We implement the skewed-t GARCH model to the above dataset and derive the
empirical copula of the standardized GARCH innovations displayed in Figure 1. The
empirical copula would converge to the true copula function under the regularity
condition; for instance, Gaenssler and Stute (1987) or van der Vaart and Wallner (1996).
Figure 1: The empirical copula of the standardized GARCH innovations Pair 1: MSCI Pacific vs MSCI Far East Pair 2: MSCI Energy vs MSCI Bank
Pair 3: CRB vs MSCI Latin America Pair 4: S&P500 vs DXY
Note: The skewed-t GARCH model is described as follows:
ri,t= μi+ εi,t,
21 hi,t= ωi+ βihi,t−1+ αiεi,t−12 ,
εi,t|ℱt −1 = hi,tzi,t , zi,t~skewed − t zi ηi, ϕi
Then the standardized GARCH innovations for each pair of underlying assets are εi,t hi,t, for 𝑖 ∈ 1,2 and 𝑡 ∈ 1, … , 𝑇 .
We observe that there exists strong positive dependence between the innovations of
first two pairs of assets, particularly in the tails. On the other hand, the innovations of
last two pairs of underlying assets are quite irrelevant and scattered all over the figure.
The dependence structures are dissimilar among these four pairs of assets.
The following Table 6 reports the maximum likelihood estimation results of the
copula-based GARCH model. The estimates of parameters for marginal skewed-t
GARCH process are presented along with the P-values in parentheses in the panel A,
where 0.0000 indicates that the P-value is less than 0.00005.
The value of ω, α and β are all nonnegative that ensure the conditional variance hi,t is positive, and the sum of α and β is close to one for all pairs of assets, which
implies the shocks between the underlying assets have high persistence in volatility and
stationary process. For the estimation of skewed-t distribution, the kurtosis parameter η
and the asymmetry parameter φ show that the conditional process of all assets is
fat-tailed and asymmetric, and this is consistence with previous studies.
The panel B of Table 6 reports the estimates of parameters for different types of
copula functions. All of these estimates are significant under 5% level, except for the
degrees of freedom ν of the last three pairs of Student’s t copulas.
22
Table 6: Copula-based GARCH Model Estimations Asset
Panel A: Estimates of marginal process
ω × 105 0.0179 0.0187 0.0115 0.0029 0.0100 0.0231 0.0030 0.0025
Panel B: Estimates of dependence process
Gaussian ρ 0.9842 0.5146 0.2801 0.0777
Note: The values in parentheses are P-values, where 0.0000 indicates that the value is less than 0.00005.
23
We compare the model goodness-of-fit through the Kolmogorov-Smirnov (K-S)
test and Anderson-Darling (A-D) test along with the P-values in parentheses in Panel A
and B of Table 7. The log-likelihood estimator and Akaike Information Criterion (AIC)
of the above copula estimates are also reported in the Panel C and D of Table 7. The
lower values of the K-S test statistics, A-D test statistics and AIC information criterion
represent better goodness-of-fit. On the contrary, the larger of log-likelihood estimates
indicate better fit.
According to both of the K-S test and A-D test, the first pair of assets, which
components are MSCI Pacific and MSCI Far East Index, has the largest P-value that
means the best model fit under the Gumbel copula function. The Frank copula is the
most ideal model for the second pair of assets, which components are MSCI Energy and
MSCI Bank Index. Although the P-values of these five dependence structures are not
significant enough for the third and fourth pair of assets, the Frank copula is still the
model that relatively describes the dependence structure best based on the empirical
distribution under the K-S and A-D test statistics.
Nevertheless, the inferred conclusion from log-likelihood estimator and AIC
information criterion indicates that the Student’s t is the best dependence structure for
all of these four pairs of asset returns, which obviously contradicted to our above
induction by the K-S and A-D tests. We still choose to apply the K-S and A-D test as our
24
basis to the following bivariate option pricing comparison.
Table 7: Goodness-of-fit test to Copula-based GARCH Models Asset Panel A: Kolmogorov-Smirnov (K-S) test
Gaussian 0.0316 0.0597 0.0886 0.1297
Gaussian 1.4626 9.0281 19.2397 43.9457
(0.1865) (0.0000) (0.0000) (0.0000)
Student’s t 1.4293 8.3982 18.8446 43.9345
(0.1951) (0.0000) (0.0000) (0.0000)
Gumbel 1.2187 10.7238 21.6726 44.4410
(0.2599) (0.0000) (0.0000) (0.0000)
Clayton 4.9203 15.9543 26.2062 46.3411
(0.0029) (0.0000) (0.0000) (0.0000)
Frank 2.3892 8.0226 17.7837 42.2574
(0.0566) (0.0000) (0.0000) (0.0000)
Panel C: Maximum Likelihood Estimation
Gaussian 2958.28 258.44 67.11 4.89
25
Student’s t 2999.23 268.76 68.08 13.49
Gumbel 2940.42 238.37 52.00 8.17
Clayton 2322.66 187.30 52.07 5.52
Frank 2854.21 244.85 62.38 5.30
Panel D: Akaike Information Criterion (AIC)
Gaussian -5914.57 -514.89 -132.23 -7.77
Student’s t -5996.46 -535.52 -134.16 -24.98
Gumbel -5878.84 -474.74 -102.00 -14.33
Clayton -4643.33 -372.59 -102.14 -9.03
Frank -5706.42 -487.69 -122.76 -8.60
Note: The test statistics of K-S test and A-D test are described as follows:
DKS= max
t FE xt − FH(xt) DAD = F[FE x −FH x ]2
H x [1−FH x ]
x dFH x .
The copula estimates in Table 5.1 are implemented in the hypothetical cumulative distribution FH and the empirical cumulative distribution FE is defined as
FE(x) =1
T Tt=1 Nj=1I(xj,t≤ xj uj∙T )
where I(∙) is the indicator function, u is a vector of marginal probabilities, and xj uj∙T is the uj∙ T th order statistic. Besides, the values in parentheses are P-values, where 0.0000 indicates that the value is less than 0.00005.
After we derive all desired parameter estimates, we simulate the asset return
process that given the parameters of skewed-t GARCH model and copula functions.
Especially we apply the parameter estimates of each marginal skew-t GARCH process
with initial conditional volatility hi,0 equals the long-term variance level ωi 1 − αi− βi . The estimates of Gaussian, Student’s t, Gumbel, Clayton and Frank
copula functions are also implemented to simulate different types of dependence
26
structure. To ensure the convergence of simulated option prices, we simulate 10,000
times and assume that the initial underlying asset return equals one for each return
process.
The following Figure 2 reports the simulation results of digital option pricing of
MSCI Pacific Index and MSCI Far East Index with different strike prices K from 0.80 to
1.20 and different time-to-maturity T from one to six months. We also fix the strike
price K to divide them into three groups of in-the-money (ITM), at-the-money (ATM)
and out-of-the-money (OTM) respectively in Figure 3.
The differences of digital option price between each type of dependence structure
tend to become wider when the maturity increases and the initial bivariate option prices
are around at-the-money. However, the tendency of price change is not very obvious,
and the same situation happens both for the spread options and rainbow options
displayed in Figure 4 to 6.
In order to confirm whether the differences among these bivariate option prices are
significant or not, we perform a paired t-test of the null hypothesis that the 10,000 times
simulated asset returns in the difference between each pairs of dependence structures are
a random sample from a normal distribution with mean zero and unknown variance,
against the alternative that the mean is not zero.
27
Figure 1:Digital Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index
A: 1 Month Maturity B: 2 Month Maturity
C: 3 Month Maturity D: 4 Month Maturity
E: 5 Month Maturity F: 6 Month Maturity
Note: The payoff function of digital option is I(X1> 𝐾 , X2> 𝐾), where I(∙) is the indicator function that equals to one if the statement in parentheses holds and zero otherwise. We assume that the number of trading days for a month is twenty days.
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
28
Figure 2: Digital Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index
ITM (𝐾 = 0.9) ATM (𝐾 = 1) OTM (𝐾 = 1.1)
Note: ITM, ATM and OTM represent that the digital option is in-the-money, at-the-money and out-of-the-money respectively. Since both the underlying asset returns are set to start at the same value 1 in the simulation process, ATM indicates that 𝐾 equals to 1, ITM indicates that K is less than 1 and OTM indicates that K is larger than 1.
Figure 3: Spread Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index
A: 1 Month Maturity B: 2 Month Maturity
C: 3 Month Maturity D: 4 Month Maturity
0.8
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
29
E: 5 Month Maturity F: 6 Month Maturity
Note: The payoff function of spread option is denoted by max(X1− X2− K , 0). We assume that the number of trading days for a month is twenty days.
Figure 4: Spread Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index
ITM (𝐾 = −0.1) ATM (𝐾 = 0) OTM (𝐾 = 0.1)
Figure 5: Rainbow Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index
A: 1 Month Maturity B: 2 Month Maturity
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.09
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
30
C: 3 Month Maturity D: 4 Month Maturity
E: 5 Month Maturity F: 6 Month Maturity
Note: The payoff function of rainbow option is max max(X1 , X2 − K, 0]. We assume that the number of trading days for a month is twenty days.
Figure 6: Rainbow Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index
ITM (𝐾 = 0.9) ATM (𝐾 = 1) OTM (𝐾 = 1.1)
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.00
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20
0.10
31
dependence structure between the returns of MSCI Pacific and MSCI Far East Index as
our basis model for the comparison.
In the scenario of one month time-to-maturity in Table 8, the difference between
each pairs of dependence structures are significant under 5% level, except that the
digital and rainbow option prices between Gumbel copula and Gaussian copula are not.
However, the price differences among each pairs of dependence structure become more
significant except for the rainbow option price between Gumbel copula and Frank
copula in the scenario of six months time-to-maturity. The results of paired t-test
support our previous observation that the differences between the bivariate options
prices are more significant as the time-to-maturity increases.
Table 8: T test for the distance between the ATM (K=1) option prices of different dependence structures with 1 month time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index
Panel 1: Digital Option
Student’s t Clayton Gumbel Frank
Gaussian -0.0009 0.0267 0.0014 0.0138
(0.0833) (0.0000) (0.1221) (0.0000)
Student’s t 0.0276 0.0023 0.0147
(0.0000) (0.0116) (0.0000)
Clayton -0.0253 -0.0129
(0.0000) (0.0000)
Gumbel 0.0124
(0.0000)
32 Panel 2: Spread Option
Student t Clayton Gumbel Frank
Gaussian 0.0001 -0.0018 -0.0003 -0.0009
(0.0000) (0.0000) (0.0000) (0.0000)
Student’s t -0.0019 -0.0003 -0.0009
(0.0000) (0.0000) (0.0000)
Clayton 0.0016 0.0010
(0.0000) (0.0000)
Gumbel -0.0006
(0.0000) Panel 3: Rainbow Option
Student t Clayton Gumbel Frank
Gaussian 0.0000 -0.0014 0.0000 -0.0002
(0.0000) (0.0000) (0.4577) (0.0000)
Student’s t -0.0014 0.0000 -0.0002
(0.0000) (0.0001) (0.0000)
Clayton 0.0014 0.0012
(0.0000) (0.0000)
Gumbel -0.0002
(0.0000)
Note: The table reports the distance between the ATM (K=1) bivariate option prices of different dependence structures with 1 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.
Table 9: T test for the distance between the ATM (K=1) option prices of different dependence structures with 6 months time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index
Panel 1: Digital Option
Student’s t Clayton Gumbel Frank
Gaussian -0.0014 0.0352 0.0048 0.0333
(0.0163) (0.0000) (0.0000) (0.0000)
33
Student’s t 0.0366 0.0062 0.0347
(0.0000) (0.0000) (0.0000)
Clayton -0.0304 -0.0019
(0.0000) (0.4016)
Gumbel 0.0285
(0.0000) Panel 2: Spread Option
Student t Clayton Gumbel Frank
Gaussian 0.0002 -0.0051 -0.0007 -0.0019
(0.0000) (0.0000) (0.0000) (0.0000)
Student’s t -0.0053 -0.0009 -0.0021
(0.0000) (0.0000) (0.0000)
Clayton 0.0044 0.0031
(0.0000) (0.0000)
Gumbel -0.0012
(0.0000) Panel 3: Rainbow Option
Student t Clayton Gumbel Frank
Gaussian 0.0001 -0.0035 -0.0001 -0.0002
(0.0000) (0.0000) (0.0000) (0.0009)
Student’s t -0.0036 -0.0002 -0.0003
(0.0000) (0.0000) (0.0000)
Clayton 0.0033 0.0033
(0.0000) (0.0000)
Gumbel -0.0001
(0.2549) Note: The table reports the distance between the ATM (K=1) bivariate option prices of different dependence structures with 6 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.
34
We also compare the options price distance with different strike prices at maturity
in Table 10 and 11. The significance of option price distance decreases when the strike
price is set to be in-the-money or out-of-the-money, expect for the rainbow options. The
differences of rainbow option prices among these copulas are almost all significant
under 5% level, which represents that the option payoff functions also determine the
importance of dependence structure of underlying asset returns.
Hence, we can prove the hypothesis that the differences of bivariate option prices
under distinct dependence structures are more significant when the option is
at-the-money. Besides, the option payoff function is also a key factor to affect the
importance of dependence structure selection.
Table 10: T test for the distance between the ITM (K=0.9) option prices of different dependence structures with 6 months time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index
Panel 1: Digital Option
Student’s t Clayton Gumbel Frank
Gaussian 0.0001 0.0039 0.0014 0.0092
(0.7815) (0.0002) (0.0433) (0.0000)
Student’s t 0.0038 0.0013 0.0091
(0.0005) (0.0741) (0.0000)
Clayton -0.0025 0.0053
(0.0433) (0.0003)
Gumbel 0.0078
(0.0000) Panel 2: Spread Option
35
Student t Clayton Gumbel Frank
Gaussian 0.0000 0.0001 -0.0001 0.0032
(0.3348) (0.7802) (0.1778) (0.0000)
Student’s t 0.0001 -0.0001 0.0032
(0.7284) (0.3229) (0.0000)
Clayton -0.0001 0.0032
(0.5916) (0.0000)
Gumbel 0.0033
(0.0000) Panel 3: Rainbow Option
Student t Clayton Gumbel Frank
Gaussian 0.0002 -0.0050 -0.0006 -0.0015
(0.0000) (0.0000) (0.0000) (0.0000)
Student’s t -0.0052 -0.0008 -0.0017
(0.0000) (0.0000) (0.0000)
Clayton 0.0044 0.0035
(0.0000) (0.0000)
Gumbel -0.0010
(0.0000) Note: The table reports the distance between the ITM (K=0.9) bivariate option prices of different dependence structures with 6 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.
Table 11: T test for the distance between the OTM (K=1.1) option prices of different dependence structures with 6 months time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index
Panel 1: Digital Option
Student’s t Clayton Gumbel Frank
Gaussian -0.0005 0.0151 -0.0009 0.0116
(0.0588) (0.0000) (0.0495) (0.0000)
Student’s t 0.0156 -0.0004 0.0121
36
(0.0000) (0.3173) (0.0000)
Clayton -0.0160 -0.0035
(0.0000) (0.0001)
Gumbel 0.0125
(0.0000) Panel 2: Spread Option
Student t Clayton Gumbel Frank
Gaussian 0.0000 -0.0001 0.0000 0.0000
(1.0000) (0.0010) (1.0000) (0.0971)
Student’s t -0.0001 0.0000 0.0000
(0.0010) (1.0000) (0.0971)
Clayton 0.0001 0.0001
(0.0010) (0.0032)
Gumbel 0.0000
(0.0971) Panel 3: Rainbow Option
Student t Clayton Gumbel Frank
Gaussian 0.0000 -0.0005 0.0000 0.0001
(0.0008) (0.0000) (0.3102) (0.0000)
Student’s t -0.0005 0.0000 0.0001
(0.0000) (0.6609) (0.0001)
Clayton 0.0005 0.0007
(0.0000) (0.0000)
Gumbel 0.0001
(0.0001) Note: The table reports the distance between the OTM (K=1.1) bivariate option prices of different dependence structures with 6 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.
37
The simulation results for the last three pairs of underlying assets are shown in the
Appendix A to C. The Frank copula is the best fit model based on our previous K-D test
and A-D test, which has neither lower nor upper tail dependency. After we compare the
distance of simulated bivariate option prices between Frank copula and other four types
of copula functions by paired t-test, the results show that the option price distinguishes
between copulas are not very significant in Table 12. Especially for the fourth pair of
asset returns that composed by S&P 500 Index and DXY currency Index, the P-values
from the paired t tests in parentheses are the highest in these four pairs of assets.
From the linear correlation estimates and the scatter plotted in Figure 1, we can
observe that the dependence is weakest for the fourth pair. Accordingly, the strength of
dependency to the underlying assets also determines the impact to bivariate option
prices. As the dependency is weaker, the distance between option prices are less
significant.
Table 12: T test for the distance between the ATM (K=1) option prices of different dependence structures with 6 months time-to-maturity – Pair 1 to 4
Panel 1: Digital Option
Pair 1 Pair 2 Pair 3 Pair 4
Gumbel Frank Frank Frank
Gaussian 0.0048 0.0021 -0.0003 -0.0004
(0.0000) (0.1937) (0.2569) (0.5164)
Student’s t 0.0062 0.0040 -0.0004 -0.0004
(0.0000) (0.0231) (0.2059) (0.7127)
38
Panel 2: Spread Option
Pair 1 Pair 2 Pair 3 Pair 4
Panel 3: Rainbow Option
Pair 1 Pair 2 Pair 3 Pair 4 copula functions, where the Gumbel copula is the best fit model for the first pair of assets, and the Frank copula are for the last three pairs. The numbers in parentheses are the p-values for the t tests.
39
Summarizing, the time-to-maturity of options, the pre-determined strike prices, the
type of payoff functions, and the dependency between underlying asset returns all have
impact to the simulated option prices. When the time-to-maturity gets longer, the initial
option price is at-the-money, the option payoff function is rainbow, and the dependency
of asset returns is stronger, the price distance between different dependence structures
would become more significant according to the t-test.
40 6.
C
ONCLUSIONWe discuss the bivariate options pricing by using the copula-based GARCH model.
Since most financial asset returns are actually non-Gaussian, that is skewed, leptokurtic,
and asymmetrically dependent, we consider that the copula-based GARCH model
would be better than the traditional linear correlation and Gaussian assumptions to price
multivariate claims.
According to the inference for the margins or IFM method proposed by Joe and Xu
(1996), we propose to select the skewed-t GARCH model to describe the marginal
distribution, and then the Gaussian, Student’s t, Gumbel, Clayton and Frank copula
functions are implemented to capture the dependence structure of underlying asset
returns. We process the goodness-of-fit tests such as Kolmogorov-Smirnov test and
Anderson-Darling test as our measurement to choose the best dependence structure as
our basis model, and then simulate the bivariate options prices with different payoff
structures.
The empirical test is based on four pairs of assets returns that composed by equity,
commodity and foreign exchange indices. The Gumbel copula that has upper tail
dependency is best fit for the pair of MSCI Pacific and MSCI Far East Index; the Frank
copula is relatively fit for the pairs of MSCI World Energy Index and MSCI World Bank
Index, Reuters/Jefferies CRB Index and MSCI Latin America Index, and S&P 500
41
Index and DXY currency Index, which means the dependence structure of these three
pairs has neither lower nor upper tail dependency.
The empirical results also show that as the time-to-maturity of options gets longer,
the initial option price is near at-the-money, and the dependency of asset returns is
stronger, the price distance between different dependence structures would tend to
become more significant under the paired t-test. Besides, we use digital, spread and
rainbow options as our payoff structures, and observe that the rainbow options are most
sensitive to the dependence structure. Hence, the dependence structure selection would
become more important when we are dealing the pricing issue of long maturity, high
dependency, and at-the-money bivariate options.
42 7.
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