurtic innovations such
as the t-distribution. To include the t innovation in the
study, we can utilize the following simple return model:
~Rt �St � St�1
St�1¼ �t þ �t"t, "t � i.i.d. Dð0, 1Þ,
�2t ¼ f�ð�k, "k; k � t� 1Þ,
8><>: ð5Þ
where �t denotes the expected simple return in the time
period [t� 1, t). In model (5), since Et�1(St)¼
St�1(1þ�t)51, the condition of the mgf is no longer
required. Although there is a positive probability of
producing negative asset prices from model (5), the
probability is infinitesimal and can be ignored in practice
when the degrees of freedom for the t-innovation are
greater than 2 and �t40. For example, when the daily
volatility �t ¼ 0:5=ffiffiffiffiffiffiffiffi250p
¼ 0:0316, the probability of
having negative asset prices in model (5) is
PðSt 5 0 j St�1 4 0Þ ¼ Pð1þ �t þ �t"t 5 0 j St�1 4 0Þ
5PðT�5 � 1=�tÞ � 0,
where "t denotes a t random variable with degrees of
freedom �42 and
T� ¼
ffiffiffiffiffiffiffiffiffiffiffi�
�� 2
r"t:
In the following, we derive the risk-neutral dynamics of
the simple return model (5) by the extended Girsanov
principle.
Proposition 3.2: Assume that the dynamic of the simple
return process f ~Rtg under the dynamic measure P satisfies
model (5), then the risk-neutral model under Q is
~Rt ¼ ~rþ1þ ~rt1þ �t
�t�t, �t � i.i.d. Dð0, 1Þ,
�2t ¼ f� �k,1þ ~r
1þ �k�k þ
~r� �k
�k, k � t� 1
� �,
8>>><>>>:where ~r is the riskless simple interest rate and the law of
�t ¼1þ �t
1þ ~r"t �
~r� �t
�t
� �under the risk-neutral measure Q is the same as that of "tunder the dynamic measure P.
4. GARCH option valuation
Once the risk-neutral models are derived, we can compute
the derivative price by the conditional expectations under
the risk-neutral probability measure. For example, the no-
arbitrage price of the European call at the initial time t0 is
e�rTEQ0 fðST � K Þþg,
where ST is the stock price at the maturity time T, K is the
strike price, and EQ0 ð�Þ denotes the conditional expectation
given F 0 under the risk-neutral measure Q. In general,
there is no closed-form expression for the multi-step
conditional expectation of the option value. To solve the
problem, Duan (1995) and Duan and Simonato (1998)
used a Monte-Carlo-based method and Huang and
Guo (2009) proposed a dynamic semiparametric
approach (DSA). The DSA method is a backward
iterative procedure that uses regression to approximate
derivative values including European options, American
options and convertible bonds. The DSA is applicable to
higher-dimensional option pricing, where the dependence
structure of the underlying assets can be modeled by
copula functions. Huang and Guo (2009) developed the
DSA method for the GARCH-normal log return model.
In the following, we extend the DSA method to condi-
tional leptokurtic GARCH models.Let 0¼B(�1)5B(0)5B(1)5� � �5B(‘)5B(‘þ1)
¼1 be a
partition of the volatility interval. Consider the European
call option at time T� 1 and given (ST�1,B(h)),
h¼ 0, . . . , ‘, the call price equals
VT�1ðST�1,BðhÞÞ ¼ E
QT�1fe
�rðST � K Þþg,
which can be evaluated either analytically or numerically.
Define the approximate option value function.
VT�1ðST�1, �TÞ for �T not on the partition curves by the
following interpolation:
VT�1ðST�1, �TÞ
¼
w�TVT�1ðST�1,BðhÞÞ þ ð1� w�T ÞVT�1ðST�1,B
ðh�1ÞÞ,
if �T 2 ðBðh�1Þ,BðhÞÞ, h ¼ 0, . . . , ‘,
w�TVT�1ðST�1,Bð‘ ÞÞ þ ð1� w�TÞVT�1ðST�1,B
ð‘�1ÞÞ,
if �T 4Bð‘ Þ,
8>>><>>>:where
w�T ¼
�T � Bðh�1Þ
BðhÞ � Bðh�1Þ, if �T 2 ðB
ðh�1Þ,BðhÞÞ, h ¼ 0, . . . , ‘,
�T � Bð‘�1Þ
Bð‘ Þ � Bð‘�1Þ, if �T 4Bð‘ Þ:
8>><>>:To compute the option value at T� 2, we need to evaluate
the conditional expectation EQT�2ðV
T�1Þ. However, since
VT�1 is a non-trivial nonlinear function of ST�1,
EQT�2ðV
T�1Þ, which does not have an analytical solution.
We use a regression method to tackle this problem. Since
given (ST�2, �T�1) the volatility �T is a function of ST�1,
the function VT�1 given (ST�2, �T�1) is a function of ST�1
only, which will be denoted by VT�1ðST�1 j FT�2Þ
throughout. Let fAð j Þ : 0¼Að0Þ5Að1Þ5 � � �5AðmÞ ¼ 2Kg
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denote a partition of the stock price interval [0, 2K ]. Ineach partition interval [A(j�1),A(j)], j¼ 1, . . . ,m, of ST�1,fit a quadratic regression function to the given set ofpoints
ðsh,VT�1ðsh j FT�2ÞÞ : sh 2 ½A
ð j�1Þ,Að j Þ�, h ¼ 1, . . . ,N� �
:
The fitted regression function is denoted by
VT�1ðST�1 j FT�2Þ ¼Xmj¼1
Qð j ÞðST�1 j FT�2Þ,
where
Qð j ÞðST�1 j FT�2Þ ¼X2k¼0
að j Þi,k S
kT�111
ð j Þ,
and 11ð j Þ ¼ 11fAð j�1Þ�ST�1 5Að j Þg, j ¼ 1, . . . ,m:
The approximate option value on the volatility partitioncurve (ST�2,B
(h)) at time tT�2 is
VT�2ðST�2,BðhÞÞ ¼ E
QT�2 e�rVT�1ðST�1 j FT�2Þ
h i¼ e�r
Xmj¼1
EQT�2 Qð j ÞðST�1 j FT�2Þ
: ð6Þ
Proceeding backward iteratively to the initial time, onecan obtain the call price. Details are given by Huang andGuo (2009). Convergence of the DSA method is provedunder the continuity assumption on the transition densi-ties of the return process. In order to obtain the derivativevalue in (6), we need to compute the conditional truncatedmoments,
EQt�1ðS
kt 11fSt�agÞ, for k ¼ 0, 1, 2, t ¼ 0, . . . ,T� 1:
For the log return model (4), since the conditionalcumulant-generating function �t is finite, the conditionaltruncated moments exist. Huang and Guo (2009, prop-osition 3.4) gives the closed-form expressions of theconditional truncated moments for the risk-neutralizedGARCH-normal log return model. In the followingproposition, we derive the conditional truncated momentsfor GARCH models with double exponentialinnovations.
Proposition 4.1: In model (4), if the innovation �t has adouble exponential distribution with zero mean and unitvariance, then �t ¼ � logð1� 0:5�2t Þ and given (St�1, �t)¼(s, b) we have
EQt�1ðS
kt 11fSt�agÞ
¼ ekm2
2�k2b2�
eðkb�ffiffi2pÞd
2�ffiffiffi2p
kb
!11fd0g þ
eðkbþffiffi2pÞd
2þffiffiffi2p
kb11fd50g
" #,
where
m ¼ log sþ rþ logð1� 0:5b2Þ, d ¼log a�m
b,
for k5ffiffiffi2p=b:
The constraint k5ffiffiffi2p=b in proposition 4.1 is used to
guarantee the existence of the conditional cumulant-
generating function of the double exponential distribution.In practice, the annual conditional standard deviation isbetween 0.2 and 0.5, thus the upper bound of k, assuming250 trading days per year, is at least
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 250p
=0:5 � 45 fordaily returns models. In the DSAmethod, k is the degree ofthe regression polynomials, which is less than or equal to 2,thus the constraint k5
ffiffiffi2p=b is satisfied in practice.
The DSA method can also be applied to solve thecomputational problem of other multi-step conditionalexpectations. For example, the delta hedging position of aEuropean call option is the rate of change of an optionvalue with respect to the changes in the price of theunderlying at time t, that is Dt¼ @Ct/@St. Since it can alsobe expressed as the following multi-step conditionalexpectation (Duan 1995, corollary 2.4):
Dt ¼ e�rðT�tÞEQt
ST
St11½STK�
� �, ð7Þ
one can apply DSA to compute the delta hedging positionjust by replacing the function VT�1(ST�1,B
(h)) definedabove by
EQT�1 e�r
ST
St11½STK�
� �:
5. Simulation study
In this section, we first investigate the model risk of theimplied GARCH-normal model in exotic option pricingwhen the underlying asset follows a conditional leptokur-tic GARCH-t model. Next, we show the effectiveness ofthe DSA for computing European GARCH option pricesand delta hedging values. All algorithms are implementedin MatLab version 7.6.0 and run on an Intel Core 2, 2.67GHz computer with 3GB RAM.
5.1. GARCH-t versus implied GARCH-normal models
The GARCH-t model proposed by Bollerslev (1987)assumes that the standardized innovations follow astandardized Student-t distribution, which relaxes theconditional normality assumption of the GARCH-normal model. Multivariate extension of GARCH-tmodels have been considered by McGuirk et al. (1993),Spanos (1994) and Heracleous and Spanos (2005), amongothers. In real-world applications, there is substantialevidence showing that the errors have a leptokurticdistribution. The extent of leptokurtosis in a GARCH-tmodel is measured by the degrees of freedom of thet-distribution, and Heracleous (2007) studied the estima-tion issue of the degrees of freedom parameter. Corhayand Rad (1994) presented empirical evidence that theGARCH-t model is appropriate for studying the behaviorof stock returns on smaller equity markets. Fofana andBrorsen (2001) used a GARCH-t process to modelChicago wheat futures price movements and found thatthe GARCH-t model with implied volatility outperformsthe implied B–S model at options close to maturity (6–15days). Based on these extensive theoretical and empirical
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advantages of the GARCH-t pricing models, we there-fore employ the GARCH-t model as the true processgoverning the dynamics of the underlying asset in oursimulation study.
The simulation procedure for investigating the modelrisk of the implied GARCH-normal model in optionpricing is as follows.
(i) Assume that the underlying asset return satisfiesthe following GARCH(1, 1)-t model with fivedegrees of freedom:
~Rt ¼ rþ ��t þ �t"t, "t �ffiffiffiffiffiffiffiffi3=5p
tð5Þ,
�2t ¼ 0:00004þ 0:1�2t�1ð"t�1 � �Þ2þ 0:8�2t�1,
(ð8Þ
where r¼ 0.05/365 is the daily riskless interest rate,the risk premium � is 0.2 and the leverage effectparameter � is 0.3.
(ii) Compute the no-arbitrage financial derivativeprices of model (8) via its risk-neutral modelgiven below (see proposition 3.2):
~Rt ¼ rþ1þ r
1þ rþ ��t�t�t, �t �
ffiffiffiffiffiffiffiffi3=5
ptð5Þ,
�2t ¼ 0:00004þ 0:1�2t�11þ r
1þ rþ ��t�1�t�1 � � � �
� �2
þ 0:8�2t�1:
8>>>>><>>>>>:The benchmark (nominal) European call optionprices are obtained by the EMS method proposedby Duan and Simonato (1998) (a brief illustrationis given in the appendix). Let Ci, i¼ 1, . . . , n,denote the nominal European call option pricescorresponding to n different sets of strike pricesand maturities.
(iii) Use the nominal call option values obtained instep (ii) to fit the following risk-neutralGARCH(1, 1)-normal model (Duan 1995):
Rt ¼ r� 0:5�2t þ �2t �t, �t � Nð0, 1Þ,
�2t ¼ 0 þ 1�2t�1ð�t�1 � �Þ
2þ 1�
2t�1:
(ð9Þ
Let �¼ (0,1,1, �) be the parameter vector, andthe estimator of � implied by the option prices isobtained by minimizing the following sum ofsquared error:
� ¼ argmin�
Xni¼1
ðCi � Cið�ÞÞ2,
where Ci is the nominal European call optionvalue obtained in (ii) and Cið�Þ, which has thesame strike price and maturity date as Ci, is theEuropean call option price obtained by the EMSmethod based on the implied GARCH-normalmodel (9).
(iv) Based on the implied risk-neutral GARCH-normal model derived in step (iii), we can thenapply the EMS method to compute the no-arbit-rage exotic option prices of the implied GARCH-normal model.
Table 1 presents the nominal European call option
prices with initial stock price S0¼ 50, maturity dates
T¼ 10, 30 and 60 and strike prices K¼ 45, 47.5, 50, 52.5
and 55 and the option prices derived by the corresponding
implied GARCH-normal model with � ¼ ð0:0000637,0:2526, 0:5995, 0:2998Þ. The results show that the bias of
the misspecified implied GARCH-normal is relatively
small for European call option pricing.Next, we consider the model risk of the implied
GARCH-normal model in exotic option pricing including
the Asian option, lookback option and barrier option.
The payoffs of the exotic options are defined as follows:
. Asian call option:
1
T
XTt¼1
St � K
!þ, where xþ ¼ maxðx, 0Þ;
. lookback call option:
ST �mT, where mT ¼ min0�t�T
St;
. barrier (up-and-out) call option:
ðST � K Þþ11ðmaxt2½0,T� St�BÞ, where B is the barrier price.
Table 2 gives the results for the Asian call option,
where the settings of the moneyness and time to
maturity are the same as in table 1. Interestingly, the
implied GARCH-normal model still gives accurate
Table 2. The GARCH-t nominal Asian call option prices andthe implied GARCH-normal option prices.
K
T 45 47.5 50 52.5 55
10 days 5.0479 2.6541 0.8043 0.1163 0.0156(5.0471) (2.6582) (0.8104) (0.1147) (0.0123)
30 days 5.2141 3.0354 1.3817 0.4737 0.1345(5.2184) (3.0424) (1.3872) (0.4750) (0.1319)
60 days 5.5241 3.5389 1.9977 0.9862 0.4345(5.5251) (3.5410) (2.0002) (0.9875) (0.4330)
The values in parentheses are derived from the implied GARCH-normal
model (9).
Table 1. The GARCH-t nominal European call option pricesand the implied GARCH-normal option prices.
K
T 45 47.5 50 52.5 55
10 days 5.1856 2.9845 1.3137 0.4254 0.1163(5.1926) (2.9933) (1.3176) (0.4238) (0.1126)
30 days 5.7926 3.8964 2.3927 1.3345 0.6895(5.7949) (3.8987) (2.3945) (1.3385) (0.6881)
60 days 6.6341 4.9251 3.5102 2.4030 1.5853(6.6287) (4.9222) (3.5095) (2.4034) (1.5848)
The values in parentheses are derived from the implied GARCH-normal
model (9).
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Asian option prices. Table 3 shows the pricing resultsof the lookback call option and the barrier call optionwith K¼ 45 and 50 and B¼ 55, 60 and 65. The resultsindicate that there are significant biases of the impliedGARCH-normal model in pricing the lookback optionas well as the barrier option. The ratios of theGARCH-normal option values to the nominal pricesfor the up-and-out option are plotted in figure 1. Sincethe ratios are higher when the barrier boundary B iscloser to the initial price S0¼ 50, e.g. the barrierboundary B¼ 55 has higher ratios than B¼ 60 and 65for fixed K, the model risk rises as the option becomeseasier to be knocked out.
Remark 1: In the simulation study we do not estimatethe parameters of the GARCH-t model, since our maininterest is in investigating the model risk in exotic optionpricing when using an incorrect implied model under theGARCH framework. Therefore, a GARCH-t model isassumed directly to depict the dynamics of the underlyingasset. In practice, the parameters of a GARCH-t modelhave to be estimated from the data and one of the mainissues is in estimating the degrees of freedom of the tdistribution (refer to Heracleous (2007) for a thoroughdiscussion).
5.2. Application of the DSA to option pricing and deltahedging
In this simulation study, we investigate the effectivenessof the proposed DSA in pricing European GARCHoptions for conditional leptokurtic GARCH models. Asbefore, the results based on the EMS method of Duanand Simonato (1998) are taken as the benchmark vales.
The log returns are simulated from model (2) with
�t ¼ rþ ��t, �2t ¼ 0 þ 1�2t�1"
2t�1 þ 1�
2t�1,
where the parameters are set to be the same as in Duan(1995), that is �¼ 0.007452, 0¼ 0.00001524, 1¼ 0.1883,2¼ 0.7162, K¼ 40, r¼ 0,
�d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0
1� 1 � 1
r¼ 0:01263 (per day, i:e: 0:2413 per annumÞ,
and the innovation "t is assumed to be normally or doubleexponentially distributed. The time to maturity T is 180days. The stock prices St are set to be 35, 40, and 45. Wecompute the European call option values and the dynamicdeltas at time t¼ 0, 90 and 150 for the GARCHmodel withnormal and double exponential innovations, respectively.In table 4, CEMS denotes the benchmark option value bythe EMSmethod based on 10,000 simulated paths and 500replicates, andCDSA denotes the option values obtained bythe DSA. The benchmarks of the delta values defined in(7), denoted by DH
t , are obtained by the EMS method,DDSA1 denotes the delta values by the DSA, and DDSA2 isthe delta obtained by the finite difference method
DtðsiÞ ffiCtðsiþ1Þ � Ctðsi�1Þ
siþ1 � si�1,
where si are the partitioned stock values and Ct are theoption prices computed by the DSA. Table 4 shows thatall of the DDSA1 and DDSA2 lie in the interval of DH
t 3standard deviation. The results suggest that DSA pro-vides a promising scheme for computing option prices anddelta hedging.
6. Discussion
The small pricing biases in tables 1 and 2 indicate that themisspecification effect of the implied GARCH-normal
Table 3. The GARCH-t nominal up-and-out and lookback call option prices and the implied GARCH-normal option prices.
B¼ 55 B¼ 60 B¼ 65Lookback
T K¼ 45 K¼ 50 K¼ 45 K¼ 50 K¼ 45 K¼ 50 option
10 days 4.2562 0.8104 5.0860 1.2454 5.1660 1.2995 1.9147(4.1177) (0.7609) (5.1398) (1.2818) (5.1905) (1.3168) (1.9523)
30 days 2.3211 0.4412 4.4064 1.5713 5.4516 2.1303 3.9657(2.2331) (0.4436) (4.5037) (1.5300) (5.4850) (2.1646) (3.8760)
60 days 1.1556 0.1994 3.2786 1.1628 4.9831 2.2439 5.9874(1.2246) (0.2660) (3.2601) (1.1961) (4.9541) (2.2514) (5.7197)
The values in parentheses are derived from the implied GARCH-normal model (9).
10 30 600.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
T (days)
GA
RC
H−
norm
al /
GA
RC
H−
t
(B,K) = (55,45)
(B,K) = (55,50)
(B,K) = (60,45)
(B,K) = (60,50)
(B,K) = (65,45)
(B,K) = (65,50)
Figure 1. The ratios of the implied GARCH-normal up-and-outoption prices to the nominal values.
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model for conditional leptokurtic GARCH underlying
can be ignored in pricing European options and Asian
options. However, significant model risk exists when
applying the implied GARCH-normal model to price
exotic options with the payoff function depending on the
extreme values of the stock price such as up-and-out and
lookback options. This is further confirmed in figure 2,
where we plot the ratios of the implied GARCH-normal
and the nominal GARCH-t option values for the plain
vanilla and exotic options. Apparently, the up-and-out
and lookback options have higher ratios for all cases of
maturity time T.The payoffs of exotic options are more complicated
than the plain vanilla option, which not only depend on
the underlying asset value at the expiration date, but also
on its path during the lifetime. For instance, the
no-arbitrage price of a lookback call option with
maturity T is
e�rTEQ0 ðST �mTÞ,
which depends on the risk-neutral joint distribution of the
random variables ST and mT. In general, moment
conditions are not sufficient to determine joint laws of
random variables. Thus the joint distribution of ST and
mT cannot be completely characterized by the moment
conditions EQ0 ðST � K Þþ of different strike price K and
maturity date T given by the plain vanilla option
information. Consequently, the implied model derived
by matching the plain vanilla options has less chance of
catching the joint distribution of ST and mT
accurately and thus produces biased prices of the look-
back option.In summary, an implied GARCH-normal model pro-
vides reasonable performance in pricing plain vanilla
options for conditional leptokurtic GARCH underlying.
However, model risk becomes significant when using a
flawed implied model to assess exotic options, as demon-
strated in this study. The reason for this is that the implied
model does not adequately capture the correct future
dynamics of the underlying asset. This biased model leads
Table 4. European call option prices and delta values of the GARCH-normal and GARCH-dexp models.
GARCH-normal GARCH-dexp
St 35 40 45 35 40 45
t¼ 0 CEMS 0.7669 2.6684 6.0181 0.7481 2.6166 5.9872(std.) (0.0079) (0.0089) (0.0081) (0.0189) (0.0220) (0.0172)CDSA 0.7640 2.6735 6.0224 0.7074 2.5774 5.9558
DHt 0.2342 0.5333 0.7877 0.2246 0.5323 0.7953
(std.) (0.0025) (0.0026) (0.0020) (0.0033) (0.0031) (0.0028)DDSA1 0.2332 0.5333 0.7880 0.2175 0.5311 0.7984DDSA2 0.2359 0.5342 0.7865 0.2232 0.5335 0.7967
t¼ 90 CEMS 0.3005 1.8741 5.4338 0.2972 1.8237 5.4202(std.) (0.0055) (0.0062) (0.0056) (0.0120) (0.0152) (0.0110)CDSA 0.2999 1.8800 5.4393 0.2800 1.8046 5.4099
DHt 0.1375 0.5237 0.8598 0.1275 0.5223 0.8697
(std.) (0.0022) (0.0025) (0.0019) (0.0027) (0.0033) (0.0024)DDSA1 0.1369 0.5236 0.8600 0.1221 0.5219 0.8728DDSA2 0.1391 0.5241 0.8590 0.1262 0.5232 0.8714
t¼ 150 CEMS 0.0393 1.0667 5.0677 0.0497 1.0271 5.0758(std.) (0.0023) (0.0035) (0.0028) (0.0052) (0.0094) (0.0053)CDSA 0.0412 1.0742 5.0731 0.0499 1.0277 5.0804
DHt 0.0302 0.5133 0.9607 0.0296 0.5119 0.9640
(std.) (0.0014) (0.0022) (0.0012) (0.0017) (0.0036) (0.0016)DDSA1 0.0300 0.5137 0.9605 0.0274 0.5125 0.9655DDSA2 0.0313 0.5136 0.9598 0.0292 0.5125 0.9646
DHt are obtained by the EMS method.
10 30 600.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
T (days)
GA
RC
H−
norm
al /
GA
RC
H−
t
Euro. K =50Asian K =50LookbackUp−and−out (B,K)=(60,50)
Figure 2. The ratios of the implied GARCH-normal optionprices to the nominal values.
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to incorrect hedging decisions when establishing a tradingportfolio for risk management. Consequently, in practice,more caution is needed when applying the implied modelobtained from plain vanilla options in pricing complexexotic derivatives. To establish the pricing model forexotic options, one needs to use the information based onthe plain vanilla option as well as the dynamics of theunderlying returns.
Acknowledgements
The authors acknowledge helpful comments from twoanonymous referees. The research of the first author wassupported by grant NSC 99-2118-M-390-003 from theNational Science Council of Taiwan. The research of thesecond author was supported by grant NSC 100-2118-M-110-003 from the National Science Council of Taiwan.
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Appendix A:
Proof of proposition 3.1
Let ~St ¼ Ste�rt and thus
~St
~St�1
¼ expf�rþ �t � �t þ �t"tg,
Mt
Mt�1�
~St= ~St�1
Et�1ð ~St= ~St�1Þ¼ expf��t þ �t"tg:
The extended Girsanov martingale measure with change
of measure density �t defined in (3) ensures that
~St
~St�1
�����Q
F t�1L¼
Mt
Mt�1
�����P
F t�1: ðA1Þ
Since �t and �t are F t�1-measurable, (A1) implies
"t �r� �t
�t
����QF t�1L¼ "t
����PF t�1:
Let
�t ¼ "t �r� �t
�t,
then given F t�1 the conditional distribution of �t undermeasure Q is the same as that of "t under measure P.
Proof of proposition 3.2
Since we are now considering the simple return, let the
discounted factor of the stock price process be ð1þ ~r Þ�1
in the time period [t� 1, t) and denote the dis-
counted stock price process by ~St ¼ ð1þ ~r Þ�tSt. We
then have
~St
~St�1
¼1þ �t þ �t"t
1þ ~r,
Mt
Mt�1�
~St= ~St�1
Et�1ð ~St= ~St�1Þ¼
1þ �t þ �t"t1þ �t
:
The extended Girsanov martingale measure with change
of measure density �t defined in (3) ensures that
~St
~St�1
�����Q
F t�1L¼
Mt
Mt�1
�����P
F t�1,
and thus
1þ �t
1þ ~r"t �
~r� �t
�t
� �����QF t�1L¼ "t
����PF t�1:
Let
�t ¼1þ �t
1þ ~r"t �
~r� �t
�t
� �,
and then the conditional distribution of �t given F t�1
under measure Q is the same as that of "t undermeasure P.
The EMS procedure for option pricing in aGARCH model
We use a European call option as an example forillustration. The no-arbitrage price of a European calloption with strike price K and maturity T at time 0 isgiven by
C0 ¼ e�rTEQðST � K Þþ:
The EMS procedure for estimating C0 is as follows.
(i) Generate n sample paths each of length T ofthe stock prices denoted by St,i, i¼ 1, . . . , n,t¼ 1, . . . ,T, by the standard Monte Carlomethod from a GARCH model.
(ii) Let bS0,i ¼ S0, i¼ 1, . . . , n, and define the empiricalmartingale stock prices bSt,i, i¼ 1, . . . , n,iteratively by
bSt,i ¼ S0Ziðt, nÞ
Z0ðt, nÞ,
where
Ziðt, nÞ ¼ bSt�1,iSt,i
St�1,i, Z0ðt, nÞ ¼
e�rt
n
Xni¼1
Ziðt, nÞ:
(iii) Compute the EMS estimator of C0 by
C0 ¼ e�rT1
n
Xni¼1
ðST,i � K Þþ:
The EMS method can also be incorporated withvariance reduction techniques such as antithetic andcontrol variate methods. We refer the reader to Duanand Simonato (1998) for further details.
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