Empirical comparison of alternative stochastic volatility option pricing models: Evidence from Korean KOSPI 200 index options market In Joon Kim, Sol Kim * Graduate School of Management, Korea Advanced Institute of Science and Technology (KAIST), 207-43 Chongyang-ni, Dongdaemoon-gu, Seoul, South Korea Received 21 October 2002; accepted 10 April 2003 Abstract This article investigates the improvement in the pricing of Korean KOSPI 200 index options when stochastic volatility is taken into account. We compare empirical performances of four classes of stochastic volatility option pricing models: (1) the ad hoc Black and Scholes procedure that fits the implied volatility surface, (2) Heston and Nandi’s [Rev. Financ. Stud. 13 (2000) 585] GARCH type model, (3) Madan et al.’s [Eur. Financ. Rev. 2 (1998) 79] variance gamma model, and (4) Heston’s [Rev. Financ. Stud. 6 (1993) 327] continuous-time stochastic volatility model. We find that Heston’s model outperforms the other models in terms of effectiveness for in-sample pricing, out-of-sample pricing and hedging. Looking at valuation errors by moneyness, pricing and hedging errors are highest for out-of-the-money options, and decrease as we move to in-the-money options in all models. The stochastic volatility models cannot mitigate the ‘‘volatility smiles’’ effects found in cross-sectional options data, but can reduce the effects better than the Black and Scholes model. Heston and Nandi’s model shows the worst performance, but the performance of the Black and Scholes model is not too far behind the stochastic volatility option pricing model. D 2003 Elsevier B.V. All rights reserved. JEL classification: G13 Keywords: Option pricing model; Out-of-sample pricing; Hedging; Volatility smiles; Stochastic volatility 0927-538X/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0927-538X(03)00042-8 * Corresponding author. Tel.: +82-2-958-3968; fax: +82-2-958-3618. E-mail address: [email protected] (S. Kim). www.elsevier.com/locate/econbase Pacific-Basin Finance Journal 12 (2004) 117 – 142
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www.elsevier.com/locate/econbase
Pacific-Basin Finance Journal 12 (2004) 117–142
Empirical comparison of alternative stochastic
volatility option pricing models: Evidence from
Korean KOSPI 200 index options market
In Joon Kim, Sol Kim*
Graduate School of Management, Korea Advanced Institute of Science and Technology (KAIST), 207-43
Chongyang-ni, Dongdaemoon-gu, Seoul, South Korea
Received 21 October 2002; accepted 10 April 2003
Abstract
This article investigates the improvement in the pricing of Korean KOSPI 200 index options
when stochastic volatility is taken into account. We compare empirical performances of four classes
of stochastic volatility option pricing models: (1) the ad hoc Black and Scholes procedure that fits the
implied volatility surface, (2) Heston and Nandi’s [Rev. Financ. Stud. 13 (2000) 585] GARCH type
model, (3) Madan et al.’s [Eur. Financ. Rev. 2 (1998) 79] variance gamma model, and (4) Heston’s
[Rev. Financ. Stud. 6 (1993) 327] continuous-time stochastic volatility model. We find that Heston’s
model outperforms the other models in terms of effectiveness for in-sample pricing, out-of-sample
pricing and hedging. Looking at valuation errors by moneyness, pricing and hedging errors are
highest for out-of-the-money options, and decrease as we move to in-the-money options in all
models. The stochastic volatility models cannot mitigate the ‘‘volatility smiles’’ effects found in
cross-sectional options data, but can reduce the effects better than the Black and Scholes model.
Heston and Nandi’s model shows the worst performance, but the performance of the Black and
Scholes model is not too far behind the stochastic volatility option pricing model.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142118
1. Introduction
Since Black and Scholes (1973) published their seminal article on option pricing, there
has been much theoretical and empirical work on option pricing. Numerous empirical
studies have found that the Black–Scholes model (henceforth BS) results in systematic
biases across moneyness and maturity. It is well known that after the October 1987 crash,
the implied volatility computed from options on the stock index in the US market inferred
from BS appears to be different across exercise prices. This is the so-called ‘‘volatility
smiles’’. Of course, given BS assumptions, all option prices on the same underlying
security with the same expiration date but with different exercise prices should have the
same implied volatility. However, the ‘‘volatility smiles’’ pattern suggests that BS tends to
misprice deep in-the-money and deep out-of-the-money options.1
There have been various attempts to deal with this apparent failure of BS. One
important direction along which the BS formula can be modified is to generalize the
geometric Brownian motion that is used as a model for the dynamics of log stock prices.
For example, Hull and White (1987), Johnson and Shanno (1987), Scott (1987), Wiggins
(1987), Melino and Turnbull (1990, 1995), Stein and Stein (1991) and Heston (1993)
suggest a continuous-time stochastic volatility model. Merton (1976), Bates (1991) and
Naik and Lee (1990) propose a jump-diffusion model. Duan (1995) and Heston and Nandi
(2000) develop an option pricing model based on the GARCH process. Recently, Madan
et al. (1998) use a three-parameter stochastic process, termed the variance gamma process,
as an alternative model for the dynamics of log stock prices.
This wide range of stochastic volatility models that account for non-constant volatility
requires comparison. Previously, Bakshi et al. (1997) evaluated the performance of
alternative models for the S&P 500 index option contracts. They examined how much
each additional feature improves the pricing and hedging performance. They showed that
the stochastic volatility term provides a first-order improvement over BS. In addition,
other factors such as the stochastic interest rate or the jump diffusion have a marginal
effect.2 To this end, we have a horse race competition among alternative stochastic
volatility models to gauge pricing and hedging performance.
Thus we are comparing the relative empirical performance of four classes of stochastic
volatility option pricing models. The first class of stochastic volatility models is the ad hoc
Black and Scholes procedure (henceforth AHBS) dealt by Dumas et al. (1998). Assuming
that option prices are given for all strikes and for all maturities, AHBS fits a volatility
function for the underlying asset price process to the prices of option contracts. Once the
volatility function is determined, it can be used to price and hedge other derivative assets.
The second class of stochastic volatility models is the GARCH type option pricing
model (henceforth GARCH) of Heston and Nandi (2000). The autoregressive structure of
the GARCH process captures empirical appearances like volatility clustering, leptokurtic
return distributions and leverage effects. We choose this model because it yields the closed
form solution.
1 In-the-money, at-the-money and out-of-the-money is henceforth ITM, ATM and OTM, respectively.2 For the KOPSI 200 index option, Jung (2001) showed the same result as Bakshi et al. (1997).
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 119
The third class of stochastic volatility models is the variance gamma option pricing
model (henceforth VG). The variance gamma process derived by Madan and Milne (1991)
is aimed at providing a model for a log-return distribution that offers physical interpre-
tation and incorporates both long-tailness and skewness characteristics in a log-return
distribution. Using this process, Madan et al. (1998) derived the closed-form solution of
the European call option.
The fourth class of stochastic volatility models is the continuous-time stochastic
volatility model (henceforth SV) of Heston (1993) which models the square of the
volatility process with mean-reverting dynamics, allowing for changes in the underlying
asset price to be contemporaneously correlated with changes in the volatility process. We
choose this model among other continuous-time stochastic models because of the
allowance of the correlation between asset returns and volatility, and it yields the closed
form solution.
Moreover, we compare alternative stochastic volatility option pricing models with the
simplest but still valuable option pricing model, BS.
This study fills two gaps. First, this study considered improvements over BS by
allowing stochastic volatility terms in pricing the KOSPI 200 index options. Although
there are several studies that have examined the performance of the stochastic volatility
option pricing models in major markets, such as S&P 500 and FTSE 100, no study has
investigated their performance in emerging markets like the Korean options market. We
doubt whether the stochastic volatility model exhibits an effective value in emerging
markets. Most market practitioners in emerging markets still use BS, and markets
reflecting this viewpoint can show that BS does not render such bad results in either
pricing or hedging performances. Moreover, an important point mentioned by Bakshi et
al. (1997) is ‘‘The volatility smiles are the strongest for short-term options (both calls
and puts), indicating that short-term options are the most severely mispriced by BS and
present perhaps the greatest challenge to any alternative option pricing model’’. Thus the
KOSPI 200 index options market with liquidity, which is concentrated in the nearest
expiration contract, will be an excellent sample market to investigate mispricing of
short-term options. Second, while there are several papers that compare the incremental
contribution of the stochastic volatility or the jump diffusion in explaining option pricing
biases, there is a paucity of studies that compare alternative stochastic volatility option
pricing models.
It has been found that SVoutperforms other models in the in-sample, out-of-sample and
hedging. Looking at the valuation errors by moneyness, the pricing and hedging errors are
highest for OTM options and decrease as we move to ITM options in all models. The
stochastic volatility models cannot mitigate the ‘‘volatility smiles’’ effects found in cross-
sectional options data, but they can reduce the effects better than BS. GARCH shows the
worst performance, but the performance of BS is not too far behind the stochastic volatility
option pricing models.
The outline of this paper is as follows. Alternative stochastic volatility option pricing
models are reviewed in Section 2. The data used for analysis are described in Section 3.
Section 4 describes estimation methods. Section 5 describes parameter estimates of each
model and evaluates pricing and hedging performances of alternative models. Section 6
concludes our study by summarizing the results.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142120
2. Model
According to the option pricing theory, European options are priced by evaluating the
expectation of the discounted terminal payoff of the option at maturity under an equivalent
risk neutral measure Q. Hence the price of a European call with a strike price of K and
maturity s is given by
Cðt; s;KÞ ¼ e�rsEQt ½maxðStþs � K; 0Þ� ð1Þ
where EtQ [�] represents the conditional expectation under the risk-neutral density.
Bakshi and Madan (2000) show that Eq. (1) can be decomposed into two components
as
C ¼ SP1 � Ke�rsP2 ð2Þ
where
P1 ¼ EQt
Stþs
St1½Stþs>K�
� �;
P2 ¼ EQt 1½Stþs>K�� �
and the indicator function 1[St + s>K] is a unity when St + s>K. The price of a European put
can be determined from the put–call parity.
In the rest of this section, we display only the probability P1 and P2 of each model.
2.1. AHBS
Since GARCH, VG and SV have more parameters than BS, they may have an unfair
advantage over BS. Therefore, we follow Dumas et al. (1998) and construct AHBS in
which each option has its own implied volatility depending on a strike price and time to
maturity. Specifically, the spot volatility of the asset that enters BS is a function of the
strike price and the time to maturity or a combination of both. However, we consider only
the function of the strike price because the liquidity of the KOPSI 200 index options
market is concentrated in the nearest expiration contract. Even if there are options with
multiple maturities in a specific day, only the function of the strike price is applied. This is
because parameters of that day must be plugged into the following day’s data, and that data
may have options with a single maturity in out-of-sample pricing and hedging.
Specifically we adopt the following specification for the BS implied volatilities:
rn ¼ b1 þ b2ðS=KnÞ þ b3ðS=KnÞ2 ð3Þ
where rn is the implied volatility for an nth option of strike Kn and spot price S.
We follow a four-step procedure. First, we abstract the BS implied volatility from each
option. Second, we estimate the bi (i = 1, 2, 3) by ordinary least squares. Third, using
estimated parameters from the second step, we plug in each option’s moneyness into the
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 121
equation, and obtain the model-implied volatility for each option. Finally, we use volatility
estimates computed in the third step to price options with the BS formula.
AHBS, although theoretically inconsistent, can be a more challenging benchmark than
the simple BS for any competing option valuation model.
2.2. GARCH
The importance of the GARCH option pricing model has recently expanded due to its
linkage with continuous-time stochastic models that are difficult to implement. The
volatility of continuous-time stochastic models is not readily identifiable with discrete
observations on the underlying asset price process. In contrast, GARCH models have the
advantage that volatility is observable from the history of underlying asset prices.
Other existing GARCH models do not have closed-form solutions for option values.
These models are typically solved by simulation (Engle and Mustafa, 1992; Amin and Ng,
1993; Duan, 1995) that require slow and computationally intensive empirical work. In
contrast, Heston and Nandi (2000) develop a closed form solution for European option
values and hedge ratios.
Under risk-neutral dynamics, the single lag version of their model takes the following
form:
lnStþ1
St
� �¼ r � 1
2htþ1 þ
ffiffiffiffiffiffiffiffihtþ1
pZtþ1; ð4Þ
htþ1 ¼ x þ bht þ aðZt � cffiffiffiffiht
pÞ2: ð5Þ
They derive risk neutral probabilities of European call option prices in a closed form,
assuming that the value of a call option with one period to expiration obeys the BS formula
as follows:
P1 ¼1
2þ 1
p
Z l
0
Ree�i/ln½K�f ð/ � iÞ
i/
� �d/; ð6Þ
P2 ¼1
2þ 1
p
Z l
0
Ree�i/ln½K�f ð/Þ
i/
� �d/; ð7Þ
where Re[�] denotes the real part of complex variables, i is the imaginary number,ffiffiffiffiffiffiffi�1
p,
f(/) = exp(A(t; T, /) +B(t; T, /)ht + 1 + i/ln[St]), A(t; s, /) and B(t; s, /) are functions of a,b, c and x.
2.3. VG
The variance gamma approach proposed by Madan and Seneta (1990), Madan and
Milne (1991) has the advantage that additional parameters in the variance gamma process
provide control over the skewness and kurtosis of the return distribution.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142122
The variance gamma process is obtained by evaluating Brownian motion with drift at a
random time given by a gamma process. Let
bðt; h; rÞ ¼ ht þ rW ðtÞ ð8Þ
where W(t) is a standard Brownian motion. The process b(t; h, r) is a Brownian motion
with drift h and volatility r. The gamma process c(t; l, m) with mean rate l and variance
rate m is the process of independent gamma increments over non-overlapping intervals. VG
process, X(t; r, m, h), is defined in terms of the Brownian motion with drift b(t; h, r) andthe gamma process with unit mean rate, c(t; 1, m) as X(t; r, m, h) = b(c(t; 1, m), h, r).
Thus, the assumed process of the underlying asset, St, is given by replacing the role of
Brownian motion in the original Black–Scholes geometric Brownian motion model by the
variance gamma process as follows:
St ¼ S0 exp½mt þ X ðt; r; m; hÞ þ wt� ð9Þ
where S0 is the initial stock price, m is the mean rate of stock return, and
w= (1/m)ln(1� hm� r2m/2).Based on the above process, Madan et al. (1998) derive risk neutral probabilities for the
price of a European option as follows:
P1 ¼ u d
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c1
m
r; ða þ rÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffim
1� c1
r;sm
" #; ð10Þ
P2 ¼ u d
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2
m
r; a
ffiffiffiffiffiffiffiffiffiffiffiffiffim
1� c2
r;sm
" #; ð11Þ
where uða; b; cÞ ¼ ml0 U affiffig
p þ bffiffiffig
p� �gc�1e�g
CðcÞ dg, U(�) is the cumulative distribution function
of a standard normal distribution, and C(�) is the gamma function.
2.4. SV
Heston (1993) provided a closed-form solution for pricing a European style option
when volatility follows a mean-reverting square-root process. The actual diffusion
processes for the underlying asset and its volatility are specified as
dS ¼ lSdt þ ffiffiffiffimt
pSdWS ; ð12Þ
dmt ¼ jðh � mtÞdt þ rffiffiffiffimt
pdWm; ð13Þ
where dWS and dWm have an arbitrary correlation q, mt is the instantaneous variance. j is
the speed of adjustment to the long-run mean h, and r is the variation coefficient of
variance.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 123
Given the dynamics in Eqs. (12) and (13), Heston (1993) shows that risk neutral
probabilities of a European call option with s periods to maturity is given by
Pj ¼1
2þ 1
p
Z l
0
Ree�i/ln½K�fjðx; mt; s;/Þ
i/
� �d/ ðj ¼ 1; 2Þ ð14Þ
where Re[�] denotes the real part of complex variables, i is the imaginary number,ffiffiffiffiffiffiffi�1
p,
fj(x,mt,s;/) = exp[C(s;/) +D(s;/)mt+ i/x] and C(s;/) and D(s;/) are functions of h, j, q, rand mt.
Because BS is already generally known, we do not display it separately.
3. Data
In July 7 1997, the Korean exchange for options introduced the KOSPI 200 index
options. The KOSPI 200 options market has become one of the fastest growing
markets in the world, despite its short history. Its daily trading volume reached 1.2
million contracts in November 2000, marking the most active index options product
internationally.
Three consecutive near-term delivery months and one additional month from the
quarterly cycle (March, June, September and December) make up four contract months.
The expiration day is the second Thursday of each contract month. Each options
contract month has at least five strike prices. The number of strike prices may, however,
increase according to the price movement. Trading in the KOSPI 200 index options is
fully automated. The exercise style of the KOSPI 200 options is European and thus
contracts can be exercised only on the expiration dates. Hence our test results are not
affected by the complication that arises due to the early exercise feature of American
options. Moreover, it is important to note that liquidity is concentrated in the nearest
expiration contract.
The sample period extends from January 3, 1999 through December 26, 2000. The
minute-by-minute transaction prices for the KOSPI 200 options are obtained from the
Korea Stock Exchange. The 3-month treasury yields were used as risk-free interest rates.3
Because KOSPI 200 contracts are European-style, index levels were adjusted for dividend
payments before each option’s expiration date. The KOSPI 200 index pays dividends only
at the end of March, June, September and December, which are used for adjustment
dates.4
The following rules are applied to filter data needed for the empirical test.
1. For each day in the sample, only the last reported transaction price, which has to occur
between 2:30 and 3:00 p.m., of each option contract is employed in the empirical test.
3 Korea does not have a liquid Treasury bill market, so the 3-month Treasury yield is used in spite of the
mismatch of maturity of options and interest rates.4 We assume that traders have perfect knowledge about future dividend payments because options in this
July 2000–Dec. 2000 Call 0.5080 0.4903 0.4806 0.4813 0.4879 0.5509
Put 0.6440 0.4944 0.5033 0.4968 0.5154 0.5302
This table reports the implied volatilities calculated by inverting the Black–Scholes model separately for each
moneyness category. The implied volatilities of individual options are then averaged within each moneyness
category and across the days in the sample. Moneyness is defined as S/K, where S denotes the spot price and K
denotes the strike price.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 125
Table 2 presents the ‘‘volatility smiles’’ effects for four consecutive subperiods. We
employ six fixed intervals for the degree of moneyness, and compute the mean over
alternative subperiods of the implied volatility. It is interesting to note that the Korean
options market seems to be ‘‘smiling’’ independent of the subperiods employed in the
estimation. So, we recognize the need of the stochastic volatility option pricing model to
mitigate this effect.
4. Estimation procedure
In applying option pricing models, one always encounters the difficulty that spot
volatility and structural parameters are unobservable. We follow the estimation method
similar to standard practices (e.g. Bakshi et al., 1997, 2000; Bates, 1991, 2000; Kirgiz, 2001;
Lin et al., 2001; Nandi, 1998), and estimate parameters of each model every sample day.
Since closed-from solutions are available for an option price, a natural candidate for the
estimation of parameters which enter the pricing and hedging formula is a non-linear least
squares procedure involving minimization of the sum of squared errors between the model
and market prices.
Estimating parameters from the physical asset returns can be an alternative, but
historical data reflect only what happened in the past. Further, the procedure using
historical data is not able to identify volatility risk premiums that have to be estimated
from the options data conditional on the estimates of other parameters. The important
advantage of using option prices to estimate parameters is to allow one to use the forward-
looking information contained in the option prices.
Let Oi(t, s; K) denote the market price of option i on day t, and let Oi*(t, s; K) denote themodel price of the option i on day t. To estimate parameters of each model, we minimize
the sum of squared percentage errors between model and market prices:
minXN Oiðt; s;KÞ � Oi*ðt; s;KÞ
� �2ðt ¼ 1; . . . ; TÞ ð15Þ
/t i¼1Oiðt; s;KÞ
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142126
where N denotes the number of options on day t, and T denotes the number of days in the
sample. Conventionally, the objective function to minimize the sum of squared errors is
used. But we adopt the above function because the fit for OTM options under the
conventional method that gives more weight to relatively expensive ITM options is the
worst.6
For BS, the volatility parameter, r, is estimated. For GARCH, we estimate the
structural parameters, {a, b, c, x} and conditional variance, ht + 1, is not estimated as an
additional parameter but determined from the daily history of index returns and the
structural parameters on day t. The starting variance, h0, is the estimate of the variance for
the past 1 year, computed from daily logarithmic returns. For VG, the unobservable
volatility parameter m with structural parameters {a, r} is estimated. For SV, we estimate
the unobservable volatility parameter mt with structural parameters {h, j, q, r}. As
mentioned before, the coefficients for AHBS are estimated via ordinary least squares.
5. Empirical findings
In this section, we compare empirical performances of alternative stochastic volatility
models with respect to three metrics: (1) in-sample performance, (2) out-of-sample
performance, and (3) hedging performance.
The analysis is based on four measures: mean absolute errors (henceforth MAEs), mean
percentage errors (henceforth MPEs), mean absolute percentage errors (henceforth
MAPEs), and mean squared errors (henceforth MSEs). MAEs and MAPEs measure the
magnitude of pricing errors, while MPEs indicate the direction of the pricing errors. MSEs
measure the volatility of errors. In the remaining sections, we mainly deal with MAPEs,
because the relative comparison considering each option price is important above all else.
5.1. In-sample pricing performance
For each model, Table 3 reports average and standard deviations (in parentheses) of
parameters, which are estimated daily. The implicit parameters are not constrained to be
constant over time. While re-estimating the parameters daily is admittedly potentially
inconsistent with the assumption of constant or slow-changing parameters used in deriving
the option pricing model, such estimation is useful for indicating market sentiment on a
daily basis.
Parameters of other stochastic volatility models except GARCH have large standard
deviations. This shows that the stability of parameters is not supported for each model.
However, as stated thereafter, the pricing performance of the model with parameters
having large standard deviations is better than that of the model with parameters having
6 There was no large difference between results using the sum of squared errors and those using the sum of
squared percentage errors in our sample.
Table 3
Parameter estimates
BS AHBS VG GARCH SV
r 0.4712
(0.0880)
b1 2.3432
(5.4914)
a 2.2670
(6.7975)
a 2.98e–6
(1.81e� 7)
h 2.8375
(9.8256)
b2 � 3.8371
(10.943)
r 0.4310
(0.1391)
b 0.6420
(0.0595)
j 9.5723
(29.9060)
b3 1.9693
(5.4493)
m 0.0311
(0.0836)
c 316.2083
(23.6614)
q � 0.0348
(0.7867)
x 7.00e� 6
(4.57e� 6)
r 0.5674
(0.3040)
m 0.2856
(0.3456)
The table reports the mean and standard deviation (in parentheses) of the parameter estimates for each model. BS
is the Black–Scholes model in which a single implied volatility is estimated across all strikes and maturities on a
given day. AHBS is the ad hoc BS procedure in which regression specification is estimated as follows:
rn ¼ b1 þ b2S=Kn þ b3ðS=KnÞ2:
VG, GARCH and SV are Madan et al.’s (1998) variance gamma model, Heston and Nandi’s (2000) GARCH
type discrete model and Heston’s (1993) continuous-time stochastic volatility model in which each parameter
is estimated by minimizing the sum of percentage squared errors between model and market option prices
every day.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 127
small ones, i.e. it is found that the stability of interdependence among parameters is more
important than that of parameters in option pricing and hedging.
The implied correlation coefficient is negative as we expected.7 This is consistent with
the leverage effect documented by Black (1976) and Christie (1982), whereby lower
overall firm values increase the volatility of equity returns, and the volatility feedback
effects of Poterba and Summers (1986) whereby higher volatility assessments lead to
heavier discounting of future expected dividends and thereby lower equity price.
We evaluate the in-sample performance of each model by comparing market prices with
model prices computed by using the parameter estimates from the current day. Table 4
reports the in-sample valuation errors for alternative models computed over the whole
sample of options as well as across six moneyness and two option type categories. Results
from the analysis are as follows.
First, with respect to MAPEs and MAEs, SV shows the best performance followed by
VG. However, according to MSEs, the order has been changed. For call options, AHBS
has the fewest errors followed by SV. For put options, BS has the best performance
followed by SV. On the whole, SV is the best for the in-sample pricing.
Unexpectedly, AHBS is not better than BS although AHBS has more parameters than
BS does. This result can be explained by the lower R2 compared to advanced markets. In
our study, the R2 of AHBS is 22% on average, which is quite low. In the study of Kirgiz
7 The positive a and c of VG and GARCH indicate a negative correlation, respectively.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142128
(2001) on the S&P 500, the R2 was 93%. Because of the low R2, AHBS seems to lead to a
relatively large in-sample errors.
In the moneyness-based error, for call options, SV has the fewest errors except deep
ITM options where AHBS does. For put options, AHBS in deep ITM, GARCH in ITM
Table 4
In-sample pricing errors
S/K < 0.94 0.94–0.97 0.97–1.00 1.00–1.03 1.03–1.06 z 1.06 All
where en.t is the absolute percentage error of the option n on day t; St/Kn, sn and rt respectively represent the
moneyness, the time-to-maturity of the option contract and the risk-free interest rate at date t. The estimated
regression coefficients are presented in this table together with their p-values, which are shown in parenthesis.
Adjusted R2 values and F-statistics for the linear regression model are reported in the last two rows of the table.
BS denotes the Black and Scholes model, AHBS denotes the ad hoc Black and Scholes procedure that fits to the
implied volatility surface, GARCH denotes Heston and Nandi’s (2000) GARCH type discrete model, VG denotes
Madan et al.’s (1998) variance gamma model, and SV denotes Heston’s (1993) continuous-time stochastic
volatility model.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 135
underlying asset price volatility alone, and carry out a delta-neutral hedge, employing
only the underlying asset as the hedging instrument.
We implement hedging with the following method. Consider hedging a short position
in an option, O(t, s; K) with s periods to maturity and strike price of K. Let DS(t) be the
number of shares of the underlying asset to be purchased, and D0( =O(t, s; K)�DS(t)St) be
the residual cash positions. We consider the delta hedging strategy of DS =BO(t, s; K)/BSt)and D0(t). The delta, for a put option, is negative, which means that a short position in put
options should be hedged with a short position in the underlying stock.
To examine the hedging performance, we use the following steps. First, on day t, we
short an option, and construct a hedging portfolio by buying DS(t) shares of the underlying
asset,8 and investing D0(t) in a risk-free bond. To compute DS(t), we use estimated
structural parameters from the previous trading day and the current day’s asset price.
Second, we liquidate the position after the next trading day or the next week. Then we
compute the hedging error as the difference between the value of the replicating portfolio
and the option price at the time of liquidation:
et ¼ DSStþDt þ D0erDt � Oðt þ Dt; s � Dt;KÞ: ð17Þ
Tables 8 and 9 present 1 day and 1 week hedging errors over alternative moneyness
categories, respectively. SV has the best hedging performance irrespective of the
8 In case of put options, some shares of the underlying asset are shorted because Ds(t) is negative.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142136
rebalancing period. Exceptionally, for 1 week ahead hedging errors of put options, BS
shows the best performance. But the difference among models is not so large. GARCH
shows similar results with other models contrary to the case of pricing. We recognize that
GARCH shows a weak point in pricing but a strong point in hedging, i.e. forecasting
volatilities. In each moneyness category, the hedging errors are highest for OTM options
and get smaller as we move to ITM options. This pattern is true for every model and for
each rebalancing frequency.
All models show positive mean hedging errors on average. This can be interpreted
with the negative risk premium associated with the volatility risk (Bakshi and
Kapadia, 2001). Purchased options are hedged against significant market declines.
The reason is that increased realized volatility coincides with downward market
moves. One economic interpretation is that buyers of market volatility are willing
to pay a premium for downside protection. In our hedging implementation, the
replicating portfolio holds a short position in volatility and the effect of change in
volatility is not taken into account. So, the replicating portfolio ( = delta-hedged
portfolio) has positive gains because of a negative risk premium of the volatility risk.
6. Conclusion
We have studied pricing and hedging performances of alternative stochastic
volatility option pricing models: Black and Scholes (1973) model, the ad hoc Black
and Scholes procedure that fits to the implied volatility surface, Heston and Nandi’s
(2000) GARCH type discrete model, Madan et al.’s (1998) variance gamma model
and Heston’s (1993) continuous-time stochastic volatility model. We estimate each
model from the daily cross-section of the KOSPI 200 index option prices. Our results
are as follows.
First, SV outperforms other models in terms of in-sample pricing, out-of-sample
pricing and hedging performances. Second, the addition of the stochastic volatility term
does not resolve the ‘‘volatility smiles’’ effects, but it reduces the effects. Third, BS is
competent in option pricing with the advantage of simplicity. This result reflects the
actuality that most market practitioners in the emerging markets still use BS. Finally,
Notes to Table 8:
This table reports 1 day ahead hedging error for the KOSPI 200 option with respect to moneyness. Only the
underlying asset is used as the hedging instrument. Parameters and spot volatility implied by all options of the
previous day are used to establish the current day’s hedge portfolio, which is then liquidated the following day.
For each option, its hedging error is the difference between the replicating portfolio value and its market price.
Denoting en hedging errors, hedging performance is evaluated by: (1) mean percentage error (MPE), ðPN
n¼1 en=OnÞ=N , (2) mean absolute percentage error (MAPE), ð
PNn¼1 AenA=OnÞ=N , (3) mean absolute error (MAE),
ðPN
n¼1 AenAÞ=N, and (4) mean squared error (MSE), ðPN
n¼1ðenÞ2Þ=N, where N is the total number of options in a
particular moneyness category. BS denotes the Black and Scholes model, AHBS denotes the ad hoc Black and
Scholes procedure that fits to the implied volatility surface, GARCH denotes Heston and Nandi’s (2000) GARCH
type discrete model, VG denotes Madan et al.’s (1998) variance gamma model, and SV denotes Heston’s (1993)
continuous-time stochastic volatility model.
I.J. Kim, S. Kim / Pacific-Basin Finance Journal 12 (2004) 117–142 137
GARCH is the worst performer. In the emerging markets such as the Korean KOSPI 200
index options market with liquidity concentrated in deep OTM options, GARCH does
not perform appropriately.
Table 8
One day ahead hedging errors
S/K < 0.94 0.94–0.97 0.97–1.00 1.00–1.03 1.03–1.06 z 1.06 All